Consider an asset paying dividends D over an infinite horizon. Assume D is a geometric Brownian motion:

dD D = μdt + σ dB for constants μ and σ and a Brownian motion B. Assume the instantaneous risk-free rate r is constant, and assume there is an SDF process M such that

dD D

dM M



= −σ λdt (13.58)

for a constant λ. Assume μ−σλ< r, and assume there are no bubbles in the price of the asset.

(a) Show that the asset price is Pt = Dt r +σ λ −μ .

Show that the Sharpe ratio of the asset is λ. Note: This is a continuous-time version of the Gordon growth model (Section 10.4).

This exercise is continued in Exercise 15.2.

(b) Assume (13.55) is an SDF for constants δ > 0 and ρ > 0, where C = D.

Show that (13.58) holds. What is λ? Referencing Exercise 12.4, calibrate to the following statistics reported by Mehra and Prescott (1985):

r = log1.008, Et[Ct+1/Ct] = 1.018,stdevt(Ct+1/Ct) = 0.036, Et[(Pt+1 +Ct+1)/Pt] = 1.0698, and stdevt(Pt+1 +Ct+1)/Pt) = 0.1654.

Calculate ρ and δ.