EXERCISE  1:   (10  Marks)  Consider a representative agent who lives for two periods. This agent is endowed with income y0 and consumes c0 and purchases shares π at price po  in the first period.  This agent does not receive an endowment in the second period and therefore must use the shares purchased in the first period to fund their consumption in the second period. In the second period, this agent can sell their shares at price p1 and they receive a risky divided d1 from the asset. The agent enjoys utility u : R+  → R+, where u\  > 0 and u\\  < 0 and therefore the agents problem is given by

max   {u(c0 ) + βE0u(c1 )}

{c0 ,c1 ,π}

Subject to

c0 + πp0  ≤ y0     and   c1  ≤ π (d1 + p1 )

Where β ∈ (0, 1) is the subjective discount factor (which captures the agents prefer- ence of time), and E0 is the expectations operator conditional on information available at time 0

Show that the solution of the agent’s problem delivers the two-period version of the fundamental asset pricing equation, i.e.

p0 = E0m1g1

Where

u\ (c1 )

m1  = β u\ (c0 )    and   g1  = d1 + p1

EXERCISE 2:  (5 Marks) We can generalise the fundamental asset pricing equation in the following way

pt = Etmt+1gt+1

Where

mt+1 = β  u\ (ct)     and   gt+1 = dt+1 + pt+1

mt+1  is known as the stochastic discount factor. Let τ := ct+1/ct  be the growth rate of consumption. Given the following CRRA utility function

c1−γ − 1

γ → 1     1 − γ

Explain how the correlation between consumption growth τ and gt+1 effects the price of the asset and what this implies about how this agent thinks about risk.

EXERCISE 3:  (5 Marks) Consider again the the fundamental asset pricing equa- tion

pt = Etmt+1gt+1

But, in this case

mt+1 = β    and   gt+1 = dt+1 + pt+1

Suppose we have a non-random dividend stream i.e.  dt  = d > 0 for all t.  Show that the equilibrium asset price in this case is given by

Hints: Use this closed form of an infinite horizon geometric series

 = 1 + r + r2 + ...

And also note

lim βk−1pt+k  = 0

EXERCISE 4:  (10 Marks) Again, given the the fundamental asset pricing equation

pt = Etmt+1gt+1

Again, where

mt+1 = β    and   gt+1 = dt+1 + pt+1

And defining the price dividend ratio vt  := pt/dt .  Consider a growing, non-random dividend process dt+1  = κdt  where 0 < κβ < 1.  Show that if vt  = v that the price dividend ratio can be given by

βκ   

v =

Also, if we let κ = 1 + ϕ and let ρ := 1/β − 1, show that the price is given by the

Gordon formula

1 + ϕ

ρ − ϕ

EXERCISE 5:  (10 Marks) Write a Python script that

1. Plots the price given by the Gordon formula as a function of dt .  Suppose that β = 0.96 and ϕ = −0.1 (4 Marks)

2. Plots how the Gordon formula price changes for ϕ values −0.1, −0.2, ..., −0.9 as a function of dt  (3 Marks)

3. Plots how the Gordon formula price changes for β values 0.91, 0.92, ..., 0.99 as a function of dt  (3 Marks)

Comment on your results

EXERCISE 6:  (20 Marks) Consider a consumer that has preferences over con- sumption stream that are ordered by the utility functional

E0  [  βtu(ct )]

Where Et  is the mathematical expectation conditioned on the consumer’s time t in- formation. ct is time t consumption. u is a strictly concave one-period utility function and β ∈ (0, 1) is the subjective discount factor. For the purposes of this problem, let us assume (1 + r)−1  = β .  The consumer maximises this stream of utility by choos- ing a consumption, borrowing plan {ct,bt+1} subject to the sequence of budget constraints

ct + bt = bt+1 + yt,    t ≥ 0

Where yt  is an exogenous endowment process, r > 0 is a time-invariant risk-free net interest rate, bt  is one period risk-free debt maturing at t.  The consumer also faces initial conditions b0  and y0 , which can be fixed or random.