Financial Risk Manager Part 1
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A stock has an initial market price of $80. Exactly one year from now, its price will be given by: P = 80 * exp(i) where i is the rate of return. i is normally distributed with mean 0.2 and standard deviation 0.3. Construct a 95% confidence interval for the price of the stock after one year.
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NNikitesh
A
($54.27, $80)
B
($80, $175.91)
C
($54.27, $175.91)
D
($54.27, $140)
Explanation:
Explanation
Given:
- Initial price =
$80 - Price after one year: P = 80 * exp(i)
- i ~ N(μ=0.2, σ=0.3)
- Need 95% confidence interval for P
Step 1: Find 95% CI for i For a normal distribution, the 95% confidence interval is: CI = μ ± Z_{α/2} * σ where α = 0.05 Z_{0.025} = 1.96
CI for i = 0.2 ± 1.96 * 0.3 = 0.2 ± 0.588 = (-0.388, 0.788)
Step 2: Transform to CI for P
P = 80 * exp(i)
Lower bound: P_lower = 80 * exp(-0.388) = 80 * 0.6784 ≈ $54.27
Upper bound: P_upper = 80 * exp(0.788) = 80 * 2.1989 ≈ $175.91
Step 3: Final CI
95% CI for P = ($54.27, $175.91)
Key Points:
- The rate of return i is normally distributed, so we first construct the CI for i
- We use the exponential transformation to get the CI for the price
- The correct answer is C: (
$54.27,$175.91)
Why other options are incorrect:
- A: Only includes lower bound up to initial price
- B: Starts at initial price, missing lower bound below
$80 - D: Upper bound is too low (
$140vs correct$175.91)
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