Financial Risk Manager Part 1

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