Take Home Assignment for Eco460
Due at or before 9:10am on November 26, 2013
No late work will be accepted
Instructions:
• Group discussion is allowed, but each individual has to write her/his own
answer to the assignment questions. Just copying someone else’s answer
is unacceptable and will be considered as an academic offence.
• Please hand in your assignment either in the morning class at NE172 or
to the economics department office, Room 257 Kaneff Building.
I. (57 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
0. 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
N
X
πi u(ci )
i=1
1
Here u(c) = 1−γ
(c1−γ − 1) and β is the time discount factor, 0 < β < 1.
That is, household value the utility from future consumption at a discount.
be the household’s consumption in period 1. 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 (1 + rf ) in period 1 or (3) invest in a risky asset that yields a risky
return of 1 + re = (1 + r1 , ..., 1 + rN ) 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 = (1 + rf )b + (1 + ri )s, i = 1, ..., N
1. (9 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.
1
2. (4 points) Let gi = ci /c(0) − 1 be the consumption growth rate in state i,
and g = (g1 , ..., gN ). Use the first-order conditions to prove that
1 + rf =
1
βE [(1 + g)−γ ]
3. (4 points) Let rie = ri − rf be the excess return of the stock in state i and
e
re = (r1e , ..., rN
). Again use the first-order conditions to prove that
E (1 + g)−γ re = 0
4. (4 points) Given that the equations in 2 and 3 hold, prove that the expected excess return of the stock (risk premium) E [re ] satisfies the following condition:
E [re ] = −β(1 + rf )Cov((1 + g)−γ , re )
5. (8 points) Assume that β = (1 + rf )−1 and use the approximation formula
(1 + gi )−γ = 1 − γgi . Prove the following formula for the risk premium of
the stock and provide an economic explanation for the formula:
E [re ] = γCov(g, re )
6. (20 points) Consider the stock as the S&P 500 index in the US and the
bond as the one-year US treasury bond. Find historical data (at least
50 years) on the annual return of the S&P 500 index, the one-year US
treasury rate, and annual consumption growth rate. (1) Calculate the
average excess return and the covariance between consumption growth
rate and excess return for the whole sample period and for the first and
second halves of the sample period. (2) Present the results in a table. Also
indicate clearly the data sources you used to calculate these statistics in
your table.
7. (8 points) Empirical evidence suggests that a typical household’s relative
risk-aversion coefficient γ is around 2. For this value of γ and use the
average excess return as an estimation of E [re ], are the data consistent
with the equation in 5? If not, what is the discrepancy and what is the
value of γ that is needed for equation in 5 to hold?
II. (23 points) Consider an Arrow-Debreu economy with two states. Assume
that the prices of the two basic Arrow-Debreu securities are both 0.5. Suppose
that a firm has potential income y = (10, 20) and outstanding debt R = 14.
1. (3 points) Show that the firm can hedge so that its income is 15 in both
states.
2
2. (10 points) Suppose that the tax rate on equity income is 30% and that
the bond holder will suffer a fixed cost of 5 if the firm defaults on its
debt. Calculate the values of equity and debt under both hedging and no
hedging, respectively. Which case generates a higher value of the firm,
hedging or no hedging?
3. (10 points) For the cases of hedging and no hedging, find the optimal debt
level R that will maximize the firm’s value in each case.
3