I. The relative risk aversion paradox

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Dependent Utility

By Andreas L

offler 1

1Fachbereich Wirtschaftswissenschaft (Graduiertenkolleg), Freie Universitat Berlin, Boltzmannstr. 20, 14195 Berlin, Germany. Research support by Deutsche Forschungsgemeinschaft, Graduiertenkolleg 'Applied Microeconomics', is gratefully acknowledged.

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Decreasing Relative Risk Aversion with

{

{preferences? A Comment on Jack Meyer and Hans{Werner Sinn

We report a surprising property of{{preferences: the assumption of nonincreasing relative risk aversion implies the optimal portfolio being riskless. We discuss a solution of that paradox in detail. (JEL D80, G11, D10)

The mean variance calculus was developed by Markowitz (1952) to explain why stock market portfolios are diversied. Some years later Sharpe (1964) showed that mean variance behavior implies a simple equation in equilibrium, the capital market line. Because of its simplicity the mean variance calculus has many applications today.

Mean variance calculus derived from an expected utility model with nite state space requires a quadratic utility function. This utility function exhibits increasing absolute risk aversion, a rather unreasonable feature. In the case of an innite state space mean variance analysis and expected utility model are consistent if the random variables are normally distributed or per- fectly correlated (two{moment decision models). Chipman (1973) investigated assets with normally distributed cash{ows. The case of two{moment decision models was undertaken by Sinn (1983) and Meyer (1987).

We do not want to base the mean variance calculus on an expected utility model. We rather consider it as a certain specication of the functional form of the utility function. In this case a foundation of the{{calculus can be given using assumptions on preferences, seeLoffler (1996).

Sinn(1983) andMeyer(1987) have (re)dened the concept of absolute and relative risk aversion for{{preferences within two{moment decision models. Risk aversion allows comparative statics of a so{called simple portfolio problem, where an investor maximizes his utility using only one riskless and one risky asset. Absolute risk aversion determines the size of the absolute share of the risky asset among dierent wealth levels. Relative risk aversion considers the same problem with the quotient of share and wealth instead of absolute share (cf. Ingersoll (1987), p. 71, and denition 1 below).

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We will answer the question whether the results of SinnandMeyercan be extended to a general (i.e. not necessarily consistent with expected utility) mean variance calculus. A rst step was undertaken byLajeri & Nielsen (1994). Both authors investigated the concept of absolute risk aversion.

Yet, as we will show, the attempt to dene decreasing or constant relative risk aversion for{{preferences leads to a paradox since the optimal portfolio has to be riskless. Furthermore, any neighborhood of the zero portfolio

x= 0 contains portfolios with arbitrary large utility.

Experiments (see Levy (1994)) as well as empirical studies (seeDalal &

Arshanapalli (1993)) have shown that constant relative risk aversion is very plausible. But now we are facing a dicult situation: we observe a behavior of investors that can not be explained using mean variance analysis. Further discussions will show whether the pathbreaking ideas of Markowitz should be modied. Especially we ask whether the CAPM can be regarded as a realistic model since it can handle only the case of increasing relative risk aversion.

The paper is organized as follows: the paradox is stated in the next section, the third section sketches a solution and its problems are discussed in the

last section. Chiu und Markowitz 1952b Ausfuhrlicher

I. The relative risk aversion paradox

There are ^N (^N ^>2) linearly independent securities over some probability space, one asset being riskless. Portfolios are vectors^x ²^R^N of securities.

Short sales are possible and hence the entries in ^xmay be negative.

We consider a utility function ^U depending only on mean and standard deviation,

U(Std[^x]^;E[^x])^:

U is dened on ^R²⁺ⁿ^f0^g. We do not yet assume that^U is dened for the zero portfolio ^x= 0.

How should nonincreasing relative risk aversion be dened? Before doing so, we make the following assumptions on^U.

Assumption 1.

The utility function^U is strictly quasiconcave.

This already goes back to Sharpe (1964). Quasiconcavity is the weakest assumption ensuring uniqueness of the optimal portfolio for arbitrary prices

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and wealth levels. Without a unique solution the concept of relative risk aversion would become meaningless since comparative statics were impossible.

Assumption 2.

The utility function ^U is continuously dierentiable on

R 2

++ and continuous on^R²⁺ⁿ^f0^g.

This is rather a technical assumption to make use of the dierential calculus.

Notice that quasiconcave functions are almost everywhere dierentiable (see Crouzeix (1981), proposition 18).

Assumption 3.

The utility function exhibits risk aversion, i.e. for the deriva- tives of ^U we have:

U

2

>0^; ^U¹ ^<0^: (1)

We assume that with increasing expectation and decreasing standard deviation the utility increases. This is equivalent to risk aversion, i.e. the investor refuses additional variance risk.

Mean variance analysis implies that the optimal portfolio is a linear combination of the riskless asset and the price portfolio (Ingersoll(1987), p.88).

The optimal portfolio ^x lies in the{{diagram on a ray E[^x] = ^w

p

1

⁺⁰^Std[^x^] ⁽²⁾

which has a slope ⁰ depending only on the prices of the securities and not on the wealth^w. Std[^x] is the share of the risky asset.

Now let us dene nonincreasing relative risk aversion.¹

Denition 1.

A utility function exhibits nonincreasing relative risk aversion if the quotient of the riskless asset's share and wealth does not increase with the wealth level^w.

We prove in the appendix

1Usually comparative statics terminology refers to a simple portfolio problem, i.e. a situation with only one risky asset. But we have assumed^N^>2. However, this is consistent with the philosophy of risk aversion since due to the^Tobinseparation theorem the optimal portfolio is always a linear combination of the riskless and the (wealth independent) price portfolio. This is a simple portfolio problem.

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Theorem 1.

Assume ^U satises the assumptions 1, 2, and 3. If^U exhibits nonincreasing relative risk aversion than the optimal portfolio for nonnega- tive wealth is always riskless. Furthermore, every neighborhood of the zero portfolio contains portfolios with arbitrary large utility.

If the optimal portfolio is always riskless the denition of risk aversion be- comes meaningless. Surprisingly enough, the zero portfolio does not have a determined utility level. It is paradoxical to believe that with almost nothing one can achieve an arbitrary high utility level.

We have always considered the case^N ^>2. What happens with the relative risk aversion paradox in two{moment decision models where only two assets are available? Mean variance utility is then dened by

U(^;) :=

Z

suppX

u(+^X)^dF(^X)^;

with ^X being risky and û monotone and concave. ^X has zero expectation and therefore, the support supp^X contains negative elements with posi- tive probability. Let us consider the case of constant relative risk aversion, i.e. û(^x) = log(^x) or û(^x) = ^x (with ² (0^;1)). For a suciently small quotient some realizations of+^Xare negative and the expected utility is not dened.² Thus in two{moment decision models the mean variance utility function is not dened on the set^R²⁺ⁿ^f0^g.

II. Wealth dependent utility and the paradox

We have learned from the last theorem that decreasing relative risk aversion is meaningless in mean variance analysis. A rule of behavior which can not explain well{established empirical and experimental results must be rejected.

In this section we discuss a new approach not precluding the advantages of Markowitz' contributions.

Let us introduce the following `wealth dependent (mean variance) utility' function

U(Std[^x]^;E[^x]^;^w)^: (3)

2It can be shown that the weaker assumption of nonincreasing relative risk aversion also implies

lim

x!+0 u

0(^x) = +^1:

Hence, in this case^uis also not dened for^x^<0.

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As the following example shows, this type of `utility' avoids the paradoxical results of theorem 1.

There are two assets,^X being riskless with expectation 1 and^Y being risky with zero expectation and standard deviation 1. A portfolio ^x= (^x¹^;^x²) is a vector that species the shares of the assets. Consider for an asset price vector^p= (^p¹^;^p²) and for the wealth^w the maximization problem

max

pxw

E[^x]^; 1

w

(Std[^x])²^: The straightforward solution yields the optimal portfolio

(^x¹^;^x²) =^w

1

p

1 + 12^;^;2^p^p¹²

: (4)

Notice that the relative shares in the optimal portfolio do not change with wealth. Hence, we have constant relative risk aversion behavior.

Another advantage of wealth dependent mean variance utility (3) is that all results from the CAPM are still valid. In an individual maximization problem the wealth^wis xed and therefore theTobinseparation holds. In an equilibrium where investors behave according to (3) the capital market line holds.

However, in a strict sense ^U(Std[^x]^;E[^x]^;^w) is not a utility function since it contains the wealth ^w. Furthermore, we will show in the next theorem (for a proof see again appendix) that our approach abandons the context of classical utility theory.

Theorem 2.

Let^U(Std[^x]^;E[^x]^;^w) be a function such that the solution of the optimization problem

x

(^p;^w) = arg max

pxw

U(Std[^x]^;E[^x]^;^w)

exhibits nonincreasing relative risk aversion. Then there is no dierentiable, strictly monotonic, strictly quasiconcave utility function ^V(^x) such that ^x is the corresponding demand, i.e.

x

(^p;^w) = arg max

pxw

V(^x)^:

Any solution to the relative risk aversion paradox using wealth dependent utility is not a demand function, i.e. violates the strong axiom of revealed preferences. If we want (3) to be a solution of the relative risk aversion paradox we have to explain what an 'wealth dependent utility' is. The last section is devoted to that problem.

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III. Revealed Preference for Income dependent util- ity

Any utility concept should be related to economic behavior to avoid tauto- logical explanations. AsSamuelsonhas put it:

\Prior to the mid{1930's utility theory showed signs of degen- erating into a sterile tautology. Psychic utility as satisfaction could scarcely be dened, let alone be measured. Austrian eco- nomics would insist that people acted to maximize their utility, but when challenged as to what that was, they found themselves replying circularly that however people behaved, they would pre- sumably not have done so unless it maximizes their satisfaction

::: "Samuelson (1970), p.280.

Utility can not be understood as a philosphical or psychological concept.

Following Samuelson's suggestion let us turn to the economic behavior revealed by wealth dependent utility.

The above theorem already said that this type of utility violates the strong axiom of revealed preference. Hence, we can not expect any rational behavior revealing wealth dependent utility. We are facing some bounded rationality. Those who insist upon rational behavior must sacrice the relative risk aversion paradox.

We ask for refutable hypotheses on functions ^x(^p;^w) which are solutions to a maximization problem with an wealth dependent utility function,

x

(^p;^w) = arg max

pxw

V(^x;^w)^: (5)

It turns out that the missing link is the axiom of Wald.³ This axiom dif- fers from the axiom of revealed preference by considering only equal wealth levels.

Denition 2.

A function ^x(^p;^w) satises Wald's axiom if

8w8p

1

;p

2 p

2

x(^p¹^;^w)^w & ^x(^p¹^;^w)⁶=^x(^p²^;^w)

=⁾^p¹^x(^p²^;^w)^>^{w :} (6)

3The terminology followsHildenbrand (1994) and^John(1995).

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A function satises the strong version of Wald's axiom if for all ^k2

8w8p

1

;:::;p

k p

2

x(^p¹^;^w)^{w ;}^:^:^:^;^p^k^x(^p^{k ;1}^;^w)^w &

x(^p¹^;^w)⁶=^x(^p^k^;^w) =⁾ ^p¹^x(^p^k^;^w)^{w :} (7) What is the economic interpretation of Wald's Axiom? The strong axiom of revealed preference is logically equivalent to the strong version of Wald's axiom and homogeneity

x(^p;^w) =^x(^p;^w) ⁸^>0^; (8)

(see John (1995), theorem 2). Seen in this light, Wald's axiom allows for money illusion. If in the strictly rational calculus prices and wealth are multiplied by the same number the amounts taken will remain the same.

However, supposing only Wald's axiom can yield rearranging of the optimal portfolio. We have already mentioned the bounded rationality of that behavior.

Any objections made againstWald's axiom should take into account that this theory is not concerned with the idealised homo{economicus. The answer wether Wald's axiom or the axiom of revealed preference are more reasonable demands experiments and empirical studies.⁴

The following theorem (for a proof see again the appendix) establishes functions of type (3) as utility functions.

Theorem 3.

A continuous function ^x(^p;^w) is a demand function for a wealth dependent utility function i ^x(^p;^w) satises the strong version of Wald's axiom and Walras' law

px(^p;^w) =^{w :} (9)

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Orth (1994) reported an experiment on the law of demand. 30 persons made 10 consumtion decisions with 9 dierent goods. Since the person's "wealth" in the experiment were xed the data of^Orthcan be used to analyze wether^W^ald's axiom is supported.

It turns out that only 3 persons violated ^W^ald's axiom (the weak and therefore also the strong version). Hence, the experiment strongly supports wealth dependent utility functions.

On the other hand,^Sippel(1994) investigated the weak axiom of revealed preferences (WARP) in an experimental study. 12 persons made 10 consumtion decisions with 9 dierent goods. He nds a considerable number of violations of WARP, 11 persons did not satisfy the axiom. In particluar, these 11 persons had inhomogeneous demands. ^Sippel closes his study that the result of his experiment "contradicts the neoclassical theory of the consumer maximizing utility under a given budget constraint".

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If we accept money illusion then the relative risk aversion paradox of mean variance preferences disappears. Future research must show to what extend this solution is economically tractable.

IV. Conclusion

We considered {{preferences that were not based on an expected utility model. We tried to dene the concept of decreasing or constant relative risk aversion and showed that a reasonable denition would imply the optimal portfolio being riskless. Hence, the concept of relative risk aversion is meaningless in {{analysis.

The paradox disappeared when we considered wealth dependent utility functions. A revealed preference theory for wealth dependent utility showed that such a utility is equivalent toWalras' law and the strong version ofWald's axiom. The relative risk aversion paradox can be avoided if we accept money illusion.

V. Appendix

Proof of theorem 1. Let the slope of the indierence curves be dened by

S(^;) :=^;^U¹(^;)

U

2(^;)^: (10)

We start with the following lemma.

Lemma 1. If ^U exhibits nonincreasing relative risk aversion then the slope satises⁵

S(^;)^S(^;) ⁸1^: (11)

Proof: Assume (11) does not hold. With wealth^wthe optimal portfolio has standard deviation^x. Then the assumption of nonincreasing relative risk aversion requires that with wealth ^w⁰ ^> ^w the optimal portfolio has standard deviation^x⁰ ^w^w⁰^x. Since ^U is strictly quasiconcave the indierence curves must intersect (see gure 1) which is impossible.

We now show that with zero wealth the optimal portfolio is always riskless:

5

Meyer(1987), theorem 6, shows this result for two{moment decision models. His proof is not applicable.

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p1 w

0

p1

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0

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x

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Figure 1: Proof of lemma 1

Lemma 2. If wealth is zero (^w= 0) then any optimal portfolio is riskless.

Proof: Assume, there is a risky optimal portfolio (^x^;^x) with^x^>0.

Then the indierence curve is tangent to the budget line at (^x^;^x).

Since^w= 0 the budget line (2) crosses the origin.

Lemma 1 implies that for ^< ^x the slopes of indierence curves crossing the budget line are greater or equal to the slope of the budget line. This is a situation similarto gure 1 with^w⁰= 0. The indierence curves intersect and this is a contradiction.

The following lemma restricts the behavior of indierence curves near the {axis.

For this purpose let

(^;^U) (12)

be the functional form of an indierence curve corresponding to the utility level

U. Since the function ^U(^;) is continuous and strictly monotonic in and , this function exists and is continuous. Using the implicit function theorem (12) is furthermore continuously dierentiable.

Lemma 3. If an indierence curve (^;^U)is dened for all^>0then lim

!+0

(^;^U) = 0^: (13)

Proof: Assume the contrary. We verify the following inequality

d(^;^U)

d

>

: (14)

Consider a point (^x^;^U^x) such that (14) is not true. The level set

:= ( ) ( ) (15)