Assume the continuous-time CAPM holds: (i r)dt = dSi Si dWm Wm for each asset

Assume the continuous-time CAPM holds:

(μi −r)dt = ρ

dSi Si

dWm Wm



for each asset i, where Wm denotes the value of the market portfolio, ρ = αWm, and α denotes the aggregate absolute risk aversion. Define σi = 

e



iei to be the volatility of asset i, as described in Section 13.1, so we have dSi Si

= μi dt +σi dZi for a Brownian motion Zi. Likewise, the return on the market portfolio is dWm Wm

= μm dt +σm dZm for some μm, σm, and Brownian motion Zm. Let φim denote the correlation process of the Brownian motions Zi and Zm.

(a) Using the fact that the market return must also satisfy the continuous-time CAPM, show that the continuous-time CAPM can be written as

μi −r = σiσmφim

σ2 m

(μm − r).

(b) Suppose r, μi, μm, σi, σm, and ρi are constant over a time interval t, so both Si and Wm are geometric Brownian motions over the time interval.

Define the annualized continuously compounded rates of return over the time interval:
ri = logSi t and rm = logWm t .
Let ri and rm denote the expected values of ri and rm. Show that the continuous-time CAPM implies ri −r = cov(ri,rm)
var(rm) (rm − r) +
1 2 [cov(ri,rm)− var(ri)]t .