Portfolio Management
Grenoble Ecole de Management
MSc Finance
2011 Exercises chapter 3
2
How to determine optimal weights
We are in a two-asset world: stock AA and stock BB. Stock AA has a mean
return of 6% and a standard deviation of 18%. Stock BB has a mean return of
12% and a standard deviation of 27%. Correlation is 0.4. Graph the efficient
frontier and point the Global Minimum Variance portfolio.
Your customer, Miss Jones, would like a portfolio with a return of 9%. Which
portfolio (weights) do you propose ? What do you say about risk to Miss Jones
? Mr Jones would like 11% of return but with risk below 20%. Which portfolio
(weights) do you propose ?
13%
AA
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
BB
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Sd-dev
27,00%
25,07%
23,28%
21,63%
20,19%
18,99%
18,07%
17,49%
17,28%
17,46%
18,00%
Returns
12,0%
11,4%
10,8%
10,2%
9,6%
9,0%
8,4%
7,8%
7,2%
6,6%
6,0%
12%
11%
10%
9%
8%
Miss Jones 50%-50%
portfolio with return of 9%
and volatility of 18.99%
Mr Jones' objectives
must be revised,
volatility cannot be
below 20% for a return
of 11%
7%
6%
5%
Non efficient portfolios
4%
14.00% 16.00% 18.00% 20.00% 22.00% 24.00% 26.00% 28.00%
Global Minimum Variance Portfolio
3
How to determine optimal weights
We are in a two-asset world: stock AA and stock BB. Stock AA has a mean
return of 6% and a standard deviation of 18%. Stock BB has a mean return of
12% and a standard deviation of 27%. Correlation is -0.3. Graph the efficient
frontier and point the Global Minimum Variance portfolio.
Your customer, Miss Jones, would like a portfolio with a return of 9%. Which
portfolio (weights) do you propose ? What do you say about risk to Miss Jones
? Mr Jones would like 11% of return but with risk below 20%. Which portfolio
(weights) do you propose ?
13%
AA
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
BB
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Sd-dev
27,00%
23,81%
20,77%
17,98%
15,57%
13,72%
12,70%
12,70%
13,73%
15,58%
18,00%
Returns
12,0%
11,4%
10,8%
10,2%
9,6%
9,0%
8,4%
7,8%
7,2%
6,6%
6,0%
Global Minimum Variance Portfolio
12%
11%
Miss Jones 50%-50% portfolio
with return of 9% and volatility
of 13.7%
Mr Jones'
objectives must
be revised,
volatility cannot
be below 20%
for a return of
11%
10%
9%
8%
7%
6%
Non efficient portfolios
5%
4%
10.00%
15.00%
20.00%
25.00%
30.00%
4
Lending and Borrowing
S has an expected return of 15% and a Sd-dev of 25%. T-bill offer a riskfree rate (rf) of 5%. If you invest half your money in T-bill and half in S.
What is the expected return of your portfolio ? Its st-dev ?
Then you borrow at rf an amount initial to your original wealth and you
invest everything in portfolio S. What is the expected return of your
portfolio ? Its st-dev ?
5
CAL calculations
The risk-free rate is 5%, the expected return to an investor’s tangency portfolio is
15% and the St-dev of the tangency portfolio is 25%.
1) How much return does this investor demand in order to take on an extra unit
of risk?
2) The investors wants a portfolio sd-dev of 10%. Which are the weights of the
risk-free rate and the tangency portfolio in his own portfolio ?
3) The investor wants to put 40% of the portfolio in the risk free asset. What is
the return and the sd-dev of this portfolio ?
4) What return can expect the investor for a portfolio with sd-dev of 35% ?
5) If the investor has EUR10 million to invest, how much she borrow at the riskfree rate to have a portfolio with an expected return of 19% ?
6
CAL calculations
1)
2)
3)
7
CAL calculations
4)
5) The investor must borrow EUR 4 million at the risk-free rate to increase
the holdings of the tangency portfolio to EUR 14 million.
8
CAPM calculations
• the market has an expected return of 8% and a variance of returns of 18%. The
risk-free rate stands at 3.0%.
• there are 3 assets, AA with covariance with the market of 0.130; BB with
covariance with the market of 0.230 ; CC with covariance with the market of 0.190.
• what are the β of these assets ? βAA = 0.72; βBB = 1.27 ; βCC = 1.05
• what can you say in term of risk ? βAA < βCC < βBB
• what are the expected returns ? RAA = 6.61%; RBB = 9.29 %; RCC = 8.28%
• what is the β of a portfolio P mixing 50% of AA and 50% of BB ? Βp = 1
• what is the marginal risk to add CC to a portfolio that mimics the market ? 0.37%
9
APT for a single factor representation
We are in a 3-asset world: A, B, C with the following characteristics.
18%
16%
14%
stocks
E(Ri)
βi
12%
A
7%
0,5
10%
B
9%
1
C
17%
1,5
AC
8%
6%
4%
B
A
C
cost-free
opportunity
2%
0%
0.5
1
1.5
β
For λ = 0.66 the portfolio AC has a β of 1 and an expected return of
10.4%. With the same β, stock B has a return of 9%.
Therefore one can take profit of this situation by selling EUR 100 of stock B
and buying the equivalent of portfolio AC. This is an arbitrage because the
operation is cost-free. The return is EUR 1.4 or 1.4%.
10
APT calculations
We are in a 3-asset world: A, B, C with the following characteristics.
18%
stocks
E(Ri)
βi
A
7%
0,5
B
15%
0,8
C
17%
1,5
C
B
16%
14%
12%
10%
8%
AC
A
cost-free
opportunity
6%
4%
0.5
0.8
1.5
For λ = 0.7 the portfolio AC has a β of 0,8 and an expected return of
10.0%. With the same β, stock B has a return of 15%.
Therefore one can take profit of this situation by buying EUR 100 of stock
B and selling the equivalent of portfolio AC. This is an arbitrage because
the operation is cost-free. The return is EUR 5 or 5%.
β