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:

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

Tel: 617 576-7360, Email:

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

Tel: 617 234-9462, Email:

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

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.

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.

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 = eln(133.33) x .50 + ln(75) x .50

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

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.
Compute the certainty equivalent of this expected utility maximizing portfolio.
Identify the portfolio along the mean-variance efficient frontier that maximizes the mean-variance approximation of expected utility and calculate its true utility.
Calculate the certainty equivalent of this portfolio.
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.

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

where,

U = utility

=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.

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.

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

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.

We identify the portfolio that yields the highest log wealth utility based on the original sample, and we compute its certainty equivalent.
We select 192 monthly return vectors with replacement from the original sample and scale the returns to produce means that conform to consensus expectations.
We identify the portfolio that yields the highest log wealth utility of the bootstrapped sample, and we compute its certainty equivalent.
We compute the gratuitous dead weight loss of the bootstrapped sample relative to the original sample.
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:

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.
We select 192 monthly return vectors with replacement from the original sample.
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.
We compute the gratuitous dead weight loss of the bootstrapped sample relative to the original sample.
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.