Arbitrage,State Prices and Portfolio Theory Handbook of the Economics of Finance Philip Dybvig Stephen A.Ross Washington University in Saint Louis MIT First draft:September,2001 This draft:September 19,2002
Arbitrage, State Prices and Portfolio Theory Handbook of the Economics of Finance Philip Dybvig Washington University in Saint Louis Stephen A. Ross MIT First draft: September, 2001 This draft: September 19, 2002
Abstract Neoclassical financial models provide the foundation for our understanding of finance.This chapter introduces the main ideas of neoclassical finance in a single- period context that avoids the technical difficulties of continuous-time models,but preserves the principal intuitions of the subject.The starting point of the analysis is the formulation of standard portfolio choice problems. A central conceptual result is the Fundamental Theorem of Asset Pricing,which asserts the equivalence of absence of arbitrage,the existence of a positive linear pricing rule,and the existence of an optimum for some agent who prefers more to less.A related conceptual result is the Pricing Rule Representation Theorem, which asserts that a positive linear pricing rule can be represented as using state prices,risk-neutral expectations,or a state-price density.Different equivalent rep- resentations are useful in different contexts. Many applied results can be derived from the first-order conditions of the portfolio choice problem.The first-order conditions say that marginal utility in each state is proportional to a consistent state-price density,where the constant of proportion- ality is determined by the budget constaint.If markets are complete,the implicit state-price density is uniquely determined by investment opportunities and must be the same as viewed by all agents,thus simplifying the choice problem.Solv- ing first-order conditions for quantities gives us optimal portfolio choice,solving them for prices gives us asset pricing models,solving them for utilities gives us preferences,and solving them for for probabilities gives us beliefs. We look at two popular asset pricing models,the CAPM and the APT,as well as complete-markets pricing.In the case of the CAPM,the first-order conditions link nicely to the traditional measures of portfolio performance. Further conceptual results include aggregation and mutual fund separation theory, both of which are useful for understanding equilibrium and asset pricing
Abstract Neoclassical financial models provide the foundation for our understanding of finance. This chapter introduces the main ideas of neoclassical finance in a singleperiod context that avoids the technical difficulties of continuous-time models, but preserves the principal intuitions of the subject. The starting point of the analysis is the formulation of standard portfolio choice problems. A central conceptual result is the Fundamental Theorem of Asset Pricing, which asserts the equivalence of absence of arbitrage, the existence of a positive linear pricing rule, and the existence of an optimum for some agent who prefers more to less. A related conceptual result is the Pricing Rule Representation Theorem, which asserts that a positive linear pricing rule can be represented as using state prices, risk-neutral expectations, or a state-price density. Different equivalent representations are useful in different contexts. Many applied results can be derived from the first-order conditions of the portfolio choice problem. The first-order conditions say that marginal utility in each state is proportional to a consistent state-price density, where the constant of proportionality is determined by the budget constaint. If markets are complete, the implicit state-price density is uniquely determined by investment opportunities and must be the same as viewed by all agents, thus simplifying the choice problem. Solving first-order conditions for quantities gives us optimal portfolio choice, solving them for prices gives us asset pricing models, solving them for utilities gives us preferences, and solving them for for probabilities gives us beliefs. We look at two popular asset pricing models, the CAPM and the APT, as well as complete-markets pricing. In the case of the CAPM, the first-order conditions link nicely to the traditional measures of portfolio performance. Further conceptual results include aggregation and mutual fund separation theory, both of which are useful for understanding equilibrium and asset pricing
The modern quantitative approach to finance has its original roots in neoclassical economics.Neoclassical economics studies an idealized world in which markets work smoothly without impediments such as transaction costs,taxes,asymme- try of information,or indivisibilities.This chapter considers what we learn from single-period neoclassical models in finance.While dynamic models are becom- ing more and more common,single-period models contain a surprisingly large amount of the intuition and intellectual content of modern finance,and are also commonly used by investment practitioners for the construction of optimal port- folios and communication of investment results.Focusing on a single period is also consistent with an important theme.While general equilibrium theory seeks great generality and abstraction,finance has work to be done and seeks specific models with strong assumptions and definite implications that can be tested and implemented in practice. 1 Portfolio Problems In our analysis,there are two points of time,0 and 1,with an interval of time in between during which nothing happens.At time zero,our champion (the agent) is making decisions that will affect the allocation of consumption between non- random consumption,co,at time 0,and random consumption fco}across states o=1,2,...,revealed at time 1.At time 0 and in each state at time 1,there is a single consumption good,and therefore consumption at time 0 or in a state at time 1 is a real number.This abstraction of a single good is obviously not"true" in any literal sense,but this is not a problem,and indeed any useful theoretical model is much simpler than reality.The abstraction does,however face us with the question of how to interpret our simple model(in this case with a single good) in a practical context that is more complex (has multiple goods).In using a single- good model,there are two usual practices:either use nominal values and measure consumption in dollars,or use real values and measure consumption in inflation- adjusted dollars.Depending on the context,one or the other can make the most sense.In this article,we will normally think of the consumption units as being the numeraire,so that“cash flows'”or“claims to consumption”have the same meaning. Following the usual practice from general equilibrium theory of thinking of units 2
The modern quantitative approach to finance has its original roots in neoclassical economics. Neoclassical economics studies an idealized world in which markets work smoothly without impediments such as transaction costs, taxes, asymmetry of information, or indivisibilities. This chapter considers what we learn from single-period neoclassical models in finance. While dynamic models are becoming more and more common, single-period models contain a surprisingly large amount of the intuition and intellectual content of modern finance, and are also commonly used by investment practitioners for the construction of optimal portfolios and communication of investment results. Focusing on a single period is also consistent with an important theme. While general equilibrium theory seeks great generality and abstraction, finance has work to be done and seeks specific models with strong assumptions and definite implications that can be tested and implemented in practice. 1 Portfolio Problems In our analysis, there are two points of time, 0 and 1, with an interval of time in between during which nothing happens. At time zero, our champion (the agent) is making decisions that will affect the allocation of consumption between nonrandom consumption, c0, at time 0, and random consumption cω ✁ across states ω ✂ 1 ✄ 2 ✄✆☎✝☎✝☎✞✄ Ω revealed at time 1. At time 0 and in each state at time 1, there is a single consumption good, and therefore consumption at time 0 or in a state at time 1 is a real number. This abstraction of a single good is obviously not “true” in any literal sense, but this is not a problem, and indeed any useful theoretical model is much simpler than reality. The abstraction does, however face us with the question of how to interpret our simple model (in this case with a single good) in a practical context that is more complex (has multiple goods). In using a singlegood model, there are two usual practices: either use nominal values and measure consumption in dollars, or use real values and measure consumption in inflationadjusted dollars. Depending on the context, one or the other can make the most sense. In this article, we will normally think of the consumption units as being the numeraire, so that “cash flows” or “claims to consumption” have the same meaning. Following the usual practice from general equilibrium theory of thinking of units 2
of consumption at various times and in different states of nature as different goods, a typical consumption vector is C fco,c1,..,co,where the real number co de- notes consumption of the single gdod at time zero,and the vector c tc1,...,co of real numbers c1,...,co denotes random consumption of the single good in each state 1,...,at time 1. If this were a typical exercise in general equilibrium theory,we would have a price vector for consumption across goods.For example,we might have the following choice problem,which is named after two great pioneers of general equilibrium theory,Kenneth Arrow and Gerard Debreu: Problem 1 Arrow-Debreu Problem Choose consumptions C co,c1,.,co to maximize utility of consumption U'C-subject to the budget constraint 2 (1) Poco=W. 0.1 Here,U'.-is the utility function that represents preferences,p is the price vector, and W is wealth,which might be replaced by the market value of an endowment. We are taking consumption at time 0 to be the numeraire,and p is the price of the Arrow-Debreu security which is a claim to one unit of consumption at time 1 in state o The first-order condition for Problem 1 is the existence of a positive Lagrangian multiplierA(the marginal utility of wealth)such that UcoA,and for all= 1,,2 Uoco-hpo: This is the usual result from neoclassical economics that the gradient of the util- ity function is proportional to prices.Specializing to the leading case in finance of time-separable von Neumann-Morgenstern preferences,named after John von Neumann and Oscar Morgenstern,two great pioneers of utility theory,we have that U'c-cWe will take v and u to be differentiable, strictly increasing (more is preferred to less),and strictly concave(risk averse). 3
of consumption at varioustimes and in different states of nature as different goods, a typical consumption vector is C c0 ✄ c1 ✄ ☎✞☎✝☎✞✄ cΩ ✁ , where the real number c0 denotes consumption of the single good at time zero, and the vector c c1 ✄ ☎✝☎✞☎✝✄ cΩ ✁ of real numbers c1 ✄ ☎✝☎✞☎✝✄ cΩ denotes random consumption of the single good in each state 1 ✄ ☎✞☎✝☎✞✄ Ω at time 1. If this were a typical exercise in general equilibrium theory, we would have a price vector for consumption across goods. For example, we might have the following choice problem, which is named after two great pioneers of general equilibrium theory, Kenneth Arrow and Gerard Debreu: Problem 1 Arrow-Debreu Problem Choose consumptions C c0 ✄ c1 ✄ ☎✝☎✞☎✝✄ cΩ ✁ to maximize utility of consumption U ✁ C ✂ subject to the budget constraint c0 ✄ Ω ∑ ω☎1 (1) pωcω ✂ W ☎ Here, U ✁✝✆ ✂ is the utility function that represents preferences, p is the price vector, and W is wealth, which might be replaced by the market value of an endowment. We are taking consumption at time 0 to be the numeraire, and pω is the price of the Arrow-Debreu security which is a claim to one unit of consumption at time 1 in state ω. The first-order condition for Problem 1 is the existence of a positive Lagrangian multiplier λ (the marginal utility of wealth) such that U ✞ 0 ✁ c0 ✂ ✂ λ, and for all ω ✂ 1 ✄✆☎✝☎✝☎✞✄ Ω, U ✞ ω ✁ cω ✂ ✂ λpω ☎ This is the usual result from neoclassical economics that the gradient of the utility function is proportional to prices. Specializing to the leading case in finance of time-separable von Neumann-Morgenstern preferences, named after John von Neumann and Oscar Morgenstern, two great pioneers of utility theory, we have that U ✁ C ✂ ✂ v ✁ c0 ✂ ✄ ∑ Ω ω☎1 πωu ✁ cω ✂ . We will take v and u to be differentiable, strictly increasing (more is preferred to less), and strictly concave (risk averse). 3
Here,nt is the probability of state o.In this case,the first-order condition is the existence of A such that (2)v'(co)=入， and for all o=1,2,…,n, (3)元od(co)=po or equivalently (4)u(co)=APo; where po=po/nt is the state-price density (also called the stochastic discount factor or pricing kernel),which is a measure of priced relative scarcity in state of nature o.Therefore,the marginal utility of consumption in a state is pro- portional to the relative scarcity.There is a solution if the problem is feasible. prices and probabilities are positive,the von Neumann-Morgenstern utility func- tion is increasing and strictly concave,and there is satisfied the Inada condition limet(c)-0.There are different motivations of von Neumann-Morgenstern preferences in the literature and the probabilities may be objective or subjective. What is important for us that the von Neumann-Morgenstern utility function rep- resents preferences in the sense that expected utility is higher for more preferred consumption patterns.2 Using von Neumann-Morganstern preferences has been popular in part because of axiomatic derivations of the theory (see,for example,Herstein and Milnor [1953]or Luce and Raiffa [1957],chapter 2).There is also a large literature on alternatives and extensions to von Neumann-Morgenstern preferences.For Proving the existence of an equilibrium requires more assumptions in continuous-state mod- els. 2Later,when we look at multiple-agent results,we will also make the neoclassical assumption of identical beliefs,which is probably most naturally motivated by symmetric objective informa- tion. 4
Here, πω is the probability of state ω. In this case, the first-order condition is the existence of λ such that v✞ ✁ (2) c0 ✂ ✂ λ ✄ and for all ω ✂ 1 ✄ 2 ✄✆☎✝☎✝☎✞✄ n, πωu✞ ✁ (3) cω ✂ ✂ λpω or equivalently u✞ ✁ (4) cω ✂ ✂ λρω ✄ where ρω pω πω is the state-price density (also called the stochastic discount factor or pricing kernel), which is a measure of priced relative scarcity in state of nature ω. Therefore, the marginal utility of consumption in a state is proportional to the relative scarcity. There is a solution if the problem is feasible, prices and probabilities are positive, the von Neumann-Morgenstern utility function is increasing and strictly concave, and there is satisfied the Inada condition limc✁∞ u✞ ✁ c ✂ ✂ 0.1 There are different motivations of von Neumann-Morgenstern preferences in the literature and the probabilities may be objective or subjective. What is important for us that the von Neumann-Morgenstern utility function represents preferences in the sense that expected utility is higher for more preferred consumption patterns.2 Using von Neumann-Morganstern preferences has been popular in part because of axiomatic derivations of the theory (see, for example, Herstein and Milnor [1953] or Luce and Raiffa [1957], chapter 2). There is also a large literature on alternatives and extensions to von Neumann-Morgenstern preferences. For 1Proving the existence of an equilibrium requires more assumptions in continuous-state models. 2Later, when we look at multiple-agent results, we will also make the neoclassical assumption of identical beliefs, which is probably most naturally motivated by symmetric objective information. 4
single-period models,see Knight [1921],Bewley [1988],Machina [1982],Blume, Brandenburger,and Dekel [19911,and Fishburn [19881.There is an even richer set of models in multiple periods,for example,time-separable von Neumann- Morgenstern(the traditional standard),habit formation(e.g.Duesenberry [1949] Pollak [1970],Abel [1990],Constantinides [1991],and Dybvig [19951),local substitutability over time (Hindy and Huang [1992]),interpersonal dependence (Duesenberry [1949]and Abel [19901),preference for resolution of uncertainty (Kreps and Porteus[1978]),time preference dependent on consumption(Bergman [1985]),and general recursive utility (Epstein and Zin [1989]). Recently,there have also been some attempts to revive the age-old idea of study- ing financial situations using psychological theories(like prospect theory,Kahne- man and Tversky [1979]1).Unfortunately,these models do not translate well to financial markets.For example,in prospect theory framing matters,that is,the observed phenomenon of an agent making different decisions when facing identi- cal decision problems described differently.However,this is an alien concept for financial economists and when they proxy for it in models they substitute some- thing more familiar(for example,some history dependence as in Barberis,Huang, and Santos [2001]).Another problem with the psychological theories is that they tend to be isolated stories rather than a general specification,and they are often hard to generalize.For example,prospect theory says that agents put extra weight on very unlikely outcomes,but it is not at all clear what this means in a model with a continuum of states.This literature also has problems with using ex post explanations(positive correlations of returns are underreaction and negative cor- relations are overreactions)and a lack of clarity of how much is going on that cannot be explained by traditional models(and much of it can). In actual financial markets,Arrow-Debreu securities do not trade directly,even if they can be constructed indirectly using a portfolio of securities.A security is characterized by its cash flows.This description would not be adequate for analysis of taxes,since different sources of cash flow might have very different tax treatment,but we are looking at models without taxes.For an asset like a common stock or a bond,the cash flow might be negative at time 0,from payment of the price,and positive or zero in each state at time 1,the positive amount coming from any repayment of principal,dividends,coupons,or proceeds from sale of the asset.For a futures contract,the cash flow would be 0 at time 0,and the cash flow in different states at time 1 could be positive,negative,or zero, depending on news about the value of the underlying commodity.In general,we 5
single-period models,see Knight [1921], Bewley [1988], Machina [1982], Blume, Brandenburger, and Dekel [1991], and Fishburn [1988]. There is an even richer set of models in multiple periods, for example, time-separable von NeumannMorgenstern (the traditional standard), habit formation (e.g. Duesenberry [1949], Pollak [1970], Abel [1990], Constantinides [1991], and Dybvig [1995]), local substitutability over time (Hindy and Huang [1992]), interpersonal dependence (Duesenberry [1949] and Abel [1990]), preference for resolution of uncertainty (Kreps and Porteus[1978]), time preference dependent on consumption (Bergman [1985]), and general recursive utility (Epstein and Zin [1989]). Recently, there have also been some attempts to revive the age-old idea of studying financial situations using psychological theories (like prospect theory, Kahneman and Tversky [1979]). Unfortunately, these models do not translate well to financial markets. For example, in prospect theory framing matters, that is, the observed phenomenon of an agent making different decisions when facing identical decision problems described differently. However, this is an alien concept for financial economists and when they proxy for it in models they substitute something more familiar (for example, some history dependence as in Barberis, Huang, and Santos [2001]). Another problem with the psychological theories is that they tend to be isolated stories rather than a general specification, and they are often hard to generalize. For example, prospect theory says that agents put extra weight on very unlikely outcomes, but it is not at all clear what this means in a model with a continuum of states. This literature also has problems with using ex post explanations (positive correlations of returns are underreaction and negative correlations are overreactions) and a lack of clarity of how much is going on that cannot be explained by traditional models (and much of it can). In actual financial markets, Arrow-Debreu securities do not trade directly, even if they can be constructed indirectly using a portfolio of securities. A security is characterized by its cash flows. This description would not be adequate for analysis of taxes, since different sources of cash flow might have very different tax treatment, but we are looking at models without taxes. For an asset like a common stock or a bond, the cash flow might be negative at time 0, from payment of the price, and positive or zero in each state at time 1, the positive amount coming from any repayment of principal, dividends, coupons, or proceeds from sale of the asset. For a futures contract, the cash flow would be 0 at time 0, and the cash flow in different states at time 1 could be positive, negative, or zero, depending on news about the value of the underlying commodity. In general, we 5
think of the negative of the initial cash flow as the price of a security.We denote by P=[P1,...,P}the vector of prices of the N securities 1,...,N,and we denote by X the payoff matrix.We have that P is the price we pay for one unit of security n and Xon is the payoff per unit of security n at time 1 in the single state of nature 0 With the choice of a portfolio of assets,our choice problem might become Problem 2 First Portfolio Choice Problem Choose portfolio holdingsΘ三{Θ1，，Θn}and consumptions C三{co,,ca}to maximize utility of consumption U(C)subject to portfolio payoffs c≡{c1,,co}=XΘand budget constraint co+P'-W. Here,is the vector of portfolio weights.Time 0 consumption is the numeraire, and wealth W is now chosen in time 0 consumption units and the entire endow- ment is received at time 0.In the budget constraint,the term P is the cost of the portfolio holding,which is the sum across securities n of the price P times the number of shares or other unit n.The matrix product X says that the consump- tion in state ois co=nonn,i.e.the sum across securities n of the payoff Xon of security n in state o,times the number of shares or other units n of security n our champion is holding. The first-order condition for Problem 2 is the existence of a vector of shadow prices p and a Lagrangian multiplier A such that (5) 元ol(co)=入pw where (6)P=pX. The first equation is the same as in the Arrow-Debreu model,with an implicit shadow price vector in place of the given Arrow-Debreu prices.The second equa- tion is a pricing equation that says the prices of all assets must be consistent with 6
think of the negative of the initial cash flow as the price of a security. We denote by P ✂ P1 ✄ ☎✝☎✞☎✝✄ PN ✁ the vector of prices of the N securities 1 ✄ ☎✞☎✝☎✝✄ N, and we denote by X the payoff matrix. We have that Pn is the price we pay for one unit of security n and Xωn is the payoff per unit of security n at time 1 in the single state of nature ω. With the choice of a portfolio of assets, our choice problem might become Problem 2 First Portfolio Choice Problem Choose portfolio holdings Θ Θ1 ✄ ☎✞☎✝☎✞✄ Θn ✁ and consumptionsC c0 ✄ ☎✝☎✞☎✝✄ cΩ ✁ to maximize utility of consumption U ✁ C ✂ subject to portfolio payoffs c c1 ✄ ☎✞☎✝☎✝✄ cω ✁ ✂ XΘ and budget constraint c0 ✄ P✞ Θ ✂ W. Here, Θ is the vector of portfolio weights. Time 0 consumption is the numeraire, and wealth W is now chosen in time 0 consumption units and the entire endowment is received at time 0. In the budget constraint, the term P ✞ Θ is the cost of the portfolio holding, which is the sum across securities n of the price Pn times the number of shares or other unit Θn. The matrix product XΘ says that the consumption in state ω is cω ✂ ∑n XωnΘn, i.e. the sum across securities n of the payoff Xωn of security n in state ω, times the number of shares or other units Θn of security n our champion is holding. The first-order condition for Problem 2 is the existence of a vector of shadow prices p and a Lagrangian multiplier λ such that πωu✞ ✁ (5) cω ✂ ✂ λpω where (6) P✞ ✂ pX ☎ The first equation is the same as in the Arrow-Debreu model, with an implicit shadow price vector in place of the given Arrow-Debreu prices. The second equation is a pricing equation that says the prices of all assets must be consistent with 6
the shadow prices of the states.For the Arrow-Debreu model itself,the state- space tableau X is I,the identity matrix,and the price vector P is p,the vector of Arrow-Debreu state prices.For the Arrow-Debreu model,the pricing equation determines the shadow prices as equal to the state prices. Even if the assets are not the Arrow-Debreu securities,Problem 2 may be essen- tially equivalent to the Arrow-Debreu model in Problem 1.In economic terms, the important feature of the Arrow-Debreu problem is that all payoff patterns are spanned,i.e.,each potential payoff pattern can be generated at some price by some portolio of assets.Linear algebra tells us that all payoff patterns can be generated if the payoff matrix X has full row rank.If X has full row rank,p is determined (or over-determined)by(6).If p is uniquely determined by the pricing equation (and therefore also all Arrow-Debreu assets can be purchased as portfolios of assets in the economy),we say that markets are complete,and for all practical purposes we are in an Arrow-Debreu world. For the choice problem to have a solution for any agent who prefers more to less, we also need for the price of each payoff pattern to be unique (the "law of one price")and positive,or else there would be arbitrage (i.e.,a "money pump"or a "free lunch").If there is no arbitrage,then there is at least one vector of positive state prices p solving the pricing equation(6).There is an arbitrage if the vector of state prices is overdetermined or if all consistent vectors of state prices assign a negative or zero price to some state.The notion of absence of arbitrage is a central concept in finance,and we develop its implications more fully in the section on preference-free results. So far,we have been stating portfolio problems in prices and quantities,as we would in general equilibrium theory.However,it is also common to describe assets in terms of rates ofreturn,which are relative price changes (often expressed as percentages).The return to security n,which is the relative change in total value (including any dividends,splits,warrant issues,coupons,stock issues,and the like as well as change in the price).There is not an absolute standard of what is meant by return,in different contexts this can be the rate of rate of return,one plus the rate of return,or the difference between two rates of return.It is necessary to figure which is intended by asking or from context.Using the notation above,the rate of return in state is rn=n-P3Often,consumption at the outset 3One unfortunate thing about returns is that they are not defined for contracts(like futures)that 7
the shadow prices of the states. For the Arrow-Debreu model itself, the statespace tableau X is I, the identity matrix, and the price vector P is p, the vector of Arrow-Debreu state prices. For the Arrow-Debreu model, the pricing equation determines the shadow prices as equal to the state prices. Even if the assets are not the Arrow-Debreu securities, Problem 2 may be essentially equivalent to the Arrow-Debreu model in Problem 1. In economic terms, the important feature of the Arrow-Debreu problem is that all payoff patterns are spanned, i.e., each potential payoff pattern can be generated atsome price by some portolio of assets. Linear algebra tells us that all payoff patterns can be generated if the payoff matrix X has full row rank. If X has full row rank, p is determined (or over-determined) by (6). If p is uniquely determined by the pricing equation (and therefore also all Arrow-Debreu assets can be purchased as portfolios of assets in the economy), we say that markets are complete, and for all practical purposes we are in an Arrow-Debreu world. For the choice problem to have a solution for any agent who prefers more to less, we also need for the price of each payoff pattern to be unique (the “law of one price”) and positive, or else there would be arbitrage (i.e., a “money pump” or a “free lunch”). If there is no arbitrage, then there is at least one vector of positive state prices p solving the pricing equation (6). There is an arbitrage if the vector of state prices is overdetermined or if all consistent vectors of state prices assign a negative or zero price to some state. The notion of absence of arbitrage is a central concept in finance, and we develop its implications more fully in the section on preference-free results. So far, we have been stating portfolio problems in prices and quantities, as we would in general equilibrium theory. However, it is also common to describe assets in terms of rates of return, which are relative price changes (often expressed as percentages). The return to security n, which isthe relative change in total value (including any dividends,splits, warrant issues, coupons, stock issues, and the like as well as change in the price). There is not an absolute standard of what is meant by return, in different contexts this can be the rate of rate of return, one plus the rate of return, or the difference between two rates of return. It is necessary to figure which is intended by asking or from context. Using the notation above, the rate of return in state ω is rωn = ✁ Xωn Pn ✂ Pn. 3 Often, consumption at the outset 3One unfortunate thing about returns is that they are not defined for contracts (like futures) that 7
is suppressed,and we specialize to von Neumann-Morgenstern expected utility. In this case,we have the following common form of portfolio problem. Problem 3 Portfolio Problem using Returns Choose portfolio proportions6三{θ1，，θn}and consumptions c={c1,,cQ} to maximize expected utility of consumption subject to the consumption equation c=We'(1+r)and the budget constraint 0'1 =1. Here,n=1,...,o}is a vector of state probabilities,u()is the von Neumann- Morgenstern utility function,and 1 is a vector of 1's.The dimensionality of 1 is determined implicitly from the context,here the dimensionality is the number of assets.The first-order condition for an optimum is the existence of shadow state price density vector p and shadow marginal utility of wealth A such that (7 l(co)=λpo and (8)1=E[(1+r)p] These equations say that the state-price density is consistent with the marginal valuation by the agent and with pricing in the market. As our final typical problem,let us consider a mean-variance optimization.This optimization is predicated on the assumption that investors care only about mean and variance (typically preferring more mean and less variance),so we have a utility function V(m,v)in mean m and variance v.For this problem,suppose there is a risk-free asset paying a return r(although the market-level implications of mean-variance analysis can also be derived in a general model without a risky have zero price.However,this can be finessed formally by bundling a futures with a bond or other asset in defining the securities and unbundling them when interpreting the results.Bundling and unbundling does not change the underlying economics due to the linearity of consumptions and constraints in the portfolio choice problem. 8
is suppressed, and we specialize to von Neumann-Morgenstern expected utility. In this case, we have the following common form of portfolio problem. Problem 3 Portfolio Problem using Returns Choose portfolio proportions θ θ1 ✄ ☎✞☎✝☎✞✄ θn ✁ and consumptions c c1 ✄ ☎✝☎✞☎✝✄ cΩ ✁ to maximize expected utility of consumption ∑ Ω ω☎1 πθu ✁ cω ✂ subject to the consumption equation c ✂ Wθ ✞ ✁ 1 ✄ r ✂ and the budget constraint θ ✞ 1 ✂ 1. Here, π ✂ π1 ✄ ☎✝☎✞☎✝✄ πΩ ✁ is a vector of state probabilities, u ✁ ✆ ✂ is the von NeumannMorgenstern utility function, and 1 is a vector of 1’s. The dimensionality of 1 is determined implicitly from the context, here the dimensionality is the number of assets. The first-order condition for an optimum is the existence of shadow state price density vector ρ and shadow marginal utility of wealth λ such that u✞ ✁ (7) cω ✂ ✂ λρω and 1 ✂ E ✁ (8) 1 ✄ r ✂ ρ✁ ☎ These equations say that the state-price density is consistent with the marginal valuation by the agent and with pricing in the market. As our final typical problem, let us consider a mean-variance optimization. This optimization is predicated on the assumption that investors care only about mean and variance (typically preferring more mean and less variance), so we have a utility function V ✁ m ✄ v✂ in mean m and variance v. For this problem, suppose there is a risk-free asset paying a return r (although the market-level implications of mean-variance analysis can also be derived in a general model without a risky have zero price. However, this can be finessed formally by bundling a futures with a bond or other asset in defining the securities and unbundling them when interpreting the results. Bundling and unbundling does not change the underlying economics due to the linearity of consumptions and constraints in the portfolio choice problem. 8
asset).In this case,portfolio proportions in the risky assets are unconstrained (need not sum to 1)because the slack can be taken up by the risk-free asset.We denote by u the vector of mean risky asset returns and by o the covariance matrix of risky returns.Then our champion solves the following choice problem. Problem 4 Mean-variance optimization Choose portfolio proportions0=[θ1，，6nlto naximize the mean--variance utility function V(c+(u.rl)'0,θ'∑Σ0) The first-order condition for the problem is (9)4,r1=入∑0， whereis twice the marginal rate of substitution (m,v)(m,v),evaluated at m=r+(u r1)'0 and v=0'0,where 0 is the optimal choice of portfolio pro- portions.The first-order condition(9)says that mean excess return for each asset is proportional to the marginal contribution of volatility to the agent's optimal portfolio. We have seen a few of the typical types of portfolio problem.There are a lot of variations.The problem might be stated in terms of excess returns(rate of return less a risk-free rate)or total return (one plus the rate of return).Or,we might constrain portfolio holdings to be positive (no short sales)or we might require consumption to be nonnegative (limited liability).Many other variations adapt the basic portfolio problem to handle institutional features not present in a neoclassical formulation,such as transaction costs,bid-ask spreads,or taxes. These extensions are very interesting,but beyond the scope of what we are doing here,which is to explore the neoclassical foundations. 2 Absence of Arbitrage and Preference-free Results Before considering specific solutions and applications,let us consider some gen- eral results that are useful for thinking about portfolio choice.These results are 9
asset). In this case, portfolio proportions in the risky assets are unconstrained (need not sum to 1) because the slack can be taken up by the risk-free asset. We denote by µ the vector of mean risky asset returns and by σ the covariance matrix of risky returns. Then our champion solves the following choice problem. Problem 4 Mean-variance optimization Choose portfolio proportions θ θ1 ✄ ☎✞☎✝☎✞✄ θn ✁ to maximize the mean-variance utility function V ✁ r ✄ ✁ µ r1 ✂ ✞ θ ✄ θ✞ Σθ✂ . The first-order condition for the problem is (9) µ r1 ✂ λΣθ ✄ where λ is twice the marginal rate of substitution V ✞ v ✁ m ✄ v ✂ Vm✞ ✁ m ✄ v ✂ , evaluated at m ✂ r ✄ ✁ µ r1 ✂ ✞ θ and v ✂ θ✞ Σθ, where θ is the optimal choice of portfolio proportions. The first-order condition (9) says that mean excess return for each asset is proportional to the marginal contribution of volatility to the agent’s optimal portfolio. We have seen a few of the typical types of portfolio problem. There are a lot of variations. The problem might be stated in terms of excess returns (rate of return less a risk-free rate) or total return (one plus the rate of return). Or, we might constrain portfolio holdings to be positive (no short sales) or we might require consumption to be nonnegative (limited liability). Many other variations adapt the basic portfolio problem to handle institutional features not present in a neoclassical formulation, such as transaction costs, bid-ask spreads, or taxes. These extensions are very interesting, but beyond the scope of what we are doing here, which is to explore the neoclassical foundations. 2 Absence of Arbitrage and Preference-free Results Before considering specific solutions and applications, let us consider some general results that are useful for thinking about portfolio choice. These results are 9