Question

Suppose the universe of available securities include only two risky stock funds, a and b, and T-bills.

() The correlation between fund a and fund b is 0.20 (i.e., = . ).

Fund a	12%	20%
Fund b	20%	40%
T-bills	2%	0%

(2) If you invest in the two risky funds, what is the best “reward-to-volatility ratio” you can achieve? (Hint: Compute the weights and for the optimal risky portfolio ; then use the weights to calculate E(r), ∂, and then finally the Sharpe ratio.

Answer 1

Please refer to below spreadsheet for calculation and answer. Cell reference also provided.

A В C D E F 1 (t-bills) 2 Fund-a Fund-b Expected return Std. Deviation of return 3 12.00% 20.00% 2.00% 40.00% 4 20.00% Correlation 0.20 6 7 Optimum risky Portfolio (fund-a+fund-b) 8 Portfolio invested in fund-a 0.7009 Portfolio invested in fund-b 0.2991 10 14.39% 20.17% Expected return 11 Std. deviation of return 12 0.6145 Sharpe ratio 13 14 15 16

Cell reference -

A В C E (t-bills) 0.02 Fund-a Fund-b 0.2 0.4 Expected return 0.12 0.2 Std. Deviation of return Correlation 0.2 Optimum risky Portfolio (fund-a fund-b) |-{(C3-E3)*D4^2-(D3-E3)*C5*C4*D4)/((C3-E3)*D4^2+(D3-E3)*C4^2-((C3-E3)+(D3-E3))* C5*C4*D4) - 1-C9 =C9*C3+C10* D3 -SQRT((C9*C4)^2+(C10*D4)^2+2*C9*C10*C4*D4*C5) |нс11-Е3)/с12 Portfolio invested in fund-a Portfolio invested in fund-b Expected return Std. deviation of return Sharpe ratio 15

Please note:

We can calculate optimal portfolio weight with following formula

(E(RB) Rf)o3 - (E(Rs) - Rf)PBSOBOS E(RB) Rf)a+ E(Rs) - Rf)og- (E(RB) Rf E(Rs) - Rf)PBSSOBOS WB

(1 WB Ws )

where,

B = Bond

S = Stock

E(R) = Expected Return

Standard deviation

Rf = risk free rate

= Correlation Bond and stock

BS

Hope this will help, please do comment if you need any further explanation. Your feedback would be highly appreciated.