Financial Econometrics – Final Exam
Professor Doron E. Avramov
Answer all Questions below – the Maximal Grade is 100.
Due Date: February 28, 2013.
Send an electronic file of your exam to both Uri Perlman at
u.perelman@gmail.com as well as myself at davramov@huji.ac.il
It is strictly prohibited to cooperate directly or indirectly with peer
students or anyone else. The exam must be solved on an individual basis.
Good luck!!!!
Question number 1 (10 points)
The relative risk aversion is typically computed as  
M  R f
. This quantity
 M2
is the excess return on the market portfolio divided by the variance of the
market portfolio. The relative risk aversion is similar to the Sharpe ratio
with the only difference that the denominator displays the variance of the
market rate of return rather than the standard deviation.
[a.] Derive the distribution of the sample relative risk aversion under two
assumptions.
First, assume that the market returns are IID normal through time.
Second, assume that the market returns deviate from normality.
In essence, this part is quite similar to what you did in Problem Set 1 in
deriving statistics for the sample Sharpe ratio.
[b.] Based on the sample of monthly market returns starting from January
1970 and using both statistics derived above, test whether you can reject the
null hypothesis that the relative risk aversion during the sample period is
equal to 2.
Question number 2 (20 points)
Consider two actively managed mutual funds A and B while the benchmark
for performance evaluation is the market portfolio M. It is assumed that the
joint distribution of LOG annual returns (natural log of gross returns NOT
log excess returns) of A, B, and M is of the MULTIVARIATE Normal form
given by
     A2 ,  AB, AM 
rA,t 
A

  IID    
2
rB ,t  ~ N    B ,  AB, B ,  BM , 
  
2 
r 
 M ,t 
  M   AM , BM , M ,
where  A  8% , B  13% , M  12% ,  A  0.15 ,  B  0.30 ,  M  0.20 , and the
coefficient of correlations between A and B, A and M, and B and M are 0.5,
0.4, and 0.8, respectively. In addition, the investment horizon is T=10 years
and the continuously compounded (log) risk free rate is 1% per year.
Assume that you start the investment with only $1.
A shortfall is defined as the event where over the investment horizon the
value of the mutual fund holding is smaller than the square root of the value
of an investment in the market portfolio. That is, shortfall is defined as

Pr (1  R A,1 )(1  R A, 2 )...(1  R A,T )  (1  RM ,1 )(1  RM , 2 )...(1  RM ,T )

Comments:
i. Notice that the upper case letters denote gross returns while the lower
case letters denote continuously compounded returns. For instance,
rA,t  ln( 1  R A,t ) .
ii. In solving parts b and c below assume that the variance covariance
matrix of log returns is equal to the variance covariance matrix of
gross returns.
iii. In solving part c below you have to compute the expected value of the
gross return – use properties of the log normal distribution.
[a.] What is the shortfall probability of mutual funds A and B?
[b.] You combine mutual funds A and B to form the Global Minimum
Variance Portfolio (GMVP) – what is the shortfall probability of that
GMVP?
[c.] You combine mutual funds A and B to form the tangency portfolio –
what is the shortfall probability of that tangency portfolio?
[d.] Repeat a, b, and c above but rather than shortfall probability compute
the VaR at the 5% level. Or put differently, compute a threshold such than in
5% of the times the invested wealth in mutual funds A and B (corresponding
to part a above) as well as the combined funds (corresponding to parts b and
c) would be below that threshold. Again recall that the value of the fund has
the lognormal distribution.
[e.] Discuss diversification gains based on these particular examples as well
as in general. You first need to identify what would be considered as a
diversification gain.
Question number 3 (15 points)
a. Visit yahoo/finance and collect data on call and put option prices on
the SPY (which tracks the S&P index). More specifically, choose
three periods to expiration, with each period exceeding 3 months.
Then for both call and put options as well as for each period plot the
implied volatility (IV) as a function of the strike price using all strikes
available on this date.
Notice that you have six plots – three for call and three for put
options.
Based on your plots would you say that the IV is constant across
maturities as assumed by the B&S formula? If not, what is the
identified pattern?
b. Now pick three strikes of out of the money call and put options. For
each of the strike price plot the IV as a function of time to expiration.
Do you recognize any pattern here? Again you have six plots – three
for call and three for put options.
c. Design a non-parametric test based on failure rate to test whether IV is
constant across maturities.
d. Same as above but across strike prices.
Remark 1: to solve those questions you must pick one particular day –
please write down date and time.
Remark 2: You will need to input into the B&S formula the dividend
yield. The gross dividend yield is 2.18% per year as the yahoo site
indicates. You need to transform it into log yield: ln(1+0.0218).
Remark 3: You also need to use the annual short (over-night) risk free
rate – use Bloomberg or any other web site to recover the short rate.
You need to input into the B&S formula the continuously
compounded annual risk free rate.
Question 4 (10 points)
A mean variance investor is one who minimizes portfolio volatility subject
to some expected return target. Consider instead an investor who minimizes
expected shortfall subject to an expected return target. Expected shortfall is
,
where
•
- an N- vector of portfolio weights;
•
- an N-vector of expected returns;
•
- is the VaR as outlined in the class notes;
• There is no risk free asset.
Thus, the investor is facing the optimization
subject to two constraints. First, the sum of weights is equal to one. Second,
the expected return on the portfolio is equal to the pre-specified target.
Assume further that asset returns obey multivariate normal distribution with
mean r and covariance matrix .
Show that, under the above formulation, the optimal solution of a down size
risk investor is identical to that of a mean-variance investor. Under which
circumstances the solutions are identical?
Question number 5 (15 points)
Visit the data library of Professor Ken French:
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Download monthly value weighted returns on 25 size book to market
portfolios. Assume that returns are normally distributed. Use the first 60
monthly observations to compute several downsize risk measures for each of
the 25 portfolios (computing drawdown requires only 12 months). The
measures are:
i. Monthly shortfall probability when shortfall is defined as negative
return.
ii. VaR – The threshold level such that 5% of the returns are below that
level.
iii. Expected Shortfall
iv. Monthly Target Semi Variance (target in the risk free rate).
v.
Downside beta (focusing on the first out of the three measures)
vi. Annual Drawdown
Extract one eigen vector (a six by one vector) to aggregate, for each
industry portfolio, the information in these seven measures and form
one aggregate measure that captures the risk of any of the industry
portfolios. Invest in five industries with the lowest risk based on this
aggregate measure. Rebalance every six month using a rolling
scheme. Compare performance of the resulting strategy relative to the
market portfolio using conventional performance measures considered
in class (e.g, alpha, sharpe ratio, information ratio, etc).
Question 6 (10 points)
a. For the above 25 portfolios compute the J1, J2, J3, and J4 statistics
in order to test the one factor CAPM.
b. Compute the Sharpe ratio for the market portfolio and the Sharpe
ratio for the tangency portfolio that consists of the market portfolio
plus the 25 test assets. Link asset mispricing  '  1 to the
difference in Sharpe ratios.
Question number 7 (10 points)
Stock returns evolve according to the conditional factor model
rt e     t rmte  et
 t   0  1 z t 1
z t   0   1 z t 1   t
cov( et ,  t )  0
E[ rme | z t 1 ]   me
where zt is a predictive variable realized at time t. The last equality
above implies that the market expected return does not vary with the
predictive variable. Rather it is constant through time.
You are currently at time T and you would like to form conditional
moments for time T+H for some H>0. More specifically, express
general expressions for the conditional mean E[rTe K | zT ]
as a function of zT , H , me ,  ,  0 , 1 ,  0 ,  1
Question number 8 (10 points)
Use the excel file momentum_crash to answer this question. The file
contains the following columns:
Columns 1 and 2 – the date (year and month)
Column 3 – market state – a dummy variable taking the value 1 if the past
24 month market return is negative. Zero otherwise.
Column 4 – Market volatility
Column 5 – Market illiquidity
Column 6 – Future momentum payoff (over the next month)
Part I
a. Run time series regression of momentum payoff on the market state
and report significance of the market state variable in predicting
momentum payoffs.
b. Same as above but the explanatory variable is market volatility.
c. Same as above but the explanatory variable is market illiquidity.
d. Same as above but using all three variables.
Describe which of the variables could predict momentum payoff as well as
which is the strongest predictor?
Part II
Momentum is often facing market crashes. A good paper on market crashes
is that of Daniel and Moskowitz (DM) (see paper attached). Describe the
strength of the above specified three variables to predict momentum crashes.
You can use some methods advocated by DM, or you can design your own
intuition about the proper examination.