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Answer: 6.53%
## Explanation The expectation E[g(A,B)] is calculated as the probability-weighted average of all possible outcomes of the function g(A,B) = 0.6A + 0.4B. From the joint probability table: - When A = 8% and B = 1%: P = 0.15 - When A = 8% and B = 3%: P = 0.40 - When A = 11% and B = 1%: P = 0.20 - When A = 11% and B = 3%: P = 0.25 **Calculations:** 1. **Case 1 (A=8%, B=1%, P=0.15):** g(A,B) = 0.6(8%) + 0.4(1%) = 4.8% + 0.4% = 5.2% Contribution = 0.15 × 5.2% = 0.78% 2. **Case 2 (A=8%, B=3%, P=0.40):** g(A,B) = 0.6(8%) + 0.4(3%) = 4.8% + 1.2% = 6.0% Contribution = 0.40 × 6.0% = 2.40% 3. **Case 3 (A=11%, B=1%, P=0.20):** g(A,B) = 0.6(11%) + 0.4(1%) = 6.6% + 0.4% = 7.0% Contribution = 0.20 × 7.0% = 1.40% 4. **Case 4 (A=11%, B=3%, P=0.25):** g(A,B) = 0.6(11%) + 0.4(3%) = 6.6% + 1.2% = 7.8% Contribution = 0.25 × 7.8% = 1.95% **Total Expected Value:** E[g(A,B)] = 0.78% + 2.40% + 1.40% + 1.95% = **6.53%** This represents the expected portfolio return based on the given joint probability distribution of the two funds' returns.
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A senior investment advisor at a wealth management firm manages an investment portfolio consisting of two main assets, fund A and fund B, for a group of clients. The annual returns of these assets can be represented as random variables, denoted as A and B, respectively. The advisor wants to estimate future portfolio performance by calculating the expectation of a specific function that is based on the anticipated returns of both funds A and B. The function is given by:
g(A, B) = 0.6A + 0.4B
where the coefficients reflect the portfolio allocation of 60% to fund A and 40% to fund B.
In the analysis, the advisor assumes that the returns of both fund A and fund B can take on only two possible year-end values, and constructs the joint probability mass function (PMF) as follows:
| A | |||
|---|---|---|---|
| 8% | 11% | ||
| B | 1% | 0.15 | 0.20 |
| 3% | 0.40 | 0.25 |
What is the correct expectation, E[g(A,B)], of the function?
A
2.30%
B
5.12%
C
6.53%
D
9.35%
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