Asset pricing: A structural theory and its applications

ASSETA Structural Theoryand Its Applications

PRICING

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N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

ASSET

Bing ChengChinese Academy of Science, China

Howell TongLondon School of Economics, UK

A Structural Theoryand Its Applications

PRICING

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ASSET PRICINGA Structural Theory and Its Applications

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Preface

Asset pricing theory plays a central role in finance theory and applications.Every asset, liability or cash flow has a value but an essential problem is,how to price it. In recent years, people even started talking about pric-ing an idea as an intangible asset! During the past half century, thereappeared several methodologies to solve this problem. The first method-ology is based on various economic partial (or general) equilibrium pricingmodels. Among these, the best known is Lucas’ consumption-based as-set pricing model. It links the asset pricing problem with the dynamicalmacro-economic theory. However, it is challenged by the well-known equitypremium puzzle as pointed out by Mehra and Prescott in 1985. The issueis unresolved to-date. The second methodology is based on the so-calledFirst Theorem in Finance due to Ross in 1976. Specifically, it values anasset by invoking the no-arbitrage principle in a complete capital market.This methodology/approach leads to, the well-known Arrow-Debreu secu-rity, the risk-neutral pricing, the APT (arbitrage pricing theory), and theequivalent-martingale measure. Interestingly, Sharpe’s CAPM model canbe derived from both methodologies referred to above. The third method-ology involves the production-based pricing models. It links, in the longterm, the firm economic growth theory with the asset pricing problem. Inrecent years, there has been a trend to unify the above methodologies andviews under the title of stochastic discount factor (SDF) pricing models.Cochrane (2000) has summarized this trend.

In this book, we develop a new theory, which we call the StructuralTheory, for asset pricing thereby putting the SDF pricing model firmly ona mathematical foundation. It includes a series of original results. Here welist some of the important ones. We separate the problem of finding a betterasset pricing model from the problem of searching for “no equity” premiumpuzzle. The uniqueness theorem and the dual theorem of asset pricingindicate that, given market traded prices, a necessary and sufficient condi-

v

vi Preface

tion for the pricing functional space (or the SDF space) to have a uniquecorrectly pricing functional (or correctly pricing SDF) is that, the space isisometric to the asset payoff space. The orthogonal projection operator, in-troduced in the dual theorem, provides a bijective and valuation-preservingmapping between the two spaces. A new explanation for the Mehra andPrescott’s puzzle can be described as follows: (1) The structure of the con-sumption growth power space is not rich enough to provide an SDF whichis capable of pricing every portfolio in the asset space correctly (i.e., thetwo spaces are not isometric), within feasible ranges of the economic pa-rameters. For example, when the risk aversion is chosen to be less than5, a big pricing error appears. (2) In order to have no pricing error, giveninsufficient structure of the SDF space, the estimated parameter has to beexaggerated to an unreasonable level. For example, the risk aversion mustbe beyond 50 for the U.S market. The structure theory indicates that theappearance of the equity premium puzzle is relative and it depends on amatching (or rather a valuation-preserving isometric mapping) between anSDF space and an asset space. For matching pairs of the two spaces, thereexists a unique SDF to price every portfolio in the asset space correctly.If the correctly pricing SDF is with sensible economic parameters, thenthere is no puzzle. However, if the correctly pricing SDF is with infeasibleeconomic parameters, we say that the puzzle appears to this SDF space.Theoretically, given the asset space, we can remove this puzzle by enlargingthe SDF space to one with sensible new economic parameters, for example,by adding new economic state variables to span a bigger SDF space ratherthan extending the range of the parameters in the original (smaller) SDFspace to an unreasonable level. If the augmented SDF space is matchedto the asset space, there is a new SDF that may provide the tool to priceevery portfolio in the asset space correctly, with the result that there is nopricing error and hence no puzzle to the asset space. Alternatively, giventhe SDF space, we may, by dropping some assets from the asset space, finda new correctly pricing SDF with sensible parameters to the reduced assetspace, with the result that we would then see no pricing error and hence nopuzzle to the reduced asset space. In general, mis-matching an SDF spacewith an asset space will definitely create some pricing error. The puzzleis in fact the result of an improper attempt to remove the pricing error.Using the above theory, it is easy to see why the Epstein-Zin model is lessprone to cause the puzzle. The SDF space generated by the Epstein-Zinmodel is richer than the CRRA (constant-relative-risk aversion) based SDF

Preface vii

space used by Mehra and Prescott in 1985. The Epstein-Zin SDF space isspanned by two state variables, namely the consumption growth and themarket return. But the CRRA (constant-relative-risk aversion) based SDFspace is only spanned by the single consumption growth state variable.So, within a relatively reasonable range of the parameters, the Epstein-Zinbased model is capable of achieving a smaller pricing error for the sameasset space.

The symmetric theorem of asset pricing provides a way to value non-tradable factors, such as economic indices, by reflexively using market as-sets and their corresponding market prices. The expanding theorem of assetpricing provides a bottom-up way to construct a correctly pricing SDF foran asset space. Based on correctly pricing SDFs for subspaces of the assetspace and other covariance information, we can find a unique SDF, in aminimum complete expansion of the sum of SDF subspaces, to price wholeportfolios in the asset space correctly. The compression theorem of assetpricing provides a top-down way to construct asset pricing models. To pricewell-diversified asset portfolios with K-factor structures correctly, a neces-sary and sufficient condition for an SDF space to have a unique correctlypricing SDF is that the SDF space possesses a K-factor structure as well. Inother words, both spaces have no idiosyncratic risk and only K factor risksare left to be considered. Cochrane (2000) has used this fact without givinga rigorous proof. A combination of the expanding theorem and the com-pression theorem can provide a routine way to value portfolios at differentlevels. Based on the theory of corporate finance, the theory of interest rateand the theory of derivative pricing, the valuation of an individual securityhas been well developed. However, portfolio valuation, in particular, riskarbitrage portfolio valuation, well-diversified portfolio valuation or indexvaluation, is less well developed. The pricing error theorem of asset pricingindicates a way of measuring how well a given SDF does the pricing job,by first projecting it and the unknown correctly pricing SDF into the assetspace, and then measuring the closeness between the two projected proxiesby using, for example, the Hansen-Jagannathan distance. There are threepossible sources of pricing errors: the difference in the means of the proxies,the difference in the volatilities of the two proxies, and the imperfect cor-relation between the two proxies. Empirical results suggest that the maincontribution to the pricing error is the difference in volatility.

In a multi-period framework, we propose to link CPPI (constant pro-portion portfolio insurance) with Merton’s consumption pricing model withminimal constraint on consumption.

viii Preface

Throughout this book, various real examples are used to illustrate ideasand applications in practice.

Finally, we are most grateful to the National Science Foundation ofChina for their support to Bing Cheng under the Risk Measurement Project(No. 70321001) and the Scientist Group on Uncertainty, and the LondonSchool of Economics and Political Science for granting Howell Tong a sab-batical term to work on the book. Bing Cheng wishes to thank his wife,Corin, for the assistance in preparing the figures in this book and for hersupport when he did the research in 2005 and wrote the book in 2007.

Contents

Preface v

1 Introduction to Modern Asset Pricing 11.1 A Brief History of Modern Asset Pricing Models . . . . . . 11.2 The Equity Premium Puzzle . . . . . . . . . . . . . . . . . . 5

2 A Structural Theory of Asset Pricing 132.1 Construction of Continuous Linear Pricing Functionals . . . 132.2 The Structural Theory of Asset Pricing – Part I . . . . . . . 172.3 Is the Equity Premium Puzzle Really a Puzzle or not a Puzzle? 222.4 Conclusions and Summary . . . . . . . . . . . . . . . . . . . 28

3 Algebra of Stochastic Discount Factors 313.1 Symmetric Theorem of Asset Pricing . . . . . . . . . . . . . 313.2 Compounding Asset Pricing Models . . . . . . . . . . . . . 343.3 Compression of Asset Pricing Models . . . . . . . . . . . . . 413.4 Decomposition of Errors in Asset Pricing Models . . . . . . 463.5 Empirical Analysis of the Asset Pricing Models . . . . . . . 51

3.5.1 The data set and utility forms . . . . . . . . . . . . 513.5.2 Three sources of pricing errors . . . . . . . . . . . . 523.5.3 Decomposition of the pricing errors . . . . . . . . . . 54

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Investment and Consumption in a Multi-period Framework 574.1 Review of Merton’s Asset Pricing Model . . . . . . . . . . . 574.2 Optimal Decisions of Investment and Consumption . . . . . 60

4.2.1 Martingale approach to the asset pricing model with-out consumption habit constraints . . . . . . . . . . 60

ix

x Contents

4.2.2 Martingale approach to the asset pricing model withconsumption habit . . . . . . . . . . . . . . . . . . . 61

4.3 Optimal Investment Behavior . . . . . . . . . . . . . . . . . 644.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 71

Index 75

List of Figures

2.1 The Uniqueness Theorem of the minimum correctly pricingfunctional space. . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 The Dual Theorem between an SDF Space and an asset pay-off space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Implication of the Dual Theorem. . . . . . . . . . . . . . . . 232.4 Plot of the SDF equation given by the CRRA model. . . . . 262.5 Plot of the SDF equation given by the Epstein-Zin model. . 28

3.1 Symmetric Theorem between SDF space and asset space. . 343.2 Expanding Theorem for SDF spaces and asset spaces. . . . 403.3 Compression Theorem for SDF spaces and asset spaces. . . 453.4 Pricing Error Theorem. . . . . . . . . . . . . . . . . . . . . 493.5 Comparing correlation error in the Pricing Error Theorem. 50

xi


List of Tables

2.1 Values of E[mR] given by the Epstein-Zin model (β = 0.95) 28

3.1 Three sources of pricing errors in the CRRA model(β = 0.95) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Three sources of pricing errors in the Abel model (β = 0.95) 533.3 Three sources of pricing errors in the Constantinides model

(β = 0.95, γ = 5) . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Three sources of pricing errors in the Epstein-Zin model

(β = 0.95, γ = 5) . . . . . . . . . . . . . . . . . . . . . . . . 533.5 Representation errors for various utilities for a single risky

asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Representation errors for Constantinides utility . . . . . . . 543.7 Representation errors for Epstein-Zin various utility . . . . 553.8 Representation errors for a portfolio of risky assets

(The utility is CRRA with β = 0.95 and γ = 5) . . . . . . . 55

xiii


Chapter 1

Introduction to Modern

Asset Pricing

1.1 A Brief History of Modern Asset Pricing Models

The history of asset pricing is more than three hundred years old but Mod-ern Finance started only about half a century ago. It is generally acceptedthat Arrow’s paper entitled Optimal Allocation of Securities in Risk Bear-ing in 1953 marked the starting point of Modern Finance. In 1874, theFrench economist, L. Warlas, introduced the concept of general equilib-rium, making it the first notion of economic equilibrium in the history ofeconomics study, which was followed by numerous contributions from manyfamous economists. It was only in 1954 that Arrow and Debreu finally gavea proof to the existence of a general equilibrium.

In Arrow’s paper in 1953, he interpreted a financial security as a seriesof commodities in various future states with different values. This inter-pretation was later refined by Debreu, who incorporated an equilibriummodel over a state space to deal with a financial market so that a securitywas nothing but a commodity with different values in different states andat different times. His notions of a state price and a state security (orArrow security) are very popular now. All of these notions are based onthe assumption of a complete financial market, that is, corresponding toeach contingent state, there is an Arrow security to be traded in a financialmarket.

When it comes to stock investment, what is its utility? In his doctoraldissertation entitled Theory of Investment Values in 1937, William proposedan appraisal model of stock that asks investor to do a long-term forecast of

1

2 Chapter 1. Introduction to Modern Asset Pricing

a firm’s future dividends and check the accuracy of the forecasting. Later,William described how to combine the forecasting with its accuracy toestimate the intrinsic value of a stock, leading to the well-known DDM(discount dividend model).

When Markowitz studied the DDM model, he found that if all investorsfollow a DDM model, they would all purchase the stock with the highest ex-pected return and avoid other stocks. This was obviously counter-intuitive.So in his seminal paper in 1952, he proposed that investors need to considera balance between the return and the risk; indeed he used the mean to de-scribe the expected return and the variance to describe the risk. With thisframework, he developed a mean-variance efficient frontier in that giventhe risk level, an optimal portfolio with the highest expected return is ob-tained, or equivalently given the expected return level, an optimal portfoliowith the minimum risk is obtained. Based on this efficient frontier, thewell known two-fund separation theorem was developed to help rationalinvestors to develop optimal investment strategy.

In 1958, Tobin pointed out that when there is a riskless asset, the frontierbecomes a straight line and an optimal portfolio is then a combination ofa risky asset and the riskless asset. In the 1960s, with all investors havingthe same expectation, Sharpe, Lintner and Mossin developed the CapitalAsset Pricing Model (CAPM) for a financial market at an equilibrium state.One of the important contributions from the CAPM is that it links excessreturn with the so-called market return. Merton in 1973 developed an Inter-temporal Capital Asset Pricing Model, which links the excess return of arisky asset to not only the market return but also several state variablesthat will eventually result in multi-factors. In 1976, Ross proposed theArbitrage Pricing Theory: given a financial market spanned by a numberof factors, asset pricing of no-arbitrage is based on the results of the factorpremiums and factor sensitivities.

In the 1970s, Lucas developed the consumption-based asset pricingmodel, and in the 1990s all the above models were merged into a moregeneral pricing model, namely the stochastic discount factor (SDF) pricingmodel, which we discuss in detail as follows.

Modern consumption theory started in the 1930s, when Keynes in hisfamous book The General Theory of Employment, Interest, and Moneyproposed an Absolute Income Hypothesis based on the FundamentalPsychological Law. Specifically,

• there exists a stable functional relationship between real consumptionand income;

1.1. A Brief History of Modern Asset Pricing Models 3

• marginal propensity to consumption is bigger than 0 but less than 1;

• average propensity to consumption is decreasing along with increaseof income.

Compared with Keynes’ absolute income hypothesis, Duesenberry’s rela-tive income hypothesis in 1949 was an advancement. Duesenberry stressedthe effect of consumption habit. Later, Modiglian and Brumberg (1954)proposed the Life Cycle Hypotheses and Friedman (1957) proposed thePermanent Income Hypotheses. The above work laid the foundation ofmodern consumption theory.

Modiglian and Brumberg assumed that there is a utility of aggregateconsumption depending on historic and future consumption paths. Fried-man (1957) divided income into two parts: a predictable income (called apermanent income) and an unpredictable income (called a temporary in-come). The Permanent Income Hypothesis claims that an individual con-sumption is not decided by the income of that period but by a life-longincome i.e. the permanent income. The mathematical tools used by theseauthors were deterministic.

In the 1970s, there were two important events that influenced the de-velopment of the consumption theory: one is Lucas’s (1976) critique onrational expectation and another is Hall’s martingale model of consump-tion (1978). Both stressed expectation and uncertainty. Lucas argued thatconsumption depends on expected income and Hall proved that consump-tion follows a martingale process if the preference of the consumer is time-separable, the utility is of a quadratic form and the interest rate is constant.Hall’s result shows that the Life Cycle Hypothesis and the Permanent In-come Hypothesis follow a random walk. Let ct denote the consumption attime t and Et the conditional expectation given the information set up totime t. Then Hall’s conclusion is

Etct+1 = ct, (1.1)

or equivalently by using the representation of a martingale difference, aconsumption dynamic process follows

ct+1 = ct + ηt+1 (1.2)

with ηt+1 being a zero-mean normal white noise. Campbell and Cochrane(2000) stated, ‘The development of consumption-based asset pricing theoryranks as one of the major advances in Financial Economics during the last


two decades.’ This comes from a very intuitive construction dealing withthe tradeoff between investment and consumption. Specifically, let ei,t bethe endowment of the i-th agent at time t = 0, 1, cji,t be his consumption ofthe j-th commodity (in physical unit or in monetary unit) at time t = 0, 1,j = 1, 2, · · · . Let xj be the payoff of a security which is a financial contractenforced among agents at time t = 0; the contract promises to pay back xj

units of commodity j at time t = 1. Here, xj is often a random variable.The action of entering a financial contract is called an investment. Then atradeoff is attained, for each agent, if endowment ei,0 is allocated betweenthe current consumption cji,0 and the investmentwj,i, which is the number ofcontracts (bought or sold) for payoff xj . People with great patience tend toconsume less now and invest more. People with low risk aversion tend to bemore involved in highly risky investments, in the hope of obtaining higherlevel of consumption in the future. The equilibrium allocation for eachagent is to maximize his utility, given a budget constraint. Mathematicallythis is

maxwj,i

E[ui(c0, c1)] (1.3)

with the budget constraint

c0 =∑

j

cji,0 = ei,0 −∑

j

wi,jpj, (1.4)

c1 =∑

j

cji,1 = ei,1 +∑

j

wi,jxj , (1.5)

where pj is the price of the financial contract for commodity j at time t = 0,ui = ui(c0, c1) is a utility function for agent i, and E is the expectation ofa random variable. By a simple calculation, we derive an equation of thefirst condition for utility maximization - the so-called Euler equation:

pj = E[IMRSixj ], j = 1, 2, · · · , (1.6)

where IMRSi is the inter-temporal marginal rate of substitution given byIMRSi = ∂ui/∂c0

∂ui/∂c1, where ∂ui/∂ci, i = 0, 1, denotes the partial derivative

of the utility with respect to the consumptions. The pioneers Lucas (1978),Breeden (1979), Grossman and Shiller (1981), and Hansen and Singleton(1982, 1983) studied the Euler equation in two ways: (i) To determine theassets’ prices given the agent’s utility function and the assets’ payoffs; (ii) Todetermine, as an inverse problem, the parameters such as the risk aversion in

1.2. The Equity Premium Puzzle 5

IMRS by the GMM estimation procedure in Hansen and Singleton (1992,1983), given the market prices and asset payoffs.

In recent years, people have started to consider a more general form ofthe Euler equation by introducing a random variable m satisfying

p = E[mx]. (1.7)

Then m is called a stochastic discount factor (SDF). This framework in-cludes the consumption-based asset models, the CAPM, the Ross arbitragepricing theorem, the option pricing models and many other popular assetpricing models as special cases. For further details, the readers may wishto consult Cochrane (2001).

Alternatively m is called a state price density (SPD), which is a verypopular name in the risk-neutral pricing world.

1.2 The Equity Premium Puzzle

The so-called risk premium is a monetary cost of uncertainty. Its generaldefinition is as follows.

Definition 1.2.1 Suppose we are given an initial wealth w0 and investmentl = 〈p1, x1; p2, x2; · · · ; pn, xn〉, in which the investment has an outcome xi

with probability pi, i = 1, 2, · · · , n; here xi could be money, a commodity orsome other type. Let w = w0 + x be the terminal wealth. Let r be a realnumber satisfying

u(Ew − r) = E[u(w)], (1.8)

where u(·) is a utility function, E is the expectation, and x =x1, x2, · · · , xn. Then r is called the risk premium.

An intuitive interpretation of the risk premium is that an investor withutility u is indifferent to risky investment w and a deterministic investment(such as bank’s deposit) with a fixed payoff Ew − r. To a risk averseinvestor, we expect that r > 0. By the no arbitrage principle, we can seeEw−r = Rf , where Rf is a risk-free interest rate. That is, for a risk averseinvestor to enter the risky trade w, the expected return of w must be higherthan the risk-free interest rate by

Ew = Rf + r,

so r is an expected excess return to compensate the investor for enteringthe risky trading. When x is an equity, the risk premium becomes an equity


premium. It is the difference between the expected equity return and therisk-free interest rate.

Mehra and Prescott (1985) announces a surprising discovery to the fi-nancial economics community, which later becomes the well-known equitypremium puzzle. The puzzle says that there exists a substantial differencebetween the equity premium estimation by using the U.S. historical stockmarket data and that by a slight variation of the Lucas model (1978) basedon the U.S. aggregate consumption data, unless the risk aversion parameteris raised to an implausibly high level. In the above, we have described brieflya justification for the consumption-based asset pricing modelling. Now, thepuzzle represents a serious challenge to the development of Financial Eco-nomics. Since its announcement, it has attracted the attention of manyfinancial economists. Specifically, Mehra and Prescott (1985) starts withthe problem of seeking an optimal solution for the following maximizationproblem, which is a variation of the Lucas model (1978) in a pure exchangeeconomy:

maxct

E0

[ ∞∑t=0

βtu(ct)

], 0 < β < 1, (1.9)

subject to the budget constraints above, where β is a subjective discountfactor. The utility function is restricted to be of constant relative riskaversion(CRRA) and takes the form

u(c) =c1−γ − 1

1 − γ, (1.10)

where γ is the coefficient of relative risk aversion. After some calculations,the following Euler equation emerges

pt = Et

[β

(ct+1

ct

)−γ

xt+1

], (1.11)

where pt is the market price at time t, xt+1 the stock’s payoff at time t+ 1with xt+1 = pt+1 + dt+1, and dt+1 the stock’s dividend.

Turning to empirical analysis, the paper finds that the average realreturn on the riskless short-term securities over 1889-1978 period is 0.8%and the average real return on the S&P 500 composite stock index over thesame period is 6.98% per annum. This leads to an average equity premiumof 6.18%. By varying γ between 0 and 10, and β between 0 and 1, they findthat it is not possible to obtain a matching sizable equity premium. The


largest premium from the simulation is only 0.35%! The paper concludesthat their model of asset returns is inconsistent with the U.S. data onconsumption and asset returns.

Given the market traded prices of some risky assets and the riskfreeasset, the aim is to check if an SDF can price them correctly. Insteadof checking the Euler equation directly, which is often complex, let m bethe SDF that, according to the Euler equation, incurs no pricing error tothe risky assets and the riskfree asset. Then Hansen and Jagannathan(1991) shows that there exists a relationship between m and the assets byintroducing an upper bound for the so-called Sharpe ratio of the returns ofrisky assets, namely

|ER −Rf |σR

≤ SRmax, (1.12)

where R is the (aggregate) return of the risky assets, ER the expectationof R, Rf the return of the riskfree asset, σR the volatility of R andSRmax the theoretically maximal Sharpe ratio. Here, SRmax = σm/Em,where σm is the volatility of m and Em the expectation of m. This latterratio is called the Hansen-Jagannathan bound, or the H-J bound for short.There are two ways to exploit it. First, the absolute value of the Sharperatio of the return of risky assets has an upper bound given the SDF. Inother words, any risky asset with a Sharpe ratio higher than the SRmax

cannot be correctly priced (i.e. priced with no pricing error) by the SDFm. Alternatively, a necessary condition for an SDF candidate to be capableof pricing the assets correctly is that its SRmax must not violate a lowerbound which is given by the maximum of the Sharpe ratios of the assets tobe priced correctly. Either way the H-J bound provides us with a necessaryand quick way to check whether the Euler equation is satisfied.

When the SDF m is given by the CRRA utility, we have m = β( c1c0

)−γ .Using the yearly U.S. economic and stock data, the inequality (1.12) isoverwhelmingly violated, unless the risk aversion parameter γ reaches anextremely high level, say between 25 and 50 or even high. Mathematically,the puzzle can be stated as follow.

Given a series of payoffs xi for risky assets and their correspondingmarket prices pi, let candidate SDF m be a series of SDFs given bym = mθ, θ ∈ Θ, where Θ is a parameter space in L. Let Θ1 be asubspace of Θ whose range is feasible economically. For example, Θ can bedefined by Θ = γ |0 < γ < 1000 and Θ1 by Θ1 = γ |0 < γ < 5. Then


a ‘parameterized’ equity premium puzzle takes the form

pi = E[mθxi] for some i and ∀θ ∈ Θ1, (1.13)

but

pi = E[mθxi] for all i and for some θ ∈ Θ. (1.14)

Hansen and Jagannathan (1997) introduces a distance to measure pric-ing errors for given asset pricing models. The distance becomes the well-known Hansen-Jagannathan distance. Both the H-J bound and the Hansen-Jagannathan distance are powerful ways to evaluate asset pricing models.

Kocherlakota (1996) points out that there are three possible ways toresolve the equity premium puzzle.

• Introduce progressively a more complex form of the utility function,e.g. by some judicious modification of the standard power utilityfunction if time and state are separable, or by adopting a completelynew form of the utility function if time and state cannot be separated.

• Traditional asset pricing models make the strong assumption that as-set markets are complete, but the real world is not like that. Acknowl-edging that the market is incomplete in one’s asset pricing model isone way to resolve the equity premium puzzle; this will be a newdirection.

• Trading costs, such as taxes and brokerage fees, should be included inthe asset pricing model because their effects on the equity premiumcannot be ignored.

Mehra (2003) points out that the equity premium puzzle is a quantitativepuzzle. The puzzle arises from the fact that a quantitative prediction of eq-uity premium is much different from what has been historically documented.The puzzle cannot be dismissed lightly because our economic intuition isoften based on our acceptance of models such as CCAPM (Consumption-based CAPM). Therefore, the validity of using such models for any quan-titative assessment has also become an issue.

Over the past 20 years, attempts to resolve the puzzle have become a ma-jor research activity in Finance and Economics. Many different approacheshave been adopted. These include, among many others, the recursive util-ity model by Epstein and Zin (1989,1991) and Weil (1989,1990), the habit


formation model by Constatinides (1990), Abel (1990), and Campbell andCochrane (1999), the three-state economy by Rietz (1988), the survivor-ship bias by Brown, Goetzmann, and Ross (1995), the idiosyncratic riskand incomplete markets considered by Mankiw (1986), Lucas (1994) andTelmer (1993), the generalized heterogeneous consumers specified by Con-stantinides and Duffie (1996) and Brav, Constantinides and Geczy (2002),the market imperfections models by Aiyagari and Gertler (1991), Alvarezand Jermann (2000), Bansal and Coleman (1996), Constantinides, Don-aldson and Mehra (2002), Heaton and Lucas (1996), and McGrattan andPrescott (2001). In fact, hundreds of papers have been published in the Fi-nance and Economics literature that are devoted to the puzzle. The puzzleis not only relevant to the foundation of the theory of Financial Economicsbut also crucial to the financial industry for such activities as the long-termasset allocation for pension funds, whose market size currently exceeds onetrillion U.S. dollars. In the following, we give a brief introduction to someof the recent developments.

I. Preference Modifications

I.a Generalized Expected Utility

The standard preference class used in macroeconomics consists of time-and-state-separable utility functions. From the empirical analysis of Mehraand Prescott (1985), we know that the CRRA preference can match theobserved equity premium only if the coefficient of relative risk aversion isimplausibly large.

Epstein and Zin (1989,1991) present the notion of generalized expectedutility preference (GEU) that allows independent parameterizations of thecoefficient of risk aversion and the elasticity of inter-temporal substitution.The recursive utility, Ut, is given by

Ut = [c1−ρt + β(EtU

1−αt+1 )(1−ρ)/(1−α)]1/(1−ρ), (1.15)

where α measures the relative risk aversion and 1/ρ the elasticity of inter-temporal substitution. In contrast to the historical average risk premiumof 6.2% and the largest premium obtained by Mehra and Prescott (1985)of 0.35%, Epstein and Zin’s results in 1991 show a low riskfree rate andan average equity premium of roughly 2%. In their words, their GEUspecification only partially resolves the puzzle. Weil’s model in 1990 is verymuch like that in Epstein and Zin (1991). It is very interesting to note that


the Epstein-Zin model gives a two-factor representation for the premiumby

Etrt+1 − rf,t+1 +σ2

t

2=θ

ψcovt(rt+1,∆ct+1)

+ (1 − θ)covt(rt+1, rp,t+1). (1.16)

For detailed derivation, see, e.g., Campbell and Viceira (2002, p. 45). Weshall give a new result later in the form of (Theorem (3.3.5)), which showsthat this two-factor structure of the Epstein-Zin model is the key to thesuperior performance of the model over models of the CRRA type.

I.b Habit Formation

Abel (1990) and Constantinides (1990) consider the effects of consumptionhabit on the decision making of individuals. Constatinides specifies a utilityas follows:

U(c) = Et

∞∑0

βs (ct+s − λct+s−1)1−α

1 − α, λ > 0, (1.17)

where λ is a parameter that captures the effect (or habit) of past con-sumption. This preference ordering makes individuals extremely averse toconsumption risk even when the risk aversion is small.

Abel (1990) provides another kind of habit formation by defining utilityof consumption relative to the average per capita consumption. The idea isthat one’s utility depends not only on the absolute level of consumption butalso on how one is doing relative to others. In contrast to Constantinides(1990), the per capita consumption can be regarded as the ‘external’ habitformation for each individual. Again we shall give new results later to showwhy such a setup is to do with a more complex utility that goes back toKocherlakota’s point above.

Campbell and Cochrane (1999) specifies habit formation as external,similar to Abel (1990), and took the possibility of recession as a statevariable so that a high equity premium could be generated.

As it turns out the habit formation models have only limited success asfar as resolving the equity premium puzzle is concerned because effectiverisk aversion and prudence become improbably large in these models.


II. Incomplete Markets

Instead of assuming all the agents are homogeneous, Mankiw (1986) arguesas follows. There are infinitely many consumers who are identical ex ante,but their consumptions are not the same ex post. The aggregate shocks toconsumption are assumed to be not dispersed equally across all individualsbut only affect some of them ex post. Under the assumption of incompletemarkets, Mankiw shows that representative agent models are not effec-tive as approximations to a complex economy with ex post heterogeneousindividuals.

Lucas (1994) proposes a more general model than Mankiw (1986) byassuming undiversified shocks to income and borrowing, and short saleconstraints at an infinite time horizon. In her model, individuals cannotinsure, ex ante, against future idiosyncratic shocks to their income. Sheshowed that individuals with a bad idiosyncratic shock can effectively self-insure by selling financial assets to individuals with good luck by trading.Hence, idiosyncratic risks to income are largely irrelevant to asset priceswith trading even when the borrowing constraints are severe. To resolvethe equity premium puzzle requires more than closing forward market forlabour income. Telmer (1993) considers much a similar incomplete marketmodel as Lucas (1994) by assuming two heterogeneous consumers withdifferent consumption stream in the economy. His research supports Lucas’conclusion.

Constantinides and Duffie (1996) assumes that consumers are heteroge-neous because of uninsurable, persistent and heteroscedastic labour incomeshocks at each time period. The paper constructs a model in which the Eu-ler equations depend on not only the marginal rate of substitution (MRS)at the aggregate level but also the cross-sectional variance of the individ-ual consumption growth. Continuing the work of Contandinides and Duffie(1996), Brav, Constantides and Geczy (2002) considers the case of assetpricing with heterogeneous agents and limited participation of householdsin capital markets. Their empirical analysis reveals that relaxation of theassumption of complete consumption insurance is helpful in resolving theequity premium puzzle.

III. Liquidity Premium and Trading Costs

Bansal and Coleman (1996) gives a monetary explanation for the equitypremium puzzle. In their model, assets other than money play a keyfeature by facilitating transactions. Using empirical evidence, the paper


claims that half of the equity premium can be captured by their model.Some economists try to explain the equity premium puzzle via transactioncosts. For example, McGrattan and Prescott (2001) proposes an explana-tion based on changes in tax rates. Heaton and Lucas (1996) finds that thedifferences in trading costs across stocks and bond markets have to be veryhigh in order to resolve the equity premium puzzle with transaction costs.Kocherlakota (1996) even shows that to match the real equity premium,the trading cost have to be implausibly high.

Mehra (2003) gives a comprehensive summary of recent developments;the puzzle remains open.

Chapter 2

A Structural Theory of Asset Pricing

and the Equity Premium Puzzle

In this chapter we introduce a new theory of asset pricing, which we callthe structural theory. Applying it to the equity premium puzzle, we shallbe able to see when the puzzle is a puzzle and when it is not.

2.1 Construction of Continuous LinearPricing Functionals and No-arbitrageConditions

Let (Ω, F, P ) be a probability space and H = L2(Ω, F, P ) be a contingentclaim space. Let x1, x2, · · · , xn, · · · be a sequence of random payoffs inH .Let n be the linear space spanned by x1, · · · , xn in H . Let = ∪∞

n=1n,

so that any x in is a random portfolio return from some finite subset ofthe assets. Let be the closure of in H . It is well known that His a Hilbert space , separable or not separable, under the inner product< x, y >= E[xy]. Therefore the asset payoff space (or the asset spacefor short) is a Hilbert space as well by recalling the fact that any closedlinear subspace in a Hilbert space is still a Hilbert space. For n ≥ 1, let Σn

be the covariance matrix of xn = (x1, · · · , xn) and we assume that Σn isnon-singular. Therefore Σn is positive-definite. Given payoffs x1, · · · , xn,a Gram-Schmit orthogonalization procedure leading to e1, · · · , en is

13

14 Chapter 2. A Structural Theory of Asset Pricing

given by

y1 = x1 and setting e1 = x1/||y1||y2 = x2− < x2, e1 > e1 and setting e2 = y2/||y2||

· · ·yn = xn −

n−1∑i=1

< xn, ei > ei and setting en = yn/||yn||.(2.1)

The sequence e1, · · · , en has the property that < ei, ej >= 0 for i = j and< ei, ei >= ||ei||2 = 1 for i = 1, · · ·n. Interested readers may consult Lax(2002, Chapter 6) for more detail.

Lemma 2.1.1 The linear subspace n is spanned by the orthonormal basise1, · · · , en, in which Σe

n, the covariance matrix of e1, · · · , en, is non-singular if and only if Σn is nonsingular.

Proof: We prove sufficiency first. When Σn is nonsingular, we know thatfor each xi, < xi, xi >= ||xi||2 > 0 because otherwise Σn is singular.Similarly if yi = 0 for some i, then xi is linearly dependent on x1, · · · , xi−1.This implies that Σn is singular. We must have ||yi|| > 0 for i = 1, · · ·n.Therefore we can use the Gram-Schmit orthogonalization procedure aboveto produce an orthonormal base e1, · · · , en for n, satisfying

(e1, · · · , en)T = An(x1, · · · , xn)T , (2.2)

where T refers to the usual transpose operation in matrix algebra. An =(ai,j) has the property

ai,j =

0 if j > i1/||xi|| if i = j. (2.3)

We turn to necessity next. If n is spanned by an orthonor-mal basis e1, · · · , en, there are two matrices A and B satisfying(e1, · · · , en)T = An(x1, · · · , xn)T and (x1, · · · , xn)T = Bn(e1, · · · , en)T .Hence (e1, · · · , en)T = AnBn(e1, · · · , en)T . By the orthogonality ofe1, · · · , en, we haveAnBn = I, where I is the n×n identity matrix. We haveΣn = BnΣe

nBTn . Since both B and Σe

n are nonsingular, Σn is nonsingular.

Q.E.D.

Lemma 2.1.2 Let e1, · · · , en, · · · be the orthonormal payoffs generatedfrom the Gram-Schmit orthogonalization procedure above. Let H1 denote

2.1. Construction of Continuous Linear Pricing Functionals 15

the space x ∈ H | x =∑∞

i=1 aiei,∑∞

i=1 a2i < ∞. Then we have = H1.

In other words, is spanned by the orthonormal base e1, e2, · · · en, · · · .Furthermore let S = x |x =

∑∞i=1 aixi with

∑∞i=1 a

2i < ∞. Then S is a

linear subspace of .1

Proof: Obviously H1 is a Hilbert space. For each x in H1 with x =∑∞i=1 aiei, define xn =

∑ni=1 aiei. By Lemma 2.1.1, xn ∈ n. Hence

x = limn→∞ xn ∈ . Therefore H1 ⊂ . On the other hand, if thereexists an x ∈ but x /∈ H1, then since H1 is a closed subspace of , wehave x = y + ε with y ∈ H1, ε ∈ −H1, and ε = 0. Then, for i = 1, 2, · · · ,< ε, ei >= 0. However, since ε ∈ , there is a vector εn in n satisfyinglimn→∞εn = ε. Since εn =

∑ni=1 aiei, < ε, εn >=

∑ni=1 ai < ε, ei >= 0,

therefore < ε, ε >= limn→∞ < ε, εn >= 0. This implies ε = 0, which is acontradiction. Thus, we must have ⊂ H1. Hence, we have = H1. Letx ∈ S with x =

∑∞i=1 aixi and define xn =

∑ni=1 aixi. Then xn → x in

H . On the other hand, by Lemma 2.1.1, xn ∈ n and since is closed, weobtain x ∈ . Hence S ⊂ .

Q.E.D.

Definition 2.1.3 Pricing Functional: Let I be an ordered set such asI = 1, · · · , n, or I = 1, 2, · · · . Let xi, i ∈ I be a sequence of payoffsfrom the assets and pi, i ∈ I be a corresponding sequence of observedmarket prices in the asset market. Then a pricing functional π is a mappingfrom SI = spanxi, i ∈ I = x |x =

∑i∈I wixi with

∑i∈I w

2i <∞ to

(the real line) given by

π(x) = p |p =∑i∈I

piwi in l2 for some (2.4)

w ∈ l2 such that x =∑

i

wixi in H. (2.5)

Definition 2.1.4 Law of One Price: All portfolios with the same payoffhave the same price. That is,

if∑i∈I

wixi =∑i∈I

w′ixi then

∑i∈I

wipi =∑i∈I

w′ipi. (2.6)

for any two portfolios w = wi, i ∈ I and w′ = w′i, i ∈ I.

1We will have S = only under very stringent conditions.


Lemma 2.1.5 The Law of One Price holds if and only if the pricing func-tional π is a linear functional on SI .

Proof: If the Law of One Price holds, then the mapping π is single valued.To prove linearity, consider payoffs x, x′ ∈ SI such that x =

∑i∈I wixi

and x′ =∑

i∈I w′ixi for some portfolios w and w′. For arbitrary a, b ∈ ,

then payoff ax+ bx′ = (axi + bx′i, i ∈ I) can be generated by the portfolioaw+ bw′ = (awi + bw′

i, i ∈ I) with the price p(aw+ bw′) =∑

i∈I pi(awi +bw′

i) = a∑

i∈I piwi + b∑

i∈I piw′i. Because π is single valued, we have

π(ax + bx′) = p(aw + bw′) = apw + bpw′ = aπ(x) + bπ(x′).

Thus π is linear. Conversely, if π is a linear functional, then the Law ofOne Price holds by definition.

Q.E.D.

The continuity of π is linked to no-arbitrage. Kreps (1981) gives arelationship between arbitrage and continuity of π in a very general setting.Here we use Chamberlain and Rothschild’s no-arbitrage assumption (1983,assumption A (ii)).

Assumption 2.1.1 Let xn be a sequence of finite portfolios in . LetV (xn) denote the variance of xn and E(xn) the expectation of xn. IfV (xn) → 0, π(xn) → 1, and E(xn) → α, then α > 0.

Lemma 2.1.6 If Assumption 2.1.1 holds, then π is continuous.

We omit the proof, which is straightforward, but refer to Chamberlain andRothschild (1983, Proposition 1).

Since π is continuous on , it is easy to prove that π is bounded on. Then according to the Hahn-Banach Theorem (e.g. Lax 2002, Chapter3), π can be extended to a bounded linear functional π on H such that on, π ≡ π. Hence by the Riesz Representation Theorem (e.g. Lax 2002,Chapter 6), there is a unique element m ∈ H satisfying

∀x ∈ H, π(x) =< m, x >= E[mx]. (2.7)

Since is a closed linear subspace of H , we have an orthogonal decompo-sition such that

m = x∗ +m1 where x∗ ∈ and m1 ⊥ . (2.8)

2.2. The Structural Theory of Asset Pricing – Part I 17

Therefore we have ∀x ∈ ,

π(x) = π(x) =< m,x >=< x∗, x >= E[x∗x]. (2.9)

Since π is a continuous extension of π, in the following, we will omit thenotation π and use simply π only.

Clearlym is a stochastic discount factor (SDF). In the following, we willuse the following terms and acronyms interchangeably: pricing functional,SDF, state price density and SPD.

2.2 The Structural Theory of Asset Pricing – Part I

Given the traded prices of assets in a market, the popular steps to search aproper SDF are as follows. First, a family of SDF candidates SDF (θ)θ∈Θ

is formed, where Θ is a parameter space. For example, in the constantrelative risk aversion (CRRA)’s case, SDF (θ) = IRMS(gc, λ, β), where gc

is the growth of the aggregate consumption, λ the risk aversion parameterand β an impatience parameter. Second, some estimation method, such asthe GMM estimation procedure, is used to estimate the parameters. Hansenand Singleton’s treatment (1982) for the CRRA model is an example. Insection 1, we demonstrated that λ ≈ 30, which led to the so-called equitypremium puzzle.

Now, we first define what we really mean by correct pricing and thendiscuss conditions which determine whether correct pricing will result ornot.

Let X = n or X = according as the asset space for assets is finiteor infinite. Let p = p1, · · · , pn or p = p1, · · · , pn, · · · be the marketprice vector depending on whether we have a finite number of assets or aninfinite number of assets. Let π denote the linear continuous functional onX satisfying

pi = π(xi) with pi = 0 for i = 1, 2, · · · ,2 (2.10)

and for each i. Here we assume that pi cannot be obtained through pric-ing a portfolio of the payoffs xj other than xi, that is they satisfy theparsimoniously pricing condition

∀x ∈ spanx1, · · · , xi−1, xi+1, · · · , xn, pi = π(x). (2.11)

2pi = 0 means that a free contract is allowed. Here we assume that no free contractexists in the asset market.


Definition 2.2.1 Under the above setup, π is called a correctly pricingfunctional (CPF) for the asset space X, given the market prices. Let m bean SDF. If the pricing functional π (i.e., ∀x ∈ X, π(x) = E[mx]) inducedby m is a CPF, then m is called a correctly pricing SDF.

Obviously, a CPF to an asset space with feasible economic parameters willmean there is no equity premium puzzle for the asset space. According tothe definition, a CPF and a correctly pricing SDF are equivalent notions.

Define an inner product in the dual space X∗ of X by < π, π′ >=<x∗, (x∗)′ >, where x∗ and (x∗)′ are the Riesz representations in X of π andπ′ respectively. It is well known that X∗ is a Hilbert space. (See Lax (2002)for example.)

Assumption 2.2.1 When X = , limn→∞∑n

i=1 w2i,n < ∞, where wn =

(w1,n, · · · , wn,n)T for n ≥ 1 given by the equation wn = Anpn, An is the

Gram-Schmit orthogonalization matrix in (2.3) and pn is the market pricevector pn = (p1, · · · , pn)T .

Lemma 2.2.2 Existence of CPF These is a unique correctly pricingSDF in X, with the understanding that Assumption 2.2.1 applies if X = .

Proof: First consider the case X = n. Select an m ∈ X such that

pi = E[mxi], i = 1, · · ·n. (2.12)

Since m =∑n

j=1 wjej, pi =∑n

j=1 wjE[ejxi]. Since xi =∑n

k=1 bi,kek,where Bn = A−1

n and An is the Gram-Schmit orthogonalization matrixin (2.3), we obtain pi =

∑nj=1 bi,jwj . Let pn = (p1, · · · , pn)T and wn =

(w1, · · · , wn)T . Then pn = Bnwn. Therefore if we let

wn = Anpn, (2.13)

then m =∑n

j=1 wjej is a correctly pricing SDF in X . The above argumentalso indicates that for any correctly pricing SDF in X , it must always followequation (2.13). This mean that it is unique. For the case X = , definean SDF mn =

∑nk=1 wk,nek, where wn = (w1,n, · · · , wn,n)T is given by

(2.13). Under Assumption (2.2.1) and since X is complete, we see that mn

converges to an m ∈ X . For each i ≥ 1, let n ≥ i. We have E[mnxi] = pi

by using (2.12) and (2.13). Letting n → ∞, we obtain E[mxi] = pi fori = 1, 2, · · · . So m is a correctly pricing SDF in X . If there is another m′

in X which is also correctly pricing, then E[mej] = E[m′ej ], j = 1, 2, · · ·


since each ej is a linear combination of xi. ∀x ∈ X, x =∑∞

j=1 wjej .Hence E[mx] = E[m′x]. By letting x = m−m′, we have E[m−m′]2 = 0.Then m = m′ in X . Therefore the correctly pricing SDF m is unique in X .

Q.E.D.

We have the first main result of the structural theory in the form of thefollowing theorem.

Theorem 2.2.3 (Uniqueness Theorem of the minimum correctlypricing functional space) Let X∗ be the dual space of X, that is, X∗

is the set of all linear continuous functionals on X. Suppose F is a closedand linear subspace of X∗. Then F has a unique correctly pricing functional(CPF) if and only if F ≡ X∗ subject to the understanding that if X = ,Assumption 2.2.1 applies.

Proof: Consider the necessary condition first. For all x ∈ X , either x =∑nj=1 ajej or x =

∑∞j=1 ajej since X = n or X = . For simplicity

of notation, we denote x =∑

j ajej . From Lemma 2.2.1, let ej be theorthonormal base of X . Define a linear pricing functional πj by

∀x ∈ X, πj(x) = E[ejx]. (2.14)

It is easy to see that < πi, πj >=< ei, ej >= 0 if i = j and < πi, πi >=<ei, ei >= 1. Suppose that π is a pricing functional satisfying < π, πj >= 0for all j. Then ∀x =

∑j ajej ∈ X , π(x) = E[x∗x] =

∑j ajE[x∗ej ] =∑

j aj < π, πj >= 0, where x∗ is the Riesz Representation of π in X . Thisimplies that π = 0. We obtain that πj is an othornormal base for X∗.Suppose that π is a CPF. Then we have

π =∑

j

ajπj , (2.15)

in X∗. If F = X∗, then there exists a πj0 satisfying πj0 /∈ F . By the repre-sentation in (2.15), π=

∑j,j =j0

ajπj . Hence π(ej0)=∑

j,j =j0ajπj(ej0)=0.

Using the Gram-Schmit orthogonalization equation (2.1), we have pj0 =π(xj0 ) =

∑j−1i=1 ciπ(xi) = π(

∑j−1i=1 cixi). However, this violates the assump-

tion of the parsimoniously pricing condition (2.11). We must therefore haveπ(ej0) = 0. This is a contradiction. Thus, F ≡ X∗.

We turn to sufficiency next. On using Lemma 2.2.1, there is a uniquecorrectly pricing SDF m in X . Let pricing functional π be induced by m.


Market Prices Asset Space

Dual Space: pricing functional space

P2 P1 x1 x2

Unique pricing functional

Figure 2.1: The Uniqueness Theorem of the minimum correctly pricingfunctional space.

That is ∀x ∈ X,π(x) = E[mx]. Then π is a CPF in F = X∗. Suppose thereare two CPFs, say π1 and π2, in F . When X = n, since π1(xi) = π2(xi)for i = 1, · · · , n and n = spanx1, · · · , xn, we have π1 = π2 in F . WhenX = , ∀x ∈ X, let xn ∈ n satisfy xn → x in X . Since π1(xn) = π2(xn),by letting n→ ∞, we have π1(x) = π2(x). Hence π1 = π2 in F .

Q.E.D.

The implication of Theorem (2.2.3) is that, in order to price correctly,we need to search those pricing functionals in a candidate space that isisometric to a corresponding linear continuous pricing functional space.Note that a proper subspace of the linear pricing functional space does notcontain a CPF. See Figure 2.1 for an illustration.

Very often, we use some SDFs or state price densities in the claim con-tingent space H to form a candidate space of pricing kernels. We introducethe following definition.

Let M be an SDF linear subspace in H . Let asset space X = n or. Let T be an orthogonal projection operator from M to X , that is,


∀m ∈ M, T (m) ∈ X such that ∀x ∈ X, E[(m − T (m))x] = 0. We denoteit by T (m) = E[m|X ].3

Obviously the orthogonal projection operator preserves the original val-uation, namely ∀x ∈ X,m ∈M, E[mx] = E[T (m)x]. We are now ready tostate the second main result.

Theorem 2.2.4 (Dual theorem for correctly pricing SDF spaceand asset payoff space) M has a unique correctly SDF if and only ifthe orthogonal operator T is a continuous linear bijective operator from M

to X, subject to the understanding that if X = , then Assumption 2.2.1applies.

Proof: We start with the necessity. ∀m ∈ M , define a linear functionalπm by ∀x ∈ X, πm(x) = E[mx]. Since M has a unique correctly pricingSDF, there is a unique CPF in F = πm|m ∈ M. According to Theorem2.2.2, F = X∗. So ∀y ∈ X , define a linear continuous functional πy by∀x ∈ X, πy(x) = E[yx]. Since X∗ = F , there exists a πmy ∈ F such thatπy = πmy . In other words, ∀x ∈ X, E[yx] = E[myx]. Now we define amapping S from X to M by

S : y ∈ X → my ∈M. (2.16)

First we show that S is a single-valued mapping. Suppose there are twomappings, saymy andm′

y, satisfying ∀x ∈ X, E[yx] = E[myx] and E[yx] =E[m′

yx]. This implies that E[(my −m′y)x] = 0, ∀x ∈ X . This means that

my−m′y⊥X . Letm0 be the correctly pricing SDF inM . Then we know that

m0+(my−m′y) is also correctly pricing for X. By the uniqueness of correctly

pricing SDF, we have my −m′y = 0. Hence S is a single-valued mapping.

Secondly we show that S is surjective. ∀m ∈ M , we have πm ∈ F = X∗.Using the Riesz Representation Theorem, there is an xm ∈ X satisfying∀x ∈ X, E[mx] = πm(x) = E[xmx]. Again by the uniqueness of correctlypricing SDF, we have S(xm) = mxm = m. Hence S is surjective. Supposethat y, z ∈ X, y = z, my = mz. This means that ∀x ∈ X, E[(y − z)x] =E[(my −mz)x] = 0. Let x = y− z. Then E[y− z]2 = 0, which means thaty = z. This is a contradiction. So, S must be injective. Hence S is bijective.Since for any two real numbers a and b, ∀x ∈ X,E[(ay + bz)x] = aE[yx] +bE[zx] = E[(amy + bmz)x]. We can see that S is linear. ∀m ∈ M and∀x ∈ X , E[mx] = E[S−1(m)x]. That is E[(m − S−1(m))x] = 0, ∀x ∈ X .

3An orthogonal operator does not have to be a conditional expectation operator.


Asset Space

SDF Space

SS: bijective valuation-preservingmapping

T=S-1 T: orthogonal projection

Figure 2.2: The Dual Theorem between an SDF Space and an asset payoffspace.

This means that S−1 is the orthogonal operator from M to X . ThereforeT = S−1. Clearly T is linear. Let mk → m in M as k → ∞. Forcontinuity of T , we need to show that T (mk) → T (m) in X as k → ∞.By the orthogonality of T , ∀x ∈ X, E[(mk −m)x] = E[(T (mk)−T (m))x].Let x = T (mk) − T (m). Then we have E[T (mk) − T (m)]2 = E[(mk −m)(T (mk)−T (m))] ≤√E[mk −m]2

√E[T (mk) − T (m)]2 by the Cauchy-

Schwartz Inequality. Hence√E[T (mk) − T (m)]2 ≤ √

E[mk −m]2. Thisimplies that T is continuous and ||T || = 1, completing the proof of necessity.

We turn to sufficiency next. We have the existence of a unique correctlypricing SDF in X from Lemma (2.2.2). Then we use operator S to map itinto space M . Hence there is one correctly pricing SDF in M . The unique-ness of correctly pricing SDF in M follows from the injective condition ofthe operator T and Lemma (2.2.2).

Q.E.D.

2.3 Is the Equity Premium Puzzle Really a Puzzle ornot a Puzzle?

The implications of Theorem (2.2.3) and Theorem (2.2.4) are very impor-tant. They indicate that, given market traded prices, a necessary andsufficient condition for the pricing functional space (or the SDF space)

2.3. Is the Equity Premium Puzzle Really a Puzzle or not a Puzzle? 23

Asset Space

Asset Space

SDF Space

If Then

Correctly pricing

SDF Space

Correctly pricing as well

SDF Space

But and

Not correctlypricing

Not correctlypricing

Asset Space

and

More than one correctly pricingSDFs

Bigger SDF Space

Bigger Asset Space

Smaller SDF Space

Smaller AssetSpace

Figure 2.3: Implication of the Dual Theorem.

to have a unique correctly pricing functional (or correctly pricing SDF)is that the space is isometric to the asset space. The orthogonal projec-tion operator, introduced in the dual theorem, provides a bijective andvaluation-preserving mapping between the two spaces. See Figure 2.3 foran illustration.

The structural theory indicates that whether there is an equity premiumpuzzle or not is relative; it depends on the existence of a matching (i.e. avaluation-preserving isometric mapping) between an SDF space and a re-lated asset space. For matching pairs, there always exists a unique SDF toprice correctly every portfolio in the asset space. If the correctly pricingSDF is with sensible economic parameters, then there is no puzzle. How-ever if the correctly pricing SDF is with infeasible economic parameters,we say that the puzzle appears in respect of this SDF space. Theoretically,given the asset space, we can remove the puzzle by enlarging the SDF space


to one with sensible new economic parameters, for example, by augment-ing new economic state variables to span a bigger SDF space rather thanby extending the range of the parameters in the original SDF space to anunreasonable level. If the augmented SDF space is matched to the assetspace, we may find a new SDF to price correctly every portfolio in theasset space that incurs no pricing error. In this case, we have no puzzle.Alternatively, given the SDF space, by dropping some assets from the assetspace we may find a new correctly pricing SDF with sensible parametersto the reduced asset space. In this way, we can incur no pricing error andhence there is no puzzle in respect of the smaller asset space. In general,any mis-matching of an SDF space and an asset space will definitely createsome pricing error. The puzzle can then appear as a result of an improperattempt to remove the pricing error.

We shall give two examples to illustrate the power of the new structuretheory.

For the first example (2.3.1), the structural theory offers a new ex-planation of the Mehra and Prescott’s puzzle: (1) The structure of theconsumption growth power space is not rich enough to provide an SDFthat is capable of pricing every portfolio in the asset space correctly (i.e.,the two spaces are not isometric), within feasible ranges of the economicparameters. For example, when the risk aversion is chosen to be less than5, a big pricing error appears. (2) The structure of the SDF space used isinsufficient so much so that the estimated parameter has to be exaggeratedto an unreasonable level (e.g. beyond 50 for the risk aversion in the U.S.market) in order to incur no pricing error.

Example 2.3.1 (The CRRA based SDFs) Here we consider the CRRA-based SDF family MK = m = β( c1

c0)−γ , 0 ≤ γ ≤ K, 0 < β ≤ 1 and

assume that there is only one risky asset to be priced. That is X = 1 =ax |∀a ∈ . Given market price p for payoff x, a correctly pricing SDF(also called a pricing kernel) in X is ax with a = p/E[x2]. Given an m, itsorthogonal projection on X is T (m) = E[m|x] = bx with b = E[mx]/E[x2].Then m is a correctly pricing SDF if and only b = a, and the identity holdsif and only p = E[mx]. Define gross return R by R = x/p. Then we needto check

1 = E[mR]. (2.17)

Let g be the consumption growth given by g = ln( c1c0

). Then m = βe−γg. Itis conventional to assume that the asset’s gross return R and the consump-


tion growth g are jointly normally distributed, i.e. (R, g) ∼ N(µ,Σ), wherethe mean vector µ = (µR, µg)T and the covariance matrix

Σ =(

σ2R ρσRσg

ρσRσg σ2g

).

Here, ρ is the correlation coefficient between R and g, σR and σg the volatil-ity of R and g respectively, and µR and µg the mean of R and g respectively.Using the Stein Lemma,

E[mR] = cov(e−γg, R) + E[m]E[R]

= −γβEe−γgcov(g,R) + E[m]E[R]

= (µR − γρσRσg)E[m]

= β(µR − γρσRσg)1√

2πσg

∫ ∞

−∞e−γge

− 12

(g−µg )2

σ2g dg (2.18)

= β(µR − γρσRσg)e−γµg+ 12 γ2σ2

g . (2.19)

Therefore we need to look for proper values of the parameters β and γ

satisfying

1 = β(µR − γρσRσg)e−γµg+ 12 γ2σ2

g . (2.20)

Given the consumption data and the market data, it is difficult to obtain anexplicit solution for (2.20). Mehra and Prescott (1985) uses the U.S. yearlyconsumption data and the S&P 500 data from 1889 to 1978 with averageconsumption growth µg = 0.76%, growth volatility σg = 1.54%, averageS&P500 gross return µR = 106.98%, return’s volatility σR = 16.54%, andcorrelation between consumption growth and stock return ρ = 37.56%. Wesee that the average ratio (µR − 1)/µg = 6.98%/0.76% = 9.18 and thevolatility ratio σR/σg = 16.54%/1.54% = 10.74. Mehra and Prescott (op.cit.) has considered many different choices for parameter β. Here we setβ = 0.94 for illustration. By varying γ, we have a plot of the right side of(2.20) as shown in Figure 2.4.

Then we know that when γ = 76.1, a solution is obtained for (2.20).However, this means that the U.S. investors must be extremely risk averseduring the past one hundred years, which is highly unlikely to be the case.

If we use the CRRA-based SDF to price the risky asset x and the risk-free rate Rf , and suppose that m prices risk-free rate correctly, then we


CRRA based SDF equation

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Gamma - the risk aversion parameter

Figure 2.4: Plot of the SDF equation given by the CRRA model.

have E[m] = 1/Rf . Using (2.19), we have a formula for γ namely

γ =µR −Rf

σR

1ρσg

. (2.21)

In Mehra and Prescott’s paper, the average gross real return for 90 daysU.S. treasury bill is 100.80% and the volatility is 5.67%. Thus the Sharperatio µR−Rf

σRis 0.37. This gives an estimation of γ at 64.6.

In either case, if we consider a small but economically feasible CRRA-based SDF candidate space, M5 for example, then we will not able to find acorrectly pricing SDF for the risky asset - S&P 500. However, if we enlargethe candidate space from M5 to the bigger space M75, then we will find acorrectly pricing SDF, but this creates the so-called Equity Premium Puzzle.We can envisage that if we use the CRRA-based SDF to price more complexasset spaces, larger pricing errors will ensue.

Example (2.3.1) has revealed that in order to find a correctly pricingSDF, we should not exaggerate its parameter space beyond its reasonablerange, but rather, given the structure of the asset space, we should enlargethe SDF candidat space by incorporating further appropriate economic statevariables.

The next example concerns the Epstein-Zin model. Using the struc-tural theory, we shall see why the model explains the puzzle better. TheSDF space generated from the Epstein-Zin model is more complex than theCRRA (constant-relative-risk aversion) based SDF space used by Mehra


and Prescott in 1985. In the Epstein-Zin model, the SDF space is spannedby two state variables, namely the consumption growth and the marketreturn. In contrast, the CRRA based SDF space is spanned by only onestate variable, namely the single consumption growth. So, within a rela-tively reasonable range of the parameters, the Epstein-Zin based model iscapable of providing a smaller pricing error for the same asset space.

Example 2.3.2 Epstein-Zin utility Epstein and Zin (1991) introducesa recursive utility by separating the risk aversion parameter γ and the elas-ticity of inter-temporal substitution parameter δ, namely

U0 = [(1 − β)C1−δ−1+β[E0U1−γ1 ]

1−δ−11−γ

0 ]1

1−δ−1 . (2.22)

It is well known that the Epstein-Zin’s SDF has the form

m = βδ−γδδ−1 (C1/C0)−

1−γδ−1R

1−δγδ−1 , (2.23)

where R is the gross market return. Let consumption growth g be ln(C1/C0)and stock return x = ln(R). Again we assume that the stock return x andthe consumption growth g are jointly normally distributed, i.e. (x, g) ∼N(µ,Σ) as in Example (2.3.1). Then

m = β0e−gλ1exλ2 , (2.24)

where β0 = βδ−γδδ−1 , λ1 = 1−γ

δ−1 xa and λ2 = 1−δγδ−1 . Obviously the Epstein-Zin

SDF is spanned by two variables, namely the consumption growth and themarket return. Hence the right side of the Euler equation is

E[mR] = β0E[e−gλ1e(1+λ2)x] = β0E[e(1+λ2)xE[e−gλ1 |x]]. (2.25)

The conditional distribution of g given x is N(µg|x, σg|x), where

µg|x = µg + ρσg

σx(x− µx) and σ2

g|x = σ2g(1 − ρ2). (2.26)

In the following, we use the simple result that∫ ∞

−∞eaxe−

12

(x−b)2

σ2 dx = σ√

2πeab+ 12 a2σ2

,

for any two real numbers a and b. We have E[e−gλ1 |x] = exµ1+c1 , whereµ1 = −λ1ρ

σg

σxand c1 = −λ1µg + λ1ρ

σg

σxµx + 1

2λ21σ

2g(1 − ρ2). This gives

E[mR] = β0ec1E[e(1+λ2+µ1)x]

= β0ec1+(1+λ2+µ1)µx+ 1

2 (1+λ2+µ1)2σ2x (2.27)

= β0e−λ1µg+ 1

2 λ21σ2

g(1−ρ2)+(1+λ2)µx+ 12 (σx+λ2σx−λ1ρσg)2 . (2.28)


Using the data in Merha and Prescott (1985), we have Figure 2.5 andTable 2.1 for values of E[mR] over various γ and δ. In contrast to thecase of the CRRA-based SDF, it is now quite easy to find a risk aversionparameter to satisfy the Euler equation.

!" ! !#$

!

!

"!

"!

!

!

#!

#!

" # % &

""

"#

"

"%

"&

"

#

'()' ''

'* +,- !"$

+,- !$

+,- !#$

Figure 2.5: Plot of the SDF equation given by the Epstein-Zin model.

Table 2.1: Values of E[mR] given by the Epstein-Zinmodel (β = 0.95)

γ δ = 0.1 δ = 0.2 δ = 0.32 0.994 0.996 0.9993 0.988 0.993 1.0034 0.983 0.992 1.0115 0.978 0.992 1.0236 0.973 0.994 1.0407 0.969 0.997 1.0628 0.966 1.002 1.0909 0.962 1.008 1.12210 0.959 1.016 1.161

2.4 Conclusions and Summary

In this chapter, we have developed a new theory for asset pricing, which wechristen the structural theory. We separate the problem of finding a better

2.4. Conclusions and Summary 29

asset pricing model from that of searching for no equity premium puzzle.Our first result from the unique theorem and the dual theorem indicatesthat, given market traded prices, a necessary and sufficient condition forthe pricing functional space (or the SDF space) to have a unique correctlypricing functional (or correctly pricing SDF) is that the space is isometric tothe asset space. The orthogonal projection operator, introduced in the dualtheorem, provides a bijective and valuation-preserving mapping betweenthe two spaces. The structural theory has provided a new explanationfor the Mehra and Prescott puzzle. It indicates that whether there is anequity premium puzzle or not is relative, depending on whether or not thereis a matching between a SDF space and an asset space. Using the abovetheory, we have seen why the Epstein-Zin model leads to a more satisfactoryresolution.


Chapter 3

Algebra of Stochastic Discount

Factors — The Structural Theory of

Asset Pricing

(Part II)

In this chapter, we develop the structural theory further to deal with anenlarged portfolio space that includes non-tradable assets; we shall discussasset pricing problems including both the bottom-up investment method-ology and the top-down investment methodology. Typical examples of theformer include (i) a fund manager first picking some individual stocks andforming a stock portfolio after a series of stock selections, then facing theproblem of portfolio valuation based on the individual stocks’ valuationsand (ii) an U.S. fund manager starting to invest in emerging markets andfacing the problem of whether or not he should, either partially or wholly,apply his valuation standard in the U.S. to emerging markets. Typicaltop-down examples include tactical asset allocations driven by valuation ofasset class. Mathematically speaking, these problems become one of pricingproblems. Specifically, when investment opportunities increase, which af-fects the portfolio space (enlarged or reduced), how can a new asset pricingmodel ‘learn’ from previous asset pricing models?

3.1 Symmetric Theorem of Asset Pricing withan Application to Value Economic Derivative

Here is the third main result of this book.

31

32 Chapter 3. Algebra of Stochastic Discount Factors

Theorem 3.1.1 (Symmetric theorem of correctly pricing func-tional spaces) Let H be a claim contingent space as introduced in thelast chapter. Let M be an SDF linear closed subspace of H and spanned bysome economic variables. Let M∗ be the dual space of all continuous linearfunctionals on M . Then M has a unique correctly pricing SDF for assetspace X if and only if M is priced by a unique functional in M∗ and M∗

is isometric to X, with the understanding that if X = , Assumption 2.2.1applies. More precisely, there exists a unique x∗∗ in X satisfying

pi = E[x∗∗mi], for i = 1, 2, · · · , (3.1)

where mi is an SDF in M satisfying E[mi|X ] = xi and M is spanned bym1, · · · ,mn if X = n and by the closure of ∪∞

n=1m1, · · · ,mn in H ifX = .

Proof: For necessity, using the Riesz representation theorem, we know thatthere is a bijective mapping RX∗ fromX∗ toX . Using Theorem (2.2.4), S isa bijective mapping from X to M . Therefore SRX∗ is a bijective mappingfrom X∗ to M . Similarly by using the Riesz representation theorem twice,we have a bijective mapping U from X∗∗ to M∗ by U = R−1

M∗ S RX∗ RX∗∗ . Since the Hilbert space X is reflexive, X∗∗ is isometric to X by amapping V . Therefore W = U V is a bijective mapping from X to M∗.Since all Riesz’s mappings and S are isometric, we know that mappingW is isometric as well. Let m0 be the correctly pricing SDF in M andrecall that T = S−1. Define x∗∗ = T (m0) ∈ X . Then it is easy to seethat pi = E[m0xi] = E[T (m0)xi]; since T (m0) ∈ X , then E[T (m0)xi] =E[T (m0)S(xi)] = E[T (m0)mi] = E[x∗∗mi]. That is

pi = E[x∗∗mi] for i = 1, 2, · · · . (3.2)

Now, let us first consider the case when X = n. By Theorem (2.2.4),M has a correctly pricing SDF for X if and only if the mapping S is acontinuous linear bijective operator from X to M . Let mi = S(xi), i =1, 2, · · · , n. ∀m ∈M, using Theorem (2.2.4), S−1(m) = E[m|x] ∈ X . SinceX is spanned by x1, · · · , xn, we have E[m|X ] =

∑ni=1 wixi. Since S is

linear, we obtain

m = S(E[m|x]) =n∑

i=1

wiS(xi) =n∑

i=1

wimi. (3.3)

In other words, M = spanm1, · · · ,mn.

3.1. Symmetric Theorem of Asset Pricing 33

Next, when X = , let space M1 be the closureof ∪∞

n=1spanm1, · · · ,mn. Since for n ≥ 1, spanm1, · · · ,mn ⊂ M

and M is a linear closed subspace in H , we have M1 ⊂ M . ∀m ∈ M ,since T (m) ∈ X , T (m) =

∑∞i=1 aiei with

∑∞i=1 a

2i < ∞. Define an

SDF ni in M by ni = S(ei). Then m = S(T (m)) = S(∑∞

i=1 aiei) =∑∞i=1 aini. Since each ei ∈ spanx1, · · · , xi = i, ni = S(ei) ∈

spanm1, · · · ,mi. Hence each ni ∈M1. Since M1 is closed in H , we havem ∈ M1. Therefore M ⊂ M1. Putting the two results together, we haveM = M1.

For sufficiency, we first note that there is an isometric mapping be-tween X and M∗ because of Assumption 2.2.1 and the fact that allRiesz’s mappings are isometric. Then the mapping S given by S =RM∗ W V −1 R−1

X∗∗ R−1X∗ from X to M is isometric. By Theorem

(2.2.4), we have completed the proof.

Q.E.D.

Theorems in the last chapter point out that, given an asset space, weneed to find an appropriate economic-factor-driven space that contains acorrectly pricing SDF for the asset space. Now, the above symmetric the-orem indicates that these economic factors can be traded and priced by amarket portfolio, reflexively. Robert Shiller has strongly prompted this ideain his 1993 book entitled Macro Markets: Creating Institutions for Manag-ing Society’s Large Economic Risks. He proposes to set up macro marketsfor claims on aggregate income and service flows and other economic riskfactors. The symmetric theorem indicates a way to price economic riskfactors. See Figure 3.1 for an illustration.

Example 3.1.2 Economic derivatives Deutsche Bank and GoldmanSachs started in October 2002 to provide vanilla and digital options andrange forwards on the U.S. non-farm payroll employment index, the ISMmanufacturing index and the U.S. retail sales (ex-autos) index. We re-fer interested readers to the report in Risk, August 2002 (page 13). Later,ICAP agreed to broker the parimutuel auction-based economic derivatives,including one- and three-month options on the eurozone harmonised in-dex of consumer price (ex-tobacco) inflation index (HICP) and the U.S.consumer price index (CPI) in June, 2003 (see Risk, June 2003, page13). Barclays Bank also provides options on the Halifax U.K. houseprice index.


Asset Space

Market Prices

SDF Space

Correctly pricing

Correctly pricing

Figure 3.1: Symmetric Theorem between SDF space and asset space.

3.2 Compounding Asset Pricing Models withApplications to Bottom-up InvestmentMethodology

In the following we consider the problem of compounding two SDFs intoone SDF to value a larger asset space. Suppose that we have two assetspaces X1 and X2 with SDF spaces M1 and M2, and correctly pricing SDFsm1 ∈M1 and m2 ∈M2 for X1 and X2, respectively, and we are interestedin investing in the combined asset space X = X1 +X2, where ‘+’ refers tothe sum of two sub-linear spaces in a Hilbert space. Financially speaking, itmeans combining two sub-portfolio spaces to form a large portfolio space. Anatural question is whether we can construct a correctly pricing SDF m inM1+M2 forX1+X2. We notice thatm1+m2 may not be a correctly pricingSDF for X1 +X2, though m1 and m2 are correctly pricing SDFs for X1 andX2 individually. The sources of trouble are as follows: (1) In an enlargedasset space, we may have redundant assets, though both m1 and m2 satisfy

3.2. Compounding Asset Pricing Models 35

the parsimoniously pricing condition (2.11) for X1 and X2 separately. Asan example, consider the case in which X1 = spanx1, · · · , xn and X2 =spany1, · · · , yn, but y2 is represented by a portfolio of x1, · · · , xn plus y1.Clearly, y2 is redundant and has to be removed when considering X1 +X2.(2) Now, m1 may incur some pricing error for asset space X2, and m2 forasset space X1. Even if we assume that X1 and X2 are orthogonal, we stillcannot guarantee that m1 incurs zero pricing error for X2 and m2 for X1.Just consider the extreme case of M1 = X2 and M2 = X1. (3) There mayexist some common portfolios in X1 and X2. Whether or not an arbitrageopportunity exists in the enlarged space depends on how m1 and m2 pricethese common portfolios.

A satisfactory solution of the compounding problem is relevant. For, itwill enable us to construct a complex pricing functional from a series of sim-pler pricing functionals. Further, in practice we do meet the compoundingproblem frequently. For example, in an international portfolio consistingof some U.S. stocks, say XUS , and some U.K. stocks, say XUK . Supposethat we already have correctly pricing domestic SDFs, mUS and mUK say,which are functionals of the respective domestic economic variables. Thequestion arises as to how we can price correctly the international portfolioXUS +XUK , based on information of the U.S. and U.K. economic variables.Put another way, suppose in a global fund management company there arealready several highly qualified domestic fund management teams. Howcan the company develop an efficient global fund management frameworkby pooling their existing expertise in the US and the UK investments?

Let V = X1 ∩ X2 denote the subspace of common portfolios in X1

and X2. It is easy to see that V is a closed subspace. Then we have theorthogonal decompositions of X1 and X2:

X1 = V1 ⊕ V and X2 = V ⊕ V2,1 (3.4)

and

X = V1 + V + V2. (3.5)

Assumption 3.2.1 Over a common portfolio space say V all SDFshave the same valuation, i.e. for any i and j

E[mi|V ] = E[mj |V ]. (3.6)

1It is not necessary that V1 is orthogonal to V2.


If this assumption does not hold, then an arbitrage opportunity exists. Letus suppose E[m1en+1] = q1 = q2 = E[m2en+1]. If q1 < q2, then we cantake a long position from agent m1 and a short position from agent m2,thus obtaining profits with zero-risk. Indeed, in an international portfolio,the HSBC bank is a cross-boundary stock listed in the New York StockExchange, the London Stock Exchange and the HongKong Stock Exchange.Given the exchange rates among the U.S. dollar, the H.K. dollar and thesterling and ignoring trading costs, all three markets must give the samevaluation to the HSBC bank; otherwise a typical arbitrage opportunity willexist.

Next, we introduce the notion of a complete SDF pricing space.

Definition 3.2.1 Let X be a portfolio space and M be an SDF space thatprovides pricing candidates to X. Then we say that M is complete to X

if it is agreed by all the SDF pricing candidates in M that only assets withzero payoff possess zero price, that is

for any x ∈ X,E[mx] = 0, ∀m ∈M =⇒ x ≡ 0. (3.7)

Lemma 3.2.2 M is complete to X if and only if the subspace of orthogonalprojections of M on X is X itself. That is

y = E[m|X ] ; ∀m ∈M = X. (3.8)

Proof: All we need to notice is that E[mx] = E[E[m|X ]x]. By settingE[m|X ] = m, we have our conclusion.

Q.E.D.

Here is the fourth main result of this book.

Theorem 3.2.3 (Expanding theorem of correctly pricing function-als) Given the asset space X = X1 +X2, let M be the minimum completeexpansion of M1 + M2 to X. Suppose that assumptions (2.1)-(2.2) hold.Then the SDF space M has a unique SDF, denoted by m, that prices port-folio space X correctly. Furthermore, suppose that assumption (3.2.1) holdsand V1 and V2 are orthogonal. Then the orthogonal projection, E[m|X ], ofm on X is given by

E[m|X ] = E[m1|X1] + E[m2|V2] = E[m1|V1] + E[m2|X2]. (3.9)


Proof: Let T be the orthogonal projector from M to X . First we checkthat it is bijective. By the completeness and Lemma (3.2.2), T is surjective.Second, if m and m′ ∈ M with m = m′ but T (m) = T (m′), then for anyx ∈ X with x = 0, E[(m − m′)x] = E[(T (m) − T (m′))x] = 0. Thiscontradicts the completeness of M to X . So T must be injective. Hence Tis bijective and according to the Dual Theorem, there exists a unique SDFm ∈M that prices X correctly. Furthermore, since E[m1|X ] = E[m1|V1]+E[m1|V ] + E[m1|V2] and E[m2|X ] = E[m2|V1] + E[m2|V ] + E[m2|V2], wehave

E[m1|X1] + E[m2|V2] = E[m2|V2] + E[m1|V1] + E[m1|V ]

= E[m2|V2] + E[m1|V1] + E[m2|V ]

= E[m2|V2 + V ] + E[m1|V1]

= E[m2|X2] + E[m1|V1].

In the above we have used assumption (3.2.1). Thus, we obtain

E[m1|X1] + E[m2|V2] = E[m1|V1] + E[m2|X2]. (3.10)

By the uniqueness of orthogonal projection in X , we have

T (m) = E[m|X ] = E[m1|X1] + E[m2|V2] = E[m1|V1] + E[m2|X2].

Q.E.D.

Theorem 3.2.3 highlights several interesting and important aspects of assetpricing.

1. Given a unique correctly pricing SDF m1 in M1 to X1, we havea linear and continuous valuation functional π1 satisfying π1(x) =E[m1x], ∀x ∈ X1. Similarly we have a linear and continuous valua-tion functional π2 for X2. Then we have an extended functional π,on X1 +X2, of π1 and π2 such that

π|X1 = π1 and π|X2 = π2. (3.11)

However, when we carry out some practical valuation for X1 +X2, wehave to use an SDF, say m, within some economic contexts to helpus make investment decision, since generally π takes only an abstract


form. This theorem tells us that we should search a correctly pricingm in a minimum complete expansion, say M , of M1 +M2 satisfying

π(x) = E[mx], ∀x ∈ X1 +X2, (3.12)

because we will not be able to find a correctly pricing SDF in anySDF space strictly smaller than M .

2. Since M1 has a unique SDF which prices X1 correctly, the minimumcomplete expansion of M1 to X1 is M1 itself, according to Lemma3.2.2. The same applies to the case of M2 and X2. However, M1+M2

is not necessarily complete. This implies that, though the enlargedportfolio space, X1 +X2, involves linear combinations of elements ofX1 and X2, pricing SDF does not simply follow similar linear com-binations. For example, suppose M1 is of dimension n and M2 isof dimension m. Then we have multi-factor representations such asm1 =

∑ni=1 aifi and m2 =

∑mj=1 bjgj . In general, we can expect that

an element in M is represented by

m =n∑

i=1

a′ifi +m∑

j=1

b′jgj +∑

k

ckhk, (3.13)

where hk are some new pricing factors to do with the covariancestructures between X1 and X2, M1 and M2, M1 and X2, and M2 andX1.

3. The above theorem reveals the difficulty when using the bottom-upmethodology to value a portfolio. We have to deal with new fac-tors hk at each upward step. Thus, in building Markowitz’s effi-cient portfolio, each time when we add a new asset into an existingportfolio, we have to re-run the enlarged efficient portfolio. WilliamSharpe’s CAPM model bypasses this difficulty by assuming that thereexists a top SDF space or a largest asset space, the so-called marketportfolio. And nobody can expand this market portfolio further. AllSDFs or rather valuation functionals are only restricted versions ofthe functional given this market portfolio. In practice, we often use amarket index as its proxy or an approximation since we never knowwhat the market portfolio really is.

4. When the whole asset market has no-arbitrage, we know that acorrectly pricing SDF m1 to a portfolio X1 already contains some


“hidden” information about m2 and X2. This reflects the fact thatvaluation is a ‘global’ matter within a whole asset market while in-herent riskiness for each asset is a ‘local’ matter. The bottom-upstrategy exploited by Graham, Buffet and other investors typically ac-knowledges this. For each individual stock such as the IBM computercompany or the BP oil company, analysts first identify its unique riskcharacteristics such as business models, corporate governance, indus-trial organization, growths of sale and earning, and so on. In otherwords, they attempt to build a model of the form

X1 =∑

i

aifi,

where the risk factors fi include common factors and idiosyncraticfactors which are unique to each stock. At this stage, the model hasnothing to do with the valuation process but everything to do with theanalysts’ judgements. Then we need to price fairly X1 or equivalentlyfi, based on all agents’ preferences and risk aversions to the wholeasset market; otherwise an arbitrage opportunity may appear. Hence,when the SDF pricing equation p = E[mix] ∀x ∈ Xi, i = 1, 2 is used,mi is not solely determined by the riskiness of Xi. Instead, by goingto an even bigger portfolio space X , X1 and X2 are priced in X . Itjust happens that m1’s ‘pricing agents’ do not know m2’s, and viceversa.

5. In practice, when asset marketsX1 andX2 are not connected, as is thecase with the current situation between the Chinese stock market andthe U.S. stock market, investors in the two markets have developedtheir respective correctly pricing SDFs. However, the moment whenthe two markets become connected, there are typically some initialbut limited arbitrage opportunities. In this situation, either m1 or m2

or both begin to change until the two markets reach a no-arbitragestate. Meanwhile fundamental risks of the Chinese stocks and theU.S. stocks are still there and unchanged.

See Figure 3.2 for an illustration.

Corollary 3.2.4 Expanding theorem of correctly pricing function-als in a very special case. Suppose that assumptions (2.2.1)- (3.2.1) holdand X has no redundant assets. If asset spaces X1 = spanx1, · · · , xn andX2 = spany1, · · · , yk, m1 is the correctly pricing SDF for X1, satisfying


X1

M1

If

Correctly pricing

X2

M2

Correctly pricing

1M

Then

Correctly pricing

X1+X2

1M has two correctly pricing SDFs

2M has two correctly pricing SDFs

2M

Figure 3.2: Expanding Theorem for SDF spaces and asset spaces.

pi = E[m1xi], i = 1, · · · , n, and m2 is the correctly pricing SDF for X2,satisfying qj = E[m2yj ], j = 1, · · · , k, then the SDF above satisfies m ∈M

and

pi = E[mxi], i = 1, · · · , n and qj = E[myj ], j = 1, · · · , k. (3.14)

In particular, if n = k = 1 and d = [p1q1− < m2, x1 >< m1, y1 >] = 0,then

m = w1m1 + w2m2,2 (3.15)

where w1 = 1d [p1− < m2, x1 >]q1 and w2 = 1

d [q1− < m1, y1 >]p1, is acorrectly pricing SDF.

Proof: From the proof of Theorem (3.2.3), it is easy to have pricing equa-tion (3.14). To prove pricing equation (3.15), we can solve for w1 andw2 in two equations: p1 =< m,x1 > and q1 =< m, y1 >. We havep1 =< m,x1 >= w1 < m1, x1 > +w2 < m2, x1 > and q1 = w1 <

m1, y1 > +w2 < m2, y1 >, which give w1 = 1d [p1− < m2, x1 >]q1 and

w2 = 1d [q1− < m1, y1 >]p1.

Q.E.D.

2If n > 1 or k > 1, formula (3.15) cannot provide a correctly pricing SDF for X.

3.3. Compression of Asset Pricing Models 41

Example 3.2.5 International index portfolio Suppose that x1 = theU.S. S&P 500 index and x2 = the U.K. FTSE 100 index. Suppose thatSDFS&P500index is the correctly pricing SDF for the U.S. S&P 500 in-dex tracking fund. Similarly suppose that SDFFTSE100index is the cor-rectly pricing SDF for the U.K. FTSE 100 index tracking fund. Thenin order to have correctly pricing for the international index portfolioconsisting of x1, x2, we have to generate a proper SDF from the two-dimensional SDF space (SDFS&P500index, SDFFTSE100index) by SDF =w1SDFS&P500index + w2SDFFTSE100index.

3.3 Compression of Asset Pricing Models withApplications to Top-down InvestmentMethodology

Since we have argued that valuation of a portfolio is a matter related toitself, we face a complexity problem due to the number of assets. For ex-ample, in the U.S. market, there are thousands of stocks; if we want todo an index valuation such as Russell 3000 small-cap index, it is infeasibleto value 3000 stocks one by one. We need some method of compression.In the top-down pricing methodology, all assets are priced by their finitelymany common risk factors; the assets have zero valuation for their ownidiosyncratic risk factor. (The bottom-up valuation methodology, such asBuffet’s value strategy, assigns non-zero valuation to idiosyncratic risk fac-tors.) Based on the Law of Large Number, Ross’s APT (1976) developsa compression method for asset pricing by pricing the common factors.Chamberlain and Rothschild(1983) and Chamberlain (1983) have built theAPT involving large number of assets. In the following, we will develop atop-down correctly pricing methodology.

Definition 3.3.1 Let xf be an element in . Then xf = 0 is called ariskless limit portfolio if V (xf ) = 0 and E[xf ] = 0. Let Rf = xf/E[xf ] ∈. Then Rf is called a riskless asset.

Lemma 3.3.2 If has no riskless limit portfolio, then is a Hilbert spaceunder the covariance inner product, that is, ∀x, y ∈ , we can define <

x, y >= cov(x, y), where cov(x, y) is the covariance between x and y.

Proof: If < x, x >= cov(x, x) = V (x) = 0, then we have x = E[x] in H .If E[x] = 0, then has a riskless limit portfolio. This is a contradiction.


Therefore we must have E[x] = 0 so that x = 0. We can easily verify thatother properties for an inner product are satisfied with the exception ofthe completeness of under the covariance as an inner product. Supposethere is a sequence xn in satisfying < xn − xm, xn − xm >→ 0 asn and m → ∞. Since < xn − xm, xn − xm >= Cov(xn − xm, xn − xm) =E(xn−xm)2 +(Exn−Exm)2, then ||xn−xm||2 = E(xn −xm)2 → 0. Since is complete under the norm ||.||, there is x ∈ satisfying ||xn − x|| → 0.By the Cauchy-Schwartz Inequality, (Exn−Ex)2 ≤ E[xn−x]2 = ||xn−x||2.Hence V (xn −x) → 0. This means that is complete under the covarianceas an inner product and is a Hilbert space correspondingly.

Q.E.D.

Suppose has a riskless limit portfolio, say xf . Define excess payoffs byzi = xi − µiRf , i = 1, 2, · · · , where µi = E[xi]. Define linear spacesΨn = spanz1, · · · , zn, Ψ = ∪∞

n=1Ψn and Ψ as the closure of Ψ. For eachi = 1, 2, · · · , E[zi] = E[xi] − µiE[Rf ] = 0. It is easy to see that Ψ has noriskless limit portfolio and hence the covariance can be regarded as an innerproduct for Ψ. Therefore Ψ is a Hilbert space and a closed linear space of.

Define 1 = if has no riskless limit portfolio and 1 = Ψ otherwise.

Definition 3.3.3 By a well-diversified portfolio space, we mean the spaceD = x ∈ 1| there exists a sequence of xn =

∑ni=1 ai,nzi with xn →

x,∑n

i=1 a2i,n → 0 as n→ ∞, where zi = xi −µiRf , i = 1, 2, · · · , if has

a riskless limit portfolio, and zi = xi, i = 1, 2, · · · , if has no riskless limitportfolio.

According to the Ross Arbitrage Pricing Theory, a well-diversified port-folio should only incur factor risk but no idiosyncratic risk. Normally thereexist only finitely many risk factors. Here we introduce the following defi-nition.

Definition 3.3.4 We say that the asset space 1 has a K-factor structureif dim(D) = K <∞.

This means that if the well-diversified portfolio space D has dimensionK, then D is spanned by only K orthonormal basis elements f1, · · · , fKsatisfying

cov(fi, fj) = 0 if i = j and V (fi) = 1, i, j = 1, · · · ,K. (3.16)


The payoffs are represented by

zi = βi,1f1 + · · · + βi,KfK + εi, for i = 1, 2, · · · , (3.17)

where factor loadings βi,k = cov(zi, fk) and cov(εi, fk) = 0 for k = 1, · · ·K.In particular if has no riskless limit portfolio, then

xi = µi + βi,1f′1 + · · · + βi,Kf

′K + ε′i for i = 1, 2, · · · , (3.18)

where µi = E[xi], f ′k = fk − E[fk] and ε′i = εi − E[εi]. If has a riskless

limit portfolio, then

xi = µiRf + βi,1f1 + · · · + βi,KfK + εi for i = 1, 2, · · · (3.19)

with E[fk] = 0 and E[εi] = 0.Let εn = (ε1, · · · , εn) and the second-moment matrix Sn

ε = E[εn(εn)T ]of εn. Let λn

max be the largest eigenvalue of Snε . To have diminishing

idiosyncratic risk, we have to make the following assumption.

Assumption 3.3.1 λ = maxn≥1λnmax <∞.

Here is the fifth main result of this book.

Theorem 3.3.5 (Compression theorem of correctly pricing func-tionals) Suppose that Assumptions 2.1.1, 2.2.1 and 3.3.1 hold. Let ∗

1

be the dual space of 1. Suppose that F is a linear subspace of ∗1

and has a unique correctly pricing functional on 1. Then 1 has a K-factor structure if and only if F has a K-factor structure as well. Thatis F = D∗ + (D∗)⊥, and D∗ is well diversified with K-factor structure.∀π ∈ F, π =

∑Kk=1 bkf

∗k + ε∗, where bk =< π, f∗

k >, the pricing func-tionals f∗

k ⊂ D∗ satisfy < f∗k , f

∗j >= 0 for k = j ,< f∗

k , f∗k >= 1 and

< f∗k , ε

∗ >= 0 for k = 1, · · · ,K, and ε∗ ∈ (D∗)⊥. A similar conclusionapplies to an SDF space.3

Proof: To prove necessity, suppose 1 = (i.e. no riskless limit portfolio).Define pricing functional πi by ∀x ∈ , πi(x) = E[xix]. Then we define aspace D∗ = π ∈ ∗| there exists a sequence of pricing functionals πn =∑n

i=1 ai,nπi with πn → π,∑n

i=1 a2i,n → 0 as n → ∞. Since F has one

correctly pricing functional, by Theorem 2.1, F = ∗. Then we know thatD∗ is a linear subspace of F . Since 1 has a K-factor structure, there is

3Cochrane (2000) has used this result without giving a rigorous proof.


an orthogonal decomposition 1 = D +D⊥. For all y ∈ D⊥ and πn ∈ D∗,πn(y) =

∑ni=1 ai,nπi(y) =

∑ni=1 ai,nE[xiy] = E[(

∑ni=1 ai,nεi)y]. Hence, by

the Cauchy-Schwartz inequality,

|πn(y)| = |E[(n∑

i=1

ai,nεi)y]| ≤ (an)TSnε a

n 12 E[y2] 1

2 ,

where Sε is the covariance matrix of εn = (ε1, · · · , εn)T and an =(a1,n, · · · , an,n)T . By the singular-value decomposition, we have Sn

ε =UΠUT , where U is an orthogonal matrix with UUT = I and Π is a di-agonal matrix Π = diag(λ1, · · · , λn) with eigenvalues λi. Using Assump-tion (3.3.1), we have |πn(y)| ≤ λE[y2] 1

2 (∑n

i=1 a2i,n). Letting n → ∞, for

∀π ∈ D∗, we have π(y) = limn→∞ πn(y) = 0. By the Riesz Represen-tation Theorem, for pricing functional π, there is an xπ ∈ 1 satisfying∀x ∈ 1, π(x) = E[xπx]. Since π(y) = 0 for all y ∈ D⊥, we have xπ ∈ D.According to Definition (3.3.4), D has dimension K and is spanned byf1, · · · , fK . We have xπ = b1f1 + · · · + bKfk. Define pricing functional f∗

k

by ∀x ∈ 1, f∗k (x) = E[fkx]. Then it is easy to see < f∗

k , f∗j >= 0 if k = j

and D∗ is spanned by f∗1 , · · · , f∗

K (∀π ∈ D∗, π = b1f∗1 + · · ·+bKf∗

k .) There-fore pricing functional subspace F has a K-factor structure D∗. The sametreatment applies to 1 = Ψ (i.e. there is a riskless limit portfolio). Alsothe above argument applies to an SDF space.

To prove sufficiency, note that using Theorem (3.1.1), we can regardasset space 1 as a correctly pricing SDF space for the dual space ∗

1, andthen apply the necessary condition to the ‘virtual SDF pricing’ space 1.

Q.E.D.

The implication of Theorem (3.3.5) is that if the payoffs of assets are eithercorrelated with or not hedged against risk factors (such as the growth ofconsumption, the GDP, the inflation, the interest rate, the monetary andfiscal policies, the business cycle, the average earning growth of stocks andso on), then in order to obtain a correctly pricing functional, or SDF, thecandidate space must be spanned by the above stated economic factors, ortheir equivalents under valuation-preserving isometry. For example, usingthe CRRA model and insisting that it has provided correctly pricing implythe assumption that the asset market has a one-factor structure, and usingthe Epstein and Zin’s model implies the assumption that the asset markethas a two-factor structure. In fact, even the asset market has a one-factorstructure. For example, suppose that the asset market follows the CAPM


SDF Space

Asset Space

The same factor structure

Figure 3.3: Compression Theorem for SDF spaces and asset spaces.

model, then the single factor is the market return and not consumptiongrowth. In order to have correctly pricing to the asset market driven bythe market return, we need to have zero effects of means, variances andcorrelations in equation (3.25) (to be given later) between the market re-turn and the consumption growth. However, when the asset market is notcomplete, it is hard to have such a perfect match between the market returnand the consumption growth. In this case, a pricing error can appear.

See Figure 3.3 for an illustration.

Example 3.3.6 Fama-French’s three factor In Fama and French(1992) and a series of subsequent papers by the same authors, they build athree-factor model

E[R] −Rf = β(E[Rm] −Rf ) + βSMBSMB + βHMLHML. (3.20)

If the stock market is spanned by these three factors, then according tothe compression theorem, the SDF space must be spanned by a three-factor


structure as well. For example, besides consumption, we should also con-sider unemployment, inflation, monetary and fiscal policies, technologicalprogress, human capital economic variables and so on.

Corollary 3.3.7 Suppose that payoffs xi ⊂ D, i.e. xi are well diver-sified. If pricing functional π is the correctly pricing functional for xi,then π ∈ D∗. In other words, π is a well diversified pricing functional in∗

1 as well.

3.4 Decomposition of Errors in Asset Pricing Models

Given market prices p1, · · · , pn, Hansen and Jagannathan (1997) intro-duces a space of correctly pricing functionals by

M = π ∈ H∗ | pi = π(xi), i = 1, · · · , n (3.21)

= m ∈ H |pi = E[mxi], i = 1, · · · , n.

Then M is a convex subspace but not a linear subspace of H . Let π bea correctly pricing functional and π be a continuous pricing functional.Hansen and Jagannathan (op.cit.) defines a maximum pricing error by

d = sup|π(x) − π(x)| over x ∈ n, ||x|| = 1. (3.22)

Theorem 3.4.1 Pricing error theorem. Let m be an unknown correctlypricing SDF for asset space n. Let m be an SDF proxy that often is anonlinear function of several economic variables. Then the pricing errordue to m is given by d, which may be expressed as follows:

1. d = || E[m|n] − E[m|n] || (3.23)

= [(E[E[m|n]] − E[E[m|n]])2

+ (1 + λ2)σ2E[m|n]

− 2ρλσ2E[m|n]

]12 (3.24)

= [dmean + dvariance + dcorrelation]12 , (3.25)

where E[m|n] and E[m|n] are the orthogonal projections of mand m on n respectively, ρ is the correlation coefficient betweenE[m|n] and E[m|n], and λ is volatility ratio given by λ =

σE[m|n]

σE[m|n].

3.4. Decomposition of Errors in Asset Pricing Models 47

In other words, the pricing error comes from three parts: (a) dmean,the difference in means of the orthogonal projections, which measuresthe pricing error of the inverses of riskless asset Rf by m and m;(b) dvariance, which measures the deviation of the variance σ2

E[m|n]

from the variance σ2E[m|n]

; [Ideally, given the correlation ρ > 0, thenthe optimal variance ratio λ = ρ, and a minimum pricing error isd =

√dmean + (1 − ρ2)σ2

E[m|n]]. (c) dcorrelation, which measures

the contribution of the pricing error from any correlation betweenE[m|n] and E[m|n]. [Ideally, ρ = 1].

2. Suppose that E[m|n] =∑n

i=1 aiei and, without loss of generality,we assume that E[m|n] ∈ L, where L is the subspace of n withL ≤ n, and a representation E[m|n] =

∑Li=1 biei. Then we have

d =

√√√√ L∑i=1

(ai − bi)2 +n∑

i=L+1

a2i > 0. (3.26)

3. Equivalently if we represent E[m|n] and E[m|n] by E[m|n] =∑ni=1 wixi and E[m|n] =

∑ni=1 vixi respectively, where vi = 0 for

i > L, then we have the well-known Hansen-Jagannathan distance

d =√

(w − v)TSn(w − v) (3.27)

=√

(pn − E[mxn])TS−1n (pn − E[mxn]), (3.28)

where Sn is the second moment of the payoff vector xn =(x1, · · · , xn)T , i.e. Sn = E[xn(xn)T ].

4. If we further assume that E[E[m|n]] = E[E[m|n]], then

d =√

(w − v)T Σn(w − v), (3.29)

where the weight vectors w = (w1, · · · , wn)T and v = (v1, · · · , vn)T .

Proof: The main ideas of the proof come from Hansen and Jagannathan(1997). Since ∀x ∈ n, π(x) = E[mx] = E[E[m|n]x]. Without loss ofgenerality, we can assume that m and m ∈ n. ∀x ∈ n with ||x|| =1, by the Cauchy-Schwartz inequality, |π(x) − π(x)| = |E[(m − m)x]| ≤E[m −m]2 1

2 E[x]2 12 = ||m −m||. Hence d ≤ ||m −m||. On the other


hand, since ||m −m|| > 0 (otherwise we have simply m = m and d = 0),define payoff m = (m− m)/||m− m||. We have π(m) − π(m) = E[mm] −E[mm] = E[(m − m)m] = E[m − m]2/||m − m|| = ||m − m||. Since||m|| = 1, d ≥ π(m) − π(m) = ||m − m||. Therefore we have proved thatd = || E[m|n]− E[m|n] ||. By using E[ξ− η]2 = (E[ξ]−E[η])2 + cov(ξ−η, ξ− η) = (E[ξ]−E[η])2 + σ2

ξ + σ2η − 2ρσξση for any two random variables

ξ and η, we obtain the pricing error decomposition formulas (3.24) and(3.25). Using (3.23), (3.26) and (3.27) can be calculated easily. To show(3.28), since m|n is an orthogonal projection of m on n, we have

v = S−1n E[mxn]. (3.30)

Next, since m is a correctly pricing SDF, we have pi = E[mxi] =∑nj=1 wjE[xixj ] for i = 1, · · · , n. Hence we obtain an ‘exact’ estimation

of w = (w1, · · · , wn) by

w = S−1n pn . (3.31)

Therefore,

d =√

(w − v)TSn(w − v) =√

(pn − E[mxn])TS−1n SnS

−1n (pn − E[mxn])

=√

(pn − E[mxn])TS−1n (pn − E[mxn]).

Q.E.D.

From (3.26), we see that a pricing error can come from two sources: (1) themost serious is due to the mistake of searching correctly pricing functionalin an insufficient candidate space, which gives a strictly positive pricingerror

∑ni=L+1 a

2i > 0; (2) even if we search in a proper space (i.e. L = n),

we can still incur non-negative pricing error∑L

i=1(ai − bi)2 ≥ 0, due towrong parameter estimation.

See Figures 3.4 and 3.5 for illustrations.Furthermore we have an exact decomposition of pricing error over each

asset.

Corollary 3.4.2 Define the representation error between m and m bythe coefficient vector w − v. Then there is no pricing error (E[m|n] =E[m|n]) if and only if w − v = 0. In general, we have a decomposition ofthe representation error vector w − v by

w − v = σmΣ−1n bcorr + (σm − σm)Σ−1

n σ = Iρ + Iσ,4 (3.32)

3.4. Decomposition of Errors in Asset Pricing Models 49

xmE 2

m2 m1

xmE 1

Then to check two random variables

xmEU 11

And xmEU 22

2 1

EU2 EU1

Namely to compare

(1) EU1 = EU2 ?

(2) 1 = 2 ?

Figure 3.4: Pricing Error Theorem.

where the vector of differences of correlations bcorr = ((ρm,x1 −ρm,x1)σ1, · · · , (ρm,xn − ρm,xn)σn)T , ρm,xi is the correlation coefficientbetween m and xi, ρm,xi is the correlation coefficient between m andxi, σm is the volatility of m, σm is the volatility of m, and σ =(σ1ρm,x1 , · · · , σnρm,xn) is the weighted volatility vector of payoffs. Iρ isthe error vector due to the difference between the correlation of m and pay-offs xi and the correlation of m and payoffs xi. Iσ is the error vectordue to the difference between the volatility of m and the volatility of m.

4When the equity premium puzzle is referred to, the main concern tends to bewith SDF’s Iσ volatility difference and with the Iρ correlation difference being typicallyignored.


U2

U1

Correlation -

Figure 3.5: Comparing correlation error in the Pricing Error Theorem.

Proof: It is straightforward to see that E[m|n] = E[m|n] if and onlyif w − v = 0. Without loss of generality, we can assume that m, m ∈ n.Since m =

∑nj=1 wjxj , we have m− E(m) =

∑nj=1 wj(xj − E[xj ]). Multi-

plying both sides of the equation by (xi − E[xi]) and taking expectation,we obtain

Cov(m,xi) = E[(m− E[m])(xi − E[xi])]

=n∑

j=1

wjCov(xi, xj). (3.33)

Hence we have w = Σ−1n Cov(m,x), where Σn is the covariance ma-

trix of x = (x1, · · · , xn)T and Cov(m,x) is the covariance vector(Cov(m,x1), · · · , Cov(m,xn))T . Similarly we have v = Σ−1

n Cov(m, x).Hence w − v = Σ−1

n (Cov(m,x) − Cov(m, x)). Because Cov(m,xi) −Cov(m, xi) = (ρm,xi − ρm,xi)σmσi + (σm − σm)σiρm,xi , thereforeCov(m,x)−Cov(m, x) = σmbcorr + (σm −σm))σ. We have completed theproof of the corollary.

Q.E.D.

3.5. Empirical Analysis of the Asset Pricing Models 51

3.5 Empirical Analysis of the Asset Pricing Models

3.5.1 The data set and utility forms

The data sets we use here are the quarterly real returns of S&P 500 indexand the one-month U.S. treasury bill over the period from the first quarterof 1950 to the last quarter of 2002. The U.S. real per capita consumptiondata are extracted from the nominal consumption data adjusted by theCPI data over the same period. Stock data are the quarterly real returnsof the American Electric Power Co. Inc. (AEP), the General ElectricCo. (GE), the General Motors Corp. (GM), the International BusinessMachines Corp. (IBM) and the Exxon Mobil Corp. (XOM), over theperiod from the first quarter of 1970 to the last quarter of 2002.5 Thecompanies are components of the Dow Jones industrial average index.

Let m0 be a correct but unknown pricing SDF for the asset space n =spanx1, · · ·xn. Let m be the orthogonal projection of m0 on n, i.e. m =E[m0|n] and takes the form

m = c′ · x, (3.34)

where c is a parameter vector, x is the payoff vector of asset x =x1, · · ·xn. The parameters can be estimated using the least squaresmethod.

Let m be an SDF which may be taken as a proxy for m0. Suppose thatm has one of the following forms:

• CRRA utility

mt+1 = β

(ct+1

ct

)−γ

, (3.35)

where β is a subjective discount factor, 0 < β < 1, γ is the coefficientof relative risk aversion, c is the consumption.

• Abel utility

mt+1 = β

(ct+1

ct

)−γ (ctct−1

)γ−1

, (3.36)

where γ is the coefficient of relative risk aversion.5The quarterly real returns of S&P 500 index are taken from Robert Shiller’s home-

page. The quarterly returns of the one-month treasury bill come from the Federal ReserveBank. The per capita consumption and the CPI data are taken from the Bureau of Eco-nomic Analysis (BEA). The data for the five listed companies come from the Yahoofinance website.


• Constantindes utility

mt+1 = β

(ct+1

ct

)−γ (1 − ht+1/ct+1

1 − ht/ct

)−γ

, (3.37)

where ht = ϕct−1, γ and ϕ are parameters.

• Epstein-Zin utility

mt+1 = βη−ηγη−1

(ct+1

ct

)− 1−γη−1

R1−ηγη−1

m,t+1, (3.38)

where η is the elasticity of intertemporal substitution, γ is the coeffi-cient of relative risk aversion and Rm is the gross return of the marketportfolio.

3.5.2 Three sources of pricing errors

Part 1 of Theorem (3.4.1) indicates that the pricing error d consists ofthree parts, namely the differences in the means, the differences in thevolatilities and the imperfect correlation. From Tables 3.1-3.5, we can seethat the difference in the means given by the orthogonal projections of twoSDFs is very small over various parameters. The CRRA-type pricing modelprovides the worst scenario among all candidate SDFs. It provides smallervolatility and even negative correlation. The negativeness is very stable overdifferent risk aversion parameters. This indicates that a wrong SDF hasbeen chosen, if we accept that over the past 100 years the historical equitypremium has been correct. However, by varying risk aversions, the volatilityratio is improved impressively. The Abel model and the Constantindesmodel provide a similar pattern of correlation and volatility ratio, but aslightly better result than the CRRA model. Not surprisingly, since theEpstein-Zin model is spanned by two state variables and is more complex

Table 3.1: Three sources of pricing errors in theCRRA model (β = 0.95)

γ dmean ρ λ5 0.0018 -0.66 0.01115 0.0074 -0.66 0.03130 0.0208 -0.66 0.056

3.5. Empirical Analysis of the Asset Pricing Models 53

Table 3.2: Three sources of pricing errors in the Abelmodel (β = 0.95)

γ dmean ρ λ5 0.0005 -0.69 0.01815 0.0004 -0.69 0.05730 0.0000 -0.68 0.114

Table 3.3: Three sources of pricing errors in theConstantinides model (β = 0.95, γ = 5)

θ dmean ρ λ0.1 0.0018 -0.66 0.0130.5 0.0017 -0.68 0.0310.9 0.0000 -0.66 0.169

Table 3.4: Three sources of pricing errors in theEpstein-Zin model (β = 0.95, γ = 5)

η dmean ρ λ0.1 0.0014 -0.71 0.1090.3 0.0027 0.69 0.1070.5 0.0020 0.72 0.465

Table 3.5: Representation errors for various utilities for a single risky asset

Panel A: CRRA utility (β = 0.95)

γ Iρ Iσ Total wρ(%) wσ(%)5 -0.018 -0.323 -0.341 5 9515 -0.052 -0.282 -0.334 16 8430 -0.101 -0.223 -0.325 31 69

Panel B: Abel utility (β = 0.95)

γ Iρ Iσ Total wρ(%) wσ(%)5 -0.019 -0.320 -0.338 6 9415 -0.060 -0.264 -0.324 19 8130 -0.129 -0.174 -0.303 42 58

Note: wρ = |Iρ|/(|Iρ| + |Iσ|), wσ = |Iσ |/(|Iρ| + |Iσ|).


than the previous three models, it improves both the correlation and thevolatility substantially. The correlation ρ is improved from the negativevalue of −71% to the positive value of 72%, and the volatility ratio λ isimproved from 0.109 to 0.465, which is much closer to the optimal volatilityratio of 72%. The structural theory of asset pricing, coupled with theabove empirical results, suggests that in order to find a proper SDF, themost important consideration is to determine which set of economic statevariables is proper. The next important consideration is to decide on theappropriate functional form for the utility functional for these economicstate variables.

3.5.3 Decomposition of the pricing errors

Tables (3.5) - (3.7) show the pricing errors in accordance to the decompo-sition in Corollary (3.4.2). Again, we can see that the main pricing error isdue to a mis-match of the volatilities of two SDFs, as a result of the volatil-ity of consumption staying low (especially after World War II) while thevolatility of stocks remaining high. The correlation bias only contributeson average 15% or less to the errors. Comparing the Epstein-Zin modelwith others, we can see that its correlation bias can be reduced readily ifthe elasticity of inter-temporal substitution is very small and the coefficientof relative risk aversion is no more than 5.

Table 3.6: Representation errors for various utilities

Panel C1: Constantinides utility (β = 0.95, γ = 5)

ϕ Iρ Iσ Total wρ(%) wσ(%)0.1 -0.019 -0.321 -0.340 6 940.5 -0.033 -0.301 -0.334 10 900.9 -0.218 -0.065 -0.283 77 23

Panel C2: Constantinides utility (β = 0.95, γ = 15)

ϕ Iρ Iσ Total cρ(%) cσ(%)0.1 -0.056 -0.276 -0.332 17 830.5 -0.101 -0.213 -0.314 32 680.9 -1.872 1.684 -0.188 55 47

Note: cρ = Iρ/(|Iρ| + |Iσ|), cσ = Iσ/(|Iρ| + |Iσ |).

3.6. Conclusions 55

Table 3.7: Representation errors for various utilities

Panel D1: Epstein-Zin utility (β = 0.95, γ = 5)

η Iρ Iσ Total cρ(%) cσ(%)0.1 -0.003 -0.303 -0.306 1 990.3 -0.090 -0.296 -0.385 23 770.5 -0.321 -0.180 -0.501 64 36

Panel D2: Epstein-Zin utility (β = 0.95, γ = 15)

η Iρ Iσ Total cρ(%) cσ(%)0.1 -0.089 -0.277 -0.366 24 760.3 -0.543 0.055 -0.598 91 90.5 -1.72 0.635 -1.085 73 27

Note: cρ = Iρ/(|Iρ| + |Iσ|), cσ = Iσ/(|Iρ| + |Iσ|).

Table 3.8: Representation errors for a portfolio of risky assets(The utility is CRRA with β = 0.95 and γ = 5)

Assets Iρ Iσ w − v cρ(%) cσ(%)AEP -0.019 -0.097 -0.117 16 84GE 0.018 -0.096 0.114 16 84GM 0.009 0.190 0.199 4 96IBM -0.017 -0.156 -0.173 10 90XOM -0.007 0.126 0.119 5 95Average -0.003 0.032 0.028 1 0 90

Table 3.8 shows the representation errors for a portfolio of five stocks,namely AEP, GE, GM, IBM and XOM, which are all blue chip companiesin the U.S. Again we can see that the pricing error given by CRRA’s SDFhas a small correlation bias, which contributes on average only 10%, butits volatility bias contributes on average 90%. Finally, the GE stock hasthe smallest volatility bias.

3.6 Conclusions

In this chapter, we have developed further the structural theory for assetpricing modelling. We now have, as a refinement of the SDF theory initiated


by Cochran (2001), an ‘algebra of operations’, with which we can derivepricing models.

The symmetric theorem provides a way to value non-tradable factors,such as economic indices, by reflexively using market assets and their cor-responding market prices. The expanding theorem provides a bottom-upway to construct asset pricing models. First we look for correctly pricingSDFs for simpler portfolios in subspaces of an asset space. Second we com-pound the SDFs into an advanced SDF to price correctly portfolios in theasset space, by removing excessive cross-pricing effects among the SDFs.For example, once we have correctly pricing SDFs domestically, then wecan compound them into a single SDF to price international portfolios cor-rectly. The compression theorem provides a top-down way to constructasset pricing models. To price well-diversified asset portfolios with a K-factor structure correctly, a necessary and sufficient condition for the SDFspace to have a unique correctly pricing SDF is that the SDF space pos-sesses a K-factor structure as well. In other words, both spaces have noidiosyncratic risk and only K factor risks are left to be considered. Cochrane(2000) has used this fact without giving a rigorous proof. A combinationof the expanding theorem and the compression theorem provides us with aroutine with which we can value portfolios at different levels. Based on thetheory of corporate finance, the theory of interest rate and the theory ofderivative pricing, the valuation of an individual security is well developed.However, portfolio valuation, especially risk arbitrage portfolio valuation,well-diversified portfolio valuation or index valuation, is less so. See Co-hen (2004) for example. Finally, the pricing error theorem indicates a wayto measure how well a given SDF does the pricing job, by first project-ing two SDFs (the given SDF and the unknown correctly pricing SDF)into the asset space, and then measuring the distance (using for exam-ple the Hansen-Jagannathan distance) between the two projected proxies.Three sources of pricing errors are identified as the differences in the meansof the two proxies, the differences in the volatilities of the two proxies,and the imperfect correlation between the two proxies. Our empirical re-sults show that the main contribution to the pricing error is the differencein volatilities.

Chapter 4

Investment and Consumption in a

Multi-period Framework

4.1 Review of Merton’s Asset Pricing Model

In Merton’s multi-period framework (1973), there is no restriction on theconsumption beyond its being non-negative. However, not every investorwill agree with this lack of restriction. One simple example is to do withendowment funds (Thaler & Williamson, 1994), such as the Nobel Prizefund. There are at least three reasons why the Nobel prize committeewould need to impose some consumption discipline. The first is that, everyyear, a fixed amount of cash is taken out from the fund to provide the Nobelprizes to the winners. The second is that the growth rate of the endowmentfunds needs to keep abreast of the inflation rate. The third is that it wasthe wish of Nobel that the Nobel prize fund should last forever.

In the habit models of Abel (1990), Campbell and Cochrane (2000),and many others, it is generally accepted that it is human nature to keepimproving one’s living standards. If the consumption is below somebody’shabit level due to some reason, the person will feel unhappy and attemptto overcome the relatively hard time. A reasonable assumption is, in ouropinion, to treat the consumption habit as a consumption constraint, ct ≥ht, where ct is the consumption level at time t and ht is the consumptionhabit at time t.

The relationship between the volatilities of the consumption and thestock markets has attracted attention in recent years. Poterba (2000) findsthe ‘wealth effect’ by examining the relationship between the total con-

57

58 Chapter 4. Investment and Consumption in a Multi-period Framework

sumption and the total wealth of the stock market. His result shows thatone US dollar change of the total value of the stock market will bring aboutone to two cents of change in the consumption. If we assume that the stockmarket is complete, then the consumption habit is spanned by the portfoliosin the stock market.

Financial Model

Let (Ω,F ,P) be a complete probability space, and (Ft)0≤t≤T a filtrationsatisfying the usual conditions. F0 = σ∅,Ω,FT = F , where positivenumber T is a fixed and finite time horizon. We consider a security marketwhich consists of m + 1 assets: one bond and m stocks. The price of therisk-free bond at time t is ert, where r is a non-negative constant. Forj = 1, · · · ,m, the price of stock j at time t is Sj

t and Sj is a strictlypositive semimartingale with cadlag paths. For notational convenience, weset S = (S1, · · · , Sm).

Stochastic integration is used to describe the outcomes of investmentstrategies. When dealing with processes in dimensions higher than one, itis understood that vector stochastic integration is used. Interested readersmay wish to refer to Protter (1990) for details on these matters. We usethe notation

∫ t

0 HdX or (H.X)t to stand for the integral of H with respectto (or w.r.t. for short) X over the interval (0, t]. In particular, (H.X)0 = 0.

A probability measure Q is called an equivalent martingale measure ifit is equivalent to the historical probability measure P and the discountedprice processes of stocks (e−rtSt) is a (vector-valued) Q-martingale. By Mwe denote the set of all equivalent martingale measures. We make the stan-dard assumption that M is not empty to exclude arbitrage opportunities.The security market is called complete if M is a singleton, or else we saythat the market is incomplete.

We assume that the market is complete and there is a unique probabilitymeasure Q, which has a density function φt = E

[dQdP

| Ft

], t ∈ [0, T ], in M.

Trading Strategies

A trading strategy is an Rm+1-valued Ft-predictable process α = α0, ψ,such that ψ is integrable w.r.t. the semi-martingale (S0, S), where ψ =(α1, · · · , αm) and αj

t represents the number of units of the asset j held attime t, 0 ≤ j ≤ m. The wealth Wt(α) of a trading strategy α = α0, ψat time t is Wt(α) = α0

tS0t + ψt · St, where ψt · St =

∑mj=1 α

jtS

jt . A trading

4.1. Review of Merton’s Asset Pricing Model 59

strategy α = α0, ψ is said to be self-financing if

dWt(α) = −ctdt+ αtd(S0t , St), t ∈ [0, T ], (4.1)

where ct is the consumption at time t.It is easy to see that for any given Rm-valued predictable process ψ

which is integrable w.r.t. S and any real number x, there exists a real-valued predictable process α0 such that α0, ψ is a self-financing strategywith initial wealth x.

Because we have assumed that the market is complete and there is aunique probability measure Q in M, every contingent claim in this market,therefore, can be replicated by a self-financing trading strategy.

Inter-temporal Investment and Consumption Problem

The investor endowed with an initial wealth W0 > 0 tries to find the opti-mal consumption and investment decisions to maximize his total expectedutility, i.e.

max(ct,αt),t∈[0,T ]

E

(∫ T

0

e−ρtU(ct)dt

), (4.2)

where ρ is the time preference of consumption and U(c) is the consumptionutility function satisfying

• U(c) has continuous second order derivative function on [0, T ];

• Uc = dUdc > 0, Ucc = d2U

dc2 < 0;

• Uc(0) = limc↓0

Uc(c) = ∞, Uc(∞) = limc→∞Uc(c) = 0.

The utility function

U(c) = cγ/γ (4.3)

is called the CRRA (Constant Relative Risk Aversion) utility function.

Consumption Habit Constraints

There exists a real valued Ft-predictable habit consumption process ht :[0, T ]× Ω → satisfying 1

ct ≥ ht, t ∈ [0, T ]. (4.4)

We shall refer to the above inequality as a consumption habit constraint.1Hereafter inequality is valid with probability 1.


There are several types of habit process. One broad class is proposedby Detemple and Karatzas (2001), which gives that

dht = (δht − αct)dt+ δη′tdBt, (4.5)

where h0 ≥ 0, α ≥ 0, δ ≥ 0 are given real constants, ηt : [0, T ]× Ω → Rm isa bounded, progressively measurable process and Bt = (B1

t , B2t , · · · , Bm

t )is the price driving Brownian process on the probability space (Ω,F ,P).

If we let α = 0, δηt = htσm, then the Detemple and Karatzas’s habitbecomes

dht

ht= µmdt+ σmdBt, (4.6)

which is studies in details in Cheng and Wei (2005).

4.2 Optimal Decisions of Investmentand Consumption

4.2.1 Martingale approach to the asset pricing modelwithout consumption habit constraints

In this section we review some classic results of Merton’s Model, in whichthere is no consumption habit constraint. Given time-separable preferencesdefined over consumption and initial wealth W0 > 0, the optimal portfolioand consumption problem without consumption habit is

max(ct,αt),t∈[0,T ]

E

(∫ T

0

e−ρtU(ct)dt

),

with budget constraints given by (4.1).

Martingale Approach

Cox and Huang (1989), Karatzas, Lehoczky and Shreve (1987) and Pliska(1986) have suggested a martingale approach to inter-temporal consump-tion and portfolio choice that takes advantage of the properties of thestochastic discount factor in a complete market. The martingale methodexploits the fact that, under assumptions of no-arbitrage and market com-pleteness, there exists a unique and positive state price φt satisfying

Et[φuSu] = E [φuSu|Ft] = φtSt, for u > t. (4.7)

4.2. Optimal Decisions of Investment and Consumption 61

The process φt can be interpreted as a system of Arrow-Debreu security.That is, φt is the claim to one unit of consumption contingent on theoccurrence of each state. Then the price of the asset is given by the weightedaverage of the prices of the payoffs of the states, with each weight being theprobability of the relevant state occurring. With the help of φ, the Merton’sdynamic optimization problem can be changed to a static optimizationproblem. The budget constraint (4.1) becomes a static budget constraint

E

[∫ T

0

ctφtdt

]= φ0W0, (4.8)

which says that the amount the investor allocates to consumption in eachstate multiplied by the price of consumption in that state must equal histotal wealth. By the same reasoning, for any t ∈ [0, T ] and any self-financingtrading strategy α, the following equality holds

E

[∫ T

t

cuφudu

∣∣∣∣∣Ft

]= φtWt(α). (4.9)

The optimal consumption is given by

c∗t = U−1c (λeρtφt), t ∈ [0, T ], (4.10)

where λ is determined by E

[∫ T

0c∗tφtdt

]= φ0W0 and U−1

c is the inversefunction of Uc.

4.2.2 Martingale approach to the asset pricing modelwith consumption habit

In this section we discuss the strategies of investors with consumption habitconstraints. We will show that the optimal consumption strategies arerelated with consumption insurance.

The following proposition is the main result, which gives the optimalconsumption decision to Merton’s problem with consumption habit con-straints. Before stating it, we need to guarantee that the initial endow-ment, W0 say, is affordable for the future habit consumption; at least theconsumption habit constraints should be satisfied. Specifically

W0 ≥ 1φ0

E

[∫ T

0

φshsds

], (4.11)


where the right-hand side is the total discount value of the future consump-tion. We say that W0 is affordable if it satisfies (4.11). From now on weassume that W0 is affordable.

Proposition 4.2.1 The optimal consumption is given by

c∗t (λ) =Xt(λ) if φt ≤ φh

t (λ),ht if φt > φh

t (λ), (4.12)

where φht (λ) = Uc(ht)/λeρt, Xt(λ) = U−1

c (λeρtφt), t ∈ [0, T ], and λ ≥ 0 isthe root of the equation

E

[∫ T

0

(Xt(λ)φtIφt≤φh

t (λ) + htφtIφt>φht (λ)

)dt

]= W0φ0. (4.13)

Moreover c∗t (λ) can be re-written as

c∗t (λ) = max((Xt(λ), ht

)(4.14)

and correspondingly the optimal wealth process is given by

W ∗t (λ) =

1φt

E

[∫ T

t

φs max(Xs(λ), hs)ds

∣∣∣∣∣Ft

], t ∈ [0, T ]. (4.15)

Sketch of proof: As the full proof is rather long, we give only the keypoints here. Interested readers can consult Cheng and Wei (2005) for de-tails. First we need a Lemma.

Lemma 4.2.2 For any given t ∈ [0, T ], let f(c) = e−ρtU(c) − λcφt +ηIc≥ht, where λ is given by (4.13), and

ηt = e−ρtU(U−1c (λeρtφt)) − λU−1

c (λeρtφt)φt − e−ρtU(ht) + λhtφt.

Then the function f(c) attains its maximum at c∗t defined in (4.14), that is,

max0≤c<∞

f(c) = f(c∗t ) (4.16)

and η ≥ 0.

We can prove this lemma in two steps. The first step is to prove η ≥ 0. Thefunction f(c) is a corrected concave function, which is moved upwards by ηand no longer a concave function with a jump point at c = c∗. So a possible

4.2. Optimal Decisions of Investment and Consumption 63

candidate for the maximum point will be at c = c∗ among other possiblecandidates. The maximum point will be obtained by comparing these pos-sible candidates. The second step is to prove that c∗t is the maximum point.We omit the details.

Now, assume that ct, t ∈ [0, T ], is a strategy satisfying the static budgetconstraint (4.8) and habit constraint (4.4), so that

E

[∫ T

0

c(t)φtdt

]≤W0φ0, (4.17)

and ct ≥ ht, ∀t ∈ [0, T ]. From the definition of λ we know

E

[∫ T

0

c∗tφtdt

]= W0φ0. (4.18)

Therefore we have

E

[∫ T

0

e−ρtU(c∗t )dt

]− E

[∫ T

0

e−ρtU(ct)dt

]

≥ E

[∫ T

0

e−ρtU(c∗t )dt

]− E

[∫ T

0

e−ρtU(ct)dt

]

− E

[λ

∫ T

0

c∗tφtdt

]+ E

[λ

∫ T

0

ctφtdt

]

+ E

[∫ T

0

ηtIc∗t ≥htdt

]− E

[∫ T

0

ηtIct≥htdt

]

= E

[∫ T

0

(f(c∗t ) − f(ct)

)dt

]≥ 0 (4.19)

Remark 4.2.3 From (4.10) we know that Xs(λ) is actually the optimalconsumption process without the consumption habit constraint (4.4). The


optimal wealth process (4.15) can be re-written as

W ∗t =

1φt

E

[∫ T

t

φs max(Xs, hs)ds

∣∣∣∣∣Ft

]

=∫ T

t

e−r(s−t)EQ[hs| Ft]ds

+∫ T

t

e−r(s−t)EQ[max(Xs − hs, 0)| Ft]ds. (4.20)

In words, the optimally invested wealth at time t is composed of two parts,one part being the wealth which finances the habit consumption in the futureafter time t, another being the total cost of a consecutive set of call optionsof the process Xu, u ∈ [t, T ] with the stochastic striking price hs ∈ [t, T ]at time s.

Remark 4.2.4 Let us consider again Merton’s problem with a differentinitial endowment Wu ≤ W0. In the above subsection, we have stated thatthe optimal consumption process is

c∗u,t(λu) = U−1c (λue

ρtφt),

where λu is the root of the equation

E

[∫ T

0

c∗u,t(λu)φtdt

]= φ0Wu.

Then λu is a function of the initial endowment Wu. Therefore if Wu is suit-ably chosen we can make λu = λ and then the process Xs(λ) = U−1

c (λeρtφt)is the same as the optimal consumption process c∗u,t(λu); that is Xt(λ) =c∗u,t(λu). From now on we will call Xt, t ∈ [0, T ], the optimal consumptionprocess of a relevant Merton’s problem with a relevant initial endowmentWu. It is easy to see that Wu ≤W0 because c∗t ≥ c∗u,t.

4.3 Optimal Investment Behavior

In this section we study the optimal investment behavior of an investorwith consumption habit constraints and without labour income. We willsee that the portfolio insurance strategy plays a central role in the con-sumption shortfall risk; the benchmark of a shortfall in the consumption isthe consumption habit.

4.3. Optimal Investment Behavior 65

For simplicity of explanation, we assume that the market consists ofjust two assets (i.e. m=2), one being a risky asset that prices according tothe geometric Brownian dynamics dSt = µStdt+ σStdBt, another being ariskless asset that earns a constant instantaneous rate of interest, r > 0,so that dRt = rRtdt, R0 > 0. In this case, the dynamics of the state pricedensity is

dφt = −rφtdt− κφtdBt, (4.21)

where κ = σ−1(µ− r). The budget constraint (4.1) becomes

dWt = [αtWt(µ− r) +Wtr − ct]dt+ αtWtσdBt. (4.22)

The general utility

Proposition 4.3.1 Assume that the optimal wealth process W ∗t can be

expressed as a function of St, φt, ht, so that there exists a functionf(t, x, y, z) satisfying W ∗

t = f(t, St, φt, ht). Assume that f has continuoussecond-order derivatives with respects to t, x, y, z. Then the demand forthe risky asset by the investor with Detemple and Karatzas’ consumptionhabit (4.5) is given by

α∗t =

Stfx

W ∗t

− φtfy

W ∗t

κσ−1 +ηtfz

W ∗t

σ−1δ. (4.23)

Proof: By Ito Lemma we have

dW ∗t =

(Lf + ft

)dt+

(fxStσ + fyZtκ+ fzδηt

)dBt, (4.24)

where L denotes the Merton differential generator of (S, φ, h) under P .(Lemma 5.1 in Merton (1990).) At the same time the optimal consumptionand the optimal portfolio satisfy the budget constraint (4.22). The demandfor risky asset is obtained by comparing the coefficients of the stochasticparts of (4.22) and (4.24) and the uniqueness of the Ito’s process.

Q.E.D.


In words, the proposition says that after adding the consumption habitconstraint, the demand for risky asset becomes

demand for risky asset = myopic demand based on asset′srisk premium

+ hedge demand against adverse changein investment opportunity

+ hedge demand against adverseconsumption with change of wealth.

(4.25)

The CRRA utility

In order to gain further insights on how consumption habit constraintsaffect the investment behavior, we specify the utility to be a CRRA utilityof the form (4.3) and consumption habit as (4.6). For a general utility asdiscussed in the last section, the effect of the consumption habit constraintis that the optimal investment decision needs to hedge adverse consumptionwith change of wealth.

Proposition 4.3.2 The optimal wealth process of an investor with the con-sumption habit (4.6) is given by

W ∗t = W g

t +W ft , (4.26)

where

W gt = Xt

∫ T−t

0

e−ruN(d1(u))du,

W ft = ht

∫ T−t

0

e−µmu (1 −N(d2)) du, (4.27)

r =ρ

1 − γ− γ

1 − γ

(r +

κ2

2(1 − γ)

), (4.28)

σy =1

1 − γκ− σm, µm = r + σmκ− µm, (4.29)

d1(u) =log (Xt/ht) + (µm − r + σ2

y/2)uσy

√u

, (4.30)

d2(u) = d1(u) − σy

√u, (4.31)

Xt =(e−ρt/φt

)1/(1−γ) (4.32)

and N(·) is the standard normal distribution.

4.3. Optimal Investment Behavior 67

Proof: Let

dQ

dP

∣∣∣∣FT = exp

(−κBt − 1

2κ2t

),dQY

dQ

∣∣∣∣FT =e−rTYT

Y0. (4.33)

Let u ∈ [t, T ] and s ∈ [t, u]. Define Zt 1/φt, µ∗m = µm − σmκ and

Ys = e−µ∗m(s−u)hs, which has the same terminal payoff as hs at time u and

is a martingale under measure Q. Therefore we have

e−r(u−t)EQ[hu|Ft] = e−r(u−t)EQ[hu|Ft] = e(µ∗m−r)(u−t)ht. (4.34)

It is easy to see that U−1c (x) = x−1/(1−γ). Now, let Xs = U−1

c (eρsZ−1s ) =(

e−ρuZs

)1/(1−γ), Xs = Xs

Ys. By Ito’s Formula and according to (4.21) we

have

dXs

Xs

=(r∗ + σ2

m − 11 − γ

σmκ

)ds+ σydB

Qs

= r∗ds+ σydBQYs , (4.35)

where σy = 11−γκ− σm and r∗ = 1

1−γ (r + γ2(1−γ)κ

2 − ρ).

Now, let Xs = e−r∗(s−u)Xs, which has the same payoff function as Xs

at the terminal time u. From (4.35) and by Ito’s Lemma we know that Xs isa QY -martingale and the mean part is zero. By the Black-Scholes formula,the value of a call option with underlying price process Xs, striking price 1and riskless return rate 0 under martingale measure QY is

EQY (max(Xs − 1, 0)∣∣∣Ft) = EQY (max(Xs − 1, 0)

∣∣Ft)

= XtN(d1) −N(d2),

where

Xt = e−r∗(t−u)Xt =e−r∗(t−u)

(e−ρt/φt

)1/(1−γ)

eµ∗m(u−t)ht

. (4.36)

After some tedious calculations, the proof of the proposition is completed.

Q.E.D.

Proposition 4.3.3 The demand for the risky asset at time t by the investorwith Cheng and Wei’s consumption habit (4.6) is given by

α∗t =

σ−1κ

1 − γαg

t + σ−1σmαft (4.37)

= αMt +Ht, (4.38)


where

αgt = W g

t /W∗t , αf

t = W ft /W

∗t , (4.39)

Ht = −σ−1

(κ

1 − γ− σm

)αf

t , (4.40)

σy,Wgt ,W

ft are defined as in Proposition (4.3.2) and αM

t = σ−1κ1−γ .

Proof: From result (4.26) it is not difficult to get

(e−ρt/φt

) 11−γ e−run(d1) − hte

−µmun(d2) = 0, (4.41)

where n(.) is the density function of a standard Normal distribution. Bydirect calculations and using the above equation, we have

W ∗ZZ =

11 − γ

W gt , W ∗

hh = W ft . (4.42)

Substituting the above equations in (4.23) and using the fact that the op-timal wealth is independent of the price processes we get our result.

Q.E.D.

As indicated in the comments after Proposition (4.3.2), we call αgt a growth

wealth ratio and αft a floor wealth ratio.

Remark 4.3.4 Let us examine again the investment strategy (4.37). Wecan re-write it as

α∗tW

∗t =

σ−1κ

1 − γ(W ∗

t −W ft ) + σ−1σmW

ft

= I1 + I2. (4.43)

In other words, the optimal investment strategy is a combination of twostrategies, the first one being the famous strategy CPPI (Constant Propor-tion Portfolio Insurance-Leland and Rubinstein, 1986). If σm = 0, then theinvestment strategy becomes a CPPI strategy with m = κ/(1 − γ), showingthe risk aversion of the investor. The second strategy is also needed becausethe consumption habit is uncertain too due to σm = 0; in other words, theinvestor will take the chance of investing more by being willing to have pos-sible lower consumption habit. The investors can lower their risk aversionif they are willing to lower their living standard.

4.4. Conclusions 69

4.4 Conclusions

In this chapter we first argue that for a large group of investors, their port-folio and consumption choice problem must be linked to the consumptionhabit constraint. For this new choice problem, by using the Cox and Huangmartingale approach, we can obtain an optimal wealth path and demandfor risky assets under a general utility. In the case of some special habitsand under the CRRA utility, analytic solutions are obtained. Furthermorewe have arrived at two interesting and important conclusions: (1) Beyondthe Merton’s decomposition formula for investor’s demand for risky asset,the demand has a third component to hedge against adverse change ofconsumption such as the willingness (or reluctance) to maintain the livingstandard, and others. (2) After imposing the consumption habit require-ment, even for the CRRA utility, the investor’s optimal strategy is relatedto the CPPI strategy.


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Index

Absolute Income Hypothesis, 2absolute income hypothesis, 3Arbitrage Pricing Theory, 2, 42arbitrage pricing theory, vArrow-Debreu security, v, 61asset payoff space, vi, 13, 21Asset pricing theory, vasset pricing theory, 3Asset Space, 20, 22, 23asset space, vi, vii, 20, 23, 24, 42,

44, 56asset spaces, 45

Capital Asset Pricing Model(CAPM), 2

CAPM, 2, 38, 44Compression theorem, 43compression theorem, 56constant relative risk aversion, 6consumption growth, vi, 11, 24,

25, 45consumption-based asset pricing

model, v, 2consumption-based asset pricing

modelling, 6correctly pricing functional, vi,

29, 32, 43, 44, 46, 48correctly pricing functional

(CPF), 18, 19correctly pricing functionals, 39CPF, 18–20

CPPI (Constant ProportionPortfolio Insurance, 68

CRRA, 26, 44, 55CRRA (Constant Relative Risk

Aversion) utility, 59CRRA (constant-relative-risk

aversion), vi, 26CRRA utility, 7, 53, 66

discount dividend model, 2Dual Theorem, 37dual theorem, v, vi, 23, 29

Epstein-Zin utility, 52Equity Premium Puzzle, 26equity premium puzzle, v, vi, 6,

8, 10–13, 17, 18, 29Euler equation, 4–7, 28Euler equations, 11Expanding Theorem, 40Expanding theorem, 36expanding theorem, vii, 56

habit formation models, 10Hansen-Jagannathan bound, 7Hansen-Jagannathan distance,

vii, 8, 47, 56Hilbert space, 13

idiosyncratic risk, vii, 9, 41–43,56

idiosyncratic risks, 11

75

76 Index

incomplete market, 11incomplete markets, 9Inter-temporal Capital Asset

Pricing Model, 2

K-factor structure, vii, 42, 43, 56K-factor structures, vii

Law of One Price, 16Life Cycle Hypotheses, 3Life Cycle Hypothesis and the

Permanent Income Hy-pothesis, 3

market return, 2, 27, 45martingale model of consump-

tion, 3mean-variance efficient frontier, 2Merton’s consumption pricing

model, vii

no-arbitrage principle, v

Permanent Income Hypotheses, 3Pricing Error Theorem, 49, 50Pricing error theorem, 46pricing error theorem, vii, 56Pricing Functional, 15pricing functional, vi, 15, 16, 18–

20, 23, 29, 35, 44pricing functional, SDF, state

price density, 17Pricing Functionals, 13pricing functionals, 36, 43pricing kernel, 24pricing kernels, 20

rational expectation, 3

recursive utility model, 8relative income hypothesis, 3risk aversion, vi, 4, 6, 9, 10, 17,

24, 27, 28, 51, 68risk aversions, 52risk-neutral pricing, v, 5

SDF, v–vii, 7, 17–24, 26–29, 32–34, 36–38, 40, 41, 44–46,51, 52, 54–56

SDF Space, 22SDFs, 20, 35, 38, 39, 52, 54, 56SPD, 17state price density (SPD), 5stochastic discount factor (SDF),

v, 2, 5, 17Structural Theory, v, 13, 17, 31structural theory, 19, 24, 26, 28,

54, 55Symmetric Theorem, 31, 34Symmetric theorem, 32symmetric theorem, vii, 33, 56

The Equity Premium Puzzle, 5The Law of One Price, 16The Permanent Income Hypothe-

sis, 3two-fund separation theorem, 2

Uniqueness Theorem, 19, 20uniqueness theorem, vutility, 1, 3–6, 8–10, 51, 52, 54,

59, 65, 66

valuation-preserving mapping, vi,22, 29

Technology

Asset pricing: A structural theory and its applications