Independent solution

How to solve this Deductible Payments question

Setup

Setup

Scale out b by writing U=X/b and Z=Y/b for the payment Y.

U∼Uniform⁡(0,1)U\sim\operatorname{Uniform}(0,1)
Z=(U−0.1)+Z=(U-0.1)_+

Model

Model

Integrate the first two moments of the scaled payment over the paying region.

E⁡[Z]=∫0.11(u−0.1) du\operatorname{E}[Z]=\int_{0.1}^{1}(u-0.1)\,du
E⁡[Z2]=∫0.11(u−0.1)2 du\operatorname{E}[Z^2]=\int_{0.1}^{1}(u-0.1)^2\,du

Compute

Compute

Evaluate the moments, form the variance, and compare it with the uniform variance.

E⁡[Z]=0.922=0.405,E⁡[Z2]=0.933=0.243\operatorname{E}[Z]=\frac{0.9^2}{2}=0.405,\qquad \operatorname{E}[Z^2]=\frac{0.9^3}{3}=0.243
Var⁡(Z)=0.243−0.4052=0.078975\operatorname{Var}(Z)=0.243-0.405^2=0.078975
Var⁡(Y)Var⁡(X)=0.0789751/12=0.9477\frac{\operatorname{Var}(Y)}{\operatorname{Var}(X)}=\frac{0.078975}{1/12}=0.9477

Answer

Answer

Taking the square root converts the variance ratio to the requested standard-deviation ratio.

SD⁡(Y)SD⁡(X)=0.9477=0.9734988\frac{\operatorname{SD}(Y)}{\operatorname{SD}(X)}=\sqrt{0.9477}=0.9734988
0.9735(E)\boxed{0.9735\quad\text{(E)}}