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Exercise 1 [30 points] (all parts are equally weighted)
Answer succinctly the following questions.
1. Why do we refer to the hedge fund manager’s compensation as a “free
option”? Why does it encourage risk taking?
Because the manager at the end gets (ignoring the mgmt fee) max{y(NAV(1)-
NAV(0)),0} where y is the incentive fee, and NAV(t) is the year-t net asset
value. This is exactly the value of an at-the-money call option on a y-piece of
the NAV. The manager does not pay for this option hence the term free. The
manager can control the volatility of NAV with his actions. He knows that the
value of his option increases with volatility and hence the compensation
induces the manager to take in more risk.
2. What is the hurdle rate and how do we use it?
The hurdle rate is the minimum average return of an asset Y to be added to a
portfolio P from a risk budgeting perspective. It is given by HR(Y)=Rf + (E(RP)-
Rf)*rhoYP*sigmaY/sigmaP.
3. Are alternative assets really “alternative”? Explain.
Alternative assets are not really alternative, they use the same core assets
(i.e. debt and equity) but in an alternative way, e.g. with the use of leverage,
derivatives, and short-selling.
4. What are the main categories of hedge fund strategies?
Market directional, Event Driven, Opportunistic.
5. What is the difference between the Sharpe Ratio and Treynor’s measure?
The Sharpe ratio divides by the total risk (standard deviation) while Treynor’s
y the systematic risk (market beta).
Exercise 2 [20 points] (all parts are equally weighted)
1. You are an investor contemplating investing with hedge fund XYZ. The
manager tells you that he has an exposure to the market (say the S&P 500) of
eta=0.25, and he yields an alpha=0.04 on top of that. If you think there is
only market risk in his strategy what are the expected returns if the market risk
premium is 4%?
Expected return is 4%+0.25*4%=5%.
2. You are thinking of diversifying by investing also in fund ABC. ABC has
alpha=-0.01 (minus 1%), and market beta=-0.25 (minus 25%). If you invest
50% of your wealth in each fund, what is the resulting alpha and market beta
of your portfolio?
alphaP=0.5*4%+0.5*(-1%)=1.5%, betaP =0.
3. What is an alternative way to get the same beta as the one of your portfolio in
2, but a higher alpha?
An alternative way would be to invest with XYZ and for each unit invested sell
0.25 forwards in the S&P 500 to hedge completely the market risk. This leads
to a portfolio with alpha=4% and beta=0, i.e. much better than the one in 2.
4. Assume the end of year returns before fees of the XYZ fund are +30% and
the ones of the ABC fund are -30%, so that the average between the two is
zero. Again assume that you have 50% of your wealth in each fund and also
that the compensation scheme is 0/20, i.e. no management fee and 20%
incentive fee, for both funds. What is your actual return as an investor after
fees?
After fee returns are 0.5*30%*80%+0.5*(-30%)=12%-15%=-3%.
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Exercise 3 [30 points] (all parts are equally weighted)
Company A owns company B. B accounts for 40% of A’s consolidated profits.
Today, B’s share price is 50, and A’s own profits (i.e. without B) are 40 per share. A
trades for 45.
Throughout assume that the share price of B is its fair price. The annual risk-free
ate is 6%, and short-selling profits yield 4% annually risk-free.
1. Is A “fairly priced” today? What would you do to profit if you thought A is going
to be fairly priced in the future? How would you devise a strategy so that it is
immune to the movements in the share price of B?
A’s fair price is 40+40%*50=60>45, so A is undervalued. Hence we buy A,
and sell B. In order to hedge for movement in B, we are going to buy 2.5
shares of A, for each one of B.
In what follows assume you fix the same strategy you determined in 1.
2. Three months from today, B’s share price is 85, A’s own profits are 50, and
A’s share price is 55. What is your return in this three-month period? What is
your return if you bo
owed half of the capital you used initially?
Gain on long of A 2.5×[55-45]=25
Gain on short of B 1×(50-85)=-35
Short rebate 4%×(3/12)×50=0.5
Total -9.5
Return -9.5/112.5=-8.44%
Return w/ 2:1
leverage at 6%
XXXXXXXXXX)/56.25=-18.39%!
3. Six months from today, B’s share price is 75, A’s own profits are 55, and A is
fairly priced in the market. What is your return in this six-month period? What
is your return if you bo
owed half of the capital you used initially?
Gain on long of A 2.5×[(55+0.4*75)-45]=100
Gain on short of B 1×(50-75)=-25
Short rebate 4%×(6/12)×50=1
Total 76
Return 76/112.5=67.55%
Return w/ 2:1
leverage at 6%
XXXXXXXXXX)/56.25=132.11%!
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4. Six months from today, B’s share price is 45, A’s own profits are 55, and A is
fairly priced in the market. What is your return in this six-month period? What
is your return if you bo
owed half of your capital? Are the returns the same as
in 3.? Why yes or why not?
Gain on long of A 2.5×[(55+0.4*45)-45]=70
Gain on short of B 1×(50-45)=5
Short rebate 4%×(6/12)×50=1
Total 76
Return 76/112.5=67.55%
Return w/ 2:1
leverage at 6%
XXXXXXXXXX)/56.25=132.11%!
Since we picked the hedging ratio to counteract movements in the share price
of B, the returns are the same when A is fairly valued.
5. Imagine you are in the situation of part 2. If the maximum three-month loss
you can incur is 25% before investors redeem their money, what is the
maximum amount of leverage you should have taken initially to avoid
edemptions?
The maximum amount of leverage K solves the following equation: (-9.5-
6%*3/12*112.5*K)/((1-K)*112.5)=-25%, which yields K=0.625.
Exercise 4 [20 points] (all parts are equally weighted)
You are considering investing into a fund specializing in mortgage backed securities
(MBS) a
itrage.
1. You see the following statistics
All figures
are monthly
Average
Return
Standard
Deviation
Skewness Kurtosis
MBS
a
itrage
0.68% 1.23% XXXXXXXXXX
Explain what the above statistics tell you about the return distribution. [You
may use a graph if you wish.]
The average return is positive, the distribution of returns has a downward bias
and really heavy tails.
2. You want to check whether the strategy is exposed to market risk and interest
ate risk by running a regression. What do you use to proxy for market risk
and what do you use to proxy for interest rate risk?
Proxy for market risk is S&P 500 index, and proxy for interest rate risk is the
yield of 30-year T-bills (minus the return of short-term T-bills).
3. You run the regression and you find the following
alpha market risk beta interest rate risk
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eta
value XXXXXXXXXX
t-statistic XXXXXXXXXX
Interpret the coefficient estimates and what they imply of the manager’s
hedging strategy. Are the results conclusive of managerial “skill”?
Both risk betas are insignificant, there is no market risk in MBS a
itrage so
the zero market risk beta was to be expected. The zero interest rate risk beta
is implying that the manager is hedging that risk with some swap agreement.
Alpha is positive and significant. However, we cannot conclude skill since
there might be more risks we omitted in this regression, e.g. credit risk.
4. What is the main risk with MBS a
itrage? How does it affect the return
distribution?
The main risk is the prepayment option risk, which is very hard to quantify,
and hard to hedge, it is causing the huge kurtosis in the return distribution.