capm basics - pantheon-sorbonnecermsem.univ-paris1.fr/arbitrage/paris/bohm.pdf · capm basics...

CAPM Basics

Volker Bohm ∗

Department of Economics

University of Bielefeld

P.O. Box 100 131

D–33501 Bielefeld, Germany

e-mail: [email protected]

December 2002

Discussion Paper No. 495

Abstract

This paper discusses demand behavior of consumers and existence of equilibriafor the standard capital asset pricing model (CAPM) with one riskless and finitelymany risky assets, mean variance preferences of consumers, and subjective expecta-tions. By treating individual expectations explicitly and parametrically, the modelencompasses the description of individual as well as aggregate demand behavior forheterogeneous expectations. The paper provides a basic factorization formula forindividual asset demand which implies the mutual fund property for agents withhomogeneous expectations. This approach unveils some of the hidden structuralfeatures of the CAPM model often not discussed in the literature. Applying notionsfrom standard static consumer theory, a characterization of the demand for riskfrom assumptions on risk preference is provided.

The paper provides sufficient conditions on preferences to generate differentiableand globally invertible asset demand behavior of consumers parameterized by wealthand arbitrary subjective expectations. In addition, the paper proves existence,uniqueness, and determinacy of equilibria for the case of arbitrary homogeneousexpectations, thus complementing, amending, and generalizing existing results. Ex-amples indicate to what extent the conditions are necessary.

Key words: Capital asset pricing, mean variance preferences, asset demand, asset marketequilibrium, existence, uniqueness, determinacy, expectations.JEL classification: E17, G12, O16.

∗Acknowledgement. This research is part of the project ‘Evolution auf Finanzmarkten’ supportedby the Deutsche Forschungsgemeinschaft under contract No. He 2592/2-1.

0

CONTENTS 1

Contents

1 Introduction 2

2 The Basic CAPM Model 3

2.1 Efficient Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Asset demand with Mean–Variance Preferences . . . . . . . . . . . . . . . 7

2.3 Budget restrictions and short sale constraints . . . . . . . . . . . . . . . . 9

2.4 Existence of Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Differentiable Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Asset Market Equilibrium 19

3.1 Existence and Uniqueness of Equilibrium . . . . . . . . . . . . . . . . . . . 21

3.2 The equilibrium set and determinacy . . . . . . . . . . . . . . . . . . . . . 27

4 Equilibria with asset endowments 30

4.1 Homogeneous Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Conclusion 33

1

1 Introduction 2

1 Introduction

The capital asset pricing model (CAPM)1 has become one of the most widely used modelsto describe demand behavior and equilibria in asset markets, deducing such well knownresults as the ’mutual fund theorem’ or the so called ’beta pricing’ relation. Although theterminology suggests that the pricing of assets is an integral part of the model, the resultsare mostly given on rates of returns rather than making asset prices fully explicit2.In ad-dition, most presentations employ a static two period version of the model invoking thenotion of a temporary asset market equilibrium within an environment of uncertainty forall agents, treating the equilibrium for given expectations which often are also not madeexplicit. In this case, the role of the expectations on demand and/or existence of tempo-rary equilibria cannot be studied. Since the static model precludes a systematic analysisof the stochastic nature of the underlying random process on returns, the relationship, be-tween the expectations, demand, and realizations cannot be studied as well. Few studiesanalyze the full stochastic embedding into an infinite sequential economy with exogenousrandom perturbations3.

In an attempt to embed the CAPM into a dynamic model with overlapping generations ofagents (see Bohm & Chiarella 2000), it was necessary to establish existence and unique-ness of temporary equilibria of the CAPM type model in an extended economic framework with explicit parameterizations of incomes, prices, and expectations. In so doing,a multiplicative factorization of asset demand was obtained as well as some additionalstructural relationship of equilibrium asset prices were derived. Altogether these resultsshed some new light on the CAPM model and its usefulness for a dynamic analysis ofasset markets.

The model used here to describe the temporary situation of an economy is (at first) asimplified version of the standard two period model of an economy with given real wealthof agents, facing uncertainty about next periods realization of the real value of theirportfolio consisting of real savings and of K assets decided on in the current period. Theprimary difference relative to the existing literature consists of making all parameters ofthe consumer decision problem i. e. prices and expectations explicit. By keeping themparametrically and conceptually separate in the formulation of the model the structuralequations reveal their separate impact on individual asset demand and on equilibriumasset prices. Thus the functional relationship between expectations and equilibrium assetprices is unlocked revealing the different impact of expectations and of preferences betweenthem. This provides an explicit formulation of efficient portfolios and of the mutualfund property for agents with homogeneous beliefs. At the same time, heterogeneousexpectations as well as constraints on budget and short sales are identified as additionalsources of inefficiencies.

For the proof of existence and of uniqueness of equilibria this parametric separation alsoproves useful presenting a version of the beta pricing relation as a function of expecta-tional parameters as well. While the method of proof here is different from most existing

1as initially studied by Sharpe (1964), Lintner (1965), Mossin (1966)2see LeRoy & Werner (2001) as one typical example.3Stapleton & Subrahmanyam (1978) is one of the most notable exceptions; for others see the discussion

in Bohm & Chiarella (2000)

2


ones4, the conditions and assumptions here have their counterparts in those of the litera-ture. Since these are made on the primitives of the agents’ characteristics, their role forexistence, uniqueness, and efficiency can be identified more directly than in the existingliterature. Moreover, the explicit parametric treatment allows the description of featuresof the equilibrium manifold.

As a consequence, this approach has several advantages and special features, which makea full dynamic analysis possible, i. e. to study, prove, and exhibit the properties of rationalexpectations equilibria, to analyze the long run behavior or prices, portfolios, and returns.In addition the impact of different and heterogeneous expectations rules, of learning, andother behavioral generalizations can be studied directly.

In other words, the beliefs on premia (qa − Rp, V a) determine the mix of risky assets(V a)−1πa of any optimal portfolio, while the level of asset demand h(πa) = ma(ρa)/ρa isdetermined by the utility function, the safe rate R and initial wealth Rea. In other words,for all πa and all λ ∈ R+, demand ϕa(λπa) and the vector of subjective conditional premia(V a)−1πa are linearly dependent. Together with concavity of the mean–variance utilityfunction one finds that individual asset demand ϕa is globally invertible (see Lemma ??in the appendix).

2 The Basic CAPM Model

The model describes markets for K ≥ 1 risky assets, a consumption commodity whichserves as numeraire and as a riskless asset with interest factor R > 0. Market interactionoccurs among finitely many investors who do not possess storage possibilities for theconsumption good directly and who attempt to maximize the expected utility of futureconsumption (wealth).

Consumers/Investors

Portfolio choices on the basis of mean–variance preferences constitute a widely acceptedframe work to analyze decisions under risk and their associated markets, inducing a muchstudied class of models successfully used in financial theory. They form the basis of theclassical capital asset pricing model (CAPM) the results of which serve as a fundamentalguideline to understanding and evaluating the trade off between returns and risk in assetmarkets

The primary importance of mean–variance preferences within the classical asset pricingtheory stems from the fact that they supply a convenient structure to analyze asset de-mand behavior explicitly. The case with quadratic utility and normally distributed returnsyields the well known standard CAPM pricing formula of Sharpe–Lintner–Mossin5. Inother more general situations, as is known with quadratic utility, mean–variance prefer-ences induce globally invertible demand functions which are often solvable algebraically.In the general equilibrium context this may yield explicit functional forms of the equilib-rium price map, which are needed if an explicit description of the asset price process is

4Without claiming to be complete, some of the most noted are, , Werner (1985), 1987, 1988, 1989,Nielsen (1988), 1990, Allingham (1991) Dana (1999), and Hens, Laitenberger & Loffler (2002)

5Sharpe (1964), Lintner (1965), and Mossin (1966)

3


the goal.

In this context, the typical consumer/investor is characterized by

• w ∈ R units of a numeraire commodity as his consumption/numeraire endowment

• a one period planning horizon

• risk preferences over the mean µ and the standard deviation σ of his future con-sumption/wealth described by a utility function

U :

{

R × R+ → R

(µ, σ) 7→ U(µ, σ)

which is increasing in the mean µ and decreasing in the standard deviation σ.

• a subjective probability distribution ν ∈ Prob(RK) for future cum-dividend assetprices (gross returns) parameterized by a pair (qe, V ) ∈ R

K ×RK2

of expected meancum dividend prices qe ∈ R

K and associated covariance matrix V ∈ RK2

.

Future Wealth

Let(

(x(1), . . . , x(K)), y)

= (x, y) ∈ RK × R denote a portfolio of K + 1 assets and let

p ∈ RK denote current asset prices in units of the numeraire commodity. Then, without

(short sale) constraints on the demand for all K+1 assets the investors budget constraintrequires

w = 〈p, x〉 + y =K∑

k=1

p(k)x(k) + y,

where 〈., .〉 denotes the scalar product of any two vectors. If qe := p′ + d ∈ RK denotes

future cum-dividend prices of the K assets his future (random) consumption/wealth be-comes

W (x,w, p, q) = R[w − 〈p, x〉] + 〈q, x〉 = Rw + 〈(q −Rp), x〉Let ν ∈ P denote a probability measure on R

K describing the distribution of future cumdividend prices, characterizing the expectations subjectively held by the investor. Then,for any asset portfolio x ∈ R

K this yields the subjectively expected value of his futurewealth

Eν [W (x,w, p, ·)] =

∫

RK

Rw + 〈(q −Rp), x〉ν(dq)

= Rw + 〈(qe −Rp), x〉with associated subjective variance

Vν [W (x,w, p, ·)] =

∫

RK+

(

W (x,w, p, q) − Eν [W (x,w, p, ·)])2ν(dq)

= 〈x, V x〉.

Concerning subjective expectations6, the following assumption will be made throughout,which implies that the consumer assumes that non of the assets is redundant.

6see Nielsen (1988)

4

2.1 Efficient Portfolios 5

Assumption 2.1The subjective expectations for future cum dividend prices are parameterized in

• qe = pe + de ∈ RK the vector of expected values of future cum dividend prices and

• V ∈ RK×K the associated covariance matrix

and satisfy:

(i) q ∈ RK,

(ii) V ∈ RK2

is symmetric and positive definite.

Define π := qe −Rp as the vector of expected risk premia of the K assets. Then, one canwrite

U(

Eν [W (x,w, p, ·)],Vν [W (x,w, p, ·)] 1

2

)

= U(

Rw + 〈π, x〉, 〈x, V x〉 1

2

)

.

As a consequence asset demand of an investor with µ− σ−preferences and expectationsν can be defined as

ϕ(w, π, V ) := arg maxx∈RK

U(

Eν [W (x,w, p, ·)],Vν [W (x,w, p, ·)] 1

2

)

(2.1)

= arg maxx∈RK

U(

Rw0 + 〈π, x〉, 〈x, V x〉 1

2

)

(2.2)

for every (w, π, V ) satisfying assumption 2.1

2.1 Efficient Portfolios

Definition 2.1An asset portfolio x ∈ R

K is called mean–variance–efficient (or µ − σ–efficient) withrespect to (π, V ), π 6= 0, if there is no x′ 6= x such that

〈π, x′〉 ≥ 〈π, x〉 (2.3)

〈x, V x〉 ≥ 〈x′V x′〉 (2.4)

with at least one strict inequality.

Note that this concept of efficiency is strictly subjective and agent specific, since it dependson individual subjective beliefs, implying that, typically, a particular portfolio x will notbe efficient for agents with heterogeneous beliefs. Let Xeff (π, V ) denote the set of allµ − σ–efficient portfolios. When asset prices p and expectations (qe, V ) are given, oneoften writes Xeff instead of Xeff (π, V ). Similarly, one simply speaks of efficiency ratherthan of µ− σ efficiency.

Lemma 2.1

Xeff (π, V ) ={

λV −1π|λ ≥ 0}

(2.5)

5

2.1 Efficient Portfolios 6

Proof: Consider λ > 0 and some x 6= λV −1π with 〈π, x〉 = 〈π, λV −1π〉. It suffices to showthat

〈λV −1π, V λV −1π〉 < 〈x, V x〉

One has

0 < 〈(x− λV −1π), V (x− λV −1π)〉= 〈x, V x〉 − λ〈V −1π, V x〉= 〈x, V x〉 − λ〈π, x〉= 〈x, V x〉 − λ〈π, λV −1π〉= 〈x, V x〉 − 〈λV −1π, V λV −1π〉

which implies 〈λV −1π, V λV −1π〉 < 〈x, V x〉 QED.

��

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�

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�

� � � �

Figure 2.1: Efficient Portfolios

Figure (2.1) displays the set of efficient portfolios Xeff for the case of two assets. Itconsists of all tangency points of the contours of the quadratic mapping 〈x, V x〉 with thecontour lines of the expected return 〈π, x〉. Thus, all efficient portfolios have the samemix of assets, but they differ in their scale. Larger asset bundles imply larger means andlarger variances, since they are positively homogeneous functions of the scale factor λ.It is immediate that the set of efficient portfolios can also be obtained as the family ofsolutions of the following standard optimization problem:

m(σ, π, V ) := maxx∈RK

{

〈π, x〉|〈x, V x〉 1

2 ≤ σ}

= 〈π, V −1π〉 1

2σ (2.6)

6

2.2 Asset demand with Mean–Variance Preferences 7

Solutions to (2.6) exist whenever V is positive definite. They are unique if π 6= 0 andgiven by

ψ(σ, π, V ) := arg maxx∈RK

{

〈π, x〉|〈x, V x〉 1

2 ≤ σ}

=σ

〈π, V −1π〉 1

2

V −1π (2.7)

which yields

Xeff =

{

x′|x′ =σ

〈π, V −1π〉 1

2

V −1π, σ ≥ 0

}

(2.8)

For given (π, V ), the function (??) defines a linear relationship between expected meanµ and the standard deviation σ among efficient portfolios. Its graph is the so calledefficiency line and its slope 〈π, V −1π〉 1

2 is the marginal or shadow return to risk.

2.2 Asset demand with Mean–Variance Preferences

Assumption 2.2Risk preferences are described by a continuous quasi concave utility function of mean andstandard deviation (µ, σ)

U :

{

R × R+ → R

(µ, σ) 7→ U(µ, σ)

which is strictly increasing in its first argument µ and strictly decreasing in its secondargument σ.

In some situations it may be convenient to include preferences which are monotonic onlyin a weak sense, i. e. non decreasing in its first argument µ and non increasing in itssecond argument σ. The next two propositions state some fundamental properties ofasset demand.

Proposition 2.1Let assumption (2.2) be satisfied. If V ∈ R

K×K is positive definite, then


(

u(Rw + 〈π, x〉, 〈x, V x〉 1

2 ))

⊂ Xeff (π, V ). (2.9)

For every x ∈ ϕ(w, π, V ), there exists λ ≥ 0, such that :

x = λV −1π (2.10)

〈π, x〉 = λ〈π, V −1π〉 (2.11)

〈x, V x〉 = λ2〈π, V −1π〉 (2.12)

〈π, x〉〈x, V x〉 1

2

= 〈π, V −1π〉 1

2 (2.13)

In other words, any demand vector is colinear to the efficient vector V −1π. This meansthat the mix of asset demand is determined by prices and expectations while the levelof demand is determined by preferences and wealth. Thus, all wealth and preference

7

2.2 Asset demand with Mean–Variance Preferences 8

characteristics must be embodied in the factor λ of equation (2.10) as a function of(w, π, V ).

Proof: Property (2.10) follows immediately from the efficiency lemma (2.1). All otherproperties are direct implications of (2.10) QED.

If demand is unique, the properties (2.10)- (2.13) imply that the demand function ϕ(w, π, V )satisfies the following properties hold for all (w, π, V ) :

ϕ(w, π, V ) = λV −1π (2.14)

〈π, ϕ(w, π, V )〉 = λ〈π, V −1π〉 (2.15)

〈ϕ(w, π, V ), V ϕ(w, π, V )〉 = λ2〈π, V −1π〉 (2.16)

〈π, ϕ(w, π, V )〉〈ϕ(w, π, V ), V ϕ(w, π, V )〉 1

2

= 〈π, V −1π〉 (2.17)

Figure (2.2) provides a geometric characterization of an efficient demand bundle x∗ for thesituation with two assets. The iso–variance lines of figure (2.1) have been superimposed

by the contours of the function of feasible utilities U(Rw0 + 〈π, x〉, 〈x, V x〉 1

2 )

��

��

��

� �

Figure 2.2: Efficient demand

Notice that the number 〈π, V −1π〉 plays an important role in these equations. It is simul-taneously equal to the mean and to the variance of the efficient portfolio V −1π. Moreover,it is also the parametrically given (linear) shadow price of the efficiency solution (2.6). Inother words, 〈π, V −1π〉 is the marginal (shadow) return to any additional risk given pricesand expectations. This is equivalent to the statement of equation (2.13) that efficientµ− σ combinations have a constant ratio of 〈π, V −1π〉, which is a homogeneous functionof degree two in prices and expectations. Combining these findings one can now derive the

8

2.3 Budget restrictions and short sale constraints 9

precise expression of the scale of asset demand λ to obtain the fundamental factorizationformula of asset demand. This result also confirms the so called mutual fund theorem, i.e. agents with identical beliefs hold identical mixes of assets.

Lemma 2.2


U(

w + 〈π, x〉, 〈x, V x〉 1

2

)

=s(w, 〈π, V −1π〉 1

2 )

〈π, V −1π〉 1

2

V −1π, (2.18)

wheres(w, 〈π, V −1π〉 1

2 ) := arg maxσ≥0

U(Rw + 〈π, V −1π〉 1

2σ, σ). (2.19)

Therefore, the individual level of asset demand is equal to the ratio of the indirect standarddeviation and that of the standardized portfolio V −1π.

Proof: One verifies easily that

maxx∈RK

U(

Rw + 〈π, x〉, 〈x, V x〉 1

2

)

= maxx∈Xeff

U(

Rw + 〈π, x〉, 〈x, V x〉 1

2

)

= maxσ≥0

U(Rw +m(σ, π, V ), σ)

= U(Rw +m(s(w, 〈π, V −1π〉 1

2 ), s(w, 〈π, V −1π〉 1

2 )

= maxσ≥0

U(Rw + 〈π, V −1π〉1

2σ, σ)

Therefore:

ϕ(w, π, V ) := ψ(s(w, 〈π, V −1π〉 1

2 ), π, V )

=s(w, 〈π, V −1π〉 1

2 )

〈π, V −1π〉 1

2

V −1π,

QED.

2.3 Budget restrictions and short sale constraints

Short sales constraints may apply when Xeff /∈ RK+ , i. e. when efficiency to buy short.

In such a case the constraints act like quantity rationing constraints. It is well knownthat they will cause spillovers in demand between different assets. In other words, de-mand with a binding short sale constraint will no longer be colinear to V −1π in general.Thus, as the geometry of Figure (2.2) suggests, demand will not be (π, V ) efficient. Asa consequence, demand an agent deviates from the efficiency line depending on individ-ual preferences, creating a wedge between the efficiency trade off and the risk trade off.Therefore, aggregate demand in any market situation with homogeneous beliefs will notbe efficient if some agent faces a binding short sales constraint. It is evident that suchsituations are also not constraint efficient in general7.

7Binding short sales constraints correspond exactly to quantity constraints as treated in the rationingliterature. Therefore all results concerning efficiency and constraint efficiency carry over directly to theasset demand here (see for example Bohm & Muller 1977)

9

2.4 Existence of Demand 10

A similar loss of efficiency occurs when the budget constraint of an agent becomes binding,i. e. when he is not allowed to go short on the safe asset to obtain credit. Figure ??describes such a situation with two assets without short sales constraint, but 〈p, x∗〉 > w.The inefficiency (i. e. x /∈ Xeff ) occurs when prices p and expectation q are not colinear.

��

�

��

�

��

��

�

PSfrag replacements

R(1)

R(0)

(a) Inefficient demand x with short saleconstraint

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��PSfrag replacements

R(1)

R(0)

(b) Inefficient demand x with binding bud-get constraint

Figure 2.3: Inefficiency of Demand x

2.4 Existence of Demand

The following four examples provide additional insight into possible properties of assetdemand. They also indicate which additional assumptions are needed to obtain welldefined demand for all prices and expectations.

Example 1: Linear mean variance preferences

Let the preferences of an investor

U :

{

R × R+ → R

(µ, σ) 7→ U(µ, σ)

be given by the function

U(µ, σ) = µ− α

2σ2, (2.20)

where α is usually interpreted as a measure of risk aversion. From the first order conditionsone calculates directly the demand as

ϕ(w, π, V ) =1

αV −1π =

1

αV −1(qe −Rp) (2.21)

Thus, asset demand is linear (homogeneous of degree one) and globally invertible in π,and independent of initial wealth. Notice that the investor’s (indirect) expected return isfrom his asset demand

m(w, π, V ) := 〈π, ϕ(w, π, V )〉 =1

α(〈π, V −1π〉) (2.22)

10


and his demand for risk (his indirect standard deviation) is

s(〈π, V −1π〉 1

2 ) := 〈ϕ(w, π, V ), V ϕ(w, π, V )〉 1

2 =1

α〈π, V −1π〉 1

2 , (2.23)

implying that he always chooses a standard deviation equal to his risk factor 1/α (sometimes called the risk tolerance) times the risk of the standardized portfolio which corre-sponds to efficient shadow return to risk. The homogeneity of demand implies that theinvestor would be willing to bear unlimited risk when the ratio of expected return to riskof the standardized portfolio becomes unbounded.

Example 2: Logarithmic mean utility

Consider next the situation with the utility function

U(µ, σ) = ln(µ) − α

2σ2 µ > 0, (2.24)

which yields the asset demand function

ϕ(w, π, V ) =1

βV −1π, (2.25)

where

β =(αRw +

√

(αRw)2 + 4α〈π, V −1π〉√

4(〈π, V −1π〉)(2.26)

β is the unique positive solution of the quadratic equation

(β2 − α)√

〈π, V −1π〉 − αβRw = 0.

One obtains as the investors expected return

m(w, π, V ) := 〈π, ϕ(w, π, V )〉 =1

β〈π, V −1π〉 1

2 (2.27)

and his demand for risk as

s(w, π, V ) := 〈ϕ(w, π, V ), V ϕ(w, π, V )〉 1

2 =1

β2(2.28)

For w = 0 the investor will always demand constant risk equal to 1/√α. Moreover, for

w > 0, one observes that β ≥ √α for all π, which implies that his demand for risk is

always less than or equal to√α, regardless of wealth, prices, and expectations. As a

consequence, his asset demand will be bounded uniformly for all (w, π, V )). Moreover,the demand for risk is decreasing in wealth w.

Example 3: Quasi concave utility

Consider next the situation with the utility function

U(µ, σ) =µα

r + σβ, r > 0, 0 < α ≤ 1 ≤ β µ ≥ 0 (2.29)

11


which is not concave, but strictly quasi concave as long as α < β. For w = 0, this yieldsthe asset demand function

ϕ(0, π, V ) =1

〈π, V −1π〉 1

2

(

rα

β − α

)1

β

V −1π (2.30)

Notice again that the investor’s demand for risk

s(0, 〈π, V −1π〉 1

2 ) =

(

rα

β − α

)1

β

is constant, i. e. independent of w and 〈π, V −1π〉 1

2 . Therefore, asset demand is uniformlybounded for all (π, V ) for α < β.

If α = β = 1, the indifference curves of U are rotating lines with higher slopes for higherutility. One finds

ϕ(R, π, V ) =

∅, if 〈π, V −1π〉 1

2 >Rw

r;

Xeff , if 〈π, V −1π〉 1

2 =Rw

r;

0, if 〈π, V −1π〉 1

2 <Rw

r.

(2.31)

Thus, demand is empty if the return to risk 〈π, V −1π〉 1

2 is larger than Rw/r. If it existsit is either zero or the whole set of efficient asset bundles Xeff .

Example 4: Unbounded Demand

Finally, consider the situation with the concave utility function

U(µ, σ) = µ− σ2

r + σ, r > 0, (2.32)

which is also strictly quasi concave. It will be shown that the demand for risk and thedemand are not defined for a large range of prices and expectations.

Given (π, V ), consider an arbitrary λ > 0 and an associated efficient bundle x(λ) =λV −1π. This induces a feasible utility

U(λ) := Rw + λ〈π, V −1π〉 − λ2〈π, V −1π〉r + λ〈π, V −1π〉 1

2

(2.33)

> Rw + λ〈π, V −1π〉 1

2

(

〈π, V −1π〉 1

2 − 1)

. (2.34)

Thus, U(λ) is strictly increasing in λ for 〈π, V −1π〉 1

2 ≥ 1. Therefore, asset demand does

not exist. For 〈π, V −1π〉 1

2 < 1, one obtains

ϕ(w, π, V ) =

r

(

1 +

√

(

1 − 〈π, V −1π〉 1

2

)−1)

〈π, V −1π〉 1

2

V −1π (2.35)

12


One observes that the demand for risk and therefore asset demand becomes unboundedas 〈π, V −1π〉 1

2 approaches 1 from below.

The last two examples combined with the factorization lemma (2.2) show that propertiesof the utility function/the preferences and not expectations are responsible for the failureof demand to exist, since asset demand becomes unbounded if and only if the demand forrisk becomes unbounded for some finite shadow return for risk. Thus, the set of convexpreferences has to be restricted if well defined demand for all prices and expectations isrequired. This leads in a natural way to the next lemma.

Definition 2.2 Let S denote a subset of Rm. The asymptotic cone of S, denoted AS is

defined as

AS :=∞⋂

k=1

C(Sk) (2.36)

where C(Sk) denotes the smallest closed cone containing Sk := {x ∈ S| |x| ≥ k}.

Lemma 2.3Let assumption 2.2 be satisfied and define

P (µ, σ) := {(µσ)|U(µ, σ) ≥ U(µ, σ)}

as the upper contour set for the utility function U(µ, σ) at the point of (µ, σ). If

AP (Rw, 0) = {λ (0, 1) | λ ≥ 0} (2.37)

then the demand for risk s(w, 〈π, V −1π〉 1

2 ) is finite for all prices and expectations satisfyingassumption (2.1).

In other words, if the asymptotic cone of the better set contains no strictly positive vectorsof R

2+, i. e. it consists of the half line through (0, 1) alone, demand is nonempty8.

Proof:Monotonicity of the utility function allows one to rewrite the demand for risk s(Rw0, 〈π, V −1π〉 1

2 )as the solution of

s(w, 〈π, V −1π〉 1

2 ) := arg maxσ≥0

u(Rw0 + µ, σ | µ ≤ 〈π, V −1π〉 1

2σ, σ ≥ 0). (2.38)

Observe that any demand pair (µ, σ) = (〈π, V −1π〉 1

2 s(w, 〈π, V −1π〉 1

2 ), s(w, 〈π, V −1π〉 1

2 ))

must lie in P (Rw, 0)∩{

(µ, σ) | µ ≤ 〈π, V −1π〉 1

2σ, σ ≥ 0}

, which is the intersection of two

closed convex sets. It is compact if the intersection of their asymptotic cones is {0}9.Since, AP (Rw, 0) = {λ (0, 1) | λ ≥ 0} it follows that

AP (Rw0, 0) ∩ A

{

(µ, σ) | µ ≤ 〈π, V −1π〉 1

2σ, σ ≥ 0}

(2.39)

= {λ(0, 1) | λ ≥ 0} ∩{

(µ, σ) | µ ≤ 〈π, V −1π〉 1

2σ, σ ≥ 0}

= {0} (2.40)

8Note the relationship to the discussion of useful trade vectors by Werner (1987)9for details see Debreu (1959) or Hildenbrand (1974)

13


Therefore, continuity of the utility function U implies the existence of demand. QED.

It follows from results of standard consumer theory, that strict convexity of preferencesimplies that demand is a continuous function. However, one additional property is re-quired to obtain non zero demand, in other words a condition is needed which inducesnon zero demand for risk for all positive shadow returns 〈π, V −1π〉 1

2 > 0.

Consider the utility contour associated with the initial wealth Rw

{(µ, σ) | U((µ, σ)) = U(Rw, 0)} .

Since U is strictly increasing in µ, there exists a function h : R × R+ → R, (Rw, σ) 7→h(Rw, σ), describing the indifference curve through (Rw, 0). h is continuous, monoton-ically increasing, and convex in σ. Moreover, h(Rw, 0) = Rw0 for all Rw. It is evidentthat the curvature of the function h is related to the properties of the asymptotic coneof P (Rw, 0). More specifically, AP (Rw, 0) is equal to the half line {λ (0, 1)} ⊂ R

2 if andonly if

limσ→∞

h(Rw, σ) −Rw

σ= ∞.

Exploiting the curvature of h near zero, it is evident that demand for risk will be positivefor all 〈π, V −1π〉 1

2 > 0 if and only if

limσ→0

h(Rw, σ) −Rw

σ= 0.

These two properties are a direct translation of the so called weak Inada condition used ingrowth models to convex functions. Combining these observations with the results fromlemmas (2.1), (2.2), and (2.3), one obtains the following theorem on existence, uniqueness,and positivity of demand under mean-variance preferences.

Theorem 2.1Let expectations satisfy (2.1) and assume that preferences are strictly quasi concave sat-isfying assumption (2.2). If, in addition, the Inada conditions

(a) limσ→0

h(Rw, σ) −Rw

σ= 0 and (b) lim

σ→∞

h(Rw, σ) −Rw

σ= ∞, (2.41)

hold, then, for all 〈π, V −1π〉 1

2 > 0,

(i) asset demand is a continuous function for all π 6= 0

(ii) the demand for risk s(w, 〈π, V −1π〉 1

2 ) > 0

(iii) asset demand ϕ(w, π, V ) =s(w, 〈π, V −1π〉 1

2 )

〈π, V −1π〉 1

2

V −1π 6= 0

Proof:Part (b) of the Inada conditions (2.41) together with strict quasi concavity implies exis-tence, uniqueness, and continuity of demand. Positive demand for risk follows from part(a) of the Inada condition, which implies a non zero asset demand. QED.

14

2.5 Differentiable Demand 15

2.5 Differentiable Demand

It is now straightforward to derive conditions under which one obtains differentiable de-mand functions by applying results from standard consumer theory. Due to the factor-ization lemma (2.2) it is sufficient to derive a differentiable demand function for risk. For

given expectations and prices define ρ := 〈π, V −1π〉 1

2 > 0 as the shadow return (or equiv-alently as the standard deviation of the standardized portfolio V −1π. Then, the demandfor risk is given by

s(w, ρ) := arg max {U(µ, σ) | µ ≤ ρσ +Rw} . (2.42)

The demand for risk induces a demand for expected return as

m(w, ρ) := Rw + ρs(w, ρ) (2.43)

This maximization corresponds to a standard consumer problem with a linear budget set,where ρ is the price/return for risk taking. Since the utility is decreasing in risk equation(2.42) corresponds to a consumer’s supply of a factor to the market (like labor with realwage ρ). Nevertheless, we will continue to speak of demand for risk rather than supply.As a consequence, all further results on individual demand for risk have their completeanalogues in consumer theory with endogenous labor supply allowing a direct applicationof all known results.

To obtain a differentiable demand function for risk the following additional assumption isnecessary and sufficient (cif. for example Bohm & Barten (1982), Debreu (1972), Katzner(1968)).

Assumption 2.3Let the utility function

U :

{

R × R+ → R

(µ, σ) 7→ U(µ, σ)

be strictly quasi concave, twice continuously differentiable and satisfy (2.2) as well as

∣

∣

∣

∣

∣

∣

U11 U12 U1

U21 U22 −U2

U1 −U2 0

∣

∣

∣

∣

∣

∣

6= 0 (2.44)

for all interior (µ, σ).

The condition (2.44) of a non vanishing bordered Hessian matrix implies that the Gaussiancurvature of all indifference curves is different from zero at all interior points, which inturn implies the differentiability of the demand function for risk s : R × R+ → R+,(w, ρ) 7→ s(w, ρ), (see Katzner (1968) and Debreu (1982)).

Lemma 2.4Given the assumptions to guarantee differentiable demand for risk and return, the follow-

15


ing properties hold for all interior (w, ρ):

∂m

∂w(w, ρ) = R + ρ

∂s

∂w(w, ρ) (2.45)

∂m

∂ρ(w, ρ) = s(w, ρ)

(

1 +ρ

s(w, ρ)

∂s

∂ρ(w, ρ)

)

(2.46)

∂s

∂ρ(w, ρ) > s(w, ρ)

∂s

∂w(w, ρ) (2.47)

Proof:The first two equations are identities induced by monotonicity and the budget equation.Equation (2.47) follows from standard duality and the Slutzky equation. QED.

Combining (2.45) – (2.47) one obtains

Eσ,ρ(w, ρ) > ρ∂s

∂w(w, ρ) =

∂m

∂w(w, ρ) −R > −R (2.48)

if ∂m/∂w > 0 where

Eσ,ρ(w, ρ) :=ρ

s(w, ρ)

∂s

∂ρ(w, ρ)

defines the elasticity of the demand for risk with respect to ρ, which is bounded below.

If wealth has a positive effect on the chosen amount of risk, ∂s/∂w > 0, this fact expressesa higher desire for risk with higher wealth. Using traditional terminology in this situa-tion, define risk to be a normal good10 if ∂s/∂w ≥ 0. By analogy from standard microeconomic terminology, define risk as an inferior good if ∂s/∂w < 011. Notice that whatis interpreted here as demand for risk is essentially the supply of a willingness to acceptrisk, which corresponds to the micro economic analogue of a household’s labor supply.Notice, that with this terminology, risk normality corresponds to the situation of laborsupply increasing in wealth implying that leisure is an inferior good (a situation usuallynot considered as typical – or normal !).

The Slutzky equation (2.47) and risk normality imply that a higher shadow return torisk implies a higher chosen amount of risk, i. e. ∂s/∂ρ > 0. Therefore, demand forrisk is globally increasing in the shadow return, a property which is necessary for uniqueequilibria (see Section ?? below). Clearly, risk inferiority is only a necessary conditionfor a decreasing demand for risk in the shadow return ρ, allowing for positive as well asnegative price effects. Moreover, risk normality may not be the most widely acceptedeconomic assumption to be used for the CAPM model. Therefore, it seems desirable toelucidate those assumption under risk inferiority for which demand for risk is still glob-ally increasing. It is difficult to obtain intuitive economic assumptions for general meanvariance preferences which imply monotonicity of demand for risk in general. However,for separable utility functions one obtains the following result.

10Hens, Laitenberger & Loffler (2002) call this property decreasing (non increasing) risk aversion.11called increasing risk aversion by Hens, Laitenberger & Loffler (2002)

16


Theorem 2.2Consider a separable concave twice continuously differentiable utility functionU : R × R+ → R of the form

U(µ, σ) = u(µ) − v(σ) (2.49)

satisfying (2.2) and

v′(0) = 0, limσ→∞

v′(σ) = ∞, and v′′(σ) > 0 for all σ > 0. (2.50)

If for all µ ∈ R:

∆ (ua)′(µ+ ∆) strictly increasing in ∆ ≥ 0 and lim∆→∞

∆u′(µ+ ∆) = ∞ (2.51)

hold, then,

(i) the demand for risk

s(w, ρ) := arg maxσ

u(Rw + ρσ) − v(σ)

is well defined and continuously differentiable in (w, ρ),

(ii)s(w, ρ) = 0 if and only if ρ = 0, (2.52)

(iii)limρ→∞

s(w, ρ) = ∞ for every w ∈ R (2.53)

(iv)∂

∂ws(w, ρ) =

Ru′′(Rw + ρs(w, ρ))

v′′(s(w, ρ)) − ρ2u′′(Rw + ρs(w, ρ)), (2.54)

i. e. risk is an inferior good, whenever u is strictly concave,

(v)∂

∂ρs(w, ρ) =

u′(Rw + ρs(w, ρ)) + ρs(w, ρ)u′′(Rw + ρs(w, ρ))

v′′(s(w, ρ)) − ρ2u′′(Rw + ρs(w, ρ))> 0 (2.55)

i. e. the demand for risk is increasing in the shadow return ρ,

These relatively strong requirements for separable µ−σ preferences are somewhat surpris-ing, but they represent the subtle balance between wealth effects and substitution effectsto induce monotonic demand. As the proof below shows, they are essentially necessary.At the same time, they integrate the quasi linear utility and the strict concavity intoone uniform set of assumptions. Regarding wealth effects they imply that the associatedexpansion paths are never up ward sloping. The two situations of quasi linearity (u′′ = 0)and (v′′ = 0) are boundary cases. Observe that the theorem does not cover the case withpiece wise quasi linear functions u, since these violate the assumptions (2.51). In suchcases, the demand for risk will not be monotonic in the shadow return (see Figure 3.5).

It is also somewhat unexpected that the sign of the price effect on the demand for risk isexclusively determined by second order properties of the function u. Standard economic

17


intuition would suggest that the function v also captures part of an agent’s attitude towardrisk, which seems counterintuitive to the result that v has no influence on the sign of thederivative of demand for risk12. The assumption (2.51) implies that u′ is convex but thatits steepness is bounded.

Proof:The function H(ρ, σ) := u(Rw+ ρσ)− v(σ) is strictly concave in σ for every ρ ≥ 0. Since

∂H

∂σ(ρ, σ) = ρu′(Rw + ρσ) − v′(σ),

one finds∂H

∂σ(0, σ) = 0 if and only if σ = 0,

since v′(0) = 0 which proves (2.52). If ρ > 0, then

∂H

∂σ(ρ, σ) = 0

has exactly one positive solution. Hence, s(w, ρ) is a function. The strict convexity of thefunction v implies assumption (2.44). Hence, s(w, ρ) is differentiable.

Property (2.53) follows from the first order conditions

0 =∂H

∂σ(ρ, s(w, rho)) = u′(Rw + ρs(w, ρ))

(

ρ− v′(s(w, ρ))

u′(Rw + ρs(w, ρ))

)

,

and from (2.51).

Inferiority of risk (2.54) follows directly from strict concavity and an application of theimplicit function theorem, while monotonicity of demand (2.55) follows from condition(2.51). QED.

Notice that no restriction for the wealth to be positive is required. If this is case one hasthe following corollary.

Corollary 2.1Consider the separable case of theorem (??) and assume conditions (2.2) and (3.14). Iffor all µ ≥ 0

µu′(µ) is strictly increasing (2.56)

andlim

µ→∞µu′(µ) = ∞ (2.57)

hold, then, for given non negative wealth w ≥ 0, the demand for risk s(w, ·) : R+ → R+

is monotonically increasing and bijective in ρ on R+.

12This confirms essentially for the separable case that measures of risk aversion which use propertiesof the function u alone are sufficient in capturing all relevant effects. Whether this is true in general stillis an open question. (compare Hens, Laitenberger & Loffler (2002)).

18


Proof:Equation (2.55) implies

∂

∂ρs(w, ρ) =


v′′(s(w, ρ)) − ρ2u′′(Rw + ρs(w, ρ))

[

1 +ρs(w, ρ)u′′(Rw + ρs(w, ρ))


]

(2.58)

≥ u′(Rw + ρs(w, ρ))

v′′(s(w, ρ)) − ρ2u′′(Rw + ρs(w, ρ))

[

1 +(Rw + ρs(w, ρ))u′′(Rw + ρs(w, ρ))


]

=u′(Rw + ρs(w, ρ))

v′′(s(w, ρ)) − ρ2u′′(Rw + ρs(w, ρ))

[

1 + Eu′(Rw + ρs(w, ρ))]

(2.59)

since the assumption (2.56) is equivalent to

Eu′(µ) :=µu′′(µ)

u′(µ)> −1 for all µ ≥ 0, (2.60)

QED.

The assumption 2.60 on the elasticity of the derivative of u was used also by Dana (1999).Notice, that the elasticity assumption alone does not guarantee the surjectivity of riskdemand13.

The method of proof also reveals that a joint condition like (2.51) is required to guaranteesurjectivity of the demand for risk (2.53). It is not sufficient to assume that the marginalutility of return tends to zero with infinite shadow return to risk, since it cannot beexcluded that the demand for risk remains bounded. Thus, the condition on the strongmarginal disutility of risk as in (2.51) is needed to generate unbounded demand for risk.

In summary, the curvature of u alone determines the sign of the price effect, and thuswhether the associated offer curve is monotonic. However, the two situations of a linearand of a strictly concave utility function u seem to require a different treatment whensurjectivity is required, since the linear case is not a direct boundary situation of thestrictly concave situation. The demand for risk is always monotonically increasing in theshadow return and tending to infinity if risk is a normal good even for non separablepreferences. This includes the quasi linear case of a (weakly) concave utility to expectedreturn. However, if the utility of expected return is not linear (strictly concave or piecewise linear) strong additional properties on the curvature of u are required, since thenrisk becomes an inferior good.

Figures 3.3, 3.2, and 3.4 display three possible offer curves violating theorem (2.2). As instandard demand theory, their curvature is restricted only by quasi concavity, implyingthe usual star shaped form and allowing for inferiority of both goods as well as Giffeneffects. Note, that an offer curve can be backward bending for a large class of concaveutility function.

3 Asset Market Equilibrium

Let each consumers a ∈ A now be characterized by mean–variance preferences U a, initialwealth wa, and beliefs (qa, V a). Let 0 6= x ∈ R

K denote aggregate asset supply. If beliefs

13for example: u(µ) := ln(r + µ), r > 0 implies (2.56) but not (2.57)

19


of consumers are not the same, then one has to distinguish for each a ∈ A a pair of firstand second moment beliefs (qa, V a)a∈A.

Following standard equilibrium terminology an asset allocation (xa)a∈A is said to be fea-sible if

∑

a xa = x. Therefore, for given beliefs, an asset market equilibrium is obtained

by a vector of equilibrium asset prices such that

∑

a∈A

ϕa(wa, qa −Rp) = x. (3.1)

Any vector of equilibrium asset prices induces a feasible asset allocation (xa)a∈A. Sinceindividual demand is a function of subjective asset premia qa − Rp, heterogeneous firstmoment beliefs enter explicitly as an argument (as an additive shift of prices) in eachdemand function while second moment beliefs (V a)a∈A determine the functional formparametrically. This fact is typically suppressed in the standard analysis.

In the context of mean–variance preferences the risk of a portfolio x is measured by its(subjective) variance or standard deviation σ(x) :=

√

〈x, V x〉. Therefore, given individ-ual second moment beliefs (V a)a∈A, an asset allocation (xa)a∈A induces a risk allocation(σa)a∈A by

σa := 〈xa, V axa〉 1

2 a ∈ A

.

Definition 3.1Given individual second moment beliefs (V a)a∈A and an aggregate stock of assets x 6= 0,an asset allocation (xa)a∈A is said to induce an efficient risk allocation (σa)a∈A, if thereis no other feasible asset allocation ya, a ∈ A such that

√

〈ya, V aya〉 ≤√σa for all a ∈ A (3.2)

with at least one strict inequality.

It is immediate that the set of feasible risk allocations

S :=

{

(σa) ∈ R|A|+ | σa := 〈xa, V axa〉 1

2 , a ∈ A,∑

a∈A

xa = x

}

(3.3)

is closed, bounded below, and convex. Moreover, if second moment beliefs are identical,the efficient boundary Seff is given by the simplex

Seff =

{

(σa) ∈ R|A|+ |

∑

a∈A

σa = 〈x, V x〉 1

2

}

(3.4)

Figure 3.1 shows the set of feasible allocations in an Edgeworth box representation for twoagents and two assets displaying the contours of the two variances σ(xa) and σ(x − xa).The ”contract curve” connecting the corners a and b is the (the projection) of the set ofefficient risk allocations in the space for agent a’ assets. If V a = V b the contract curve isa straight line.

20

3.1 Existence and Uniqueness of Equilibrium 21

�

�

��

��

� ��

Figure 3.1: Efficient Risk Allocations

If first moment expectations are identical, any equilibrium asset price vector induces thesame equilibrium asset premium, the equilibrium condition can be rewritten as π ∈ R

K if

∑

a∈A

ϕa(wa, π) = x. (3.5)

In such a case one obtains the following lemma.

Lemma 3.1If first moment beliefs are identical, any asset market equilibrium induces an efficient riskallocation.

3.1 Existence and Uniqueness of Equilibrium

Continuing with the assumption of common (identical) expectations satisfying (2.1) forall consumers14, an asset market equilibrium is achieved for a vector of subjective premiaπ, such that

∑

a∈A

ϕa(wa, π) = x. (3.6)

Exploiting the features of the factorization lemma (2.2), one finds that π is an equilibriumpremium if and only if the condition

(

∑

a∈A

sa(wa, 〈π, V −1π〉 1

2 )

)

1

〈π, V −1π〉 1

2

V −1π = x (3.7)

14Existence of equilibria with heterogeneous beliefs will be discussed in a later section.

21


holds. Let ρ := 〈π, V −1π〉 1

2 and define aggregate demand for risk as

s((wa)a∈A, ρ) :=∑

a∈A

sa(wa, ρ)

Lemma 3.2If aggregate demand for risk s((wa)a∈A, ·) : R+ → R+, ((wa)a∈A, ρ) 7→ s((wa)a∈A, ρ) iscontinuous and surjective in the shadow return ρ, there exists an asset market equilibriumfor every aggregate asset supply x 6= 0 and arbitrary homogeneous beliefs (q, V ) satisfying(2.1). If, in addition, demand for risk is strictly monotonic in ρ, then asset equilibriumis unique with an equilibrium premium given by

π =1

√

〈x, V x〉s−1(

(wa)a∈A,√

〈x, V x〉)

V x (3.8)

and associated equilibrium asset prices

p =1

R

(

q − 1√

〈x, V x〉s−1(

(wa)a∈A,√

〈x, V x〉)

V x

)

(3.9)

where s−1((wa)a∈A, ·) is the inverse of the aggregate demand function for risk with respectto ρ.

Notice that the existence of an equilibrium in the markets for the K assets is reduced tofinding a zero of the one dimensional mapping of excess demand for risk15.

Proof:From the asset market equilibrium equation (3.7)

(

∑

a∈A

sa(wa, 〈π, V −1π〉 1

2 )

)

1

〈π, V −1π〉 1

2

V −1π = x

it follows that any equilibrium premium π must be colinear to V x. Choose λ > 0 arbitraryand define π := λV x 6= 0. Then,

x =

(

∑

a∈A

sa(wa, 〈π, V −1π〉 1

2 )

)

1

〈π, V −1π〉 1

2

V −1π (3.10)

⇐⇒

x =s(

(wa)a∈A, λ〈x, V x〉1

2

)

〈x, V x〉 1

2

x (3.11)

⇐⇒s(

(wa)a∈A, λ√

〈x, V x〉)

=√

〈x, V x〉 (3.12)

In other words, π := λV x is an equilibrium premium if and only if aggregate demandfor risk is equal to aggregate risk for its induced shadow return λ

√

〈x, V x〉. Therefore,

15revealing a formal equivalence to the method of proof used by Dana (1999)

22


continuity and surjectivity of the left hand side of (3.12) in λ implies existence of an assetmarket equilibrium. Strong monotonicity implies the unique solution

λ∗ =1

√

〈x, V x〉s−1(

(wa)a∈A,√

〈x, V x〉)

(3.13)

Thus, ρ∗ = λ∗√

〈x, V x〉 is the equilibrium shadow return to risk. i. e.

∑

a∈A

sa(wa, λ∗√

〈x, V x〉) =√

〈x, V x〉,

implying that the sum of individual risk is equal total risk. This yields the equations forthe unique equilibrium premium (3.8) and for the equilibrium asset price vector (3.9).QED.

We close this section with two theorems of existence and uniqueness by stating two setsof economic assumptions on preferences, beliefs, and wealth to guarantee a unique assetmarket equilibrium.

Theorem 3.1Let beliefs of consumers be identical and satisfy (2.1). Assume that preferences of allconsumers a ∈ A fulfill assumptions (2.3), and the Inada conditions (2.41).

If risk is a normal good for all consumers and at least one consumer has quasi linearpreferences in µ, there exists a unique asset market equilibrium for any expectations (q, V )and any aggregate supply of assets x 6= 0.

Proof:In view of lemma (3.2) it is sufficient to show that aggregate demand for risk is continuous,strictly monotonic and surjective in ρ. Assumptions (2.1), (2.3), and the Inada conditions(2.41) imply that individual demand for risk exists and is a continuous function for allρ > 0 with

limρ→0

sa(wa, ρ) = 0.

Normality of risk implies that the demand for risk is increasing in ρ. Moreover, the Inadaconditions and quasi linearity for some agent a imply that

limρ→∞

sa(wa, ρ) = ∞.

Hence, aggregate demand is surjective and there exists a unique shadow return ρ∗ with

ρ∗ = s−1(

(wa)a∈A,√

〈x, V x〉)

. QED.The assumption that risk is a normal good for all consumers seems fairly restrictive. ,since there is a large set of reasonable preferences for which risk is an inferior good (as inexamples (2.32) and (2.29). Therefore, it is desirable to obtain existence and uniquenessin such cases as well. The following theorem provides a positive answer for the subclassof separable preferences.

23


Theorem 3.2Assume that beliefs of consumers are identical satisfying (2.1) and that preferences satisfyassumption (2.3). If, for all consumers a ∈ A, one has

(va)′(0) = 0, limσ→∞

(va)′(σ) = ∞, and (va)′′(σ) > 0 for all σ > 0, (3.14)

∆ (ua)′(µ+ ∆) strictly increasing in ∆ ≥ 0 for all µ ∈ R, (3.15)

and for at least one consumer

lim∆→∞

∆ (ua)′(µ+ ∆) = ∞ (3.16)

Then there exists a unique asset market equilibrium for any expectations (q, V ) and anyaggregate supply of assets x 6= 0.

Proof:Assumptions (2.1) and (2.3) for all consumers a ∈ A imply that aggregate demand forrisk is differentiable. It follows from (2.55) that the demand for risk is an increasingfunction for every agent a ∈ A. In addition, (3.16) for at least one agent implies thatlimρ→∞ sa(wa, ρ) = ∞ which yields this property for the aggregate demand for risk.Therefore, aggregate demand for risk satisfies the conditions of lemma (3.2). QED.

Observe that both of the preceding theorems include in their assumptions the case of quasilinear preferences in µ for all agents. However, for preferences with strict concavity in µ,the two sets of assumptions are mutually exclusive, since risk is an inferior good underseparability. The condition (3.16) is crucial in guaranteeing existence, while the mono-tonicity condition (3.15) assures uniqueness only. Example 3 provides a case where themonotonicity condition is satisfied but demand for risk is bounded. For such a situationit is easy to construct examples with no equilibrium and beliefs satisfying assumptions(2.1)16.

The factorization lemma (2.2) implies that individual asset demand is given by

ϕa(wa, π) =sa(wa, ρ)√

〈x, V x〉x =

sa(wa, ρ)

σx

which is a positive multiple of the market portfolio. Thus, in equilibrium agents hold thesame mix of assets as the market portfolio and they share aggregate risk proportionally,confirming the mutual fund result of the CAPM literature.

Finally, asset prices are given by

p =1

R

[

q − ρ

〈x, V x〉 1

2

V x

]

(3.17)

such that s((wa)a∈A, ρ) = 〈x, V x〉 1

2 . Thus, equilibrium asset prices are an affine map inexpected prices plus an additive constant which depends on preferences, second momentbeliefs, and aggregate asset supply. This corresponds to a form of the ’beta pricing rule’,which can be transformed directly into the usual affine relation ship between returns.Observe that equilibrium prices are bounded above by the discounted subjective first

24


�

��

��

Figure 3.2: No equilibrium x

�

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� �� !

Figure 3.3: Two equilibria

25


�

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� ��

Figure 3.4: Three equilibria

moment expectations if V x is non negative, and prices are positive only if expectationsare sufficiently optimistic. Figures (3.2), (3.3), and (3.4) provide geometric insightinto the existence and uniqueness issue. They display the aggregate offer curve for threedifferent situations for which existence or uniqueness fails. In the case of piece wisequasi linear preferences it is apparent that the demand for risk cannot be differentiableand monotonically increasing for all wealth levels. There can be multiple sections withdeceasing demand for risk. This implies the possibility of an arbitrary finite number ofequilibria, a situation depicted in Figure 3.5.

16Dana (1999) uses only (3.15) for positive µ in her proof of existence.

26

3.2 The equilibrium set and determinacy 27

�

�

��

�

Figure 3.5: Multiple equilibria with piece wise quasi linear preferences

3.2 The equilibrium set and determinacy

In the case of uniqueness we can now summarize the results as follows providing a com-plete characterization of asset prices under general and homogeneous beliefs for arbitrarypreferences, initial wealth, and aggregate stock of assets.

Let (q, V ) and x 6= 0 be given and σ := 〈x, V x〉 1

2 denote the market risk. Define ρ > 0 tobe the unique market clearing shadow return; i. e.

s((wa)a∈A, ρ) = 〈x, V x〉 1

2 . (3.18)

If the demand for risk is differentiable and regular, standard results from the theory ofregular economies ((cif. Mas-Colell 1985)) provide the parallel results for the case here.For a smooth regular aggregate demand function for risk, equilibria are determinate andgenerically finite, implying the typical structure of the equilibrium manifold for givenpreferences and second moment beliefs

Wρ :={

(w, σ, ρ) ∈ R|A| × R

2+ | s((wa)a∈A, ρ) = σ}

}

. (3.19)

Figures (3.6), (3.7), and (3.8) show the three essential cases which may occur.

Since the asset price equation (3.17) is a smooth mapping, the equilibrium manifold forasset prices

W :=

(w, q, V, x, p)

s(w, ρ) = 〈x, V x〉 1

2 , ρ > 0

p =1

R

[

q − ρ

〈x, V x〉 1

2

V x

]

(3.20)

27


�

��

��

Figure 3.6: Unique equilibrium x

��

��

Figure 3.7: Global equilibrium manifold x

28


�

��

��

Figure 3.8: Bounded equilibrium manifold x

with w := (wa)a∈A and

(w, q, V, x, p) ∈ R|A| × R

K × RK2 × R

K+ × R

K

is well behaved in general. Thus, the following proposition is an immediate consequenceof standard results from smooth equilibrium analysis (see for example Mas-Colell (1985)).

Proposition 3.1Let beliefs satisfy (2.1) and assume that aggregate asset demand s(w, ρ) is regular. Then,

W ⊂ R|A| × R

K × RK2 × R

K+ × R

K

is a smooth manifold. Thus, the set of equilibrium asset prices is finite.

Proof:The equilibrium manifold W is the graph of the smooth price map (3.17) restricted tothe smooth manifold (3.19). QED.

Summarizing the results so far, one observes that existence and uniqueness of equilibriumare guaranteed under a set of intuitive economic assumptions on preferences and beliefsfor the mean variance case. The fact that beliefs are homogeneous (identical) reduces theexistence issue to finding a zero of a one dimensional excess demand function for risk,formally similar to the reduction used by Dana (1999), but conceptually quite distinct.At the same time, such equilibria are mean variance efficient given expectations. Thus,

29

4 Equilibria with asset endowments 30

in equilibrium, asset bundles are all positive multiples of the aggregate market portfolio.As a consequence, the equilibrium shadow return to risk ρ is the same for all consumersand equal to the subjective Sharpe ratio (i. e. the ratio of expected return to standarddeviation). Thus, individual risk taking induces an efficient (ex ante) risk sharing.

4 Equilibria with asset endowments

Consider the capital asset pricing model as a pure exchange model with private ownershipof asset endowments. In this situation total wealth of agents is endogenous and pricedependent. It is straightforward to extend the above model to the case with individualasset endowments and endogenous wealth.

Let (ea, xa) ∈ RK+1+ , a ∈ A, denote agent a’s endowment of the numeraire commodity

ea ∈ R+ and of the risky assets xa ∈ RK+ , with total asset endowments x =

∑

A xa. Then,

the list {(Ua, (ea, xa))})a∈A is an exchange economy in the usual sense.

Definition 4.1A mean–variance asset exchange economy with private ownership E is a list

E := {(Ua, (ea, xa)), (qa, V a))a∈A} (4.1)

such that for all a ∈ A:

(iii) Ua satisfies assumption (2.2)

(ii) (ea, xa) ∈ RK+1+

(i) (qa, V a) satisfies assumption (2.1),

For any asset price vector p ∈ RK , the relationship between the previous model and the

private ownership situation is the usual one. For every agent a ∈ A, define the wealthfunction

Wa(p) := ea + 〈p, xa〉. (4.2)

and, for given beliefs (qa, V a), consider the equilibrium asset price equation (3.17) as ageneral price law of P : R+ → R

K defined by

p = P (ρ) :=1

R

[

q − ρ

〈x, V x〉 1

2

V x

]

(4.3)

4.1 Homogeneous Beliefs

Definition 4.2Given homogeneous beliefs (q, V ), an asset price vector p ∈ R

K is an equilibrium pricevector if there exists a shadow return ρ ≥ 0 and a wealth distribution w := (wa)a∈A ∈ R

|A|,

30

4.1 Homogeneous Beliefs 31

such that

x =∑

a∈A

xa (4.4)

〈x, V x〉 1

2 =∑

a∈A

sa(wa, ρ) (4.5)

p =1

R

[

q − ρ

〈x, V x〉 1

2

V x

]

(4.6)

ρ = 〈(q −Rp), V −1(q −Rp)〉 1

2 (4.7)

wa = Wa(p) = ea + 〈p, xa〉 a ∈ A (4.8)

In other words an asset market equilibrium is a fixed point of the four mappings givenby the right hand sides of equations (4.5)– (4.8). Therefore existence and uniqueness cannow be proved exploiting the properties derived in the previous sections. The two resultsare directly parallel of the two previous theorems 3.1 and 3.2.

Theorem 4.1Let individual endowments (ea, xa) ≥ 0 with x =

∑

a∈A xa 6= 0,

∑

a∈A ea > 0. Assume

that preferences of all consumers a ∈ A fulfill assumptions (2.3), and the Inada conditions(2.41).

If risk is a normal good for all consumers and at least one consumer has quasi linearpreferences in µ, there exists a unique equilibrium asset price p ∈ R

K for the privateownership economy for all beliefs satisfying (2.1) such that q >> 0 and V x� 0.

The second theorem which allows for risk inferiority parallels that of Theorem (3.2).

Theorem 4.2Let individual endowments (ea, xa) ≥ 0 with x =

∑

a∈A xa 6= 0,

∑

a∈A ea > 0. If, for all

consumers a ∈ A, preferences are separable with 2.3 such that:

(va)′(0) = 0, limσ→∞

(va)′(σ) = ∞, and (va)′′(σ) > 0 for all σ > 0, (4.9)

∆ (ua)′(µ+ ∆) strictly increasing in ∆ ≥ 0 for all µ ∈ R, (4.10)

and for at least one consumer

lim∆→∞

∆ (ua)′(µ+ ∆) = ∞ (4.11)

there exists a unique equilibrium asset price p for the private ownership economy for allbeliefs satisfying (2.1) such that q >> 0 and V x� 0.

Notice that the conditions (4.9)–(4.11) are the same as (3.14)–(3.16). We only give theproof of the extension for the case of Theorem (3.2).

31

4.1 Homogeneous Beliefs 32

Proof:Theorem (3.2) implies the existence of a continuous function r : R

|A| → R+ defining theunique equilibrium shadow return ρ = r(w) for every initial wealth w = (wa)a∈A, i. e. ris the solution function for equation (4.7).

〈x, V x〉 1

2 = s(w, r(w)) =∑

a

sa(wa, r(w)).

Consider the mapping

H :=

{

R|A| → R

|A|

w 7→ H(w)

defined byH(w) = (Ha(w))a∈A Ha(w) := (Wa ◦ P ◦ r)(w) (4.12)

H is continuous and each Ha(w) is bounded above by

w :=∑

ea + 〈x, 1

Rq〉

and below by

w := mina∈A

(

ea + 〈xa,1

R

[

q − r(w)

〈x, V x〉 1

2

V x

]

〉)

.

Hence, restricting H to the compact convex set

W :={

w ∈ R|A| | w ≤ wa ≤ w, a ∈ A

}

implies H : W → W . Since H is continuous, there exists a fixed point w∗ ∈ W . Thisinduces an equilibrium shadow returnρ∗ = r(w∗) and equilibrium asset prices

p∗ =1

R

[

q − ρ∗

〈x, V x〉 1

2

V x

]

(4.13)

To prove uniqueness, assume to the contrary that there exists two fixed points w1 6= w2. Ifρ1 = r(w1) = r(w2) = ρ2, there exist a unique asset price from equation (4.13). Supposeρ1 > ρ2 > 0. This yields

p1 =1

R

[

q − ρ1

〈x, V x〉 1

2

V x

]

< p2 =1

R

[

q − ρ2

〈x, V x〉 1

2

V x

]

,

implying w2 > w1 since endowments are positive. Since preferences are assumed to beseparable and strictly concave, risk is an inferior good. The assumption (??) implies thatthe demand for risk increases in the shadow return ρ. Therefore, one finds that

sa(w1, ρ1) > sa(w2, ρ1) > sa(w2, ρ2)

so that〈x, V x〉 1

2 =∑

A

sa(w2, ρ2) >∑

a

sa(w1, ρ1) = 〈x, V x〉 1

2

which is a contradiction. QED.

The following example indicates that an assumption on the limiting behavior such as(2.56) cannot be dispensed with if existence is to be guaranteed.

32

5 Conclusion 33

Example

Consider an exchange economy with two agents A = {a, b}

0 < r < 〈x, V x〉 and q >> 0

. There exists H ⊂ {1, . . . , K} such that

(V x)k < 0 k ∈ H, (V x)k > 0 k /∈ H, (4.14)

(xa)k > 0 k ∈ H, (xa)k = 0 k /∈ H, (4.15)

(xb)k = 0 k ∈ H, (xb)k > 0 k /∈ H, (4.16)

Ua(µ, σ)) = ln(r + µ) − σ2 ea > 0, (4.17)

U b(µ, σ)) = −σ2 eb = 0. (4.18)

It is apparent that consumer b will supply (x)b at any price with zero demand for risk.Notice that, because of the negative correlation for commodities k ∈ H, equilibrium pricesfor these commodities must be positive, since

p =1

R

(

q − λ

〈x, V x〉 1

2

V x

)

.

Therefore, a’s wealth will be larger than ea > 0 implying that his demand for risk will beless than r. Hence, there is excess supply of risk for all asset prices.

Finally, Figure 4.1 portrays the situation of multiple equilibria with two agents A = {a, b}where b has quasi linear preferences and a a has a piece wise linear utility in the return µ.

5 Conclusion

Under a general non redundancy assumption on expectations, the CAPM with mean–variance preferences induces demand behavior explicitly derivable from intuitive assump-tions using standard concepts like normality and the Slutzky decomposition. For ho-mogeneous expectations globally invertible demand for risk is obtained using elementarytechniques. By treating expectations parameters explicitly the interplay between expec-tations and actual equilibrium asset prices can be revealed formally, reconfirming the twofundamental findings of CAPM theory: the mutual funds theorem and the beta pricingrule. The results of the paper have shown for the CAPM model, that existence anduniqueness of equilibria are essentially determined by those properties of risk preferencesin mean in standard deviation which guarantee globally invertible demand functions forrisk, when agents hold arbitrary but homogeneous beliefs. These features have an immedi-ate impact on the equilibrium set, inducing typically smooth manifolds when preferencesare smooth.

33

5 Conclusion 34

�

�

��

��

��

Figure 4.1: Multiple Equilibria with two agents

34

REFERENCES 35

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REFERENCES 36

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