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

	E(Ri)	Stddev.i	Weightsi
Stock D	15%	20%	Wd
Stock E	20%	40%	We
Risk-free rate	2%

	E(Ri)	Std devi
Portfolio P	16.20%	18.77%
Portfolio Q	16%	19.23%

For Portfolio P

16.2% = Wd*15% + We*20%

0.162 = 0.15*Wd + 0.20*(1-Wd)

0.162 = 0.20 - 0.05Wd

0.05Wd = 0.20 - 0.162

Wd = 0.76

We = 1 - Wd = 1-0.76 = 0.24

For Portfolio P weights are Wd = 76% , We = 24%

Similarly For portfolio Q,

0.16 =0.15*Wd + 0.20*(1-Wd)

0.05Wd = 0.20 - 0.16 = 0.04

Wd = 80% We = 1-0.80 = 20%

For Portfolio Q weights are Wd = 80% , We = 20%

Tangency portfolio is the one which has highest sharpe ratio.

Sharpe ratio of a portfolio P = E(Rp) - Rf/ std devp   = (16.2% - 2%) / 18.77% = 0.76

Sharpe ratio of a portfolio Q = E(Rq) - Rf/ std devq   = (16% - 2%) / 19.23% = 0.73  

As sharpe ratio of Portfolio P is greater than sharpe ratio of portfolio Q, then Portfolio P is the Tangency portfolio which has weights Wd = 76% , We = 24%