Portfolio Selection Based on Return, Risk, and Relative Performance

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Portfolio Selection Based on Return, Risk, and Relative Performance Author(s): George Chow Source: Financial Analysts Journal, Vol. 51, No. 2 (Mar. - Apr., 1995), pp. 54-60Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4479831Accessed: 15-09-2015 19:26 UTC

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Portfolio Selection Based on Return, Risk

and Relative Performance

George Chow

Markowitz introduced the concept of portfolio selection based on return and variance. Recognition that many investors evaluate performance relative to a benchmark led to the idea of portfolio selection based on return and relative risk. For many investors, both approaches fail to yield satisfactory results. Although they acknowledge the importance of benchmarks in the decision process, they are not indifferent to the variance of absolute returns. In this situation, a utility function that measures return, variance, and tracking error is more appropriate. Analysis of this utility function shows that its set of efficient portfolios includes the mean-variance efficient set, the mean-tracking-error efficient set, and all convex combinations of these two sets. Optimization with this utility function may find solutions that investors will actually use.

An investor selects a single portfolio from an enormous set of possibilities. Portfolio optimi-

zation techniques can assist in the search for the portfolio that best suits each investor's particular objective. Markowitz described an optimization approach that has become today's standard.1 In brief, it assumes that an investor seeks portfolios with high expected returns and low expected risks. In the event that these goals are at odds with each other, the investor has a linear trade-off between portfolio variance and return. Based on this model of investor preference, the objective is to maximize the quantity:

Expected variance Expected return -

Risk tolerance

where risk tolerance sets the price of variance in terms of return.

When applied to asset allocation, the most common result of applying this formulation is that an investor rejects the optimal solution and con- sciously chooses an alternative portfolio. Why does this happen? Perhaps an investor's utility is not based on expected return and expected variance. The investor may compare performance against a benchmark portfolio and be uncomfort- able with large deviations from the benchmark.

Franks described the process of optimization relative to a benchmark. For uncertainty of absolute returns, he substituted uncertainty of returns relative to a benchmark. Harlow explored a different measure of risk, downside risk, from the perspective of relative and absolute performance.3 Vari- ance and semivariance are special cases of his formulation. These variations yield portfolios that are closer to the portfolios investors actually hold.

The substitution of relative risk for absolute risk is too extreme for many investors. Although relative risk may be very important, investors are still concerned with the prospect of losing money. All of the utility functions described above seek to maximize return and minimize one form of risk. I propose that investors seek portfolios with high return, low standard deviation, and low tracking error. This formulation implies that a utility function should measure all of these portfolio characteristics. In support of this proposal, I show that optimization with this utility function generates portfolios that more closely resemble the portfolios investors actually choose than do utility functions that do not take tracking error into account.

A BENCHMARK-SENSMVE UT1LITY FUNCTION The standard mean-variance (MV) utility function is inadequate for investors concerned about portfolios that diverge from the benchmark. For these investors, the utility function should be modified

George Chow is an associate at Windham Capital Management- Boston.

54 Financial Analysts Joumal / March-April 1995



to include dissatisfaction with deviations from the benchmark. One such modification is the addition of tracking error as a parameter in the utility function; that is,

[ExpRisk(P)]2 Maximize U(P) = ExpRet (P) - [ r

rt

[ExpTE(P)]2

tet

subject to 2Pi = 1

Pi > 0 for all i, where

ExpRet(P) = percent expected return of the portfolio

ExpRisk(P) = percent expected standard deviation of portfolio returns

ExpTE(P) = percent expected tracking error of portfolio returns

rt = risk tolerance tet = tracking-error tolerance Pi = proportion of the portfolio allo-

cated to asset i

Given this new utility function, the portfolio optimization process searches for portfolios with high expected return and low portfolio volatility on an absolute basis, as well as relative to the benchmark. Investors define the relative importance of these three goals when they assign risk tolerance and tracking-error tolerance. I refer to this formula as the mean-variance/tracking-error (MVTE) utility function.

To elaborate on the nature of the MVTE utility function, I describe the characteristics of two cases that bound this function. The first case is the subset of utility functions that ignore expected

tracking error. To create this subset, set tracking- error tolerance to infinity. The tracking-error term vanishes, and the MVTE utility function reduces to a mean-variance utility function. The second case is the subset of utility functions that ignore expected risk. The risk term disappears when risk tolerance is set to infinity. In this case, the MVTE utility function simplifies to a mean-tracking-error (MTE) utility function. The MTE utility function is appropriate for investors who accept the inherent volatility of the benchmark and view risk as return uncertainty compared with the benchmark.4

With these two cases defined, I compare them with a portfolio that seeks to avoid both portfolio risk and tracking error. In addition, I identify the domain of MVTE optimal portfolios in risk-return space.

DATA AND RESULTS The portfolio under consideration contains the following asset classes: U.S. stocks, U.S. small stocks, non-U.S. stocks, U.S. bonds, real estate, and cash. The proxies for these asset classes are the S&P 500, Wilshire 4500, MSCI EAFE, Lehman Government Corporate Bond, and Salomon Non- dollar Bond indexes; a portfolio of REITs; and a riskless return series. This information is con- tained in Asset Allocation ToolsTM.5

Expected return, expected standard deviation, expected tracking error, and portfolio weights for each of a series of portfolios are shown in Table 1. These portfolios include the minimum-risk portfolio, Portfolio A, and the minimum-tracking-error portfolio, Portfolio C. Portfolio A, which is 100 percent cash, has no uncertainty in absolute returns and a tracking-error deviation of 10.33 percent relative to the benchmark. To compare the

Table 1. Represetative Portfolios on the MVTE Efficient Surface A B C D E F C H I J

Characteristics MV MV MTE MV MTE MVTE MV MTE MV MTE

Expected return 4.00% 8.50% 8.50% 9.00% 9.00% 9.00% 10.00% 10.00% 11.00% 11.00% Standard deviation 0.00 9.27 10.33 10.30 11.26 10.78 12.56 13.35 15.14 15.76 Tracking-error deviation 10.33 4.56 0.00 4.69 1.15 1.79 6.07 4.02 8.40 7.12 Allocation

Cash 100.0 8.7 5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 International bonds 0.0 8.5 0.0 9.4 0.9 3.5 11.2 2.7 0.0 0.0 International stocks 0.0 21.4 5.0 23.9 7.9 12.7 30.6 14.6 46.4 24.8 Real estate 0.0 9.5 0.0 10.6 1.2 4.1 13.4 4.0 6.5 0.0 Small stocks 0.0 6.7 0.0 7.9 2.2 3.9 15.3 9.6 40.6 41.0 U.S. bonds 0.0 28.0 40.0 29.1 36.0 33.9 7.1 14.0 0.0 0.0 U.S. stocks 0.0 17.3 50.0 19.2 51.8 41.9 22.4 55.0 6.5 34.2

Financial Analysts Journal / March-April 1995 55



MV and MTE frontiers, portfolios from each frontier that have the same expected return are paired together. The expected return values of 8.5 percent, 9.0 percent, 10.0 percent, and 11.0 percent correspond to portfolio pairs (B,C), (D,E), (G,H), and (I,J). Portfolio F has the same expected return as D and E but does not lie on the MV or MTE frontier. It is MVTE optimal for a risk tolerance of 57.8 and a tracking-error tolerance of 25. Notice the extreme difference in portfolio weights between D and E; the allocations for Portfolio F appear to be a compromise between these two portfolios.

Figure 1 shows the surface generated from the set of all MVTE efficient portfolios. The MV and MTE efficient frontiers delineate two of the borders of this surface. The MV efficient frontier starts with the risk-return coordinates of the lowest risk portfolio. From this point, the expected return and expected risk increase. Depending on the choice of benchmark, the expected tracking error may be increasing or decreasing along this frontier. The curve ends with the coordinates of the highest expected return portfolio.

Figure 1. MVTE Efficient Surface

18

14 Surface MTE Frontier

10~~~~~~~~~~~~~~

6

2

0~~~~~~~~~~~~~~ ] 6 ~Low XLows

High

Expected Risk

The MTE efficient frontier starts with the benchmark and joins the MV frontier at the highest expected return portfolio. A line that connects the coordinates of the portfolio with the lowest absolute risk and the portfolio with the lowest relative risk (the benchmark) completes the bound- ary for the MVTE efficient surface. The return, risk, and tracking error of any optimal portfolio generated by the MVTE optimizer lie on this surface.

The top panel of Figure 2 projects this surface onto the two-dimensional space of return and risk; the lower panel is a view of this surface in return- tracking-error space. In both panels, the MV and MTE frontiers mark the boundaries of the MVTE surface. The portfolios labeled D, E, and F in Table 1 are highlighted in Figure 2. Portfolio D has the lowest risk but the highest tracking error. The opposite holds for portfolio E. Portfolio F, a compromise between the two extremes, may be the type of portfolio that investors seek.

OBSERVATIONS The MV optimal portfolio, Portfolio D, has the characteristic allocations that lead investors to reject an MV solution. Specifically, the portfolio has too much in international equities and not enough in U.S. equities. Almost all institutional investors who find this solution decide not to implement it. Portfolios D and E have the same expected return, and Portfolio E has the lowest possible tracking error.

Although an MTE solution is closer to what an investor desires, it may be inadequate in certain instances. In the examples shown in Table 1, the expected return of cash is 4 percent and the expected return of the benchmark is 8.5 percent. Suppose the expected return of cash changes to 8.84 percent. The 5 percent weight of cash in the benchmark changes the benchmark's expected return from 8.5 percent to 8.74 percent. Now, the benchmark and an all-cash portfolio have the same expected return but very different risk and tracking error. The MTE utility clearly prefers the benchmark to an all-cash portfolio because the cash portfolio has tremendous tracking-error risk and does not offer higher returns than the benchmark. The MTE efficient frontier cannot accommodate investors who are willing to bear some tracking error in order to reduce absolute risk. This extreme case points out the fact that although tracking error is an important characteristic ignored by MV utility functions, absolute portfolio uncertainty is an important component that is omitted in MTE utility functions.

ERRORS IN THE FORECASTS Perhaps the mean-variance solution has too much in international equities because its expected return is too high. Perhaps the problem with this optimization example lies in the forecasts as op- posed to the objective. The optimization algorithm finds a portfolio that is optimal for a particular set

56 Financial Analysts Journal / March-April 1995



Figure 2. MV and MTE Efficient Frontiers Ploted In Return-Risk Space and in Return-Trackng-Error Space

Return-Risk Space

12

10

* 8 75

>< 6

4

0 2 4 6 8 10 12 14 16 18

Expected Risk (%)

Return-Tracking-Error Space

12

10

08

6

4 0 2 4 6 8 10

Expected Tracking Error (%)

* MTE O MVTE n Benchmark

of expected return and covariance inputs. The number of inputs is a function of the number of assets. For an n-asset problem, one needs n ex-

n pected returns and E i expected covariances.

For example, a1 problem with six assets re- quires 27 inputs [6 + (6 + 5 + 4 + 3 + 2 + 1)]. The investor must estimate the true expected return and covariance set. Errors in the estimates lead the optimization algorithm to choose a solution that appears optimal but is, in actuality, suboptimal.

Most investors believe that their sets of forecasts contain errors that invalidate the optimal portfolio. To identify forecast errors, they can inspect the optimal portfolio for unusual allocations. An extreme allocation to a particular asset may indicate an error in its forecast of return, risk, or correlation. To correct this problem, the investor can modify the forecasts or constrain the optimizer to search among portfolios devoid of extreme allocations.

Financial Analysts Journal / March-April 1995 57



Estimation error is clearly a significant problem; investors find that small changes in the inputs translate into large changes in the resulting optimal portfolios. Although investors have difficulty discriminating among small differences in the inputs, they rarely are indifferent to the effects of these differences on the resulting output. For example, when an investor tries to forecast the mean return for stocks, he or she may be able to narrow it to a range of 11-13 percent. Optimization with a mean-variance utility function can map a 2 percent change in expected return to a 20 percent change in its asset weight in the optimal portfolio. If, for example, an 11 percent expected return for stocks yields an optimal stock allocation of 40 percent, then a 13 percent expected return could translate to a 60 percent allocation. An MV utility function implies that if a forecast of mean return cannot distinguish between 11 percent and 13 percent, then the investor should be indifferent among stock allocations within the range of 40-60 percent. This inference is not valid for most investors.

The example in this paper shows that, in a situation in which the MV utility function yields portfolios with characteristics that most investors reject, use of the MVTE utility function delivers results that approximate the actual portfolios investors hold.

THE BENCHMARK In the examples above, the benchmark is chosen to reflect the holdings of the average institutional portfolio. A rationale for the use of this benchmark is that investors often compare their results with those in similar situations. Black and Litterman pointed out that the MV utility function can be thought of as an MTE utility function in which the benchmark is a risk-free investment.6 Viewed from this perspective, the MVTE utility function is just a special case of the class of utility functions with two risk benchmarks.

A criticism of this entire analysis is that the logic used in justification of the MVTE utility function is circular. If the benchmark represents the actual portfolio that the investor currently holds, then the optimization exercise is given the solution as part of its inputs. This approach is valid if the investor chooses the benchmark, but it does not apply if the investor selects a different portfolio. In addition, the MVTE utility function allows for calculation of the price for choosing the benchmark in terms of decrease in expected return and increase in expected risk. Upon quantification of

the opportunity cost of holding a portfolio that looks like the benchmark, the investor may decide to increase his or her tolerance of tracking error.

Another issue is whether benchmarks should affect investment decisions. Concern about relative performance often comes at the expense of absolute performance. In certain circumstances, how- ever, the reduction of tracking error is a legitimate component of an investment objective. Consider an endowment fund (labeled ABC) of a private university. The objective of ABC is to meet specific ongoing liabilities and also to fund projects that improve the institution. Higher returns on ABC's portfolio translate into more projects; therefore, ABC wishes to maximize its portfolio's expected return. It also wants to ensure that the assets in the fund can meet ongoing expenses. At the same time, it does not want to upset the university's alumni-a major source of funding-with a report of an investment loss. In short, ABC is risk averse. ABC recognizes that it competes with other universities for students, faculty, and research grants. If the endowments of other universities are able to fund more projects and give out more scholar- ships, then ABC university is at a disadvantage. Assume that no institution has an advantage with respect to its endowment fund. This equality disappears if ABC's portfolio performs poorly relative to the portfolios of the other endowment funds. The situation deteriorates as the better-funded institutions begin to produce superior alumni and faculty-the source of future donations to the endowment fund. To minimize this risk, ABC can reduce the expected tracking error of its portfolio vis-a-vis the composite portfolio of other endowment funds.

Corporations, like universities, have competitors. A company with a pension plan that is less funded than its competitors' plans faces a situation quite similar to that of an endowment fund. A company with a better-funded pension plan relative to its competitors contributes less if the plan is underfunded and can increase pension benefits if the plan is overfunded. In the latter case, this ostensibly altruistic act could increase loyalty, be used as a substitute for wage increases, and put pressure on the competition to give their employ- ees the same deal.

Some pension funds do not care about what the competition is doing. Instead, they worry about fluctuations in required contributions. Sup- pose a portfolio's asset value rises by 15 percent, and because of a significant fall in interest rates, the actuary determines that the value of the fund's

58 Financial Analysts Joumal / March-April 1995



liabilities has grown by 20 percent. The corpora- tion must contribute more money to the fund, despite the portfolio's positive return. The volatility of a pension fund's surplus is much different from that of its assets. In this situation, the appropriate benchmark is the present value of the fund's liability stream.7

MVTE VERSUS A CONSTRAINED MV SOLUTION One alternative to an MVTE utility function is the combination of the MV utility function with constraints on the portfolio weights. These constraints force the optimizer to choose among portfolios that do not have undesirable allocations.

For example, assume that an investor uses the MV utility function and finds the solution is Port- folio D in Table 1. Upon review of the portfolio mix, the investor decides that an allocation in excess of 15 percent to international equities is unacceptable. The investor can change the current upper bound on international equities from 100 percent to 15 percent and run the optimizer again to arrive at a more acceptable portfolio. Portfolio A in Table 2 is the result.

Table 2 The Effect of a 15 Percent Maximum Allocation to Inteatonal Stocks on the Mean-Vriance Soludon

Portfolio A: Risk Tolerance

Identical to Portfolio B: Portfolio D in Force Expected

Characteristic Table 1 Return = 9

Expected return 8.92% 9.00% Standard deviation 10.18 10.35 Allocation

Cash 0.0 0.0 International bonds 17.7 18.3 International stocks 15.0 15.0 Real estate 13.0 13.3 Small stocks 10.0 10.8 U.S. bonds 23.4 21.3 U.S. stocks 20.9 21.3

Another possibility is to impose the constraint on international equities and find the minimum- risk portfolio with the same return as Portfolio D. That solution is Portfolio B in Table 2. Similarly, the investor can change the upper or lower bounds on each asset to remove any unacceptable allocations from the final solution. In this example, the next step could be to impose an upper bound on the amount of international bonds.

The problem with adding constraints is that it implies the investor has a discontinuous utility function. As the optimizer evaluates portfolios that approach a constraint, there is no concern until the constraint is binding. At that point, any violation of the constraint is unacceptable, regardless of its potential beneficial impact on portfolio return and risk. Although this condition may apply in the case that the constraints prevent short positions or leverage, it seems unreasonable for other circumstances. The implication for the example above is that any allocation to international equities less than or equal to 15.0 percent of the total portfolio is of no concern, but a portfolio with a 15.1 percent allocation is not allowed. If 15.1 percent is not tolerable, then many investors feel some anxiety at 15.0 percent. The MVTE utility function allows for a smoother trade-off between the portfolio's tracking error and its risk and return.

The application of constraints to the MV formulation makes their impact on the final solution difficult to quantify-and increasingly difficult as the number of binding constraints increases. For example, assume that an appropriate solution re- quires constraints on the maximum allocation to international stocks, international bonds, and real estate, as well as on the minimum allocation to U.S. stocks. Although the constraints and the objective jointly determine the resulting portfolio, how to quantify the impact of each factor is not obvious. In contrast, the MVTE utility function explicitly defines the relative importance of return, risk, and tracking error through the values of risk tolerance and tracking-error tolerance.

IMPLEMENTATION Perhaps one of the most compelling arguments for optimization with an MVTE utility function is how easily an investor can find an acceptable portfolio. The investor must still estimate the return and covariance for each asset; in addition, the benchmark must be defined. After that, the investor adjusts risk tolerance and tracking-error tolerance in order to search for an acceptable portfolio. The appeal lies in the fact that these adjustments map into portfolio changes that make intuitive sense: A decrease in tracking-error tolerance moves the optimal portfolio closer to the benchmark, and a decrease in risk tolerance shifts the portfolio away from risky assets. From an academic perspective, the appeal of this approach is that the search for an optimal portfolio involves adjustments to param- eters that are investor specific. The forecasts of

Financial Analysts Joumal / March-April 1995 59



asset returns and covariances are not manipulated in order to find an acceptable portfolio.

CONCLUSIONS Investment practitioners have implicitly sent a message that optimizers have limited relevance in real-world investment decisions. One of the best arguments for this assertion is that few investors allocate their assets in the proportions indicated by an optimization analysis. In particular, although many asset allocation solutions call for large com- mitments to international equities and real estate, few investors implement this recommendation. The decision to choose portfolios that the analysis identifies as suboptimal may result from a lack of confidence in the optimization forecasts. Clearly, if the forecasts are wrong, then optimization is irrel- evant.

Even with accurate forecasts, the results will be inadequate if the investor's objective is misspec- ified. The examples in this paper recreate MV efficient portfolios that most investors reject, as well as the MTE efficient portfolios described by others. The MTE utility function seems to yield portfolios that are closer to what investors want,

but its disregard for total portfolio risk is too extreme for many investors. The MVTE utility function is based on a broader set of criteria. In fact, the standard MV case and variations of portfolio immunization and surplus optimization prob- lems are all special cases of this more general formulation.

One benefit of optimization with an MVTE utility function is that it quantifies the relative cost of minimizing tracking error and absolute portfolio risk in units of expected return. Once investors can price these goals, they may decide to modify their tolerance for tracking error or portfolio standard deviation.

The assumption here is that investors know what they like and dislike. Although I describe situations in which an MVTE utility function is appropriate, I do not develop normative arguments as to whether one should have such a utility function. The purpose of the MVTE utility function is to model investor preference accurately and to allow an optimizer to deliver more useful results. The merit of this approach should be judged by whether investors decide to use the output from the optimization analysis.8

FOOTNOTES

1. H. Markowitz, "Portfolio Selection," The Journal of Finance, vol. 7, no. 1 (March 1952):77-91.

2. E. Franks, "Targeting Excess-of-Benchmark Returns," The Journal of Portfolio Management, vol. 18, no. 4 (Summer 1992):6-12.

3. W. V. Harlow, "Asset Allocation in a Downside-Risk Frame- work," Financial Analysts Journal, vol. 47, no. 5 (September/ October 1991):28-40.

4. Roll analyzed the MTE utility function from a mean-variance perspective. He showed that optimization relative to a benchmark can be thought of as combining the benchmark with a specific combination of the mean-variance efficient portfolios. The optimal portfolio for any level of tracking error is a linear combination of the benchmark and a particular self-financing portfolio-a portfolio whose cumulative holdings sum to zero. The set of combinations of the benchmark and this self-financing portfolio traces out an efficient frontier in mean-tracking-error space. Roll's analysis allows for short positions in the optimal portfolio. In this

paper, I do not allow for short positions and thereby find different results. R. Roll, "A Mean/Variance Analysis of Tracking Error," The Journal of Portfolio Management, vol. 18, no. 4 (Summer 1992):13-22.

5. Asset Allocation Tools (distributed by Boyd and Fraser Publishing, Danvers, Massachusetts) is a product developed by William F. Sharpe. The benchmark is 50 percent U.S. stocks, 5 percent non-U.S. stocks, 40 percent U.S. bonds, and 5 percent cash. The optimization exercises are run on AMPL/MINOS, a standard optimization program. AMPL is distributed by KFS Scientific Management Systems, South San Francisco, California.

6. F. Black and R. Litterman, "Global Portfolio Optimization," Financial Analysts Journal, vol. 48, no. 5 (September/October 1992):28-43.

7. M. Kritzman, Asset Allocation for Institutional Portfolios (Homewood, Ill.: Business One Irwin, 1990).

8. I want to acknowledge the valuable comments of Mark P. Kritzman and Jack L. Treynor.

60 Financial Analysts Journal / March-April 1995



Portfolio Selection Based on Return, Risk, and Relative Performance

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