Answer: ($54.27, $175.91)

## Explanation To construct a 95% confidence interval for the stock price after one year: 1. **Given Information:** - Initial price: $80 - Future price: P = 80 × exp(i) - i ~ N(μ=0.2, σ=0.3) - 95% confidence level 2. **Confidence Interval Formula:** For a 95% confidence interval, we use: \[ CI = μ ± Z_{α/2} × σ \] where α = 0.05, so Z_{α/2} = 1.96 3. **Calculate the interval for i:** \[ CI_i = 0.2 ± 1.96 × 0.3 = 0.2 ± 0.588 \] \[ CI_i = (0.2 - 0.588, 0.2 + 0.588) = (-0.388, 0.788) \] 4. **Convert to price interval:** \[ P_{lower} = 80 × exp(-0.388) = 80 × 0.6784 = $54.27 \] \[ P_{upper} = 80 × exp(0.788) = 80 × 2.1989 = $175.91 \] 5. **Final 95% Confidence Interval:** \[ CI_{price} = ($54.27, $175.91) \] **Key Points:** - The rate of return i is normally distributed - We construct the confidence interval for i first, then transform it to the price domain using the exponential function - The interval is asymmetric because the exponential transformation is nonlinear - Option C correctly captures both the lower and upper bounds of the confidence interval

Author: Tanishq Prabhu