Bahnemann.Chapter6

Reading: Bahnemann, D., "Distributions for Actuaries", CAS Monograph #2, Chapter 6.

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!

Premium Concepts

Most of Section 6.1 of the text should be very familiar to anyone who has worked in insurance for a while. The key points are given here but you should skim-read the source just to be safe.

The premium charged for a policy is the expected loss (including expected Allocated Loss Adjustment Expense) plus a load for general expenses, underwriting profit, and a provision for risk.

Let N be the per policy claim count random variable, m be the number of exposures, and φ be the ground-up claim frequency per exposure. Let Y be the claim size including ALAE. Then [math]E[N]=m\phi[/math] is the expected claim count and the expected loss and ALAE for a policy is given by [math]E[N]\cdot E[Y][/math]. The per policy pure premium is [math]p=\phi\cdot E[Y][/math].

Usually, the expected claim count, [math]E[N][/math], depends on the exposure base associated with the coverage. A policy may have one exposure such as in the case of a 1-year Homeowners policy, or may have multiple exposures. An example of multiple exposures on a policy would be a 6-month auto policy which covers 3 vehicles. This has an exposure of [math]0.5\cdot 3=1.5[/math] vehicle years.

Claim frequency is the expected number of claims per unit of exposure, and claim severity is the average claim size given a claim has occurred.

The risk charge (also know as the provision for risk) is extra premium collected by the insurer to cover:

1. Process risk - the random fluctuation of losses about the expected values.
2. Parameter risk - the uncertainty surrounding the selection of model parameters.

The rate per unit of exposure is given by [math]R=\frac{p+f}{1-v}[/math], where p is the pure premium, f is the fixed expense dollars, and v is the variable expense percentage.

The policy premium is given by [math]P=mR[/math].

If all expenses are variable then f = 0 and the quantity [math]\psi=\frac{1}{1-v}[/math] is called a loss cost multiplier (LCM). The LCM is used to load all other costs on top of the pure premium to get the final rate.

Increased Limit Factors (ILFs)

You can either use empirical loss data organized by the per policy limits to calculate increased limit factors for higher levels of coverage, or you can fit a distribution to the loss data. Fitting a distribution is useful for obtaining factors for higher limits where there may be a lack of credible data.

An increased limit factor is the ratio of the policy premium at limit L to the policy premium at the basic limit, b. Mathematically, [math]I(L)=\frac{P_L}{P_b}=\frac{E[Y;L]}{E[Y;b]}[/math], under the following assumptions:

• The loss cost multipliers are identical for each limit,
• Frequency and severity are independent,
• Frequency is the same across the layers (doesn't change by policy limit).

Expense Loading and ILFs:

There are a couple of ways that l(allocated) loss adjustment expenses can be incorporated into increased limit factors.

1. The policy limit applies to the total claim amount, i.e. indemnity loss plus allocated loss adjustment expense.
   • In this case, the ILF is the ratio of policy severities at limits L and b.
2. The policy limit applies only to the indemnity portion of the claim (usual situation). Letting X be the indemnity component of the claim and ε be the average per claim allocated loss adjustment expense, set [math]E[Y;L]=E[X;L]+\epsilon[/math], then the ILF is defined as usual.
3. Instead of 2. assume the allocated loss adjustment expenses have a relationship with claim size. Usually bigger claims have more loss adjustment expenses, so assume loss adjustment expenses are a fixed multiple u of the indemnity amount. Then [math]E[Y;L]=E[X;L]+u\cdot E[X;L][/math] and defined the ILF as usual.
4. Approaches 2 and 3 may be combined to get a general formula: [math]E[Y;L]=\left(E[X;L]+c\right)\cdot\left(1+u\right)[/math].
   • This approach is used by Insurance Services Office, Ltd (ISO) to load c for ALAE and then apply u for ULAE (unallocated loss adjustment expenses) on top.