Multiperiod Consumption, Portfolio Choice, And Asset Pricing

Part IIMultiperiod Consumption,Portfolio Choice, and AssetPricing139

Chapter 5A MultiperiodDiscrete-Time Model ofConsumption and PortfolioChoiceThis chapter considers an expected-utility-maximizing individual’s consumptionand portfolio choices over many periods. In contrast to our previous singleperiod or static models, here the intertemporal or dynamic nature of the problem is explicitly analyzed. Solving an individual’s multiperiod consumption andportfolio choice problem is of interest in that it provides a theory for an individual’s optimal lifetime savings and investment strategies.Hence, it hasnormative value as a guide for individual nancial planning. In addition, justas our single-period mean-variance portfolio selection model provided the theoryof asset demands for the Capital Asset Pricing Model, a multiperiod portfolio141

142CHAPTER 5. A MULTIPERIOD DISCRETE-TIME MODELchoice model provides a theory of asset demands for a general equilibrium theory of intertemporal capital asset pricing. Combining this model of individuals’preferences over consumption and securities with a model of rm productiontechnologies can lead to an equilibrium model of the economy that determinesasset price processes.1In the 1920s, Frank Ramsey (Ramsey 1928) derived optimal multiperiodconsumption-savings decisions but assumed that the individual could invest inonly a single asset paying a certain return. It was not until the late 1960s thatPaul A. Samuelson (Samuelson 1969) and Robert C. Merton (Merton 1969) wereable to solve for an individual’s multiperiod consumption and portfolio choicedecisions under uncertainty, that is, where both a consumption-savings choiceand a portfolio allocation decision involving risky assets were assumed to occureach period.2ming.Their solution technique involves stochastic dynamic program-While this dynamic programming technique is not the only approachto solving problems of this type, it can sometimes be the most convenient andintuitive way of deriving solutions.3The model we present allows an individual to make multiple consumptionand portfolio decisions over a single planning horizon. This planning horizon,which can be interpreted as the individual’s remaining lifetime, is composed ofmany decision periods, with consumption and portfolio decisions occurring onceeach period.The richness of this problem cannot be captured in the single-period models that we presented earlier. This is because with only one period,an investor’s decision period and planning horizon coincide. Still, the results1 Important examples of such models were developed by John Cox, Jonathan Ingersoll, andStephen Ross (Cox, Ingersoll, and Ross 1985a) and Robert Lucas (Lucas 1978).2 Jan Mossin (Mossin 1968) solved for an individual’s optimal multiperiod portfolio decisions but assumed the individual had no interim consumption decisions, only a utility ofterminal consumption.3 An alternative martingale approach to solving consumption and portfolio choice problemsis given by John C. Cox and Chi-Fu Huang (Cox and Huang 1989). This approach will bepresented in Chapter 12 in the context of a continuous-time consumption and portfolio choiceproblem.

5.1.ASSUMPTIONS AND NOTATION OF THE MODEL143from our single-period analysis will be useful because often we can transformmultiperiod models into a series of single-period ones, as will be illustrated next.The consumption-portfolio choice model presented in this chapter assumesthat the individual’s decision interval is a discrete time period. Later in thisbook, we change the assumption to make the interval instantaneous; that is,the individual may make consumption and portfolio choices continuously. Thislatter assumption often simpli es problems and can lead to sharper results.When we move from discrete time to continuous time, continuous-time stochastic processes are used to model security prices.The next section outlines the assumptions of the individual’s multiperiodconsumption-portfolio problem.Perhaps the strongest assumption that wemake is that utility of consumption is time separable.4The following sectionshows how this problem can be solved. It introduces an important techniquefor solving multiperiod decision problems under uncertainty, namely, stochasticdynamic programming. The beauty of this technique is that decisions over amultiperiod horizon can be broken up into a series of decisions over a singleperiod horizon. This allows us to derive the individual’s optimal consumptionand portfolio choices by starting at the end of the individual’s planning horizonand working backwards toward the present.In the last section, we completeour analysis by deriving explicit solutions for the individual’s consumption andportfolio holdings when utility is assumed to be logarithmic.5.1Assumptions and Notation of the ModelConsider an environment in which an individual chooses his level of consumptionand the proportions of his wealth invested in n risky assets plus a risk-free4 Time-inseparable utility, where current utility can depend on past or expected futureconsumption, is discussed in Chapter 14.

144CHAPTER 5. A MULTIPERIOD DISCRETE-TIME MODELasset.As was the case in our single-period models, it is assumed that theindividual takes the stochastic processes followed by the prices of the di erentassets as given. The implicit assumption is that security markets are perfectlycompetitive in the sense that the (small) individual is a price-taker in securitymarkets. An individual’s trades do not impact the price (or the return) ofthe security. For most investors trading in liquid security markets, this is areasonably realistic assumption. In addition, it is assumed that there are notransactions costs or taxes when buying or selling assets, so that security marketscan be described as “frictionless.”An individual is assumed to make consumption and portfolio choice decisionsat the start of each period during a T -period planning horizon. Each period isof unit length, with the initial date being 0 and the terminal date being T .55.1.1PreferencesThe individual is assumed to maximize an expected utility function de nedover consumption levels and a terminal bequest. Denote consumption at datet as Ct ; t 0; :::; T1, and the terminal bequest as WT , where Wt indicatesthe individual’s level of wealth at date t.A general form for a multiperiodexpected utility function would be E0 [ (C0 ; C1 ; :::; CTsimply assume that1 ; WT )],where we couldis increasing and concave in its arguments.as a starting point, we will assume thatHowever,has the following time-separable, oradditively separable, form:E0 [ (C0 ; C1 ; :::; CT1 ; WT )] E0"T 1Xt 0U (Ct ; t) B (WT ; T )#(5.1)5 The following presentation borrows liberally from Samuelson (Samuelson 1969) andRobert C. Merton’s unpublished MIT course 15.433 class notes "Portfolio Theory and CapitalMarkets."

5.1.ASSUMPTIONS AND NOTATION OF THE MODEL145where U and B are assumed to be increasing, concave functions of consumptionand wealth, respectively. Equation (5.1) restricts utility at date t, U (Ct ; t), todepend only on consumption at that date and not previous levels of consumption or expected future levels of consumption.While this is the traditionalassumption in multiperiod models, in later chapters we loosen this restrictionand investigate utility formulations that are not time separable.65.1.2The Dynamics of WealthAt date t, the value of the individual’s tangible wealth held in the form of assetsequals Wt .yt . 7In addition, the individual is assumed to receive wage income ofThis beginning-of-period wealth and wage income are divided betweenconsumption and savings, and then savings is allocated between n risky assetsas well as a risk-free asset. Let Rit be the random return on risky asset i overthe period starting at date t and ending at date t 1: Also let Rf t be the returnon an asset that pays a risk-free return over the period starting at date t andending at date t 1: Then if the proportion of date t saving allocated to riskyasset i is denoted ! it , we can write the evolution of the individual’s tangiblewealth asWt 1 (Wt ytCt ) Rf t nXi 1 St Rt! it (Rit!Rf t )(5.2)where St Wt yt Ct is the individual’s savings at date t, and Rt Rf t PnRf t ) is the total return on the individual’s invested wealth overi 1 ! it (Rit6 Dynamic programming, the solution technique presented in this chapter, can also beapplied to consumption and portfolio choice problems where an individual’s utility is timeinseparable.7 Wage income can be random. The present value of wage income, referred to as humancapital, is assumed to be a nontradeable asset. The individual can rebalance how his nancialwealth is allocated among risky assets but cannot trade his human capital.

146CHAPTER 5. A MULTIPERIOD DISCRETE-TIME /CPChoices 10.pdfFigure 5.1: Multiperiod Decisionsthe period from date t to t 1.Note that we have not restricted the distribution of asset returns in anyway.In particular, the return distribution of risky asset i could change overtime, so that the distribution of Rit could di er from the distribution of Rifor t 6 .Moreover, the one-period risk-free return could be changing, sothat Rf t 6 Rf .Asset distributions that vary from one period to the nextmean that the individual faces changing investment opportunities. Hence, in amultiperiod model, the individual’s current consumption and portfolio decisionsmay be in‡uenced not only by the asset return distribution for the currentperiod, but also by the possibility that asset return distributions could changein the future.The information and decision variables available to the individual at eachdate are illustrated in Figure 5.1. At date t, the individual knows her wealthat the start of the period, Wt ; her wage income received at date t, yt ; and

5.2. SOLVING THE MULTIPERIOD MODEL147the risk-free interest rate for investing or borrowing over the period from datet to date t 1, Rf t .Conditional on information at date t, denoted by It ,she also knows the distributions of future one-period risk-free rates and wageincome, FRf jIt and Fy jIt , respectively, for dates t 1; :::; T1. Lastly,the individual also knows the date t conditional distributions of the risky-assetreturns for dates t; :::; T1, given by FRi jIt .Date t information, It ,includes all realizations of wage income and risk-free rates for all dates up untiland including date t. It also includes all realizations of risky-asset returns forall dates up until and including date t 1. Moreover, It could include any otherstate variables known at date t that a ect the distributions of future wages, riskfree rates, and risky-asset returns. Based on this information, the individual’sdate t decision variables are consumption, Ct , and the portfolio weights for then risky assets, f! it g, for i 1; :::; n.5.2Solving the Multiperiod ModelWe begin by de ning an important concept that will help us simplify the solution to this multiperiod optimization problem. Let J (Wt ; t) denote the derivedutility-of-wealth function. It is de ned as follows:J (Wt ; It ; t)maxEtCs ;f! is g;8s;iTP1U (Cs ; s) B (WT ; T )(5.3)s twhere “max” means to choose the decision variables Cs and f! is g for s t; t 1; :::; T1 and i 1; :::; n so as to maximize the expected value of theterm in brackets. Note that J is a function of current wealth and all informationup until and including date t.This information could re‡ect state variablesdescribing a changing distribution of risky-asset returns and/or a changing riskfree interest rate, where these state variables are assumed to be exogenous to