You want your portfolio beta to be 1.10. Currently, your portfolio consists of $3,000 invested in stock A with a beta of 1.65 and $2,000 in stock B with a beta of.72. You have another $5,000 to invest and want to divide it between an asset with a beta of 1.48 and a risk-free asset. How much should you invest in the risk-free asset?

A) $0
B) $775
C) $1,885
D) $3,115
E) $5,000

Question

Tasha Henderson · Accepted Answer

Final Answer : C, Explanation :   To achieve a portfolio beta of 1.10, we first calculate the weighted average beta of the current portfolio and then determine the required investment in the risk-free asset and the asset with a beta of 1.48 to achieve the target beta. The current portfolio beta is calculated as follows:  Current Portfolio Beta = ( 3000 3000 + 2000 ) × 1.65 + ( 2000 3000 + 2000 ) × 0.72 = 1.386  	ext{Current Portfolio Beta} = \left(\frac{3000}{3000+2000}ight) 	imes 1.65 + \left(\frac{2000}{3000+2000}ight) 	imes 0.72 = 1.386  Current Portfolio Beta = ( 3000 + 2000 3000 ​ ) × 1.65 + ( 3000 + 2000 2000 ​ ) × 0.72 = 1.386  With an additional $5,000 to invest, the total portfolio value will be $10,000. To achieve a portfolio beta of 1.10, we use the formula:  Target Portfolio Beta = ( Current Investment × Current Beta + New Investment in Asset × Beta of Asset + New Investment in Risk-Free Asset × 0 Total Portfolio Value )  	ext{Target Portfolio Beta} = \left(\frac{	ext{Current Investment} 	imes 	ext{Current Beta} + 	ext{New Investment in Asset} 	imes 	ext{Beta of Asset} + 	ext{New Investment in Risk-Free Asset} 	imes 0}{	ext{Total Portfolio Value}}ight)  Target Portfolio Beta = ( Total Portfolio Value Current Investment × Current Beta + New Investment in Asset × Beta of Asset + New Investment in Risk-Free Asset × 0 ​ )  Given that the beta of the risk-free asset is 0, we rearrange the formula to solve for the new investments. Let   x  x  x   be the amount invested in the risk-free asset and   5000 − x  5000 - x  5000 − x   be the amount invested in the asset with a beta of 1.48. The equation to solve is:  1.10 = ( 3000 × 1.65 + 2000 × 0.72 + ( 5000 − x ) × 1.48 ) 10000  1.10 = \frac{(3000 	imes 1.65 + 2000 	imes 0.72 + (5000 - x) 	imes 1.48)}{10000}  1.10 = 10000 ( 3000 × 1.65 + 2000 × 0.72 + ( 5000 − x ) × 1.48 ) ​  Solving this equation gives us the value of   x  x  x  , the amount to invest in the risk-free asset, which is $1,885.

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