Bahnemann.Chapter5

Reading: Bahnemann, D., "Distributions for Actuaries", CAS Monograph #2, Chapter 5.

Synopsis: To follow...

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Estimated study time: x mins, or y hrs, or n₁-n₂ days, or 1 week,... (not including subsequent review time)

BattleTable

Based on past exams, the main things you need to know (in rough order of importance) are:

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• fact B...

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In Plain English!

Chapters 1 through 4 aren't on the exam but are worth quickly reading if you have time. You don't need to worry about the proofs but it is a great introduction to the distributions you'll be using in chapters 5 and 6.

Overview of Excess Claims

All claims are restricted in size to at least as large as some fixed amount A > 0. We do not know anything directly about claims smaller than A as these would be paid at $0, so not reported to the insurer in general. A common example of this is a deductible, but other situations can occur such as when an insurer covers a non-primary layer of claims in the case of umbrella insurance.

Mathematically, if X is an unlimited claim size random variable, we define the random variable Y by [math]Y=\begin{cases}0 &\mbox{if }0\leq X\leq A \\ X-A & \mbox{if }A\lt X\lt \infty\end{cases}[/math].

The distribution Y reflects the claim size modified by the condition that the insurer pays nothing until claims reach a certain size, A.

The cumulative distribution function for Y is [math]F_Y(y)=Pr(Y\leq y)=\begin{cases}0 & \mbox{if } -\infty \lt y\lt 0 \\ F_X(y+A) & \mbox{if } 0\leq y\lt \infty\end{cases}[/math].

If [math]E[X][/math] exists then [math]E[Y][/math] does as well. We have [math]E[Y]=E[X]-E[X;A][/math], where [math]E[X;A][/math] is the expected value of X limited to A. Since this is a positive amount, we have [math]E[Y]\leq E[X][/math].

Since insurers are mainly interested in claims which they have to pay out on, it's better to work with the variable, [math]X_A=X-A[/math], which is defined on [math]A\lt X\lt \infty[/math] and is called the excess of X over the limit A. This variable ignores claims less than or equal to A and reduces all others by amount A. When a variable X is modified in this fashion, it is said to be truncated from below and shifted by A.

Three formulas worth memorizing are:

• [math]F_{X_A}(x)=Pr(X-A\leq x \;|\; X\gt A) =\begin{cases}0 & \mbox{if } -\infty\lt x\lt 0 \\ \frac{F_X(X+A)-F_X(A)}{1-F_X(A)} & \mbox{if }0\leq x\lt \infty\end{cases}[/math]
• [math]E[X_A]=\frac{E[X]-E[X;A]}{1-F_X(A)}[/math]
• [math]E[X_A;L]=\frac{E[X:A+L]-E[X;A]}{1-F_X(A)}[/math] for some limit [math]L\geq A[/math].

Bahnemann gives formulas for [math]E[X_A^2][/math] and [math]E[X_A^3][/math]. Since they're fairly complicated, it's likely you would be given them in the exam if needed.

Since Pareto distributions are common in insurance applications, it's worth taking a look at Example 5.2 from the text.

Example 5.2: Calculate [math]E[X_d][/math] for a Pareto claim size variable X

Solution

The probability density function of a Pareto distribution is [math]f_X(x)=\frac{\alpha\beta^\alpha}{(x+\beta)^{\alpha+1}}[/math], [math]0\lt x\lt \infty[/math]. The expected value, [math]E[X][/math] of a Pareto distribution is [math]E[x]=\frac{\beta}{\alpha-1}[/math] and is defined whenever [math]\alpha\gt 1[/math].

Rather than calculating directly, let's make an observation. We have [math]f_{X_d}(x) = \frac{f_X(x+d)}{1-F_X(d)}=\frac{\alpha(d+\beta)^\alpha}{(x+d+\beta)^{\alpha+1}}[/math]. This is itself a Pareto distribution with parameters α, [math](d+\beta)[/math]. Consequently, we immediately get [math]E[X_d]=\frac{d+\beta}{\alpha-1}[/math]. Keeping an eye out for relations like this can save you a lot of time on the exam...