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Mean-variance analysis of portfolios of dependent investments: An extension
J ECO BUSN 1988; 40:147-157
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Mean-Variance Analysis of Portfolios of Dependent Investments: An Extension Mario H. Pastore
This paper extends portfolio-choice theory under risk aversion to the case of dependent investments. R establishes the circumstances in which optimal portfolios of dependent investments will be diversified or completely specialized, and reevaluates earlier statements in this connection arrived at without reference to risk preference. The model, which assumes two risky assets, focuses first on the normal case, in which one asset has both a higher mean and variance of expected returns and, second, on the perverse case, in which one of the two assets is both more profitable and less risky than the other. In the first case diversified portfolios are the rule irrespective of the degree of correlation between assets; completely specialized portfolios may only obtain as corner solutions generally resulting from low-risk aversion; complete specialization obtains when there is high-risk aversion only in the limiting normal case of equally profitable but unequally risky assets. In the second, or perverse case, a wider variety of diversified portfolios than previously thought is found to be optimal. Specifically, diversification may be optimal even if one security has an extremely higher yield and lower variance of returns than the other, provided asset correlation is sufficiently low or negative. There will be complete specialization in the more profitable and less risky asset only if the correlation coefficient between asset returns assumespositive values in an interval that will be narrower or wider depending on the assumed degree of risk aversion and other considerations.
I. Introduction When is it optimal for risk-averse individuals to hold a diversified or a specialized portfolio of dependent investments? Markowitz (1952) first stated that the right diversification would, in general, decrease the variance of expected returns of portfolios below that of any of the included assets "for
An earlier version of this paper was presented at the 36th annual convention of the New York State Economics Association, Syracuse, New York, April 1984. Assistant Professor, Department of Economics, Ithaca College. I would like to thank Subodh Mathur, Gerard Caprio, Pham Chi Thanh and an anonymous referee for their comments and suggestions, as well as Dorothy Owens for her editorial and typing assistance. Address reprint requests to Mario H. Pastore, 118 Delaware Avenue, Ithaca, NY 14850. Journal of Economics and Business © 1988 Temple University
0148-6195/88/$03.50
148
M.H. Pastov~:~ a large, presumably representative range of means, variances and covariances ol expected returns." In this view, an undiversified portfolio would not be preferred to diversified one unless one of the securities had an extremely higher expected yield and lower variance of returns than all of the others. Bierman (1968) reexamined the question sometime later, concluding that an individual may prefer a portfolio completely specialized in the less risky of two unequally risky assets, even if there is no difference between the expected returns of the two assets. All that is required is for the correlation coefficient between expected returns to be in a certain interval at the positive end of its allowed range of values. Neither Markowitz nor Bierman, however, took into account individual preference for return and risk. The question arises, therefore, whether their conclusions continue to hold when such preferences are explicitly introduced. In this paper I analyze the optimal portfolio choices of risk-averse individuals in order to ascertain the circumstances under which they will prefer to hold a diversified or, alternatively, a specialized portfolio of dependent investments. I restrict the analysis to the simple case of two risky assets. I find that on the assumption of a sufficiently high degree of risk aversion and in the normal case in which higher returns can be obtained in the market only at the cost of higher risk, a diversified portfolio will be optimal regardless of the degree of correlation between the expected returns of the included assets, as already established by Samuelson (1967). However, portfolios completely specialized in the more profitable asset can obtain, but only as corner solutions, for the most part the result of low-risk aversion. Only in the limiting case of two equally profitable but unequally risky assets analyzed by Bierman does the portfolio's complete specialization in the less risky asset result from a corner solution unrelated to low-risk aversion. On the other hand, in the perverse case, in which one of the two assets is both more profitable and less risky than the other, complete specialization in the first of the assets does not necessarily follow, as Markowitz had supposed; a wide variety of diversified portfolios can be optimal even in this case, so long as asset correlation is sufficiently low or negative and the more profitable asset is not riskless. Conversely, the optimal portfolio will be specialized in the more profitable and less risky asset if the latter is riskless; if it is not, the optimal portfolio will be specialized in the less risky asset if the correlation coefficient assumes positive values falling in an interval which, depending on the assumed degree of risk aversion and other considerations, may be wider or narrower. Unless these conditions obtain, however, the optimal portfolio of dependent investments is a diversified portfolio in the perverse case as well. The analysis produced two main conclusions other than specialization in the normal case pays only if risk aversion is too low. First, in the limiting normal case, analyzed by Bierman, a completely specialized portfolio continues to obtain after risk aversion is assumed. Both this and the remaining completely specialized portfolios that obtain in the normal case result from corner solutions, although the latter are due to low-risk aversion. Second, and perhaps more importantly, the principle of portfolio diversification is found to be applicable not only in the normal case but in a wider range of instances of the perverse case than previously believed as well. The completely specialized portfolios that obtain in this case are also the result of corner solutions, although these are not necessarily due to low-risk aversion. Mathematical proof of the above propositions is presented in Sections II and III that, respectively, present the model and its comparative statistics. The results are graphically derived in Section IV, to which readers interested in a more intuitive, but quicker, presentation may turn.
Portfolios of Dependent Investments
149
II. The Model Two assets, X and Y, can be included in the portfolio; each is assumed to have a distribution of expected returns known to the individual, so that R ~ - ( # x , Ox 2) and
Ry (Izy, 02). e~
An investment fund I may be allocated in some proportion between X and Y. Let a be the proportion of I allocated to X, (1 - o0 the proportion of I allocated to Y. A portfolio consisting of X and Y will have an expected mean return given by # p - c~/zx+ (1 -a)#y.
(1)
The variance of expected returns from the portfolio-will be 0.2#=or 2 0 2. + ( 1 - o 0 2 @ + 2 r o f f l - o 0 o x 0 . y .
(2)
Suppose now, with Tobin (1958), that a function in #o and 0.o such a s f ( R ; #o, 0.o) can describe the probability distribution of returns. Then, U ( R ) f ( R ; #o, oo).
E[U(R)] =
(3)
--0o
Normalizing R one obtains Z
R - #o 0.o
~
t
which, substituted in Equation (3) above, yields
E[U(R)]=E[~o, 0.o1=
f"
-co
U(~o+oo.z)f(z;
O, 1)
dz.
(4)
Differentiating Equation (4) with respect to e o, one obtains a mean-variance indifference curve along which expected utility is constant:
® e'(#o+0.o.Z ) [k--d-~a d#°" +z]j f(z; o, O= fj_•
1) dz
or
d#o dop
I~* zU'(R)f(zO,
=
1)
dz
-o0
>0
I** U'(R)f(z; O, 1) dz -oo
for a risk averter. Assume a quadratic utility function such as
U(R) =(1 +b)R+bR2>_O with - 1 < b < 0 for risk aversion and R _< - 1 +
(5)
b/2b required for marginal utility to
150
M.H. Pastore be positive. ~ Given Equation (5), we may investigate the implied slope and curvature of the indifference curve. So far as the slope is concerned, from Equation (3) we get
i
E[U(R)]=
oo
U(R)f(R)
d R = ( l + b)/to+ b(o~+ #2p).
Holding E[ U(R)] constant and differentiating with respect to %, the slope of the riskreturn indifference curve obtained is d/xp dop
vo
(6)
l+b 2b
#o
For proof that for a risk averter the second derivative is positive and the indifference locus is concave upwards, see Tobin (1958). Recall now the expressions for the mean and variance or the portfolio given in Equations (1) and (2). Since for a given r, #p and 0 .2P c a n only vary with ct, differentiating Equations (1) and (2) with respect to ot and taking the ratio of the two derivatives we obtain from Equation (1) d tzp
do~
=/~x -/& = A #
and from Equation
(7)
(2)
d % _ ~ 0 2 - (1 - ct)ay2 + roxoy - 2rotoxay da
oo
(8)
From Equations (7) and (8) d#p dap
% • A# a a x2 - (1 -- or)a2 + rtrxay- 2raxoy
(9)
which gives the increase in expected mean returns that accrues on assumption of greater risk. 2 Equating (6) and (9) to solve the constrained maximum problem, and solving for c~ yields a -rff ct = ~ B-2ra
(10)
Feldstein (1969) pointed out that one must necessarily assume a quadratic utility function or a normal distribution of returns if the individual is to be able to consistently rank risk-return combinations, a prerequisite for theorizing on portfolio selection on the basis of utility maximization. 2 Tobin (1958) showed that the inclusion of a risldess asset permits linearization of the risk-return trade-off available to the individual. If more than two assets are considered, the linear frontier may still be preserved if at least one of the assets is riskless. When all assets are risky, however, this is no longer the case. The frontier then becomes a hyperbola (Renshaw (1972)).
Portfolios of Dependent Investments
151
where, A=B
l+b 2---b- " AP'-#Y " A # + ° 2
(11)
= %2+ ~y+ 2 (A~)2
02)
and ~ = axtry.
(13)
A, B, and a, in Equations (11), (12) and (13) are all constants. Therefore, ot in Equation (10) is a function of r alone. Substituting Equation (10) into Equations (1) and (2), we obtain the values of the portfolio's expected mean return and variance of returns that maximize the individual's expected utility:
A-ra ( A-ra ~ #x+ 1 IZ, - B _ 2r ~ K_"~-ro/ l~y *-
(1')
and
e2*=(A-r°~2 ( A-ra'~ 2 2 P \B-2ra] °2x+ I B-2raJ ay
:A-ro 6 +2r \~-2-~a]\
B-2ra]
a.
(2')
We can now ask whether portfolio specialization may reduce risk in the normal case, in which more profitable assets are available at the expense of greater risk, that is, we ask whether a = 0 or ot = 1 for any value(s) of r. The portfolio will be completely specialized in Y, the more profitable and risky asset, if ot = 0, which can be seen to obtain from (10), if
A -ra=O that is, if l+b - 2---if- • A/~-/zy • A# + a 2 - rtr= 0.
(14)
For given coefficient of risk aversion, expected mean and expected variance of returns, Equation (14) obtains if l+b
A#
2b
a
- l <_r= - - -
<0
a >0
ax
(15)
>1
For b sufficiently close to zero (sufficiently steep risk-return indifference curves) the first term in Equation (15) is negative and greater in absolute value than the positive sum of the last two terms; therefore, the r for which ot = 0 is not in the interval [ - 1; + 1], and no specialization in Y can therefore take place. All optimal solutions yield diversified portfolios. Even if b is close to - 1 (flat risk-return indifference curve), tangency conditions may obtain for all r if the difference between expected means is small and the difference between variances is large. In this case, the last term in Expression (15) guarantees that
152
M.H. Past~tc the r that will make c~ --- 0 lies outside the interval [ - 1; + 1 l. On the other hand, il the difference between expected mean returns is large and that between expected variances is small, tangency conditions will not be feasible for b close to - I, and the portfolio may be specialized in Y. This, however, does not invalidate the conclusion of the previous paragraph, for the specialized portfolio is due to an insufficient degree of risk aversion. If portfolio specialization in Y does not occur, the portfolio cannot specialize in X either. Barring insufficient risk aversion, therefore, the optimal portfolio in this case is a diversified portfolio. Consider now Bierman's limiting case, referred to in the introduction. Bierman considered the case of two equally profitable but unequally risky assets, that is, At* = 0, o,. < o~,. In this case, Equation (15) becomes - 1 a g r = ° ~ $ t, ox since ay > ox. Consequently, no r in [ - 1; + 1] can make c~ = 0, in which case the portfolio would not be specialized in Y. Does the portfolio become specialized in X ? It does if a = 1, which, from Equation (10), obtains when -l_r
l+b At* --~" "'&tz+(A#)2+°~
(16)
When a # = 0, (16) reduces to Ox
- l_
specialized in X. This is also the result obtained by Bierman w h e n h e considered the c a s e without reference to risk-return preferences. Evidently, the results are the same b e c a u s e the term in Equation (16) containing b, the coefficient of risk aversion, disappears from the expression. Bierman's findings, therefore, are seen to result as a particular case of the more general problem studied here. 3 Consider now the perverse case, in which one asset is both more profitable and less risky, that is, tZx
.
1 +b
.
.
2b
At*
.
A~
#y " - - + ° Y < I .
o <0
o >0
(15')
o~ <1
3 Introducing the assumption of risk aversion, however, does alter some of Bierman's results. In particular, he asserted that if the correlation coefficient between two equally profitable but unequally risky assets is equal to - 1, the investment fund would be equally distributed between both assets, that is, cx = t/2. However, if r = - 1, tzx * txy and o2 = ay, we obtain
Gr
wMch is not equal to 1/2 in general. That is, under conditions of risk aversion, when the expected mean returns of the investment assets are different, but perfectly negatively correlated, and the risk attached to each asset is the same, the optimal portfolio is not a perfectly diversified one.
Portfolios of Dependent Investments
153
It is clear, therefore, that the same conclusion derived for the normal case can be derived in the perverse case, except that it applies to a more restricted interval of r. Conversely, there is, in the perverse case, a certain interval of admissible values of r for which the portfolio is completely specialized. In the next section, we turn to determining the interval of r for which that is the case.
III. T h e C o m p a r a t i v e Statics o f the M o d e l How will the optimal portfolio composition change as the covariance of returns between assets increases? The manner in which a risk averse, utility maximizing individual will modify his choice of portfolios as the correlation coefficient varies can be deduced by differentiating Equation (10) with respect to r. After some manipulation we obtain dot
(2A - B)
dr - ( B - 2ro) 2
(17)
or, alternatively,
dot
o
\ i-0(18)
(B-2ro) 2
dr
2 where A V = oy2 _ o x. The sign of Equation (18) depends on the sign of the term between brackets in the numerator. In the normal case, A/~ < 0 and A V = 0 y2 - a x2 > 0. Therefore, >0
dot
o
<0
AV---.
2b
dr
<0
<0
A~+ 2 • A~_(A#)2 ( B - 2roxo~) z
For A V and b sufficiently close to zero, dot~dr < 0, and as r rises, the proportion in X falls. However, it is possible that the first term of the expression between brackets will outweigh the others, in which case dot~dr > 0 and the portfolio will become relatively more specialized in X. That is, it is clear that in the normal case one cannot unambiguously state that the portfolio will become progressively more specialized in X or Y as r rises. In the perverse case, however, since A V < 0, dot~dr is unambiguously less than zero, and the portfolio becomes progressively more specialized in Y, the less risky asset. In this case, provided - 1 _ A / o _< 1, which is clearly possible, ot = 0 if(A - r o ) / ( B - 2ro) = 0 or, i f r = A / o . Since dot/dr is less than zero, for r > A / o the portfolio must remain completely specialized in Y. Therefore, ot = 0, that is, the portfolio is completely specialized in Y, for all r in [ A / o ; 1]. The result is the same in Bierman's case, that is, specialization in the less risky asset, that this time is X, again takes place. If in Equation (16) we assume A# = 0, dot
o[A V]
dr
04
154
M.H. Pastorc
/!A a
x
a
y
p
Figure 1. Limiting normal case, Portfolio specialization with high-risk aversion and unequally risky, but equally profitable assets.
which is positive or negative depending on the sign of A 1I. In the normal case, A V > 0, and da/dr > 0, that is, as r increases, the portfolio becomes progressively more specialized in the less risky asset X. Since, in this case, a = 0 for r = ox/oy, as r rises above this value, the portfolio must continue to be completely specialized in X, so that complete X specialization obtains for r in the interval [ax/Oy; 1].
IV. Graphical Hlustration Practically every finance text graphically illustrates the optimal portfolio chosen in the normal case under conditions of risk aversion, illustration that for reasons of space we do not reproduce here. The tangency conditions that in this case must hold for the constrained maximum problem to have a solution easily obtain, provided risk aversion is not too low, for Tobin's risk-return indifference curves and the efficient portfolio contours all slope upward. However, if risk aversion is too low, only corner solutions are possible, and the individual will opt for a portfolio specialized in the more profitable and risky of the two assets, since among corner solutions rational individuals will prefer that which yields the highest utility level. If both assets are equally profitable but unequally risky, the case studied by Bierman, it is intuitively obvious that portfolio specialization in the less risky asset will take place. In addition, graphical analysis shows that the efficient portfolio contours reduce to a horizontal line, as depicted in Figure 1, so that tangency conditions will not obtain unless
Portfolios of Dependent Investments
lapl .
lay
.
.
.
.
.
.
155
.
.
.
.
la x cr
y
CT
x
(I P
Figure 2. Perverse case. Portfolio diversification with low correlation of asset returns.
the marginal rate of risk-return substitution is itself zero. For any marginal rate of riskreturn substitution greater than zero, only corner solutions will obtain and, again, the rational individual will opt for that solution which yields the highest utility level, given by a portfolio specialized in the less risky of the two assets. Therefore, complete specialization in this asset occurs not only under the conditions Bierman envisioned, but also when risk averse preferences are introduced. In fact, Bierman's is a particular case of the general one studied here. However, the graphical analysis does not reveal the exact r-interval for which portfolio specialization occurs. For that, one must resort to the preceding mathematical analysis which shows that the r-interval in question is ; 1
,ox
Turning now to the perverse case, depicted in Figure 2, below, in which one asset is more profitable and less risky than the other, it is clear that there are two sets of efficient portfolio contours, one upward sloping, the other downward sloping. Clearly, diversified optimal portfolios are feasible for this upward sloping set of contours, but not for the set of downward sloping contours. Figure 3 isolates the set of downward sloping efficient portfolio contours. As long as
156
M.H. Pastorc
k~
~y
(7
Y
(7
x
Figure 3. Perverse case. Portfolio specialization for highly positively correlated asset returns.
risk aversion is assumed, tangency conditions cannot obtain in the r-interval I(
l+b _
2b or, what is the same, [ A / a ; 1]. In this interval, only corner solutions will result, no matter what the degree of risk aversion, and rational, risk-averse individuals will opt for that solution which yields the highest utility level, that is, a portfolio specialized in the less risk and more profitable asset. Unless the more profitable asset is riskless, however, tangency solutions will still be possible in the r-interval [ - 1; ax/Oy) and diversified portfolios will result even if one of the assets is far more profitable and less risky than the other.
References Bierman, H. 1968. Using investment portfolios to change risk. Journal o f Financial and Quantitative Analysis 3:151 - 157.
Feldstein, M. 1969. Mean-variance analysis in the theory of liquidity preference and portfolio selection. Review o f Economic Studies 36:5-12.
Portfolios of Dependent Investments
157
Markowitz, H. 1952. Portfolio selection. Journal of Finance 7:77-91. Renshaw, E. F. 1967. Portfolio balance models in perspective: Some generalizations that can be derived from the two asset case. Journal of Financial and Quantitative Analysis 2:123-140. Samuelson, P. A. 1967. General proof that diversification pays. Journal of Finance and
Quantitative Analysis 2. Tobin, J. 1958. Liquidity preference as behavior toward risk. Review of Economic Studies 25:65-86.
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Mean-Variance Analysis of Portfolios of Dependent Investments: An Extension Mario H. Pastore
This paper extends portfolio-choice theory under risk aversion to the case of dependent investments. R establishes the circumstances in which optimal portfolios of dependent investments will be diversified or completely specialized, and reevaluates earlier statements in this connection arrived at without reference to risk preference. The model, which assumes two risky assets, focuses first on the normal case, in which one asset has both a higher mean and variance of expected returns and, second, on the perverse case, in which one of the two assets is both more profitable and less risky than the other. In the first case diversified portfolios are the rule irrespective of the degree of correlation between assets; completely specialized portfolios may only obtain as corner solutions generally resulting from low-risk aversion; complete specialization obtains when there is high-risk aversion only in the limiting normal case of equally profitable but unequally risky assets. In the second, or perverse case, a wider variety of diversified portfolios than previously thought is found to be optimal. Specifically, diversification may be optimal even if one security has an extremely higher yield and lower variance of returns than the other, provided asset correlation is sufficiently low or negative. There will be complete specialization in the more profitable and less risky asset only if the correlation coefficient between asset returns assumespositive values in an interval that will be narrower or wider depending on the assumed degree of risk aversion and other considerations.
I. Introduction When is it optimal for risk-averse individuals to hold a diversified or a specialized portfolio of dependent investments? Markowitz (1952) first stated that the right diversification would, in general, decrease the variance of expected returns of portfolios below that of any of the included assets "for
An earlier version of this paper was presented at the 36th annual convention of the New York State Economics Association, Syracuse, New York, April 1984. Assistant Professor, Department of Economics, Ithaca College. I would like to thank Subodh Mathur, Gerard Caprio, Pham Chi Thanh and an anonymous referee for their comments and suggestions, as well as Dorothy Owens for her editorial and typing assistance. Address reprint requests to Mario H. Pastore, 118 Delaware Avenue, Ithaca, NY 14850. Journal of Economics and Business © 1988 Temple University
0148-6195/88/$03.50
148
M.H. Pastov~:~ a large, presumably representative range of means, variances and covariances ol expected returns." In this view, an undiversified portfolio would not be preferred to diversified one unless one of the securities had an extremely higher expected yield and lower variance of returns than all of the others. Bierman (1968) reexamined the question sometime later, concluding that an individual may prefer a portfolio completely specialized in the less risky of two unequally risky assets, even if there is no difference between the expected returns of the two assets. All that is required is for the correlation coefficient between expected returns to be in a certain interval at the positive end of its allowed range of values. Neither Markowitz nor Bierman, however, took into account individual preference for return and risk. The question arises, therefore, whether their conclusions continue to hold when such preferences are explicitly introduced. In this paper I analyze the optimal portfolio choices of risk-averse individuals in order to ascertain the circumstances under which they will prefer to hold a diversified or, alternatively, a specialized portfolio of dependent investments. I restrict the analysis to the simple case of two risky assets. I find that on the assumption of a sufficiently high degree of risk aversion and in the normal case in which higher returns can be obtained in the market only at the cost of higher risk, a diversified portfolio will be optimal regardless of the degree of correlation between the expected returns of the included assets, as already established by Samuelson (1967). However, portfolios completely specialized in the more profitable asset can obtain, but only as corner solutions, for the most part the result of low-risk aversion. Only in the limiting case of two equally profitable but unequally risky assets analyzed by Bierman does the portfolio's complete specialization in the less risky asset result from a corner solution unrelated to low-risk aversion. On the other hand, in the perverse case, in which one of the two assets is both more profitable and less risky than the other, complete specialization in the first of the assets does not necessarily follow, as Markowitz had supposed; a wide variety of diversified portfolios can be optimal even in this case, so long as asset correlation is sufficiently low or negative and the more profitable asset is not riskless. Conversely, the optimal portfolio will be specialized in the more profitable and less risky asset if the latter is riskless; if it is not, the optimal portfolio will be specialized in the less risky asset if the correlation coefficient assumes positive values falling in an interval which, depending on the assumed degree of risk aversion and other considerations, may be wider or narrower. Unless these conditions obtain, however, the optimal portfolio of dependent investments is a diversified portfolio in the perverse case as well. The analysis produced two main conclusions other than specialization in the normal case pays only if risk aversion is too low. First, in the limiting normal case, analyzed by Bierman, a completely specialized portfolio continues to obtain after risk aversion is assumed. Both this and the remaining completely specialized portfolios that obtain in the normal case result from corner solutions, although the latter are due to low-risk aversion. Second, and perhaps more importantly, the principle of portfolio diversification is found to be applicable not only in the normal case but in a wider range of instances of the perverse case than previously believed as well. The completely specialized portfolios that obtain in this case are also the result of corner solutions, although these are not necessarily due to low-risk aversion. Mathematical proof of the above propositions is presented in Sections II and III that, respectively, present the model and its comparative statistics. The results are graphically derived in Section IV, to which readers interested in a more intuitive, but quicker, presentation may turn.
Portfolios of Dependent Investments
149
II. The Model Two assets, X and Y, can be included in the portfolio; each is assumed to have a distribution of expected returns known to the individual, so that R ~ - ( # x , Ox 2) and
Ry (Izy, 02). e~
An investment fund I may be allocated in some proportion between X and Y. Let a be the proportion of I allocated to X, (1 - o0 the proportion of I allocated to Y. A portfolio consisting of X and Y will have an expected mean return given by # p - c~/zx+ (1 -a)#y.
(1)
The variance of expected returns from the portfolio-will be 0.2#=or 2 0 2. + ( 1 - o 0 2 @ + 2 r o f f l - o 0 o x 0 . y .
(2)
Suppose now, with Tobin (1958), that a function in #o and 0.o such a s f ( R ; #o, 0.o) can describe the probability distribution of returns. Then, U ( R ) f ( R ; #o, oo).
E[U(R)] =
(3)
--0o
Normalizing R one obtains Z
R - #o 0.o
~
t
which, substituted in Equation (3) above, yields
E[U(R)]=E[~o, 0.o1=
f"
-co
U(~o+oo.z)f(z;
O, 1)
dz.
(4)
Differentiating Equation (4) with respect to e o, one obtains a mean-variance indifference curve along which expected utility is constant:
® e'(#o+0.o.Z ) [k--d-~a d#°" +z]j f(z; o, O= fj_•
1) dz
or
d#o dop
I~* zU'(R)f(zO,
=
1)
dz
-o0
>0
I** U'(R)f(z; O, 1) dz -oo
for a risk averter. Assume a quadratic utility function such as
U(R) =(1 +b)R+bR2>_O with - 1 < b < 0 for risk aversion and R _< - 1 +
(5)
b/2b required for marginal utility to
150
M.H. Pastore be positive. ~ Given Equation (5), we may investigate the implied slope and curvature of the indifference curve. So far as the slope is concerned, from Equation (3) we get
i
E[U(R)]=
oo
U(R)f(R)
d R = ( l + b)/to+ b(o~+ #2p).
Holding E[ U(R)] constant and differentiating with respect to %, the slope of the riskreturn indifference curve obtained is d/xp dop
vo
(6)
l+b 2b
#o
For proof that for a risk averter the second derivative is positive and the indifference locus is concave upwards, see Tobin (1958). Recall now the expressions for the mean and variance or the portfolio given in Equations (1) and (2). Since for a given r, #p and 0 .2P c a n only vary with ct, differentiating Equations (1) and (2) with respect to ot and taking the ratio of the two derivatives we obtain from Equation (1) d tzp
do~
=/~x -/& = A #
and from Equation
(7)
(2)
d % _ ~ 0 2 - (1 - ct)ay2 + roxoy - 2rotoxay da
oo
(8)
From Equations (7) and (8) d#p dap
% • A# a a x2 - (1 -- or)a2 + rtrxay- 2raxoy
(9)
which gives the increase in expected mean returns that accrues on assumption of greater risk. 2 Equating (6) and (9) to solve the constrained maximum problem, and solving for c~ yields a -rff ct = ~ B-2ra
(10)
Feldstein (1969) pointed out that one must necessarily assume a quadratic utility function or a normal distribution of returns if the individual is to be able to consistently rank risk-return combinations, a prerequisite for theorizing on portfolio selection on the basis of utility maximization. 2 Tobin (1958) showed that the inclusion of a risldess asset permits linearization of the risk-return trade-off available to the individual. If more than two assets are considered, the linear frontier may still be preserved if at least one of the assets is riskless. When all assets are risky, however, this is no longer the case. The frontier then becomes a hyperbola (Renshaw (1972)).
Portfolios of Dependent Investments
151
where, A=B
l+b 2---b- " AP'-#Y " A # + ° 2
(11)
= %2+ ~y+ 2 (A~)2
02)
and ~ = axtry.
(13)
A, B, and a, in Equations (11), (12) and (13) are all constants. Therefore, ot in Equation (10) is a function of r alone. Substituting Equation (10) into Equations (1) and (2), we obtain the values of the portfolio's expected mean return and variance of returns that maximize the individual's expected utility:
A-ra ( A-ra ~ #x+ 1 IZ, - B _ 2r ~ K_"~-ro/ l~y *-
(1')
and
e2*=(A-r°~2 ( A-ra'~ 2 2 P \B-2ra] °2x+ I B-2raJ ay
:A-ro 6 +2r \~-2-~a]\
B-2ra]
a.
(2')
We can now ask whether portfolio specialization may reduce risk in the normal case, in which more profitable assets are available at the expense of greater risk, that is, we ask whether a = 0 or ot = 1 for any value(s) of r. The portfolio will be completely specialized in Y, the more profitable and risky asset, if ot = 0, which can be seen to obtain from (10), if
A -ra=O that is, if l+b - 2---if- • A/~-/zy • A# + a 2 - rtr= 0.
(14)
For given coefficient of risk aversion, expected mean and expected variance of returns, Equation (14) obtains if l+b
A#
2b
a
- l <_r= - - -
<0
a >0
ax
(15)
>1
For b sufficiently close to zero (sufficiently steep risk-return indifference curves) the first term in Equation (15) is negative and greater in absolute value than the positive sum of the last two terms; therefore, the r for which ot = 0 is not in the interval [ - 1; + 1], and no specialization in Y can therefore take place. All optimal solutions yield diversified portfolios. Even if b is close to - 1 (flat risk-return indifference curve), tangency conditions may obtain for all r if the difference between expected means is small and the difference between variances is large. In this case, the last term in Expression (15) guarantees that
152
M.H. Past~tc the r that will make c~ --- 0 lies outside the interval [ - 1; + 1 l. On the other hand, il the difference between expected mean returns is large and that between expected variances is small, tangency conditions will not be feasible for b close to - I, and the portfolio may be specialized in Y. This, however, does not invalidate the conclusion of the previous paragraph, for the specialized portfolio is due to an insufficient degree of risk aversion. If portfolio specialization in Y does not occur, the portfolio cannot specialize in X either. Barring insufficient risk aversion, therefore, the optimal portfolio in this case is a diversified portfolio. Consider now Bierman's limiting case, referred to in the introduction. Bierman considered the case of two equally profitable but unequally risky assets, that is, At* = 0, o,. < o~,. In this case, Equation (15) becomes - 1 a g r = ° ~ $ t, ox since ay > ox. Consequently, no r in [ - 1; + 1] can make c~ = 0, in which case the portfolio would not be specialized in Y. Does the portfolio become specialized in X ? It does if a = 1, which, from Equation (10), obtains when -l_r
l+b At* --~" "'&tz+(A#)2+°~
(16)
When a # = 0, (16) reduces to Ox
- l_
specialized in X. This is also the result obtained by Bierman w h e n h e considered the c a s e without reference to risk-return preferences. Evidently, the results are the same b e c a u s e the term in Equation (16) containing b, the coefficient of risk aversion, disappears from the expression. Bierman's findings, therefore, are seen to result as a particular case of the more general problem studied here. 3 Consider now the perverse case, in which one asset is both more profitable and less risky, that is, tZx
.
1 +b
.
.
2b
At*
.
A~
#y " - - + ° Y < I .
o <0
o >0
(15')
o~ <1
3 Introducing the assumption of risk aversion, however, does alter some of Bierman's results. In particular, he asserted that if the correlation coefficient between two equally profitable but unequally risky assets is equal to - 1, the investment fund would be equally distributed between both assets, that is, cx = t/2. However, if r = - 1, tzx * txy and o2 = ay, we obtain
Gr
wMch is not equal to 1/2 in general. That is, under conditions of risk aversion, when the expected mean returns of the investment assets are different, but perfectly negatively correlated, and the risk attached to each asset is the same, the optimal portfolio is not a perfectly diversified one.
Portfolios of Dependent Investments
153
It is clear, therefore, that the same conclusion derived for the normal case can be derived in the perverse case, except that it applies to a more restricted interval of r. Conversely, there is, in the perverse case, a certain interval of admissible values of r for which the portfolio is completely specialized. In the next section, we turn to determining the interval of r for which that is the case.
III. T h e C o m p a r a t i v e Statics o f the M o d e l How will the optimal portfolio composition change as the covariance of returns between assets increases? The manner in which a risk averse, utility maximizing individual will modify his choice of portfolios as the correlation coefficient varies can be deduced by differentiating Equation (10) with respect to r. After some manipulation we obtain dot
(2A - B)
dr - ( B - 2ro) 2
(17)
or, alternatively,
dot
o
\ i-0(18)
(B-2ro) 2
dr
2 where A V = oy2 _ o x. The sign of Equation (18) depends on the sign of the term between brackets in the numerator. In the normal case, A/~ < 0 and A V = 0 y2 - a x2 > 0. Therefore, >0
dot
o
<0
AV---.
2b
dr
<0
<0
A~+ 2 • A~_(A#)2 ( B - 2roxo~) z
For A V and b sufficiently close to zero, dot~dr < 0, and as r rises, the proportion in X falls. However, it is possible that the first term of the expression between brackets will outweigh the others, in which case dot~dr > 0 and the portfolio will become relatively more specialized in X. That is, it is clear that in the normal case one cannot unambiguously state that the portfolio will become progressively more specialized in X or Y as r rises. In the perverse case, however, since A V < 0, dot~dr is unambiguously less than zero, and the portfolio becomes progressively more specialized in Y, the less risky asset. In this case, provided - 1 _ A / o _< 1, which is clearly possible, ot = 0 if(A - r o ) / ( B - 2ro) = 0 or, i f r = A / o . Since dot/dr is less than zero, for r > A / o the portfolio must remain completely specialized in Y. Therefore, ot = 0, that is, the portfolio is completely specialized in Y, for all r in [ A / o ; 1]. The result is the same in Bierman's case, that is, specialization in the less risky asset, that this time is X, again takes place. If in Equation (16) we assume A# = 0, dot
o[A V]
dr
04
154
M.H. Pastorc
/!A a
x
a
y
p
Figure 1. Limiting normal case, Portfolio specialization with high-risk aversion and unequally risky, but equally profitable assets.
which is positive or negative depending on the sign of A 1I. In the normal case, A V > 0, and da/dr > 0, that is, as r increases, the portfolio becomes progressively more specialized in the less risky asset X. Since, in this case, a = 0 for r = ox/oy, as r rises above this value, the portfolio must continue to be completely specialized in X, so that complete X specialization obtains for r in the interval [ax/Oy; 1].
IV. Graphical Hlustration Practically every finance text graphically illustrates the optimal portfolio chosen in the normal case under conditions of risk aversion, illustration that for reasons of space we do not reproduce here. The tangency conditions that in this case must hold for the constrained maximum problem to have a solution easily obtain, provided risk aversion is not too low, for Tobin's risk-return indifference curves and the efficient portfolio contours all slope upward. However, if risk aversion is too low, only corner solutions are possible, and the individual will opt for a portfolio specialized in the more profitable and risky of the two assets, since among corner solutions rational individuals will prefer that which yields the highest utility level. If both assets are equally profitable but unequally risky, the case studied by Bierman, it is intuitively obvious that portfolio specialization in the less risky asset will take place. In addition, graphical analysis shows that the efficient portfolio contours reduce to a horizontal line, as depicted in Figure 1, so that tangency conditions will not obtain unless
Portfolios of Dependent Investments
lapl .
lay
.
.
.
.
.
.
155
.
.
.
.
la x cr
y
CT
x
(I P
Figure 2. Perverse case. Portfolio diversification with low correlation of asset returns.
the marginal rate of risk-return substitution is itself zero. For any marginal rate of riskreturn substitution greater than zero, only corner solutions will obtain and, again, the rational individual will opt for that solution which yields the highest utility level, given by a portfolio specialized in the less risky of the two assets. Therefore, complete specialization in this asset occurs not only under the conditions Bierman envisioned, but also when risk averse preferences are introduced. In fact, Bierman's is a particular case of the general one studied here. However, the graphical analysis does not reveal the exact r-interval for which portfolio specialization occurs. For that, one must resort to the preceding mathematical analysis which shows that the r-interval in question is ; 1
,ox
Turning now to the perverse case, depicted in Figure 2, below, in which one asset is more profitable and less risky than the other, it is clear that there are two sets of efficient portfolio contours, one upward sloping, the other downward sloping. Clearly, diversified optimal portfolios are feasible for this upward sloping set of contours, but not for the set of downward sloping contours. Figure 3 isolates the set of downward sloping efficient portfolio contours. As long as
156
M.H. Pastorc
k~
~y
(7
Y
(7
x
Figure 3. Perverse case. Portfolio specialization for highly positively correlated asset returns.
risk aversion is assumed, tangency conditions cannot obtain in the r-interval I(
l+b _
2b or, what is the same, [ A / a ; 1]. In this interval, only corner solutions will result, no matter what the degree of risk aversion, and rational, risk-averse individuals will opt for that solution which yields the highest utility level, that is, a portfolio specialized in the less risk and more profitable asset. Unless the more profitable asset is riskless, however, tangency solutions will still be possible in the r-interval [ - 1; ax/Oy) and diversified portfolios will result even if one of the assets is far more profitable and less risky than the other.
References Bierman, H. 1968. Using investment portfolios to change risk. Journal o f Financial and Quantitative Analysis 3:151 - 157.
Feldstein, M. 1969. Mean-variance analysis in the theory of liquidity preference and portfolio selection. Review o f Economic Studies 36:5-12.
Portfolios of Dependent Investments
157
Markowitz, H. 1952. Portfolio selection. Journal of Finance 7:77-91. Renshaw, E. F. 1967. Portfolio balance models in perspective: Some generalizations that can be derived from the two asset case. Journal of Financial and Quantitative Analysis 2:123-140. Samuelson, P. A. 1967. General proof that diversification pays. Journal of Finance and
Quantitative Analysis 2. Tobin, J. 1958. Liquidity preference as behavior toward risk. Review of Economic Studies 25:65-86.