MBA 570x
Homework 1 – Due 9/24/2014
Solution
Individual works:
1. Questions related to Chapter 11, ST
Why do you think is a fund of funds market for hedge funds, but not for mutual funds?
Answer: Investors can inexpensively recreate diversification by investing in various mutual
funds. Most investors do not have the amount of capital needed to invest in multiple hedge funds
to achieve diversification. In addition, while mutual funds are very transparent about their
investment strategies (which are relatively straight forward) and there is an entire industry
dedicated to providing investors with analysis of mutual funds, hedge funds are much more
opaque about their investment activities and their strategies are more esoteric. As a result, the
cost/reward ratio for a fund of hedge funds is much higher than for a fund of mutual funds.
2. Questions related to Chapter 12:
a. During the height of the financial crisis in late 2008, the yield curve flattened and the yield
on the 30-year Treasury bond reached a then-all-time low of 2.52%. As a hedge fund manager,
suppose you think the market has overreacted and will eventually correct itself, leading to a
steepening in the yield curve. What trades might you execute in a long/short strategy to take
advantage of the situation?
Answer: Short the 30-year Treasury; long the 1-year Treasury.
b. Why did convertible arbitrage strategies perform so poorly in 2008?
In this trade, hedge funds go long the convertible, and short the underlying stock. When the
temporary short-ban was instituted in late 2008, traders could no longer hedge their position by
shorting stock. As the stock market fell, convertible values dropped, and the inability to manage
short positions compounded the loss.
3.1 Chapter 7 (BKM): 5
Suppose that your client decides to invest in your portfolio a proportion y of the total investment
budget so that the overall portfolio will have an expected rate of return of 16%.
a. What is the proportion y?
y*0.18 + (1-y)*0.08 = 0.16. Solve the above: y = 0.8
b. What are your client’s investment proportion in your three stocks and the T-bill fun?
Stock 1: 0.8*0.25 = 0.2
Stock 2: 0.8*0.32 = 0.26
Stock 3: 0.8*0.43 = 0.34
Risk free asset: 1-0.8 = 0.2
c. What is the standard deviation of the rate of return on your client’s portfolio?
0.28*0.8 = 22.40%
3.2 Chapter 7 (BKM): 7
You client’s degree of risk aversion is A=3.5.
a. What proportion, y of the total investment should be invested in your fund?
y* = (0.18 – 0.08)/(0.01*3.5*0.282) = 0.36
b. What is the expected value and standard deviation of the rate of return on your client’s
optimized portfolio?
0.36*18% + 0.64*0.08 = 11.60%
Standard deviation of the portfolio = 0.36*28 = 10.08%
4. Chapter 8 (BKM): 1, 6
1. A pension fund manager is considering three mutual funds. The first is a stock fund, the second
is a long-term government and corporate bond fund, and the third is a T-bill money fund that
yields a rate of 8%. The probability distribution of the risky funds is as the following:
Stock fund (S)
Expected return
20% (E( r S ))
Bond fund (B)
12% (E( r B ))
standard deviation
30% (  S )
15% (  B )
What are the investment proportions in the minimum-variance portfolio of the two risky funds,
and what is the expected value and standard deviation of its rate of return?
w min 

B

2
B
2
E

 cov( r B , r E )
 2 cov( r B , r E )
2
E
(page 214, footnote 4)
Using the above information, we find the weight of the minimum variance portfolio in the stock
fund is 0.17 and the weight in the bond fund is 0.83.
E(rp) = 12*0.83 + 20*0.17 = 13.36%

2
p
 w B
2
2
B
 wS
2
2
S
 2 w s w B  s 
B
= 193.71
Thus, σp=13.92%
6. You require that your portfolio yield an expected return of 14%, and that it be efficient on the
best feasible CAL.
a. what is the standard deviation of your portfolio?
Obtaining the weights for optimal risky portfolio based equation (8.7) on page 221.
You get weight in bond is 0.548; and the weight in stocks is 0.452.
2
2
2
2
2
 p  w B  B  w S  S  2 w s w B  s  B =272.74
We have: σp=16.514%
b. what is the proportion invested in the T-bill fund and each of the two risky funds?
E(rp) = 0.548*12 + 0.452*20 = 15.6%;
15.6% = 8%*w + 15.6%*(1-w)
w: weight in the risk free asset = 0.21.
the rest in stock and bond funds
5.1 Chapter 24 (BKM): 3
A manager buys three shares of stock today, and then sells one of those shares each year for the
next three years. His actions and the price history of the stock are summarized below. The stock
pays no dividends:
Time
0
1
2
3
Price
90
100
100
100
Action
Buy 3 shares
Sell 1 share
Sell 1 share
Sell 1 share
a. calculate the time-weighted geometric average return on this “portfolio”.
b. Calculate the time-weighted arithmetic average return on this portfolio.
c. Calculate the dollar-weighted average return on this portfolio.
Solution:
a.
Time
0
1
2
3
Cash flow
3×(–$90) = –$270
$100
$100
$100
Holding period return
(100–90)/90 = 11.11%
0%
0%
Time-weighted geometric average rate of return =
(1.1111 × 1.0 × 1.0)1/3 – 1 = 0.0357 = 3.57%
b.
Time-weighted arithmetic average rate of return = (11.11% + 0 + 0)/3 = 3.70%
The arithmetic average is always greater than or equal to the geometric average; the
greater the dispersion, the greater the difference.
c.
Dollar-weighted average rate of return = IRR = 5.46%
[Using a financial calculator, enter: n = 3, PV = –270, FV = 0, PMT = 100. Then
compute the interest rate, or use the CF0=−300, CF1=100, F1=3, then compute IRR].
The IRR exceeds the other averages because the investment fund was the largest
when the highest return occurred.
5.2 Chapter 24 (BKM): 4
Based on current dividend yields and expected capital gains, the expected rates of return on
portfolios A and B are 12% and 16%, respectively. The beta of A is 0.7, while that of B is 1.4.
The T-bill rate is currently 5%, whereas the expected rate of return of the S&P 500 index is 13%.
The standard deviation of portfolio A is 12% annually, that of B is 31%, and that of the S&P 500
index is 18%.
a. If you currently hold a market-index portfolio, would you choose add either of these
portfolio to your holdings?
Compare their alpha (since alpha is relevant measure when the portfolio is already well
diversified). A is greater. (1.4% versus -0.2%)
b. If instead you could invest only in T-bills and one of these portfolio which would you
choose.
Comparing Sharpe ratios (since the Sharpe ratio is relevant measure when the portfolio is
not diversified), A is greater. (0.583 versus 0.355)
5.3 Chapter 24 (BKM): 5
Consider the two (excess return) index-model regression results for stocks A and B. the risk-free
rate over the period was 6%, and the market’s average return was 14%. Performance is
measured using an index model regression on excess returns.
a. Compute the statistics for each stock
A
B
Alpha
1%
2%
Appraisal ratio
1/10.3
2/19.1
Sharpe ratio
10.6/21.6
8.4/24.9
Treynor measure
10.6/1.2
8.4/0.8
b. Which stock is the best choice under the following circumstances?
i.
This is the only risk asset to be held by the investor – A, based on Sharpe ratio
ii.
The stock is to mixed with the rest of the investor’s portfolio, currently composed
solely of holding in the market index fund – B, based on alpha.
iii.
This is one of many stocks that the investor is analyzing to form an actively
managed stock portfolio. – A, based on the Treynor measure.
5.4 Chapter 24 (BKM): 7
Consider the following information regarding the performance of a money manager in a recent
month. The table represents the actual return of each sector of the manager’s portfolio in column
1, the fraction of the portfolio allocated to each sector in column 2, the benchmark or neutral
sector allocations in column 3, and the returns of sector indices in column 4
Equity
Bonds
Cash
Actual ret
2%
1%
0.5
actual wgt
0.70
0.20
0.10
a.
Bogey:
Actual:
b.
Security Selection:
Market
Equity
Bonds
Cash
benchmark wgt
0.60
0.30
0.10
index ret
2.5% (S&P)
1.2% (Salomon index)
0.5
(0.60 × 2.5%) + (0.30 × 1.2%) + (0.10 × 0.5%) = 1.91%
(0.70 × 2.0%) + (0.20 × 1.0%) + (0.10 × 0.5%) = 1.65%
Underperformance:
0.26%
(1)
Differential return
within market
(Manager – index)
(2)
(3) = (1) × (2)
Manager's
portfolio weight
Contribution to
performance
–0.5%
0.70
–0.2%
0.20
0.0%
0.10
Contribution of security selection:
−0.35%
–0.04%
0.00%
−0.39%
c.
Asset Allocation:
Market
(1)
Excess weight
(Manager – benchmark)
Equity
Bonds
Cash
0.10%
2.5%
–0.10%
1.2%
0.00%
0.5%
Contribution of asset allocation:
Summary:
Security selection
Asset allocation
Excess performance
(2)
Index
Return
–0.39%
0.13%
–0.26%
(3) = (1) × (2)
Contribution to
performance
0.25%
–0.12%
0.00%
0.13%