Pricing Insurance Risk. Stephen J. Mildenhall

retains losses in excess of ¤100M. (Note: ¤30M excess 20M refers to a layer with a limit of ¤30M that attaches at ¤20M.)

      Example 25 Structured finance tranches asset-backed securities (MBS, CDO, etc.) in an analogous way. The tranches are generally determined to achieve a certain ratings defined by probability of default, meaning they have a dual implicit definition.

      It is convenient to introduce the notation Ldd+l for the layer with limit l in excess of attachment d. The layer pays

       (3.6)

      for a subject loss x. The detachment or exhaustion point of the layer is d + l. The cover can be written succinctly as

      The notation Ldd+l mimics integrals, with the attachment and detachment points as sub- and super-scripts, and makes it easy to add them: L0l1+Ll1l1+l2=L0l1+l2.

      We use the two equivalent expressions L0l(X) and X∧l interchangeably for a ground-up cover.

      The expected loss and premium for a layer divided by the layer’s limit are called loss on line and rate on line, respectively.

      When applied to a random loss X, Ldd+l(X) becomes a random variable.

      Exercise 26 Using multiple layers it is possible to create any continuous indemnity function that increases with subject loss. Describe in words and plot payments from the following towers as functions of the subject loss 0≤x≤1000.

      1 L0500(x)

      2 L250∞(x)

      3 L2501000(x)

      4 0.5L250500(x)+0.75L500750+L7501000

      5 Which of (1)–(4) has the same payouts as a call option? What is its strike?

      6 Write the payout function for a put option in terms of L functions.

Solution. (1)–(4) See Figure 3.9. Cover (4) includes co-participations or coinsurance clauses, where some layers are only partially placed. (5) L250∞ has the same payout as a call option with strike 250. (6) A put option with strike k has payout function equal to k−L0k.

      Figure 3.9 Sample layering functions for Exercise 26.

      Remark 27 Limits and deductibles can be applied per claimant, claim, occurrence, or in the aggregate. We assume the reader is familiar with these concepts. The exact meaning of a limit and deductible is defined by that of X. In this book X almost always represents aggregate loss on a portfolio.

      3.5.4 Computing Layer Losses with the Lee Diagram

      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.

      3.5.5 Algorithm to Evaluate Expected Loss for Discrete Random Variables

      The algorithm in this subsection is very basic. We present it to establish an approach to working with discrete random variables that we generalize in subsequent chapters.

      We present an algorithm to compute E[X] in two ways, based on Eq. 3.1,

(3.9)

      The two integrals correspond to the areas shown in Figure 3.11, panels (a) and (b), respectively. In (a), dF equals minus the backward difference of S, and in (b) dx equals the forward difference of x.

Figure 3.11 Two ways of computing expected loss from a discrete sample.

      Follow these steps to evaluate Eq. 3.9.

      Algorithm.

      Algorithm Input: X is a discrete random variable,