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2. Consider the following expected returns, volatilities, and correlations: \be...

May 7, 2024

Solution

To ensure a risk-free investment with stocks Vital and Mital, we need to find the weights of the portfolio that eliminate risk. Since Vital and Mital have a perfect negative correlation of -1.0, a risk-free portfolio can be created by allocating the investments in a way that the weighted returns of one stock offset the weighted returns of the other

Let

w

be the weight of the investment in Vital and

(1-w)

be the weight of the investment in Mital. The portfolio's variance is given by the formula:

Var(P) = w^2\sigma_{Vital}^2 + (1-w)^2\sigma_{Mital}^2 + 2w(1-w)\sigma_{Vital}\sigma_{Mital}\rho_{Vital,Mital}

, where

\rho_{Vital,Mital}

is the correlation coefficient between Vital and Mital. Since

\rho_{Vital,Mital} = -1

, the cross-product term will cancel out the individual variances if the weights are chosen correctly

To find the weight

w

, we set the portfolio's variance to zero and solve for

w

0 = w^2\sigma_{Vital}^2 + (1-w)^2\sigma_{Mital}^2 - 2w(1-w)\sigma_{Vital}\sigma_{Mital}

. Simplifying this equation will give us the weight

w

that needs to be invested in Vital to ensure a risk-free investment

Solving the equation from step c, we get:

w = \frac{\sigma_{Mital}}{\sigma_{Vital} + \sigma_{Mital}}

. Plugging in the values

\sigma_{Vital} = 6\%

and

\sigma_{Mital} = 24\%

, we find

w = \frac{24}{6 + 24} = \frac{24}{30} = 0.8

Answer

The portion of the investment that should be allocated to Vital stock to ensure a risk-free investment is 80%.

Key Concept

Portfolio variance and correlation

Explanation

By understanding the relationship between the variances of individual stocks and their correlation, we can determine the weights of a portfolio that minimize risk, in this case, creating a risk-free portfolio due to perfect negative correlation.