risk aversion and wealth effects on portfolios with many assets: an extension

The Review of Economic Studies, Ltd.

Risk Aversion and Wealth Effects on Portfolios with Many Assets: An ExtensionAuthor(s): Ngo Van LongSource: The Review of Economic Studies, Vol. 42, No. 3 (Jul., 1975), pp. 473-477Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2296860 .

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Notes and Comments

Risk Aversion and Wealth Effects on Portfolios with Many Assets: An Extension'2

NGO VAN LONG Australian National University

In a recent paper [2], Cass and Stiglitz showed that " if there are as many securities as states of nature, each of the risky securities yields returns in only one state of nature, and all assets are held in positive amounts, then the (wealth) elasticity of demand for money is greater than, equal to, or less than unity as relative risk aversion increases, remains constant, or decreases with wealth" (Theorem 1", p. 340). They also stated: "Un- fortunately, stronger theorems do not seem available ", . . . with a slightly more complicated pattern of returns, the wealth elasticity of demand for money may be greater than or less than unity, regardless of whether there is increasing or decreasing relative risk aversion" (p. 341).

We shall show that in fact a stronger theorem is available. That is, there are sufficient conditions which are weaker than those given by Cass and Stiglitz. Before stating our theorem it is useful to recall Cass and Stiglitz's notation.

There are n securities and n states of nature. The ith security yields a return per dollar invested in it of pie in state 0. The matrix of returns is a square matrix P:

p = [Pio] i, 0 _=1, 2, ..., n. ... (1)

The first security is called money, a safe asset. By assumption Plo = PM>O for all 0. Cass and Stiglitz assumed that the pattern of returns for the (n - 1) risky assets is:

Pio = ? for 0 i (i>l) ...(2)

po= pi>0 for 0 = i (i> 1). ...(3)

Conditions (2) and (3) are rather stringent. We offer the following weaker conditions on the pattern of returns for the risky assets:

(i) There exists some state of nature (which we shall assume to be the first state of nature, without loss of generality) such that

Pii = O (i> 1). ..(4)

(ii) Let S be the submatrix obtained by deleting the first row and the first column of P. We require that S is non-singular (given (i) and the assumption that the first asset is money, (ii) is equivalent to the assumption that P is non-singular).

Obviously our assumptions are weaker than those of Cass and Stiglitz since we do not require that S be a diagonal matrix nor do we assume that the diagonal elements of S be strictly positive.

1 First version received January 1974; final version accepted May 1974 (Eds.). 2 I wish to thank Frank Milne for comments. The responsibility for errors is mine alone.

473



474 REVIEW OF ECONOMIC STUDIES

We shall prove the following theorem, which is a generalization of Cass and Stiglitz's Theorem 1":

Theorem. If there are as many securities as states of nature, there exists some state of nature in which all the (n - 1) risky securities yield zero returns, the n patterns of returns are linearly independent (i.e. rank P = n) and all assets are held in positive amounts 1 (in the optimal portfolio), then the wealth elasticity of demandfor money is greater than, equal to, or less than unity as relative risk aversion increases, remains constant, or decreases with wealth.

It might seem at first that any (n-1) x (n-1) submatrix S could be shown equivalent to the Cass-Stiglitz diagonal submatrix simply by a change of basis (a redefinition of the basic risky assets) provided that S is non-singular and/or S has n-I independent eigenvectors. This conjecture turns out to be incorrect because of the following constraints:

(i) the Cass-Stiglitz submatrix must have positive diagonal elements, and (ii) by definition of a matrix of returns, each row is the pattern of returns per dollar

invested in the corresponding assets. Hence each of the newly defined patterns of returns must be a linear combination, with weights adding to one, of the original patterns of returns.

Consider the following counter-example:

S- Q1 2) 3 4y

S is non-singular and has two distinct eigenvalues (and hence two linearly independent eigenvectors), but there are no t1, t2, r, >0, r2>0 such that:

t1(1, 2)+(1-tl)(3, 4) = (rl, 0)

t2(1, 2)+(1-t2)(3, 4) = (0, r2).

Before proving the theorem we need the following results which are hinted at in an earlier paper [1] by Cass and Stiglitz but no precise statements nor formal proofs were given.

Proposition 1. Let Q = (qi,) be an n x n semi-positive, non-singular matrix of returns. Let bij be the (i, j) entry of Q' 1. If .jbij 0 O for some i, then there exists some original security dominated by a mutual fund (i.e. linear combination, with weights adding to one, of the original securities).

As a trivial corollary of Proposition 1, we state:

Proposition 2. Let Q be an n x n semi-positive, non-singular matrix of returns. If there is no original security which is dominated by some mutual fund, then Q le>0, where e is a (conformal) column vector with all its entries equal to unity.

Proof of Proposition 1. Assume Zibij < 0 for some i. Define t = (It1, u2, *tn)

where

I, = bil, I = 2, 3, ........, n ... (5)

n

1 = bil1- Ebi ... (6) j=

1 This assumption can be weakened. See footnote 2, p. 476. 2 See Appendix I of [11, pp. 151-152, especially the sixth paragraph on page 152.



LONG RISK AVERSION AND WEALTH EFFECTS 475

PQ-( E bij, 0, 0, ...,, 0, 0) Q+(bil, bi2, E- bin)Q

. ,uQ > (bil, b.b. b.)Qin)Q

The right-hand side of (7) is the ith row of the matrix Q' Q L L Hence

uQ ?0'

with strict inequality for at least the ith component (Q' is a zero row vector in Re). Note that ji # Q' by (8) and ligi = 0 by (5) and (6). Hence Pk<0 for some k. Note that 4uQ can be regarded as a linear combination of the rows of Q. Let Qj

be the jth row of Q. From (8) n

E HjQj > ' (9) j

with strict inequality for at least the ith component. Hence, for some Uk <0, (9) gives:

E (-i' j/k)(Qj) >? Qk+( /1k)Q' = Qk* ...(10)

jo k

The security k is therefore dominated by the mutual fund described by the left-hand side of (10).

QED The proof of Proposition 2 is obvious since the statement X=> Y is equivalent to

Non Y=>Non X. We are now ready to prove our theorem.

Proof of the Theorem. The assumption that the optimal portfolio consists of positive amounts of all securities implies that none of the securities is dominated by a mutual fund. That is, no row of P is dominated by a linear combination, with weights adding to unity, of other rows of P. Hence the submatrix S (obtained by deleting the first row and the first column of P) contains no dominated row. Therefore S1'e>O by Proposition 2. Define an (n-1) x (n-1) diagonal matrix S with positive diagonal elements P-2, P3, ***P which are inverses of the elements of the column vector S 1e. In symbols:

Stle = S-'e>O. ... (11)

Partition the matrix P into four submatrices:

PM PM * PM PM '_

p- 0 P22 . P2n O S (12)

L0 Pn2 Pnn K..(2

Define the "equivalent matrix of returns " P:

PM PM 1..... PM PM

P2 P3 ......O S]. ...(13)

_ O O. .... PnM[_



476 REVIEW OF ECONOMIC STUDIES

Let a = (a,, a2, ..., an) where a, is the proportion of wealth allocated to the ith asset, with ?iai = 1, Partition a into:

a (aM I aR), where aR = (a2, a3 ... an) and

aM a,. ... (14)

For any portfolio allocation a, the resulting pattern of returns is:

aP = (aMpM I aMJ+aRS). ...(15)

We wish to show that for any allocation a, there exists an equivalent allocation a = (a1l d29 ..., a n), iii = 1 such that:

aP = ZIP. ... (16) Define:

a-(aM I aR) ... (17) where

aR-aRSS ...(18) and aM is identical to aM in (14).

Obviously by (17), (18) and (11): n

Z i= aM+aRe= aM+aRSS9 e= aM+aRe i = 1

n

= Z aa=1 ... (19) i = 1

and aP= (aMpM I amJ +RS)= aP ... (20)

by virtue of (18) and (15). By a similar argument, it is easy to show that for any allocation a(Iii = 1) one can

find a such that (16) holds. Note that if aR is a positive or semi-positive vector then so is aR as defined by (18),

since S is a semi-positive, non-singular matrix and S- 1 is a diagonal matrix with positive diagonal elements.

Our matrix P and allocation a satisfy all the conditions of Cass and Stiglitz's Theorem 1". Hence, using their theorem,1 2

dao 2 as R'[W] >0, ... (21) dWo

where R[W] --u"[W]. W/u'[W].

But ai1 a1 _ aM by (14) and (17). Hence our theorem is proved. QED

Note the following misprints in Cass and Stiglitz's paper [2]: (i) ai on lines 7 and 9 (p. 341) should be d1. (ii) Equation (16) (p. 338) should be:

dln a_ 1/RO_ d ln Wo 2(a0/a?0)

0

(This equation is the relevant part of Lemma 1 for the purpose of Theorem 1".) 2 The assumption that all assets are held in positive amounts is made so as to ensure

(i) that no security is dominated; (ii) that consumption is positive for all states of nature; and (iii) that min 'idi > 'id, (p. 341 of [2]).

With Pjaj> p1ld for some j. For (iii) the non-negative holding of all risky assets, with strict positivity for at least one risky asset, is sufficient.

Thus the assumption al > 0 for all i can be replaced by weaker assumptions (a) no dominated securities, (b) positive consumption and (c) ai ? 0 for all i> 1 with aj > O for some j> 1.



LONG RISK AVERSION AND WEALTH EFFECTS 477

REFERENCES

[1] Cass, D. and Stiglitz, J. E. " The Structure of Investor Preferences and Asset Returns, and Separability in Portfolio Allocation: A Contribution to the Pure Theory of Mutual Funds ", Journal of Economic Theory, 2 (June 1970), pp. 122-160.

[2] Cass, D. and Stiglitz, J. E. " Risk Aversion and Wealth Effects on Portfolios with Many Assets ", Review of Economic Studies, 39 (July 1972), pp. 331-354.



risk aversion and wealth effects on portfolios with many assets: an extension

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