CAS Exam 7 Spring 2013, Question 19

Study guide: Exam 9

Given the following information for an insurance company:

Loss distribution for earthquake (EQ)

Losses ( $x$ , in $000,000)	$f(x)$
0	0.91
200	0.07
400	0.02

Loss distribution for coastal property (CP)

Losses ( $y$ , in $000,000)	$g(y)$
0	0.80
250	0.20

Losses for these two perils are independent.
The Wang transform has the form $F^*(x) = \Phi\{\Phi^{-1}[F(x)] - \Phi^{-1}(\alpha)\}$ , where $\Phi$ represents the cumulative standard normal distribution.
The Wang transform risk measure for CP is $197.5 million.
The Wang transform risk measure for CP and EQ combined is $323 million.

$z$	$\Phi(z)$
0.304	0.62
0.409	0.66
0.804	0.79
0.842	0.80
1.341	0.91
1.645	0.95
2.054	0.98

Part a (1.75 points)

For EQ, calculate:

i. Tail value-at-risk (TVaR) at the 95% level.
ii. Wang transform at the 95% level.
iii. Expected policyholder deficit (EPD) at the 5% level.

Part b (0.5 points)

Evaluate whether using TVaR as a risk measure for EQ would satisfy the company's requirement to have a pricing charge for any risk taken.

Part c (0.5 points)

Evaluate whether using the Wang transform as a risk measure for EQ would satisfy the company's requirement to have a pricing charge for any risk taken.

Part d (0.5 points)

Evaluate whether using TVaR as a risk measure for EQ would reflect the company's increasing risk aversion to larger losses.

Part e (0.5 points)

Evaluate whether using the Wang transform as a risk measure for EQ would reflect the company's increasing risk aversion to larger losses.

Part f (2.5 points)

Calculate the TVaR and Wang transform risk measures at the 95% level for the combined CP and EQ book of business. Evaluate whether these measures exhibit the diversification benefits of writing both coverages.

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