Question

Question 7. Solution needs to be handed in Suppose CAPM holds and that the market portfolio has an expected return of 11% and a volatility of 24%. Also, assume that the risk-free rate is 1%; What is the volatility of the portfolio that has the lowest possible volatility while having an expected return equal to 7%2 (Use two decimal digits in your final answer) .

hi guys how can i solve this step by step?

Answer 1

We have following information in question -

Expected Return of Market Portfolio (Rm) = 11% or 0.11

Volatility of Market Portfolio ($\sigma _{m}$) = 24% or 0.24

Risk-free rate(Rr) = 1% or 0.01

Volatility of a risk-free assets is always zero(0) , that's why its called risk free.

Therefore, volatility of Risk-free assets ($\sigma _{r}$) = 0.

According to CAPM, there is always a higher return for a higher level of risk which means a portfolio having zero risk is consisting only risk-free assets and as risky assets added to this portfolio, expected return would increase with increase in risk of portfolio and vice versa. A portfolio having lower return with high level of risk does not lies on optimal portfolio frontier.

Thus, If we need to reduce risk of market portfolio then we have to add risk free assets to this portfolio however, it will reduce the overall expected return of the portfolio.

Expected Return of Portfolio is weighted average of individual asset's return in the portfolio -

$E(R_{p}) = \sum_{i=1}^{n}R_{i}*W_{i}$

Computation of weights of portfolio having expected return 7% with lowest possible volatility

We add risk-assets to this market portfolio till we get expected return of portfolio - 7%, as CAPM holds good here, so this mix would have lowest possible volatility of the portfolio .

Lets assume,

Weight of Market portfolio = W₁

Weight of Market Portfolio(W₂) =(1-W₁)

E(R_p) = Rm*W₁ + Rr*(1-W₁)

7% = 11%*W₁ + 1%*(1-W₁)

7% = 11%*W₁ + 1% - 1%W₁

7% = 10%*W₁ + 1%

10%*W₁ = 6%

W₁ = 6/10

W₁ = 0.6

W₂ = (1 - 0.6) = 0.4

Computation of volatility of Portfolio

Volatility of Portfolio ($\sigma _{p}$) = { w²₁*σ²_m + w²₂*σ²_r + 2*(w₁)*(w₂)*Cov(R_m, R_r)}^1/2

Where,

Cov(R_m, R_r) = Covariance between Market portfolio and risk free assets, which is Zero(0)

W₁ = 0.6

W₂ = 0.4

$\sigma _{m}$ = 24% or 0.24

$\sigma _{r}$ = 0

Thus,

$\sigma _{p} = \sqrt{\sigma _{m}^{2}*w_{1}^{2}}$

$\sigma _{p} = \sqrt{\0.24^{2}*0.6^{2}}$

$\sigma _{p} = \sqrt{\0.0576*0.36}$

$\sigma _{p} = \sqrt{0.020736}$

$\sigma _{p} = 0.144$

Thus, Lowest possible volatility of portfolio with expected return of 7% is 14.4%