Financial Risk Manager Part 1

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Use the regression equation " $\overline{\text{WPO}} = -3.2\% + 0.49(\text{S&P 500})$ " to calculate a 95% confidence interval on the predicted value of WPO. You have been given that $n = 30$ , 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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(1.7%, 9.37%)

(-5.97%, 1.7%)

(4.9%, 9.37%)

(-5.97%, 9.37%)

Explanation:

Explanation

Step 1: Calculate the Predicted Value

Using the regression equation:

\overline{\text{WPO}} = -3.2\% + 0.49 \times 10\% = 1.7\%

Step 2: Determine the Critical t-value

Sample size: n = 30
Degrees of freedom = n - 2 = 28
For 95% confidence interval, α = 0.05, α/2 = 0.025
t-value for 28 degrees of freedom at 0.025 significance level: t₀.₀₂₅,₂₈ = 2.04

Step 3: Calculate the Margin of Error

Margin of error = t-value × Standard error of forecast

\text{Margin of Error} = 2.04 \times 3.76\% = 7.67\%

Step 4: Construct the Confidence Interval

\text{CI}_{95\%} = \overline{\text{WPO}} \pm \text{Margin of Error} = 1.7\% \pm 7.67\%

\text{Lower Bound} = 1.7\% - 7.67\% = -5.97\%

\text{Upper Bound} = 1.7\% + 7.67\% = 9.37\%

Therefore, the 95% confidence interval is (-5.97%, 9.37%).

Key Points

The confidence interval is centered around the predicted value (1.7%)
The width of the interval reflects the uncertainty in the prediction
The interval includes negative values, indicating that WPO could potentially decrease even when S&P 500 has a positive excess return
The standard error of forecast accounts for both the uncertainty in the regression coefficients and the variability of individual observations around the regression line

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