# Theory of risk/risk management exam

# Theory of risk/risk management exam

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## Job Description

FINAL EXAM - DUE 11/29, 2:30 PM

FLORIN BIDIAN

1. Testing CAPM (45p)

Download the weekly prices (adj. close) for the 30 stocks in the (current) composition

of the Dow Jones Industrial Average (DJIA)

1

and for the SP500, for the period 1/7/2002-

11/18/2012. You can obtain the data at finance.yahoo.com.

(a) Compute the beta of each stock by running, for each stock, the time series regres-

sions

rjt = j + jrM t + "jt

;

where rjt represents the return of stock j at period t, rM t represents the market

return (SP500) at t, and "jt

is the regression residual.

(b) Report the minumum and maximum R-squares in these regressions? What does

this suggest about the noise in the beta estimates you obtained?

(c) Using the beta estimates obtained before, run the cross-sectional regression

rj =

0 +

1

j + "j ;

where rj is the weekly average return on stock j,

j is the previous estimate for

stock j's beta, and "j is the regression residual.

(d) Are

0 and

1 statistically signicant? Relate them to the security market line

(SML) if CAPM were true, and discuss.

(e) Analyze the presence of time-varying betas in the data. Focus only on BA and

BAC, and plot their \rolling betas" over time, computed using the previous 5 years

of weakly returns. Discuss your ndings in connection to CAPM.

Date: November 19, 2012.

1MMM AA AXP T BAC BA CAT CVX CSCO KO DD XOM GE HPQ HD INTC IBM JNJ JPM MCD

MRK MSFT PFE PG TRV UNH UTX VZ WMT DIS

12 FLORIN BIDIAN

2. Mean variance frontier and Markowitz optimization (45p)

Consider a subset of the data from the previous problem involving T, BAC, BA, CAT,

CVX, CSCO, KO.

(a) Calculate the variance-covariance matrix using the 2002-2006 returns.

(b) Derive (plot) the ecient frontier, and show on the same graph the position of the

seven stocks.

(c) Derive and plot in the same graph the restricted minimum variance frontier, in

which short-selling is prohibited. Thus you are allowed to hold only long (positive)

positions in the available assets.

(d) Determine the average (weakly) risk-free rate for 2002-2006. For this, get the 3-

month treasury rate from the St. Louis Fed, at monthly frequency (name of the

series is GS3M), average it over the 2002-2006 and divide it by 53 (as it was in

annual terms, rather than weekly).

(e) Report the minimum variance portfolio and the optimal risky portfolio.

(f) At the end of 2006 (last trading day of 2006 in your sample) you invest $ 7000 in

the 7 stocks, in the proportions required by the optimal risky portfolio (you allow

short-selling, that is, negative weights). What is the realized Sharpe ratio of your

portfolio between 2007-2012 (use the average risk-free rate for 1/2007-10/2012, in

weekly terms)? What is the nal value of your portfolio (as of the last day in your

sample, 11/12/2012)?

(g) Assume instead that at the end of 2006 you invest the $ 7000 in equal proportions

in the 7 stocks ($ 1000 in each). What is the realized Sharpe ratio of your portfolio

between 2007-2012 and the nal value of your portfolio?FINAL EXAM - DUE 11/29, 2:30 PM 3

3. Pricing Internet Stocks (10p)

Consider the usual stochastic T-period economy in which agents have common utilities

E

XT

t=0

t

ln(c

i

t

);

where 2 (0; 1). In this economy, equilibrium aggregate consumption is given by a se-

quence of random variables (stochastic process) (Ct)

T

t=0

, and agents have access to a com-

plete set of markets.

(a) For each period t < T, what is the stochastic discount factor mt+1 resulting from

agents' utility maximization problem? You do not need to prove your answer, but

you need to justify it, by referring to our results derived for CBAPM in class.

(b) In this economy, for each period t < T, derive an expression for the price pt at t of

a stock that has a random payo equal to CT in period T, where 0 < 1, and

payo equal to zero in all other periods. (HINT: Remember that according to the

valuation equation implied by CBAPM, the value at t of a security paying XT at

T and zero at all the other periods is Et

[mt+1mt+2 : : : mT XT ].)

(c) It has been argued that stocks like Amazon are overvalued, since they hardly pay

any dividends, and yet their stock prices keep growing relative to GDP (gross

domestic product). What does your above analysis tell you about this argument?

(HINT: Base your discussion on the evolution of pt=Ct as a function of time t.)

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Skills: management, amazon, analysis