Author Draft For Review Only

Option Theory, Time Diversification, and Index Funds

JACEK NIKLEWSKI

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is lecturer in finance at the Coventry University Business School in Coventry, UK. [email protected]

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iegel [2002] conducted a study of U.S. markets covering the period from 1802 to 2001 and found that the relative risk of stocks compared to bonds and U.S. Treasury-bills diminished with the length of the time, for example over 20-year periods stocks outperformed bonds 91.7% and T-bills 94.5% of the time. Bennyhoff [2009] showed a chart depicting the lowest annualized real returns from stocks over varying periods using stock market data drawn from the period 1926 to 2007. For investment periods of 15 years and more stocks had never shown a negative real annual rate of return. The chance of losing money over 15 years or more appeared to be zero. The lowest, experienced, annual, average, real rate of return continued to improve as the investment period extended. In the cases of bonds and cash, the lowest, experienced, real return remained negative even for periods as long as 30 years. Another illustration comes from Bogle [2008] who pointed out that in the United States, there had been a 9.6% average annual return from stocks over the century to 2007. This resulted from average dividend yields of 4.5% per annum (pa) and average earnings growth of 5% pa (justifying stock price increases averaging 5% pa). So over a long period, the effects of temporary f luctuations amounted to just 0.1% pa. Over one year, the effects of temporary f luctuations can be

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massive, but over the long term they even out and become very small. Further empirical evidence in support of time diversification has been provided by Lee [1990] and Thorley [1995], who showed that the relative riskiness of stocks over bonds diminished with the length of the horizon. It is often recommended that asset allocation, in particular the proportion of stocks in a portfolio, should be inf luenced by the time for which the portfolio is expected to be held. Short investment horizons are seen as unsuitable for stocks. Long-term portfolios are regarded as suitable for high proportions of stocks. This is partly because long investment periods are seen as moderating the relative risk of stocks without distracting from the relatively high expected returns of stocks. Time diversification is one factor that ameliorates the long-run risk of stocks. Over the long term, there will be periods of relatively good returns and periods of relatively poor returns; good periods and bad periods have a tendency to offset each other over long-time spans. In consequence, risk increases less than proportionately with time, whereas returns have a compounding effect over time. This could be regarded as time diversification that reduces the probability of loss as the investment horizon extends. Consideration of time diversification leads to the standard recommendation of financial advisors; bank deposits and bonds

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is principal lecturer in finance at the Coventry University Business School in Coventry, UK. [email protected]

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K EITH R EDHEAD

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KEITH REDHEAD AND JACEK NIKLEWSKI

THE JOURNAL OF INDEX INVESTING

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Objective risk can be measured in a number of ways. These include the:

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1. Probability that outcomes fall within a particular range of values; 2. Standard deviation of outcomes; 3. Standard deviation divided by expected outcome (risk per unit of return); 4. Probability of a loss; 5. Expected size of loss, in the event that a loss occurs.

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MEASURES OF RISK AND THE EFFECTS OF TIME DIVERSIFICATION

are high. If an investor holds assets for a long time, it might be expected that good periods would tend to offset bad periods so that long-term risk is moderated. It is tempting to treat a succession of independent times as equivalent to a portfolio of randomly chosen stocks and to arrive at the conclusion that over time a diversification effect would reduce standard deviation toward zero. However, as demonstrated by Samuelson [1963] that would be a fallacious use of the law of large numbers. Rather than the probability of outcomes falling within a narrow range increasing toward one with the passage of time, the probability declines. Adding more stocks, without reducing the numbers of shares of existing stocks, reduces percentage standard deviation but raises absolute standard deviation. Adding more periods is closer to adding stocks to a growing portfolio. Although the standard deviation of percentage returns declines, the absolute standard deviation (measured in dollar terms) increases. The annualized return gets closer to the mean— that is to say, the expected—return as the investment horizon gets longer, which is to say that the probability of the annualized return falling within a narrow range increases. The likelihood of annualized returns falling below zero thus declines as long as growth exceeds zero. See Exhibit 1. The broken lines show the dispersion of annualized rates of return around the mean (the expected rate of return). For example, the broken lines may provide 95% confidence intervals such that there is a 95% probability that the realized rate of return will fall between the broken lines. With a short investment

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for short-term and medium-term investments and stocks for long-term investments. An increase in the investment horizon reduces the probability of making losses from equity investments; in other words it reduces the likelihood of stocks underperforming an investment with zero return. Likewise a long investment horizon reduces the probability of stocks underperforming other investments with lower returns than stocks, such as deposits and bonds (although the probability of underperformance is greater than in the case of a zero-return investment).

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Time diversification can be evaluated in relation to each of those measures. The probability that outcomes fall within a particular range has an apparent parallel with Markowitz diversification. According to the theory of Markowitz diversification, adding more stocks to a portfolio would progressively reduce the standard deviation of expected outcomes such that, with enough stocks, nonsystematic risk is E X H I B I T 1 reduced to zero. In the absence of systemThe Probability of Losing Money vs. the Investment Horizon atic risk, diversification can reduce total risk to zero. As more stocks are added to the portfolio, the probability that outcomes (percentage rates of return) fall within a particular range will increase. If diversification across different investments reduces risk by way of bad performances from some investments being offset by good performances from others, it might be expected that over time bad periods could be offset by good periods. The idea of time diversification is that over time there are periods when returns are low and periods when returns 2

OPTION THEORY, TIME DIVERSIFICATION, AND INDEX FUNDS

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on that measure of risk in evaluating the effect of any time diversification. However, the risk that inf luences financial decisions is not objective. Subjective risk is the relevant measure of risk for decision making and subjective risk can differ substantially from all five objective risk measures described previously. Choices appear to be better explained by perceived risk (subjective risk) than by volatility (see Jia, Dyer, and Butler [1999]). Unfortunately for the descriptive and predictive value of models that use standard deviation as the measure of risk (e.g., mean–variance diversification) standard deviation does not appear to correlate closely with subjective or perceived risk. Probability of loss and mean excess loss appear to be related to perceived risk, but with biases to the estimates (e.g., Klos, Weber, and Weber [2005]). These considerations suggest that option prices are relatively good measures of risk since they ref lect the probability of losing money and the prospective size of any loss. The variable on the horizontal axis of Exhibit 2 is the excess of the outcome (asset value) over the benchmark. If the strike price were set at zero, underperformance relative to the benchmark would entail the option being in the money. If the benchmark were the initial value of the investment, there is a first period probability of 0.5 that the benchmark would be underperformed. In the absence of any change in the expected value of the investment, the time value would rise as the expiry date of the option became more distant (the investment horizon became longer). Although the probability of loss would remain at 50%, the size of the potential loss would increase. This increase in the potential shortfall ref lects a rise in the average size of deviations below the benchmark, and would entail an increase in the cost of the option. The increase in the (statistical) expectation of size of shortfall corresponds to the increase in standard deviation resulting from an increase in investment horizon. The mean excess loss increases as the investment horizon lengthens with the effect that the theoretical option price becomes greater. A positive expected return from the investment would produce a positive expected excess of the outcome over the benchmark. That excess would be expected to grow at an accelerating rate. This growth would be represented by a movement to the right along the horizontal axis of Exhibit 2. The option would continuously move deeper out of the money. So as the broken line depicting option prices rises, there is a rightward movement along

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horizon, there is a wide dispersion and more than 2.5% probability that the rate of return will fall below zero. With a long investment horizon, there is less than 2.5% probability that the realized return will fall below zero. The chance of losing money declines as the investment horizon increases. Some researchers (e.g., Olsen [1997]) report that most investors define risk as the chance of losing money. When risk is defined in this way, stock market risk may decline with the length of the investment horizon. However, as the number of time periods increases, the dispersion of possible outcomes (terminal levels of wealth) becomes wider. Although the dispersion of annual rates of return declines, the increase in number of years causes the dispersion of terminal levels of wealth to increase. The standard deviation of outcomes rises in proportion to the square root of the number of periods if returns follow a random walk with each period’s percentage return being independent of that of other periods. Risk, defined as standard deviation divided by expected outcome, may also decline. This is partly because of the exponential rise in the expected values of investments over time and partly because of the behavior of standard deviation, which ref lects the effects of time diversification and rises asymptotically as a function of the square root of time (see Mukherji [2002]). Whilst the standard deviation increases in proportion to the square root of time, the expected outcome increases as a function of a power of time. In consequence, whilst risk increases asymptotically, the expected outcome increases exponentially. Time diversification, in the sense that the probability of loss declines with the passage of time, will occur if the expected rate of growth is above zero. However, it is not just the probability of loss, but the prospective size of any loss, that needs to be considered in a measure of risk. Whilst the probability of loss may fall, the expected size of a loss (the mean excess loss) may increase. The passage of time could reduce risk in one sense, but increase it in the other sense. One means of combining those two measures of risk is to consider option prices (see Bodie [1995]). The price of a put option is the cost of hedging against risk, and hence could be used as a value of the risk. This price ref lects both the probability of loss and the potential size of any loss. The analysis of this article uses option prices as the measure of risk, and considers the effect of time

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EXHIBIT 2

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and the volatility. High growth and low volatility may enhance the likelihood of the dominance. In Exhibit 3, the curve A depicts option prices after a short time and curve B depicts option prices after a longer period. It can be seen that if there is no investment growth the option remains at the money with the result that the option price increases from X to Y. However, if the investment value were to grow so that there is a movement along the horizontal axis to V, there would be no net effect on the option price. If growth were sufficient to achieve a value above V, the net result would be that the option price falls below X. The value

EXHIBIT 3

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the line. The upward movement of the line increases the potential size of a loss, whereas the rightward movement along the line reduces the probability of a loss. If there were no expected growth in the value of the investment, longer time horizons would increase risk. There would be no change in the probability of shortfall but there would be an increase in the expected size of any shortfall. However, an expected rate of growth in the investment’s value and hence in the excess of the expected investment value over the benchmark, suggests the possibility that the fall in the probability of loss could dominate the increase in the potential size of loss. This dominance depends upon the expected rate of growth

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An Options Perspective on Time Diversification

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Short-Term Option Prices vs. Long-Term Option Prices

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P = –Se (b-r)TN(-d1) + Ke-rtN(-d2 ) where

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σ T ⎛S⎞ ⎛ ln ⎜ ⎟ + b ⎝K⎠ ⎝ σ T

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EXHIBIT 4 The Relationship Between Put-Option Price, Expected Growth, and Investment Horizon Based on the Generalized Black–Scholes Model. (σ = 0.2; r = 0)

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d2 =

σ2 ⎞ T 2⎠

(1)

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d1 =

⎛S⎞ ⎛ ln ⎜ ⎟ + b ⎝K⎠ ⎝

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A generalized Black–Scholes option-pricing formula for put options is shown in Equation (1) (see Haug [1998]; Bogdan and Villiger [2010]; Avance [2010]):

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A GENERALIZED BLACK–SCHOLES MODEL

value equaled 1. In the zero-growth cases, the estimated put-option prices were 0.0797 for a one-year put option and 0.2482 for a ten-year put option. Those values were in line with the estimates cited by Bodie. Allowing for 10% pa expected growth reduced the estimated price of the one-year put option to 0.0415 and the estimated price of the ten-year put option to 0.0244. The longer investment horizon is associated with lower put-option prices, which indicates lower risk. This is consistent with the operation of time diversification. Exhibit 4 generalizes the relationship between putoption price, expected growth, and investment horizon (again based on the generalized Black–Scholes model). It can be seen that at low rates of investment growth risk, as measured by the cost of hedging with a put option, increases as the investment horizon extends. This is consistent with Bodie’s findings. But, at high rates of investment growth the reverse occurs; risk declines as the investment horizon extends. The latter is consistent with time diversification.

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V corresponds to the tipping-point growth rate. Above V, time diversification occurs. Bodie [1995] focused exclusively on a zero-growth case and concluded that the risk from stocks would always increase as the investment horizon lengthened. In the zero-growth case, the cost of a put option increases with a lengthening of the investment horizon. There is an increase in the value of risk, when measured as the cost of hedging the risk. Bodie took that as evidence against time diversification. But, incorporation of the positive investment growth could result in falling putoption prices, as would be expected from the presence of time diversification (Redhead and Shutes [2010]). The following analysis shows that variations in volatility also have significance for the presence of time diversification.

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P is the put-option price, S the initial value of the asset, b the expected growth rate in the value of the asset, r is the riskfree rate of interest, T is the investment horizon, K is the strike price, and σ is the expected volatility of the asset value. The present authors used the generalized Black–Scholes model to estimate option prices with zero-expected investment growth and with 10% pa expected investment growth (see Appendix 1). The benchmark was set at the initial asset value, expected volatility was 0.2 (20% pa), and the riskfree rate of interest was 0% pa The initial asset

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EXHIBIT 5

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Depiction of Holding the Growth Rate Constant. (b = 0.05; r = 0)

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support of such a view, some theoretical considerations have pointed in the opposite direction. Risk measured as the cost of hedging the risk by means of put options may increase as the investment horizon extends. This article has demonstrated that the results based on put-option prices are dependent on the assumption of low expected rates of return on stocks and/or high volatility. When higher rates of return, and/or low volatility, are assumed the behavior of put-option prices is consistent with time diversification so option theory (in the form of the generalized Black–Scholes model) does not refute time diversification; but it does indicate that it is conditional on stock returns being sufficiently high and/or volatility being sufficiently low. One interesting implication is that, from the perspective of long-term investing, there is not necessarily a trade-off between risk and return. For the long-term investor, low return is associated with high risk. Such an inverse relationship between expectations of return and risk has been observed by behavioral theorists. Shefrin [2001] investigated the relationship between risk factors and expected returns. There appeared to be a negative relationship between beta and the expected return. More risk was associated with reduced expectations of return; the opposite of the prediction from the capital asset pricing model. Likewise book-to-market ratios and firm size had relationships to expected returns, which contradicted the predictions of the Fama–French three-factor capital asset pricing model (Fama and French [1993, 1996]). Shefrin proposed that the negative relationship between perceived risk and expected return arose from the representativeness bias (Tversky and Kahneman [1974]). The stocks of good companies were seen as good investments and good investments were characterized by both high returns and low risk. This is similar to the halo effect, which is the tendency to see something (or someone) with one good feature as exhibiting other good features and something with a bad characteristic as having other bad characteristics. Further evidence of an inverse relationship between perceived risk and expected return comes from MacGregor, et al [1999]; Ganzach [2000]; Diacon and Ennew [2001]; Jordan and Kaas [2002]; and Byrne [2005]. The analysis of the present article shows that an inverse relationship between expectations of risk and return can be explained without behavioral heuristics.

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In terms of Exhibit 2, the movement to being deeper out of the money fails to offset the effect of increased time to maturity when the investment growth is slow. When the investment growth is high, the movement to being further out of the money dominates the effect of increased time to expiry on the time value of the option. There appears to be a tipping-point growth rate. At lower growth rates, risk becomes greater as the investment horizon extends. At higher growth rates, risk declines as the investment horizon extends. Time diversification operates to reduce risk when investment growth exceeds the tipping-point level. Exhibit 5 holds the growth rate constant (at 5% pa), while allowing volatility to vary. At low levels of volatility, put-option prices decline as the time to expiry (the investment horizon) extends, which is to say that when volatility is relatively low, time diversification occurs, but when volatility is high, risk increases as the investment horizon extends. In summary, whereas practitioners typically recommend that longer investment horizons allow higher asset allocations to stocks because of time diversification, and much empirical evidence has been amassed in

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are better than bonds or T-bills, when there is a long investment horizon. There has been a tendency in some time diversification studies to treat stock returns as being derived solely from stock price movements. This is understandable because dividends are difficult to handle in empirical research that uses stock returns. But, when considering long-term risk, the role of dividends is important. Bogle [2008] pointed out that dividends have accounted for nearly half of average annual returns on stocks. There is also evidence that dividends are much more stable than stock prices (Shiller [1981]). The contribution of dividends, and their reinvestment in further shares, adds an element of stability to the long-term growth in the value of stock holdings. It seems plausible that the contribution of dividends to investment growth reduces volatility. Stock returns may be much less volatile than would be supposed simply on the basis of stock price fluctuations. The reduced volatility would enhance the likelihood of time diversification being effective. It is not only risk that needs to be interpreted from the perspective of the mind of the investor, but also future levels of wealth. Economics and finance normally assume diminishing marginal utility of wealth; each increment to wealth is valued less than the previous increment of the same size. Samuelson [1963, 1969, 1989] has suggested that this effect offsets the effects of exponential investment growth (see Kritzman [1994] for a very clear exposition).

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Another interesting implication is that it is return that determines risk, rather than expected risk being the determinant of expected return. The capital asset pricing model is contradicted not only in terms of the sign of the relationship between risk and return but also in terms of the direction of causation. Another interesting behavioral insight relevant to the present analysis comes from Klos, Weber and Weber [2005]. They carried out experiments using repeated gambles. Repeated gambles simulate investing over times of varying length. A large number of repeated gambles parallels investing for a long period. They found that perceived risk was not closely related to standard deviation. Instead, it was related to the anticipated probability of a loss, and the expected size of any such loss. These are the risk dimensions captured by option prices. The other risk measure that was found to be related to perceived risk was the coefficient of variation (standard deviation divided by expected outcome). This sits comfortably with the analysis used in this article wherein the mean of the distribution of returns is seen as growing over time, and in which that growth in the mean is capable of dominating an increase in the standard deviation of returns arising from an increase in the investment horizon. Swank, Rosen, and Goebel [2002] argued that the standard deviation of returns rises less than proportionately to the square root of time. They provided evidence that returns on stocks had some tendency toward mean reversion, which means that there is some tendency for price movements to subsequently reverse. It is slightly more likely that a good period will be followed by a bad period, and vice versa. Proportionality to the square root of time is based on the price change in each period being unrelated to that of previous periods. A tendency towards mean reversion increases the time diversification effect, such that risk (standard deviation) rises more slowly than the proportionality suggests. Guo and Darnell [2005] confirmed that stocks showed mean reversion. They also found that bonds and T-bills tended to follow the opposite time series pattern, such that good periods tended to be followed by good periods and bad periods had a tendency to be followed by other bad periods (there was momentum). An implication of momentum is that the standard deviation of returns rises more than in proportion to the square root of time. With the passage of time, mean reversion reduces stock price risk and momentum increases bond and T-bill risk. This further supports the view that stocks

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BENEFITS OF INDEX FUNDS

One advantage of index funds, to the individual investor in collective investments, is that they avoid active risk. The performance of actively managed funds can, during any period of time, vary considerably. Some will outperform a stock index and others will underperform it. The difference between the best and the worst can be considerable. Active risk is the risk that a particular mutual fund (or other investment product) will perform worse than the average fund. For example amongst U.K. mutual funds in the U.K. equity sector over the ten years to July 2010, the mean compound rate of return was 1.9% pa with the best performer providing 10% pa and the weakest –6.8% pa (Money Management, September 2010). If the direction and size of the deviations from the index occur by chance, as empirical evidence seems to suggest (Redhead [2008]), the individual investor faces an active risk. Individual investors


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special analysis) that gave them an advantage in attempts to out-perform stock indices. There is also evidence that high portfolio turnover (and hence high brokerage costs) is associated with lower net returns (such a relationship was found by Carhart [1997] and by Haslem, Baker, and Smith [2008], but not by Carhart, et al [2002]). Actively managed funds have much higher stock turnovers than index funds. Not all the empirical evidence favors index funds. Blanchett and Israelsen [2007] found that a valueweighted approach produced a higher average return from actively managed funds than the head-count average frequently used. Carosa [2005] used a value-weighted average and concluded that U.S. equity funds showed outperformance relative to the S&P 500 index between 1975 and 2004. An interesting additional finding was that there appeared to be long periods during which actively managed funds outperformed the index, and other long periods characterized by underperformance by actively managed funds.

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run the risk that their chosen funds will be relatively poor performers. By investing in index funds, they avoid that active risk. Since index funds do not bear active risk, they are likely to be less risky than actively managed funds. In consequence, index funds are more likely to provide time diversification. Furthermore, there is evidence to indicate that index funds tend to have lower costs and produce higher net-of-costs returns, so there are two reasons to suppose that index funds are more likely to provide time diversification. A number of studies have found that actively managed mutual funds produce higher returns than indices before charges and costs are considered, but underperform indices after allowance is made for such expenses (for example Shukla [2004]; Wermers [2000]; Daniel, et al [1997]). Many studies have suggested that funds with high expenses tend to provide investors with lower returns (for example, Reichenstein [1999]; Indro, et al [1999]; Bogle [1998]; Carhart [1997]; Carhart, et al [2002]; Haslem, Baker and Smith [2008]). Index funds tend to have lower charges than actively managed funds. Bogle [2002] compared the performance of highcost U.S. mutual funds (top quartile for annual costs, 1.8%) against the performance of low-cost mutual funds (bottom quartile, 0.6% pa) over 1991–2001 and found that the low-cost funds outperformed the high-cost funds by more than the cost differential (by 2.2% pa). The low-cost funds also exhibited lower risk than the high-cost funds (the strongly performing low-cost funds included index funds). Holmes [2007] allocated U.S. funds to nine categories using the dimensions value, blend, growth, large-cap, mid-cap, and small-cap. In each case, the performance of the actively managed funds in the category was compared to the performance of a stock index relating to the same category. Index management was found to outperform active management in all three large-cap categories. The results were mixed for the mid-cap categories, with mid-cap blend and mid-cap growth characterized by index funds showing superior performance. Even two of the small-cap categories, value and growth, entailed index funds showing superior performance to actively managed funds. So even among relatively neglected stocks actively managed funds tend not to outperform index funds. There was no clear evidence that fund managers, on average, had any special information (including 8


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APPENDIX One-Year Option with Zero-Expected Investment Growth P = –Se (g-r)TN(–d1) + Ke-rtN(–d2 ) where

d1 =

d2 =

⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

σ2 ⎞ T 2⎠

σ T ⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

σ2 ⎞ T 2⎠

σ T

assuming that S = 1, K = 1, r = 0 and b = g = 0 we get: P = –N(–d1) + N(–d2 )

d1 =

⎛ σ2 ⎞ ⎜⎝ 2 ⎟⎠ T σ T

=

σ T 2

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d2 =

⎛ σ2 ⎞ ⎜⎝ − 2 ⎟⎠ T σ T

=−

Ten-Year Option with Zero-Expected Investment Growth

σ T 2

P = –Se (g-r)TN(–d1) + Ke-rtN(–d2 )

when σ = 0.2 and T = 1

where

d1 = 0.1 d2 = –0.1 P = –N(–0.1) + N(0.1) = –0.460172 + 0.539828 = 0.079656

One-Year Option with Expected Investment Growth of 10% pa

⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

σ2 ⎞ T 2⎠

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σ T

P = –N(–d1) + N(–d2 )

d1 =

σ2 ⎞ T 2⎠

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⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

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where

σ T

d2 =

⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

σ2 ⎞ T 2⎠

σ T

P = –e 0.1TN(–d1) + N(–d2 )

d2 =

⎛ σ2 ⎞ 0 1 − T ⎜⎝ 2 ⎟⎠ σ T

⎛ σ2 ⎞ 0 1 − T ⎜⎝ 2 ⎟⎠ = σ

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when σ = 0.2 and T = 1

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σ T

⎛ σ2 ⎞ 0 1 + T ⎜⎝ 2 ⎟⎠ = σ

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d1 =

⎛ σ2 ⎞ 0 1 + T ⎜⎝ 2 ⎟⎠

d1 = 0.6 d2 = 0.4 P = –e 0.1*1N(–0.6) + N(–0.4) = –1.105171*0.274253 + 0.344578 = 0.041482 The expected increase in the value of the investment moves the put option out of the money and reduces the price of the put option from 0.0797 to 0.0415.

d1 =

d2 =

⎛ σ2 ⎞ ⎜⎝ 2 ⎟⎠ T σ T

⎛ σ2 ⎞ ⎜⎝ − 2 ⎟⎠ T σ T

=

σ T 2

=−

σ T 2

when σ = 0.2 and T = 10

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assuming that S = 1, K = 1, r = 0 and b = g = 0.1 we get:

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d2 =

σ T

assuming that S = 1, K = 1, r = 0 and b = g = 0 we get:

P = –Se (g-r)TN(–d1) + Ke-rtN(–d2 )

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d1 =

⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

d1 = 0.316228 d2 = –0.316228 P = –N(–0.316228) + N(0.316228) = –0.375915 + 0.624085 = 0.24817

Ten-Year Option with 10% pa Expected Investment Growth P = –Se (g-r)TN(–d1) + Ke-rtN(–d2 ) where

d1 =

d2 =

⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

σ2 ⎞ T 2⎠

σ T ⎛S⎞ ⎛ ln ⎜ ⎟ + g ⎝K⎠ ⎝

σ2 ⎞ T 2⎠

σ T


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assuming that S = 1, K = 1, r = 0 and b = g = 0.1 we get:

——. “Black Monday and Black Swans.” Financial Analysts Journal, Vol. 64, No. 2 (2008), pp. 30-40.

P = –e 0.1TN(–d1) + N(–d2 )

d2 =

σ T ⎛ σ2 ⎞ 0 1 − T ⎜⎝ 2 ⎟⎠ σ T

Byrne, K. “How Do Consumers Evaluate Risk in Financial Products?” Journal of Financial Services Marketing, Vol. 10, No. 1 (2005), pp. 21-36.

⎛ σ2 ⎞ 0 1 + T ⎜⎝ 2 ⎟⎠ = σ

Carhart, M. “On Persistence in Mutual Fund Performance.” Journal of Finance, Vol. 52, No. 1 (1997), pp. 57-82.

⎛ σ2 ⎞ 0 1 − T ⎜⎝ 2 ⎟⎠ = σ

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d1 =

⎛ σ2 ⎞ 0 1 + T ⎜⎝ 2 ⎟⎠

Carhart, M.M., J.N. Carpenter, A.W. Lynch, and D.K. Musto. “Mutual Fund Survivorship.” Review of Financial Studies, Vol. 15, No. 5 (2002), pp. 1439-1463.

when σ = 0.2 and T = 10

REFERENCES

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Avance. Real Options: Black–Scholes? Available at http:// www.avance.ch/downloads/avance_on_BlackScholes.pdf, Accessed October 10, 2010.

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