Financial Risk Manager Part 1
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Use the regression equation "WPO = -3.2% + 0.49(S&P 500)" to calculate a 95% confidence interval on the predicted value of WPO. You have been given that n = 32, the standard error of the forecast is 3.76%, and the forecasted value of S&P 500 excess return is 10%.
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Community
NNikitesh
A
(1.7%, 9.37%)
B
(-5.97%, 1.7%)
C
(4.9%, 9.37%)
D
(-5.97%, 9.37%)
Explanation:
Step 1: Calculate the predicted value of WPO
Using the regression equation: WPO = -3.2% + 0.49(S&P 500)
Given S&P 500 excess return = 10%: WPO = -3.2% + 0.49 × 10% = -3.2% + 4.9% = 1.7%
Step 2: Determine the critical t-value
- Sample size n = 32
- Degrees of freedom = n - 2 = 32 - 2 = 30
- For a 95% confidence interval with two-tailed test, α = 0.05, α/2 = 0.025
- t₀.₀₂₅,₃₀ = 2.04 (from t-distribution table)
Step 3: Calculate the margin of error
Margin of error = t-value × standard error of forecast = 2.04 × 3.76% = 7.67%
Step 4: Construct the confidence interval
CI = Predicted value ± Margin of error = 1.7% ± 7.67% = (1.7% - 7.67%, 1.7% + 7.67%) = (-5.97%, 9.37%)
Key concepts:
- Confidence intervals for regression predictions account for both parameter uncertainty and prediction error
- The standard error of forecast incorporates both the regression coefficient uncertainty and the residual variance
- Degrees of freedom for simple linear regression is n-2 (one for intercept, one for slope)
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