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Portfolio Management
Pr Philippe Dupuy
MSc Finance.
Closed book. Calculators are allowed, but not PC.
Internet connexion, and all communication system between students are forbidden. This would
generate a zero to the test.
Question 1 (3.5 points)
Please, answer the following questions:
1. The investment policy statement enables one to know the fund manager’s objectives in
terms of risk?
a. Yes
b. No
2. The slope of the SML is always positive?
a. Yes
b. No
3. The risk of a portfolio might be lower than the risk of the less risky asset in the portfolio. Is
it true?
a. Yes
b. No
4. In order driven markets:
a. There is no intermediation by a dealer
b. There is intermediation by a dealer
5. Benchmarks might be equally weighted?
a. Yes
b. No
6. The concept of alpha that measures managers’ skills is directly derived from the CAPM?
a. Yes
b. No
7. The Bid-Ask spread might increase due to poor liquidity in the market?
a. Yes
b. No
Question 2 (1.5 points).
Please explain, in 10 lines maximum, why the expected return of an asset is more useful for
portfolio management than its expected price.
Returns are distributed as the Normal distribution or close to the normal distribution. It is not the
case for prices…
Exercise 1 (3 points)
A portfolio has recorded the following monthly returns: 6% in month 1 then 8% in month 2 and
finally -2% in month 3. The portfolio has a net asset value of 100 at the beginning of month 1.
During the same months, the benchmark has recorded returns of 2% in month 1, 5% in month 2 and
1% in month 3.
What is the final value of the portfolio at the end of month 3?
(1+6%)*(1+8%)*(1+(-2%))*100 = 112.1904
What is the non-annualized geometric mean return over these three months?
Sqr/3 (1+6%)*(1+8%)+(1+(-2%))-1 = 3.908%
What is the non-annualized arithmetic mean return over these three months?
(6%+8%+(-2%))/3 = 4%
What is the annualized mean over(under)performance of the portfolio relatively to the benchmark?
12*((6%-2%)+(8%-5%)+((-2%)-1%))/3=16%
What are the tracking error and active return of this portfolio?
Active return = 16%
Tracking error = 0.1311if you use 1/(n-1) or 0.1070 if you use 1/n
What is the information ratio of this portfolio?
16%/13.11% = 1.2199 or 16%/10.7% = 1.4941
Exercice 2 (5 points)
You are a portfolio manager. The economic research team has given you the following expectations
for the risk free rate, 2%, and the market return 8% with a variance of 18%. You can invest in a
stock AA which has a covariance to the market portfolio of 0.56.
1) Graph the CML and situate the stock. What can you say about the behavior of this stock
relatively to the market return?
Beta = 0.56/0.18 = 3.11. The stock amplifies the return of the market.
2) Calculate the Adjusted beta. What should be the expected return of this stock now? Adjusted
beta = 0.33 + 0.66* 3.11 = 2.39. Expected return 2%+2.39*(8%-2%) = 16.36%
3) The P/E and Price/Book ratio for this stock are above the market P/E and Price/Book.
Define these ratios and conclude for the expected beta of the stock. Would you increase or
decrease your position on this stock?
The beta might decrease because these stocks are expensive. I might underweight them in a
portfolio.
4) A customer wants to invest 40% in the risk free asset and 60% in the stock. What are the
return and the Beta of this portfolio?
As the beta of the cash is zero, beta = 60%*2.39 = 1.195; ER = 40%*2%+60%*16.36% =
10.62%
5) A second stock, BB, has a beta of 1.6 and an expected return of 25%. What should you do
using only the market portfolio? What are the name and the return of this operation not
accounting for cash cost or return?
One solution is X1*1.6-(1-x1)*1=0 x1= (1/2.6)=0.3846 I buy 38.46% of stock BB and sell
61.54% of the market index. Return = 4.69%. The name of this operation is arbitrage.
Exercise 3 (5 points)
A well diversified portfolio has recorded a non-annualized one-month return of 3.21% while the
benchmark has recorded a non-annualized one-month return of 3.15%. There are three assets in this
portfolio. Please, using the weights and returns in the table attribute performances according to the
three usual effects. Rebalance the portfolio as to maintain a constant proportion of each asset in the
portfolio at the end of month 1. Provide the % to be bought or sold for each asset relatively to the
value of the portfolio at the end of month 1.
Sectors
Industrials
Financial
Technology
Total return
Portfolio
Weights in %
35%
23%
42%
100%
Returns
2,00%
3,00%
7,00%
4,33%
Benchmark
Weights in % Returns
24%
1,00%
39%
5,00%
37%
4,00%
100%
3,67%
Answers
Sectors
Pure allocation Sector selection
Industrials
-0,29%
0,24%
Pharmaceutical
-0,21%
-0,78%
Technology
0,02%
1,11%
Total
-0,49%
0,57%
Mix
0,11%
0,32%
0,15%
0,58%
Total
0,06%
-0,67%
1,28%
0,66%
Rebalancing
Weights
35%
23%
42%
100%
Value 1-m
35,70
23,69
44,94
104
Rebased
34,22%
22,71%
43,07%
100%
Delta
0,78%
0,29%
-1,07%
0%
Exercise 4 (2 points)
You buy an equity that has a beta of 1.6 and a bond with a modified duration of 4.2. What are the
beta and modified duration of a balanced portfolio mixing 50% of each asset.
Beta is 50%*1.6 = 0.8 and modified duration is 50%*4.2 = 2.1.
On top of this portfolio you add a long-short position (arbitrage). You sell Equity A with beta of 0.8
and buy equity B with beta of 1.2 for 10% of the asset value of the portfolio. What is the new beta
of the fund? And what is the cash position of this fund?
10% * 1.2 = +0.12 and 10% * 0.8 = 0.08 ; the beta of the position is then 0.12-0.08 = 0.04 and the
beta of the portfolio = Beta fund + Beta LS = 0.8+0.04 = 0.84. The cash position is unchanged.
If you want to introduce a zero-beta position, what is the cash position of the fund? There is no one
unique answer.
For instance if I sell -15% of the asset with beta 0.8 and buy 10% of the asset with beta 1.2, I build
a zero-beta position. As I sell 15% but buy only 10% I have a positive cash position of 5%.