Chapter 31 Measuring investment risk.
Measuring investment risk is a critical but difficult aspect of investment planning. This chapter describes and explains the principal risk measures, such as variance and standard deviation, semi-variance and semi-deviation, skewness, kurtosis, correlation and covariance, and beta. A brief review of probability distributions is useful before examining these ways of measuring risk.
Measuring investment returns, including risk-adjusted returns, is discussed in Chapter 33, "Measuring Investment Return." Some performance measures for measuring and managing a portfolio are discussed in Chapter 37, "Portfolio Management and Measurement."
To deal with the uncertainty of returns, investors need to think explicitly about a security's distribution of probable total returns. In other words, investors need to keep in mind that, although they may expect a security to return 10%, for example, this is only a one-point estimate of the entire range of possibilities. Given that investors must deal with the uncertain future, a number of possible returns could occur.
In the case of a Treasury bond paying a fixed rate of interest, the interest payment will be made with (almost) 100% certainty, barring a financial collapse of the economy. The probability of occurrence is close to 1.0, because no other outcome is possible (but nothing is ever really certain).
Of all the ways to describe risk, the simplest and possibly most accurate is "the uncertainty of a future outcome." The anticipated return for some future period is known as the expected return. The actual return over some past period is known as the realized return. The simple fact that dominates investing is that the realized return on an asset with any risk attached to it may be different from what was expected.
Financial economists define volatility as the range of movement (or price fluctuation) from the expected level of return. The more a stock, for example, goes up and down in price, the more volatile that stock is. Because wide price swings create more uncertainty of an eventual outcome, increased volatility can be equated with increased risk. Being able to measure and determine the past volatility of a security is important in that it provides some insight into the riskiness of that security as an investment.
Investors and analysts should be at least somewhat familiar with the study of probability distributions. Since the return an investor will earn from investing is not known, investors must estimate it. An investor may expect the total return on a particular security to be 10% for the coming year, but in truth this is only a point estimate.
With the possibility of two or more outcomes, which is the norm for common stocks and virtually all other investments, investors must consider each possible likely outcome and assess the probability of its occurrence. The result of considering these outcomes and their probabilities together is a probability distribution consisting of the specification of the likely returns that may occur and the probabilities associated with these likely returns.
Probabilities represent the likelihood of various outcomes and are typically expressed as a decimal (sometimes fractions are used). The sum of the probabilities of all possible outcomes must be 1.0, because they must completely describe all the (perceived) likely occurrences.
How are these probabilities and associated outcomes obtained? In the final analysis, investing for some future period involves uncertainty, and therefore subjective estimates. Although investors may rely heavily on past occurrences (frequencies) to estimate the probabilities, they must adjust these estimates based on the past for any changes expected in the future.
Probability distributions can be either discrete or continuous. With a discrete probability distribution, a probability is assigned to each possible outcome. With a continuous probability distribution an infinite number of possible outcomes exist. The most familiar continuous distribution is the normal distribution depicted by the well-known bell-shaped curve often used in statistics. It is called a two-parameter distribution because one only needs to know the mean and the variance to fully describe the distribution.
To describe the single most likely outcome from a particular probability distribution, it is necessary to calculate its expected value. The expected value is the average of all possible return outcomes, where each outcome is weighted by its respective probability of occurrence. Investors typically use variance or standard deviation, at least as a first approximation, to calculate the total risk associated with the expected return.
STANDARD DEVIATION AND VARIANCE (1)
What is the standard deviation and what relevance does it have?
Technically, the standard deviation is a statistical measure defined as the square root of the variance of returns. The variance of returns is the expected value of the average squared differences from the mean of the distribution. (2) Technicalities aside, it is simply a measure of how much, on average, any particular observation of a randomly distributed variable (in this case, the returns on various asset classes) will differ from the average or mean value of the distribution. Since variance, volatility, and risk, in this context, can be used synonymously, remember that the larger is the standard deviation, the more uncertain is the outcome.
For example, assume balls with numbers ranging from zero to 100 are placed in a sack. If a person picks a ball from the sack, each ball is equally likely to be drawn. If balls are repeatedly picked from the sack, replacing each ball after it is selected, one would expect the average value of the balls selected to be close to 50, since 50 is the average value. If a person had to guess what value will turn up in any particular draw and wanted to minimize the potential error of the guess, the best guess would be 50, the expected or mean value of the possible outcomes. The greatest possible error one can make by guessing the mean (50) is 50, since the minimum and maximum possible values that could actually be drawn are zero or 100. If a person guessed any value other than 50, the error could exceed 50. For example, if a person guessed 20 and the ball actually drawn had the number 100, the error would be 80.
Even though the mean value (50) is the best possible guess, it is not necessarily a good guess in the sense of being anywhere near the value that will actually be drawn. Even though 50 is the best guess, one would expect most picks to differ substantially from 50. As explained above, by guessing 50, the result could still be wrong by as much as 50. What the standard deviation measures is by how much one would expect, on average, any given draw to differ from the best guess of 50.
Suppose that a person knows that the value of a ball drawn from the sack has a value equal to or less than 50. What would then be this person's best guess? Clearly, the answer is 25, since it is the average of the values from zero to 50. By guessing 25, this person could never be off by more than 25. The analysis is similar if a person knows the value of the drawn ball is equal to or greater than 50 before making the guess as to its value. In this case, the best guess is 75, since that is the mean value over the range from 50 to 100. By guessing 75, one could never be wrong by more than 25. Therefore, even when a person does not know whether the ball drawn is either above or below 50, the person does know that if it is above 50, it will be above 50 by an average of 25; if it is below 50, it will be below 50 by an average of 25. In other words, the guesser knows that the best guess of the mean value 50 will differ from the actual number drawn by an average absolute value of 25. For this distribution, 25 is the mean absolute deviation, (3) the usual or average amount by which the values drawn differ from the mean of the distribution, which is closely related to the standard deviation. (4)
It is generally held that investments whose distributions of returns have large standard deviations are riskier than those having smaller standard deviations since the future outcomes are much less certain. This idea is not entirely inappropriate, but it is also not entirely correct, since it does not encompass the entire scope of the issue.
For example, assume now that all the balls with values below 40 and above 60 are thrown out of the sack. The best guess regarding the value of a ball drawn from the sack is still 50, since the mean value has not changed. However, the mean absolute deviation is now just five, not 25 (standard deviation is 5.77). What does this tell a person about the best guess? It tells one nothing about what the best guess should be. It is still 50. What it tells a person is the relative goodness of the best guess. In the first case with numbers ranging from zero to 100, 50 is the best guess, but it is not a very good guess, since a person could be wrong by 50. In the second case, with numbers ranging from 40 to 60, the best guess is still 50, and it is a relatively good guess, since a person could be wrong by only 10, and a person would expect to be wrong by only five, on average.
Suppose now that as a ball is drawn from either sack, a person will receive or pay an amount based on the outcome. If the value of the drawn ball exceeds 50, the person will receive the difference between the value on the drawn ball and $50. If the value of the drawn ball is less than 50, the person will have to pay the amount by which the draw is less than $50. Regardless of from which sack the ball is drawn, the expected payoff/payout is $0. However, most people would agree that drawing balls from the sack containing balls numbered from zero to 100 is riskier than drawing balls from the sack with balls numbered from 40 to 60. In this sense, the standard deviation is a good indicator of risk.
However, now suppose the sack containing the balls from 40 to 60 is reconstituted to include only the balls numbered from zero to 20. The expected or mean value is now 10, but the mean absolute deviation is still five. The payoff/payout rules are also now changed to pay the person the amount by which the number on any drawn ball exceeds 10 and for the person to pay back the amount by which the number of any drawn ball falls below 10. In this case, would draws from the sack containing balls numbered zero to 100 still be considered riskier?
No one would say that drawing from the sack containing balls numbered zero to 100 is riskier than drawing from the sack containing balls numbered zero to 20, even though the mean absolute deviation (or standard deviation) of the number of outcomes for the sack containing balls numbered zero to 100 remains 25, or five times greater than the mean absolute deviation of draws from the sack containing balls numbered from zero to 20. Draws from the zero-to-20 sack still have an expected payoff of $0; the person has a 50% chance of losing money on any particular draw, and could lose as much as $10 [ball 0-10] if the worst occurs.
In contrast to draws from the zero-to-20 sack, draws from the zero-to-100 sack have an expected payoff of $40 [50 average value of a ball minus 10] and the person has only a 10% chance of losing money on any particular draw [the chance of losing money is limited to balls 0 to 9; 10 is breakeven]. In addition, the maximum potential loss on a draw from the zero-to-100 sack is still $10, exactly the same as the maximum potential loss on a draw from the zero-to-20 sack. Therefore, drawing from the zero-to-100 sack presents one with the possibility of losing, at most, $10 [ball 0-10] (the same as drawing from the zero-to-20 sack), only a 10% chance of losing any money (as compared to a 50% chance from the zero-to-20 sack), an expected gain of $40 (as compared to $0 from the zero-to-20 sack), and a maximum potential gain of up to $90 [ball 100-10] (as compared to only $10 from the zero-to-20 sack).
Clearly, draws from the zero-to-100 sack are no more risky (in terms of loss) than draws from the zero-to-20 sack, even though the uncertainty regarding the payoff of draws from the zero-to-100 sack as measured by the mean absolute deviation is five times greater than the uncertainty regarding the payoff from the zero-to-20 sack (the standard deviation is 5.006 times larger). Therefore, the standard deviation alone is an incomplete and sometimes misleading indicator of risk. In the area of investment analysis, risk assessment must also include differences in expected values and the downside or loss potentials of the alternative investments.
Consequently, the standard deviation can best be described as a measure of the "goodness" or confidence one can place in a best guess estimate (mean value) of the outcome of a random variable. As applied to investment returns, the expected value may be the best guess of future returns, but if the standard deviation of returns is large, the best guess still may not be a very good guess.
Also, the standard deviation is at best only a partial or incomplete and, sometimes misleading, measure of the riskiness of an investment relative to another investment. In order for standard deviations to take on any real meaning when comparing alternative investments, investors must use standard deviations relative to the investments' expected returns. That is, standard deviations are useful in conjunction with expected returns to measure each investment's return per unit of risk--its return bang per risk buck, so to speak. The standard deviation of equity returns (i.e., S&P stocks and small-capitalization stocks) is greater than the standard deviation of fixed-income investment returns (i.e., corporate and government bonds and Treasury bills) for all investment horizons. However, at longer investment horizons, equities dominate fixed income investments. Despite greater standard deviations, equities can and should be viewed as less risky investments than bonds and Treasury bills for longer investment horizons.
One of the nice characteristics of the standard deviation is that if returns are normally distributed, investors can predict that actual realized returns will fall within one standard deviation above or below the mean about 68% of the time and within 2 standard deviations above or below the mean about 95% of the time. Based on history, investors can expect to earn average annual returns of between 7.7% and 14.9% on a 20-year investment in S&P stocks with about a 68% probability, or odds of two in three. They can expect the average annual returns to range from 4.1% to 18.5% with odds of about 19 out of 20. Investors have only about three chances in 1,000 that the realized average annual compound return would fall outside three standard deviations from the mean (below 0.5% or above 22.1%).
Calculating a standard deviation using probability distributions involves making subjective estimates of the probabilities and the likely returns. However, investors cannot avoid such estimates because future returns are uncertain. The prices of securities are based on investors' expectations about the future. The relevant standard deviation in this situation is the ex ante standard deviation and not the ex post standard deviation based on past realized returns.
Although standard deviations based on past realized returns are often used as proxies for ex ante standard deviations, investors should be careful to remember that the past cannot always be extrapolated into the future without modifications. Ex post standard deviations may be convenient, but they are subject to errors.
One important point about the estimation of standard deviation is the distinction between individual securities and portfolios. Standard deviations for well-diversified portfolios tend to be reasonably steady over time and, therefore, historical calculations may be fairly reliable in projecting the future. Moving from well-diversified portfolios to individual securities, however, makes historical calculations much less reliable. Fortunately, the number one rule of portfolio management is to diversify by holding a broad portfolio of securities, and the standard deviations of well-diversified portfolios may be more stable.
One further very important point is that the standard deviation is a measure of the total risk of an asset or a portfolio, including therefore both systematic and unsystematic risk. It captures the total variability in the asset's or portfolio's return, whatever the sources of that variability.
For an example of how to calculate the standard deviation, see "How It Is Done--An Example," below.
The use of standard deviation as a measure of risk has been criticized because it considers the possibility of returns above the expected return as well as below the expected return. Investors, however, do not view possible returns above the expected return as an unfavorable outcome. Many investors think that only deviations below the expected return really matter.
Harry Markowitz, the father of modern portfolio theory, recognized this limitation and suggested a measure of downside risk--the risk of realizing an outcome below the expected return--called the semi-variance. The semi-variance is similar to variance except that in making this calculation, no consideration is given to returns above the expected return. (5)
In some cases, investors may be more concerned with the chance that an investment's return will fall below some benchmark or target return, such as T-bill rates, rather than the expected return of the investment. In such cases, investors may use a measure called the downside volatility or semi-volatility. The semi-volatility is computed in exactly the same way as the semi-variance, except that all returns above a benchmark or target return (sometimes called the minimal acceptable rate of return, or mar), such as the risk-free rate, rather than the expected return are ignored.
The square root of the semi-variance is sometimes called the semi-deviation, although there does not appear to be any universally accepted term for this concept. Semi-deviation is to semi-variance as the standard deviation is to variance. The term downside deviation is sometimes used to refer to the square root of the semi-variance, but more frequently seems to be used to refer to the square root of the downside volatility, which is computed with respect to a minimal acceptable rate of return different from the expected return.
Even though semi-variance is conceptually superior to standard deviation, in practice, because of some difficult math problems, most researchers and analysts use the standard deviation as the risk measure of choice. As it turns out, as long as the return distribution is symmetrical (the same shape above and below the mean), then the standard deviation gives exactly the same answers in a portfolio context as the semi-variance. With notable exceptions, stock return distributions do seem to be reasonably symmetrical, especially for longer investment horizons.
However, especially for shorter-term investment horizons and for certain types of investments, such as options and other derivatives, technology stocks, media stocks, telecom stocks, or hedge funds, the distributions may not be normally distributed. In these cases, the standard deviation can be an incomplete and misleading measure of risk, even when used in conjunction with expected returns.
For an example of how to calculate the semi-variance and the semi-deviation, see "How It Is Done--An Example," below.
COVARIANCE AND CORRELATION
The principal conclusion of Harry Markowitz's seminal book, Portfolio Selection, which heralded the era of modern portfolio and capital market theory, was that portfolios of risky stocks could be put together in such a way that the portfolio as a whole could actually be less risky than any one of the individual stocks in it. This result follows in part from the concept of covariance among securities, or lack of it.
As the term suggests, covariance means varying together. Covariance is a measure of the degree to which two variables move in a systematic or predictable way, either positively or negatively. If two variables move in perfect lockstep, up and down, then they are said to exhibit perfect positive covariance. For example, the pressure and temperature of a fixed volume of gas at sea level exhibit perfect covariance. Double the pressure of a given volume of gas and the temperature will change proportionally, and vice-versa. Or, halve the temperature of a gas and the pressure will also decline proportionally, and vice versa.
Two variables that move perfectly in lockstep, but in opposite directions, are said to exhibit perfect negative covariance. For instance, if one doubles (halves) the pressure on a gas and wants to keep the temperature unchanged, the volume for the gas must increase (decrease) in direct proportion to the change in pressure.
If two variables are completely independent, showing no systematic relationship, their covariance is zero.
Correlation is a standardized version of covariance where values can range from -1 (perfect negative covariance) to +1 (perfect positive covariance). Letting [[sigma].sub.12] represent the covariance between 2 variables, [[sigma].sub.1] the standard deviation of variable 1, [[sigma].sub.2] the standard deviation of variable 2, and [[rho].sub.12] the correlation of the two variables, the relation between correlation and covariance is as follows:
[[rho].sub.12] = [[sigma].sub.12]/[[sigma].sub.1] x [[sigma].sub.2] 12 o1 x o2
Expressed as a function of covariance, covariance is equal to the correlation times the standard deviations of the two variables:
[[sigma].sub.12] = [[rho].sub.12] x [[sigma].sub.1] x [[sigma].sub.2]
Markowitz showed that if investors added stocks that do not exhibit perfect covariance to their portfolio, the total risk of the portfolio as measured by variance or standard deviation would decline. For example, assume an investor holds just one stock, [S.sub.1], with a standard deviation of [[sigma].sub.1]. Thus, the standard deviation of the investor's initial "portfolio" of one security is [[sigma].sub.1]. If the investor sells off part of his interest in S1 and uses the proceeds to buy another stock, [S.sub.2] with a standard deviation of [[sigma].sub.2], the variance per dollar of investment in his portfolio, [[sigma].sub.2], is given by the following equation:
[[sigma].sub.2.sub.p] = ([w.sup.2.sub.1] x [[sigma].sub.2.sub.1]) + ([w.sup.2.sub.2] x [[sigma].sub.2.sub.2]) + (2 x [w.sub.1] x [w.sub.2] x [[sigma].sub.12])
Here, [[sigma].sub.12] is the covariance between [S.sub.1] and [S.sub.2], and [w.sub.i] is the proportion of the portfolio invested in stock [S.sub.i].
Alternatively, one can express the variance of the portfolio in terms of correlation rather than covariance:
[[sigma].sup.2.sub.p] = ([w.sup.2.sub.1] x [[sigma].sup.2.sub.1]) + ([w.sup.2.sub.2] x [[sigma].sub.2.sub.2]) + (2 x [w.sub.1] x [w.sub.2] x [[rho].sub.12] x [[sigma].sub.1] x [[sigma].sub.2])
Assume, first, for illustration, that the weight invested in each stock is equal, 50%, that the standard deviations of the stocks are equal, [[sigma].sub.1] = [[sigma].sub.2], and that the returns on the two stocks are independent, that is, the covariance and correlation are zero, then, the variance of the portfolio is:
[[sigma].sup.2.sub.p] = (0.[5.sup.2] x [[sigma].sup.2.sub.1]) + (0.[5.sup.2] x [[sigma].sup.2.sub.2])
[[sigma].sup.2.sub.p] = (0.25 x [[sigma].sup.2.sub.1]) + (0.25 x [[sigma].sub.2.sup.2])
[[sigma].sub.p.sup.2] = (0.25 x [[sigma].sub.1.sup.2]) + (0.25 x [[sigma].sub.1.sup.2])
[[sigma].sub.b.sup.2] = 0.5 x [[sigma].sub.1.sup.2]
By splitting the portfolio equally between two independently distributed stocks, the investor cuts the variance of the portfolio in half! In fact, for each additional independent stock added to the portfolio, the variance declines in proportion to the number of stocks, that is
[[sigma].sub.p.sup.2] = [[sigma].sub.1.sup.2]/n
This equation explains the principle of diversification. By splitting a portfolio into more and more securities, the variance attributable to the nonsystematic (uncorrelated) risks approaches zero as the number of securities increases.
Taking the square root of each side of the equation to derive the standard deviation, one can see that the standard deviation of the portfolio declines by the square root of the number of independently distributed securities in the portfolio:
[[sigma].sub.p] = [[sigma].sub.1]/[square root of n]
In other words, the standard deviation of a portfolio split into 100 independently distributed securities is 10 times less than the standard deviation of any one of the securities alone.
Now, assume instead that the investor starts once again with just one stock, S1, but then splits his portfolio evenly between stock [S.sub.1] and stock [S.sub.3], which, once again, for illustration, is assumed to have a standard deviation, [[sigma].sub.3], equal to [[sigma].sub.1]. However, in this case assume [S.sub.3]'s return is perfectly negatively correlated with [S.sub.1]'s return. Then the variance of the portfolio is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If two securities are perfectly negatively correlated, splitting one's investment equally between the two securities completely eliminates all variability!
Finally, assume instead the investor starts once again with just S1 in his portfolio, but then puts half his money into security [S.sub.4], with a standard deviation equal to S1's and a correlation with [S.sub.1] of +1. The variance of the port-folio is, substituting immediately for [[sigma].sub.1] = [[sigma].sub.4] and [[rho].sub.14] = 1:
[[sigma].sub.p.sup.2] = (0.25 x [[sigma].sub.1.sup.2]) + (0.25 x [[sigma].sub.1.sup.2]) + (2 x 0.25 x 1 x [[sigma].sub.1.sup.2])
[[sigma].sub.p.sup.2] = (0.25 x [[sigma].sub.1.sup.2]) + (0.25 x [[sigma].sub.1.sup.2]) + (0.5 x [[sigma].sub.1.sup.2])
[[sigma].sub.p.sup.2] = [[sigma].sub.1.sup.2]
The portfolio's variance remains unchanged. There is no risk reduction advantage to adding perfectly positively correlated assets to one's portfolio.
In the real world, instances of either perfectly negatively correlated or perfectly positively correlated securities are extremely rare, if not nonexistent. In fact, instances of correlations between securities that are negative at any level do exist, but are relatively few. However, investors do not need negative correlations between securities for them to benefit by adding securities to their portfolio. As long as a security added to a portfolio is not perfectly positively correlated with the existing portfolio, the addition of the security will reduce the portfolio's risk as measured by variance or standard deviation.
This concept is a central principle of modern portfolio and asset allocation theory. As long as there are any classes of assets whose returns are not perfectly correlated with investors' current portfolios, these investors can further reduce the risk of their portfolios by adding securities from those asset classes.
While standard deviation determines the volatility of a security or fund according to the disparity of its returns over a period of time, beta, another useful statistical measure, determines the volatility, or risk, of a security or fund in relation to that of its index or benchmark. In the single factor Capital Asset Pricing Model, the index or benchmark is the "market" portfolio, often measured by the S&P 500 index. When beta is used to compare funds or to measure an investment manager's performance, the benchmark is frequently the average of the funds in that mutual fund category.
A fund with a beta very close to 1 means the fund's performance closely matches the index or benchmark--a beta greater than 1 indicates greater volatility than the overall market, and a beta less than 1 indicates less volatility than the benchmark. If, for example, a fund has a beta of 1.05 in relation to the S&P 500, the fund has been moving 5% more than the index. Therefore, if the S&P 500 increased 15%, the fund would be expected to increase 15.75%. On the other hand, a fund with a beta of 2.4 would be expected to move 2.4 times more than its corresponding index. If the S&P 500 moved 10%, the fund would be expected to rise 24%, and, if the S&P 500 declined 10%, the fund would be expected to lose 24%.
Investors expecting the market to be bullish may choose funds exhibiting high betas, which increase investors' chances of earning high returns in up markets. If an investor expects the market to be bearish in the near future, the funds that have betas less than 1 are a good choice because they would be expected to decline less in value than the index. For example, if a fund had a beta of 0.5 and the S&P 500 declined 6%, the fund would be expected to decline only 3%. Be aware of the fact that beta by itself is limited and can be skewed due to factors other than the market risk affecting the fund's volatility.
The R-squared of a fund (also called the coefficient of determination) is a measure of what proportion of a security's or portfolio's total variability is explained by its relationship to a benchmark or index and how much is its independent risk unrelated to the benchmark or index. When used in conjunction with ratings of mutual funds or the performance of professional managers it advises investors if the beta of a mutual fund is measured against an appropriate benchmark. Measuring the correlation of a fund's movements to that of an index, R-squared describes the level of association between the fund's volatility and market risk, or more specifically, the degree to which a fund's volatility is a result of the day-to-day fluctuations experienced by the overall market.
R-squared values range between 0 and 100, where 0 represents the least correlation and 100 represents full correlation. If a fund's beta has an R-squared value that is close to 100, the beta of the fund should be trusted. On the other hand, an R-squared value that is close to 0 indicates that the beta is not particularly useful because the fund is being compared against an inappropriate benchmark.
If, for example, a bond fund were judged against the S&P 500, the R-squared value would be very low. A bond index such as the Lehman Brothers Aggregate Bond Index would be a much more appropriate benchmark for a bond fund, so the resulting R-squared value would be higher. Obviously the risks apparent in the stock market are different than the risks associated with the bond market. Therefore, if the beta for a bond were calculated using a stock index, the beta would not be trustworthy.
An inappropriate benchmark will skew more than just beta. Alpha (see Chapter 32) is calculated using beta, so if the R-squared value of a fund is low, it is also wise not to trust the figure given for alpha.
Skewness measures the coefficient of asymmetry of a distribution. (7) In contrast with the normal distribution, or bell curve, where both tails of the distribution are symmetrical mirror images of each other, skewed distributions have one tail of the distribution that is longer than the other. Figure 31.1 shows two distributions with equal means and standard deviations, but one has positive skewness and the other has negative skewness.
[FIGURE 31.1 OMITTED]
If investors base their investment decisions solely upon means and risk as measured by standard deviations, they should be indifferent between these two distributions since the means and standard deviations are equal. However, a risk-averse investor does not like negative skewness. In this case, the investment with negative skewness has a substantial downside tail exposing the investor to low or negative returns below the worst potential returns on the investment with positive skewness. In addition, the investment with positive skewness offers investors the potential for upside returns far above any they could ever expect from the negatively skewed investment.
It would seem almost inescapable that any risk adverse investor would prefer the positively skewed investment to the negatively skewed investment. Although this is in fact true, it may not be quite as evident as it first appears. Figure 31.1 also shows the medians of the two distributions. In symmetric distributions, such as the normal distribution, the mean and median are equal, but in skewed distributions, they are different.
The median is the point where there is a 50% probability that realized returns will fall below (or above) that value. In positively skewed distributions, the median is below the mean. In negatively skewed distributions, the median is above the mean. What this means is that an investor actually has less than a 50% chance of earning the mean return in a positively skewed investment, but more than a 50% chance of earning the mean return on a negatively skewed investment. So although the means and standard deviations of these two investments are identical, an investor actually has a higher chance of earning at least the mean return with the negative skewed investment than with the positively skewed investment. Therefore, although all risk adverse investors will prefer the positively skewed to the negatively skewed investment when their means and standard deviations are identical, it does not mean that they would always prefer a positively skewed investment to a symmetrically distributed investment with the same mean and standard deviation, or other possible combinations of means, standard deviations, and skewness.
For an example of how to calculate skewness, see "How It Is Done--An Example," below.
Kurtosis measures the degree of "fatness" in the tails of a distribution. Figure 31.2 shows an example of a symmetric distribution that has a positive coefficient of excess kurtosis relative to a normal distribution. (8) The distributions have equal means and standard deviations.
It is easy to see that risk-averse investors prefer a distribution with low kurtosis (that is, where tails are thin and returns are more likely to fall closer to the mean). Although the extra fatness in the downside tail is exactly matched by the excess fatness in the upside tail, risk averse investors will always weight the potential downside returns heavier than the potential upside returns. Consequently, they will always prefer a lower excess kurtosis. If investors use only means and standard deviations to choose among investments and some of those investments have either positive or negative excess kurtosis, they will be ignoring an important factor in making the best choice.
[FIGURE 31.2 OMITTED]
One of the reasons posited for the small-stock premium being higher than it should be in theory under the mean-variance framework of the capital-asset pricing model is that small cap stocks exhibit greater excess kurtosis and negative skewness than larger stocks. For instance, the risk of total failure and bankruptcy is unquestionably greater for most small business than it is for most large ones. Negative skewness does not get any greater than -100%! Since the mean-variance framework of much of modern portfolio and capital market theory does not account for these factors, it cannot explain this excess return anomaly.
For an example of how to calculate kurtosis, see "How It Is Done--An Example," below.
SKEWNESS AND KURTOSIS COMBINED
Distributions can exhibit both skewness and excess kurtosis. Figure 31.3 shows an example of two negatively skewed distributions, but one also has a positive excess kurtosis.
[FIGURE 31.3 OMITTED]
As soon as skewness begins to be negative the impact of a high excess kurtosis is significant for a risk-averse investor. The most interesting situation is when the return distribution has a negative skewness of -1 (or below) and an excess kurtosis higher than 1. In this case, the probability of having a huge negative return increases dramatically. For a distribution with a skewness of -1 and an excess kurtosis of -5 (technology stocks, media stocks, telecom stocks or hedge funds may have this kind of excess kurtosis level), a mean-variance approach will conclude that an investor will not lose more than -3.5% in the next one day with 99% probability. However, an approach that accounts for skewness and kurtosis gives instead a 99% probability of not losing more than 7.4% in the next one day. The difference is substantial: more than a 100% underestimation of the potential one-day loss when using just the mean-variance approach that ignores skewness and kurtosis.
In conclusion, for optimization, simulation, and investment selection, the investor's approach should account not only for volatility, but also for skewness and kurtosis when the return distribution over the relevant period is likely to be significantly different than the normal distribution.
HOW IT IS DONE--AN EXAMPLE
The following example will illustrate how to calculate the standard deviation, the semi-variance, skewness, and kurtosis.
Figure 31.4 illustrates calculation of the standard deviation, along with measurements of skewness and kurtosis, in the following steps.
1. Sample total returns for 16 periods are shown, sorted in order of lowest return (biggest loss) to highest return. [Returns have been sorted to help illustrate determination of the median, and to help illustrate calculation of the semi-variance (which follows).] Total returns could be for a single investment, a portfolio of investments, an entire market segment, or the entire market. Total returns could be historical or projected.
2. The total return is summed up and then divided by the number of values (or periods) to determine the average return [66.0 + 16) = 4.1].
3. The deviation for each period is determined by subtracting from the total return for each period the average return [e.g., -25.0-4.1 = -29.1].
4. The deviation for each period is squared [e.g., -29.[1.sup.2] = 848.3].
5. The squared deviations for each period are summed up and then divided by the number of values (or periods) minus one to determine the average squared deviation, or variance [2,705.8 + (16-1) = 180.4]. Where the whole universe of values is used instead of just a sample, the number of values may be used instead of the number of values minus one.
6. The square root of the variance is the standard deviation [180.[4.sup.1/2] = 13.4].
7. Skewness and kurtosis are then calculated as shown.
[FIGURE 31.5 OMITTED]
The median is the middle value; one-half of returns are above the median and one-half of returns are below the median. Where there is an even number of values, as here, the median is equal to one-half the sum of the two middle values [(1 + 5) + 2 = 3.0]. Note that the median [3.0] is less than the average [4.1]. This is consistent with the slightly positively-skewed distribution [0.330821]. See also Figure 31.5.
Figure 31.6 illustrates calculation of the semi-variance and semi-deviation, along with measurements of skewness and kurtosis, in the following steps.
1. Sample total returns for 16 periods are shown (the same as above), sorted in order of lowest return (biggest loss) to highest return. [Returns have been sorted to help illustrate determination of the median, and to help illustrate calculation of the semi-variance).] Total returns could be for a single investment, a portfolio of investments, an entire market segment, or the entire market. Total returns could be historical or projected.
2. The total return is summed up and then divided by the number of values (or periods) to determine the average return [66.0 +- 16 = 4.1].
3. The deviation for each period is determined by subtracting from the total return for each period the average return [e.g., -25.0--4.1 = -29.1]. A target return, such as the risk-free rate of return or a return of zero, could be used instead of the average return.
4. Only those deviations with a negative result are then used.
5. The deviation for each period is squared [e.g., -29.[1.sup.2] = 848.3].
6. The squared deviations for each period are summed up and then divided by the number of values (or periods) used minus one to determine the average squared deviation, or semi-variance [1,210.9 + (8-1) = 173.0]. Where the whole universe of values is used instead of just a sample, the number of values may be used instead of the number of values minus one.
7. The square root of the semi-variance is the semi-deviation [173.[0.sup.1/2] = 13.2].
8. Skewness and kurtosis are then calculated as shown.
The median is the middle value (for the numbers used); one-half of returns are above the median and one-half of returns are below the median. Where there is an even number of values, as here, the median is equal to one-half the sum of the two middle values [(-1 + 0) + 2 = -0.5]. Note that the median [-0.5] is greater than the average of the values used [-4.9]. This is consistent with the negatively-skewed distribution [-2.396096].
WHERE CAN I FIND OUT MORE?
1. Brown, Robert A., Ph.D., CFA, "Avoiding 5 Common Mistakes in Overestimating Future Equity Returns," FPA Journal, May 2002.
2. Campbell, John Y., "Forecasting U.S. Equity Returns in the 21st Century," Paper presented to the Social Security Advisory Board, August 2001.
3. Diamond, Peter A., "What Stock Market Returns to Expect for the Future: An Update," Paper presented to the Social Security Advisory Board, August 2001.
4. Diamond, Peter A., "What Stock Market Returns to Expect for the Future," Social Security Bulletin, Vol. 63, No. 2, 2000.
5. Evanson, Steven, "Stocks for the Long Run?" Evanson Asset Management, June 2001.
6. Fama, Eugene F., and Kenneth R. French, "The Equity Premium," The Center for Research in Security Prices Working Paper No. 522, The University of Chicago, Gradute School of Business April 2001.
7. Ibbotson, Roger G. and Peng Chen, "The Supply of Stock Market Returns," Ibbotson Associates, Chigcago, IL June 2001.
8. Reichenstein, William, "The Investment Implications of Lower Stock Return Prospects," AAII Journal, October 2001.
9. Shoven, John B., "What Are Reasonable LongRun Rates of Return to Expect on Equities," Paper presented to the Social Security Advisory Board, August 2001.
(1.) Standard deviation and variance are discussed as concepts here. Actual calculations of standard deviation and related subjects are rather complex. Do not be concerned with the actual calculations here. Actual calculations of standard deviation and related subjects are discussed in detail later in this chapter, under "How It Is Done--An Example."
(2.) Variance [equivalent to] [[sigma].sup.2] = E[[(R-[mu]).sup.2]], where [sigma] is the standard deviation, [mu] is the mean value, R is the return variable, and E is the expectation operator. For continuous distributions, F(R), with probability density function f(R), [[sigma].sup.2] = [integral]c[(R-[mu]).sup.2] x f(R)dR. For discreet distributions, with probabilities [p.sub.i] for each possible [outcome.sub.i] in the set of all possible outcomes, [[sigma].sup.2] = [SIGMA] [p.sub.i] [([R.sub.i]-[mu]).sup.2]. Variances and standard deviations are generally estimated based upon historical returns which are computed as follows: [[sigma].sup.2] = [SIGMA] [([R.sub.i] - [mu]).sup.2] /(n -1), where [R.sub.i] are each of the historical observations, n is the number of observations, and [mu] * is the average return of the observations used to compute the estimate of variance.
(3.) The mean absolute deviation is the expected value of the absolute value of the differences from the mean: MAD = E([absolute value of R - [mu]]), where R is the return variable and [mu] is the expected value of R. It is estimated from data using the formula: MAD = [SIGMA]/[absolute value of [R.sub.i] - [mu] *]/n, where Ri is the ith observation of R from the dataset, n is the number of data points in the dataset, and [mu] * is the average value of the n observation of R.
(4.) The standard deviation is actually 28.9 for this distribution, slightly higher than the mean absolute deviation, but closely related. For nonuniform distributions, other than those like picking balls from an urn, the standard deviation is generally considered the better measure of dispersion around the mean of the distribution.
(5.) Estimates of semi-variance are calculated in the same manner as variances, except only the historical returns below the average return are used in the calculation. Specifically, semi-variance = [[sigma].sub.sv.sup.2] = [SIGMA] [([R.sub.i.sup.<[mu] *] - [mu]*).sup.2], where [R.sub.i.sup.<[mu]] is the ith return less than the average return, [mu] *, and n<u is the number of observations of [Ri that are less than [mu] *. Similar to the standard deviation, which is the square root of the variance, the semi-standard-deviation or semi-deviation is the square root of the semi-variance, or [[sigma].sub.sv].
(6.) Beta is described and discussed in more detail in relation to the Capital Asset Pricing Model in Chapter 36, "Asset Pricing Models."
(7.) To estimate the skewness from sample data use the following formula:
S = n/(n - 1)(n - 2) [SIGMA] [([R.sub.i] - [mu] */[sigma]).sup.3]
Here, [sigma] * is the sample standard deviation and [mu] * is the sample mean.
(8.) Since the normal distribution has a kurtosis of 3, the measure of kurtosis is generally expressed as the coefficient of excess kurtosis, which is simply K-3. Therefore, a measure of less than zero means the distribution has skinnier tales and a fatter center than the normal distribution while an excess kurtosis greater than zero means the distribution has fatter tales and a skinnier center than the normal distribution. To estimate the sample coefficient of excess kurtosis from sample data, use the following formula:
EK = n(n + 1)/(n - 1)(n - 2)(n - 3) [SIGMA][([R.sub.i] - [mu] */[sigma]).sup.4] - 3[(n -).sup.2]/(n - 2)(n -3)
Here, [mu] * is the sample mean and a* is the sample standard deviation.
Figure 31.4 STANDARD DEVIATION (SD) Total Deviation (D / SD) (D / SD) Period Return (D) D (2) (3) (4) 1 -25.0 -29.1 848.3 -10.1977 22.1142 2 -10.0 -14.1 199.5 -1.1632 1.2234 3 -5.0 -9.1 83.3 -0.3136 0.2131 4 -1.0 -5.1 26.3 -0.0556 0.0212 5 0.0 -4.1 17.0 -0.0290 0.0089 6 0.0 -4.1 17.0 -0.0290 0.0089 7 1.0 -3.1 9.8 -0.0126 0.0029 8 1.0 -3.1 9.8 -0.0126 0.0029 9 5.0 0.9 0.8 0.0003 0.0000 10 5.0 0.9 0.8 0.0003 0.0000 11 5.0 0.9 0.8 0.0003 0.0000 12 10.0 5.9 34.5 0.0837 0.0366 13 10.0 5.9 34.5 0.0837 0.0366 14 10.0 5.9 34.5 0.0837 0.0366 15 25.0 20.9 435.8 3.7548 5.8360 16 35.0 30.9 953.3 12.1486 27.9277 Sum 66.0 2,705.8 4.3420 57.4691 Average 4.1 180.4 Average = 4.1 Median = (1 + 5) / 2 = 3.0 Variance = 180.4 Standard Deviation = [square root of 180.4] = 13.4 Skewness = 16/(16 - 1)(16 - 2) x 4.342029 Skewness = 0.076190 x 4.342029 Skewness = 0.330821 Kurtosis = 16 (16 + 1)/(16 - 1)(16 - 2)(16 - 3) x 57.469059 - 3(16 - 1)(2)/ (16 - 2)(16 - 3) Kurtosis = 0.099634 x 57.469059 - 3.708791 Kurtosis = 5.725855 - 3.708791 Kurtosis = 2.017064 Figure 31.6 SEMI-VARIANCE Total Deviation (D / SD) (D / SD) Period Return (D) D (2) (3) (4) 1 -25.0 -29.1 848.3 -10.8592 24.0470 2 -10.0 -14.1 199.5 -1.2387 1.3303 3 -5.0 -9.1 83.3 -0.3340 0.2317 4 -1.0 -5.1 26.3 -0.0592 0.0231 5 0.0 -4.1 17.0 -0.0309 0.0097 6 0.0 -4.1 17.0 -0.0309 0.0097 7 1.0 -3.1 9.8 -0.0134 0.0032 8 1.0 -3.1 9.8 -0.0134 0.0032 9 5.0 10 5.0 11 5.0 12 10.0 13 10.0 14 10.0 15 25.0 16 35.0 Sum 66.0 1,210.9 -12.5795 25.6578 Average 4.1 173.0 Average = -4.9 Median = (-1 + 0) / 2 = -0.5 Semi-Variance = 173.0 Semi-Deviation (SD) = [square root of 173] = 13.2 Skewness = 8/(8 - 1) (8 - 2) x -12.579502 Skewness = 0.190476 x -12.579502 Skewness = -2.396096 Kurtosis = 8(8 + 1)/(8 - 1)(8 - 2)(8 - 3) x 25.657804 - 3(8 - 1) (2)/(8 -2)(8 - 3) Kurtosis = 0.342857 x 25.657804 - 4.9 Kurtosis = 3.896961
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|Title Annotation:||Techniques of Investment Planning|
|Publication:||Tools & Techniques of Investment Planning, 2nd ed.|
|Date:||Jan 1, 2006|
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