# compound rate of return

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This is because for a geometric random walk, the average compounded rate of return is not the short-term (or one-period) return m (0 here), but is g = m − s 2 /2. This follows from the general formula for compounded growth g(f ) given in the appendix to this chapter, with the leverage f set to 1 and risk-free rate r set to 0. This is also consistent with P1: JYS c06 JWBK321-Chan September 24, 2008 13:57 98 Printer: Yet to come QUANTITATIVE TRADING the fact that the geometric mean of a set of numbers is always smaller than the arithmetic mean (unless the numbers are identical, in which case the two means are the same). When we assume, as I did, that the arithmetic mean of the returns is zero, the geometric mean, which gives the average compounded rate of return, must be negative. The take-away lesson here is that risk always decreases long-term growth rate—hence the importance of risk management!

The take-away lesson here is that risk always decreases long-term growth rate—hence the importance of risk management! *This example was reproduced with corrections from my blog article “Maximizing Compounded Rate of Return,” which you can ﬁnd at epchan.blogspot.com/2006/ 10/maximizing-compounded-rate-of-return.html. Often, because of uncertainties in parameter estimations, and also because return distributions are not really Gaussian, traders prefer to cut this recommended leverage in half for safety. This is called “half-Kelly” betting. If you have a retail trading account, your maximum overall leverage l will be restricted to either 2 or 4, depending on whether you hold the positions overnight or just intraday. In this situation, you would have to reduce each fi by the same factor l/(| f1 | + | f2 | + · · · + | fn|), where | f1 | + | f2 | + · · · + | fn| is the total unrestricted leverage of the portfolio.

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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

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FIGURE 1.2 Graham Simple Value Strategy Performance Chart (1976 to 2011) Table 1.2 presents the results from our study of the simple Graham value strategy. Graham's strategy turns \$100 invested on January 1, 1976, into \$36,354 by December 31, 2011, which represents an average yearly compound rate of return of 17.80 percent—outperforming even Graham's estimate of approximately 15 percent per year. This compares favorably with the performance of the S&P 500 over the same period, which would have turned \$100 invested on January 1, 1976, into \$4,351 by December 31, 2011, an average yearly compound rate of return of 11.05 percent. The performance of the Graham strategy is attended by very high volatility, 23.92 percent versus 15.40 percent for the total return on the S&P 500. The strategy would also have required a cast-iron gut because only a few stocks qualified at any given time, and the back-test assumed that we invested all our capital in those stocks.

We also weight the stocks in the portfolio by market capitalization to make the returns comparable to the market capitalization–weighted S&P 500, while Greenblatt equally weights the stocks in his portfolios (we discuss our back-test procedures in detail in Chapter 11). Importantly, the Magic Formula's performance does compare favorably with the performance of the S&P 500 over the same period, which would have turned \$100 invested on January 1, 1964, into \$7,871 by December 31, 2011, an average yearly compound rate of return of 9.52 percent. Table 2.1 confirms that Greenblatt's Magic Formula was a better risk-adjusted bet: Sharpe, Sortino, and drawdowns are all better than the S&P 500. TABLE 2.1 Performance Statistics for the Magic Formula Strategy (1964 to 2011) Figures 2.2(a) and 2.2(b) show the rolling 1-year and 10-year returns for the Magic Formula for the period 1964 to 2011. As Figure 2.2 illustrates, Greenblatt's Magic Formula strategy has underperformed in many single-year periods; however, over longer periods of time, it has proven to perform exceptionally well.

Figure 2.5 shows the cumulative performance of the Magic Formula and the Quality and Price strategies for the period 1964 to 2011. FIGURE 2.5 Magic Formula and Quality and Price Strategies Comparative Performance Chart (1964 to 2011) Table 2.4 sets out the summary annual performance statistics for Quality and Price. Quality and Price handily outpaces the Magic Formula, turning \$100 invested on January 1, 1964, into \$93,135 by December 31, 2011, which represents an average yearly compound rate of return of 15.31 percent. Recall that the Magic Formula turned \$100 invested on January 1, 1964, into \$32,313 by December 31, 2011, which represents a CAGR of 12.79 percent. As you can see in Table 2.4, while much improved, Quality and Price is not a perfect strategy: the better returns are attended by higher volatility and worse drawdowns. Even so, on a risk-adjusted basis, Quality and Price is the winner.

Analysis of Financial Time Series by Ruey S. Tsay

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. √ The standard deviation of the price 6 months from now is 241.92 = 15.55. Next, let r be the continuously compounded rate of return per annum from time t to T . Then we have PT = Pt exp[r (T − t)], where T and t are measured in years. Therefore, PT 1 r= ln . T −t Pt By Eq. (6.9), we have ln PT Pt ∼N σ2 µ− 2 (T − t), σ 2 (T − t) . Consequently, the distribution of the continuously compounded rate of return per annum is σ2 σ2 r ∼ N µ− , . 2 T −t The continuously compounded rate of return √ is, therefore, normally distributed with mean µ − σ 2 /2 and standard deviation σ/ T − t. Consider a stock with an expected rate of return of 15% per annum and a volatility of 10% per annum. The distribution of the continuously compounded rate of return of the stock over two years is normal√with mean 0.15 − 0.01/2 = 0.145 or 14.5% per annum and standard deviation 0.1/ 2 = 0.071 or 7.1% per annum.

Obtain the mean and standard deviation of the distribution and construct a 95% confidence interval for the stock price. 7. A stock price is currently \$60 per share and follows the geometric Brownian motion d Pt = µPt dt +σ Pt dt. Assume that the expected return µ from the stock is 20% per annum and its volatility is 40% per annum. What is the probability distribution for the continuously compounded rate of return of the stock over 2 years? Obtain the mean and standard deviation of the distribution. 8. Suppose that the current price of Stock A is \$70 per share and the price follows the jump diffusion model in Eq. (6.26). Assume that the risk-free interest rate is 8% per annum and the stock volatility is 30% per annum. In addition, the price on average has about 15 jumps per year with average jump size −2% and jump volatility 3%.

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Mathematics for Economics and Finance by Michael Harrison, Patrick Waldron

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. , wn ) return on the portfolio w the investor’s desired expected return Table 6.2: Notation for portfolio choice problem In general, the polynomial defining the IRR has T (complex) roots. Conditions have been derived under which there is only one meaningful real root to this polynomial equation, in other words one corresponding to a positive IRR.1 Consider a quadratic example. Simple rates of return are additive across portfolios, so we use them in one period cross sectional studies, in particular in this chapter. Continuously compounded rates of return are additive across time, so we use them in multi-period single variable studies, such as in Chapter 7. Consider as an example the problem of calculating mortgage repayments. 6.2.2 Notation The investment opportunity set for the portfolio choice problem will generally consist of N risky assets. From time to time, we will add a riskfree asset. The notation used throughout this chapter is set out in Table 6.2.

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

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The median incentive fee for this study’s sample is 20.00 percent and only 15.00 percent in Golec’s sample. Finally, the average assets under management in this study were \$34.68 million compared to \$5.01 million in TABLE 13.3 Summary of CTA Average Attributes, February 1974–February 1998, 974 CTA Programs Attribute Mean Std. Error Min Max Months listed Average monthly return (%) Margin to equity ratio (%) Annual compounded rate of return (%) Annual standard deviation (%) Maximum drawdown Management fee (%) Incentive fee (%) Assets (Millions \$) 65.14 1.31 19.40 45.91 1.34 10.58 5.00 −3.14 1.03 278.00 13.47 100.00 12.75 26.24 −0.27 2.46 20.27 34.68 15.14 18.41 0.18 1.31 4.45 186.95 −47.51 0.79 −0.99 0.00 0.00 0.10 139.00 142.89 0.10 6.00 50.00 2,954.00 253 The Effect of Management and Incentive Fees on the Performance of CTAs Golec’s sample.

CTA COMPENSATION PARAMETERS AND PERFORMANCE In this section we empirically explore the relationship between CTA returns and the standard deviation of returns to their compensation parameters by replicating Golec’s (1993) analysis. We examined the issue by fitting two ordinary least squares (OLS) cross-sectional regressions on the means and standard deviations of returns of the CTAs on their fee parameters as follows: ARORj = b0 + b1km + b2ki + b3ln(At − 1) + ej (13.3) sj = a0 + a1km + a2ki + a3ln(At − 1) + uj (13.4) where ARORj = annual compounded rate of return for CTAj sj = annual standard deviation of CTAj returns ej, uj = error terms. Because the distribution of assets under management is clearly skewed, we use the natural logarithm of assets under management as the “size” variable. Significance tests use White’s (see Greene 2000) heteroskedasticity consistent standard errors. Table 13.4 presents OLS estimates of regression TABLE 13.4 Estimation of the Relationship between Compensation Parameters and CTA Mean Annual Compounded Returns and Standard Deviation of Returns Independent Variables Dependent Variables Intercept km ki ln(At − 1) Mean Annual Returns −0.255* (0.075) 0.229* (0.057) 0.580 (0.583) 1.424* (0.482) 0.693* (0.259) 0.654* (0.156) 0.016* (0.003) −0.009* (0.003) Standard Deviation *Significant at the 1 percent level under H0 = 0. 254 MANAGED FUTURES INVESTING, FEES, AND REGULATION coefficients from equations 13.3 and 13.4, along with white standard errors in parentheses.

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

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Twelve years later they repeated the study, using the same stocks they had used in their previous study. This time the returns were even higher despite the fact that they made no adjustment for any of the new firms or new industries that had surfaced in the interim. They wrote: If a portfolio of common stocks selected by such obviously foolish methods as were employed in this study will show an annual compound rate of return as high as 14.2 percent, then a small investor with limited knowledge of market conditions can place his savings in a diversified list of common stocks with some assurance that, given time, his holding will provide him with safety of principal and an adequate annual yield.21 Many dismissed the Eiteman and Smith study because it did not include the Great Crash of 1929 to 1932. But in 1964, two professors from the University of Chicago, Lawrence Fisher and James H.

Jones, “A Century of Stock Market Liquidity and Trading Costs,” working paper, May 23, 2002. 11 The cost of some index funds for even small investors is only 0.1 percent per year. See Chapter 20. 130 PART 2 Valuation, Style Investing, and Global Markets discount to such safe and liquid assets as government bonds. As stocks become more liquid, their valuation relative to earnings and dividends should rise.12 The Equity Risk Premium Over the past 200 years the average compound rate of return on stocks in comparison to safe long-term government bonds—the equity premium—has been between 3 and 31⁄2 percent.13 In 1985, economists Rajnish Mehra and Edward Prescott published a paper entitled “The Equity Premium: A Puzzle.”14 In their work they showed that given the standard models of risk and return that economists had developed over the years, one could not explain the large gap between the returns on equities and fixed-income assets found in the historical data.

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The Simple Path to Wealth: Your Road Map to Financial Independence and a Rich, Free Life by J L Collins

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True enough and so far so good. But he goes on to say this means “buy and hold investing doesn’t work anymore.” The magazine interviewer then points out, and good for him, that even during the “lost decade” of the 2000s, the buy and hold strategy of stock investing would have returned 4%. The professor responds: “Think about how that person earned 4%. He lost 30%, saw a big bounce back, and so on, and the compound rate of return….was 4%. But most investors did not wait for the dust to settle. After the first 25% loss, they probably reduced their holdings, and only got part way back in after the market somewhat recovered. It’s human behavior.” Hold the bloody phone! Correct premise, wrong conclusion. We’ll come back to this in a moment. Magazine: “So what choice do I have instead?” Professor: “We’re in an awkward period of our industry where we haven’t developed good alternatives.

Beat the Market by Edward Thorp

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Figure 7.1 graphs the performance of the basic system from 1946 through 1966. This is not as revealing as Figure 7.2, which contains this information on a semi-log grid. There, equal vertical distances represent equal percentage changes and a straight line represents a constant percentage increase, compounded annually. The greater the slope of the line, the greater the compound rate of increase. Since we are interested in compound rate of return, a *This is the arithmetic average. For investors interested mainly in long-term growth, the equivalent annual compounding rate, which is 26% before taxes, is a more important figure. Elsewhere in the book we have referred to these figures of 26% and 30% by citing “more than 25% for seventeen years.” †It is customary in stock market literature to figure rates of return before taxes, since the effect of taxes will vary with the type of investment, the investor’s situation, and with the tax laws. 94 semi-log grid makes it easier to compare various investments.

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Investing Demystified: How to Invest Without Speculation and Sleepless Nights by Lars Kroijer

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This does not mean that stock markets will be particularly poor or attractive right now; it means that investors historically have demanded a premium for investing in risky equities, as opposed to less-riskier assets. We also assume that investors expect to be paid a similar premium for investing in equities over safe government bonds in future as they have historically. The size of the equity risk premium is subject to much debate, but numbers in the order of 4–5% are often quoted. If you study the returns of the world equity markets over the past 100 years (see Table 5.1) the annual compounding rate of return for this period is close to this range. Of course it is impossible to know if the markets over that period have been particularly attractive or poor for equityholders compared to what the future has in store. The equity risk premium is not a law of nature, but simply an expectation of future returns, in this case based on what those markets achieved in the past, including the significant drawdowns that occurred.

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Derivatives Markets by David Goldenberg

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In our model without 194 FORWARD CONTRACTS AND FUTURES CONTRACTS carrying charges the equilibrium futures price is F0=P0*(1+r[0,1]). The only difference between these two results is the compounding method for interest and the time interval. Suppose that [0,1] corresponds to one year and therefore =1.0. Then, under simple interest compounding, \$1 grows to \$1*(1+rA) where rA is the annual interest rate under simple compounding. On the other hand, \$1 invested in an account that grows continuously at a continuously compounded rate of return rc for one year is \$1*erc. If we equate these two terminal amounts, we get the continuously compounded rate rc that is equivalent to the simple interest rate rA. It is that rate that gives the same terminal amount as the simple rate, \$1*erc =\$1*(1+rA). We can carry out the exact same procedure when time to maturity is . We use the interval [0,] for simple compounding to obtain that \$1 grows to \$1*(1+r [0,]) under simple compounding, and to \$1*erc. under continuous compounding.

There are several ways to estimate it. The ﬁrst method will be called the historical volatility estimator method. It is described below, A. The Historical Volatility Estimator Method 1. Collect historical data, say daily closing prices, for a given stock over a given historical period. 2. Calculate the log price relatives which are deﬁned as ln(Si /Si–1). This represents the continuously compounded rate of return of the stock over the period [i–1,i]. 3. Calculate the mean of these log price relatives in the ordinary manner as the sum of the log price relatives divided by the number of log price relatives. Call this quantity E{ln(Si /Si–1)}. 584 OPTIONS 4. The next step is to calculate the standard deviation of these log price relatives over the entire period, ͡ daily.This is deﬁned as,  daily = ⎡ ⎛S ⎞ ⎧⎪ ⎛ S ⎞⎫⎪⎤ i ⎢ ln − E ⎨ ln ⎜ i ⎟⎬⎥ ⎜ ⎟ ∑i=1⎢ S ⎩⎪ ⎝ Si −1 ⎠⎭⎪⎥⎦ ⎣ ⎝ i −1 ⎠ 2 N (N − 1) where N is the number of log price relatives.

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The Essays of Warren Buffett: Lessons for Corporate America by Warren E. Buffett, Lawrence A. Cunningham

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Further illuminating the folly of junk bonds is an essay in this collection by Charlie Munger that discusses Michael Milken's approach to finance. Wall Street tends to embrace ideas based on revenue-generating power, rather than on financial sense, a tendency that often perverts good ideas to bad ones. In a history of zero-coupon bonds, for example, Buffett shows that they can enable a purchaser to lock in a compound rate of return equal to a coupon rate that a normal bond paying periodic interest would not provide. Using zero-coupons thus for a time enabled a borrower to borrow more without need of additional free cash flow to pay the interest expense. Problems arose, however, when zero-coupon bonds started to be issued by weaker and weaker credits whose free cash flow could not sustain increasing debt obligations.

Monte Carlo Simulation and Finance by Don L. McLeish

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SIMULATING THE VALUE OF OPTIONS Suppose these portfolios (or their managers) have been selected retrospectively from a list of “survivors” which is such that the low of the portfolio value never crossed a barrier at l = Oe−a (bankruptcy of fund or termination or demotion of manager, for example) and the high never crossed an upper barrier at h = Oeb . However, for the moment let us assume that the upper barrier is so high that its influence can be neglected, so that the only absorbtion with any substantial probability is at the lower barrier. We interested in the estimate of return from the two portfolios, and a preliminary estimate indicates a continuously compounded rate of return from portfolio 1 of R1 = ln(56.625/40) = 35% and from portfolio two of R2 = ln(56.25/40) = 34%. Is this diﬀerence significant and are these returns reasonably accurate in view of the survivorship bias? We assume a geometric Brownian motion for both portfolios, (5.34) dSt = µSt dt + σSt dWt , and define O = S(0), C = S(T ), H = max S(t), 0 t T L = min S(t) 0 t T with parameters µ, σ possibly diﬀerent.

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

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TOTAL ANNUAL RETURNS FOR BASIC ASSET CLASSES, 1926–2009 Average Annual Return Risk Index (Year-to-Year Volatility of Returns) Small-company common stocks 11.9% 32.8% Large-company common stocks 9.8 20.5 Long-term government bonds 5.4 9.6 U.S. Treasury bills 3.7 3.1 Common stocks have clearly provided very generous long-run rates of return. It has been estimated that if George Washington had put just one dollar aside from his first presidential salary and invested it in common stocks, his heirs would have been millionaires more than ten times over by 2010. Roger Ibbotson estimates that stocks have provided a compounded rate of return of more than 8 percent per year since 1790. (As the table above shows, returns have been even more generous since 1926, when common stocks of large companies earned almost 10 percent.) But this return came only at substantial risk to investors. Total returns were negative in about three years out of ten. So as you reach for higher returns, never forget the saying “There ain’t no such thing as a free lunch.”

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Since this method doesn’t have an exit point, it would be difficult to improve upon it through position sizing. Futures Market Models Kaufman Adaptive Moving-Average Approach Kaufman doesn’t really discuss position sizing in his book, Smarter Trading. He does discuss some of the results of position sizing such as risk and reward. By risk he means the annualized standard deviation of the equity changes and by reward he means the annualized compounded rate of return. He suggests that when two systems have the same returns, the rational investor will choose the system with the lowest risk. Kaufman also brings up another interesting point in his discussion-the 50.year rule. He says that levees were built along the Mississippi River to protect them from the largest flood that has occurred in the last 50 years. This means that water will rise above the levee, but not very often-perhaps once in a lifetime.

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The Sovereign Individual: How to Survive and Thrive During the Collapse of the Welfare State by James Dale Davidson, Rees Mogg

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But remember, that assumes an annual tax payment of \$45,000. 156 Compared to a tax haven like Bermuda, where the income tax is zero, the lifetime loss for paying taxes at American rates would be about \$1.1 billion. You may object that an annual return of 20 percent is a high rate of return. No doubt you would be right. But given the startling growth in Asia in recent decades, many investors in the world have achieved that and better. The compound rate of return in Hong Kong real estate since 1950 has been more than 20 percent per annum. Even some economies that are less widely known for growth have afforded easy opportunities for high profits. You could have pocketed an average real return of more than 30 percent annually in U.S. dollar deposits in Paraguayan banks over the last three decades. High Investment returns are easier to realize in some places than others, but skilled investors can certainly achieve profits of 20 percent or more in good years, even if they do not consistently match the performances of George Soros or Warren Buffet.

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A Man for All Markets by Edward O. Thorp

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Appendix D * * * PERFORMANCE OF PRINCETON NEWPORT PARTNERS, LP Table 14: Annual Return in Percent Period Beginning and Ending Princeton Newport Partners, LP (1) Princeton Newport Partners, LP (2) S&P 500 Index (3) 3 Month US T-Bill Total Return 11/01/69—12/31/69 +4.0 +3.2 -4.7 +3.0 01/01/70—12/31/70 +16.3 +13.0 +4.0 +6.2 01/01/71—12/31/71 +33.3 +26.7 +14.3 +4.4 01/01/72—12/31/72 +15.1 +12.1 +19.0 +4.6 01/01/73—12/31/73 +8.1 +6.5 -14.7 +7.5 01/01/74—12/31/74 +11.3 +9.0 -26.5 +7.9 01/01/75—10/31/75* +13.1 +10.5 +34.3 +5.1 11/01/75—10/31/76 +20.2 +16.1 +20.1 +5.2 11/01/76—10/31/77 +18.1 +14.1 -6.2 +5.5 11/01/77—10/31/78 +15.5 +12.4 +6.4 +7.4 11/01/78—10/31/79 +19.1 +15.3 +15.3 +10.9 11/01/79—10/31/80 +26.7 +21.4 +32.1 +12.0 11/01/80—10/31/81 +28.3 +22.6 +0.5 +16.0 11/01/81—10/31/82 +27.3 +21.8 +16.2 +12.1 11/01/82—10/31/83 +13.1 +10.5 +27.9 +9.1 11/01/83—10/31/84 +14.5 +11.6 +6.5 +10.4 11/01/84—10/31/85 +14.3 +11.4 +19.6 +8.0 11/01/85—10/31/86 +29.5 +24.5 +33.1 +6.3 11/01/86—12/31/87** +33.3 +26.7 +5.1 +7.1 01/01/88—12/31/88 +4.0 +3.2 +16.8 +7.4 Total Percentage Increase1 2,734% +1,382% 545% 345% Annual Compound Rate of Return1 19.1% 15.1% 10.2% 8.1% * Fiscal year changed to November 1 start date from January 1 start date. ** Fiscal year changed back to January 1 start date. 1 These figures are for the period from inception through 12/31/88. The period 01/01/89 through 05/15/89 is omitted because: (a) the partnership was liquidating and distributing its capital in a series of payments, (b) it was no longer engaged in its traditional business and the return on capital was complex to calculate, (c) available figures are estimates.

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

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The same conclusion can be found in the statistics provided by Hulbert’s ﬁnancial digest, which currently follows the performance of over 500 investment portfolios recommended by newsletters. In one Hulbert study, 57 newsletters were tracked for the 10year period from August 1987 through August 1998. During that time, less than 10 percent of the newsletters beat the Wilshire 5000 Index’s compound rate of return. Armstrong also contends that expertise, beyond a minimal level, adds little in the way of predictive accuracy. Thus, consumers would be better off buying the least expensive predictions, which are likely to be as accurate as the most expensive, or investing the modest effort required to achieve a level of accuracy that would be comparable to the most expensive experts. Recently, there have been signs that sophisticated consumers of Wall Street advice are unwilling to pay for traditional TA.

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Money Changes Everything: How Finance Made Civilization Possible by William N. Goetzmann

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For example, what happened to Edgar Lawrence Smith’s Investors Management Company? That fund is actually alive today, and you can trace its daily fluctuations from 1932 to the present. The sponsoring company, American Funds, still maintains the daily record of prices and dividends. Reinvesting dividends (and not having to pay taxes), each dollar invested in the fund in 1932 would have grown to \$2,747 by 2010; an annual, compound rate of return of about 10.7%. This is just about what an investment in a broad index of large US stocks would have earned—10.9%. You might not have beaten the market, but you would have made a great return over nearly eighty years, just as Edgar Lawrence Smith predicted. Those eight decades included four major US wars (Second World War, Korean War, Vietnam War, and the Gulf Wars), and they included most of the Great Depression and the Great Recession.

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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

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Well-accepted statistical principles dictate that the best unbiased estimator of the mean (expectation) for any random variable is the arithmetic average. Therefore, to determine a security’s expected return for one period, the best unbiased predictor is the arithmetic average of many one-period returns. A one-period risk premium, however, can’t value a company with many years of cash flow. Instead, long-dated cash flows must be discounted using a compounded rate of return. But when compounded, the arithmetic average will generate a discount factor that is biased upward (too high). APPENDIX F 853 The cause of the bias is quite technical, so we provide only a summary here. There are two reasons why compounding the historical arithmetic average leads to a biased discount factor. First, the arithmetic average may be measured with error. Although this estimation error will not affect a one-period forecast (the error has an expectation of zero), squaring the estimate (as you do in compounding) in effect squares the measurement error, causing the error to be positive.