The opportunity cost of E-V optimal portfolios.

Tompkins, Daniel L.

ABSTRACT

This paper examines empirical common stock data sets to investigate the opportunity costs of mean--variance optimal portfolios as compared to an investor's direct expected utility maximization. We investigate whether the opportunity cost can be attributed to (1) the investor's utility function, (2) the number of securities included in the data sets, (3) the number of observation used to determine the optimal portfolios, or (4) the investor's preferences for skewness and kurtosis. We have found, that the E-V model provides a very low cost approximation to the actual optimal portfolio. Eighty-one percent of the mean-variance optimal portfolios all the cases have an opportunity cost that equal zero.

INTRODUCTION

Investing has two stages. The first is to determine which assets to invest in and the second is to determine how much should be put in each asset. Markowitz' (1952) developed the Mean Variance model (hereafter, E-V) portfolio optimization model to solve the second stage. Using this model, investors maximize their portfolio by finding the efficient frontier--selecting securities with the highest return for the given amount of risk or the lowest risk for the given amount of return.

However, many argue that those who use the E-V model might not be maximizing their utility. Some argue that for E-V to hold one of two conditions must exist: (1) the individual must have a quadratic utility function, or (2) security returns must be normally distributed. The first condition is argued to be invalid because the quadratic utility function possesses undesirable properties, such as increasing absolute risk aversion. The second condition doesn't hold since securities aren't normally distributed.

Though these conditions may not hold, existing evidence shows that the E-V model accurately approximates an investor's optimal portfolio. Evidence from Levy and Markowitz (1979); Pulley (1981 and 1982); Kroll, Levy, and Markowitz (1984); and Tew, Reid, and Witt (1991) shows that E-V performs well, even when the investor's utility function differs from the quadratic. To test how closely the results of the E-V model approximated the direct expected utility maximization model (EUM) results Levy and Markowitz used the correlation coefficient; Pulley developed the Pulley Score; and Kroll, Levy, and Markowitz developed the KLM Index. However, it has been shown that the Pulley Score is not invariant to a linear transformation of the utility function. Also, the Kroll, Levy, and Markowitz Index is not linearly invariant to a transformation of the naive portfolio.

Another method used to find the goodness of fit for the E-V approximation is Simaan's (1986) Opportunity Cost Approach. This method can be explained as follows: suppose an investor has a choice between the free use of the E-V approximation to the optimal portfolio or the use of a costly direct EUM optimal portfolio. Let E(r) equal the return on the EUM portfolio and E(r') equal the E-V portfolio's expected return. Then it follows that the investor will be indifferent between the choice if:

E (r - O) = E (r') (1)

where O equals the opportunity cost of the E-V solution. Tew, Reid, and Witt (1989) simulated farm data to find the opportunity cost of an E-V approximation. They found that for 76% of their samples, the opportunity cost equaled zero. In only six percent of the cases did the costs exceed $0.00025 and the highest opportunity cost amounted to $0.04175. They concluded that an E-V approximation adequately represented the actual situation.

The purpose of this research is to provide information on the opportunity cost of E-V optimal common stock portfolios. By examining empirical data, the research investigates whether the opportunity cost can be attributed to (1) the investor's utility function, (2) the number of securities included in the data sets, (3) the number of observations used to determine the optimal portfolios, or (4) the investor's preferences for skewness and kurtosis.

DATA AND METHODOLOGY

As previously mentioned, this paper examines the opportunity cost of the E-V approximation to the direct EUM optimal solution. Thus, for each data set we need to find the EUM optimal portfolio and the E-V approximate solution for each utility function. Direct expected utility maximization will be computed with the use of non-linear mathematical programming. The general form of the model to maximize EU:

= SUM (U (SUM [r.sub.jt] [X.sub.j])) (2) subject to,

X, a > b

X > = 0

Sum [X.sub.j] = 1.0

Where [r.sub.jt] is the daily return of asset j for observation t; [X.sub.j] is the percentage of available capital devoted to asset j; n is the number of securities from which a portfolio is selected; U(Sum([r.sub.jt][X.sub.j])) is the value of the utility function for the portfolio of assets for observation t where U is the specific utility function; a is a (q * n) matrix of constraint coefficients. This constraint, a budget constraint, requires that all available funds are spent. Thus, the program is put in the return form. Also, b is the q-dimensional column vector of constraint resource levels; x is the n-dimensional column vector of [X.sub.j,], the weight for each asset in the portfolio. Finally, 0 is a n-dimensional column vector of zeros.

Thus the user specifies the right-hand side of the constraint matrix (-1.0), the return matrix, the upper bounds (maximum investment allowed for investment in security j), the lower bounds (the minimum amount to be invested in each security), and the initial starting position.

We will compare the optimal portfolios obtained through direct utility maximization with the optimal portfolios obtained through an E-V approximation. The algorithm used to determine the E-V optimal portfolio involved several steps. First, it used a linear programming subroutine to solve for the top half of the efficient frontier. The second step involves finding the minimum variance for each expected value. Thus, it traced out the E-V efficient frontier by each combination of E and minimized variance. During this search, the program calculated the expected utility of each of the utility functions to see if it had reached a maximum along the frontier. The solution from this second search was determined to be the optimal E-V utility (denoted as E*U).

In both the direct EUM and the E-V approximation we will find the optimal solutions for several different utility functions. We use the same utility functions as the papers by Levy and Markowitz (1979), Pulley (1981), and Kroll, Levy, and Markowitz (1979). They are:

Logarithmic: U(TR) = LN(TR) (3)

Power: U(TR) = [W.sup-a] (4)

Negative Exponential: U(TR) = [-e.sup.-aW] (5)

where "TR" equals the terminal total return value and "a" equals the risk aversion parameter. We will calculate the risk aversion parameter over the same range as used in the previous studies. As given in Pulley (1981) these ranges for the Power and Negative Exponential functions are:

Power: a = .1, .5, .9

Negative Exponential: a = 1, 2, 3, 4, 5

The objective function for the programming will be from a Taylor's series expansion. To represent investors who consider the mean, variance, skewness, and kurtosis, four forms of the Taylor's series expansion will be used. The first one represents investors who only consider the security's expected return:

F(x) = f(x)/0! +(f'(x)/1!) (x-[x.sub.0]) (6)

The second form represents those who consider the expected return and variance:

F(x) = f(x)/0! + (f'(x)/1!) (x-[x.sub.0]) + (f"(x)/2!) [(x-[x.sub.0]).sub.2] (7)

The third form adds consideration of skewness:

F(x) = f(x)/0! + (f'(x)/1!) (x-x0) + (f"(x)/2!) (x-x0)2 + (f'"(x)/3!)([(x-[x.sub.0]).sub.3] (8)

Finally, the last form considers expected return, variance, skewness, and kurtosis:

F(x) = f(x)/0! + (f'(x)/1!) (x-[x.sub.0])+ (f"(x)/2!) [(x-[x.sub.0]).sup.2] + (f;'"(x)/3!)[(x-[x.sub.0]).sup.3] + f""(x)/4!)[(x-[x.sub.0]).sup.4] (9)

Thus, after the user inputs the relative risk aversion level (a) and the initial weights, the procedure checks to find out what the value of the objective function is (it seeks to maximize the investor's utility). Then, through an iterative process, it changes the weight of each security. With each iteration, it calculates what the value of the objective function is, as it tries to find the weights that provide the highest utility to the investor.

In collecting the data we had several considerations. We wanted to examine the issue empirically.. We also wanted to examine whether the number of securities or observations, or the investor's preferences for skewness or kurtosis provides for differences in the opportunity cost. The empirical data sets are taken from the returns of those firms that make up the Standard & Poor's 500 Stock Index. We chose these securities for several reasons: (1) many large institutional investors are restricted to investing in these securities because of size or other constraints such as the Prudent Investor rule, and (2) Since these are among the largest securities (in terms of market value), they are among the most liquid securities. The time period for the data was chosen randomly. We gathered daily returns, including dividends, from Nov. 1987 to Dec. 1988.

To examine if the number of securities makes a difference in the opportunity cost of the E-V approximation, we vary the number of firms in the data sets. Though they have not explicitly explored this issue, other studies such as Burgess and Bey (1988), have also varied the number of firms included in the data set. We randomly selected firms for portfolios of size ten (an amateur portfolio), twenty, thirty, and forty firms (a small professional portfolio). Since as many as three portfolios are selected randomly for each size, the portfolios are not necessarily mutually exclusive. We also examine whether the number of observations will matter. To examine this issue, we vary the number of observations for each firm in the portfolio. We will use data sets with 100, 200, and 300 observations. We have chosen these observations for the following reasons. First, as Lau and Wingerter (1989) point out, a problem with calculating skewness and kurtosis is the large sampling errors. With less than a few hundred observations, Lau and Wingerter say, skewness and kurtosis may have misleading and erratic behavior. And, as Ederington (1988) mentions, using a small sample of annual returns may not include any observations from the tails of the distribution. However, by expanding a data set of annual returns (if enough data would be available), Ederington adds there may be a problem in that the distribution has changed from the original one. Thus to get large enough samples from within the same distribution, we have chosen to use daily data. Table 1 shows the number of firms and observations that are used in each of the data sets.

The measurement method to determine the quality of the E-V solution is the opportunity cost approach. For a constant relative risk aversion function, we can directly solve for the opportunity cost. For other utility functions we employ a search routine to find the opportunity cost. The search routine involves specifying the returns for the data set, the E-V and EU portfolio weights for each asset, the utility function, risk aversion level, and initial value for the opportunity cost. The routine calculates and compares the value of EU(R-C) and EU(R'). Then it adjusts the value of C in small increments until the difference between EU(R-C) and EU(R') becomes negligible.

RESULTS

The results show that generally E-V provides a very good approximation of the direct EUM optimal portfolio. Table 2 provides the opportunity costs of the E-V approximation for the data sets. Eighty-one percent of the portfolios have an opportunity cost of zero.

Next, we broke the results down to examine whether the investor's utility function would account for any differences in the opportunity costs. As seen in Table 3, the utility function does not make a significant difference in the opportunity cost of the E-V portfolio. Though the results for the negative exponential utility function with a risk parameter of 1, have the lowest percent of cases with an opportunity cost of zero (fifty-nine percent), it still has a high percentage with an opportunity cost of $0.0000025 or less (eighty-two percent). And all of the other utility functions have higher percentages of cases with an opportunity cost of $0.0000025 or less (and greater percentage of case with a zero costs).

Since one criticism of E-V is that it assumes the investor only considers a security's mean and variance, we wanted to see if inclusion of skewness and kurtosis would result in a larger opportunity cost for the E-V optimal portfolio. Table 4 summarizes the opportunity cost results as a function of the number of moments included in the utility function. The results show that the E-V approximations with the fewest cases with a zero opportunity cost are the utility functions that only include the first moment. For the first moment investors, seventy-f our percent of the cases have an opportunity cost of zero. However, eighty-eight percent of the cases for the first moment category have an opportunity cost of $0.0000025 or less per dollar invested, similar to the other categories.

With ten percent of the cases having an opportunity cost more than $0.00025 per dollar invested or more, the utility functions that contain the first three moments have the most cases in the highest cost category. However, the other ninety percent of the cases have opportunity costs of $0.0000025 or less, similar to the other categories. The E-V approximation performs best for the utility functions that include all four moments. In this category, eighty-three percent of the cases have an opportunity cost of zero with an additional nine percent having an opportunity cost of $0.000025 or less.

Table 5 breaks down the results by the number of securities included in the data sets. This table shows there is little difference among the ten, twenty, and forty securities sets. These categories respectively have ninety-six, ninety-five, and one hundred percent of their cases with an opportunity cost of $0.000025 or less. The twenty securities category has ninety-two percent of its cases with an opportunity cost of zero. The forty securities category is close behind with ninety percent of the cases with an opportunity cost of zero. The ten security data set has a total of seventy-nine percent of its cases with an opportunity cost of zero. Though most of the thirty security cases (sixty-six percent) have an opportunity cost of zero, twenty-nine percent of the cases have a cost between zero and $0.00025 per dollar invested. In general, the results show that the E-V model results in a low cost approximation of the investors direct expected utility maximization portfolio at a low cost.

Finally, we examined the empirical results of the opportunity cost as a function of the number of observations. Table 6 summarizes this information. The 200 and 300 observation categories each provide E-V approximations with very low opportunity cost. Respectively, Ninety-seven and ninety-five percent of the cases have an opportunity cost of $0.0000025 or less. Eighty-three percent of the cases for the 200-observation category have an opportunity cost of zero, with just two cases having a cost of $0.00025 or more. For the 300-observation category, ninety-three percent of the cases have a cost of zero, with a single case that has an opportunity cost of $0.00025 or more. The 100-observation category has a majority of cases that have an opportunity cost of zero, sixty-nine percent; however, a greater percentage of cases, sixteen percent, have an opportunity cost of $0.00025 per dollar invested or higher.

CONCLUSIONS

We have used the opportunity cost approach to examine common stock portfolios to evaluate how closely the E-V model approximates the direct EUM optimal portfolio. We find that the E-V model provides a very low cost approximation to the actual optimal portfolio. Eighty-four percent of all the cases have an opportunity cost that equal zero.

These results are similar to the results reported by Tew, Reid, and Witt, as well as the other studies using different comparison methods. Thus we conclude that investors can use the E-V model and closely approximate their actual optimal portfolio.

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Daniel L. Tompkins, Niagara University

TABLE 1:
Number of Firms and Observations

Data Set number of firms in portfolio number of observations

1 10 100
2 10 200
3 10 300
4 20 100
5 20 200
6 20 300
7 30 100
8 30 200
9 30 300
10 40 100
11 40 200
12 40 300

Table 2:
Summary of Opportunity Cost Results

Opportunity Cost

 0.00000 to 0.0000025 to greater
0.00000 0.0000025 0.00025 than 0.00025 Totals

321 (81%) 34 (8%) 6 (2%) 35 (9%) 396 (100%)

Opportunity Cost is per dollar invested.

Table 3:
Summary of Opportunity Cost Results As a Function of Investor's
Utility Function

 Opportunity Cost

Utility Function
Negative 0.00000 0.00000 to 0.0000025 to
Exponential 0.0000025 0.00025

1 26 (59%) 10 (23%) 3 (7%)
2 30 (68%) 7 (16%) 3 (7%)
3 30 (68%) 9 (20%) 0 (0%)
4 39 (89%) 1 (2%) 0 (0%)
5 41 (93%) 0 (0%) 0 (0%)

Power

.1 40 (91%) 0 (0%) 0 (0%)
.5 40 (91%) 0 (0%) 0 (0%)
.9 40 (91%) 0 (0%) 0 (0%)

Logarithmic 35 (80%) 7 (16%) 0 (0%)

Totals 321 (81%) 34 (8%) 6 (2%)

 Opportunity Cost

Utility Function
Negative greater than Totals
Exponential 0.00025

1 5 (11%) 44 (100%)
2 4 (9%) 44 (100%)
3 5 (11%) 44 (100%)
4 4 (9%) 44 (100%)
5 3 (7%) 44 (100%)

Power

.1 4 (9%) 44 (100%)
.5 4 (9%) 44 (100%)
.9 4 (9%) 44 (100%)

Logarithmic 2 (4%) 44 (100%)

Totals 35 (9%) 396 (100%)

Opportunity Cost is per dollar invested.

Table 4:
Summary of Opportunity Cost Results As a Function of Number of
Moments Included in Utility Function

 Opportunity Cost

Number of 0.00000 0.00000 to 0.0000025 to
Moments 0.0000025 0.00025

1 73 (74%) 14 (14%) 5 (5%)
2 81 (82%) 8 (8%) 1 (1%)
3 83 (84%) 6 (6%) 0 (0%)
4 84 (85%) 6 (6%) 0 (0%)
Totals 321 (81%) 34 (8%) 6 (2%)

 Opportunity Cost

Number of greater than Totals
Moments 0.00025

1 7 (7%) 99 (100%)
2 9 (9%) 99 (100%)
3 10 (10%) 99 (100%)
4 10 (9%) 99 (100%)
Totals 35 (9%) 396 (100%)

Opportunity Cost is per dollar invested.

Table 5:
Summary of Opportunity Cost Results As a Function of Number of
Securities Included in Data Set

 Opportunity Cost

Number of 0.00000 0.00000 to 0.0000025 to
Securities 0.0000025 0.00025

10 85 (79%) 19 (17%) 0 (0%)
20 100 (92%) 3 (3%) 5 (5%)
30 71 (66%) 5 (4%) 1 (1%)
40 65 (90%) 7 (10%) 0 (0%)
Totals 321 (81%) 34 (8%) 6 (2%)

 Opportunity Cost

Number of greater than Totals
Securities 0.00025

10 4 (4%) 108 (100%)
20 0 (0%) 108 (100%)
30 31 (29%) 108 (100%)
40 0 (0%) 72 (100%)
Totals 35 (9%) 396 (100%)

Opportunity Cost is per dollar invested.

Table 6:
Summary of Opportunity Cost Results As a Function of Number of
Observations

 Opportunity Cost

Number of 0.00000 0.00000 to 0.0000025 to
Observations 0.0000025 0.00025

100 100 (69%) 12 (6%) 0 (0%)
200 120 (83%) 20 (14%) 2 (1.5%)
300 101 (93%) 2 (2%) 4 (4%)
Totals 321 (81%) 34 (8%) 6 (2%)

 Opportunity Cost

Number of greater than Totals
Observations 0.00025

100 32 (16%) 144 (100%)
200 2 (1.5%) 144 (100%)
300 1 (1%) 108 (100%)
Totals 35 (9%) 396 (100%)

Opportunity Cost is per dollar invested.