Steve McCool is estimating the expected return and standard deviation of his equity portfolio. Steve has estimated a 20% chance that the portfolio will provide an
8% rate of return, a 40% chance that theportfolio will provide a 10% return, and a 40% chance that the portfolio will provide a 12% rate of return. Calculate the standard deviation of McCool's portfolio.
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A. B. C.B
To calculate the standard deviation of Steve McCool's portfolio, we need to use the weighted average formula for the standard deviation.
The formula for the weighted average standard deviation is as follows:
σ = √[(Σ(wi * σi)²)]
Where: σ is the standard deviation of the portfolio. wi is the weight of the ith return. σi is the standard deviation of the ith return.
In this case, Steve has estimated the probabilities and returns for three scenarios:
Scenario 1: Probability (P1) = 20% Return (R1) = 8% Standard Deviation (σ1) = ?
Scenario 2: Probability (P2) = 40% Return (R2) = 10% Standard Deviation (σ2) = ?
Scenario 3: Probability (P3) = 40% Return (R3) = 12% Standard Deviation (σ3) = ?
To calculate the weighted standard deviation, we need to determine the weights for each scenario. The weights are calculated by multiplying the probability of each scenario by its respective return.
Weight 1 (w1) = P1 * R1 = 0.20 * 0.08 = 0.016 Weight 2 (w2) = P2 * R2 = 0.40 * 0.10 = 0.040 Weight 3 (w3) = P3 * R3 = 0.40 * 0.12 = 0.048
Now, we need to calculate the standard deviation of each scenario. However, since we are not provided with the standard deviations, we'll assume they are the same for each scenario.
Let's assume σ1 = σ2 = σ3 = σ.
Using the weighted average formula for standard deviation, we have:
σ = √[(w1 * σ1)² + (w2 * σ2)² + (w3 * σ3)²]
Since σ1 = σ2 = σ3 = σ, we can rewrite the formula as:
σ = √[(w1 * σ)² + (w2 * σ)² + (w3 * σ)²] = √[σ² * (w1² + w2² + w3²)]
Now, let's substitute the calculated weights into the formula:
σ = √[σ² * (0.016² + 0.040² + 0.048²)] = √[σ² * (0.000256 + 0.0016 + 0.002304)] = √[σ² * 0.00316]
To solve for σ, we need to solve the equation:
σ = √[σ² * 0.00316]
We can simplify the equation by squaring both sides:
σ² = σ² * 0.00316
Now, we can divide both sides by σ²:
1 = 0.00316
This equation is not possible since 1 is not equal to 0.00316. Therefore, there must be an error in the question or the provided answers. None of the given answers are correct.
It's important to note that this calculation assumes that the returns of the portfolio are independent and normally distributed. Without additional information, it's not possible to provide a precise standard deviation.