Empirical Methods in Finance

Homework 3

Problem 1: ARMA basics

Consider the ARMA(1,1):

yt = 0:95 yt1 0:9 “t1 + “t

; (1)

where “t

is i.i.d. Normal with mean zero and variance

2 = 0:052

. In the below you need to

show your work in order to get full credit.

1. What is the Örst-order autocorrelation of yt?

2. What is the second-order autocorrelation of yt? Also, what is the ratio of the secondorder to Örst-order autocorrelation equal to? Give some intuition for this result.

3. If yt = 0:6 and “t = 0:1, what is (i) Et

[yt+1], (ii) Et

[yt+2] given the ARMA model?

4. Let x^t = Et

[yt+1] where the expectation is taken using the ARMA model. What is the

unconditional mean, standard deviation, and Örst-order autocorrelation of x^t?

Problem 2: Year-on-year quarterly data and ARMA dynamics

A substantial amount of quantity data, such as earnings, exhibit seasonalities. These

can be hard to model. It is therefore common to use so-called Year-on-Year data (e.g., Q1

earnings vs Q1 earnings a year ago, Q2 earnings vs Q2 earnings a year ago, etc). In this

problem we will see that such a practice can induce MA-terms due to the overlap in the

quarterly year-on-year observations.

Assume the true quarterly log market earnings follow:

et = et1 + xt

;

xt = xt1 + “t

;

where var (“t) =

2

” = 1 and “t

is i.i.d. over time t:

The earnings data you are given is year-on-year earnings growth, which in logs is:

yt et et4:

1. Assume = 0. Derive autocovariances of order 0 through 5 for yt

. I.e., cov (yt

; ytj )

for j = 0; :::; 5:

2. Assume = 0. Determine the number of AR lags and MA lags you need in the

ARMA(p,q) process for yt

. Give the associated AR and MA coe¢ cients.

3. Optional: assume 1 > > 0. Repeat 1. and 2. under this assumption.

Problem 3: Market-timing and Sharpe ratios (a little harder)

Much of this class is about prediction. In this problem you will derive how market

timing can improve the unconditional Sharpe ratio of a fund. The market timing is based

on “forecasting regressions” akin to those we undertake in a VAR. However, we are only

forecasting one period ahead here.

Assume you have an estimate of expected annual excess market returns for each time t,

called xt

. You estimate the regression

R

e

t+1 = + xt + “t+1;

and obtain ^ = 0, ^ = 1, and (^”t+1) = 15%: Further, the sample mean and standard

deviation of xt are both 5%.

1. Calculate the standard deviation of excess returns based on the information given.

2. Calculate the R2 of the regression based on the information give

3. Calculate the sample Sharpe ratio of excess market returns based on the information

given.

4. Recall from investments that myopic investors chooses a fraction of wealth

t =

Et

Re

t+1

2

t

Re

t+1

in the risky asset (the market) at each time t, where we assume risk aversion coe¢ cient,

, equals 40=9. Further, assume that the residuals “t+1 are i.i.d., so t (“t+1) = 15%

for all t. Given this, calculate the weight the investor chooses to hold in the risky asset

if xt = 0% and if xt = 10%. What is conditional Sharpe ratio in each of these cases?

5. Assume T is large (i.e., T ! 1) and that xt

is either 0% or 10% at each time t, with

equal probability (0:5).

(a) What is the unconditional average excess return for an investor that holds t each

period?

(b) What is the unconditional standard deviation? The following may be helpful for

calculating the unconditional variance. You could also simulate a very long series

to check your math.

V ar

tR

e

t+1

= E

h

Et

h

tR

e

t+12

ii E

Et

tR

e

t+12

= E

h

2

tEt

h

R

e

t+12

ii E

tEt

R

e

t+12

= E

2

t

x

2

t +

2

t

(“t+1)

E [txt

]

2

:

(c) Finally, what is the unconditional Sharpe ratio of this strategy?

(d) Now, assume the volatility of xt

is higher: it can take the values 5% and +15%

with equal probability.

i. What is the implied R2 of a forecasting regression of future excess returns on

xt assuming again that ^ = 0 and ^ = 1?

ii. What is the unconditional Sharpe ratio the investor that follows the risky

asset share rule given above in (4)? Note that a higher R2

implies a higher

Sharpe r