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 as well as myself at

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. This quantity 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 isquite 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 logof gross returns NOTlog excessreturns) of A, B, and M is of the MULTIVARIATENormal form given by

where, ,,,,, 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

Comments:

  1. Notice that the upper case letters denote gross returns while the lower case letters denote continuously compounded returns. For instance, .
  2. 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.
  3. 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)

  1. 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?

  1. 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.
  2. Design a non-parametric test based on failure rate to test whether IV is constant across maturities.
  3. 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 BS 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 rand 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:

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:

  1. Monthly shortfall probability when shortfall is defined as negative return.
  2. VaR – The threshold level such that 5% of the returns are below that level.
  3. Expected Shortfall
  4. Monthly Target Semi Variance (target in the risk free rate).
  5. Downside beta (focusing on the first out of the three measures)
  6. 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)

  1. For the above 25 portfolios compute the J1, J2, J3, and J4 statistics in order to test the one factor CAPM.
  2. 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 to the difference in Sharpe ratios.

Question number 7 (10 points)

Stock returns evolve according to the conditional factor model

where 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 timeT+H for some H>0. More specifically, express general expressions for the conditional mean

as a function of

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

  1. Run time series regression of momentum payoff on the market state and report significance of the market state variable in predicting momentum payoffs.
  2. Same as above but the explanatory variable is market volatility.
  3. Same as above but the explanatory variable is market illiquidity.
  4. 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.