Question

Consider two stocks, D and E, with expected returns and volatilities given by E[rD]=15%, sD=20%, E[rE]=20%, sE=40%. The riskless rate is 2%. Consider now two portfolios P and Q with the following expected returns and standard deviations: E[rP]=16.2%, sP=18.77% and E[rQ]=16%, sQ=19.23%. These portfolios are formed by investing in stocks D and E and the riskless asset. It is known that one of these portfolios is a tangency portfolio for the efficient frontier constructed by investing in stocks D and E and the riskless asset. Determine the tangency portfolio and its portfolio weights.

Answer 1

Based on trail and error the Tangency portfolio is the Q as the extra 0.2 in P cannot be achieved as the standard deviation of E is higher than D and it would result in the value of SD going up rather than down, hence the risk free rate would have to be a part of the portfolio mix.

When trying to arrive at the weights we use trial and error method. The weights are 0.8 in D and 0.2 in E

For return - 0.8*15% + 0.2*20% = 16%

For SD - SQRT((wD^2)*(sD1^2))+((wE^2)*(sE^2))+(2*wD*wE*sE*sD*1)

SQRT((0.8^2)*(0.2^2))+((0.2^2)*(0.4^2))+(2*0.8*0.2*0.4*0.2*1) = 19.23%