Consider a discrete-time economy in which asset prices are described by an unconditional linear factor model t+1

Consider a discrete-time economy in which asset prices are described by an unconditional linear factor model

ζt+1

ζt

= a + b · xt+1, t = 0, 1, ... , T − 1, where the conditional mean and second moments of the factor are constant, that is Et[xt+1] = μ and Et[xt+1x t+1] =  for all t.

You want to value an uncertain stream of dividends D = (Dt). You are told that dividends evolve as Dt+1 Dt

= m + ψ · xt+1 + εt+1, t = 0, 1, ... , T − 1, where Et[εt+1] = 0 and Et[εt+1xt+1] = 0 for all t.

The first three questions will lead you through the valuation based directly on the pricing condition of the state-price deflator.

(a) Show that, for any t = 0, 1, ... , T − 1, Et



Dt+1 Dt

ζt+1

ζt



= ma + (mb + aψ) · μ + ψb ≡ A.

(b) Show that Et



Dt+s Dt

ζt+s

ζt



= As

.

(c) Show that the value at time t of the future dividends is Pt = Dt A 1 − A (
1 − AT−t )
.
Next, consider an alternative valuation technique. From Eq. (4.26) we know that the dividends can be valued by the formula Pt = T −t s=1 Et[Dt+s] − βt 
Dt+s, ζt+s ζt 
ηt,t+s (1 + yˆ
t+s t )s , (*)
where βt 
Dt+s, ζt+s ζt 
= Covt 
Dt+s, ζt+s ζt 
/ Vart 
ζt+s ζt 
, ηt,t+s = − Vart 
ζt+s ζt 
/ Et 
ζt+s ζt 
.
In the next questions you have to compute the ingredients to this valuation formula.

(d) What is the one-period risk-free rate of return r f t? What is the time t annualized yield yˆ
t+s t on a zero-coupon bond maturing at time t + s? (If Bt+s t denotes the price of the bond, the annualized gross yield is defined by the equation Bt+s t = (1 + yˆ
t+s t )
−s .)

(e) Show that Et[Dt+s] = Dt (m + ψ · μ)s .

(f) Compute Covt 
Dt+s, ζt+s ζt 
.
(g) Compute βt 
Dt+s, ζt+s ζt 
.
(h) Compute ηt,t+s.
(i) Verify that the time t value of the future dividends satisfies (*).