LECTURE 12: INTERTEMPORAL-CAPM

1. LECTURE 12:INTERTEMPORAL-CAPM

2. I-CAPM Suppose investors require a different expected return in each future period in order that they will willingly hold a particular stock (Merton 1973). We have therefore EtRt+1= kt+1 (**)

3. I-CAPM where the time subscript on kindicates a time-varying return. Forward substitution, gives Pt=Et[δt+1Dt+1 + δt+1δt+2Dt+2 + ··+ δt+N−1δt+N(Dt+N + Pt+N)] which leads to (assuming the TVC holds) : Pt= Et∑(j=1,∞)δt,t+jDt+j(***) whereδt+i= 1/(1+kt+i)

4. I-CAPM • The current stock price, therefore, depends on expectations of future discount rates and dividends. Note that 0 < δt+j < 1 • for all periods and hence expected dividends $-for-$ have less influence on the current stock price the further they accrue in the future.

5. I-CAPM It is possible that an event announced today (e.g. a merger with another company) could be expected to have a substantial impact on dividends starting in, say, 5 years’ time. In this case, the announcement could have a large effect on the current stock price even though it is relatively heavily discounted. Note that in a well-informed (‘efficient’) market, one expects the stock price to respond immediately and completely to the announcement even though no dividends will actually be paid for 5 years.

6. I-CAPM In contrast, if the market is inefficient (e.g. noise traders are present), then the price might rise not only in the current period but also in subsequent periods. Tests of the stock price response to announcements are known as event studies.

7. Market Portfolio The one-period CAPM predicts that in equilibrium, all investors will hold the market portfolio (i.e. all risky assets will be held in the same proportions in each individual’s portfolio) Merton (1973) developed this idea in an inter-temporal framework and showed that the excess return over the risk-free rate, on the market portfolio, is proportional to the expected variance of returns on the market portfolio. EtRm,t+1− rt = λ(Etσ2m,t+1) λ proxies for the ARRA of market investors

8. Market Portfolio The expected return can be defined as comprising a risk-free return plus a risk premium rpt: EtRm,t+1= rt + rpt where rpt = λEtσ2m,t+1 *** Comparing (**) and (***), we see that according to the CAPM, the required rate of return on the market portfolio is given by kt= rt + λ(Etσ2m,t+1)

9. Market Portfolio The equilibrium required return depends positively on the risk-free interest rate rt and on the (non-diversifiable) risk of the market portfolio, as measured by its conditional varianceEtσ2m,t+1

10. Market Portfolio Onlyif • agents do not perceive the market as risky (Etσ2m,t+1 = 0)or • agents are risk-neutral (i.e. λ = 0) then the appropriate discount factor used by investors is the risk-free rate rt . Note that in this model to determine the price using the RVF, investors must determine ktand hence forecast future values of the risk-free rate and the risk premium.

11. IndividualAssetReturns Consider now the price of an individual security or a portfolio of assets, which is a subset of the market portfolio (e.g. shares of either all industrial companies or all banking firms). The CAPM implies that to be held as part of a diversified portfolio, the expected return on portfolio-i is given by EtRit+1 = rt + βit(EtRm,t+1 − rt); βit = Et(σim/σ2m)t+1 Substitutingfrom (***), wehaveEtRit+1 = rt+ λEt(σim )t+1 • where σim is the covariance between returns on asset-i and the market portfolio. Again • comparing (21) and (28), the equilibrium required rate of return on asset-i is • kt+1 = rt+ λEt(σim)t+1

12. IndividualAssetReturns Substitutingfrom (***), wehave EtRit+1 =rt + λEt(σim)t+1 where σim is the covariance between returns on asset-i and the market portfolio. Again, the equilibrium required rate of return on asset-iis kt+1 = rt + λEt(σim)t+1