Question

You are trying to build the best possible risky portfolio for your investment clients. You have two risky assets available to you: A risky stock with an expected excess return of 0.222 and a standard deviation of 0.91, and a risky bond with an expected excess return of 0.034, and a standard deviation of 0.663. If these two assets have a coefficient of correlation of -0.27, what proportion of the money you invest in risky assets should you put in the bond? An answer of 0 means invest no money in the bond, an answer of 1 means put all of your money in the bond.

Answer 1

Let's assume:

Weight of Stock fund in optimal risky portfolio = Ws

Weight of Bond fund in optimal risky portfolio = Wb

We can compute the weights of optimal risky portfolio with following equation:

$\large W_s = \frac{R_s*\sigma _{b}^{2}-R_b*\rho _{sb}*\sigma _s*\sigma _b}{R_s*\sigma _{b}^{2}+R_b*\sigma _{s}^{2}-(R_s+R_b)*\rho _{sb}*\sigma _s*\sigma _b}$

where,

R = Excess Return

S = stock fund

B = Bond fund

$\small \rho (sb) =\textup{ coefficient of correlation}$

Putting the values:

$W_s = \frac{0.222*0.91^{2}-0.034*-0.27*0.91*0.663}{0.222*0.663^{2}+0.034*0.91^{2}-(0.222-0.034)*-0.27*0.91*0.663}$

$W_s = \frac{0.1031228874}{0.1674418876}$

$\large W_s = 0.6159$

Weight of Bond Fund:

$W_b = 1 - 0.6159$

$\bg_green \large W_b = 0.3841$

Portion risky asset to be put in Bond fund = 0.3841

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