Financial Risk Manager Part 1
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At Capital Bank, the compensation framework is made up of a basic salary plus bonuses. The average salary among sales employees is $30,000 per year, and they are also entitled to a bonus of $0.05 for every dollar of sales brought in. Average sales amount to $300,000 per year with a variance of 5,000,000. Determine the standard deviation of compensation received by employees.
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TTanishq
A
$165
B
$450
C
$222
D
$112
Explanation:
Explanation
To compute the standard deviation of compensation, we apply linear transformation principles to sales data:
Step 1: Define the Compensation Formula
Compensation (Y) is given by:
Y = α + βX
Y = 30,000 + 0.05X
Y = α + βX
Y = 30,000 + 0.05X
where:
- Y = compensation (transformed variable)
- X = sales variable
- β = scale constant (
$0.05/$1) - α = shift constant (
$30,000)
Step 2: Calculate Variance of Compensation
For linear transformations, the variance of Y is:
Var(Y) = Var(α + βX) = β²Var(X) = β²σ²
Var(Y) = Var(α + βX) = β²Var(X) = β²σ²
Given:
- Var(X) = σ² = 5,000,000
- β = 0.05
Var(Y) = (0.05)² × 5,000,000 = 0.0025 × 5,000,000 = 12,500
Var(Y) = (0.05)² × 5,000,000 = 0.0025 × 5,000,000 = 12,500
Step 3: Calculate Standard Deviation
Standard deviation is the square root of variance:
SD(Y) = √Var(Y) = √12,500 = `$111.8` ≈ `$112`
SD(Y) = √Var(Y) = √12,500 = `$111.8` ≈ `$112`
Key Insight: The basic salary ($30,000) doesn't affect the variance or standard deviation since it's a constant shift that gets canceled out in variance calculations. Only the variable component (bonus) contributes to the dispersion of compensation.
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