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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: [email protected]
Windham Capital Management Boston, 5 Revere Street, Cambridge, MA 02138
Tel: 617 576-7360, Email: [email protected]
State Street Associates, 136 Mt. Auburn Street, Cambridge, MA 02138
Tel: 617 234-9462, Email: [email protected]
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 UTILITY1
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
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 responsible for any errors.
2
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
3
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
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.
2
4
implies greater risk aversion.3 These utility functions, along with a quadratic utility
function, are shown in Figure 1.
Figure 1: Utility Functions
7.0
6.0
ln(W)
Utility
5.0
Sqrt( W )
4.0
1-1/( W )
3.0
Quadratic
2.0
1.0
0.0
0
5
10
15
20
Wealth
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
3
When γ equals -1, utility is expressed as 1 – W-1.
5
1987 through December 2002. A normal distribution has skewness equal to 0 and
kurtosis equal to 3.
Table 1: Non-normality (monthly returns from 1/87 - 12/02)
Asset Class
U.S. equity
International equity
U.S. bonds
Real Estate
Private equity
Index
S&P 500
EAFE
Lehman govt. & corp.
NRIXTR
Wilshire LBO
Skewness Kurtosis
-0.84
5.64
-0.15
3.31
-0.10
3.00
-0.22
4.70
-0.71
4.97
JB Test
Failed
Passed
Passed
Failed
Failed
A Jarque-Bera test4 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
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.
6
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 meanvariance 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 meanvariance 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.
7
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
5
This notion was introduced by the famous mathematician, Daniel Bernoulli in 1738. For an English
translation, see Bernoulli (1954).
8
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 meanvariance 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
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takes into account all of the features of the empirical sample, including possible
skewness, kurtosis, and any other peculiarities of the distribution.
Table 2: Utility of Annual Stock and Bond Returns
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
Stock
Returns
10.06%
1.32%
37.53%
22.93%
33.34%
28.60%
20.89%
-9.09%
-11.94%
-22.10%
Bond
Returns
16.16%
-7.10%
29.95%
0.14%
14.52%
11.76%
-7.64%
16.14%
7.26%
14.83%
Stock
Weight
57.13%
57.13%
57.13%
57.13%
57.13%
57.13%
57.13%
57.13%
57.13%
57.13%
Bond
Weight
42.87%
42.87%
42.87%
42.87%
42.87%
42.87%
42.87%
42.87%
42.87%
42.87%
Portfolio
Utility
ln[(1+.1006)x.5713+(1+.1616)x.4287]x1/10
ln[(1+.0132)x.5713+(1 -.0710)x.4287]x1/10
ln[(1+.3753)x.5713+(1+.2995)x.4287]x1/10
ln[(1+.2293)x.5713+(1+.0014)x.4287]x1/10
ln[(1+.3334)x.5713+(1+.1452)x.4287]x1/10
ln[(1+.2860)x.5713+(1+.1176)x.4287]x1/10
ln[(1+.2089)x.5713+(1 -.0764)x.4287]x1/10
ln[(1 -.0909)x.5713+(1+.1614)x.4287]x1/10
ln[(1 -.1194)x.5713+(1+.0726)x.4287]x1/10
ln[(1 -.2210)x.5713+(1+.1483)x.4287]x1/10
Expected Utility
=
=
=
=
=
=
=
=
=
=
1.1931%
-0.2317%
2.9477%
1.2367%
2.2533%
1.9375%
0.8305%
0.1715%
-0.3777%
-0.6470%
9.3138%
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.
6
Levy and Markowitz (1979) describe how to derive these approximations with Taylor Series expansions.
10
Table 3: Mean-Variance Approximations of Power Utility
Utility Function
Approximate Utility
Certainty
Equivalent
U  ln 1  r 
1 2 2
Uˆ  ln 1    
1   2
eU
U  1 r
1 8 2
Uˆ  1   
1   3 2
U2
1
2
ˆ
U  1

1   1   3
1
1U
U  1  1  r 
1
where,
U = utility
Û =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 meanvariance 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
11
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.
Table 4: Standard Deviations and Correlations
Standard
Asset Class
Index
Deviation
U.S. equity
S&P 500
16.05%
International equity
EAFE
17.56%
U.S. bonds
Lehman govt. & corp. 7.65%
Real Estate
NRIXTR
11.96%
Private Equity
Wilshire LBO
44.97%
International
Equities
60.93%
U.S. bonds
14.24%
0.05%
Real
Estate
42.66%
25.31%
15.49%
Private
Equity
80.94%
53.56%
10.68%
46.60%
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.
Total Market Size and Asset Mix – United States, Greenwich Associates, February 11, 2003,
http://www.greenwich.com.
7
12
Table 5: Implied Returns
Asset Class
U.S. equity
International equity
U.S. bonds
Real Estate
Private equity
Average
Weights
50.70%
12.13%
30.12%
3.68%
3.36%
Index
S&P 500
EAFE
Lehman govt. & corp.
NRIXTR
Wilshire LBO
Implied
Returns
9.00%
8.61%
7.48%
7.86%
11.58%
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 6: Gratuitous Dead Weight Loss
Asset Class
U.S. equity
International equity
U.S. bonds
Real Estate
Private Equity
Expected Utility
Certainty equivalent
Gratuitous dead weight loss
Monthly loss per $100 million
Logwealth
Full
MeanSample
Variance
49.06%
49.78%
11.95%
12.12%
33.18%
30.92%
3.94%
5.08%
1.87%
2.09%
0.00627169
1.00629140
0.00627120
1.00629090
0.00000050
$50
 = -1
Full
Sample
33.01%
11.62%
45.76%
13.29%
-3.68%
0.00593103
1.00596642
MeanVariance
33.03%
11.88%
44.83%
13.79%
-3.53%
0.00593084
1.00596622
0.00000019
$19
 = 1/2
Full
MeanSample
Variance
80.44%
83.35%
12.66%
12.61%
9.59%
3.06%
-15.50%
-12.38%
12.81%
13.36%
1.00334738
1.00670596
1.00334627
1.00670373
0.00000222
$222
Table 7 presents the same information, but for portfolios that are restricted to long
only exposures.
13
Table 7: Gratuitous Dead Weight Loss without Short Positions
Asset Class
U.S. equity
International equity
U.S. bonds
Real Estate
Private Equity
Expected Utility
Certainty equivalent
Gratuitous dead weight loss
Monthly loss per $100 million
γ = -1
Log Wealth
Full
MeanSample
Variance
49.06%
49.78%
11.95%
12.12%
33.18%
30.92%
3.94%
5.08%
1.87%
2.09%
0.00627169
1.00629140
0.00627120
1.00629090
0.00000050
$50
Full
Sample
25.54%
11.67%
50.30%
12.49%
0.00%
MeanVariance
25.90%
11.95%
49.18%
12.96%
0.00%
0.00592247
1.00595776
0.00592229
1.00595757
0.00000019
$19
γ = 1/2
Full
MeanSample
Variance
76.1%
75.2%
11.5%
10.6%
0.2%
0.0%
0.0%
0.0%
12.2%
14.1%
1.00334408
1.00669934
1.00334354
1.00669826
0.00000108
$108
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.
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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
15
combined approximation error and sampling error associated with mean-variance
optimization. These results assume the portfolios may include short positions.
Table 8: Sampling Error with Short Positions (log wealth utility)
Full Sample
Maximization
Mean-Variance
Approximation
Full-Sample
Advantage
Mean
per $100 million
0.00009282
$9,282
0.00009912
$9,912
0.00000630
$630
Standard deviation
per $100 million
0.00014797
$14,797
0.00016734
$16,734
0.00001937
$1,937
Table 9 shows similar results for portfolios that preclude short positions.
Table 9: Sampling Error without Short Positions (log wealth utility)
Full Sample
Maximization
Mean-Variance
Approximation
Full-Sample
Advantage
Mean
per $100 million
0.00006653
$6,653
0.00006231
$6,231
0.00000422
$422
Standard deviation
per $100 million
0.00006561
$6,561
0.00006051
$6,051
0.0000051
$510
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.
16
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
17
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.
8
See the appendix for detail.
18
Table 10: Gratuitous Dead Weight Loss with Manufactured Skewness (monthly per $100 million, log wealth)
Correlation
Three
normal
0%
33%
67%
$0
$1
$2
One with
positive
skewness
One with
negative
skewness
Skewness about 1.00
Skewness about 3.50
Skewness about -1.00
Skewness about -3.50
$33
$192
$319
$324
$1,999
$3,480
$43
$323
$894
$656
$5,662
$11,879
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.
Table 11: Gratuitous Dead Weight Loss with Manufactured Kurtosis
(monthly per $100 million, log wealth)
One with
Three
non-normal
Correlation
normal
kurtosis
0%
33%
67%
$2
$2
$9
Kurtosis about 4.50
Kurtosis about 6.00
$1
$16
$57
$0
$49
$173
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 mean9
See the appendix for detail.
19
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 nonquadratic 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
20
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!
21
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
1
 2  Log  x   2
1
f  x |  ,  
e 2
 2 x

e  1 2  e
2
has skewness
2
 and kurtosis e
4 2
 3e 2  2e 3  3 . The
2
2
double gamma distribution
1
1
 1   x
f x |  ,   
x
e
2    
22
has standard deviation  2   2
   4 
  2
and kurtosis
, where   is the
2
 
  2 
gamma function.
23
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.
24