Empirical Methods in Finance
Homework 6
Prof. Lars A. Lochstoer
February 22, 2023
Problem 1: A simulated economy In this problem you will simulate an economy that incorporates many of the things we have discussed over the last couple of weeks. The intention is that playing around with this will help you understand both the math and econometrics as well as the economic lessons
we can draw. In this economy, agents have risk aversion parameter t which evolves according to an AR(1) process: 7%3+(L-I-7L)o+L=7L (1)
and can be at times very high (bad times, recessions, disasters) or very low (good times, booms, bubbles) The market's real log dividend growth is assumed to be i.i.d.:
d=+Ed,t
(2)
The conditional expected log excess market returns is given by
Et(r++1-r)=to3,
(3)
where r is the log stock market return and r + is the risk-free rate. Thus, the risk premium is
1
higher when risk aversion is higher. Assume the log real risk-free rate is constant and equal to rf = 1%. From the class notes, we have that:
pd,=constant +Dpi-1Et(d+i) -D pi-1Et (r++i). j=1 j=1
(4)
1. Given the process assumed for dividend growth and risk aversion, as well as the specification of the risk premium and the risk-free rate, derive the following expression for the log price-dividend ratio:
pdt = constant +CF-DRt,
CFt = 1-p rf+Yo? 02 DRt = (-L) 1-p 1 - 0
(5)
Suggested solution: Using the risk-free rate r f is constant and iterated expectations
E(rt+j)=E(E+j-1(rt+j)) = Et (rf+Yt+j-1o3) =rf+o3Et(Yt+j-1)
The conditional expectation of the risk aversion is the following
j-1
Et(++j)=(1-)L$n+$y
=(1-)+ =+ (yt-)
Then we replace in the previous expression
Et (r++j)=rf+02+$-13(xt-)
2
The conditional expectation for the log dividend growth is
Et(t+j)=
Therefore, replacing in (4) we finally obtain
pdt = constant +
(2-t)PO,-t$+PO+f)1
oL+fa 1-p
= constant
1-0
-+ 1 - po
2. We can decompose the variance of the pd ratio as follows:
Var(CFt) Var(DRt) Cov(CFt,DRt) Var (pdt) Var(pdt)
(6)
How much of the variance is due to the stock market expected return (the discount
Suggested solution: Since CFt is constant, Var (pdt) = Var (DRt). Therefore, the
coefficient is 1.
3. From class, log market returns can be written:
Tt+1 = constant + p pdt+1 - pd, + dt+1.
(7)
Given our assumptions, the variance of stock returns is equal to:
Var(rt) = Var(dt)+Var(pDRt-DRt-1)
= 0a+04o
(8)
1-0
Optional: derive this equation). Let = 2%, = 0.9, = 1.5, = 5, oa = 12%
= 0.96. How much of the variance of stockreturns, as opposed to the pd-ratio, in this economy comes from shocks to discount rates (future expected returns)? That is, what is:
1-2
03