1. Portfolio Management Grenoble Ecole de Management MSc Finance 2011 Exercises chapter 3

2. How to determine optimal weights 2 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 ? Global Minimum Variance Portfolio AABBSd-devReturns 0%100%27,00%12,0% 10%90%25,07%11,4% 20%80%23,28%10,8% 30%70%21,63%10,2% 40%60%20,19%9,6% 50% 18,99%9,0% 60%40%18,07%8,4% 70%30%17,49%7,8% 80%20%17,28%7,2% 90%10%17,46%6,6% 100%0%18,00%6,0%

3. How to determine optimal weights 3 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 ? Global Minimum Variance Portfolio AABBSd-devReturns 0%100%27,00%12,0% 10%90%23,81%11,4% 20%80%20,77%10,8% 30%70%17,98%10,2% 40%60%15,57%9,6% 50% 13,72%9,0% 60%40%12,70%8,4% 70%30%12,70%7,8% 80%20%13,73%7,2% 90%10%15,58%6,6% 100%0%18,00%6,0%

4. Lending and Borrowing 4 S has an expected return of 15% and a Sd-dev of 25%. T-bill offer a risk- free rate (r f ) 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 r f 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. 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 risk- free rate to have a portfolio with an expected return of 19% ?

6. 6 CAL calculations 1) 2) 3)

7. 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 8 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 ? β AA = 0.72; β BB = 1.27 ; β CC = 1.05 what can you say in term of risk ? β AA < β CC < β BB what are the expected returns ? R AA = 6.61%; R BB = 9.29 %; R CC = 8.28% what is the β of a portfolio P mixing 50% of AA and 50% of BB ? Β p = 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 9 stocksE(R i )βiβi A7%0,5 B9%1 C17%1,5 We are in a 3-asset world: A, B, C with the following characteristics. 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%. β

10. APT calculations 10 We are in a 3-asset world: A, B, C with the following characteristics. 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 stocksE(R i )βiβi A7%0,5 B15%0,8 C17%1,5