Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 51

The Lee diagram makes it easy to visualize different loss layers and write down their expected values using survival-function form expectations. We use a for a height on the vertical axis because it usually represents assets available to Ins Co. for paying claims. Alternatively it can represent an attachment point.

Figure 3.10 illustrates several significant actuarial quantities in a Lee diagram.

Figure 3.10 Insurance variables in a Lee diagram.

 The area E[(X−a)+] equals the unconditional excess loss cost for losses in excess of the attachment a. It is called the insurance charge in US retrospective rating plans. When a represents assets it is called the insolvency put or expected policyholder deficit (EPD). In finance, the excess loss cost corresponds to the expected payout of a call option and a is called the strike price.

 The limited expected value up to a level a is the expected value with a limit a on losses. It is given by (3.7)the sum of the two shaded areas at the bottom, to the right of the curve. The integral is a Riemann-Stieltjes integral if X has a mixed distribution. Limited expected values are used to compute increased limits factors.

 The expected loss, E[X] is the sum of the three shaded areas under the curve.

 E[(a−X)+] is the insurance savings or the investor’s residual value, or, in finance, the expected value of a put option.

The diagram makes obvious a put-call parity-like relationship

(3.8)

The shaded area under the curve plus the put equals the strike plus the call. (Put-call parity in finance is more subtle as it relates to prices not mathematical expectations.) In insurance retrospective rating terminology Eq. 3.8 says that the expected losses plus the savings equals the attachment (entry ratio) plus the expense (Kallop 1975).

Exercise 28 Show that E[(a−X)+]=∫0aF(x)dx by computing (a−X)+=a−(X∧a) and by using integration by parts. Identify the relevant areas on the Lee diagram.