# risk free rate

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Python for Finance by Yuxing Yan

The second issue is that the deviation is from the average instead of a fixed benchmark, such as a risk-free rate. To overcome these shortcomings, Sortino (1983) suggests the lower partial standard deviation, which is defined as the average of squared deviation from the risk-free rate conditional on negative excess returns, as shown in the following formula: m LPSD = ∑(R − R ) i =1 i n −1 f 2 , where Ri − R f > 0 (5) Because we need the risk-free rate in this equation, we could generate a Fama-French dataset that includes the risk-free rate as one of their time series. First, download their daily factors from http://mba.tuck.dartmouth.edu/pages/faculty/ken. french/data_library.html.Then, unzip it and delete the non-data part at the end of the text file.

What are the definitions of effective annual rate, effect semi-annual rate, and riskfree rate for the call option model? Assuming that the current annual risk-free rate is 5 percent, compounded semi-annually, which value should we use as our input value for the Black-Scholes call option model? 7. What is the call premium when the stock is traded at \$39, the exercise price is \$40, the maturity date is three months, the risk-free rate is 3.5 percent (compounding continuously), and the volatility is 0.15 per year? [ 77 ] 13 Lines of Python to Price a Call Option 8. Repeat the previous exercise if the risk-free rate is still 3.5 percent per year but compounded semiannually. 9. What are the advantages and disadvantages of using other programs?

The corresponding output is given as follows: Assume that we are interested in estimating the market risk (beta) for IBM using daily data downloaded from Yahoo! Finance. The beta is defined by the following linear regression: 5L W 5 I EL 5PNW W 5 I W W (10) Here, Ri,t is the stock return for stock i, Rf is the risk-free rate, Rmkt,t is the market return, and E L is the beta for stock i. Since the impact of the risk-free rate is quite small on the beta estimation, we could use the following formula for an approximation: 5L W D EL 5PNW W H W (11) The following Python program is used to download the IBM and S&P500 daily price data and estimate IBM's beta in 2013: import numpy as np import numpy as np import statsmodels.api as sm from matplotlib.finance import quotes_historical_yahoo def ret_f(ticker,begdate, enddate): p = quotes_historical_yahoo(ticker, begdate, enddate,asobject=True, adjusted=True) return((p.aclose[1:] - p.aclose[:-1])/p.aclose[:-1]) begdate=(2013,1,1) enddate=(2013,11,9) [ 201 ] Statistical Analysis of Time Series y=ret_f('IBM',begdate,enddate) x=ret_f('^GSPC',begdate,enddate) x=sm.add_constant(x) model=sm.OLS(y,x) results=model.fit() print results.summary() In the following program, we use a module called matplotlib, which is discussed in the previous chapter.

Risk Management in Trading by Davis Edwards

The simplest metric is called a Sharpe Ratio. The Sharpe Ratio measures average excess returns (returns above the risk-free rate) divided by the standard deviation of returns. (See Equation 4.1, Sharpe Ratio.) Sharpe Ratio = P * Average (Excess Daily Return) P * StdDev (Excess Daily Returns) (4.1) where P Excess Daily Return The time period adjustment for the Sharpe Ratio The average daily return minus an appropriately scaled risk free rate. For example: Excess Daily Return = daily return − risk free rate 109 Backtesting and Trade Forensics Most commonly, Sharpe Ratios are calculated based on daily returns and then annualized.

Used in the Merton model, this is the continuous yield of the underlying security. Option holders are typically not paid dividends or other payments until they exercise the option. As a result, this factor decreases the value of an option. rf Foreign Risk‐Free Rate. Used in the Garman Kohlhagen model, this is the risk‐free rate of the foreign currency. Each currency will have a risk‐free rate. N(x) Cumulative Normal Distribution Function. This is a common mathematical formula describing the probability of some event within a standard normal distribution. A standard normal distribution is a normal distribution with mean = 0 and standard deviation = 1.

This leads to the relationship between risk adjusted discount rates and risk‐free rates. (See Equation 9.5, CDS‐Based Default Probability.) A Credit Default Swap (CDS) is a financial derivative where the issuer of the swap will compensate the CDS buyer in the event of a default. PD = (1 − CDSSpread) / LGD (9.5) where PD CDS Spread LGD Probability of Default CDS Spread. The price of the Credit Default Swap Loss Given Default A similar calculation can be performed on bond prices. When this is done, the bond rate (a credit adjusted rate) is compared to risk‐free rates. The spread between the bond rate and the risk‐free rate is due to two factors— the risk of default and a liquidity premium—since the corporate bond will be more difficult to trade than the risk‐free bond.

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Derivatives Markets by David Goldenberg

Since p′ and 1–p′ act as probabilities, the numerator is the expected value (the probability-weighted average) of the option’s payoffs in the up state and in the down state, using the risk-neutral probabilities p′ and 1–p′ to weight the payoffs. The denominator is one plus the risk-free rate r=1+r′. The intuitive rationale for the term ‘risk-neutral’ is that p′ and 1–p′ are the probabilities that a risk-neutral investor would use in pricing the option. They must be, because the expected value in the numerator of (BOPM, N=1) is being discounted by the risk-free rate. Therefore, (BOPM, N=1) is the discounted expected value of the option, where the expectation is based on the risk-neutral probabilities p′ and 1–p′, not on the real-world ones p and 1–p, and discounting is at the risk-free rate, not at a risk-adjusted discount rate, as would be customary for risky assets like stocks.

Therefore, its percentage return is similarly defined as, Now, Therefore, Noting that C0–Δ*S0=B0 we obtain a weighted-average expression for RH, where XC=[C0/B0]<0 and XS=–Δ[S0/B0]>0 because we are shorting the option in the reverse hedge. The portfolio weights add up to 1.0 as they should, and so RH represents the return to the reverse hedge (short the option and long Delta units of stock). It is the reverse hedge (long the risk-free bond) that earns the risk-free rate. The normal hedge (short the risk-free bond) pays the risk-free rate (earns negative the risk-free rate). 17.5 ANALYSIS OF THE RELATIVE RISKS OF THE HEDGE PORTFOLIO’S RETURN 17.5.1 An Initial Look at Risk Neutrality in the Hedge Portfolio Now, in analyzing RH there is no reason to believe, nor to assume, that either of its component returns RC and RS are riskless.

Currency Forward Pricing We have already priced forward contracts on underlyings without and with a dividend yield in Chapters 3 and 4 respectively. The pricing formula for the equilibrium forward price on underlyings with a dividend yield ρ, a risk-free rate r, and time to expiration τ is . It is a theoretical result, beyond the scope of this text, that if interest rates are non-stochastic, then equilibrium futures prices are equal to equilibrium forward prices. A constant risk-free rate is non-stochastic therefore the equilibrium futures price must also equal where St is the underlying stock price at time t. 7.4 STOCK INDEX FUTURES 7.4.1 The S&P 500 Spot Index Stock index futures are financial derivative innovations that allow portfolio investors (for example, mutual fund managers and their customers) to hedge the market risk of their portfolios.

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Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

We also depict the opportunity set without risk free borrowing and lending (the dotted curve) and the efficient frontier when there is borrowing and lending at the risk free rate (the bold line). Under the five assumptions stated in the introduction to this section all investors will agree on the optimal allocation between all risky assets when there is no lending or borrowing. This allocation is called the market portfolio, and we denote it M. Efficient portfolios with riskless lending and borrowing Expected Return M P Rf Slope is Sharpe ratio for portfolio P StDev Figure I.6.10 Market portfolio Some investors may choose to borrow at the risk free rate and others may choose to lend at the risk free rate depending on their preferences, but the net allocation over all investors to the risk free asset must be zero.

For simplicity we refer to this risk measure as the CAPM 21 The expected excess return on a risky asset (or portfolio) is the difference between the expected return on the asset (or portfolio) and the risk free rate of return. Introduction to Portfolio Theory 253 beta of the asset. The term inside the bracket on the right-hand side is called the market risk premium: it is the additional return that investors can expect, above the risk free rate, to compensate them for the risk of holding the market portfolio. If we know the asset’s beta, the expected return on the market portfolio and the risk free rate of return then model (I.6.44) establishes the equilibrium value for the expected return on the asset, ERi .

Then by (I.5.36) the transition probability would be exp 0 05 × 0 25 − 0 8 = 0 4724 1 25 − 0 8 It can also be shown that the expression (I.5.36) for the transition probability is equivalent to the assumption that the underlying asset price S grows at the risk free rate.21 In other words, we have a risk neutral setting where assets earn the risk free rate of return. The relationship (I.5.35) illustrates that arbitrage free pricing leads to the risk neutral valuation principle, i.e. that the option price at any time t is equal to the discounted value of its expected future price. The expected value in (I.5.35) must refer to some probability distribution and we call this distribution the risk neutral measure.22 The fundamental theorem of arbitrage states that there is no arbitrage if and only if there is a risk neutral measure that is equivalent to the measure defined by the price process (e.g. a lognormal measure when the price process is a standard geometric Brownian motion).

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Valuation: Measuring and Managing the Value of Companies by Tim Koller, McKinsey, Company Inc., Marc Goedhart, David Wessels, Barbara Schwimmer, Franziska Manoury

If market prices eventually rise to reflect low interest rates (or interest rates rise to reflect market prices), make sure to reevaluate your perspective. Estimating the risk-free rate in typical times Once Treasury yields better reflect market prices, use the current yield on long-term government bonds to estimate the risk-free rate, and consequently the cost of equity. In choosing the bond’s duration, the most theoretically sound approach is to discount each year’s cash flow at a cost of equity that matches the maturity of the cash flow. In other words, year 1 cash flows would be discounted at a cost of equity based on a one-year risk-free rate, while year 10 cash flows would be discounted at a cost of equity based on a 10-year discount rate.

Capital asset pricing model Because the CAPM is discussed at length in modern finance textbooks,17 we focus only on the key ideas. The CAPM postulates that the expected rate of return on any security equals the risk-free rate plus the security’s beta times the market risk premium: E(Ri ) = rf + 𝛽i [E(Rm ) − rf ] where E(Ri ) = expected return of security i rf = risk-free rate 𝛽i = security i’s sensitivity to the market E(Rm ) = expected return of the market In the CAPM, the risk-free rate and the market risk premium, which is defined as the difference between E(Rm ) and rf , are common to all companies; only beta varies across companies. Beta represents a stock’s incremental risk to a diversified investor, where risk is defined as the extent to which the stock moves up and down in conjunction with the aggregate stock market.

The solution is to use excess returns ESTIMATING THE COST OF CAPITAL 503 over the risk-free rate, rather than total returns.13 Beta estimates are consistent across currencies when the stock’s excess returns are regressed against the excess return of a global market portfolio, as follows for any period ending at time t: ( ) ( ) A A A rA − r = 𝛽 r − r j M,t j,t f ,t f ,t where rA = realized return for stock j in currency A j,t rA = risk-free rate in currency A f ,t A rM,t = realized return for global market portfolio in currency A If the international Fisher relation and purchasing power parity would hold, differences in international interest rates would reflect differences in inflation across countries; and differences in inflation across countries would also be reflected in changes in exchange rates. In that case, the risk-free rate for each currency should equal the U.S. dollar risk-free return and the change in the exchange rate: ( ) ( )X t−1 1 + rA = 1 + r\$f ,t f ,t Xt (23.5) where r\$f ,t = risk-free rate in U.S. dollars Xt = exchange rate at time t of currency A expressed in U.S. dollars If risk-free rates across currencies are tied to changes in exchange rates in this way, beta estimates based on excess returns will be the same whether we use U.S. dollars or Swiss francs (or any other currency, for that matter).

Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

At time t • cash the dividend div and invest it at the risk-free rate for the remaining time T − t. At time T • sell the share for F (0, T ); • pay S(0)erT to clear the loan with interest and collect er(T −t) div. The ﬁnal balance will be positive: F (0, T ) − S(0)erT + er(T −t) div > 0, a contradiction with the No-Arbitrage Principle. On the other hand, suppose that F (0, T ) < [S(0) − e−rt div]erT . In this case, at time 0 • enter into a long forward contract with forward price F (0, T ) and delivery at time T ; • sell short one share and invest the proceeds S(0) at the risk-free rate. At time t • borrow div dollars and pay a dividend to the stock owner.

Given S(0), ﬁnd the price S(1) of the stock after one day such that the marking to market of futures with delivery in 3 months is zero on that day. This exercise shows an important benchmark for the proﬁtability of a futures position: An investor who wants to take advantage of a predicted increase in the price of stock above the risk-free rate should enter into a long futures position. A short futures position will bring a proﬁt should the stock price go down or increase below the risk-free rate. 6.2.2 Hedging with Futures The Basis. One relatively simple way to hedge an exposure to stock price variations is to enter a forward contract. However, a contract of this kind may not be readily available, not to mention the risk of default.

Another possibility is to hedge using the futures market, which is well-developed, liquid and protected from the risk of default. Example 6.2 Let S(0) = 100 dollars and let the risk-free rate be constant at r = 8%. Assume that marking to market takes place once a month, the time step being τ = 1/12. Suppose that we wish to sell the stock after 3 months. To hedge the exposure to stock price variations we enter into one short futures contract on the stock with delivery in 3 months. The payments resulting from marking to market are invested (or borrowed), attracting interest at the risk-free rate. The results for two diﬀerent stock price scenarios are displayed below. The column labelled 6.

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The Investopedia Guide to Wall Speak: The Terms You Need to Know to Talk Like Cramer, Think Like Soros, and Buy Like Buffett by Jack (edited By) Guinan

Related Terms: • Beta • Risk-Return Trade-Off • Unsystematic Risk • Correlation • Systematic Risk Risk-Free Rate of Return What Does Risk-Free Rate of Return Mean? The theoretical rate of return for an investment that has zero risk. The risk-free rate represents the expected return from an absolutely risk-free investment over a specified period. Investopedia explains Risk-Free Rate of Return In theory, the risk-free rate of return is the minimum return an investor expects for any investment because he or she will not accept additional risk unless the potential rate of return is greater than the risk-free rate. In practice, however, the risk-free rate does not exist 260 The Investopedia Guide to Wall Speak because even the safest investments carry a very small amount of risk.

Related Terms: • Bull Market • Capital Market Line—CML • Stock • Capital • Return on Assets Capital Market Line (CML) What Does Capital Market Line (CML) Mean? A line used in the capital asset pricing model that plots the rates of return for efficient portfolios, depending on the risk-free rate of return and the level of risk (standard deviation) for a particular portfolio. Investopedia explains Capital Market Line (CML) The CML is derived by drawing a tangent line from the intercept point on the efficient frontier to the point where the expected return equals the risk-free rate of return. The CML is considered superior to the efficient frontier because it takes into account the inclusion of a risk-free asset in the portfolio.

The return provided by an individual stock or the overall stock market in excess of the risk-free rate. This excess return compensates investors for taking on the relatively higher risk of the equity market. The size of the risk premium will vary as the risk in a particular stock, or in the stock market as a whole, changes; high-risk investments are compensated with a higher premium. Also referred to as the equity premium. Investopedia explains Equity Risk Premium The risk premium is the result of the risk-return trade-off, in which investors require a higher rate of return on riskier investments. The risk-free rate in the market often is quoted as the rate on longerterm U.S. government bonds, which are considered risk-free because of the unlikelihood that the government will default on its loans.

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Beyond Diversification: What Every Investor Needs to Know About Asset Allocation by Sebastien Page

In addition to beta, expected returns depend on the current risk-free rate and the expected market risk premium (market expected return – risk-free rate). Purists will point out that there’s no such thing as a risk-free rate. That’s correct, as even cash has nonzero volatility, and in theory, there’s default risk associated with even the safest of government bonds. However, practitioners aren’t purists, and US Treasury bonds are generally used as a proxy for the risk-free asset. (As an interesting side note, for liability-focused investors such as defined benefit plan sponsors, cash is far from the risk-free rate. In fact, on a surplus basis, cash is a high-volatility asset compared with a long-duration bond that matches the duration of the liabilities.

In fact, on a surplus basis, cash is a high-volatility asset compared with a long-duration bond that matches the duration of the liabilities. I’ll discuss liability-driven investing in Part Three of this book.) To account for the “historical data” critique, our estimate of the risk-free rate must be forward-looking. We can think of the CAPM as a simple building block approach: we start with the current risk-free rate and add a risk premium (scaled by the asset’s beta). Suppose we use the three-month US Treasury bill rate as our risk-free rate, and we want to calculate expected return on an asset class with a beta of 1. Also, suppose we estimate the total market risk premium to be 1.9%, and we assume it doesn’t change over time.

To calculate an asset’s beta, we multiply (a) the ratio of its volatility to the market’s volatility by (b) its correlation with the market. So the formula is Beta = (asset volatility/market volatility) × correlation where volatility is the standard deviation of returns, and the correlation is calculated between the asset and the market. Then we define expected return as follows: Expected return = risk-free rate + beta × (market expected return – risk-free rate) In their 2004 critique of the CAPM, Fama and French show that high-beta stocks can underperform low-beta stocks during long periods of time, which invalidates this equation. On the other hand, in his excellent monograph, titled Expected Returns on Major Asset Classes, Antti Ilmanen (2012) points out that from 1962 to 2009, there is a positive relationship between beta and the relative returns of stocks and bonds.

A Primer for the Mathematics of Financial Engineering by Dan Stefanica

OPTIONS. For continuously compounded interest, the value B(t2) at time t2 > tl of B(tl) cash at time tl is 1.9. THE PUT-CALL PARITY FOR EUROPEAN OPTIONS From Lemma 1.7, we conclude that C(t) P(t) (1.42) where r is the risk free rate between time tl and t2. The value B(tl) at time tl < t2 of B(t2) cash at time t2 is 37 O', K e -r(T-t) , where r is the constant risk free rate. (1.45) D (1.43) More details on interest rates are given in section 2.6. Formulas (1.42) and (1.43) are the same as formulas (2.46) and (2.48) from section 2.6. Lemma 1.7. If the value V(T) of a portfolio at time T in the future is independent of the state of the market at time T J then V(t) = V(T) e-r(T-t) , 1.9 The Put-Call parity for European options Let C(t) and P(t) be the values at time t of a European call and put option, respectively, with maturity T and strike K, on the same non-dividend paying asset with spot price S(t).

The four months at-the-money put and call options on this asset are trading at \$2 and \$4, respectively. The risk-free rate is constant and equal to 5% for all times. Show that the Put-Call parity is not satisfied and explain how would you take advantage of this arbitrage opportunity. 16. The bid and ask prices for a six months European call option with strike 40 on a non-dividend-paying stock with spot price 42 are \$5 and \$5.5, respectively. The bid and ask prices for a six months European put option with strike 40 on the same underlying asset are \$2.75 and \$3.25, respectively. Assume that the risk free rate is equal to O. Is there an arbitrage opportunity present?

If the value V(T) of a portfolio at time T in the future is independent of the state of the market at time T J then V(t) = V(T) e-r(T-t) , 1.9 The Put-Call parity for European options Let C(t) and P(t) be the values at time t of a European call and put option, respectively, with maturity T and strike K, on the same non-dividend paying asset with spot price S(t). The Put-Call parity states that P(t) (1.44) + S(t) - C(t) = K e-r(T-t). (1.46) If the underlying asset pays dividends continuously at the rate q, the Put-Call parity has the form where t < T and r is the constant risk free rate. Proof. For clarity purposes, let z = V(T) be the value of the portfolio at time T. Consider a portfolio made of l;2(t) = ze-r(T-t) cash at time t. The value l;2(T) of this portfolio at time T is l;2(T) = er(T-t) l;2(t) = er(T-t) (ze-r(T-t)) z', cf. (1.42) for tl = t, t2 = T, and B(t) = l;2(t).

Solutions Manual - a Primer for the Mathematics of Financial Engineering, Second Edition by Dan Stefanica

. , with T - t large) ATJVI calls on an underlying asset paying dividends continuously at a rate equal to the constant risk-free rate , i.e. , with q = r , show that the Theta may be positive. θ(vega(P))θ(vega(C)) = z Volga(C); θσθσ θ(vega(P))θ(vega(C)) Solution: vama(P)===vanm(C)? θSθs where volga(C) and van叫 C) are given by (3.8) and (3.9) , respectively. 口 (i) 丑ecall that Sσe-q(T-t) ~ @ ( C ) = - e 2 十 qSe-q(T-t)N(d 1 ) 2 、/2π(T - - t) r K e-r(T-t) N(d 2 ). Problem 10: Show that an ATM call on an underlying asset paying dividends continuously at rate q is worth more than an ATM put with the same maturity if and only if q 三飞 where r is the constant risk free rate. Use the Put-Call parity, and then use the Black-Scholes formula to prove this result.

Problem 2: Assume that an asset with spot price 50 paying dividends continuously at rate q = 0.02 has lognormal distribution with mean μ= 0.08 and volatility σ= 0.3. Assume that the risk-free rates are constant and equal to r = 0.05. (i) Find 95% and 99% confidence intervals for the spot price of the asset in 15 days , 1 month , 2 months , 6 months , and 1 year. CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING. 108 (ii) Find 95% and 99% risk-neutral con且dence intervals for the spot price of in 15 days , 1 month , 2 months , 6 months , and 1 year , i.e. , assuming that the drift of the asset is equal to the risk-free rate. theωset Solution: If the asset has lognormal distribution , then S(t) = = S(O) 叫( (μ-q-f)t 十 σ0z }) 2 } 飞飞 TXT 50 exp(O.Ol 日十 0.30z)

Then , the short position in e- q1 shares at time 0 will become a short position in one share 1 at time T. The value of the portfolio at maturity is V(T) Problem 15: A stock with spot price 40 pays dividends continuously at a rate of 3%. The four months at-the-money put and call options on this asset are trading at \$2 and \$4 , respectively. The risk-free rate is constant and equal to 5% for all times. Show that the Put-Call parity is 丑at satis丑ed and explain how would you take advantage of this arbitrage opportunity. Solution: The followi吨 values are give口 : S = 40; K = 40; T = 1/3; r = 0.05; P(T) + S(T) - C(T) - C - 39.5821 > 39.3389 - Ke- = max(K - S(T) , 0) + S(T) - max(S(T) - K , 0) = K , regardless of the value S(T) of the underlying asset at maturity.

Then the excess return used in calculating the Sharpe ratio is R + rF – rF = R. So, essentially, you can ignore the risk-free rate in the whole calculation and just focus on the returns due to your stock positions. Similarly, if you have a long-only day-trading strategy that does not hold positions overnight, you again have no need to subtract the risk-free rate from the strategy return in order to obtain the excess returns, since you do not have financing costs in this case, either. In general, you need to subtract the risk-free rate from your strategy returns in calculating the Sharpe ratio only if your strategy incurs financing cost.

The Sharpe ratio is actually a special case of the information ratio, suitable when we have a dollar-neutral strategy, so that the benchmark to use is always the risk-free rate. In practice, most traders use the Sharpe ratio even when they are trading a directional (long or short only) strategy, simply because it facilitates comparison across different strategies. Everyone agrees on what the riskfree rate is, but each trader can use a different market index to come up with their own favorite information ratio, rendering comparison difficult. (Actually, there are some subtleties in calculating the Sharpe ratio related to whether and how to subtract the risk-free rate, how to annualize your Sharpe ratio for ease of comparison, and so on.

There is one subtlety that often confounds even seasoned portfolio managers when they calculate Sharpe ratios: should we or shouldn’t we subtract the risk-free rate from the returns of a dollarneutral portfolio? The answer is no. A dollar-neutral portfolio is P1: JYS c03 JWBK321-Chan September 24, 2008 13:52 44 Printer: Yet to come QUANTITATIVE TRADING self-financing, meaning the cash you get from selling short pays for the purchase of the long securities, so the financing cost (due to the spread between the credit and debit interest rates) is small and can be neglected for many backtesting purposes. Meanwhile, the margin balance you have to maintain earns a credit interest close to the risk-free rate rF . So let’s say the strategy return (the portfolio return minus the contribution from the credit interest) is R, and the riskfree rate is rF .

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

Wilmott magazine March 66-78 What is the Market Price of Risk? Short Answer The market price of risk is the return in excess of the risk-free rate that the market wants as compensation for taking risk. Example Historically a stock has grown by an average of 20% per annum when the risk-free rate of interest was 5%. The volatility over this period was 30%. Therefore, for each unit of risk this stock returns on average an extra 0.5 return above the risk-free rate. This is the market price of risk. Long Answer In classical economic theory no rational person would invest in a risky asset unless they expect to beat the return from holding a risk-free asset.

Sharpe Ratio The Sharpe ratio is probably the most important non-trivial risk-adjusted performance measure. It is calculated as where µ is the return on the strategy over some specified period, r is the risk-free rate over that period and σ is the standard deviation of returns. The Sharpe ratio will be quoted in annualized terms. A high Sharpe ratio is intended to be a sign of a good strategy. If returns are normally distributed then the Sharpe ratio is related to the probability of making a return in excess of the risk-free rate. In the expected return versus risk diagram of Modern Portfolio Theory the Sharpe ratio is the slope of the line joining each investment to the risk-free investment.

The latter term can be thought of as compensation for taking risk. But the asset and its option are perfectly correlated, so the compensation in excess of the risk-free rate for taking unit amount of risk must be the same for each. For the stock, the expected return (dividing by dt) is µ. Its risk is σ. From Itô we have Therefore the expected return on the option is and the risk is Since both the underlying and the option must have the same compensation, in excess of the risk-free rate, for unit risk Now rearrange this. The µ drops out and we are left with the Black-Scholes equation. Utility Theory The utility theory approach is probably one of the least useful of the ten derivation methods, requiring that we value from the perspective of an investor with a utility function that is a power law.

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The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

However, they had already concluded that the stock’s expected return should not feature in the equation, so they recast Sprenkle’s solution in the special case in which the risk of the option is exactly offset by the risk of the shorted stock, which was what the optimal hedge weighting was designed to do. As such, they set the expected return of the stock to the risk-free rate of The Black-Scholes Options Pricing Theory 113 return, and likewise the discount rate to the risk-free rate. From this, they calculated the value of an option that would result if the underlying stock were risk-free. The resulting solution also satisfied Black’s differential equation, so their hunch turned out to be the back door solution to the equation.

Assumptions for Merton’s derivation Merton’s intuition took a leaf out of the book of the Chicago School. In the absence of transaction costs, the correct combination of any two of the following instruments should be able to predict the third if arbitrage exists: the risk-free rate of return, a stock price, an option written on the stock price. In this case, his dynamic (continuous) trading strategy using just the stock and the risk-free rate of return should price the option as predicted by the Black-Scholes equation in the absence of arbitrage opportunities. Merton demonstrated the accuracy of Black-Scholes through the application of techniques he had already developed in his work in 154 The Rise of the Quants continuous-time portfolio selection.

Return – the expected surplus offered to entice individuals to hold a financial instrument. Rho ␳ – the effect on the option price for a single percentage point change in the risk-free rate of return. Risk – in finance, the degree of uncertainty associated with exchanging a known sum for a larger future but less certain sum. Risk-averse – a property that states an agent would prefer less risk to more for an equal return. Risk-free asset – an asset that yields a certain return over all possible states Risk-free rate of return – the return offered by an asset that does not vary over future states. Risk–reward trade-off – an individual’s determination of the required reward to compensate for additional risk.

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Asset and Risk Management: Risk Oriented Finance by Louis Esch, Robert Kieffer, Thierry Lopez

Example Let us consider a call option of seven months’ duration, relating to an equity with a current value of ¤100 and an exercise price of ¤110. It is assumed that its volatility is σR = 0.25, calculated on an annual basis, and that the risk-free rate is 4 % per annum. We will assess the value of this call at t = 0 by constructing a binomial tree diagram with the month as the basic period. The equivalent volatility and risk-free rate as given by: 1 · 0.25 = 0.07219 12 √ 12 RF = 1.04 − 1 = 0.003274 σR = We therefore have u − 1/u = 0.1443, for which the only positive root is13 u = 1.07477 (and therefore d = 0.93043). The risk-neutral probability is: q= 1.003274 − 0.93043 = 0.5047 1.07477 − 0.93043 13 If we had chosen α = 1/3 instead of 1/2, we would have found that u = 1.0795, that is, a relatively small difference; the estimation of u therefore only depends relatively little on α.

Under this hypothesis, it can therefore be said that for a portfolio of value V put together at moment 0, if VT = 0, no ﬁnancial movement occurs in that portfolio between 0 and T and the interest rate does not vary during that period and is valid for any maturity date (ﬂat, constant rate curve), then Vt = 0 for any t ∈ [0; T ]. This hypothesis of absence of arbitrage can be expressed as follows: in the context mentioned above, a portfolio which has been put together so as not to contain any random element will always present a return equal to the risk-free rate of interest. The concept of ‘valuation model’ A valuation model for a ﬁnancial asset is a relation that expresses quite generally the price p (or the return) for the asset according to a number of explanatory variables3 2 See for example Miller and Modigliani, Dividend policy, growth and the valuation of shares, Journal of Business, 1961.

In addition, even the basic model gives good results, as do the applications that arise from it (see Sections 3.3.3, 3.3.4 and 3.3.5). 3.3.1.2 Separation theorem This theorem states that under the conditions speciﬁed above, all the portfolios held by the investors are, in terms of equilibrium, combinations of a risk-free asset and a market portfolio. According to the hypotheses, all the investors have the same efﬁcient frontier for the equities and the same risk-free rate RF . Therefore, according to the study of Markowitz’s model with the risk-free security (Section 3.2.5), each investor’s portfolio is located on the straight line issuing from point (0, RF ) and tangential to the efﬁcient frontier. This portfolio consists (see Figure 3.25) of: • The risk-free equity, in proportion X. • The portfolio A, corresponding to the tangent contact point, in proportion 1 − X.

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The Big Secret for the Small Investor: A New Route to Long-Term Investment Success by Joel Greenblatt

So first we compare a potential investment against what we could earn risk-free with our money (for purposes of our discussion and for reasons that will be detailed later, we have set the minimum risk-free rate that we will have to beat at a 6 percent annual return). If we have high confidence in our estimates and our investment appears to offer a significantly higher annual return over the long term than the risk-free rate, we’ve passed the first hurdle. Next, we compare our potential investment with our other investment alternatives. In our example, Bad Bob’s offers a higher expected annual return and a higher growth rate, and we are even more confident in our estimates than we are for Candy’s Candies.

Instead, when we evaluate the purchase price of a company, we make sure that our investment will return more than the 6 percent per year we could earn risk-free from the U.S. government (see the box on this page for further explanation). If our investment appears to offer a significantly higher annual return over the long term than the risk-free rate and we have high confidence in our estimates, we’ve passed the first hurdle. Next, we compare the expected annual returns of our potential investment and our level of confidence in those returns to our other investment alternatives. If we can’t make a good estimate of the future earnings for a particular company, we skip that one and find a company we can evaluate.

Even after we come up with these guesses, very small changes in our estimates for future earnings, growth rates, and discount rates (reflecting our level of confidence in our earnings estimates) make a huge difference in our ultimate estimates of value. Other methods, such as relative value, acquisition value, liquidation value, and sum-of-the-parts analysis, are difficult to use and can often lead to seriously inaccurate estimates of value. Even when we simplify things and compare our investment alternatives to the risk-free rate and to each other, we still must rely on the accuracy of our estimates and our ability to assess our level of confidence in those estimates. Needless to say, making earnings estimates, risk assessments, and comparisons for not just one but dozens or even hundreds of companies, and doing it well, must be really hard.

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Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined by Lasse Heje Pedersen

., Rb = 0), so they report the IR simply as which is always higher than the SR, since it does not subtract the risk-free rate. Even if many hedge funds report this number, I view it as an unreasonable number as it gives the hedge fund credit for earning the risk-free rate (and depends on the level of interest rates). The IR is almost always reported as an annualized number, as discussed further below. Both the SR and the IR are ways of calculating risk-adjusted returns, but some traders and investors say, You can’t eat risk-adjusted returns. Suppose, for instance, that a hedge fund beats the risk-free rate by 3% at a tiny risk of 2%, realizing an excellent SR of 1.5.

The performance of the market-neutralized strategy is then Since the idiosyncratic risk εt is zero on average, the expected excess return of the market-neutral strategy is We see that the expected return in excess of the risk-free rate and the exposure to the market is given by the alpha, α. Alpha is clearly the sexiest term in the regression: It is the Holy Grail all active managers seek. Alpha measures the strategy’s value added above and beyond the market exposure due to the hedge fund’s trading skill (or luck, given that alpha is estimated based on realized returns). If a hedge fund has a beta of zero (i.e., a market-neutral hedge fund) and an alpha of 6% per year, this means that the hedge fund is expected to make the risk-free return plus 6% per year. For instance, if the risk-free rate is 2% per year, the hedge fund is expected to make 8%, but the actual realization could be far above or below that, depending on the realized idiosyncratic shock.

He has not only delivered a high Sharpe ratio, he has also delivered much higher absolute returns than the overall stock market, on average beating the risk-free rate by 19%, about three times the overall stock market’s excess return of 6.1% per year. Berkshire’s volatility of 25% is significantly higher than that of the market, in part because Buffett has leveraged his equity investments about 1.6-to-1. Buffett’s leverage comes from several sources. First, Berkshire has issued highly rated bonds at low yields, enjoying a AAA rating from 1989 to 2009. Second, Berkshire has financed about a third of its liabilities with its insurance float at an average cost below the risk-free rate. To understand this usually cheap and stable source of financing, note that Berkshire operates insurance and reinsurance companies and, when these companies sell insurance, they collect the premiums up front and pay a diversified set of claims later, which is like getting a loan.

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Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

It is called the capital market line (CML) and represents an optimal set of portfolios, made of the efficient portfolio of stocks B and of risk-free instruments in various proportions. In particular: at A: the portfolio is 100% invested in the risk-free rate; at B: 100% investment in an efficient portfolio of stocks; between A and B: mixed portfolio, invested at x% in the risk-free rate and (1 − x)% in the efficient portfolio of stocks; beyond B: leveraged portfolio, assuming the investor has borrowed money (at the rf rate) and has then invested >100% of his available resources in an efficient portfolio. For a given investor, characterized by some utility function U, representing his well-being, assuming his wealth as a portfolio P, if the portfolio return were certain (i.e., deterministic), we would have but, more realistically (even if simplified, in the spirit of this theory), if the portfolio P value is normally distributed in returns, with some rP and σP, where f is some function, often considered as a quadratic curve.4 So that, given the property of the CML (i.e., tangent to the efficient frontier), and some U = f(P) curve, the optimal portfolio must be located at the tangent of U to CML, determining the adequate proportion between B and risk-free instrument.

Fortunately, for the present chapter, it is enough to know that bonds and swaps are used here only as “sources” of interest rates, without being concerned by how they run. To build a term structure you first need to determine the market and the kind of debtors the curve will refer to. Historically speaking, one determined a yield curve referring to risk-less Organisation for Economic Co-operation and Development (OECD) government bonds, 2 hence using risk-free rates. For non-risk-less debtors, of lower rating, a spread was added upon, depending on the maturity and on the degree of risk taken on the issuer's name. This procedure was justified for two reasons: 1. On mature markets, the government counterparty risk is the only fully objective and clearly identified (non-defaultable sovereign risk of OECD countries). 2.

There is actually a homogeneity in swap market rates, although their counterpart risk level – the big banks of OECD countries – remains rather heterogeneous: AAA rates cohabit with various AA sub-classes, or lower. Altogether, this has not prevented the swap market rates from superseding government/risk-free rates as reference or benchmark rates, except – up to now – in the US. As a result, except for the USD yield curve, market practitioners prefer to start from a swap yield curve and, for each maturity, deduct some spread to obtain the corresponding risk-free yield curve, or add a spread to quote corporate bonds of other issuers of lower rating, or to penalize a restricted liquidity.

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Investment Banking: Valuation, Leveraged Buyouts, and Mergers and Acquisitions by Joshua Rosenbaum, Joshua Pearl, Joseph R. Perella

EXHIBIT 3.15 Calculation of CAPM where: rf = risk-free rate βL = levered beta rm = expected return on the market rm - rf = market risk premium Risk-Free Rate (rf) The risk-free rate is the expected rate of return obtained by investing in a “riskless” security. U.S. government securities such as T-bills, T-notes, and T-bonds88 are accepted by the market as “risk-free” because they are backed by the full faith of the U.S. federal government. Interpolated yields89 for government securities can be located on Bloomberg90 as well as the U.S. Department of Treasury website,91 among others. The actual risk-free rate used in CAPM varies with the prevailing yields for the chosen security.

Investment banks may differ on accepted proxies for the appropriate risk-free rate, with some using the yield on the 10-year U.S. Treasury note and others preferring the yield on longer-term Treasuries. The general goal is to use as long dated an instrument as possible to match the expected life of the company (assuming a going concern), but practical considerations also need to be taken into account. Due to the moratorium on the issuance of 30-year Treasury bonds92 and shortage of securities with 30-year maturities, Ibbotson Associates (“Ibbotson”)93 uses an interpolated yield for a 20-year bond as the basis for the risk-free rate.94,95 Market Risk Premium (rm - rf or mrp) The market risk premium is the spread of the expected market return96 over the risk-free rate.

Due to the moratorium on the issuance of 30-year Treasury bonds92 and shortage of securities with 30-year maturities, Ibbotson Associates (“Ibbotson”)93 uses an interpolated yield for a 20-year bond as the basis for the risk-free rate.94,95 Market Risk Premium (rm - rf or mrp) The market risk premium is the spread of the expected market return96 over the risk-free rate. Finance professionals, as well as academics, often differ over which historical time period is most relevant for observing the market risk premium. Some believe that more recent periods, such as the last ten years or the post-World War II era are more appropriate, while others prefer to examine the pre-Great Depression era to the present. Ibbotson tracks data on the equity risk premium dating back to 1926. Depending on which time period is referenced, the premium of the market return over the risk-free rate (rm - rf) may vary substantially. For the 1926 to 2007 period, Ibbotson calculates a market risk premium of 7.1 %.97 Many investment banks have a firm-wide policy governing market risk premium in order to ensure consistency in valuation work across their various projects and departments.

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The Asian Financial Crisis 1995–98: Birth of the Age of Debt by Russell Napier

Of course they had added a premium to that rate to account for the greater risks associated with investment in Asia, but the building block had still been the risk-free rate of the world’s largest economy. They were using the risk-free rate of an economy with a sophisticated commercial infrastructure, a high-quality rule of law, an independent judiciary and a democracy that had weathered many economic cycles and two world wars. To import that US risk-free rate into an economy such as Indonesia was nuts, but it is what happened, and none of the risk premiums added to that risk-free rate came anywhere close to the rupiah interest rates that developed once the currency was allowed to float.

The good news about events in Thailand is that policy makers are increasingly likely to move proactively. The downside of the current monetary regimes is now as obvious to politicians as it is to investors. Thus investors in Asia must now begin to factor in the domestic risk-free rate when valuing equities rather than the US dollar risk-free rate. Hopefully yesterday’s events will mean that you will never again have to listen to any equity sales people expound on why Indonesian equities are ‘cheaper’ than Singapore equities because the PE is lower. “One small step for ….” Market movements across Asia suggest that this new reality is already beginning to be priced into equity markets.

That regime was now threatened with extinction by the very organisation some investors thought had arrived to save it. The luxury and excitement of Southeast Asia 9 September 1997, New Asia At this point in history, the investor is in a luxurious position in regard to analysis using the domestic risk-free rate. Equities in Southeast Asia (ex Singapore) are so overvalued that risk-free rates would have to fall significantly below the pre-’Mekong’ crisis levels before fair value could begin to be established. At this stage the investor’s position is straightforward. With interest rates very unlikely to return even to pre-Mekong crisis levels over the next 12 months, these markets are sells now.

Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

For instance, there is the question of how to extract the appreciation rate a(t) and volatility σ(t) from the time series of prices. Interpretation of historical data in view of continuous time models is studied in Chapter 9. 5.15 Problems Below, a is the appreciation rate, σ is the volatility, r is the risk-free rate, S(t) is the stock price, X(t) is the wealth, is the discounted stock price, is the discounted wealth, (β(·), γ(·)) is a self-financing strategy, where γ is the quantity of the stock shares, β is the quantity of the bonds. Self-financing strategies for a continuous time market Problem 5.69 Let r(t)≡0.

Problem 5.76 Find an initial wealth and a strategy that replicates the claim (Hint: use Theorem 5.38 and find the solution V of the boundary value problem 5.15 from the proof of Theorem 5.38 for φ(x, t)=e−rT ert x/T, Ψ≡0; V can be found explicitly as Problem 5.77 Consider an option with payoff where K>0 (it is the so-called digital option). Express the option price via an integral with the probability density function of a Gaussian random variable. Problem 5.78 (Bachelier’s model). Consider a market model where the risk-free rate r≥0 is constant and known, and where the stock price evolves as dP(t)=adt+σ dw(t), where σ>0 is a given constant, w(t) is a Wiener process. Assume that the fair price for a call option is e−rTE* max(P(T)−K, 0), where K is the strike price, T is termination time. Here E* is the expectation defined by the risk-neutral probability measure (this measure gives the same probability distribution of P(·) as the original measure for the case when r=a).

© 2007 Nikolai Dokuchaev 6 American options and binomial trees This chapter introduces numerical methods for option pricing based on the so-called binomial trees. The binomial trees method is important since it can be used for complicated cases when the Black-Scholes formula is not applicable: • for American options and exotic options; • for models with time-variable random volatility or the risk-free rate. We will first demonstrate how to apply the binomial trees for European options, and then this method will be extended for American options. 6.1 The binomial tree for stock prices 6.1.1 General description Binomial trees are used to approximate the distributions of continuous time random processes of the stock price S(t) via discrete time processes.

Commodity Trading Advisors: Risk, Performance Analysis, and Selection by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant, Fabrice Douglas Rouah

The long OEX put option is used to capture the synthetic long put option exposure. The long S&P 100 index is used to capture any residual market risk that exists when the market performs positively. Last, we use the risk-free rate to measure the option premium that must be paid by CTAs to the right-hand side of the threshold value (when the stock market performs positively). We use the coefficient estimates from equation 9.3 to construct the mimicking portfolio. Long OEX Put Option Strike = OEX index × (1 + Threshold + risk-free rate) Volatility = VIX index The number of options bought = (blow − bhigh) Short the S&P 100 4 The number of S&P 100 to buy is = bhigh Long Risk-Free Security The number of risk-free securities to buy = 1 − blow Figures 9.5 through 9.8 present the results from our mimicking portfolios.

We use the Extreme metrics software available on the www.alternativesoft.com web site to compute the results using a 99 percent VaR probability, and we assume that we are able to borrow at a risk-free rate of 0 percent. The difference between the traditional and modified Sharpe ratio is that, in the latter, the standard deviation is replaced by the modified VaR in the denominator. The traditional Sharpe ratio, generally defined as the excess return per unit of standard deviation, is represented by this equation: Sharpe Ratio = Rp − RF σ where RP = return of the portfolio RF = risk-free rate and s = standard deviation of the portfolio (22.1) 380 PROGRAM EVALUATION, SELECTION, AND RETURNS A modified Sharpe ratio can be defined in terms of modified VaR: Modified Sharpe Ratio = Rp − RF MVaR (22.2) The derivation of the formula for the modified VaR is beyond the scope of this chapter.

Return Maximum Uninterrupted Loss Excess Kurtosis Skewness % of Winning Months Average Winning Return % of Losing Months Average Losing Return 0.73% 0.65% 6.91% −5.43% −5.43% −0.10 0.15 56.79% 2.52% 43.21% −1.62% S&P 500 Lehman Global Bond Index 0.50% 0.76% 9.67% −14.58% −20.55% −0.28 −0.43 55.56% 4.32% 44.44% −4.27% 0.06% 0.12% 2.15% −3.94% −6.75% 1.44 −0.76 54.32% 0.83% 45.68% −0.85% 9.17% 0.84% 0.39% 0.49% 17.94% 3.22% 1.76% 1.85% 3.75% 0.14% 0.08% 0.12% VaR (99%) Modified VaR (99%) −6.89% −6.52% −12.55% −13.49% −2.58% −3.31% Sharpe Ratio Sortino Ratio (MAR = Rf*) 0.72 11.01 Monthly Std Deviation Ann’d Monthly Variance Ann’d Monthly Semivariance Ann’d Monthly Downside Risk (MAR = Rf*)** 0.21 1.05 −0.39 −8.11 **The risk-free rate is calculated as the 3-month LIBOR average over the period January 1997 to September 2003, namely 4.35 percent. **This indicator is also referred to as the lower partial moment of order 2. and −8.11) due to a limited downside risk (i.e., 0.49 percent versus 1.85 percent for the S&P 500). The Edhec CTA Index posts positive returns in about 57 percent of months, with an average gain of 2.52 percent versus an average loss of −1.62 percent in 43 percent of the cases.

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Triumph of the Optimists: 101 Years of Global Investment Returns by Elroy Dimson, Paul Marsh, Mike Staunton

This required return should be the risk free rate plus a premium for risk. What overall cost of equity should companies be using? Perhaps the simplest case is to consider a hypothetical investment in a global project, one that spans all the countries represented in this book, and whose nature is not related to the sovereignty of the investing company. The required rate of return for such a capital investment should reflect the attributes of the project, not those of the country in which the corporation happens to be headquartered. As can be seen in Figure 15-1, the real risk free rate of return is essentially the same everywhere (except in two countries, South Africa and the United Kingdom, where it would be necessary to adjust respectively for default risk and tax advantages).

From Figure 4-2, we saw that this was equivalent to 0.9 percent in real terms. UK treasury bills gave a higher nominal return of 5.1 percent (see Figure 4-3), and also a slightly higher real return of 1.0 percent (see Figure 4-4). Figure 5-4 shows the path of US bill rates over time. From 1900–30, the short-term risk free rate averaged 4.6 percent, with interest rates at their highest around the First World War, peaking at 7.6 percent in 1920. From the early 1930s until the mid-1950s, interest rates were very low, averaging around 0.5 percent. During the 1930s, rates were understandably very low, as this was a largely deflationary period.

Switzerland was a low inflation country throughout the twentieth century. US inflation was higher in the second half of the twentieth century than in the first, although the US inflation rate was below average in both halves of the century. Treasury bills are an important asset class since they tell us the return on cash, and provide a benchmark for the risk free rate. US bill investors earned an annualized real return of 0.9 percent from 1900–2000, while UK investors earned a virtually identical 1.0 percent. Over this 101-year period, investors in five countries, Germany, France, Italy, Belgium, and Japan, earned negative real returns on bills. In 1923, German bill (and bond) investors lost everything, reminding us that, although we can generally regard short-dated government bills as risk free, in extreme circumstances this ceases to be the case.

Principles of Corporate Finance by Richard A. Brealey, Stewart C. Myers, Franklin Allen

The stock market provided an average return of 11.3%, a premium of 7.3% over the safe rate of interest. This gives us two benchmarks for the opportunity cost of capital. If we are evaluating a safe project, we discount at the current risk-free rate of interest. If we are evaluating a project of average risk, we discount at the expected return on the average common stock. Historical evidence suggests that this return is 7.3% above the risk-free rate, but many financial managers and economists opt for a lower figure. That still leaves us with a lot of assets that don’t fit these simple cases. Before we can deal with them, we need to learn how to measure risk.

Treasury bonds were higher still, at about 2.7% on 20-year bonds. The CAPM is a short-term model. It works period by period and calls for a short-term interest rate. But could a .02% three-month risk-free rate give the right discount rate for cash flows 10 or 20 years in the future? Well, now that you mention it, probably not. Financial managers muddle through this problem in one of two ways. The first way simply uses a long-term risk-free rate in the CAPM formula. If this short-cut is used, then the market risk premium must be restated as the average difference between market returns and returns on long-term Treasuries.10 The second way retains the usual definition of the market risk premium as the difference between market returns and returns on short-term Treasury bill rates.

Its only remaining asset is 20 million reals in cash. If Bahia’s managers do nothing, the bondholders will receive all of this cash, which is earning interest at the risk-free rate of 15%. Bahia can undertake the following projects. Project X: Cost now is 20 million reals. Payoff one year from now is 160 million reals with 5% probability and zero reals (complete wipe-out) with 95% probability. Beta is zero (βX = 0), so the opportunity cost of capital is the risk-free rate. Project Y: Cost now is 20 million reals. Payoff one year from now is 30 million reals with 100% probability. Project Z: Cost now is 20 million reals.

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The New Science of Asset Allocation: Risk Management in a Multi-Asset World by Thomas Schneeweis, Garry B. Crowder, Hossein Kazemi

This leads to the so-called CAPM: E ( Ri ) − Rf = [ E ( Rm ) − Rf ] βi βi = Corr ( Ri , Rm ) × σi σm where Rf = Return on the riskless asset E(Rm) and E(Ri) = Expected returns on the market portfolio and a security σm and σi = Standard deviations of the market portfolio and the security Corr(Ri,Rm) = Correlation between the market portfolio and the security CML Expected Return Markowitz Efficient Frontier Market Portfolio Risk-Free Rate Standard Deviation EXHIBIT 1.2 Capital Market Line 6 THE NEW SCIENCE OF ASSET ALLOCATION Expected Return SML Market Portfolio Risk-Free Rate 0 1 2 Beta EXHIBIT 1.3 Security Market Line Thus, in the world of the CAPM all the assets are theoretically located on the same straight line that passes through the point representing the market portfolio with beta equal to 1.

CLASSIC SHARPE RATIO For much of this and the previous chapter we have emphasized the wide range of risks involved in asset allocation and security return estimation; however, for many, when the choice is between two (or more) assets, one way of ranking investments (the Sharpe Ratio) is based on simplifying risk into a single parameter (e.g., standard deviation). This ratio essentially divides the return of the security (after first subtracting the risk-free rate of return) by the price risk (standard deviation of return) of the security. The higher the ratio, the more favorable the assumed risk-return characteristics of the investment. The Sharpe Ratio is computed as: Si = (Ri − Rf ) σi where R̄i is the estimated mean rate of return of the asset, Rf is the risk-free rate of return, and σi is the estimated standard deviation. This measure can be taken to show return obtained per unit of risk. While the Sharpe Ratio does offer the ability to rank assets with different return and risk (measured as standard deviation), its use may be limited to comparing portfolios that may realistically be viewed as alternatives to one another.

For example, the Sharpe Ratio, defined as: Si = (Ri − Rf ) σi was meant to provide evidence of the relative benefit of two efficient risky portfolios on the capital market line and became the performance measurement vehicle of choice. Note that the Sharpe Ratio for an individual asset or portfolio merely provides evidence of the number of standard deviations the mean return of a portfolio/asset is from the risk-free rate. 8. It is hard to remember the importance of the initial studies which demonstrated the return to risk benefits of international investment. However the studies failed to emphasize the point that if the two international financial markets were separated to any great detail, the historical risk relationships may not tell us much about the expected return to risk relationships after the two countries became integrated (e.g., new market portfolio).

Trading Risk: Enhanced Profitability Through Risk Control by Kenneth L. Grant

The second will tell you what kind of portfolio volatility you should target in order to yield a specific return (again given a Sustainable Sharpe). Both equations can be extremely useful in the determination of appropriate target ranges of exposure. To illustrate, consider a portfolio with the following characteristics: Target Return: Risk-Free Rate: Return Risk-Free Rate: Sustainable Sharpe: 25% 5% 20% 2.0 We take the target return, the risk-free rate, and the Sustainable Sharpe as constants in this analysis, meaning in the case of the Sustainable Setting Appropriate Exposure Levels (Rule 1) 113 Sharpe that the account in question believes it can generate more than \$2 in return for \$1 of exposure it takes.

Our “Sustainable Sharpe” is one that uses the historical Sharpe (and perhaps other inputs) to derive a lower bound for what we can confidently achieve as a Sharpe Ratio on a going forward basis. The results of this algebraic maneuvering are as follows: 1. (Return Risk-Free Rate) Sustainable Sharpe Portfolio Volatility 2. Portfolio Volatility (Return Risk-Free Rate)/Sustainable Sharpe These equations define a concept that I will refer to as the Inverted Sharpe Methodology for setting exposure levels. The first equation is designed to provide you with some idea of the amount of return you can expect given your Sustainable Sharpe and your current level of volatility.

Inserting this figure, along with the Sustainable Sharpe and Risk-Free Rate parameters into equation 1, and solving for Return, we find that this portfolio is likely to produce returns of approximately 19%—respectable, but significantly below our target return of 25%. This analysis begs the question of what portfolio volatility is consistent with a 25% return target, again assuming the portfolio in question is able to sustain a Sharpe Ratio of 2.0. We can arrive at this answer quite simply using the second equation by inputting Return, Risk-Free Rate, and Sustainable Sharpe, and solving for Portfolio Volatility.

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In Pursuit of the Perfect Portfolio: The Stories, Voices, and Key Insights of the Pioneers Who Shaped the Way We Invest by Andrew W. Lo, Stephen R. Foerster

Recall that Markowitz had been able to identify various portfolios of risky securities that met this criterion, which he referred to as “efficient portfolios.” FIGURE 3.3: Lending and borrowing at the risk-free rate extends investment possibilities. The optimal risky portfolio—the one closest to the “most desirable” area in the graph— is the portfolio M, which is also the market portfolio. The capital market line shows combinations of borrowing or lending at the risk-free rate and investing in M. By combining risk-free lending (or borrowing) with investing in risky assets, it turns out that among the various efficient portfolios that Markowitz had identified, there was only one special portfolio of risky assets that all investors would want to hold: the market portfolio.

The greater the sensitivity of a stock’s return to the market’s return (a variable now known as beta, or β), the greater the expected return for that stock, as shown in figure 3.4. The equation for the security market line is also the now-famous CAPM equation: E(R) = Rf + β × (Rm − Rf), where E(R) is a stock’s expected return, Rf is the risk-free rate of return, β is a stock’s riskiness relative to the overall market, and (Rm − Rf) is the expected return on the market in excess of the risk-free rate of return, also known as the market risk premium, or MRP.29 FIGURE 3.4: The security market line compares a stock’s expected return with its risk as measured by beta (β). According to the CAPM, all stocks (held in a diversified portfolio) should fall along the security market line.

Since every theoretical model is a simplification of the real world, Sharpe started by making assumptions. Sharpe assumed that investors could not only invest in risky securities but could also borrow or lend at the same riskless rate, such as the Treasury bill rate, the rate at which the U.S. government can borrow money over the short term. Lending and borrowing at the risk-free rate extended the investment possibilities for an investor. Lending was the same as buying a Treasury bill. Borrowed money was then invested in risky assets. Another assumption was that in the theoretical world Sharpe created, everyone would want to hold the “best” possible portfolio of securities, in the Markowitz sense.

The Concepts and Practice of Mathematical Finance by Mark S. Joshi

(8.20) 210 Continuous barrier options The property that Brownian motion starts anew at stopping times is sometimes called the strong Markov property. 8.5 Girsanov's theorem revisited We have derived a formula for the joint law of minimum and terminal value for a driftless Brownian motion. Unfortunately this is a not a great deal of help, as a stock in a non-zero interest rate environment will have drift equal to the risk-free rate and the log of the stock will have drift depending on the risk-free rate and the volatility. Note that working with discounted prices will not help here, as the barrier will depend upon the actual price, not the discounted price. The standard tool for changing the drift of a Brownian motion is, of course, Girsanov's theorem. Previously, we have treated Girsanov's theorem as a black box and avoided looking at how the measure is changed.

The argument remains valid regardless of what the probability of an up-jump is. If the reader thinks `Ah, but what if the probability of an up-jump is 1?' we observe that a probability of 1 would lead to a simple arbitrage opportunity - the value of the stock tomorrow cannot grow faster than the risk-free interest rate. Otherwise, one could borrow at the risk-free rate and use the money to buy a stock and 46 Trees and option pricing achieve a certain profit. Thus in an arbitrage-free world a probability of 1 is not possible. The only role of probabilities is thus to ensure that both world states are possible. Suppose we did try a probabilistic approach. In particular, suppose that world A occurred with probability p, and world B occurred with probability 1 - p.

As in the interest-rate free world, given a derivative that pays f (S) at time At, we can construct a portfolio which precisely replicates it by considering a multiple of the stock and a multiple of the bond. However, rather than repeating that argument we look at the risk-neutral valuation approach. We need to find the probability that makes the stock grow on average at the risk-free rate. In other words, we must find p such that IE(S°t) = pS+ + (1 - p)S_ = Soe'°t, (3.9) p(S+ - S_) = Soe' °t - S_. (3.10) or We thus deduce that p _ Soe''°t S - S_ S_ (3 . 11 ) It follows from (3.8) that p lies strictly between zero and one. We previously justified risk-neutral valuation by saying that the expectation value of every possible portfolio was equal to today's value, so no portfolio could be an arbitrage portfolio; a portfolio of zero value today with possible positive value tomorrow and no possibility of negative value would not have zero expectation. 3.6 Putting interest rates in 59 Let IERN denote expectation with the risk-neutral probability p.

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A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing by Burton G. Malkiel

*Those who remember their high school algebra will recall that any straight line can be written as an equation. The equation for the straight line in the diagram is: Rate of Return = Risk-free Rate + Beta (Return from Market – Risk-free Rate). Alternatively, the equation can be written as an expression for the risk premium, that is, the rate of return on a portfolio of stocks or any individual stock over and above the risk-free rate of interest: Rate of Return – Risk-free Rate = Beta (Return from Market – Risk-free Rate). The equation says that the risk premium you get on any stock or portfolio increases directly with the beta value you assume.

If an investor’s portfolio has a beta of zero, as might be the case if all her funds were invested in a government-guaranteed bank savings certificate (beta would be zero because the returns from the certificate would not vary at all with swings in the stock market), the investor would receive some modest rate of return, which is generally called the risk-free rate of interest. As the individual takes on more risk, however, the return should increase. If the investor holds a portfolio with a beta of 1 (as, for example, holding a share in a broad stock-market index fund), her return will equal the general return from common stocks. This return has over long periods of time exceeded the risk-free rate of interest, but the investment is a risky one. In certain periods, the return is much less than the risk-free rate and involves taking substantial losses. This, is precisely what is meant by risk.

Some readers may wonder what relationship beta has to the covariance concept that was so critical in our discussion of portfolio theory. The beta for any security is essentially the same thing as the covariance between that security and the market index as measured on the basis of past experience. *Assuming expected market return is 15 percent and risk-free rate is 10 percent. * “Quant” is the Wall Street nickname for the quantitatively inclined financial analyst who devotes attention largely to the new investment technology. *Final value of investing in all equity funds vs. investing in only surviving funds. * In 2010 anyone, regardless of income, was allowed to convert a regular IRA into a Roth IRA

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Understanding Asset Allocation: An Intuitive Approach to Maximizing Your Portfolio by Victor A. Canto

Mean return can be considered the average return an investment or investment class is expected to deliver over time, while variance can be considered the average range of asset performance around the mean return. To calculate the Sharpe ratio, subtract the risk-free rate returns (that is, Treasury bill [T-bill] returns) from the asset returns in question and divide that result by the standard deviation of the return of the asset class in question less that of the risk-free rate. In this manner, risk is pinpointed. One way to think of this is to consider a person who borrows money to invest. After doing so, that person’s net gain is the difference between the return of the investment and the funds borrowed; the greater the difference (on the positive side), the greater the reward.

Chapter 9 Active Versus Passive Management 177 Table 9.7 Average annual returns and standard deviation of alternative combinations of the cap- and equal-weighted S&P 500 equity portfolios: 1990–2004. 100% 90% 80% 70% 60% 50% Average Annual Return over the Risk Free Rate 9.16% 8.99% 8.81% 8.63% 8.46% 8.28% Standard Deviation 15.74% 14.51% 13.45% 12.60% 12.01% 11.71% Sharpe Ratio 0.582 0.619 0.655 0.685 0.704 0.707 40% 30% 20% 10% 0% Best Average Annual Return over the Risk Free Rate 8.10% 7.93% 7.75% 7.57% 7.40% 11.03% Standard Deviation 11.73% 12.06% 12.68% 13.56% 14.64% 14.23% Sharpe Ratio 0.691 0.657 0.611 0.559 0.505 0.775 Source: Research Insight The numbers show a portfolio consisting solely of the equal-weighted stocks generated an average of 9.16 percent risk-adjusted excess returns per year (Table 9.7, first row, “100%”), while the cap-weighted portfolio returned 7.40 percent per year.

After doing so, that person’s net gain is the difference between the return of the investment and the funds borrowed; the greater the difference (on the positive side), the greater the reward. Similarly, the higher the Sharpe ratio, the lower the risk in relation to the reward. The Sharpe ratio is calculated using the mean and standard deviation of an excess return. That is the net of the asset class return and the risk free rate (that is, three months’ T-bill yields). A related measure is obtained when the ratio is calculated based on the mean and return of a single investment. This ratio is also known as the information ratio. Then, there’s the capital asset pricing model (CAPM), which similarly looks at the relationship between an investment’s risk and its expected market return— or, more specifically, the ways investment risk should affect its expected return.3 2 One major insight of the CAPM is that not all risks should affect asset prices.

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A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing (Eleventh Edition) by Burton G. Malkiel

*Those who remember their high school algebra will recall that any straight line can be written as an equation. The equation for the straight line in the diagram is: Rate of Return = Risk-free Rate + Beta (Return from Market – Risk-free Rate). Alternatively, the equation can be written as an expression for the risk premium, that is, the rate of return on a portfolio of stocks or any individual stock over and above the risk-free rate of interest: Rate of Return – Risk-free Rate = Beta (Return from Market – Risk-free Rate). The equation says that the risk premium you get on any stock or portfolio increases directly with the beta value you assume. Some readers may wonder what relationship beta has to the covariance concept that was so critical in our discussion of portfolio theory.

If an investor’s portfolio has a beta of zero, as might be the case if all her funds were invested in a government-guaranteed bank savings certificate (beta would be zero because the returns from the certificate would not vary at all with swings in the stock market), the investor would receive some modest rate of return, which is generally called the risk-free rate of interest. As the individual takes on more risk, however, the return should increase. If the investor holds a portfolio with a beta of 1 (as, for example, holding a share in a broad stock-market index fund), her return will equal the general return from common stocks. This return has over long periods of time exceeded the risk-free rate of interest, but the investment is a risky one. In certain periods, the return is much less than the risk-free rate and involves taking substantial losses. This is precisely what is meant by risk.

ILLUSTRATION OF PORTFOLIO BUILDING* Desired Beta Composition of Portfolio Expected Return from Portfolio 0 \$1 in risk-free asset 10% ½ \$.50 in risk-free asset \$.50 in market portfolio ½ (0.10) + ½ (0.15) = 0.125, or 12½%† 1 \$1 in market portfolio 15% 1½ \$1.50 in market portfolio borrowing \$.50 at an assumed rate of 10 percent 1½ (0.15) – ½ (0.10) = 0.175,or 17½% * Assuming expected market return is 15 percent and risk-free rate is 10 percent. † We can also derive the figure for expected return using directly the formula that accompanies the preceding chart: Rate of Return = 0.10 + ½ (0.15 – 0.10) = 0.125 or 12½%. Just as stocks had their fads, so beta came into high fashion in the early 1970s. Institutional Investor, the prestigious magazine that spent most of its pages chronicling the accomplishments of professional money managers, put its imprimatur on the movement by featuring on its cover the letter beta on top of a temple and including as its lead story “The Beta Cult!

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High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems by Irene Aldridge

Rising labor costs, on the other hand, may signal tough business conditions and decrease earnings expectations as a result. The discount rate in classical finance is, at its bare minimum, determined by the level of the risk-free rate and the idiosyncratic riskiness of a particular equity share. The risk-free rate pertinent to U.S. equities is often proxied by the 3-month bill issued by the U.S. Treasury; the risk-free rate significant to equities in another country is taken as the short-term target interest rate announced by that country’s central bank. The lower the riskfree rate, the lower the discount rate of equity earnings and the higher the theoretical prices of equities.

The Sharpe ratio was designed in 1966 by William Sharpe, later a winner of the Nobel Memorial Prize in Economics; it is a remarkably enduring concept used in the study and practice of finance. A textbook definition of F , where R̄ is the annualized average return the Sharpe ratio is SR = R̄−R σR from trading, σ R is the annualized standard deviation of trading returns, and RF is the risk-free rate (e.g., Fed Funds) that is included to capture the opportunity cost as well as the position carrying costs associated with 52 HIGH-FREQUENCY TRADING TABLE 5.1 Performance Measure Summary Sharpe Ratio (Sharpe [1966]) SR = E [r] E [r]−r f , where σ [r] r1 +···+rT = T σ [r] = Adequate if returns are normally distributed.

According to Eling and Schuhmacher (2007), more risk-averse investors should use higher order n. LPMs consider only negative deviations of returns from a minimal acceptable return. As such, LPMs are deemed to be a better measure of risk than standard deviation, which considers both positive and negative deviations (Sortino and van der Meer [1991]). Minimal acceptable return can be 0, risk-free rate, or average return. 53 Evaluating Performance of High-Frequency Strategies TABLE 5.1 (Continued) E [ri ]−τ Omega (Shadwick and Keating [2002]), (Kaplan and Knowles [2004]) i = Sortino Ratio (Sortino and van der Meer [1991]) Sortinoi = Kappa 3 (Kaplan and Knowles [2004]) K3i = Upside Potential Ratio (Sortino, van der Meer, and Plantinga [1999]) UPRi = LPM1i (τ ) +1 E [ri ] − τ is the average return in excess of the benchmark rate.

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The Long Good Buy: Analysing Cycles in Markets by Peter Oppenheimer

Exhibit 4.6 Equity/bond correlation can turn positive with higher yields (12m rolling US equity correlation with US 10-year bonds since 1981, weekly) SOURCE: Goldman Sachs Global Investment Research. Hence, while there is a cycle in equity and equity valuations, and part of this reflects the interplay between growth expectations and the bond yield (the ‘risk-free rate’), cycles can be complicated by the changing relationships between bond yields (prices) and equities over time that might be affected by structural factors, such as the prevailing inflationary environment and the level of interest rates. Structural Shifts in the Value of Equities and Bonds Although this chapter has focused mainly on the cyclical drivers that determine the relationship between bond and equity performance, the shifts in correlation that have occurred since the end of the 20th century, and in particular since the financial crisis, also demonstrate some of the secular or structural changes in the relationship.

The bubble in this cycle was more evident in the real estate market in the US and parts of southern Europe than it was in equity prices. What really sets the 2007–2009 bear market apart from other structural bear markets is the policy response. The rapid cuts in interest rates and adoption of QE resulted in a sharper rebound in equity (as well as other financial asset) prices than we have seen in the past. Lower risk-free rates triggered a search for yield in nominal assets such as bonds while also pushing up the present value of future income streams. This unusual backdrop, supported by very low inflation, has paved the way for an extended cycle and a rise in valuations. I look at this particular cycle in more detail in chapter 9.

Exhibit 9.5 Unlike after the 1930s crisis in the US or the 1990s crisis in Japan, US markets quickly recovered their losses after 2009 (nominal price returns; US: S&P 500; Japan: TOPIX) SOURCE: Goldman Sachs Global Investment Research. All Boats Were Lifted by the Liquidity Wave Part of the success of financial assets over the past 10 years has been that they have all been driven by a common factor – falling risk-free rates, which have contributed to rising valuations. Although equities have achieved higher returns than bonds, the impact of loose monetary policy has been felt across all asset classes. The impact of aggressive policy easing (including QE) post the crisis has been meaningful for asset returns. Indeed, the gap between ‘inflation’ measured in the real economy and that of financial assets has also been notable in this cycle (exhibit 9.6).

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Tools for Computational Finance by Rüdiger Seydel

An important application of options is hedging (−→ Appendix A2). The value of V (S, t) also depends on other factors. Dependence on the strike K and the maturity T is evident. Market parameters aﬀecting the price are the interest rate r, the volatility σ of the price St , and dividends in case of a dividend-paying asset. The interest rate r is the risk-free rate, which applies to zero bonds or to other investments that are considered free of risks (−→ Appendices A1, A2). The important volatility parameter σ can be deﬁned as standard deviation of the ﬂuctuations in St , for scaling divided by the square root of the observed time period. The larger the ﬂuctuations, respresented by large values of σ, the harder is to predict a future value of the asset.

Recall that in Section 1.4 we showed the implication E(ST ) = S0 erT =⇒ p = P(up) = erT − d , u−d whereas in this section we arrive at the implication p = P(up) = erT − d u−d =⇒ E(ST ) = S0 erT . So both statements must be equivalent. Setting the probability of the up movement equal to p is equivalent to assuming that the expected return on the asset equals the risk-free rate. This can be rewritten as e−rT EP (ST ) = S0 . (1.20) The important property expressed by equation (1.20) is that of a martingale: The random variable e−rT ST of the left-hand side has the tendency to remain at the same level. That is why a martingale is also called “fair game.” A martingale displays no trend, where the trend is measured with respect to EP .

We will return to this in Section 1.8. 1.7.3 Risk-Neutral Valuation As Appendix A4 shows for the continuous case, an option can be modeled independent of individual subjective expectations on the growth rate µ. For modeling of V (St , t), a risk-neutral world is assumed which allows to replace µ by the risk-free rate r. This was discussed for the discrete one-period model in Section 1.5. For the continuous-time setting, we start from GBM in (1.33), and get dSt = rSt dt + (µ − r)St dt + σSt dWt µ−r dt + dWt = rSt dt + σSt σ (1.35) In the reality of the market, an investor expects µ > r as compensation for the risk that is higher for stocks than for bonds.

Capital Ideas Evolving by Peter L. Bernstein

So the frontier beyond that point is composed only of portfolios with increasing amounts of cash. Bill Sharpe’s 1964 paper on the Capital Asset Pricing Model adds one more assumption to what Markowitz and Tobin had developed: that you can borrow as well as lend at the risk-free rate—all you want, in either direction. Under these conditions, the investor selects only one risky portfolio—the market portfolio—which is then mixed with lending or borrowing at the risk-free rate in order to create the frontier. These are the conditions in which Markowitz, Tobin, and Sharpe had worked out the structure of the efficient frontier. But now Markowitz declares the market portfolio is not an efficient portfolio!

He is no longer the same Harry Markowitz whose view of these matters first put Bill Sharpe to work on the relation between individual stocks and the market as a whole—a step that led to the Single-Index Model and then the Capital Asset Pricing Model. Markowitz has lost faith in what he terms the traditional neoclassical “equilibrium models.” These models, he claims, “make unrealistic—absurd—assumptions about the actors. For example, they can borrow all they want at the risk-free rate. Or they can revise their portfolios continuously. It would be nice to think through systems in which there would be more recognizable economic agents.”1 Furthermore, at a time when the world changes so rapidly and the markets are so dynamic, the equilibrium at the foundation of Capital Ideas will never come about or will stand still for too short a time to matter.

In the September/ October 2005 issue of the Financial Analysts Journal, Markowitz takes aim at two of the underlying assumptions of the Capital Asset Pricing Model as stock prices and portfolios move toward overall equilibrium.4 First, CAPM assumes investors can borrow infinite amounts of money at the risk-free rate—and without any regard to their current resources, bern_c08.qxd 3/23/07 9:05 AM Page 105 Harry Markowitz 105 which are obviously a matter of high importance to any lender. Second, investors can sell short without limit and use the proceeds to take on long positions—which means any investor can deposit \$1,000 with a broker, sell short \$1 million worth of one security, and buy long \$1,001,000 of another security.

The Intelligent Asset Allocator: How to Build Your Portfolio to Maximize Returns and Minimize Risk by William J. Bernstein

This simple, yet powerful construct is extraordinarily useful in understanding long-term returns in markets around the globe. Simply put, any stock asset class earns four different returns: ■ The risk-free rate, that is, the time value of money. Usually set at the short-term T-bill rate. ■ The market-risk premium. That additional return earned by exposing yourself to the stock market. ■ The size premium. The additional return earned by owning smallcompany stocks. ■ The value premium. The additional return earned by owning value stocks. Everyone earns the risk-free rate. So in the Fama-French universe, the only important decision you have to make is how much exposure you want to the other three factors.

So the dividend growth rate going forward may be quite a bit lower than it has been in the past. Decreasing dividend growth by 2.5% has the same effect as increasing the DR by the same amount—Dow 5172. So what determines the appropriate DR? It is very simply two things: the cost of money (or the risk-free rate) plus an additional amount to compensate for risk. 130 The Intelligent Asset Allocator Think of the DR as the interest rate a reasonable lender would charge a given loan applicant. The world’s safest borrower is the U.S. Treasury. If Uncle Sam comes my way and wants a long-term loan, I’ll charge him just 6%.

(Treasury securities of 1–10 years’ maturity are called notes.) Book value: A company’s assets minus intangible assets and liabilities; very roughly speaking, a company’s net assets. Capital asset pricing model (CAPM): A theory relating risk and expected return. Basically, it states that the return of a security or portfolio is equal to the risk-free rate plus a risk premium defined by Glossary 189 its beta. This theory contains a large number of unrealistic assumptions and has been shown to be inconsistent with empirical data (i.e., in the real world it turns out that high-beta stocks do not have higher returns than low-beta stocks). Capital gain: The amount of profit made on the sale of a security or fund.

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Stigum's Money Market, 4E by Marcia Stigum, Anthony Crescenzi

Passage of one week has a greater effect on the price of an option that is one month from expiration than an option that is one year from expiration. The intuition for this result is that the former option becomes 25% closer to expiration, while the latter becomes only 2% closer to expiration. Risk-Free Rate (Rho) Rho is a measure of how the value of an option changes with the risk-free rate. It is positive for a call and negative for a put. A rough intuition for this result is that if the risk premium for the underlying stock does not change and the risk-free rate increases, the total expected return of the underlying stock increases.8 A greater expected increase in the underlying stock price would increase the price of a call option and decrease the price of a put option.

Thus, the implicit assumption is that this increase in volatility is idiosyncratic volatility and not volatility that we should be compensated for. volatility of the underlying stock, and the risk-free rate. Four of these parameters are unambiguous and easy to find. The strike price and time to maturity of an option are defined in an option’s terms. The price of the underlying stock and the risk-free rate can be readily obtained from financial news sources. However, there is no clear way to find the volatility of the underlying stock. How exactly can we define this value? Historical Volatility A typical way that we might calculate volatility is to use the historical volatility of the underlying stock.

Frictionless, competitive, continuous markets, and no constraints on short sales. 2. A constant risk-free rate of interest. 3. No dividends or coupons paid by the underlying asset, although this assumption can be relaxed. 4. The underlying asset follows a geometric Brownian motion. This means that percentage returns are normally distributed. Let C = the price of a call option St = the price of the underlying asset today t = current date T = expiration date K = the strike price r = the risk-free rate σ = the volatility of the underlying asset Where and and N(.) is the cumulative distribution function for the standard normal distribution.

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The Big Short: Inside the Doomsday Machine by Michael Lewis

At the end of February a Bear Stearns analyst named Gyan Sinha published a long treatise arguing that the recent declines in subprime mortgage bonds had nothing to do with the quality of the bonds and everything to do with "market sentiment." Charlie read it thinking that the person who wrote it had no idea what was actually happening in the market. According to the Bear Stearns analyst, double-A CDOs were trading at 75 basis points above the risk-free rate--that is, Charlie should have been able to buy credit default swaps for 0.75 percent in premiums a year. The Bear Stearns traders, by contrast, weren't willing to sell them to him for five times that price. "I called the guy up and said, 'What the fuck are you talking about?' He said, "Well, this is where the deals are printing.'

By early 2005 Howie Hubler had found a sufficient number of fools in the market to acquire 2 billion dollars' worth of these bespoke credit default swaps. From the point of view of the fools, the credit default swaps Howie Hubler was looking to buy must have looked like free money: Morgan Stanley would pay them 2.5 percent a year over the risk-free rate to own, in effect, investment-grade (triple-B-rated) asset-backed bonds. The idea appealed especially to German institutional investors, who either failed to read the fine print or took the ratings at face value. By the spring of 2005, Howie Hubler and his traders believed, with reason, that these diabolical insurance policies they'd created were dead certain to pay off.

He'd collect a tiny bit of interest...for nothing. He wasn't alone in this belief, of course. Hubler and a trader at Merrill Lynch argued back and forth about a possible purchase by Morgan Stanley, from Merrill Lynch, of \$2 billion in triple-A CDOs. Hubler wanted Merrill Lynch to pay him 28 basis points (0.28 percent) over the risk-free rate, while Merrill Lynch only wanted to pay 24. On a \$2 billion trade--a trade that would, in the end, have transferred a \$2 billion loss from Merrill Lynch to Morgan Stanley--the two traders were arguing over interest payments amounting to \$800,000 a year. Over that sum the deal fell apart. Hubler had the same nit-picking argument with Deutsche Bank, with a difference.

Monte Carlo Simulation and Finance by Don L. McLeish

. ⎝ S0N ⎞ ⎛ S11 ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜ S1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ , S1 = ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎠ ⎝ S1N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ where at time 0, S0 is known and S1 is random. Assume also there is a riskless asset (a bond) paying interest rate r over one unit of time. Suppose we borrow money (this is the same as shorting bonds) at the risk-free rate to buy wj units P of stock j at time 0 for a total cost of wj S0j . The value of this portfolio at P time t = 1 is T (w) = wj (S1j − (1 + r)S0j ). If there are weights wj so that this sum is always non-negative, and P (T (w) > 0) > 0, then this is an arbitrage opportunity. Similarly, by replacing the weights wj by their negative −wj , there is an arbitrage opportunity if for some weights the sum is non-positive and negative with positive probability.

Similarly, covariances between returns for individual stocks and the return of the portfolio Π are given by exactly the same quantity, namely cov(Ri (t + 1), RΠ (t + 1)|Ht ) = 1 . S 0 (t)Σ−1 t S(t) Let us summarize our findings so far. We assume that the conditional covariance matrix Σt of the vector of stock prices is non-singular. Under the risk neutral measure, all stocks have exactly the same expected returns equal to the risk-free rate. There is a unique self-financing minimum-variance portfolio Π(t) and all stocks have exactly the same conditional covariance β with Π. All stocks have exactly the same regression coeﬃcient β when we regress on the minimum variance portfolio. Are other minimum variance portfolios conditionally uncorrelated with the portfolio we obtained above.

The above analysis assumes that our objective is minimizing the variance of the portfolio under the risk-neutral distribution Q. Two objections could be made. First we argued earlier that the performance of an investment should be made through the returns , not through the stock prices. Since under the risk neutral measure Q, the expected return from every stock is the risk-free rate of 56 CHAPTER 2. SOME BASIC THEORY OF FINANCE return, we are left with the problem of minimizing the variance of the portfolio return. By our earlier analysis, this is achieved when the proportion of our total investment at each time period in stock i is chosen as the corresponding component of the vector Σ−1 t 1 10 Σ−1 t 1 where now Σt is the conditional covariance matrix of the stock returns.

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Capital Ideas: The Improbable Origins of Modern Wall Street by Peter L. Bernstein

The primary role of the Capital Asset Pricing Model (CAPM) is to predict expected returns, or to place a valuation on risky assets. The expected returns come in three parts. First, a stock should be expected to earn at least as much as the risk-free rate of interest available on Treasury bills or a government-guaranteed savings account. Second, as stocks are a risky asset, the market as a whole should actually earn a premium over the risk-free rate. Third, an individual stock’s beta—its volatility relative to the portfolio’s volatility—will then determine how much higher or lower the expected returns of that stock will be relative to what investors expect from the market as a whole.

Sharpe declares at the end of his paper that “only the responsiveness of an asset’s rate of return to the level of economic activity is relevant in assessing its risk.”21 Yet both Sharpe and Treynor emphasize that identifying the dominant factor is beyond the limits of what they set out to demonstrate. Their more modest objective is to show that the covariance of any asset with existing holdings is what determines its risk premium—that is, the extra return an investor demands over the risk-free rate. When the beta concept came to the attention of professional portfolio managers, they reacted with confusion, then disgust, then annoyance, and finally total skepticism. Trading risk for higher reward seemed more complicated, more subtle, and more intuitive than just judging the volatility of the asset relative to the market.

Taxes and transaction costs often prevent them from trading freely enough to price all assets precisely as they should be priced. Information is not always freely available, nor does everyone understand it in the same way and act on it as soon as it appears. That is not all. In a world where the future purchasing power of money is uncertain, how does one determine a risk-free rate of interest? Is the capital market the only “basic underlying factor” that influences the value of the assets traded there? What about inflation and the distribution of wealth, to name just two possible factors that might come into play as well? And how do we define and measure “the market” when capital assets exist all over the world and include everything from cash, stocks, bonds, and real estate to art, gold, venture capital, and even intellectual capital like education and acquired skills?

All About Asset Allocation, Second Edition by Richard Ferri

The maturity value of these bonds increases in direct proportion to an FIGURE 2-2 Cumulative 30-day T-bill Return after Taxes and Inflation, Assuming a 25 Percent Tax Rate, from December 31, 1954 \$150 \$125 \$100.39 \$100 \$93.40 \$75 \$75.26 \$50 \$25 2009 2004 1999 1994 1989 1984 1979 1974 1969 1964 1959 1954 \$0 Understanding Investment Risk 29 increase in the inflation rate. The interest paid during the period also increases with the inflation rate. Some people argue that TIPS are a better representation of a risk-free rate than T-bills because inflation is factored out. But TIPS are not without their own risks. First, TIPS are publicly traded securities, and, as such, they fluctuate in value as interest rates rise and fall. The Barclays Capital U.S. Treasury Inflation Protected Securities Index fell by 2.4 percent in 2008 while the Consumer Price Index (CPI), a proxy for inflation, was up by 0.1 percent.

Calculating an expected return for any asset class starts with T-bill risk and return because that is the safest investment you can own. The T-bill return can be divided into two parts: an expected inflation component and a real risk-free return component. T-bill yields ⫽ expected inflation over the maturity period ⫹ real risk-free rate Since Treasury bills are the safest investment available, all other investments must earn at least the T-bill’s return. In addition, all other investments must have an extra return to compensate for the risks inherent in the particular investment. The extra return is known as a risk premium. Asset class expected investment return ⫽ T-bill yield ⫹ risk premium Every investment has an expected risk premium above and beyond T-bills that is based on that investment’s unique risks.

There are many other risks that are unique to individual investments that also deserve a return premium. Some of those risk premiums are used to calculate the long-term market return estimates at the end of this chapter. TA B L E 11-4 Examples of Expected Returns Derived by Layering Risk Premiums T-Bills IntermediateTerm Treasury Notes IntermediateTerm LargeCorporate Cap Bonds Stocks Real risk-free rate Term risk premium (intermediate) Credit risk premium (intermediate) Equity risk premium Value stock risk premium Small stock risk premium 0.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 1.0 1.0 2.0 1.0 2.0 Real expected return Inflation 0.5 3.0 2.0 3.0 3.0 3.0 5.0 3.0 8.0 3.0 Total expected return 3.5 5.0 6.0 8.0 11.0 SmallCap Stocks 2.0 1.0 Realistic Market Expectations 233 MODEL 2: ECONOMIC FACTOR FORECASTING A second method for calculating expected market returns is through a “top-down” approach using an economic growth assumption.

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Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures by Frank J. Fabozzi

We will define an up market precisely as one where the average excess return (market return over the risk-free rate or (rM - rft)) for the prior three months is greater than zero. ThenDt = 1 if the average (rMt - rft) for the prior three months > 0 Dt = 0 otherwise The independent variable will then beDtxt = xt if (rM - rft) for the prior three months > 0 Dtxt = 0 otherwise We use the S&P 500 Stock Index as a proxy for the market returns and the 90-day Treasury rate as a proxy for the risk-free rate. The data are presented in Table 21.1, which shows each observation for the variable Dtxt. TABLE 21.1 Data for Estimating Mutual Fund Characteristic Line with a Dummy Variable TABLE 21.2 Regression Results for the Mutual Fund Characteristic Line with Dummy Variable The regression results for the two mutual funds are shown in Table 21.2.

., t = 0) value of some payment X in the future, say at t = T, is equal to X ⋅ e − rf ⋅T which is less than X if rf and T are positive, where rf is the risk-free interest rate over the relevant time period. In other words, the future payment is discounted at the risk-free rate. Here, we use compounded interest with the constant risk-free rate rf which implies that interest is paid continuously and each payment of interest itself is immediately being paid interest on.307 At maturity t = τ, the call is worthCτ ( Sτ ) = ( Sτ − K, 0)+ where the expression on the right side is the abbreviated notation for the maximum of Sτ - K and 0.308 To understand this final value, consider that the option provides the option buyer with the right—but not the obligation—to purchase one unit of stock A at time τ.

See Spearman’s rho Right-continuity, demonstration Right-continuous empirical distribution Right-continuous function Right-continuous value Right-point rule Right-skewed distribution, location parameters Right skewed indication Risk analysis Risk-free asset, excess Risk-free interest rate composition Risk-free position, addition Risk-free rate implication. See Constant risk-free rate market return, relationship Risk-free return Risk management, variables consideration Risk manager, hedge position Risk measures role Risky assets Rounded monthly GE stock returns, marginal relative frequencies Row vector form R-squared (R2).

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The Devil's Derivatives: The Untold Story of the Slick Traders and Hapless Regulators Who Almost Blew Up Wall Street . . . And Are Ready to Do It Again by Nicholas Dunbar

The market is likely to sniff out problems before a ratings agency chalks up another default, and margin calls will quickly force people to sell. Default or bankruptcy is still going to be a problem if you own a bond, but rather than waiting to record a loss the way an insurance company does, the question is whether you can afford to stay in the game. In this price-driven environment, the spread (the return above risk-free rates) paid by a bond or loan is no longer an actuarial insurance premium for long-term default risk. Instead, it is compensation for price risk, which changes to reflect the day-to-day opinion of the market. Suppose that after you have relied upon the ratings agency “health check” and made a large investment in a company, the market turns against the company so much that no one will buy its bonds.

But what really rocked both of these worlds was a radical financial innovation: credit derivatives. Imagine a bank looking to make corporate loans or to own bonds—but without the credit risk. How does it strip out the risk? Easy: think of the loan as two separate parts. Pretend the loan is made to a borrower as safe as the government, which will repay the money without fail, and pays a “risk-free” rate of interest in compensation. Then there is an “insurance policy” or indemnity, for which the risky borrower pays an additional premium to compensate the lender for the possibility of not repaying the loan (although they might have to hand over some collateral). Bundled together, the risk-free loan plus the insurance policy amount to a risky corporate bond or loan.

The problem was that the Japanese didn’t like the direction in which banks such as Citi were going. Rather than a bank promising to pay shareholders 15 or 20 percent returns on their investment every year, the Swiss and Japanese wanted something safer and more boring. Just a couple of percentage points more than the risk-free rate was fine for them. Sossidis and Partridge-Hicks had a brain wave. If they couldn’t find a real bank for the Japanese and Swiss to invest in, they would create one. It would be safe and boring: it would restrict itself to investing in the very safest assets—the debt of highly rated banks, for example, and the topmost slices of the new kinds of securitizations that Citi was doing with its credit card portfolio.

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Red-Blooded Risk: The Secret History of Wall Street by Aaron Brown, Eric Kim

It is a measure of risk-adjusted return. A strategy with an annualized Sharpe ratio of 1 will make more than the risk-free rate about five years out of six. A strategy with an annualized Sharpe ratio of 2 will make more than the risk-free rate about 39 years out of 40. However, it’s hard to find Sharpe ratios near or above 1 in high-capacity, liquid strategies that are inexpensive to run. You don’t need a Sharpe ratio near or at 1 to get rich. For a Kelly investor, the long-term growth of capital above the risk-free rate is approximately equal to the Sharpe ratio squared (it’s actually always higher than this, substantially so for high Sharpe ratios, but that doesn’t affect the points I want to make).

It happens to be 3.322 to 1 leverage, meaning take positions in \$3,322 total notional absolute value of commodities. At the end, after we’ve paid back our loan, we have \$8,145 after inflation. Note that all this happened during a period of generally declining real commodity prices, when six of the seven commodities returned less than the risk-free rate and the one positive return was only 10.6 percent for the entire 40 years. Moreover, we beat the market with no work, no fundamental analysis, and no active trading. I hope you can start to see why hedge fund managers get rich. I haven’t yet explained how they do it without genies, but that makes it only a little harder.

When you think about the market for emerging market real estate, IGT makes more sense. In the real world in which MPT was the dominant theory, the dominant model of market equilibrium was the capital asset pricing model (CAPM). It held that the expected excess return of any asset—remember excess means the return above a risk-free rate of interest—is equal to the asset’s beta times the expected excess return of the market. This follows from MPT and EMH, with some specific assumptions about the market and investors. Investment Growth Theory In the parallel universe of IGT, we get a different formula. I present it here to expand consciousness, not to make a serious argument for it.

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Beyond the Random Walk: A Guide to Stock Market Anomalies and Low Risk Investing by Vijay Singal

Weisbenner (2001), and Haugen and Jorion (1996). 2. For example, see the Wall Street Journal, December 6, 2001. 3. Excluding the last day of the year is common in many studies because of low trading volume on that day. 4. The abnormal return is calculated as the actual return minus the expected return. Expected return is the risk-free rate plus market risk premium times the beta of the stock. 5. The difference in the return to SPDRs and futures may be attributable to differences in closing times. 39 40 Beyond the Random Walk 3 The Weekend Effect The weekend effect refers to relatively large returns on Fridays compared to those on Mondays.

In the case of stock mergers, shares of the acquiring firm are short-sold in the correct proportion and the position is liquidated upon merger completion or upon withdrawal of the offer. This is the raw return earned from merger arbitrage. The abnormal return is obtained after adjusting for return commensurate with this kind of risk. If the risk is assumed to be zero, then the abnormal return is simply the raw return minus the risk free rate. If risk is assumed to be market risk, then the abnormal return is the raw return minus the market return. The systematic risk in the case of merger arbitrage is usually less than the market risk because the risk of failure of the merger depends on many factors besides stock market movements.

If no merger deals are available in a particular period, then the abnormal return for that period is zero. Based on these considerations, the raw return to an arbitrage portfolio is 1.5 percent per month for all offers. If only the first offer is considered, then the return is 1.6 205 206 Beyond the Random Walk percent per month. By comparison, the market earns 1.2 percent per month, and the risk-free rate is 0.6 percent per month. Adjusted for risk, the excess return is about 0.8 percent per month or 9.6 percent annually. The other study of cash and stock mergers has the largest sample of bids (4,750 offers) over the longest period (1963–98). The sample includes both pure cash and pure stock bids, and bids that seek less than 100 percent of the target as long as the acquisition will result in the acquirer holding the entire 100 percent.

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Stocks for the Long Run 5/E: the Definitive Guide to Financial Market Returns & Long-Term Investment Strategies by Jeremy Siegel

The reasons investors discount the future are (1) the existence of a risk-free rate, a yield on a safe alternative asset such as government or other AAA-rated securities, which allows investors the ability to transform a dollar invested today into a greater sum tomorrow; (2) inflation, which reduces the purchasing power of cash received in the future, and (3) the risk associated with the magnitudes of expected cash flows, which induces investors of risky assets, such as stocks, to demand a premium to that on safe securities. The sum of these three factors—the risk-free rate, the inflation premium, and the equity risk premium—determines the discount rate for equities.

The Gordon Dividend Growth Model of Stock Valuation To show how dividend policy impacts the price of a stock, we use the Gordon dividend growth model developed by Roger Gordon in 1962.4 Since the price of a stock is the present value of all future dividends, it can be shown that if future dividends per share grow at a constant rate g, then the price per share of a stock P, which is the discounted value of all future dividends, can be written as follows: where d is the dividend per share, g is the rate of growth of future dividends per share, and r is the required return on equity, which is the sum of the risk-free rate, the expected rate of inflation, and the equity risk premium. Since the Gordon model formula is a function of the per share dividend and the per share dividend growth rate, it appears that dividend policy is crucial to determining the value of the stock. But as long as one specific condition holds—that the firm earns the same return on its retained earnings as its required return on equity—then future dividend policy does not impact the price of the stock or the market value of the firm.5 This is because dividends not paid today become retained earnings that generate higher dividends in the future, and it can be shown that the present value of those dividends is unchanged, no matter when they are paid.

Short-run volatility has always been part of the stock market, and the flash crash had no lasting effect on the recovery from the 2007–2009 bear market. THE NATURE OF MARKET VOLATILITY Although most investors express a strong distaste for market fluctuations, volatility must be accepted to reap the superior returns offered by stocks. Accepting risk is required for above-average returns: investors cannot make any more than the risk-free rate unless there is some possibility that they can make less. While the volatility of the stock market deters many investors, it fascinates others. The ability to monitor a position on a minute-by-minute basis fulfills the need of many to quickly validate their judgment. For many the stock market is truly the world’s largest casino.

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Safe Haven: Investing for Financial Storms by Mark Spitznagel

(“It is better to sleep well than to eat well.”) And academics claim that lower returns actually accompany lower risk or volatility as a consequence, ipso facto. Their story goes, as you hedge and diversify away all of your correlated and uncorrelated risks, respectively, your returns will approach the lowly risk‐free rate. Or going the other way, the academics argue that in order to induce an investor to hold an asset that is relatively volatile, its price drops until its expected return becomes high enough to justify the additional risk. In their world, it sounds reasonable, and it's a comfortable story, asserted but not proved: You've got to take more risk to make higher returns; sleeping well comes at a cost.

., 16 evaluating strategies for, 16–17 as getting your fate right, 200 as goal of investing, 26–27 inconsistency of, 187 irony of, 109–111 need for, 11–13 and net portfolio effect, 86 and safe haven frontier, 182–189 theory vs. possibility of, 19–21 Cramer, Gabriel, 35 Crash‐bang‐for‐the‐buck, 109, 133, 140 Crash payoffs: and buying into crashes, 149–151 effects of, 196 and safe haven frontier, 185 for safe haven imposters, 112–114 for safe haven phenotypes, 105, 106 Crisis alpha, 176 Cryptocurrencies, 181–182 CTA strategies, see Commodity trading advisor strategies Curve, concavity of, 54–56, 95, 198 D Dao of Capital, 109 and roundabout investing, 12 Darwin, Charles, 102 Debt, reckless accumulation of, 11 Deductive reasoning: modus tollens in, 18–22 in risk mitigation, 22–26 Defense, offense and, 148–151 Demon thought experiments: Nietzsche's eternal return, 58–63, 69–73, 75–76, 79, 89, 90, 193, 199 Schrödinger's multiverse, 64–69, 74–76, 79, 89, 90 the whole vs. parts in, 124–125 Denying the antecedent, 21 Dichotomy of control, 13, 196 Die(‐ce): 120‐sided, 118–122, 127–129 as deductive tool, 22–25 expected outcome of rolling, 72–73 geometry of, 18 law of large numbers for, 30 Die fröhliche Wissenschaft (Nietzsche), 58–59 Diminishing marginal utility, 35–36, 53 Dionysian ideal, 69 Discrete probability distribution: real‐world, 117–122 in Schrödinger's demon, 66–67 Disdyakis tricontahedron, 118, 127, 157. See also 120‐sided die (d120) Diversification: Bernoulli on, 44–45 Buffett on, 115–116 and diworsifier havens, 112, 115–117 as “only free lunch in finance,” 59 and risk‐free rate, 17 Diversity, within species, 102 Diworsification, 175 Peter Lynch on, 115 Diworsifier havens, 112, 115–117 Dogma of diversification, 115–117 d120, see 120‐sided die Dylan, Bob, v, 76, 203 E Economically dominant strategy, 94, 139–140, 186 Economic meaning, quantitative meaning vs., 59 Economics: diminishing marginal utility in, 35–36 Neumann–Morgenstern utility function in, 37 Efficiency, 202 Efficient markets, 150 Einstein, Albert, 14, 73, 167 Emergent properties, 124–126 Emolumentum medium, 35–39, 198 Ensemble average, 75, 80–81 Ephemeralization to the limit, 157 Epictetus, 199 Ergodic processes, 75 Errors: of omission and commission, 154, 170 type I and type II, 169–170 Essentialism, 99–103 Eternal return thought experiment, see under Nietzsche, Friedrich Expectations: in choosing investing path, 62–63 in gold investing, 179 in rolling die, 72–73 of wealth not realized vs. of growth rate of wealth, 75 Expected value: Bernoulli's, see Bernoulli's expected value (BEV) geometric, 42, 48 infinite, 32–33 with Nietzsche's demon, 72 in Petersburg wager, 34–35 with Schrödinger's demon, 67, 68 when reshuffling returns, 143 Experiments.

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Cryptoassets: The Innovative Investor's Guide to Bitcoin and Beyond: The Innovative Investor's Guide to Bitcoin and Beyond by Chris Burniske, Jack Tatar

Sharpe Ratio Similar to the concepts behind MPT, the Sharpe ratio was also created by a Nobel Prize winner, William F. Sharpe. The Sharpe ratio differs from the standard deviation of returns in that it calibrates returns per the unit of risk taken. The ratio divides the average expected return of an asset (minus the risk-free rate) by its standard deviation of returns. For example, if the expected return is 8 percent, and the standard deviation of returns is 5 percent, then its Sharpe ratio is 1.6. The higher the Sharpe ratio, the better an asset is compensating an investor for the associated risk. An asset with a negative Sharpe ratio is punishing the investor with negative returns and volatility.

An asset with lower absolute returns can have a higher Sharpe ratio than a high-flying asset that experiences extreme volatility. For example, consider an equity asset that has an expected return of 12 percent with a volatility of 10 percent, versus a bond with an expected return of 5 percent but volatility of 3 percent. The former has a Sharpe ratio of 1.2 while the latter of 1.67 (assuming a risk-free rate of 0 percent). The ratio provides a mathematical method to compare how different assets compensate the investor for the risk taken, making bonds and equities, or apples and oranges, more comparable. Correlation of Returns and the Efficient Frontier One of the key breakthroughs of modern portfolio theory was to show that a riskier asset can be added to a portfolio, and if its behavior differs significantly from the preexisting assets in that portfolio, it can actually decrease the overall risk of the portfolio.

To represent U.S. bonds, U.S. real estate, gold, and oil, we used the Bloomberg Barclays US Aggregate Bond Index, the Morgan Stanley Capital International US Real Estate Investment Trust Index, the gold index underlying the SPDR Gold Shares ETF, and crude oil futures, respectively. 8. Minus the risk-free rate. 9. Using weekly returns to standardize for # of days scalar multiplier. All previous charts have used daily data. 10. http://www.coindesk.com/bitcoin-price-2014-year-review/. 11. http://corporate.morningstar.com/U.S./documents/MethodologyDocuments/MethodologyPapers/StandardDeviationSharpeRatio_Definition.pdf. 12.

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Quantitative Value: A Practitioner's Guide to Automating Intelligent Investment and Eliminating Behavioral Errors by Wesley R. Gray, Tobias E. Carlisle

William Sharpe created the Sharpe ratio in 1966, intending it to be used to measure the risk-adjusted performance of mutual funds.5 Sharpe was interested in the extent to which managers took on extra risk to generate additional return. He wanted to find some measure that would adjust the return for the risk taken to generate it. He created the Sharpe ratio, which does this by examining the historical relationship between excess return—the return in excess of the risk-free rate—and volatility, which stands in for risk. The higher the Sharpe ratio, the more return is generated for each additional unit of volatility, and the better the price metric. The Sortino ratio, like the Sharpe ratio, measures risk-adjusted return. The difference is that the Sortino ratio only measures downside volatility, while the Sharpe ratio measures both upside and downside volatility.

The MW Index's returns represent a passive investment in the universe of all stocks we analyze. CAGR Compound annual growth rate. Standard Deviation Sample standard deviation (annualized by square root of 12). Downside Deviation Sample standard deviation of all negative observations (annualized by square root of 12). Sharpe Ratio Monthly return minus risk-free rate divided by standard deviation (annualized by square root of 12). Sortino Ratio (MAR = 5%) Monthly return minus minimum acceptable return (MAR/12) divided by downside deviation (annualized by square root of 12). Worst Drawdown Worst peak-to-trough performance. Worst Month Return Worst monthly performance.

The index's returns represent a passive investment in the universe of all stocks we analyze. CAGR Compound annual growth rate. Standard Deviation Sample standard deviation (annualized by square root of 12). Downside Deviation Sample standard deviation of all negative observations (annualized by square root of 12). Sharpe Ratio Monthly return minus risk free rate divided by standard deviation (annualized by square root of 12). Sortino Ratio (MAR=5%) Monthly return minus minimum acceptable return (MAR/12) divided by downside deviation (annualized by square root of 12). Worst Drawdown Worst peak-to-trough performance. Worst Month Return Worst monthly performance.

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Smarter Investing by Tim Hale

But as soon as you step away from these types of investments, you enter the realm of market risk either by branching out into bonds, or becoming an owner by way of equities. Stepping away from risk-free assets Imagine that you decide that you need to generate a higher return on your assets than just the risk-free rate (low risk = low return). You could go out and buy the shares of just one company. Most people would consider that to be pretty risky and they would be right. The risks associated with that one company would be high, with its share price and fortunes gyrating wildly over time (think about BP’s journey over the past few years).

Over the longer term the returns for an equity investor weigh up the true value of ownership being the cash dividends received and the real growth in earnings that the company has delivered. This is the weighing machine. This longer-term return for taking on the voting machine risks of being an owner, above the risk-free rate, is known as the equity risk premium (see Figure 6.4). Smaller company (size) and value risks The Fama and French research referred to above, breaks down the risk that an investor receives into three key components: Equity market risk: being in the equity market as a whole, as opposed to simply holding ‘safe’ cash, for which you receive the equity risk premium, or ERP for short.

If yields fall, bond prices rise. This is sometimes referred to as the bond see-saw, with yields at one end and prices at the other (Figure 6.6). The components of a bond’s yield At its simplest, the yield on a bond can be broken down into six components, some of which relate to all bonds (the risk-free rate) and some of which relate to the specific circumstances of the issuer of the bond and the characteristics of the bond. Let us start by looking at a risk-free investment i.e. lending on a short-term basis (say for one month) to a government with an AAA credit rating (the highest there is). The risk of not receiving the interest due and your money back is negligible.

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Inside the House of Money: Top Hedge Fund Traders on Profiting in a Global Market by Steven Drobny

Regardless of what happens with inflation, real interest rates will come down a lot and nominal rates should come down as well. Look at the yield on 30-year Treasury inflation-protected securities (TIPS). It got as low as 1.6 percent recently, which is the lowest it has ever been. (See Figure 9.5.) That means the risk-free rate of return for the next 30 years is currently being valued at 1.6 percent per year. That rate has been declining every year since the bubble burst, which shows that the bubble unwind process is still going strong. If TIPS are priced correctly, and we think they are, they’re telling us that equity markets are extremely overvalued.

If repeated year in and year out, the expected average gain of 30 percent (0.50 × 0.60) minus the expected average loss of 20 percent (0.50 × 0.40) would produce an average annual return of 10 percent. (See Figure A.1.) Big Bet Performance Analysis Hedge fund managers are often judged by their Sharpe ratios, which are calculated as the fund’s return minus the risk-free rate divided by the volatility of returns. The Sharpe ratio is also known as the reward-tovolatility ratio and provides a sense of the quality of the managers’ returns per unit of risk. It allows for some element of comparison across managers. Hedge funds often strive for a Sharpe ratio of at least 1.0 such that their 70% 60% Probability 50% 40% 30% 20% 10% 0% –50% –40% –30% –20% –10% 0% Return FIGURE A.1 Hypothetical Big Bet Portfolio 10% 20% 30% 40% 50% WHY GLOBAL MACRO IS THE WAY TO GO 345 performance is commensurate with the amount of risk assumed.

Hedge funds often strive for a Sharpe ratio of at least 1.0 such that their 70% 60% Probability 50% 40% 30% 20% 10% 0% –50% –40% –30% –20% –10% 0% Return FIGURE A.1 Hypothetical Big Bet Portfolio 10% 20% 30% 40% 50% WHY GLOBAL MACRO IS THE WAY TO GO 345 performance is commensurate with the amount of risk assumed. Hedge funds that produce a Sharpe ratio well over 1.0 attract investor interest while those with a Sharpe ratio well below 1.0 do not. In the example of the single big bet, with a volatility of approximately 50 percent, the strategy delivers a Sharpe ratio of 0.1 assuming a risk-free rate of 5 percent [(10 – 5)/50 = 0.1]. While investors would probably be pleased with a 50 percent return in the good years, it is unlikely that they would tolerate the negative 50 percent years and remain invested long enough to witness a 10 percent average annual return. In sum, a hedge fund manager taking the big-bet approach may prove lucky for a spell, but that luck would eventually run out in the longer term and the manager would be forced out of business.

Stocks for the Long Run, 4th Edition: The Definitive Guide to Financial Market Returns & Long Term Investment Strategies by Jeremy J. Siegel

CHAPTER 8 The Impact of Economic Growth on Market Valuation and the Coming Age Wave 133 longer be appropriate in today’s market. If the real risk-free rate of interest on long-term TIPS bonds is 2 percent, then a 3 percent equity premium will yield a 5 percent real return on equities, equivalent to a price-to-earnings ratio of 20. If the equity premium shrinks to 2 percent, then the price-to-earnings ratio can rise to 25 to yield a 4 percent real, forward-looking return on equities. If the real risk-free rate rises to 3 percent, the real return on equities will be 5 percent with a 2 percent risk premium, implying a P-E ratio of 20.

But the “best” risk-return portfolio, called the efficient portfolio, is not the one with the lowest risk but the one that optimally balances risk and return. This “best” portfolio is found at a much higher 37.8 percent foreign stock allocation.7 For comparison, in July 2007 the EAFE stocks FIGURE 10–2 Portfolio Allocation between U.S. and EAFE Stocks 13.80% Risk-free rate is 5.0% 0% U.S., 100% EAFE 13.60% 10% U.S., 90% EAFE 20% U.S., 80% EAFE 13.40% Return 13.20% 30% U.S., 70% EAFE 40% U.S., 60% EAFE 13.00% 50% U.S., 50% EAFE 12.80% 12.60% 60% U.S., 40% EAFE 70% U.S., 30% EAFE 80% U.S., 20% EAFE 12.40% 90% U.S., 10% EAFE 12.20% 12.00% 16% U.S. Portfolio EAFE Portfolio Minimum Risk Efficient Portfolio Risk 17.100% 21.928% 16.585% 16.825% Return 12.207% 13.641% 12.529% 12.749% % U.S. 100.0% 0.0% 77.5% 62.2% % EAFE 0.0% 100.0% 22.5% 37.8% 100% U.S., 0% EAFE 17% 18% 19% 20% Risk 21% 22% 23% CHAPTER 10 Global Investing and the Rise of China, India, and the Emerging Markets 171 represented about 57 percent, and the U.S. stocks represented 43 percent of this world portfolio based on market values.

This sometimes leads to an acceleration of price declines toward the price limits, thereby increasing short-term volatility, as occurred when prices fell to the limits on October 27, 1997.8 THE NATURE OF MARKET VOLATILITY Although most investors express a strong distaste for market fluctuations, volatility must be accepted to reap the superior returns offered by stocks. For risk is the essence of above-average returns: investors cannot make any more than the risk-free rate of return unless there is some possibility that they can make less. While the volatility of the stock market deters many investors, it fascinates others. The ability to monitor a position on a minute-byminute basis fulfills the need of many to quickly validate their judgment. For many the stock market is truly the world’s largest casino.

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The 100-Year Life: Living and Working in an Age of Longevity by Lynda Gratton, Andrew Scott

One variable is your appetite for risk; risky assets pay a higher return than safe assets in order to persuade investors to take on that risk. So the return on investment has three components: the risk-free rate of return (usually the interest rate on government bonds, assuming that it is a government that doesn’t default); the risk premium (how much extra an investor earns from risky assets); and the balance of the portfolio between risk-free and risky assets. The risk-free rate and the risk premium often vary from one decade to another and different portfolio mixes bring about very different investment returns. Hence there is no one single ‘golden’ number on which everyone agrees and so a voluminous debate exists in the finance literature on this topic.

Hence there is no one single ‘golden’ number on which everyone agrees and so a voluminous debate exists in the finance literature on this topic. An important source of data is the work of Elroy Dimson, Paul Marsh and Mike Staunton.3 Every year they produce estimates for a range of countries going back more than 100 years. For the US between 1900 and 2014 they report that, adjusting for inflation, the risk-free rate was 2 per cent and the risk premium 4.4 per cent – in other words, equities earn 4.4 per cent a year more than risk-free assets such as government bonds. So if an investor split their portfolio 50:50 between safe and risky assets, then they earned a return of 4.2 per cent (0.5 x 2 + 0.5 x 6.4) over and above inflation.

Deep Value by Tobias E. Carlisle

On the other hand, if the yield on the government bond rises to say 20 percent, the value of the bad business falls to no more than one-quarter its invested capital (5 percent ÷ 20 percent = 0.25×), and the value of the good business falls to no more than one times its invested capital (20 percent ÷ 20 percent = 1×). “The rates of return that investors need from any kind of investment,” wrote Buffett in 1999, “are directly tied to the risk-free rate that they can earn from government securities:”38 The basic proposition is this: What an investor should pay today for a dollar to be received tomorrow can only be determined by first looking at the risk-free interest rate. Consequently, every time the risk-free rate moves by one basis point—by 0.01%—the value of every investment in the country changes. People can see this easily in the case of bonds, whose value is normally affected only by interest rates.

Buffett attributed the apparently unusual behavior of the stock market to the relationship between two important variables: interest rates and valuation. “Interest rates,” wrote Buffett, “. . . act on financial valuations the way gravity acts on matter: The higher the rate, the greater the downward pull.”13 That’s because the rates of return that investors need from any kind of investment are directly tied to the risk-free rate that they can earn from government securities. So if the government rate rises, the prices of all other investments must adjust downward, to a level that brings their expected rates of return into line. Conversely, if government interest rates fall, the move pushes the prices of all other investments upward.

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Early Retirement Extreme by Jacob Lund Fisker

You can calculate whether this is the case by considering the net asset value (NAV) NAV = (annual rental income ×(1-upkeep))/(risk-free rate + risk premium). Here the upkeep ratio is typically 30-40% of the rental income. The risk free rate is the going rate of 30-year bonds. The risk premium is typically 1-2%. Calculate the capitalization rate (cap rate) given by cap rate = (annual rental income ×(1-upkeep))/(price). The cap rate should be a few percent over the risk-free rate to adjust for the possibility of the house burning down. As you can see, it's essentially the same calculation. You should understand these calculations and be able to modify them to your situation.

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It should be a large multiple, to ensure that the strategy will survive worse-than-expected execution. 14.7 Efficiency Until now, all performance statistics considered profits, losses, and costs. In this section, we account for the risks involved in achieving those results. 14.7.1 The Sharpe Ratio Suppose that a strategy's excess returns (in excess of the risk-free rate), {rt}t = 1, …, T, are IID Gaussian with mean μ and variance σ2. The Sharpe ratio (SR) is defined as The purpose of SR is to evaluate the skills of a particular strategy or investor. Since μ, σ are usually unknown, the true SR value cannot be known for certain. The inevitable consequence is that Sharpe ratio calculations may be the subject of substantial estimation errors. 14.7.2 The Probabilistic Sharpe Ratio The probabilistic Sharpe ratio (PSR) provides an adjusted estimate of SR, by removing the inflationary effect caused by short series with skewed and/or fat-tailed returns.

HHI index on negative returns. HHI index on time between bars. The 95-percentile DD. The 95-percentile TuW. Annualized average return. Average returns from hits (positive returns). Average return from misses (negative returns). Annualized SR. Information ratio, where the benchmark is the risk-free rate. PSR. DSR, where we assume there were 100 trials, and the variance of the trials’ SR was 0.5. Consider a strategy that is long one futures contract on even years, and is short one futures contract on odd years. Repeat the calculations from exercise 2. What is the correlation to the underlying?

In words, the transaction costs associated with each asset are the sum of the square roots of the changes in capital allocations, re-scaled by an asset-specific factor Ch = {cn, h}n = 1, …, N that changes with h. Thus, Ch is an Nx1 vector that determines the relative transaction cost across assets. The Sharpe Ratio (Chapter 14) associated with r can be computed as (μh being net of the risk-free rate) 21.4 The Problem We would like to compute the optimal trading trajectory that solves the problem This problem attempts to compute a global dynamic optimum, in contrast to the static optimum derived by mean-variance optimizers (see Chapter 16). Note that non-continuous transaction costs are embedded in r.

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The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal by Ludwig B. Chincarini

The higher the ratio, the better the portfolio manager. (See Box 2.1.) Table 2.1 shows that, before 1998, the LTCM fund had a Sharpe ratio five times that of the standard returns of U.S. Treasury bills and bonds. Box 2.1 The Sharpe Ratio The Sharpe ratio measures the return of a portfolio minus the risk-free rate divided by the portfolio’s standard deviation. It is a risk-adjusted return measure that assists in comparing different portfolios or investments, even in the presence of leverage. If portfolio A has a higher Sharpe ratio than portfolio B, then there is no amount of leverage that can make portfolio B as good as A.

Measure LTCM’s returns against one of the standard academic models for assessing hedge funds’ excess performance and you’ll find that LTCM provided an excess return or alpha of 2.93% per month. The commonly used academic model for hedge funds is the Fung-Hsieh model (2001, 2004), which regresses fund returns against a series of benchmark returns. The model is of the form , where is the net-of-fee return on a hedge fund portfolio in excess of the risk-free rate, RMRF is the excess return on value-weighted aggregate market proxy, SMB, HML, and MOM are the returns on a value-weighted, zero-investment, factor-mimicking portfolios for size, book-to-market equity, and one-year momentum in stock returns as computed by Fama and French, 10 yr is the Lehman U.S. ten-year bellwether total return, CS is the Lehman aggregate intermediate BAA corporate bond index return minus the Lehman 10-year bond return, BdOpt is the look-back straddle for bonds, FXOpt is the look-back straddle for foreign exchange, ComOpt is the look-back straddle for commodities, and EE is the total return from an emerging market equity index. 11.

This scheme was a compromise. 6. The ratio of the option’s price to the notional value was roughly 39%. The fund’s historical volatility was around 8% to 9%, LTCM partners recall, but they priced the option using a volatility around 14%. Using a Black-Scholes model to price an option with a 14% volatility, risk-free rate of 7.5% (Treasury Yield + 50 basis points of swap spread + 100 basis points borrowing fee), with 100 as the current fund value and 110 as the strike price (slightly in the money), and seven years to maturity gives a price of 35.93 and a ratio of 36%. 7. Even if UBS had tried to dynamically hedge the position, it wouldn’t have worked.

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The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall

Nonetheless, they yield a certain rate of return, so that if you invest in Treasury bonds, you are guaranteed to make money at a fixed rate. Most investments, however, are inherently risky. Treynor realized that it would be crazy to put your money into one of these risky investments, unless you could expect the risky investments to have a higher rate of return, at least on average, than the risk-free rate. Treynor called this additional return a risk premium because it represented the additional income an investor would demand before buying a risky asset. CAPM was a model that allowed you to link risk and return, via a cost-benefit analysis of risk premiums. When Black learned about CAPM, he was immediately hooked.

(Indeed, if CAPM-style reasoning is correct, you shouldn’t be able to both eliminate risk and still make a substantial profit.) Black’s approach was to find a portfolio consisting of stocks and options that was risk-free, and then argue by CAPM reasoning that this portfolio should be expected to earn the risk-free rate of return. Black’s strategy of building a risk-free asset from stocks and options is now called dynamic hedging. Black had read Cootner’s collection of essays on the randomness of markets, and so he was familiar with Bachelier’s and Osborne’s work on the random walk hypothesis. This gave him a way to model how the underlying stock prices changed over time — which in turn gave him a way to understand how options prices must change over time, given the link he had discovered between options prices and stock prices.

Black and Scholes, meanwhile, worked in the opposite direction. They started with a hedging strategy, by observing that it should always be possible to construct a risk-free portfolio from a combination of stocks and options. They then applied CAPM to say what the rate of return on this portfolio should be — that is, the risk-free rate — and worked backward to figure out how options prices would have to depend on stock prices in order to realize this risk-free return. The distinction may seem inconsequential — after all, the two arguments are different paths to the same model of options prices. But in practice, it was crucial.

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A Mathematician Plays the Stock Market by John Allen Paulos

There are many complications to portfolio selection theory. As the Sharpe measure suggests, an important one is the existence of risk-free investments, such as U.S. treasury bills. These pay a fixed rate of return and have essentially zero volatility. Investors can always invest in such risk-free assets and can borrow at the risk-free rates as well. Moreover, they can combine risk-free investment in treasury bills with a risky stock portfolio. Variation Two of portfolio theory claims that there is one and only one optimal stock portfolio on the efficient frontier with the property that some combination of it and a risk-free investment (ignoring inflation) constitute a set of investments having the highest rates of return for any given level of risk.

Alternatively, if you want to divide your money between the two, you put p% into the risk-free treasury bills and (100 - p)% into the optimal risky stock portfolio for an expected rate of return of [p × (risk-free return) + (1 - p) × (stock portfolio)]. An investor can also invest more money than he has by borrowing at the risk-free rate and putting this borrowed money into the risky portfolio. In this refinement of portfolio selection, all investors choose the same optimal stock portfolio and then adjust how much risk they’re willing to take by increasing or decreasing the percentage, p, of their holdings that they put into risk-free treasury bills.

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The Invisible Hands: Top Hedge Fund Traders on Bubbles, Crashes, and Real Money by Steven Drobny

If you went to play golf for the next 10 years and had to put all your money in one trade, what would it be? That would be a terrible punishment but I would probably buy inflation-linked debt. I do not believe in a free lunch in that you can earn excess returns risklessly. Therefore, I will take the risk-free rate unless I am allowed to manage it actively myself. There is nothing that I can predict over the next 10 years. I take it you do not believe that diversification is the only free lunch in finance? No, of course not. Diversification is mainly a method for reducing volatility and the confusion comes from the fact that finance professors teach that volatility and risk mean the same thing.

The biggest risk to this strategy is that the market will shift, resulting in a futures price above the spot price, a condition is known as contango. Backwardation—The market condition in which the spot price is above the futures price. This is also known an inverted sloping forward curve. This is said to occur due to the convenience yield being higher than the prevailing risk-free rate. The phenomenon of backwardation in commodity futures contracts was originally called “normal backwardation” by economist John Maynard Keynes. Keynes believed that commodity producers are more prone to hedge their price risk than consumers, creating additional demand to sell forward commodities.

Right now, for example, opportunities abound in highly liquid instruments that you can trade short-term, and the liquidity premium attached to longer term illiquid trades offers really attractive valuations for those with a longer horizon. We are in a lower return environment going forward—returns around the world will come down. The risk-free rate is close to zero and long end interest rates around the world are converging toward levels significantly lower than historical norms. Going forward, everyone will be forced to accept lower returns on all assets, from stocks to bonds to businesses. We will also have slower growth stabilizing somewhere between 0 and 3 percent globally.

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The Greed Merchants: How the Investment Banks Exploited the System by Philip Augar

Margins had to come down from unreasonably high levels during the eighties and early nineties, but the trend turned in the mid nineties and they rebuilt nicely for several years. Over this period, return on equity, the figure the industry likes to use to judge itself, compared very favourably with the risk-free rate of return available in government bond markets and with the returns achieved in corporate America at large. When adjusted for inflation, real returns were very high. There is one further important adjustment to make: compensation. In practice, investment banking is still a partnership between the owners of the business and the staff.

., p. 11 for 1980–2000; author calculations for 1990–2003 trend line; SIA Fact Book 2003, p. 31, ‘Major Firms’, for 2001–3 margins. 15. www.sternstewart.com, ‘About EVA’. 16. Author interview, 2003, 17. Morgan Stanley Annual Report 2002, p. 3. 18. SIA Fact Book 2003, p. 36. 19. SIA Fact Book 2003, pp. 36 and 109; 2004, pp. 32 and 105. 20. Hintz, op. cit., p. 10. 21. Under the CAPM cost of equity is calculated by adding the risk free rate of return (the yield on ten-year Treasuries, say 4 per cent) to the equity risk premium (the amount equity investors require to compensate for the extra risk they accept, a number calculated by academics, say 3 per cent) and multiplying the total by a factor known as beta. Beta measures the volatility of a company’s share price and its correlation with the market as a whole.

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Finding Alphas: A Quantitative Approach to Building Trading Strategies by Igor Tulchinsky

According to CAPM, a stock’s expected return is the investor’s reward for the stock’s market risk: Expected return Risk-free rate Stock’s market beta * Market risk premium Since its birth, CAPM has been challenged for its restrictive assumptions and inconsistency with empirical data. The arbitrage pricing theory (APT), developed chiefly by Ross (1976), does not require the stringent assumptions of CAPM. APT states that in a market with perfect competition, a stock’s expected return is a linear function of its sensitivities to multiple unspecified factors: Expected return Risk-free rate Stock’s factor beta * Factor’s risk premium CAPM and APT provided the theoretical foundation of stock return analysis and alpha evaluation.

Investment: A History by Norton Reamer, Jesse Downing

The second contribution Black-Scholes made was to one of the models that immediately preceded it: that of James Boness, whose work has since been largely forgotten by practitioners. Boness’s model had a few vital errors that made the difference between his relative obscurity and a chance for enduring acclaim. The ﬁrst error was the discount rate used.20 The discount rate in the Black-Scholes model is the risk-free rate, again because of the no-arbitrage condition. Boness, though, used the expected return of the stock as the discount rate, but this is not logical in light of the dynamic hedging strategy. Also, Boness tried to incorporate risk preferences into his work, but Black-Scholes imposed the assumption of risk neutrality and did not distinguish between the risk characteristics of various parties.21 As for the equation itself, it is a partial differential equation—with partial derivatives that are now known as the Greeks: delta, gamma, vega, theta, and rho—that bears some semblance to thermodynamic equations.

A line is drawn from this point marking off the risk-free asset and is dropped down upon the efficient frontier to ﬁnd the most efficient point on the frontier. This line is known as the capital allocation line and represents all possible combinations of the market portfolio and the risk-free asset. It is here that Tobin’s famous separation theorem arises: the agent should hold some linear combination of the risk-free rate and the assets on the point of the efficient frontier that intersects with the capital allocation line (see ﬁgure 7.2).31 There is a remarkable implication of Tobin’s separation theorem: the only difference in the assets every agent in the market should hold is just in the combination of risk-free assets and the tangency portfolio (assuming, of course, that everyone agrees on the risk and return Higher expected return The efficient frontier Expected return The tangency portfolio Assets Capital allocation line Lower expected return Lower risk Higher risk Risk/volatility (standard deviation) Figure 7.2 Tobin’s Separation Theorem Source: “The Capital Asset Pricing Model—Fundamental Analysis,” EDinformatics, accessed 2013, http://edinformatics.com/investor _education/capital_asset_pricing_model.htm.

To consider an alternative example, merger arbitrage spread—the premium earned by buying a ﬁrm that another company intends to acquire and holding it until the deal closes—sometimes grows when the volume of available deals grows. This is not because the deals have become more risky but, rather, because there is a limited amount of capital that buys in to merger arbitrage deals. In theory, the price of risk should be a function of just a few variables according to the capital asset pricing model, namely, the risk-free rate, the equity premium, the beta of the asset, and factor premiums like size and value. However, opportunities that arise because of agents’ mandate fragmentation are fundamentally mispriced risk. These opportunities simply have nothing to do with these variables and instead arise from market structure.

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Walk Away by Douglas E. French

Edelman the hot-shot financial advisor claimed we should all stay in hock up to our necks and invest whatever money we might use to pay down the mortgage just in case home prices actually fell. While Edelman advised this, the stock market crashed, commodity markets crashed and interest rates on Treasuries and bank CDs went to virtually zero. During no time period could a person earn a risk-free rate of return higher than even the tax-advantaged rate of a 30-year mortgage. * * * CHAPTER SIX * * * Social Conscience, Fiduciary Duty and Libertarian Ethics The average Joe and Jane are quick to sign on the dotted line with what they think is the American dream clearly in sight.

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Guide to business modelling by John Tennent, Graham Friend, Economist Group

A rational investor would prefer \$100 today rather than \$100 in a year’s time as the \$100 today could be invested in a bank where it would earn interest and grow to an amount greater than the \$100 received in a year’s time. The discount rate, used in dcf analysis, incorporates the time value of money by including the risk-free rate of return that could be earned on \$100 invested risk-free at, say, a bank or in a government bond. Risk premium A rational investor, if given the choice between investing in a bank or bond promising to pay 5% interest a year and a project to build a series of holiday homes on Mars, which also promises to pay 5%, would probably prefer to invest in a bank.

They demand a higher reward or return for assuming higher levels of risk. Any project which is deemed more risky than a risk-free investment, such as investing in a bank, will require an additional return, which is called the risk premium, to compensate for the additional risk. 180 15. PROJECT APPRAISAL AND COMPANY VALUATION Discount rate The combination of the risk-free rate of return and the risk premium gives the discount rate, which is used in dcf analysis to adjust future cash flows so that they appear as if they were received at the start of the project. Short time intervals For projects that conclude in less than five years, it may be preferable to lay out the cash flows using more frequent time intervals than the typical annual layout.

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The Power of Passive Investing: More Wealth With Less Work by Richard A. Ferri

Treynor created a formula for fund evaluation using beta in 1965, which became known as the Treynor Ratio. His findings were published in the Harvard Business Review.11 The article was entitled “How to Rate Management of Investment Funds.” Here is how the Treynor Ratio works: Take any portfolio’s return and subtract a risk-free rate of return (usually the Treasury bill yield); then divide the result by the portfolio’s beta. The result is the ratio of a portfolio’s excess return to market risk as measured by the portfolio’s beta. The Treynor Ratio can be used to compare many portfolios to one another and sort the results by the best risk-adjusted returns.

investment advisor A person or organization that makes the day-to-day decisions regarding the investments in a portfolio. Also called a portfolio manager. investment-grade bond A bond whose credit quality is considered to be among the highest by independent bond-rating agencies. Jensen’s alpha A ratio created by Michael Jensen that measures the return earned in excess of the risk free rate on a portfolio to the portfolio’s total risk as measured by the standard deviation in its returns over the measurement period. junk bond A bond with a credit rating of BB or lower. Also known as a high-yield bond because of the potential rewards offered to those who are willing to take on the additional risk of a lower-quality bond.

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Python for Algorithmic Trading: From Idea to Cloud Deployment by Yves Hilpisch

Under these assumptions, the daily values scale up to the yearly ones from before and one gets the following: One now has to maximize the following quantity to achieve maximum long-term wealth when investing in the stock: Using a Taylor series expansion, one finally arrives at the following: Or for infinitely many trading points in time (that is, for continuous trading), one arrives at the following: The optimal fraction then is given through the first order condition by the following expression: This represents the expected excess return of the stock over the risk-free rate divided by the variance of the returns. This expression looks similar to the Sharpe ratio but is different. A real-world example shall illustrate the application of the preceding formula and its role in leveraging equity deployed to trading strategies. The trading strategy under consideration is simply a passive long position in the S&P 500 index.

Everything being equal, the Kelly criterion implies a higher leverage when the expected return is higher and the volatility (variance) is lower: In [22]: mu = data['return'].mean() * 252 In [23]: mu Out[23]: 0.09992181916534204 In [24]: sigma = data['return'].std() * 252 ** 0.5 In [25]: sigma Out[25]: 0.14761569775486563 In [26]: r = 0.0 In [27]: f = (mu - r) / sigma ** 2 In [28]: f Out[28]: 4.585590244019818 Calculates the annualized return. Calculates the annualized volatility. Sets the risk-free rate to 0 (for simplicity). Calculates the optimal Kelly fraction to be invested in the strategy. The following Python code simulates the application of the Kelly criterion and the optimal leverage ratio. For simplicity and comparison reasons, the initial equity is set to 1 while the initially invested total capital is set to .

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The Misbehavior of Markets: A Fractal View of Financial Turbulence by Benoit Mandelbrot, Richard L. Hudson

For instance, the expectation of a fair coin-tossing game is 0, because you have a 0.5 chance of winning 1 point and a 0.5 chance of losing 1 point (0.5 times 1 plus 0.5 times–1 equals 0). So the present equation is saying that the expected return r on security i equals the sum of two numbers. The first is the “risk-free rate” that you would expect to get from something safe like a Treasury bill. The second is Sharpe’s beta times the “market premium”—that is, however much better you expect the market M to perform over the Treasury rate. Beta is the key to it. Each stock has its own beta, or degree to which its price movements correlate to that of the market overall.

The other two functions, N(d1) and N(d2), are the probabilities of a random number, d, that follows a bell-curve distribution, being less than the quantities below: where σ is the standard deviation of the stock price and ln is the natural logarithm. They are, in essence, the probabilities of the option expiring “in the money,” that is, paying off. As illustration, we can borrow an example from a popular textbook, Bodie, Kane and Marcus 2002: Assume the current stock price S0 is \$100, the exercise price X is \$95, the risk-free rate is 10 percent, the time to expiration T is a quarter-year, and the stock’s standard deviation is 50 percent. A calculator quickly shows d1 is 0.43 and d2 is 0.18. A bell-curve probability table shows N(d1) is 0.6664 and N(d2) is 0.5714. Finally, plugging those values into the full equation, we find the fair price of the call option C0 is \$13.70.

Layered Money: From Gold and Dollars to Bitcoin and Central Bank Digital Currencies by Nik Bhatia

Instead, the Fed shifted to a monetary policy regime focused on managing short-term interest rates. The Dollar’s Suite of Reference Rates Reference rates are pivotal in understanding how the dollar system broke in 2007. A reference rate is the interest rate of a credit instrument considered risk-free within financial academic theory. Financial theory uses the concept of a “risk-free rate” as the reference point for quantifying the risk of an investment. But credit instruments, by definition, have counterparty risk; no such thing can truly be wholly risk-free. Any borrower, no matter how mighty, can theoretically default. In reality, however, an entity like the United States Treasury has never defaulted on its debt obligations and has its own central bank to implicitly back any and all of its issuance.

The Future of Money by Bernard Lietaer

(All values are rounded to the nearest dollar for illustration purposes, since carrying lots of decimals would not modify the argument presented.) We all know that money in the future is worth less than money today. How much less depends critically on the 'discount rate' applied to the project. Our analyst knows he could deposit \$91 in a bank today at a 10% risk- free rate of return, and automatically get \$100 a year from now. Therefore the \$100 a year from now is identical to \$91 today. By the same reasoning, the second year's 8100 would only be worth \$83, the third's \$75, etc. By the tenth year, the \$100 inflow only represents to him \$39; and in the fifteenth year a paltry \$24.

Reduction in value of one currency in terms of other currencies. Discounted Cash Flow: Calculates the value of a future cash flow in terms of an equivalent value today. For instance \$100 a year from now is the same as 890.909 today if one uses a discount rate of 10%, because \$90.909 dollars invested for one year at a risk-free rate of 10% will yield \$100. Euro: Single European currency replacing the national currencies of 11 European countries as of January 1, 1999. Bills in euros will be circulating from 2002 onwards. The transition mechanism from the previous national currencies towards the euro was the ECU, a currency unit which was defined in 1979 as a basket of European currencies.

Evidence-Based Technical Analysis: Applying the Scientific Method and Statistical Inference to Trading Signals by David Aronson

In ﬁnancial market terms, compensation for accepting increased risk is called a risk premium or economic rent. Financial markets offer several kinds of risk premiums: • The equity market risk premium: Investors provide working capital for new business formation and are compensated by receiving a return that is above the risk-free rate for incurring the risk of business failure, economic downturns, and such. • Commodity and currency hedge risk transfer premium: Speculators assume long and short positions in futures to give commercial hedgers (users and producers of the commodity) the ability to shed the risk of price change.

The commodities futures markets perform an economic function that is fun- Theories of Nonrandom Price Motion 381 damentally different from the stock and bond markets. The stock and bond markets provide companies with a mechanism to obtain equity and debt ﬁnancing99 and provide investors with a way to invest their capital. Because stocks and corporate bond investments expose investors to risks that exceed the risk-free rate (government treasury bills), investors are compensated with a risk premium—the equity risk premium and the corporate-bond risk premium. The economic function of the futures markets has nothing to do with raising capital and everything to do with price risk. Price changes, especially large ones, are a source of risk and uncertainty to businesses that produce or use commodities.

This implies that a statistic is itself a random variable. Kachigan, Statistical Analysis, 101. R.S. Witte and J.S. Witte, Statistics, 7th ed. (New York: John Wiley & Sons, 2004), 230. Other statistics include: Z, t, F-ratio, chi-square, and so forth. There are many. The Sharpe ratio is deﬁned as the annualized average return in excess of the risk-free rate (e.g., 90-day treasury bills) divided by the standard deviation of the returns stated on an annualized basis. The proﬁt factor is deﬁned as a ratio of the sum of gains of proﬁtable transactions to the sum of all losses on unproﬁtable transactions. The denominator is stated as an absolute value (no sign) so the value of the proﬁt factor is always positive.

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Hedge Fund Market Wizards by Jack D. Schwager

He and his brothers started a managed money business, while I managed my individual accounts for a while. When did you develop your own option-pricing model? In 1967, I took some of the ideas about how to price warrants in the Random Character of Stock Prices by Paul Cootner and thought I could derive a formula if I made the simplifying assumption that all investments grew at the risk-free rate. Since the purchase or sale of warrants combined with delta neutral hedging led to a portfolio with very little risk, it seemed very plausible to me that the risk-free assumption would lead to the correct formula. The result was an equation that was equivalent to the future Black-Scholes formula.

The Dow companies would be in the best position to pass along higher prices. There is also an interest rate bet implicit in buying long-term options that can be quite interesting when interest rates are very low, as they are now. By being long 10-year call options, we are taking exposure on the risk-free rate implicit in the option pricing models. If interest rates go up, the value of the options can go up dramatically. Are there other option-pricing inconsistencies you look for? Option models generally assume that forward prices are predictive of the future movements in the spot price. Academic research and common sense suggest that this relationship is often invalid.

No, I ended up doing forward conversions, which are a riskless arbitrage.2 The idea was to put on these arbitrage trades and earn 18 to 19 percent annualized. The option market was that inefficient at the time? No, interest rates were that high at the time. I think the arbitrage added about 5 percent to 6 percent to the risk-free rate. Frankly, the trading was kind of mechanical. At the time, you didn’t have the option prices on the screen in front of you. I had to run to the other side of floor to get a printout of option prices to see what options were setting up attractively relative to the stock. Then I would run back to my desk to try to execute the trade.

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Toward Rational Exuberance: The Evolution of the Modern Stock Market by B. Mark Smith

By 1964, Sharpe’s idea had evolved to the point where it could be used to create a comprehensive method for valuing financial assets such as stocks. In this Capital Asset Pricing Model, the expected return earned by a stock consisted of the sum of the “risk free” rate of return (defined as the interest rate paid on a riskless asset like a U.S. government bond) and a risk premium (representing the additional return, over the risk-free rate, that an investor must expect to receive as compensation for taking the extra risk involved in purchasing the risky asset). The risk premium, in turn, was determined by two factors: an “alpha” factor, which represented the specific return due to the unique characteristics of the stock itself, and a “beta” factor, which represented the influence of the overall market’s return on the individual stock.

Stock Market Wizards: Interviews With America's Top Stock Traders by Jack D. Schwager

I used the risk-free interest rate [T-bill rate] to generate theoretical option values, and I used a commercial interest rate to reflect the perspective of an option buyer who had a cost of borrowing funds that was greater than the risk-free rate. As a consequence of using two different rates, trading opportunities appeared. What precisely was the anomaly you found? The market was pricing options based on a theoretical model that assumed a risk-free rate. For most investors, however, the relevant interest rate was the cost of borrowing, which was higher. For example, the option-pricing model might assume a 7 percent interest rate while the investor might have an 8 percent cost of borrowing.

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Portfolio Design: A Modern Approach to Asset Allocation by R. Marston

To compare stocks and bonds, it’s useful to ask the following question: If an investor were to choose between these two assets, which would be more attractive in risk-adjusted terms?7 To assess the return on stocks versus the return on bonds, the first step is to adjust each return for risk. The Sharpe ratio adjusts each return by first subtracting the risk-free rate, rF , then dividing by the standard deviation. If rj is the return on asset j and σ j is its standard deviation, then the Sharpe ratio for asset j is [rj − rF ]/sj To calculate the Sharpe ratio, we use the arithmetic returns and standard deviations in Table 2.1. It’s possible to define a Sharpe ratio using geometric returns, but only if the standard deviation is defined over a similar long horizon.8 Using the returns in Table 1, the Sharpe ratios are defined by Stocks : [0.113 − 0.047]/0.146 = 0.45 Bonds : [0.063 − 0.047]/0.095 = 0.17 The Sharpe ratio for stocks is more than twice the size of the ratio for bonds.

To avoid rounding errors, the returns are stated with more precision in this calculation. 10. Strangely enough, the real return on the medium-term series is actually higher than that of the long-term series. The arithmetic average for long-term bonds is higher than that of medium-term bonds. 11. Arnott and Bernstein (2002), for example, conclude that the equity premium over the risk-free rate is zero and that a sensible expectation for future real returns on stocks and bonds is 2 to 4 percent. 12. The stock price and earnings data are from www.econ.yale.edu/∼shiller which updates series reported in Shiller (2000). 13. The dividend data are also from www.econ.yale.edu/∼shiller. The dividend yield is calculated by dividing the dividend for a given year by the S&P price on the last day of the preceding year.

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Cryptoeconomics: Fundamental Principles of Bitcoin by Eric Voskuil, James Chiang, Amir Taaki

Management of disparate loan maturities [842] and rates of interest is a risk management strategy. While capital reservation is also a risk management strategy, the distinction of a reserve is that reserved capital is “present”, having a maturity of zero. Risk Free Return Fallacy The hypothetical concept of risk free rate of return [843] is the economic interest rate obtainable with a guaranteed return of loan principal. There is a theory that Bitcoin allows this to exist in actual practice by enforcing principle return. A corollary to the theory is that this capability can limit credit expansion [844] generally.

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Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

The fair value of future variance is deﬁned as following (Eq. (26) in Demeterﬁ et al. (1999)): S∗ 2 P(T , K ) S∗ S0 rT rT +e e − 1 − log dK rT − Kvar = T S∗ S0 K2 0 ∞ C(T , K ) rT dK , (5.1) +e K2 S∗ where S0 is the current asset price, C and P are call and put prices, respectively, r is the risk-free rate, T and K are option maturity and strike price, respectively, and S∗ is an arbitrary stock price which is typically chosen to be close to the forward price. The calculation of VIX by CBOE follows Equation 5.1, as explained in the CBOE white paper see CBOE (2003). CBOE calculates the VIX index using the following formula: 2 2 1 F0 Ki rT 2 e Q(T , Ki ) − −1 , (5.2) σVIX = T i Ki2 T K0 where • • • • • σvix is the VIX 100 .

The quadrinomial tree method outperforms other approximating techniques (with the exception of analytical solutions) in terms of both error and time. In this work, we use this methodology to approximate an underlying asset price process following: ϕ2 dXt = r − t 2 dt + ϕt St dWt , (5.4) 100 CHAPTER 5 Construction of Volatility Indices where Xt = logSt and St is the asset price, r is the short-term risk-free rate of interest, and Wt a standard Brownian motions. ϕt models the stochastic volatility process. It has been proved that for any proxy of the current stochastic volatility distribution at t the option prices calculated at time t converge to the true option prices (Theorem 4.6 in Florescu and Viens (2008)).

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The New Depression: The Breakdown of the Paper Money Economy by Richard Duncan

Had it not done so, it is likely that the government would have had to pay much higher interest rates to fund its 2011 budget deficit. Higher interest rates on government bonds would have pushed up all the other interest rates throughout the economy since government bonds act as the benchmark “risk-free” rate. Consequently, higher interest rates would have negatively impacted most sectors of the economy. QE2 also pushed up stock prices, however. By buying \$600 billion worth of government bonds, the Fed effectively financed the entire government budget deficit during those seven months. By doing so, it forced other investors who would have bought those bonds to buy something else.

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Security Analysis by Benjamin Graham, David Dodd

They were probably very good examples of what Graham and Dodd most disapproved of in the financial markets. In the ensuing collapse, most of them had no recovery at all. Graham and Dodd had the insight that the difference between the risk-free rate of return and the yields offered by securities of varying risk created investment opportunities, especially if a diversified portfolio could lock in high returns while reducing the overall risk. In almost any kind of investing, returns have at least some (if not a mathematically exact) connection to the risk-free rate of return, with investors demanding higher returns for greater risk. The premium that investors demand for high yield bonds over the safety of Fed Funds offers a good snapshot for the market’s appetite for risk, as seen in this two-decade survey: ARE YOU GETTING PAID TO TAKE RISK?

Unlike, say, venture capital, with which the investor is seeking a pot of gold (albeit with some diversification so that the occasional winner offsets the losers), the chart is relevant to the large segment of any portfolio designed to create a rate of return for opportunistic capital. That rate of return available is effectively tethered to the risk-free rate, and the spread shown in the chart is the simplest, and perhaps best, measure of where markets have priced it over the last two decades. The yields move in tandem most of the time, but the real opportunities come when they move in opposite directions—that is, when the spread expands, it is time to buy; when it compresses, to sell.

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Dead Aid: Why Aid Is Not Working and How There Is a Better Way for Africa by Dambisa Moyo

In the past, research has found that emerging-market debt (broadly as a group, as well as for individual countries) has low (and sometimes even negative) correlations with other major asset classes. To put it simply, emerging-market investments tend to fare well when other asset classes (say, developed-market stocks and bonds) fare less well. Indeed, the correlation of key emerging-market spreads (the difference between the risk-free rate and the rate charged to a riskier concern) and US bond returns is typically negative – moving in the same direction when the global economy is universally bad. Emerging-market debt has the advantage of being countercyclical to the developed business cycle, since, in a global recession, poor countries can find it cheaper to repay their debts.

Analysis of Financial Time Series by Ruey S. Tsay

A nice consequence of this property is that one can assume that investors are risk-neutral. In a risk-neutral world, we have the following results: • The expected return on all securities is the risk-free interest rate r , and • The present value of any cash flow can be obtained by discounting its expected value at the risk-free rate. 6.6.2 Formulas The expected value of a European call option at maturity in a risk-neutral world is E ∗ [max(PT − K , 0)], where E ∗ denotes expected value in a risk-neutral world. The price of the call option at time t is ct = exp[−r (T − t)]E ∗ [max(PT − K , 0)]. (6.18) Yet in a risk-neutral world, we have µ = r , and by Eq. (6.10), ln(PT ) is normally distributed as σ2 2 ln(PT ) ∼ N ln(Pt ) + r − (T − t), σ (T − t) . 2 Let g(PT ) be the probability density function of PT .

But we can still derive an option pricing formula that does not depend on attitudes toward risk by assuming that the number of securities available is very large so that the risk of the 249 JUMP DIFFUSION MODELS sudden jumps is diversifiable and the market will therefore pay no risk premium over the risk-free rate for bearing this risk. Alternatively, for a given set of risk premiums, one can consider a risk-neutral measure P ∗ such that nt d Pt = [r − λE(J − 1)]dt + σ dwt + d (Ji − 1) Pt i=1 nt = (r − λψ)dt + σ dwt + d (Ji − 1) , i=1 where r is the risk-free interest rate, J = exp(X ) such that X follows the double exponential distribution of Eq. (6.27), ψ = eκ /(1 − η2 ) − 1, 0 < η < 1, and the parameters κ, η, ψ, and σ become risk-neutral parameters taking consideration of the risk premiums; see Kou (2000) for more details.

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Extreme Money: Masters of the Universe and the Cult of Risk by Satyajit Das

The CAPM is one of modern finance’s iconic equations: E[Ri] = Rf + Beta [E[Rm] – Rf] where: E[Ri] is the expected return on the asset. Rf is the risk-free rate of interest on government bonds. Beta is the sensitivity of the asset returns to market returns. E[Rm] is the expected return of the market. [E[Rm] – Rf] is sometimes known as the market premium or risk premium (the difference between the expected market rate of return and the risk-free rate of return). The CAPM’s insight was that the general risk of markets (systematic risk) could be reduced by diversification but the unique risk of a security (unsystematic risk) could not.

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The Education of a Value Investor: My Transformative Quest for Wealth, Wisdom, and Enlightenment by Guy Spier

Buffett had a huge investment in Freddie Mac and a substantial stake in Fannie Mae. Both companies would subsequently lose their way. But at the time, Freddie and Fannie were great businesses. Their key asset was the implied faith, backing, and credit of the US government, which meant they could borrow at virtually risk-free rates. I looked for a firm with a similar advantage and found Farmer Mac—a tiny government-sponsored enterprise in the US farm sector. It struck me as an undiscovered gem of the same ilk. In 2003, I invited the company’s management to give a presentation to the Posse. Whitney Tilson, who is a well-known hedge fund manager, author, and TV commentator, later shared the idea with Bill Ackman.

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Money Mavericks: Confessions of a Hedge Fund Manager by Lars Kroijer

Instead we could create a mini-portfolio of essentially riskless government bonds (if such a thing exists today), and combine this with a portfolio of risk-taking bonds (both corporate and government) and our equity portfolio. For the technically minded you could argue that our cash and riskless bonds constitute the risk-free rate and our different levels of allocations to that versus our riskier assets represent different points on the capital market line. For those with a desire for risk beyond the tangency portfolio, this kind of liquid and transparent portfolio should be fairly easy to borrow against. Concretely our portfolio allocations could take something like the following table.

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The Rise of Carry: The Dangerous Consequences of Volatility Suppression and the New Financial Order of Decaying Growth and Recurring Crisis by Tim Lee, Jamie Lee, Kevin Coldiron

By contrast, asset marketability, money, and credit conditions and the economy as a whole deteriorate suddenly and substantially during carry crashes. Carry trades are “short volatility.” This means they benefit from falling levels of variation in financial asset prices. More concretely, carry trades will provide a positive return above the risk-free rate as long as the volatility of the underlying asset, currency, or commodity price does not end up being higher than expected. Indeed, there exists a range of sophisticated carry trades that use financial derivatives to gain an income that depends directly on a relative lack of volatility of the underlying asset prices.

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Debunking Economics - Revised, Expanded and Integrated Edition: The Naked Emperor Dethroned? by Steve Keen

At this time, his confidence in the soundness of the American economy was complete’ (Barber 1997). 9 Barber observed that among the other reasons was the fact that ‘In the 1930s, his insistence on the urgency of “quick fix” solutions generated frictions between Fisher and other professional economists’ (ibid.). 10 Almost 90 percent of the over 1,200 citations of Fisher in academic journals from 1956 were references to his pre-Great Depression works (Feher 1999). 11 Strictly speaking, this was supposed to be anything in which one could invest, but practically the theory was applied as if the investments were restricted to shares. 12 Since diversification reduces risk, all investments along this edge must be portfolios rather than individual shares. This concept is important in Sharpe’s analysis of the valuation of a single investment, which I don’t consider in this summary. 13 In words, this formula asserts that the expected return on a share will equal the risk-free rate (P), plus ‘beta’ times the difference between the overall market return and the risk-free rate. Beta itself is a measure of the ratio of the variability of a given share’s return to the variability of the market index, and the degree of correlation between the share’s return and the market index return. 14 There are three variations on this, known as the weak, semi-strong and strong forms of the EMH. 15 As I have explained, however, to Fisher’s credit, his failure led to an epiphany that resulted in him renouncing neoclassical thinking, and making a major contribution to the alternative approach to economics that Minsky later developed into the Financial Instability Hypothesis.

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Market Sense and Nonsense by Jack D. Schwager

Yet the standard deviation, the risk component of the Sharpe ratio, is 30 percent higher for Manager B. As a result, even though both Managers A and B have equal cumulative returns and Manager A has much larger equity retracements, Manager A also has a significantly higher Sharpe ratio: 0.71 versus 0.58 (assuming a 2 percent risk-free rate). Why does this occur? Because Manager B has a number of very large gain months, and it is these months that strongly push up Manager B’s standard deviation, thereby reducing the Sharpe ratio. Although most investors would clearly prefer the return profile of Manager B, the Sharpe ratio decisively indicates the reverse ranking.

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When the Money Runs Out: The End of Western Affluence by Stephen D. King

If the financial sector's role is to allocate resources efficiently over time – to meet the interests of savers and investors – we can only know tomorrow whether today's financial decisions will pay off. And, if they don't pay off, that need not be the fault of the financial system alone. Another option – one that is now routinely used in the construction of national accounts – is to treat the income earned by banks through the gap between the ‘risk-free’ rate at which the more creditworthy of their ilk can raise funds and the rate at which they can then lend to customers as a reward for risk-taking. The bigger the gap, the greater is the supposed value of banking services. Yet, in the aftermath of the financial crisis, this looks distinctly odd: those banks that took more risk were apparently adding to national income yet, in hindsight, were also partly responsible for its subsequent collapse.

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MONEY Master the Game: 7 Simple Steps to Financial Freedom by Tony Robbins

KB: Well, the two things to take into account for the options pricing model are (1) the risk-free rate and the (2) volatility of the underlying asset. So imagine if the turkey used this theory. If he were gauging his risk [of being killed] based upon the historical volatility of his life, it would be zero risk. TR: Right. KB: Until Thanksgiving Day. TR: Until it’s too late. KB: When you think about Japan, there’s been ten years of suppressed prices and subdued volatility. The volatility is mid-single digits. It’s as low as any asset class in the world. The risk-free rate is one-tenth of 1%. So when you ask the price on an option, the formula basically tells you it should be free.

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Liar's Poker by Michael Lewis

But no one even on Wall Street could put a price on the homeowners' option (and people still can't, though they're getting closer). Being a trader, Ranieri figured, and argued, that since no one was buying mortgages and everyone was selling them, they must be cheap. More exactly, he claimed that the rate of interest paid by a mortgage bond over and above the government, or risk-free, rate more than compensated the mortgage bondholder for the option he was granting to the horneowner. Ranieri cast himself in an odd role for a Wall Street salesman. He personified mortgage bonds. When people didn't buy them, he appeared wounded. It was as if Ranieri himself were being sold short.

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Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb

When a company shows an increase in earnings once, it draws no immediate attention. Twice, and the name starts showing up on computer screens. Three times, and the company will merit some buy recommendation. Just as with the track record problem, consider a cohort of 10,000 companies that are assumed on average to barely return the risk-free rate (i.e., Treasury bonds). They engage in all forms of volatile business. At the end of the first year, we will have 5,000 “star” companies showing an increase in profits (assuming no inflation), and 5,000 “dogs.” After three years, we will have 1,250 “stars.” The stock review committee at the investment house will give your broker their names as “strong buys.”

Systematic Trading: A Unique New Method for Designing Trading and Investing Systems by Robert Carver

Whilst this might seem unrealistic a trading strategy with a large number of instruments, covering a group of half a dozen asset classes, the returns of which will be relatively uncorrelated, can have returns which are two to three times those for a single asset. 33. Actually it’s the information ratio, which is identical to the Sharpe ratio except that the numerator is the return relative to a benchmark rather than the risk free rate. For our purpose however the distinction isn’t important. 42 Chapter Two. Systematic Trading Rules Medium (average holding period: a few hours or days, to several months) Again because of the law of active management you get more attractive returns as you reduce your holding period to a few months or less.

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The Butterfly Defect: How Globalization Creates Systemic Risks, and What to Do About It by Ian Goldin, Mike Mariathasan

In finance, however, it is often difficult for a consumer to distinguish careless monitoring from exogenous risks outside the trader’s control, leading to a dilution or avoidance of accountability. This is because, to a certain extent, the risk of failure is an implicit feature of any financial dealing that is paying interest above the “risk-free” rate. Lenders accept the risk that borrowers might not be able to repay their loans and thus demand interest payments in return. In a complex world, however, it is increasingly hard to determine whether a failure to repay occurs as a result of exogenous variables or because of reckless behavior on the part of the borrower.

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Fool's Gold: How the Bold Dream of a Small Tribe at J.P. Morgan Was Corrupted by Wall Street Greed and Unleashed a Catastrophe by Gillian Tett

“Quite honestly, [the] funding ordinarily never kept us awake at night,” David Littlewood, one of Frost’s partners, later explained. “[It] required very little maintenance apart from agreeing on a daily funding strategy with QSR.” In mid-July, however, QSR told Cairn that the commercial paper sector was freezing up. By late July, Cairn could sell notes only at the ruinously high cost of 100 basis points over the risk-free rate of borrowing. Frantically, Frost and Littlewood tried to discuss with investors about what action Cairn could take. They were convinced that their fund was far better than those of rivals. But investors did not wish to hear about all the data that Cairn had to back that up. They were losing any ability to discriminate.

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Value Investing: From Graham to Buffett and Beyond by Bruce C. N. Greenwald, Judd Kahn, Paul D. Sonkin, Michael van Biema

From Adjusted Earnings to Earnings Power Value We have expressed skepticism at analysts' ability to derive with great precision the rate at which the earnings of a company should be discounted to calculate the EPV. In this case, we are looking at Intel's earnings over a period in which long-term interest rates fell from around 12 percent to under 7 percent, and the risk-free rate fell from more than 13 percent to under 6 percent. Because the company rarely had any net debt on its books (i.e., debt left after deducting cash and short-term investments), its weighted average cost of capital would not have benefited from the fact that it costs less to borrow than to raise equity, especially after the deductibility of interest payments for tax purposes is taken into account.

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The Road to Ruin: The Global Elites' Secret Plan for the Next Financial Crisis by James Rickards

How is the risk embedded in this leverage being managed? The prevailing theory is called value at risk, or VaR. This theory assumes that risk in long and short positions is netted, the degree distribution of price movements is normal, extreme events are exceedingly rare, and derivatives can be properly priced using a “risk-free” rate. In fact, when AIG was on the brink of default in 2008, no counterparty cared about its net position; AIG was about to default on the gross position to each counterparty. Data show that the time series of price moves is distributed along a power curve, not a normal curve. Extreme events are not rare at all; they happen every seven years or so.

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Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

For the most part, active investors will be paying a percent or two in fees and more in commissions and taxes. (Hedge fund investors pay much more in fees when the fund does well.) This is something like 2 percent of capital, per year, and must be deducted from the return. Two percent is no trifle. In the twentieth century, the average stock market return was something like 5 percent more than the risk-free rate. Yet an active investor has to earn about two percentage points more than average just to keep up with the passive investors. Do some active investors do that? Absolutely. They’re the smart or lucky few who fall at the upper end of the spectrum of returns. The majority of active investors do not achieve that break-even point.

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How the City Really Works: The Definitive Guide to Money and Investing in London's Square Mile by Alexander Davidson

The premium consists of both intrinsic and time value, both of which can change constantly. These are factors used in the Black–Scholes model, which was developed in 1973 and is widely used in ﬁnancial markets for valuing options. Other factors used in the model are volatility, the underlying stock price, and the risk-free rate of return. But Black–Scholes makes key assumptions that are not always tenable, including a constant risk-free interest rate, continuous trading and no transaction costs. Equity options tend to come in the standard contract size of 1,000 shares. To ﬁnd the cost of an option contract, multiply the option price by 1,000.

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Madoff Talks: Uncovering the Untold Story Behind the Most Notorious Ponzi Scheme in History by Jim Campbell

It is the total owed by the customer to a broker for funds advanced to purchase securities. * The Hadassah Foundation is a charitable organization that invests in social change to empower girls and women in Israel and the United States. * The “Sharpe ratio” is the return earned in excess of the risk-free rate, per unit of volatility or total risk. 8 THE MADOFF FAMILY Did They Know? Bernie Madoff: “Andy and Catherine. I’m so sorry for everything. Dad”1 (From prison, apologizing in a one-sentence letter. See Figure 8.1.) Ruth Madoff: “What’s a Ponzi scheme?”2 (Ruth’s first words uttered upon Bernie’s confession to the family, as revealed exclusively to the author.)

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Nerds on Wall Street: Math, Machines and Wired Markets by David J. Leinweber

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Anatomy of the Bear: Lessons From Wall Street's Four Great Bottoms by Russell Napier

However this collapse in the price of government bonds, during a period of historically high deflation, must have considerably disrupted investor perceptions of what represented “fair value” in financial assets. The impact on financial asset valuation in periods of declining prices and earnings but rising risk-free rates is particularly negative. Even in the worst deflationary times a collapse in the price of government bonds was thus proved possible. What other unexpected reactions might there be in financial markets? Investors in the corporate bond market were also to be shocked by the scale of the bear. The performance of this market provides perhaps the best indication that investors expected a normal economic contraction to follow after the crash of October 1929.

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Other People's Money: Masters of the Universe or Servants of the People? by John Kay

Since the value of all shares is around \$50 trillion, a single basis point is \$5 billion, and so the cost of intermediation reduces individual savings pots substantially (on the one hand) and pays for limousines and private jets (on the other). Today real rates of return on low-risk, long-term investments – such as the indexed bonds of the British, German or US government – hover around zero. The anticipated risk premium – the amount by which the return on equities exceeds the risk-free rate – is unlikely to be more than 3 to 4 per cent. After charges, many users of the investment channel are now unlikely to earn any real return on their savings at all. This is the epitome of a financial system designed for the needs of financial market participants rather than the users of finance.

Hedgehogging by Barton Biggs

Equity holders get paid only if there is money left after the bondholders get paid, and they, therefore, require a risk premium.To assume a zero premium is to ignore the fact that corporate bonds do default (unlike the U.S.Treasury). Discounting uncertain and cyclical corporate cash flows at a risk-free rate just does not make sense. During our debate, Jim Glassman said that it was different this time and emphasized the power of the new era in which “fresh, unfettered thinking and new rules apply.” However, it absolutely overwhelmed my imagination that the new world could be so different that fair value for U.S. stocks would be 100 times earnings when, for almost two centuries as a young country exploded with fantastic progress and growth, the multiple of the S&P 500 had averaged 14 times.

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The Ascent of Money: A Financial History of the World by Niall Ferguson

Work out that option price accurately, rather than just relying on guesswork, and you truly deserve the title ‘rocket scientist’. Black and Scholes reasoned that the option’s value depended on five variables: the current market price of the stock (S), the agreed future price at which the option could be exercised (X), the expiration date of the option (T), the risk-free rate of return in the economy as a whole (r) and - the crucial variable - the expected annual volatility of the stock, that is, the likely fluctuations of its price between the time of purchase and the expiration date (σ - the Greek letter sigma). With wonderful mathematical wizardry, Black and Scholes reduced the price of the option (C) to this formula: where Feeling a bit baffled?

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Misbehaving: The Making of Behavioral Economics by Richard H. Thaler

Simply texting patients to remind them to take their prescribed medications (in this study, for lowering blood pressure or cholesterol levels) reduced the number of patients who forgot or otherwise failed to take their medications from 25% to 9% (Wald et al., 2014). † They were able to do this because, for technical reasons, the standard theory makes a prediction about the relation between the equity premium and the risk-free rate of return. It turns out that in the conventional economics world, when the real (inflation-adjusted) interest rate on risk-free assets is low, the equity premium cannot be very large. And in the time period they studied, the real rate of return on Treasury bills was less than 1%. ‡ That might not look like a big difference, but it is huge.

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The Bank That Lived a Little: Barclays in the Age of the Very Free Market by Philip Augar

Philip Augar, The Death of Gentlemanly Capitalism, Penguin Books, London, 2001, pp. 181–2 9. Vander Weyer, Falling Eagle, p. 215. Shareholders compare return on capital to the rate of return on a risk-free investment such as the yield on government stock and to the cost of capital, a complex calculation based on interest rates and the company’s share price. In 1993, the risk-free rate of return was 8 per cent and Barclays’ cost of capital was a couple of percentage points higher. 3. The Scholar’s Tale, 1986–1993 1. Geoffrey Owen, The Rise and Fall of Great Companies: Courtaulds and the Reshaping of the Man-Made Fibres Industry, Oxford University Press, Oxford, 2010, p. 119 2.

How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter

Gordon is the developer of the Gordon Growth Model; but one of his less conventional contributions to finance is a model of the development of capitalist societies via simulated investment strategies. In his JWPR007-Lindsey May 7, 2007 17:9 Julian Shaw 229 model, Gordon assumed that the only available investments are a stock portfolio with parameters similar to that of the S&P 500, and a bank account. People are free to adjust the portfolio mix, but their consumption exceeds the risk-free rate so they must invest some fraction of their wealth in the risky asset in order to survive in the long run. Those whose wealth falls below a threshold have to work for a living. From this work emerged a new investment criterion, maximization of the probability of long-run survival, which is qualitatively different from the other famous normative2 investment criterion, the Kelly (log optimal) criterion.

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Traders, Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives by Satyajit Das

N The value of ⌬ must be equal to a number that makes sure that the value of the portfolio is the same for both an increase and a decrease in the share price: DAS_C07.QXP 8/7/06 4:45 PM Page 197 6 N Super models – derivative algorithms 197 \$110 ⌬ – \$5 = \$90 ⌬ ⌬ = 0.25 This means that to hedge or replicate one sold call you hold .25 units of the share. Irrespective of whether the share price moves up or down, the portfolio has a value of \$22.5 at the expiry of the option. A riskless portfolio should earn the risk free rate of interest (money in the bank). It is now possible to derive the fair value of the option based on the following steps: N The riskless portfolio in present value terms, that is today, is worth: e–Rf.T 22.5 = e–.10 × 1 22.5 = \$ 20.36 N The value of the option is calculated using the known value of the portfolio of shares at commencement of the transaction (.25 × \$ 100 = \$ 25): Value of share portfolio – option premium = value of riskless portfolio \$ 25 – option premium = \$ 20.36 Option premium = \$4.64 Cox, Ross and Rubinstein showed that you could combine millions of paths like this to work out the option price.

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Endless Money: The Moral Hazards of Socialism by William Baker, Addison Wiggin

The aftermath is now ugly, with nest eggs reduced, taxes increasing, inflation probable, and the risk of a generational depression palpable but now downplayed by a financial community whose forecasting and analysis blows with the latest breeze. Consider the risk that no one wishes to discuss, which is the failure of the U.S. government. Today the only safe haven is investing in debt obligations of the U.S. Treasury, which define the risk-free rate. This solidarity has been unquestioned since the passage of the Sixteenth Amendment in 1913, which granted the Treasury Department unlimited power to confiscate the private property of its citizens. The next chapter, “Faux Class Warfare,” will argue that taxation may have reached its upper limit of productivity in the United States, just as it has in 152 ENDLESS MONEY other socialist regimes.

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The Euro and the Battle of Ideas by Markus K. Brunnermeier, Harold James, Jean-Pierre Landau

All of these arguments were among the key motivations underlying the initial calls for government stimulus back in 2008. Interest Rate Credit Spreads There is yet an important dimension to the interest rate debate when a country borrows in a foreign or common currency: borrowing interest rates on government debt are not necessarily risk-free rates. With excessive government borrowing, sovereign debt may actually be viewed as risky, and so yields on government debt could spike even in a ZLB environment—something the Keynesian analysis presented above simply does not consider—and potentially even flip the sign of the multiplier. In light of this, talk about confidence effects gained traction within policy circles after 2010.

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Rentier Capitalism: Who Owns the Economy, and Who Pays for It? by Brett Christophers

The ratio of banks’ net interest income to total interest-bearing assets (the ‘net interest margin’) therefore declined – an effect to which, according to Alessandri and Nelson, the downward drift in market rates charted in Figure 1.5 ‘contributed strongly’.55 Figure 1.6 Lending spreads, UK-based banks, 1997–2017 Household spread represents weighted average of mortgage spread (quoted mortgage rates over risk-free rates) and unsecured lending spread (spreads on credit cards, overdrafts and personal loans, relative to Bank Rate). Corporate spread represents weighted average of: SME lending rates over Bank Rate; CRE average senior loan margins over Bank Rate; and, as a proxy for the rate at which banks lend to large, non-CRE corporates, UK investment-grade company bond spreads over maturity-matched government bond yield.

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The Joys of Compounding: The Passionate Pursuit of Lifelong Learning, Revised and Updated by Gautam Baid

At all times, in all markets, in all parts of the world, the tiniest change in rates changes the value of every financial asset. You see that clearly with the fluctuating prices of bonds. But the rule applies as well to farmland, oil reserves, stocks, and every other financial asset. And the effects can be huge on values.4 The rates of return that investors need from any kind of investment are directly tied to the risk-free rate that they can earn from government securities. So if the government rate rises, the prices of all other investments must adjust downward, to a level that brings their expected rates of return into line…. In the case of equities or real estate or farms or whatever, other very important variables are almost always at work, and that means the effect of interest rate changes is usually obscured.

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Big Debt Crises by Ray Dalio

., banks), like an avalanche. Both lenders’ and depositors’ justified fears feed on themselves, leading to runs on financial institutions that typically don’t have the cash to meet them unless they are under the umbrella of government protections. Cutting interest rates doesn’t work adequately because the floors on risk-free rates have already been hit and because as credit spreads rise, the interest rates on risky loans go up, making it difficult for those debts to be serviced. Interest rate cuts also don’t do much to help lending institutions that have liquidity problems and are suffering from runs. At this phase of the cycle, debt defaults and austerity (i.e., the forces of deflation) dominate, and are not sufficiently balanced with the stimulative and inflationary forces of printing money to cover debts (i.e., debt monetization).

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The Social Life of Money by Nigel Dodd

Instead of a gold standard, the post–Bretton Woods system has relied on a “Treasury bill standard” whereby central banks no longer cash in their dollar inflows for gold but rather Treasury bills at rates of interest that are near zero. These flows have been fuelled by a U.S. balance of payment deficit that has been growing more or less constantly. This is the so-called “risk-free” rate, although it looks increasingly like an accounting fiction: the corresponding debt is perpetually rolled over. But the unique privilege that the United States appears to enjoy as a debtor cannot be explained by economics alone. It is underpinned by a political logic and reinforced by a military power: The essence of U.S. military predominance in the world is, ultimately, the fact that it can, at will, drop bombs, with only a few hours’ notice, at absolutely any point on the surface of the planet.

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Unelected Power: The Quest for Legitimacy in Central Banking and the Regulatory State by Paul Tucker

Pure Credit Policy: Steering Supply to Stimulate Aggregate Demand What central bankers and economists call “credit policy” goes a lot further than MMLR operations.23 The elemental question is why deliberately large outright purchases of risky paper might ever be contemplated. While the obvious motive is to drive down the cost of credit in the capital markets, that would need to be over and above what could be delivered, directly or indirectly, by lowering the expected path of risk-free rates and by using basic QE to squeeze investors out of government paper into private securities. In other words, more regular central bank operations should be exhausted first. That suggests the following minimum substantive criteria before the use of credit policy options becomes a live issue: the monetary policy rate is at or very close to the effective lower bound and is expected to stay there; vanilla quantitative easing and guidance on the prospective path of the policy rate will not suffice or will entail even more unacceptable risks; repo operations in private sector paper will not suffice, even if eligible counterparties were extended beyond banks and maturities lengthened; in consequence, there is a serious risk of a deep and protracted recession that would create powerful disinflationary forces or even deflation becoming embedded in people’s expectations.

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Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

Here is a recap on option pricing using the replication argument: in the classic Black–Scholes–Merton (BSM) world [1] option traders can continuously delta-hedge the market-directional (delta) risk of their options by trading offsetting amounts of the underlying security, and thus create a synthetic risk-free asset. No-arbitrage arguments imply that the option should be priced so that the expected return of the delta-hedged option position equals the risk-free rate. This is the reason that derivative pricing is consistent with risk-neutral probabilities even when real-world investors may be risk averse. Specifically, risk-neutral pricing of a derivative off its underlying asset does not require assuming that real-world investors are risk neutral. Risk aversion does not matter if positions could be hedged.

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Trading and Exchanges: Market Microstructure for Practitioners by Larry Harris

• Futile traders believe that they are profit-motivated traders, but they cannot trade successfully enough to profit in the long run. • Pseudo-informed traders trade on stale information. 8.6 QUESTIONS FOR THOUGHT • Besides saving for retirement and borrowing for education, what other intertemporal cash flow timing problems do people commonly face? • Can the real risk-free rate of interest ever be negative? • Suppose a wheat farmer sells a wholesale baker a forward contract. What happens if hail destroys the farmer’s crop or if the baker loses a large contract to deliver bread in the future? Will their hedges protect them against these risks? How can they manage these risks?

The First Tycoon by T.J. Stiles

It is worth twenty-five percent a year without any risk.” Given the size of Vanderbilt's business operations, the \$11 million figure rang true. It would have made him one of a half-dozen or so of the richest men in America; only William B. Astor and very few others could boast notably larger estates. The risk-free rate of return he cited was clearly hyperbole, but his point was clear: he had taken great care to put his affairs in order. To Van Pelt, Vanderbilt seemed very much like a man preoccupied by his own death—and incapable of accepting it. “Commodore,” he once asked, “suppose anything should happen, what are you going to do with your property?”