Posted: February 9th, 2015

Financial Risk Management

Paper, Order, or Assignment Requirements

 

 

Take Home Assignment for Eco460

Due at 6:10pm on November 25, 2014

No late work will be accepted

Note: Please hand in your assignment in class.

1. (40 points) Consider a two-period extension of the model in Chapter 4 of

the Lecture Notes. The two periods are indexed by t = 0 and 1. There is no

uncertainty in period 0 and let c(0) be the household’s consumption in period

1. There is uncertainly in the second period and the household’s consumption

plan is denoted by c(1) = (c1, c2, …, cN ), where N is the number of states. The

household’s preferences for consumption in two periods are summarized by the

following utility function:

U(c(0), c(1)) = u(c(0)) + β

X

N

i=1

πiu(ci)

Here u(c) = 1 −

1

1−γ

(c

1−γ − 1) and β is the time discount factor, 0 < β < 1.

That is, household value the utility from future consumption at a discount.

The household is endowed with a certain income y(0) in period 0, which the

household can be used for (1) consumption in period 0, (2) invest in a risk-free

bond that yields a risk-free return of e

rf

in period 1 or (3) invest in a risky asset

that yields a risky return of e

er = (e

r1

, …, erN ) in period 1. So the household’s

budget constraint in period 0 is:

c(0) + b + s = y0

where b is the amount invested in the risk-free bond and s is the amount invested

in the risky asset (stock). There is no endowment income in period 1. Since

this is a two-period economy, there would be no investment in period 1 either.

So the household’s budget constraint in period 1 is

ci = berf + seri

, i = 1, …, N

1. (10 points) The household’s optimal investment problem is to choose c(0),

b and s to maximize U(c(0), c(1)) subject to the budget constraints in both

periods. Use the Lagrangian method to derive the first-order conditions

for optimal choices of c(0), b and s. [Hint: you can substitute the budget

constraints for ci

into the expected utility function.]

2. (10 points) Let gi = ln [ci/c(0)] be the consumption growth rate in state

i, and g = (g1, …, gN ). Use the first-order conditions to prove that

1 = βE

e

rf −γg

,

and

1 = βE

e

r˜−γg

.

1