Financial Risk Manager Part 1

Explanation

To compute the bonus, we apply the following linear transformation to each employee's salary:

$Y = \alpha + \beta X$ $Y = 1,000 + 0.15X$

where:

• Y is the transformed variable (bonus)
• X is the original variable (salary)
• β = 0.15 (scale constant)
• α = 1,000 (shift constant)

The variance of Y (bonuses) is given by:

$Var(Y) = Var(\alpha + \beta X) = \beta^2 Var(X) = \beta^2 \sigma^2$

Given:

• Variance of salary $Var(X) = 6,000,000$
• β = 0.15

Therefore: $Var(Y) = 0.15^2 \times 6,000,000 = 0.0225 \times 6,000,000 = 135,000$

Standard deviation is the square root of variance: $SD(Y) = \sqrt{135,000} = \$367.42 \approx \$367$

Key Insight: When applying a linear transformation Y = α + βX:

• The shift constant α affects the mean but not the variance
• The scale constant β affects both mean and variance
• Standard deviation scales by |β|, while variance scales by β²