Unformatted text preview:

“chap01” — 2004/10/7 — page3—#31Consumption-Based Modeland OverviewAn investor must decide how much to save and how much to consume,and what portfolio of assets to hold. The most basic pricing equation comesfrom the first-order condition for that decision. The marginal utility lossof consuming a little less today and buying a little more of the asset shouldequal the marginal utility gain of consuming a little more of the asset’s payoffin the future. If the price and payoff do not satisfy this relation, the investorshould buy more or less of the asset. It follows that the asset’s price shouldequal the expected discounted value of the asset’s payoff, using the investor’smarginal utility to discount the payoff. With this simple idea, I present manyclassic issues in finance.Interest rates are related to expected marginal utility growth, and henceto the expected path of consumption. In a time of high real interest rates, itmakes sense to save, buy bonds, and then consume more tomorrow. There-fore, high real interest rates should be associated with an expectation ofgrowing consumption.Most importantly, risk corrections to asset prices should be driven by thecovariance of asset payoffs with marginal utility and hence by the covarianceof asset payoffs with consumption. Other things equal, an asset that doesbadly in states of nature like a recession, in which the investor feels poor andis consuming little, is less desirable than an asset that does badly in states ofnature like a boom in which the investor feels wealthy and is consuming agreat deal. The former asset will sell for a lower price; its price will reflect adiscount for its ‘‘riskiness,’’ and this riskiness depends on a co-variance, nota variance.Marginal utility, not consumption, is the fundamental measure of howyou feel. Most of the theory of asset pricing is about how to go from marginalutility to observable indicators. Consumption is low when marginal utilityis high, of course, so consumption may be a useful indicator. Consumptionis also low and marginal utility is high when the investor’s other assets havedone poorly; thus we may expect that prices are low for assets that covary3“chap01” — 2004/10/7 — page4—#44 1. Consumption-Based Model and Overviewpositively with a large index such as the market portfolio. This is a CapitalAsset Pricing Model. We will see a wide variety of additional indicators formarginal utility, things against which to compute a covariance in order topredict the risk-adjustment for prices.1.1 Basic Pricing EquationAn investor’s first-order conditions give the basic consumption-basedmodel,pt= Etβu(ct+1)u(ct)xt+1.Our basic objective is to figure out the value of any stream of uncertaincash flows. I start with an apparently simple case, which turns out to capturevery general situations.Let us find the value at time t of a payoff xt+1. If you buy a stock today,the payoff next period is the stock price plus dividend, xt+1= pt+1+ dt+1.xt+1is a random variable: an investor does not know exactly how much hewill get from his investment, but he can assess the probability of variouspossible outcomes. Do not confuse the payoff xt+1with the profit or return;xt+1is the value of the investment at time t + 1, without subtracting ordividing by the cost of the investment.We find the value of this payoff by asking what it is worth to a typicalinvestor. To do this, we need a convenient mathematical formalism to cap-ture what an investor wants. We model investors by a utility function definedover current and future values of consumption,U (ct, ct+1) = u(ct) + βEt u(ct+1),where ctdenotes consumption at date t. We often use a convenient powerutility form,u(ct) =11 − γc1−γt.The limit as γ → 1is1u(c) = ln(c).1To think about this limit precisely, add a constant to the utility function and write it asu(ct) =c1−γt− 11 − γ.“chap01” — 2004/10/7 — page5—#51.1. Basic Pricing Equation 5The utility function captures the fundamental desire for moreconsumption, rather than posit a desire for intermediate objectives such asmean and variance of portfolio returns. Consumption ct+1is also random;the investor does not know his wealth tomorrow, and hence how much hewill decide to consume tomorrow. The period utility function u(·) is increas-ing, reflecting a desire for more consumption, and concave, reflecting thedeclining marginal value of additional consumption. The last bite is neveras satisfying as the first.This formalism captures investors’ impatience and their aversion to risk,so we can quantitatively correct for the risk and delay of cash flows. Discount-ing the future by β captures impatience, and β is called the subjective discountfactor. The curvature of the utility function generates aversion to risk and tointertemporal substitution: The investor prefers a consumption stream thatis steady over time and across states of nature.Now, assume that the investor can freely buy or sell as much of the payoffxt+1as he wishes, at a price pt. How much will he buy or sell? To find theanswer, denote by e the original consumption level (if the investor boughtnone of the asset), and denote by ξ the amount of the asset he chooses tobuy. Then, his problem ismax{ξ}u(ct) + Et βu(ct+1)s.t.ct= et− ptξ,ct+1= et+1+ xt+1ξ.Substituting the constraints into the objective, and setting the derivativewith respect to ξ equal to zero, we obtain the first-order condition for anoptimal consumption and portfolio choice,ptu(ct) = Et βu(ct+1)xt+1, (1.1)orpt= Etβu(ct+1)u(ct)xt+1. (1.2)The investor buys more or less of the asset until this first-order conditionholds.Equation (1.1) expresses the standard marginal condition for an opti-mum: ptu(ct) is the loss in utility if the investor buys another unit of the asset;Et βu(ct+1)xt+1is the increase in (discounted, expected) utility he obtainsfrom the extra payoff at t + 1. The investor continues to buy or sell the assetuntil the marginal loss equals the marginal gain.“chap01” — 2004/10/7 — page6—#66 1. Consumption-Based Model and OverviewEquation (1.2) is the central asset pricing formula. Given the payoffxt+1and given the investor’s consumption choice ct, ct+1, it tells you whatmarket price ptto expect. Its economic content is simply the first-orderconditions for optimal consumption and portfolio formation. Most of thetheory of asset pricing just consists of specializations and


View Full Document

Chicago Booth BUSF 35150 - LECTURE NOTES

Documents in this Course
CLONES

CLONES

8 pages

Load more
Download LECTURE NOTES
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view LECTURE NOTES and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view LECTURE NOTES 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?