Asset Pricing

Department of Business and Management Esben Bjørn Christensen March 25, 2022 Asset Pricing

Assignment 1

Disclaimer The assignment must be solved in groups of 2-3 students. Be sure to include all the

group members’ names and birthdates on the front page of the assignment. If a group member

wishes an individual assessment of the assignment it must be clearly stated who is responsible

for which part. The assignment must be handed in as a single PDF-file via itslearning no later

than Friday 29th April at 10:00 (the start of the lecture).

Assignment The assignment consists of the following two exercises. Be sure to show your

calculations in a suitable level of detail.

Exercise 1 Consider the following 1-period economy with a single representative agent. The

agent is at time t = 0 endowed with e0 = 1.5 and her future endowment at time t = 1 depends

on the outcome of three possible economic scenarios as shown in Figure 1. Assume that the

financial market is complete, i.e. the agent can obtain any future consumption plan given her

budget constraint.

e1(ω1) = 3

e1(ω2) = 2

e1(ω3) = 1

Figure 1: State-contingent endowment of the agent. Each of the three states are equally likely.

The agent’s preferences are characterized by the utility function u(c) = c 1−γ

1−γ , with γ > 0

and for γ = 1 she has log utility. In addition she has time-additive expected utility with a time

preference parameter δ = 0.05.

(a) What is the agent’s optimization problem in the complete market? Write the Lagrangian in

terms of the consumption today, the state-contingent consumption and state-price deflator

ζ at time t = 1.

(b) Find the unique equilibrium state-price deflator ζ(ω) for any γ.

(c) Find the equilibrium risk-free rate Rf for different parameters γ ∈ [0.5, 2]. Present your results in a graph.

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Consider the zero-net supply Asset 1 with state-contingent dividends given in Figure 2.

D1(ω1) = 2

D1(ω2) = 1

D1(ω3) = 0

Figure 2: State-contingent dividends of Asset 1.

(d) What is the equilibrium price of Asset 1 for any parameter γ?

(e) Find the equilibrium expected excess return of Asset 1, E[R1]−Rf, for different parameters γ ∈ [0.5, 2]. Present your results in a graph. Explain the economic intuition behind your results.

Consider the zero-net supply Asset 2 with state-contingent dividends given in Figure 3.

D2(ω1) = 1

D2(ω2) = 2

D2(ω3) = 0

Figure 3: State-contingent dividends of Asset 2.

(f) Find the equilibrium expected excess return of Asset 2, E[R2]−Rf, for different parameters γ ∈ [0.5, 2]. Compare your results to Asset 1 economically, using the assets’ correlation with the agent’s future endowment.

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Exercise 2 Consider the following discrete-time multiple period economy with a single repre-

sentative agent. There is no terminal period so the economy continues forever. The agent is

endowed with one unit of an asset paying dividends Dt which follows from the recursion

ln Dt+∆t = ln Dt + µ∆t + σ √ ∆tεt+∆t, (1)

where µ, σ are constants and the noise terms εt+∆t have a mean of zero, a variance of 1, and are

mutually independent for all t, and hence independent of Dt. All other assets are in zero-net

supply.

The agent’s preferences are characterized by the utility function u(c) = c 1−γ

1−γ , with γ > 0

and for γ = 1 she has log utility. In addition she has time-additive expected utility with a time

preference parameter δ.

(a) Argue that in equilibrium, the agent’s optimal consumption must be equal to the dividend

of the asset, i.e. Ct = Dt for all t.

(b) Show/argue that the relative state-price deflator induced by the agent’s preferences and

optimal consumption are given as

ζs ζt

= e−δ(s−t) ( Cs Ct

)−γ , (2)

over any time period [t, s].

(c) Find the equilibrium one-period risk-free rate R f t .

To find the equilibrium price of the risky asset, we conjecture that it is given as Pt = DtA

for some constant A > 0.

(d) Argue that the price of the risky asset must follow the recursive expectation

Pt = E

[ ζt+∆t ζt

(Pt+∆t + Dt+∆t)

] (3)

(e) Show that the constant A > 0 is given as

A = 1

e(δ−(1−γ)µ− 1 2 (1−γ)2σ2)∆t − 1

and therefore we must have that

δ − (1 − γ)µ − 1

2 (1 − γ)2σ2 > 0

(f) Argue that the one-period return on the risky asset is given as

Rt+∆t = Dt+∆t Dt

1 + A

A (4)

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Now consider a zero-net supply asset called the variance derivative introduced at time t,

with a payoff at time T = t + 4∆t related to the realized one-period log returns of the

risky asset over the following 4 periods, given as

VT ≡ Vt+4∆t = 10 000 · 1

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t+4∆t∑ s=t+∆t

(ln Rs) 2 (5)

(g) Argue that the time-t price of the variance derivative must be given as

Vt = Et

[ ζt+4∆t ζt

VT

] (6)

(h) Estimate the time-t price of the variance derivative by simulating the 4-quarter economy

M = 10 000 times with Dt = 100, ∆t = 1 4 , µ = 0.02, σ = 0.05, δ = 0.02 and for two different

γ = 2 and γ = 10. Explain in detail how you simulate the economy and how you find the

final random variable you are using in your Monte-Carlo estimation.

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