Financial Risk Manager Part 1
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Living in a certain city is really expensive. The mean spent by a citizen monthly is $3,000 with a variance of $500. Nonetheless, the city mayor decided to put a tax in order to make people regulate their monthly expenses adding 20% to all daily living articles. Find the new variance of this city.
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Community
NNikitesh
A
520
B
580
C
600
D
720
Explanation:
Explanation
Let X be the original living costs with:
- Mean(X) =
$3,000 - Variance(X) =
$500
The new living costs Y after adding 20% tax is:
Y = 1.2X
Y = 1.2X
Using the variance property: Var(aX) = a²Var(X)
Var(Y) = Var(1.2X)
= (1.2)² × Var(X)
= 1.44 × 500
= 720
Var(Y) = Var(1.2X)
= (1.2)² × Var(X)
= 1.44 × 500
= 720
Key Points:
- Variance is affected by scaling factor squared
- Mean would also change: Mean(Y) = 1.2 × 3000 =
$3,600 - The original mean of
$3,000 is not needed for variance calculation - Standard deviation would be: σ_Y = 1.2 × σ_X = 1.2 × √500 ≈ 1.2 × 22.36 = 26.83
Therefore, the new variance is 720.
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