BALANCING ACTIVE AND PASSIVE MANAGEMENT



Revere Street Working Paper Series

Financial Economics 272-12

PORTFOLIO FORMATION

WITH HIGHER MOMENTS AND PLAUSIBLE UTILITY

Jan-Hein Cremers, Mark Kritzman, and Sebastien Page

State Street Associates, 136 Mt. Auburn Street, Cambridge, MA 02138

Tel: 617 234-9482, Email: jcremers@

Windham Capital Management Boston, 5 Revere Street, Cambridge, MA 02138

Tel: 617 576-7360, Email: mkritzman@

State Street Associates, 136 Mt. Auburn Street, Cambridge, MA 02138

Tel: 617 234-9462, Email: spage@

THIS VERSION : NOVEMBER 22, 2003

Abstract

Most serious investors use mean-variance optimization to form portfolios, in part, because it requires knowledge only of a portfolio’s expected return and variance. Yet this convenience comes at some expense, because the legitimacy of mean-variance optimization depends on questionable assumptions. Either investors have quadratic utility or portfolio returns are normally distributed. Neither of these assumptions is literally true. Quadratic utility assumes that investors are equally averse to deviations above the mean as they are to deviations below the mean and that they sometimes prefer less wealth to more wealth. Moreover, asset returns have been shown to exhibit significant departures from normality. The question is: does it matter? Levy and Markowitz (1979) demonstrate that mean-variance approximations of utility based on plausible utility functions and empirical return distributions correlate very strongly with true utility. Samuelson (2003) argues that investors now have sufficient computational power to maximize expected utility based on plausible utility functions and the entire distribution of returns from empirical samples, and he introduces a different metric to determine the robustness of mean-variance approximation. We apply Samuelson’s metric to measure the approximation error of mean-variance optimization based on a sample of representative asset returns. Moreover, we apply Samuelson’s metric to compare the sampling error of these alternative approaches. Finally, we introduce a hybrid approach to portfolio formation, which enables investors to maximize expected utility based on plausible utility functions, but which relies on theoretical rather than empirical return distributions.

PORTFOLIO FORMATION

WITH HIGHER MOMENTS AND PLAUSIBLE UTILITY[1]

I. Introduction

Most serious investors use mean-variance optimization to form portfolios, in part, because it requires knowledge only of a portfolio’s expected return and variance. Yet this convenience comes at some expense, because the legitimacy of mean-variance optimization depends on questionable assumptions. Either investors have quadratic utility or portfolio returns are normally distributed. Neither of these assumptions is literally true. Quadratic utility assumes that investors are equally averse to deviations above the mean as they are to deviations below the mean and that they sometimes prefer less wealth to more wealth. Moreover, asset returns have been shown to exhibit significant departures from normality. The question is: does it matter? Levy and Markowitz (1979) demonstrate that mean-variance approximations of utility based on plausible utility functions and empirical return distributions correlate very strongly with true utility. Samuelson (2003) argues that investors now have sufficient computational power to maximize expected utility based on plausible utility functions and the entire distribution of returns from empirical samples, and he introduces a different metric to determine the robustness of mean-variance approximation. We apply Samuelson’s metric to measure the approximation error of mean-variance optimization based on a sample of representative asset returns. Moreover, we apply Samuelson’s metric to compare the sampling error of these alternative approaches. Finally, we introduce a hybrid approach to portfolio formation, which enables investors to maximize expected utility based on plausible utility functions, but which relies on theoretical rather than empirical return distributions.

We organize the paper as follows. In Part II we discuss the theoretical limitations of mean-variance optimization, and we review Levy and Markowitz’s position. We then maximize expected utility according to Samuelson based on the full sample of returns for a representative set of assets, and we apply his metric to measure the approximation error of mean-variance optimization. In Part III we bootstrap an empirical return sample to measure the sampling error of these two approaches. In Part IV we introduce a hybrid approach to portfolio formation. We maximize power utility based on a full sample of returns as Samuelson suggests, but one that is determined theoretically rather than empirically. We summarize the paper in Part V.

II. Approximation Error of Mean-Variance Optimization

Mean-variance optimization identifies portfolios that offer the highest expected returns for given levels of risk, defined as variance, based on assumptions about the means, variances, and covariances of the component assets. It does not use information about the assets’ specific periodic returns or even about other features of their distributions such as skewness or kurtosis. This approach to portfolio formation is sufficient for maximizing expected utility if at least one of two conditions prevails. Either portfolio returns are normally distributed or investors have quadratic utility, which is defined as E(U) = μ – λ σ2, where μ equals portfolio expected return, λ equals risk aversion, and σ2 equals portfolio variance. If returns are normally distributed, investors can infer the entire distribution of returns from its mean and variance; hence the irrelevance of specific periodic returns or higher moments. And even if returns are not normally distributed, quadratic utility assumes that investors are indifferent to other features of the distribution.

It is easy to understand the historical appeal of mean-variance optimization by considering the alternative. In order to identify portfolios that maximize expected utility based on more plausible utility functions in the presence of non-normal return distributions, investors would need to compute expected utility for every period and for every possible combination of assets. For example, with 20 assets to choose from and shifting the asset weights by increments of 1%, one would need to compare 4,910,371,215,196,100,000,000 different portfolios for every set of periodic returns to find the one that maximizes expected utility -- a daunting task even for a summer intern.[2]

However, quadratic utility is not a realistic description of a typical investor’s attitude toward risk. Financial economists usually assume that investors have power utility functions, which define utility as 1/γ x Wealthγ. A log wealth utility function is a special case of power utility. As γ approaches 0, utility approaches the natural logarithm of wealth. A γ equal to ½ implies less risk aversion than log wealth, while a γ equal to -1 implies greater risk aversion.[3] These utility functions, along with a quadratic utility function, are shown in Figure 1.

[pic]

Notice that as wealth increases, the increments to utility become progressively smaller. This concavity indicates that investors derive less and less satisfaction with each subsequent unit of incremental wealth. Also notice that power utility functions, unlike quadratic utility functions, never slope downward, which would reflect a preference to reduce wealth.

It is also the case that many return distributions are not normal. Table 1 shows the skewness and kurtosis of five asset classes based on monthly returns from January 1987 through December 2002. A normal distribution has skewness equal to 0 and kurtosis equal to 3.

[pic]

A Jarque-Bera test[4] of normality shows that U.S. equities, real estate, and private equity are significantly non-normal.

Levy and Markowitz (1979) attacked this problem head on. They showed how to approximate log wealth utility and other variations of power utility functions using only a portfolio’s mean and variance. They demonstrated that mean-variance approximations to utility based on plausible power utility functions performed exceptionally well for returns ranging from -30% to +60%. For a sample of mutual funds they found correlations in excess of 99% between true power utility and mean-variance approximated utility. Moreover, they demonstrated that the mean-variance efficient frontier contained the portfolio that maximized true power utility. This result is extremely important if it holds broadly, because it allows investors to maximize expected utility based only on mean and variance, even if they have power utility functions and even if return distributions are not normal.

Samuelson’s Recommendation

Samuelson (2003) suggests that investors should maximize expected utility based on plausible utility functions as well as the entire distribution of returns from empirical samples, rather than just their means and variances. Today there are search algorithms that direct us efficiently toward the solution in a matter of seconds.

Samuelson also proposes a new metric to measure the robustness of mean-variance approximation. Rather than focus on the correlation between true utility and approximate utility à la Levy and Markowitz, he argues that investors should compute the difference between the certainty equivalents of the true utility maximizing portfolio and the mean-variance approximation to the truth. The portfolio that maximizes the mean-variance approximation to true expected utility will always lie on the efficient frontier as determined by quadratic programming. Samuelson refers to the difference in these certainty equivalents as “gratuitous dead weight loss.” Now let us digress briefly to review the notion of certainty equivalent.

A Digression on Certainty Equivalents

A certainty equivalent is the value of a certain prospect that yields the same utility as the expected utility of an uncertain prospect.[5] Consider an investor who has log wealth utility and is faced with a risky investment that has an equal probability of increasing by 1/3 or falling by 1/4. The utility of this investment equals the sum of the probability weighted utilities of the two outcomes. If the initial investment is $100.00 the expected utility of this investment equals 4.60517 as shown.

4.60517 = ln(133.33) x .50 + ln(75.00) x .50

This investment has an expected value of $104.17, but this expectation is uncertain. How much less should the investor be willing to accept for sure such that she would be indifferent between this amount and an uncertain value of $104.17? It turns out that $100.00 also yields utility of 4.6052 (ln(100) = 4.06517). Therefore, if her utility function equals the logarithm of wealth, she would be indifferent between receiving $100.00 for sure and an equal probability of receiving $133.33 or $75.00.

For a log wealth utility function, we find the certainty equivalent by raising e, the base of the natural logarithm, to the power of expected utility.

100.00 = e ln(133.33) x .50 + ln(75) x .50

According to Samuelson, investors should evaluate the accuracy of mean-variance optimization as follows.

1. Calculate a portfolio’s utility for as many asset mixes as necessary, including those not on the mean-variance efficient frontier, in order to identify the weights that yield the highest possible expected utility, given a plausible utility function such as log wealth.

2. Compute the certainty equivalent of this expected utility maximizing portfolio.

3. Identify the portfolio along the mean-variance efficient frontier that maximizes the mean-variance approximation of expected utility and calculate its true utility.

4. Calculate the certainty equivalent of this portfolio.

5. Compute the “gratuitous dead weight loss,” which equals the certainty equivalent of the true utility maximizing portfolio less the certainty equivalent of the mean-variance approximated portfolio.

We illustrate Samuelson’s approach with a sample of stock and bond returns, which are shown in Table 2. For an investor with log wealth utility, we compute utility each period as ln[(1+RS) x WS + (1+RB) x WB], where RS and RB equal the stock and bond returns, and WS and WB equal the stock and bond weights.

We then shift the stock and bond weights until we find the combination that maximizes expected utility, which for this example equals a 57.13% allocation to stocks and a 42.87% allocation to bonds. The expected utility of the portfolio equals 9.3138345%, and its certainty equivalent equals 1.097613574. This approach implicitly takes into account all of the features of the empirical sample, including possible skewness, kurtosis, and any other peculiarities of the distribution.

[pic]

Mean-Variance Approximation

In order to find the portfolio that maximizes the mean-variance approximation to expected utility, we first identify the portfolios along the efficient frontier as prescribed by Markowitz (1952). We then approximate their expected utility using an approximation based only on mean and variance.[6] Table 3 shows the approximations for the three power utility functions described earlier, along with the certainty equivalents of these utility functions. In this example, we assume the investor has log wealth utility; therefore, we use the first approximation formula.

Table 3: Mean-Variance Approximations of Power Utility

|Utility Function |Approximate Utility |Certainty Equivalent |

|[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |

where,

U = utility

[pic]=approximated utility

ln = natural logarithm

( = arithmetic average of yearly returns of unranked portfolios

( = annualized standard deviation of unranked portfolios

We next identify the portfolio weights from those portfolios along the efficient frontier that maximize approximate expected utility and substitute these weights into Table 2 to find this portfolio’s true expected utility. Based on the means, variances, and correlation of the returns shown in Table 2, a 59.29% allocation to stocks and a 31.71% allocation to bonds yield the highest approximate expected utility (9.3041%). The true expected utility of this portfolio, however, is 9.3128%, and its certainty equivalent is 1.097603. The difference between the true utility maximizing portfolio and the mean-variance approximation equals 0.00001086; that is, $1,086 for every $100 million of investment.

This example is merely illustrative of the issue raised by Samuelson. It contains only two assets and is based on annual returns, which may mask departures from normality in higher frequency returns. Next we consider a more realistic sample in which a U.S. based fund is allocated among the five asset classes listed in Table 1: U.S. equities, international equities, U.S. bonds, real estate, and private equity; and its allocation is based on their monthly returns from January 1987 through December 2002. Table 4 shows the standard deviations, and correlations of these asset classes computed from the monthly returns of representative indexes. Recall that three of these asset classes have significantly non-normal distributions.

Sophisticated investors seldom use historical means as expectations for future means. Moreover, the means do not affect the approximation error of mean-variance optimization; only the shape of the distribution matters. Therefore, we scale each of the returns to produce means that conform to the revealed expectations of institutional investors. Specifically, we use the expected returns that are implied by the average asset weights of 1,729 U.S. pension funds as of 2002 as reported by Greenwich Associates.[7] These returns render the average weights efficient; they are thus a good approximation of consensus expectations.

[pic]

Based on these samples of monthly returns, we identify the true expected utility maximizing portfolio weights for three power utility functions: log wealth utility, a more risk averse utility function (γ = -1), and a less risk averse utility function (γ = ½). Then we identify approximate weights using mean-variance optimization. Table 6 shows these weights along with expected utility, certainty equivalent, and gratuitous dead weight loss, assuming short positions are permitted.

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Table 7 presents the same information, but for portfolios that are restricted to long only exposures.

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These results suggest that mean-variance optimization performs extremely well, at least based on the selected sample of returns. However, these returns are but one pass through history and not necessarily characteristic of future returns. Therefore, in the next section we bootstrap our historical sample to generate 1,000 different histories in order to gauge the impact of sampling error on expected utility for both full-sample utility maximization and mean-variance optimization.

III. Sampling Error

We estimate sampling error for full-sample utility maximization as follows.

1. We identify the portfolio that yields the highest log wealth utility based on the original sample, and we compute its certainty equivalent.

2. We select 192 monthly return vectors with replacement from the original sample and scale the returns to produce means that conform to consensus expectations.

3. We identify the portfolio that yields the highest log wealth utility of the bootstrapped sample, and we compute its certainty equivalent.

4. We compute the gratuitous dead weight loss of the bootstrapped sample relative to the original sample.

5. We repeat this process 999 additional times to generate a distribution of the sampling error associated with full-sample utility maximization.

We estimate combined approximation and sampling error for mean-variance optimization as follows:

1. We identify the portfolio along the mean-variance efficient frontier that maximizes the mean-variance approximation of log wealth utility based on the original sample, and we compute its true utility and certainty equivalent.

2. We select 192 monthly return vectors with replacement from the original sample.

3. We identify the portfolio along the mean-variance efficient frontier that maximizes the mean-variance approximation of log wealth utility based on the bootstrapped sample, and we compute its true utility and certainty equivalent.

4. We compute the gratuitous dead weight loss of the bootstrapped sample relative to the original sample.

5. We repeat this process 999 additional times to generate a distribution of the combined approximation error and sampling error associated with mean-variance optimization.

Table 8 presents the mean and standard deviation of the sampling error associated with full-sample utility maximization as well as the mean and standard deviation of the combined approximation error and sampling error associated with mean-variance optimization. These results assume the portfolios may include short positions.

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Table 9 shows similar results for portfolios that preclude short positions.

[pic]

Tables 8 and 9 reveal that sampling error associated with full-sample utility maximization is only slightly lower than the combined approximation error and sampling error associated with mean-variance optimization, and that it is more than 10 times as great as approximation error by itself. Again, it seems there is little motivation to abandon mean-variance optimization.

IV. A Hybrid Solution

In the absence of theory it is dangerous to base predictions on empirical results, even if we perturb these results to account for sampling error. Instead, it might make sense to hypothesize a return generating process and to model the shapes of return distributions based on theory rather than empirical precedent. For example, if markets are efficient and new information arrives randomly, we should expect returns to be independent from period to period. This serial independence, together with the effect of compounding, provides a theoretical basis for a lognormal distribution. If markets are instead characterized by frictions, which cause prices to respond abruptly to accumulated information, we have a theoretical basis to expect fat tails. If we engage in dynamic trading strategies that shift exposure between assets based on relative price changes, we have a theoretical basis to expect a skewed distribution.

If theory leads us to believe that return distributions are significantly non normal, and we have a non-quadratic utility function, then we should employ a hybrid approach to form portfolios. We should use simulation to generate distributions that conform to our theory of the return generating process, and we should use the entire simulated distribution to maximize utility based on a plausible utility function. This approach differs from mean-variance optimization in two ways: we use the entire distribution of simulated returns rather than just its mean and variance, and we assume non-quadratic utility.

This approach also differs from Samuelson’s recommendation. Rather than maximize expected utility based on empirical return distributions whose essential features may be an artifact of sampling error, we simulate the distributions to accord with our theory of the return generating process.

In order to introduce skewness to a return distribution, we randomly select monthly continuous returns from a normal distribution and convert them to discrete returns. Then we compound the discrete returns forward to exacerbate the skewness. We next rescale the multi-period cumulative returns back to monthly returns, but we preserve the skewness of the multi-period cumulative returns.[8]

Table 10 compares mean-variance optimization to full-sample utility maximization for portfolios with manufactured skewness. Although we induce skewness directly into only one of the assets, when we randomly select positively correlated returns, some of this skewness leaks into the other assets’ distributions. For example, when we introduce positive skewness of about 3.50 in one asset, we produce positive skewness in the other assets of about 1.00, and when we introduce negative skewness of about -3.50 in one asset, we produce negative skewness of about -1.00 in the other assets.

Table 10 reveals that mean-variance optimization performs extremely well for portfolios in which the components are not highly correlated. Even with non-normality at the individual asset level, the central limit theorem shifts the portfolio returns back toward normality. As the correlation increases, the central limit theorem is less effective; hence we observe greater gratuitous dead weight loss. Even so, it is still less than 1/10th of a basis point for moderate skewness and only slightly more than one basis point for hedge fund type skewness of about -3.50.

[pic]

Next we manufacture kurtosis by sampling from a double gamma distribution rather than a normal distribution.[9] Again, we find that mean-variance optimization performs extremely well in the presence of kurtosis. In the worst case, gratuitous dead weight loss is less than 2/100th of a basis point; however, a cautionary note is in order. We have assumed log wealth utility throughout in these examples. For investors who are more risk averse, gratuitous dead weight loss will likely be higher.

[pic]

V. Summary

Investors have long worried about the approximation error of mean-variance optimization, which assumes implicitly that they either have quadratic utility or that returns are normally distributed. Levy and Markowitz have previously shown that mean-variance approximations to power utility are remarkably accurate, based on a sample of mutual fund returns. Samuelson proposes a new metric, which he calls gratuitous dead weight loss, to gauge the accuracy of mean-variance optimization in the presence of non-quadratic utility and non-normality. This issue is critically important to many investors given the widespread use of hedge funds, which often display significant departures from normality.

We follow Samuelson’s advice and measure the gratuitous dead weight loss of mean-variance optimization for a variety of plausible utility functions based on representative empirical distributions. For the most part, the gratuitous dead weight loss of mean-variance optimization is negligible – only a fraction of a basis point.

We also measure the gratuitous dead weight loss associated with sampling error for both mean-variance optimization and full-sample utility maximization. We find that the gratuitous dead weight loss associated with sampling error for both mean-variance optimization and full-sample utility maximization is still quite small, but more than 10 times as great as the gratuitous dead weight loss associated with mean-variance approximation error.

Finally, we introduce a hybrid approach to portfolio formation in which we simulate theoretically based non-normal distributions. We then use the entire sample to maximize expected utility based on plausible utility functions. Our simulations suggest that, except in unusual circumstances, mean-variance optimization performs well by virtually any standard.

We close with three caveats. First, we assume throughout our analyses that investors have some variation of power utility. Even though mean-variance optimization produces portfolio weights that vary by as much as 20% from full-sample utility maximization in our examples with manufactured skewness and kurtosis, their certainty equivalents are not particularly different. Higher moments seem not to matter very much for investors with power utility. However, some investors might have very different utility functions. Perhaps certain investors are faced with thresholds, which if they were to breach, would sharply reduce their utility. Given this type of bilinear utility function, it is quite likely that mean-variance optimization would generate substantial gratuitous dead weight loss. We strongly recommend Samuelson’s full-sample approach or our hybrid approach in such situations.

Second, we have only considered allocation among a few assets of moderate risk. Mean-variance optimization may not perform as well when applied to portfolios comprising a large number of relatively risky securities, especially if these securities are highly correlated.

Third, even if we choose to ignore higher moments for the purpose of forming portfolios, we should not ignore them when we assess exposure to loss or manager skill. Value at risk, for example, is typically much higher for portfolios with significant negative skewness and kurtosis than we would infer from a normal distribution. In addition, some investment strategies may appear to add value based on mean and variance, when in fact, the apparent value added is an artifact of higher moments. Higher moments do matter, but typically not for choosing portfolios!

Appendix: Manufacturing Skewness and Kurtosis

Our goal is to generate samples from a theoretical multivariate distribution, where each sample is a vector of returns, in such a way that we can independently specify the mean and covariance as well as higher moments, skewness and kurtosis.

There is a standard method to sample from the multivariate Gaussian distribution of dimension N. First a row vector Rindep of N independent random draws are made from the 1-dimensional Gaussian distribution. To introduce covariance described by a matrix (, we use the Cholesky factorization (=C’C. The multivariate returns are then given by Rmult = RC. To introduce skewness and kurtosis we follow the same procedure, but instead of using independent draws from Gaussian distributions in step 1, we draw from distributions with skewness and kurtosis. To get realistic distributions we draw from the lognormal distribution to generate skewness, and from a “double gamma” distribution to generate kurtosis without skewness. In the latter case we draw from the regular gamma distribution and give each draw equal chances to be positive or negative. The lognormal distribution with probability density function

[pic]

[pic]

has skewness [pic] and kurtosis [pic]. The double gamma distribution

[pic]

has standard deviation [pic] and kurtosis [pic], where [pic] is the gamma function.

References

Bernoulli, Daniel, “Exposition of a New Theory on the Measurement of Risk,” Econometrica, January 1954.

Levy, Haim and Harry M. Markowitz, “Approximating Expected Utility by a Function of Mean and Variance,” American Economic Review, June 1979, Vol. 69, No. 3.

Markowitz, Harry, M., “Portfolio Selection,” Journal of Finance, March 1952.

Samuelson, Paul, A., “When and Why Mean-Variance Analysis Generically Fails,” forthcoming in the American Economic Review, 2003.

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[1] We thank Paul A. Samuelson for encouraging us to engage in this research and for his helpful comments. We have also benefited from discussions with Jordan Alexiev, Javier Estrada, Kenneth Froot, Harry Markowitz, Lesley McAdams, Frederyk Ngantung, Pasha Roberts, Stephen Satchell, William Sharpe, and Lee Thomas. We are responsibordan Alexiev, Javier Estrada, Kenneth Froot, Harry Markowitz, Lesley McAdams, Frederyk Ngantung, Pasha Roberts, Stephen Satchell, William Sharpe, and Lee Thomas. We are responsible for any errors.

[2] This value is given by the following formula: (δ+n-2)! / [(n-1)! x (1/δ+1)!], where δ = increment by which portfolio weights are changed and n = number of assets.

[3] When γ equals -1, utility is expressed as 1 – W-1.

[4] The Jarque-Bera test is a non-parametric test of normality that is based on skewness and kurtosis. It is non-parametric in the sense that it tests normality without specification of a particular mean or variance. The actual statistic is a sum of two independent parts, one built from the third moment, the other from the fourth moment. Each of these parts is distributed as a standard normal. The Jarque-Bera statistic is their sum of squares and thus should be compared to a chi-square distribution with two degrees of freedom.

[5] This notion was introduced by the famous mathematician, Daniel Bernoulli in 1738. For an English translation, see Bernoulli (1954).

[6] Levy and Markowitz (1979) describe how to derive these approximations with Taylor Series expansions.

[7] Total Market Size and Asset Mix – United States, Greenwich Associates, February 11, 2003, .

[8] See the appendix for detail.

[9] See the appendix for detail.

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