Econ 745 Fall 2013
Simon Gilchrist
Problem Set 1 - Due Tuesday, Nov 6th
Asset pricing basics (the “standard model”)
Consider an endowment economy with a representative agent who has expected utility preferences
and power utility (CRRA):
X C 1−γ
U =E
βt t .
1−γ
t≥0
Assume that the endowment follows the process
∆ log Ct+1 = µc −
σc2
+ σc εt+1 ,
2
where εt+1 is iid N (0, 1).
Consider N assets, i = 1...N which pay respectively dividends Dit where each Dit follows a different
process:
χ2
λ2i
− i + λi εt+1 + χi ηi.t+1 ,
2
2
with ηit+1 iid N (0, 1) and χi , λi , µi characterize this process. ηi,t+1 is uncorrelated with εt+1 at all leads
and lags, i.e.
E(εt+1−k ηi,t+1 ) = 0 f or all k ≥ 0, and all k ≤ 0
∆ log Dit+1 = µi −
Note: you will need the log-normal formula, i.e. if X is N (µ, σ 2 ) then E (exp(X)) = exp µ +
1. Compute the mean of
Ct+1
Ct
and
Dit+1
Dit .
(So you can see why I added the terms −
σc2
2 ,
−
λ2i
2 ,
σ2
2
.
etc.).
2. Write down a portfolio problem for this Lucas tree economy. Derive the optimality conditions for
this problem.
3. Consider the equilibrium of this economy:
(a) Compute the risk-free rate.
(b) Compute the price-dividend ratio on asset i. Explain intuitively how it depends on µi , λi and
χi .
(c) Compute the expected return and expected excess return (i.e. return less the risk-free rate) on
asset i. Explain intuitively how it depends on µi , λi and χi . Discuss the statement “idiosyncratic risk is not priced.”
(d) Is it true that more volatile assets (“more risky assets”) have higher average returns?
(e) Plot (roughly) the effect of a shock εt+1 on consumption, dividends (both for a low λ and a
high λ asset), returns, and the price-dividend ratio. (This is akin to an “impulse response
function”)
4. Define the asset i’s “consumption beta” βi,c as the slope of the time-series regression of the asset
return on consumption growth:
i
Rt+1
= αi + βi,c ∆ log Ct+1 + νi,t+1 .
Compute βi,c . What is the cross-sectional (i.e., across i) relation between βi,c and expected returns?
[Hint: you can use the following approximation: if (U, V ) is jointly normal, and g is a smooth
function, then: Cov(g(U ), V ) = E(g 0 (U )) × Cov(U, V ). (this is known as Stein’s lemma).]
1
5. In the US, starting around 1985, the volatility of consumption growth fell (the “Great Moderation”).
What would a standard asset pricing model predict for the risk-free rate and the equity premium,
if people realize immediately in 1985 the decrease in volatility?
6. True, false or uncertain: according to theory, countries with more volatile consumption and dividends should have more volatile stock prices.
Comment: this problem set is important because this simple example includes many of the principles
of asset pricing - how P/D ratios are related to expected returns, the role of βi , the fact that idiosyncratic
risk does not matter, etc.; so make sure you are comfortable with this. However, keep in mind that some
assumptions are strong, e.g. the iid assumption is restrictive. Problem 2 considers variation in expected
growth rates and volatilities.
2