Financial Risk Manager Part 1

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A regression analysis of monthly returns of a sales company on the market return over ten years gives an intercept of $\hat{\beta}_0 = 0.65$ , the slope $\hat{\beta} = 1.65$ . Other quantities include: $s^2 = 20.45$ , $\hat{\sigma}_X^2 = 18.65$ and $\hat{\mu}_X = 0.61$ . What is the standard error estimate of $\hat{\beta}_0$ ?

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NNikitesh

Last updated: February 2, 2026 at 10:22

0.5463

0.56435

0.4552

0.4169

Explanation:

The standard error of the intercept $\hat{\beta}_0$ is calculated using the formula:

\text{SEE}_{\beta_0} = \sqrt{\frac{s^2(\hat{\mu}_X^2 + \hat{\sigma}_X^2)}{n\hat{\sigma}_X^2}}

Given:

$s^2 = 20.45$ (estimated error variance)
$\hat{\mu}_X = 0.61$ (sample mean of X)
$\hat{\sigma}_X^2 = 18.65$ (sample variance of X)
$n = 120$ (10 years × 12 months = 120 observations)

Plugging in the values:

\text{SEE}_{\beta_0} = \sqrt{\frac{20.45(0.61^2 + 18.65)}{120 \times 18.65}} = \sqrt{\frac{20.45(0.3721 + 18.65)}{120 \times 18.65}} = \sqrt{\frac{20.45 \times 19.0221}{2238}} = \sqrt{\frac{389.002}{2238}} = \sqrt{0.1738} = 0.4169

Thus, the standard error estimate of $\hat{\beta}_0$ is 0.4169, which corresponds to option D.

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