Empirical Methods in Finance

Homework 8

Please use Matlab/R to solve these problems. [The quality of the write-up matters for

your grade. Please imagine that youíre writing a report for your boss at Goldman when

drafting answers these questions. Try to be clear and precise.]

Working with factor models

Assume you have been given three assets to invest in, in addition to the market portfolio.

From a historical regression of excess asset returns on the excess market return for t = 1; :::; T,

you have:

R

e

1t = 0:01 + 0:9R

e

mt + ^”1t

;

R

e

2t = 0:015 + 1:2R

e

mt + ^”2t

;

R

e

3t = 0:005 + 1:0R

e

mt + ^”3t

:

The sample mean excess return on the market is, Re

m = 0:05; the sample standard deviation

of excess market returns is 15%. Thus, the market Sharpe ratio is 1=3. Finally, the sample

variance-covariance matrix of residual returns, “t = [“1t “2t “3t

]

0

, is:

var (^”t) = ^

” =

2

6

4

0:1

2

0

0

0

0:152

0

0

0

0:052

3

7

5.

1. What is the sample mean, standard deviation, and Sharpe ratio of the excess returns

these three assets?

2. For each of the three assets, construct the market-neutral versions by hedging out the

market risk (R

jt). For each of these three hedged asset returns, give the sample average

return, standard deviation, and Sharpe ratio.

3. Calculate the maximum Sharpe ratio you can obtain by optimally combining the three

hedged assets. Give the math behind this calculation.

4. Given your result in (3), what is the maximum Sharpe ratio you can obtain by combining these three assets with the market portfolio?

5. You have been told to form a portfolio today, assuming the historical estimates given

above are the true values also going forward, that (a) provides the maximum (expected)

Sharpe ratio of returns and (b) has an (expected) volatility of 15%. You can invest in

the three assets, as well as the market portfolio.

(a) Give the portfolio weights (really, the loadings on each of these in total four assets

since each asset is an excess return) that achieves objectives (a) and (b).

Hint: Recall that the mean-variance weights are proportional to the

1Re where

is the 4 4 covariance matrix of all excess returns (inlcuding the market) and

R is the 4 1 vector of expected asset returns. Thus, we have w

MV E = k

1Re

,

for some constant k. Next, recall that the variance of the mean-variance e¢ cient

portfolio is k

2

Re

0

1Re

. Thus, we can Önd k by setting k

q

Re

0

1Re = 15%.

(b) Give the expected excess return, standard deviation, and Sharpe ratio of this

portfolio. When you are evaluating variance and covariances, recall that the

variance of each asset includes a systemic component (relative to i

) in addition

to the residual covariance matrix given above.

6. (Optional) Next, you run Fama-MacBeth regressions of the three asset returns on their

market betas and an intercept.

(a) Give the mean return, standard deviation, and Sharpe ratio of the factor-mimicking

portfolio that is implied by the regressi

(b) What is the correlation between this factor-mimicking portfolio return and the

market portfolio? You are given enough information in the above to make this

calculation, given your answer in (a).

(c) Do a PCA on the three assets. Again, when calculating the variance-covariance

matrix of returns, remember to account for both systemic and residual return

variation. What is the variance of each of the three PCs, relative to the sum of

their variances?

(d) What are the portfolio weights of each PC?

(e) Di¢ cult: In (b) you created a factor-mimicking portfolio that, at least intuitively,

should deliver the market factor. Yet, it doesnít. The same is true in the data:

a market beta sort does not give you the market factor, but something with

relatively low correlation with the market factor. But, why is that? Explain your

reasoning.

Hint: The simplest way to understand whatís going on is to consider, for illustrative purposes, the two factor model:

R

e

it = 1iMKT1t + 2iF2t + “it; (1)

where the Örst factor is the market factor, the other is a long-short factor (e.g.,

a long-short industry factor long auto industry and short mining). Assume the

second factor is uncorrelated with the market factor. However, assume that crosssectionally, the market betas are correlated with the long-short factor betas, i.e.

Corr (1i

; 2i

) 6= 0. In this case, a cross-sectional regression with only market

betas (1i

) on the right-hand-side su§ers from an omitted variable bias.

Going back to the initial factor model considered in this homework, and given (5c),

and (5d), note that all three PCs explain a non-trivial amount of total variance,

and also note that the second principal component have weights that look akin

to a long-short beta portfolio.

3