Financial Risk Manager Part 1
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A renowned economist has calculated that the Canadian economy will be in one of 3 possible states in the coming year: Boom, Normal, or Slow. The following table gives the returns of stocks A and B under each economic state.
| State | Probability State | Return for Stock A | Return for Stock B |
|---|---|---|---|
| Boom | 40% | 12% | 18% |
| Normal | 35% | 10% | 15% |
| Slow | 25% | 8% | 12% |
Which of the following is closest to the covariance of the returns for stocks A and B?
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TTanishq
A
0.103
B
0.0001734
C
0.1545
D
0.0003765
Explanation:
Covariance Calculation Explanation
To calculate the covariance between stocks A and B, we use the formula:
Cov(A, B) = Σ P(s) × [R_A – E(R_A)] × [R_B – E(R_B)]
Step 1: Calculate Expected Returns
E(R_A) = Σ P(s) × R_A
- Boom: 0.4 × 0.12 = 0.048
- Normal: 0.35 × 0.10 = 0.035
- Slow: 0.25 × 0.08 = 0.020
- E(R_A) = 0.048 + 0.035 + 0.020 = 0.103
E(R_B) = Σ P(s) × R_B
- Boom: 0.4 × 0.18 = 0.072
- Normal: 0.35 × 0.15 = 0.0525
- Slow: 0.25 × 0.12 = 0.030
- E(R_B) = 0.072 + 0.0525 + 0.030 = 0.1545
Step 2: Calculate Covariance Components
| State | P(s) | [R_A – E(R_A)] | [R_B – E(R_B)] | P(s) × [R_A – E(R_A)] × [R_B – E(R_B)] |
|---|---|---|---|---|
| Boom | 0.40 | 0.12 - 0.103 = 0.017 | 0.18 - 0.1545 = 0.0255 | 0.4 × 0.017 × 0.0255 = 0.0001734 |
| Normal | 0.35 | 0.10 - 0.103 = -0.003 | 0.15 - 0.1545 = -0.0045 | 0.35 × (-0.003) × (-0.0045) = 0.000004725 |
| Slow | 0.25 | 0.08 - 0.103 = -0.023 | 0.12 - 0.1545 = -0.0345 | 0.25 × (-0.023) × (-0.0345) = 0.00001984 |
Step 3: Sum the Components
Cov(A, B) = 0.0001734 + 0.000004725 + 0.00001984 = 0.0003765
Therefore, the covariance between stocks A and B is 0.0003765, which corresponds to option D.
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