You want to create a portfolio equally as risky as the market, and you have $500

Question

You want to create a portfolio equally as risky as the market, and you have $500,000 to invest. Information about the possible investments is given below: Asset Investment Beta Stock A $ 142,000 .87 Stock B $ 138,000 1.32 Stock C 1.47 Risk-free asset Requirement 1: How much will you invest in Stock C? (Do not round intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).) Requirement 2: How much will you invest in the risk-free asset? (Do not round intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

Explanation / Answer

Answer

Your goal is "risky as the market". Suppose The market beta is 1.0. Risk-free asset has a beta of 0. So it's a weighted average problem. Let's use:

SA = Stock A investment
BA = Beta of Stock A
SB = Stock B investment
BB = Beta of Stock B
SC = Stock C investment
BC = Beta of Stock C
RFA = investment in risk-free asset

We know that we want the portfolio beta to be 1.0:

((SA * BA) + (SB * BB) + (SC * BC)) / 500000 = 1.0

We also know the sum of the investments + RFA = 500K:

SA + SB + SC + RFA = 500,000

We don't know SC or RFA.

Solve for SC first:

(1) Multiply both sides by 500000

(SA * BA) + (SB * BB) + (SC * BC) = 500000

(2) Subtract "(SA * BA) + (SB * BB)" from both sides:

(SC * BC) = 500000 - (SA * BA) - (SB * BB)

(3) Divide both sides by BC:

SC = (500000 - (SA * BA) - (SB * BB)) / BC

(4) Plug in the numbers:

SC = ( 500000 - (1,42,000 * 0.87) - (1,38,000 * 1.32)) / 1.47

SC = 1,32,176.90

Then, solve for RFA:

SA + SB + SC + RFA = 500,000

RFA = 500000 – 1,42,000 – 1,38,000 – 1,32,176.90

RFA = 87,823.90

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