Question

7

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You invest $100 in a T-bill with a rate of return of 0.05, and in a risky asset with an expected rate of return of 0.12 and a standard deviation of 0.15.

What percentages of your money must be invested in the risk-free asset and the risky asset, respectively, to form a portfolio with a standard

deviation of 0.06?

Answer 1

Return on T-bill = E[R₁] = 0.05, T-bills are risk-free asset so, standard deviation of T-bill = σ₁ = 0

Expected Return on Risky asset = E[R₂] = 0.12, Standard deviation of risky-asset = σ₂ = 0.15

Portfolio

weight of risk-free asset or T-bill in portfolio = w₁, weight of risky-asset in portfolio = w₂

Variance of a portfolio consisting of T-bill and the risky assets is given by:

Variance of the portfolio = σ_P² = w₁²*σ₁² + w₂²*σ₂² + 2*w₁*w₂*ρ*σ₁*σ₂

_Since, σ₁ = 0

so, σ_P² = w₁²*σ₁²

Standard deviation is the square-root of the variance of the portfolio

Standard deviation of the portfolio = σ_P = [w₂²*σ₂²]^1/2 = w₂*σ₂

It is given that the standard deviation of the portfolio is 0.06 and σ₂ = 0.15

σ_P = w₂*σ₂

0.06 = w₂*0.15

w₂ = 0.06/0.15 = 0.4 = 40%

w₁ = 1-w₂ = 1-0.4 = 0.6 = 60%

Answer

Percentage of money must be invested in the risk-free asset = 60%

Percentage of money must be invested in the risky-asset = 40%