The expected return on the market is 12% and the risk free rate is 7%. The standard deviation of the return on the market is 15%. Ones investor creates a portfolio on the efficient frontier with an expected return of 10%. Another creates a portfolio on the efficient frontier with an expected return of 20%. What is the standard deviation of the return of the two portfolios?.
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To find the standard deviation of the return of the two portfolios, we need to use the formula for portfolio standard deviation when combining two assets.
The formula for the portfolio standard deviation (σp) is:
σp = √(w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ)
Where:
- w₁ and w₂ are the weights of assets 1 and 2 in the portfolio (the sum of weights should equal 1)
- σ₁ and σ₂ are the standard deviations of the returns of assets 1 and 2
- ρ is the correlation coefficient between the returns of assets 1 and 2
In this case, we have two portfolios:
Portfolio 1:
- Expected return (Rp₁) = 10%
- Weights: w₁ = 1 and w₂ = 0 (since it is a single-asset portfolio)
- Standard deviation of the market (σ₁) = 15%
Portfolio 2:
- Expected return (Rp₂) = 20%
- Weights: w₁ = ? and w₂ = ? (these need to be determined)
- Standard deviation of the market (σ₂) = 15%
Now, we need to find the weights for Portfolio 2 that satisfy the given expected return. We can use the formula for the expected return of a portfolio:
Rp = w₁R₁ + w₂R₂
For Portfolio 2:
20% = w₁(12%) + w₂(7%)
Solving this equation will give us the weights for Portfolio 2.
Once we have the weights for both portfolios, we can substitute the values into the formula for portfolio standard deviation to calculate the standard deviation of the return of each portfolio.