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Mean-variance efficiency when investors are not required to invest all their money
JOURNAL
OF ECONOMIC
THEORY
3,
214-218 (1990)
Mean-Variance Efficiency When Investors Are Not Required to Invest All Their Money HANS EHRBAR* Economics Depurtment, Salt Lake City, Received
March
University Utah 84112
29, 1988; revised
qf Utah.
March
24. 1989
A segment consisting of the lowest-return portfolios on the rising branch of the mean-variance boundary should not be considered efftcient, since the wealth constraint is binding from below rather than from above. A definition of mean-variance dominance is proposed which excludes these portfolios, and a comparison is made with other approaches that also exclude the lowest-return portfolios. Journal of Economic Literature Classitication Numbers: 026, 3 13. ‘( 1 1990 Academic Press, Inc.
Assume portfolio p has expected rate of return rp = 7% with standard deviation rrP= 9%, and portfolio q has expected rate of return ry = 20% with standard deviation gy = 10%. Which portfolio is better? The usual answer is that neither portfolio dominates the other. q has a higher expected return, but it also has a higher variance; therefore it depends on the attitude of the investor towards risk whether he choosesp or q. Neither portfolio dominates the other according to the usual concept of “mean-variance dominance,” which goes back to Markowitz [4, p. 221: I.
DEFINITION
A portfolio
q
ry 3 rp
“dominates” a portfolio and
p
if it satisfies
(r, d up
(1)
with at least one of these inequalities strict. Nevertheless, I will argue that q is unambiguously better than p. Compare two investors, each with $100. One invests all his money in p. At maturity he will obtain a cash flow with mean $107 and standard deviation $9. The other investor spends $90 on portfolio q and uses the remaining $10 for his private consumption. If he likes dramatic gestures, he may even * I thank
Shmuel
Kandel,
Hal Varian,
and an anonymous
214 0022-0531/90
$3.00
Copyright ‘C_I 1990 by Academ,c Press, Inc All rights of reproduclron in any form reserved.
referee
for helpful
hints
MEAN-VARIANCE
215
EFFICIENCY
burn the $10. Still he is better off than the first investor. At maturity he looks forward to a cash flow with mean $108 and standard deviation $9, i.e., he has a higher mean than the first investor, with equal standard deviation. Since the wealth constraint limits the amount investors can spend, but does not prevent their spending less than the full amount, this example suggests that definition I should be replaced by the following criterion: DEFINITION II. Proposed Alternative Definition of Mean-Variance Dominance. A portfolio q “dominates” a portfolio p if there is an investment amount ~30 such that
with at least one of these three inequalities If one assumes that portfolio > - lOO%, then an equivalent, that rp
p is “better than nothing,” i.e., that but mathematically simpler criterion is
and
rq 2 rp
strict.
g <‘+r,
(3)
“‘l+v,“p
with at least one inequality strict. Figure 1 shows the expected rates of return r and standard deviations (T of all portfolios dominating a given portfolio p under Definitions I and II. If Definition II is adopted, only a subset of all formerly efficient portfolios remains efficient. With unlimited short sales and no riskless asset, the expected rates of return and standard deviations of all portfolios with a given net worth w form a region bounded by a hyperbola, as illustrated ,
r
a
II
b I’
FIG. 1. Locus Definition II.
of portfolios
dominating
p: (a) according
to Definition
I; (b) according
to
216
HANS
FIG.
2.
Feasible
EHRBAR
sets for several
values
of H’.
by curve w w in Fig. 2. According to the usual definition, all portfolios on the rising branch of the boundary are efficient. If the alternative definition is used, only those portfolios located at or above the “tangency portfolio” t are efficient. The tangent in t intersects the r-axis at the point r = - lOO%, at which the expected payoff of principal plus interest is zero. Figure 2 shows that the portfolios on the boundary between the “global minimum variance portfolio” g and the tangency portfolio t are dominated by investments which have less than the full endowment u’ invested. (For ease of exposition, a set of portfolios was used with very high standard deviations. In real-world asset markets, the tangent to the feasible set through the origin is almost vertical, therefore portfolios t and g are very close together.) The result that the part of the mean-standard deviation frontier with low expected return, low standard deviation, and a steep slope is not efficient can also be obtained with different assumptions. Baumol [I] argues that few investors would give up several percentage points of expected return in order to gain one percentage point of standard deviation, because this would mean insuring themselves against a potential loss by accepting the same or a greater loss with certainty. Instead of standard deviation, Baumol proposes therefore the lower limit of a 95% confidence interval for the return as a measure of risk. This criterion makes the set of portfolios dominating p almost look like Fig. 1b. The only difference is that the ray ph has a slope of 1.96 (because the 95% confidence interval has a halfwidth of 1.96 standard deviations), instead of having an intercept fixed at - 100%. Portfolios efficient under this definition are also mean-standard deviation efficient, but the efficient set stops where the slope of the boundary exceeds 1.96.
MEAN-VARIANCE
EFFICIENCY
217
Fishburn [3] places Baumol’s argument on a more systematic footing. He emphasizes that the question whether or not returns are bounded, i.e., whether investors can be certain that returns will never be below a certain lower limit, is an important factor affecting their attitude toward standard deviation. If returns are bounded, the general criterion of stochastic dominance for all risk-averse utility functions yields the mean-standard deviation efficient set, again minus a segment containing its lowest-return portfolios. Although both results are very similar to the one presented here, the arguments are quite different. Both worked with alternative definitions of risk, while the present approach depends on the detinition of risk only indirectly. The argument presented here matters whenever risk is defined in such a way that decreasing one’s portfolio may turn out to be advantageous. Baumol’s definition does contain the possibility that the wealth constraint may be binding from below. In the fictitious set of portfolios shown in Fig. 2, in which the slope of the boundary in t is less than 1.96, Baumol’s efficient set would stop at t under an inequality wealth constraint. If the rate of return has a floor of - loo%, however, then it is true for any investment that more is better. In Fishburn’s model, therefore, investors would always invest all their funds. Even if one remains in the usual mean-variance model and maintains the usual Definition I, one can arrive at the conclusion that the portfolios between g and t are not efficient. For this, one only has to specify as one of the options open to investors that they might “burn” their money, i.e., lend it at the riskfree rate of return of - 100%. The shape of the efficient set in the case of riskfree lending but not borrowing has been discussed by Black [Z]. It consists of the straight line through the riskfree return (here - 100%) to the “tangency point” on the rising branch of the boundary hyperbola, plus the part of the hyperbola above this tangency point. This is the envelope of the hyperbolas shown in Fig. 2. The advantage of maintaining the familiar definition of mean-variance dominance, however, has its price. The model requires investors to invest all their money, even if additional investments do not add to their utility, and then compensates for this unrealistic assumption. by offering investors the pathological “investment” opportunity of burning part of their money. It is not only more straightforward to specify the wealth constraint as an inequality constraint, it also allows investors to find better uses for the money they cannot invest.
218
HANS
EHRBAR
REFERENCES J. BAUMOL. An expected gain+onlidence limit criterion for portfolio selection, MaRage .Sci. 10 (1963), 174182. 2. F. BLACK, Capital market equilibrium with restricted borrowing, J. Bus. 45 (1972) 444454. 3. P. C. FISHBURN, Stochastic dominance and the foundations of Mean-variance analysis. Res. Finance 2 (1980), 69-97. “Portfolio Selection,” Cowles Foundation Monograph 16, Wiley. 4. H. MARKOWITZ, New York, 1959. 1. W.
OF ECONOMIC
THEORY
3,
214-218 (1990)
Mean-Variance Efficiency When Investors Are Not Required to Invest All Their Money HANS EHRBAR* Economics Depurtment, Salt Lake City, Received
March
University Utah 84112
29, 1988; revised
qf Utah.
March
24. 1989
A segment consisting of the lowest-return portfolios on the rising branch of the mean-variance boundary should not be considered efftcient, since the wealth constraint is binding from below rather than from above. A definition of mean-variance dominance is proposed which excludes these portfolios, and a comparison is made with other approaches that also exclude the lowest-return portfolios. Journal of Economic Literature Classitication Numbers: 026, 3 13. ‘( 1 1990 Academic Press, Inc.
Assume portfolio p has expected rate of return rp = 7% with standard deviation rrP= 9%, and portfolio q has expected rate of return ry = 20% with standard deviation gy = 10%. Which portfolio is better? The usual answer is that neither portfolio dominates the other. q has a higher expected return, but it also has a higher variance; therefore it depends on the attitude of the investor towards risk whether he choosesp or q. Neither portfolio dominates the other according to the usual concept of “mean-variance dominance,” which goes back to Markowitz [4, p. 221: I.
DEFINITION
A portfolio
q
ry 3 rp
“dominates” a portfolio and
p
if it satisfies
(r, d up
(1)
with at least one of these inequalities strict. Nevertheless, I will argue that q is unambiguously better than p. Compare two investors, each with $100. One invests all his money in p. At maturity he will obtain a cash flow with mean $107 and standard deviation $9. The other investor spends $90 on portfolio q and uses the remaining $10 for his private consumption. If he likes dramatic gestures, he may even * I thank
Shmuel
Kandel,
Hal Varian,
and an anonymous
214 0022-0531/90
$3.00
Copyright ‘C_I 1990 by Academ,c Press, Inc All rights of reproduclron in any form reserved.
referee
for helpful
hints
MEAN-VARIANCE
215
EFFICIENCY
burn the $10. Still he is better off than the first investor. At maturity he looks forward to a cash flow with mean $108 and standard deviation $9, i.e., he has a higher mean than the first investor, with equal standard deviation. Since the wealth constraint limits the amount investors can spend, but does not prevent their spending less than the full amount, this example suggests that definition I should be replaced by the following criterion: DEFINITION II. Proposed Alternative Definition of Mean-Variance Dominance. A portfolio q “dominates” a portfolio p if there is an investment amount ~30 such that
with at least one of these three inequalities If one assumes that portfolio > - lOO%, then an equivalent, that rp
p is “better than nothing,” i.e., that but mathematically simpler criterion is
and
rq 2 rp
strict.
g <‘+r,
(3)
“‘l+v,“p
with at least one inequality strict. Figure 1 shows the expected rates of return r and standard deviations (T of all portfolios dominating a given portfolio p under Definitions I and II. If Definition II is adopted, only a subset of all formerly efficient portfolios remains efficient. With unlimited short sales and no riskless asset, the expected rates of return and standard deviations of all portfolios with a given net worth w form a region bounded by a hyperbola, as illustrated ,
r
a
II
b I’
FIG. 1. Locus Definition II.
of portfolios
dominating
p: (a) according
to Definition
I; (b) according
to
216
HANS
FIG.
2.
Feasible
EHRBAR
sets for several
values
of H’.
by curve w w in Fig. 2. According to the usual definition, all portfolios on the rising branch of the boundary are efficient. If the alternative definition is used, only those portfolios located at or above the “tangency portfolio” t are efficient. The tangent in t intersects the r-axis at the point r = - lOO%, at which the expected payoff of principal plus interest is zero. Figure 2 shows that the portfolios on the boundary between the “global minimum variance portfolio” g and the tangency portfolio t are dominated by investments which have less than the full endowment u’ invested. (For ease of exposition, a set of portfolios was used with very high standard deviations. In real-world asset markets, the tangent to the feasible set through the origin is almost vertical, therefore portfolios t and g are very close together.) The result that the part of the mean-standard deviation frontier with low expected return, low standard deviation, and a steep slope is not efficient can also be obtained with different assumptions. Baumol [I] argues that few investors would give up several percentage points of expected return in order to gain one percentage point of standard deviation, because this would mean insuring themselves against a potential loss by accepting the same or a greater loss with certainty. Instead of standard deviation, Baumol proposes therefore the lower limit of a 95% confidence interval for the return as a measure of risk. This criterion makes the set of portfolios dominating p almost look like Fig. 1b. The only difference is that the ray ph has a slope of 1.96 (because the 95% confidence interval has a halfwidth of 1.96 standard deviations), instead of having an intercept fixed at - 100%. Portfolios efficient under this definition are also mean-standard deviation efficient, but the efficient set stops where the slope of the boundary exceeds 1.96.
MEAN-VARIANCE
EFFICIENCY
217
Fishburn [3] places Baumol’s argument on a more systematic footing. He emphasizes that the question whether or not returns are bounded, i.e., whether investors can be certain that returns will never be below a certain lower limit, is an important factor affecting their attitude toward standard deviation. If returns are bounded, the general criterion of stochastic dominance for all risk-averse utility functions yields the mean-standard deviation efficient set, again minus a segment containing its lowest-return portfolios. Although both results are very similar to the one presented here, the arguments are quite different. Both worked with alternative definitions of risk, while the present approach depends on the detinition of risk only indirectly. The argument presented here matters whenever risk is defined in such a way that decreasing one’s portfolio may turn out to be advantageous. Baumol’s definition does contain the possibility that the wealth constraint may be binding from below. In the fictitious set of portfolios shown in Fig. 2, in which the slope of the boundary in t is less than 1.96, Baumol’s efficient set would stop at t under an inequality wealth constraint. If the rate of return has a floor of - loo%, however, then it is true for any investment that more is better. In Fishburn’s model, therefore, investors would always invest all their funds. Even if one remains in the usual mean-variance model and maintains the usual Definition I, one can arrive at the conclusion that the portfolios between g and t are not efficient. For this, one only has to specify as one of the options open to investors that they might “burn” their money, i.e., lend it at the riskfree rate of return of - 100%. The shape of the efficient set in the case of riskfree lending but not borrowing has been discussed by Black [Z]. It consists of the straight line through the riskfree return (here - 100%) to the “tangency point” on the rising branch of the boundary hyperbola, plus the part of the hyperbola above this tangency point. This is the envelope of the hyperbolas shown in Fig. 2. The advantage of maintaining the familiar definition of mean-variance dominance, however, has its price. The model requires investors to invest all their money, even if additional investments do not add to their utility, and then compensates for this unrealistic assumption. by offering investors the pathological “investment” opportunity of burning part of their money. It is not only more straightforward to specify the wealth constraint as an inequality constraint, it also allows investors to find better uses for the money they cannot invest.
218
HANS
EHRBAR
REFERENCES J. BAUMOL. An expected gain+onlidence limit criterion for portfolio selection, MaRage .Sci. 10 (1963), 174182. 2. F. BLACK, Capital market equilibrium with restricted borrowing, J. Bus. 45 (1972) 444454. 3. P. C. FISHBURN, Stochastic dominance and the foundations of Mean-variance analysis. Res. Finance 2 (1980), 69-97. “Portfolio Selection,” Cowles Foundation Monograph 16, Wiley. 4. H. MARKOWITZ, New York, 1959. 1. W.