Department of Economics Microeconomic TheoryUniversity of California, Berkeley Economics 101AApril 25, 2007 Spring 2007Economics 101A: Microeconomic TheorySection Notes for Week 14: CAPM1 Asset Demand in GeneralIn order to think about the problem of the demand for financial assets in general we considera consumer who derives utility through consumption and lives for two periods. A risky assetis available in the first period at a price pt. The risky asset can then be sold in the secondperiod at price pt+1which is a random variable. The consumer begins the first period withWtin initial wealth and must choose the quantity of the asset to purchase,s in order tomaximize her total expected utility. This consumer’s objective function can be written asmaxsu(ct) + δEt[u(ct+1)]where 0 < δ < 1 is the consumer’s dis count factor. The consumer faces two budget con-straints. In the first period, she can either consume her wealth or invest in the asset and inthe second period she can only consume as muct as the asset pays off.ct= Wt− ptsct+1= pt+1sher first order condition with respect to the quantity of asset to buy is thus∂∂s(u(Wt− pts) + δEt[u(pt+1s)]) = −ptu0(Wt− pts) + δEt[pt+1u0(pt+1s)]= −ptu0(ct) + δEt[pt+1u0(ct+1)] = 0orptu0(ct) = δEt[pt+1u0(ct+1)]pt= Etδu0(ct+1)u0(ct)pt+1Note, in general, both u0(ct+1) and pt+1are random variables. The content of the CAPM orany asset pricing model is in specifyingmt+1≡ δu0(ct+1)u0(ct)which is known as the stochastic discount factor (SDF).1We can also see what the first order condition implies about a risk-free asset.pt= Etδu0(ct+1)u0(ct)pt(1 + Rf)= pt(1 + Rf)Etδu0(ct+1)u0(ct)11 + Rf= Etδu0(ct+1)u0(ct)= Et[mt+1]showing that the expected value of the SDF is equal to the inverse of the gross risk-free rateof return.Also note that the pricing equation can be written as by divding both sides by pt.1 = Et[mt+1(1 + Ri,t+1)]2 Content of The Capital Asset Pricing ModelThe Capital Asset Pricing Model (CAPM) says two things1. A linear factor pricing model holds.2. The single factor in the model is the portfolio containing the entire asset market.Quick Reminder: Assuming there are no dividend payments, the expected net return onasset i over the next year can isEt[Ri,t+1] =Et[Pi,t+1] − Pi,tPi,tre-arranging, we getPi,t=Et[Pi,t+1]1 + Et[Ri,t+1]thus an asset pricing model is equivalent to a model of expected returns.3 Derivations of the CAPMThe following derivations are taken from John Cochrane’s book, Asset Pricing (pp.152-1671st edition). In lecture we saw a derivation of the CAPM in which investors had preferencesfor minimum variance portfolios given a target expected return. The f ollowing derivationsassume that investors have more primative preferences defined over consumption. Note,these are all some sort of vN-M expected utilility preferences.3.1 Two Per iod Quadratic UtilityIn this derivation, investors have quadratic preferences over consumtion u(c) = −12(c − c∗)2and live for two periods.U(ct, ct+1) = u(ct) + δEt[u(ct+1)]= −12(ct− c∗)2− δ12Et[(ct+1− c∗)2]2where 0 < δ < 1 is a discount factor. Note that c∗is what is referred to as the bliss point ofthe quadratic utility function. Remember, that it makes sense for consumers to prefer morewealth to less. The quadratic utility function looks like a parabola with the open part facingdown, thus we have to be careful to only use the section of the parabola that is increasing.The bliss point, c∗represents the level of consumption at which the parabola acheives itsmaximum, thus when working with a quadratic utility function we want to think of c∗assome level of consumption far above anything that the consumer might actually realize.In this model the SDF ismt+1= δu0(ct+1)u0(ct)= δct+1− c∗ct− c∗= −δc∗ct− c∗+δct− c∗ct+1= a + bct+1which is a linear factor model where ct+1is the factor. Thus we have shown part 1 of theCAPM. Next, we need to identify the factor. Since all remaining wealth is consumed inperiod twoct+1= (1 + Rp,t+1)(Wt− ct)where the Rp,t+1is the return on the optimal portfolio from the investor’s FOC.Rp,t+1=Xiα∗iRi,t+1s.t.Xiα∗i= 1so the SDF can be written asmt+1=δ(Wt− ct) − δc∗ct− c∗+δ(Wt− ct)ct− c∗Rp,t+1= a0+ b0Rp,t+1Since all consumers have that same preferences, they will all hold the same portfolio, p.Since everyone wants to hold p, then in equilibrium all these shares of p must sum to be theentire asset market thus the portfolio p is a share of the entire market, so Rp,t+1= RM,t+1the market return. We have shown part two of the CAPM. The factor has been identifiedas the market portfolio.We can pin down the parameters a0and b0by requiring that the model correctly pricetwo assets such as the risk-free asset and the market