Question at position 22 [table]
State | Probability | Return Boom | 0.30 | 20% Normal | 0.50 | 10% Recession | 0.20 | −5%
[/table] Standard deviation equals closest to: 13-15%9-10%8-9%10-12%

Question

BlackTom AI · Accepted Answer

Expected return = 0.30(20%) + 0.50(10%) + 0.20(−5%) = 6% + 5% − 1% = 10%.

Relevant knowledge: For a discrete probability distribution of returns, the standard deviation is the square root of the variance, where variance = Σ p_i (R_i − E[R])^2. Probabilities given sum to 1, so we use the population variance formula.

Step-by-step calculation:
1) Compute deviations from the mean (10%):
- Boom: 20% − 10% = 10% = 0.10
- Normal: 10% − 10% = 0 = 0.00
- Recession: −5% − 10% = −15% = −0.15

2) Square each deviation:
- Boom: (0.10)^2 = 0.0100
- Normal: (0.00)^2 = 0.0000
- Recession: (−0.15)^2 = 0.0225

3) Multiply squared deviations by their probabilities and sum (variance):
- Boom contribution: 0.30 × 0.0100 = 0.003000
- Normal contribution: 0.50 × 0.0000 = 0.000000
- Recession contribution: 0.20 × 0.0225 = 0.004500
Variance = 0.003000 + 0.000000 + 0.004500 = 0.007500

4) Standard deviation = sqrt(0.007500) ≈ 0.0866025 = 8.66%

Why this answer is correct: The computed standard deviation of approximately 8.66% falls in the 8–9% range, so the closest choice is 8-9%.

Why other options are incorrect:
- 13-15% and 10-12% are substantially higher than 8.66% and would imply a much larger variance than calculated.
- 9-10% is slightly above 8.66% and not the closest interval; the value lies clearly within 8–9%.

Conclusion: The standard deviation is about 8.66%, so the correct choice is 8-9%, matching the provided answer.

Question at position 22 [table] State | Probability | Return Boom | 0.30 | 20% Normal | 0.50 | 10% Recession | 0.20 | −5% [/table] Standard deviation equals closest to: 13-15%9-10%8-9%10-12%单项选择题

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