compound rate of return

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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! *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.

If you buy this stock, are you most likely—in the long run and ignoring ﬁnancing costs—to make money, lose money, or be ﬂat? Most traders will blurt out the answer “Flat!,” and that is wrong. The correct answer is that you will lose money, at the rate of 0.005 percent (or 0.5 basis point) every minute! 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).

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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

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.

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.

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.

Analysis of Financial Time Series by Ruey S. Tsay

. √ 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.

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. These results allow us to construct confidence intervals (C.I.) for r . For instance, a 95% C.I. for r is 0.145±1.96 × 0.071 per annum (i.e., 0.6%, 28.4%). 6.5 DERIVATION OF BLACK–SCHOLES DIFFERENTIAL EQUATION In this section, we use Ito’s lemma and assume no arbitrage to derive the Black– Scholes differential equation for the price of a derivative contingent to a stock valued at Pt .

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.

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The Outsiders: Eight Unconventional CEOs and Their Radically Rational Blueprint for Success by William Thorndike

Under the leadership of Jim Hale and Phil Meek, Capital Cities evolved an approach to the newspaper business that grew out of its experience in operating TV stations, with an emphasis on careful cost control and maximizing advertising market share. What is remarkable in looking at the company’s four major newspaper operations is the consistent year-after-year-after-year growth in revenues and operating cash flow. Amazingly, these properties, which were sold to Knight Ridder in 1997, collectively produced a 25 percent compound rate of return over an average twenty-year holding period. According to the Kansas City Star’s publisher Bob Woodworth (subsequently the CEO of Pulitzer Inc.), the operating margin at the Star, the company’s largest paper, expanded from the single digits in the mid-1970s to a high of 35 percent in 1996, while cash flow grew from \$12.5 million to \$68 million.

In 1993, Chabraja joined the company as general counsel and senior vice president, with the implicit understanding that he would become Mellor’s successor. Chabraja set ambitious goals for himself when he became CEO. Specifically, he wanted to quadruple the company’s stock price over his first ten years as CEO (a 15 percent compound rate of return). He looked back into S&P records and found that this was an appropriately difficult target: fewer than 5 percent of all Fortune 500 companies had achieved that benchmark in the prior ten-year period. Chabraja looked coolly at the company’s prospects for the next ten years and concluded that he could get about two-thirds of the way there through market growth and improved operating margins.

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Mathematics for Economics and Finance by Michael Harrison, Patrick Waldron

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.

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

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.

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.

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

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.

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.

Beat the Market by Edward Thorp

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.”

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

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.

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The Decline and Fall of IBM: End of an American Icon? by Robert X. Cringely

Sure, productivity has gone up, but that can be done through automation or by beating more work out of employees (more on that later). Jensen and Meckling created the very problem they purported to solve—a problem that really hadn’t existed in the first place. Maximizing shareholder return dropped the compounded rate of return on the S&P 500 from 7.5 percent annually from 1933-76, to 6.5 percent annually from 1977 to today. That one percent may not look like much, but from the point of view of the lady at the bank the loss of so much compound interest may well have led to our corporate malaise of today. Profits are high—but are they real?

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Derivatives Markets by David Goldenberg

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 τ=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 . 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, . We can carry out the exact same procedure when time to maturity is τ.

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)}. 4. The next step is to calculate the standard deviation of these log price relatives over the entire period, .

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The Behavioral Investor by Daniel Crosby

O’Shaughnessy used the now-familiar methodology of dividing stocks into deciles and observing returns from 1963 to the end of 2009. His results highlight the efficacy of value investing and the power of slightly improved annualized returns to greatly compound wealth. Looking at price-to-earnings (P/E) ratios, he found that the cheapest decile of stocks with respect to P/E ratios turned \$10,000 into \$10,202,345 for a compound rate of return of 16.25% per year. Compare that to the index return of 11.22% that would have turned that same \$10,000 into a mere \$1,329,513. Buying cheap stocks would have made you \$9,000,000 dollars more and done so with less volatility, defying the efficient market notion that more risk is required for great returns.117 But what of the most expensive decile of stocks, the glamour names?

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

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.

The Simple Living Guide by Janet Luhrs

Either way, she was going to be 44 years old. This got her very excited. How \$100 Invested Monthly Will Grow at Various Annual Compound Rates of Return YEARS 5% 7% 9% 5 \$6,801 \$7,159 \$7,542 10 15,528 17,308 19,351 15 26,729 31,696 37,841 20 41,103 52,093 66,789 25 59,551 81,007 112,112 30 83,226 121,997 183,074 35 113,609 180,105 294,178 40 152,602 262,481 468,132 For all of you who never saw a compound interest chart, please refer to the box entitled: How \$100 Invested Monthly Will Grow at Various Annual Compound Rates of Return. Make a copy and put it on your wall in a prominent place if that will help you to stick with your pay-yourself goal.

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

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

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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Lying for Money: How Fraud Makes the World Go Round by Daniel Davies

If you want to steal a lot of money, you have to keep the fraud going. You also have to keep the fraud going if you haven’t figured out your escape route yet, or if you just blundered into it and don’t have a plan at all. But while the fraud is going, it has to be growing; the returns and repayments you owe to other people are growing at a compound rate of return, so you have to commit ever-increasing amounts of new fraud to stand still. This snowball property is the main challenge in managing an ongoing fraud. The Pigeon King Modern versions of Ponzi’s scheme tend to follow him in trying to avoid dealing with even lightly regulated markets. One of Ponzi’s highest priorities from the start of the scheme was to be sure that his dealings in postal reply coupons were not covered by the Commonwealth of Massachussetts’ ‘blue-sky laws’, which had been enacted to regulate the activities of stock promoters who would ‘sell shares in the blue sky’ unless prevented from doing so.

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

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.

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

The Dow Jones Industrials broke through the December 1961 high in 1963, and eventually established a new high slightly over 1,000 in early 1966. Thus an investor who bought the Dow Industrial stocks at the “overpriced” high of 734.51 in December 1961 would, over the next four years, achieve a compounded rate of return (including dividends) of approximately 11%, roughly the historic average rate of return for stocks over the century. Had he simply ignored the unpleasant volatility of 1962, the long-term investor would have made out just fine. Perhaps even more surprisingly, the investor who owned the highmultiple “glamour” stocks before the 1962 break also did quite well over the long term, if he had the stomach to ride out the severe downdrafts of 1962.

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The Investment Checklist: The Art of In-Depth Research by Michael Shearn

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

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 2014. 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 about 10 percent.) But this return came only at substantial risk to investors. Total returns were negative in about three years out of ten.

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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

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.22 Many dismissed the Eiteman and Smith study because the period studied did not include the Great Crash of 1929 to 1932.

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

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.

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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

He managed the endowment of his alma mater, Cambridge University, from 1921 until his death in 1946, and a recent study by David Chambers, Elroy Dimson, and Justin Foo has painstakingly reconstructed the investment returns of Keynes’s portfolio.37 From the end of August 1921 to the end of August 1946, the annual compound return on his discretionary portfolio was 14.41 percent, versus 8.96 percent for the equally weighted UK equity market index during the same period. But Chambers and Dimson discovered a fact far more remarkable than Keynes’s overall performance: Keynes made a sharp improvement in his investment approach in 1932. From 1921 to 1931 he generated a compound rate of return of only 8.06 percent, only marginally better than the equally weighted UK equity market index return of 6.67 percent. But from 1931 to 1946 Keynes produced a compound return of 18.84 percent, far outstripping the equally weighted UK index return of 10.52 percent during this fifteen-year interval.

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, using the academic definitions of the terms. 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 lower 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.

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

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.

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

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.

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

Our only hope of gaining insights from historical rates of return is to look at a very long period.5 Arithmetic Averages and Compound Annual Returns Notice that the average returns shown in Table 7.1 are arithmetic averages. In other words, we simply added the 112 annual returns and divided by 112. The arithmetic average is higher than the compound annual return over the period. The 112-year compound annual return for common stocks was 9.3%.6 The proper uses of arithmetic and compound rates of return from past investments are often misunderstood. Therefore, we call a brief time-out for a clarifying example. Suppose that the price of Big Oil’s common stock is \$100. There is an equal chance that at the end of the year the stock will be worth \$90, \$110, or \$130. Therefore, the return could be –10%, +10%, or +30% (we assume that Big Oil does not pay a dividend).

This bias would be small in most corporate-finance applications, however. 10Some of the disagreements simply reflect the fact that the risk premium is sometimes defined in different ways. Some measure the average difference between stock returns and the returns (or yields) on long-term bonds. Others measure the difference between the compound rate of return on stocks and the interest rate. As we explained above, this is not an appropriate measure of the cost of capital. 11There is some theory behind this instinct. The high risk premium earned in the market seems to imply that investors are extremely risk-averse. If that is true, investors ought to cut back their consumption when stock prices fall and wealth decreases.

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Trading and Exchanges: Market Microstructure for Practitioners by Larry Harris

Capital additions would inflate the performance and capital distributions would deflate it. Analysts use two approaches to address the problem of capital additions and distributions. The most common approach is to compute the internal rate of return for the portfolio. The internal rate of return (IRR) is the compounded rate of return that a savings account would have to earn to exactly replicate the capital flows into and out of the portfolio. The IRR calculation assumes that beginning and ending savings account balances are equal to the beginning and ending portfolio values. The IRR is approximately a time- and value-weighted geometric average of the total returns measured between each capital addition and distribution.

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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

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.