Question

You invest $1,200 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 13% and a standard deviation of 20% and a Treasury bill with a rate of return of 4%. __________ of your complete portfolio should be invested in the risky portfolio if you want your complete portfolio to have a standard deviation of 7%.

Answer 1

The formula to calculate standard deviation of a portfolio comprising two assets is given as:

$\sigma _{portfolio}=\left (w_{1}^{2}\sigma _{1}^{2} + w_{2}^{2}\sigma _{2}^{2} + 2w_{1}w_{2}Cov(1,2) \right )^{\frac{1}{2}}$

Where,

• $\sigma _{1}$ and $\sigma _{2}$ are the standard deviations of the two assets
• $w _{1}$ and $w _{2}$ are the respective weights or proportions of the two assets in the portfolio
• $Cov(1,2)$ represents the covariance between the two assets

Now, in this case since one asset is risk free, its standard deviation and its covariance with the other asset are zero. That means, for this portfolio the formula reduces to:

$\sigma _{portfolio}=w_{1}\sigma _{1}$

The requirement is that $\sigma _{portfolio}$ must be 7%. Substituting the value we get

$0.07=w_{1}*0.2$

or, $w_{1}=0.35$

or 35% of your complete portfolio....