Can you please help. I am stuck on this problem, (in my textbook I see there is a formula given which also take into account the correlation coefficient & Beta but in this problem none of this is given and it looks like I should be calculating the SD independently of this relationship as the two stocks are not part of a portfolio). Thank you in advance for your help!! If you also direct me to material that can help me understand this, it would be much appreciated. The expected return on the market is 12% and the risk-free rate is 7%. The standard deviation of the return on the market is 15%. One investor creates a portfolio on the efficient frontier with an expected return of 10%. Another creates a portfolio on the efficient frontier with an expected return of 20%. What is the standard deviation of the returns of the two portfolios?
I’m not sure but I guess this is how you should get it.
Since all three portfolios are on the efficent frontier, then they all have the same risk-adjusted returns.
Therefore, the risk premium (12%-7%) per unit risk (15%) of the market is 1/3. I guess that’s the Sharpe ratio though. which should hold constant throughout all portfolios on the frontier if I remember correctly.
That gives you 3%* 3 = 9% for the first portfolio, and 13% * 3 = 39% for the second portfolio.
That will hold for all portfolios on the Capital Market Line (CML), which is, in effect, the efficient frontier when you include the risk-free asset. It’s not true for an efficient frontier comprising only risky assets.
20% – 7% = _ 13% _, not 12%.
The first thing you need to do is calculate the weights of the risk-free asset and the market portfolio for each of the investors’ portfolio; you do this by writing the portfolio’s return as a weighted average of the returns of the risk-free asset and the market portfolio, then solving for the weights. If we let w_rf_ be the weight on the risk-free asset and w_m_ the weight on the market portfolio, then:
- For the first investor, w_rf_(7%) + (1 – w_rf_)(12%) = 10%, so w_rf_ = 0.4 and w_m_ = 0.6.
- For the second investor, w_rf_(7%) + (1 – w_rf_)(12%) = 20%, so w_rf_ = -1.6 and w_m_ = 2.6.
With the weights, and the fact that σ_rf_ = 0, we have:
- For the first investor, σ²_port_ = w²_rf_σ_²rf_ + w²_m_σ_²m_ + 2w_rf_w_m_σ_rfσm_ρ(rf,m) = 0.4²(0) + 0.6²15%² +2(0.4)(0.6)(0)(15%)ρ(rf,m) = 0.0081, so σ_port_ = √0.0081 = 0.09 = 9%.
- For the second investor, σ²_port_ = (-1.6)²(0) + 2.6²15%² +2(-1.6)(2.6)(0)(15%)ρ(rf,m) = 0.1521, so σ_port_ = √0.1521 = 0.39 = 39%.
Oops 