Financial Risk Manager Part 1
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Assumed that asset prices are normally distributed. The expected value of an asset price is $80 with daily volatility of 2%. Compute the 95% confidence interval of the asset price at the end of 4 days.
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NNikitesh
A
$80 \pm 2.000$
B
$80 \pm 3.200$
C
$80 \pm 6.272$
D
$80 \pm 3.136$
Explanation:
Step-by-step calculation:
- Calculate 4-day volatility:
Daily volatility = 2% = 0.02
For 4 days, volatility scales with the square root of time:
\sigma_{4\text{-day}} = \sigma_{\text{daily}} \times \sqrt{4} = 0.02 \times 2 = 0.04 \text{ or } 4\%$`$2`. **Calculate one standard deviation move in dollar terms:**
`80` \times 0.04 = \`3.20
$3`. **Calculate 95% confidence interval:** For a normal distribution, the 95% confidence interval corresponds to ±1.96 standard deviations: $$\`80\pm 1.96 \times \3.20` = \`80\pm \$6.27`2$$
Why other options are incorrect:
- A (
$80± 2.000): This would be approximately one daily standard deviation move, not accounting for the 4-day horizon or the 95% confidence level. - B (
$80± 3.200): This is exactly one standard deviation move over 4 days, but for 95% confidence we need ±1.96 standard deviations. - D (
$80± 3.136): This appears to be$80± 0.98 × 3.20, which would correspond to approximately a 67% confidence interval (one standard deviation), but the calculation is slightly off.
Key concepts:
- Volatility scales with the square root of time
- For normally distributed returns, 95% confidence interval = mean ± 1.96 × standard deviation
- The standard deviation in dollar terms = asset price × volatility
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