Independent solution

How to solve this Limited Loss Variables question

Setup

Setup

Let B be the payment obtained by capping the loss Y at 10. The density is supported above 1, so losses below the cap and losses above the cap must be handled separately.

B=min⁡(Y,10)B=\min(Y,10)
fY(y)=2y−3,y>1f_Y(y)=2y^{-3},\qquad y>1

Model

Model

For losses between 1 and 10 the payment equals the loss, whereas above 10 the payment is exactly 10. Splitting the expectation at the cap accounts for both regions.

E[B]=∫110y 2y−3 dy+10∫10∞2y−3 dyE[B]=\int_1^{10}y\,2y^{-3}\,dy+10\int_{10}^{\infty}2y^{-3}\,dy

Compute

Compute

The below-cap first-moment integral contributes 1.8. The probability above the cap is 0.01, so capped losses contribute a further 10 × 0.01 = 0.1.

∫1102y−2 dy=1.8\int_1^{10}2y^{-2}\,dy=1.8
10Pr⁡(Y>10)=10(10−2)=0.110\Pr(Y>10)=10(10^{-2})=0.1
E[B]=1.8+0.1=1.9E[B]=1.8+0.1=1.9

Answer

Answer

Adding the two disjoint contributions gives an expected capped payment of 1.9, which is choice D.

1.9(D)\boxed{1.9\quad\text{(D)}}