Fisher.AggExcess

Reading: Fisher, G. et al, "Individual Risk Study Note," CAS Study Note, Version 3, October 2019. Chapter 3. Section 1

Synopsis: This is an overview of the key terminology that comes up in the next couple of readings. Although the definitions are somewhat formula heavy, the probability of being asked for one of the formulas is relatively low.

Study Tips

This is an article to quickly read a couple of times before revisiting the BattleQuiz periodically to make sure you've memorized all the details. Don't worry if it seems abstract at first, the next couple of wiki articles will help.

Estimated study time: 3 hours (not including subsequent review time)

BattleTable

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

• How to work with both a per-occurrence limit and an aggregate limit.
• Definitions for: entry ratio, Table M charge, Table M savings, Limited Table M charge and savings, Table L charge and savings, excess ratio, net insurance charge.

Questions held out from most recent exam: #X. (Skip these for now to have a fresh exam to practice on later. For links to these questions see Exam Summaries.)

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In Plain English!

When modelling an insurance contract you need to know who pays and how much they pay out. Risk sharing at an aggregate level makes it challenging to determine overall coverage responsibilities. A familiar example (to US students at least) of aggregate loss cost coverage is health insurance. An insured has a deductible, a co-payment (another type of deductible), and a maximum annual out of pocket amount (the aggregate limit for your deductible and co-pay).

Let's look at the following example from the text. Suppose we have a Commercial General Liability large deductible policy with a per-occurrence deductible of $100,000 and an aggregate limit on the deductible of $500,000. The claim payment pattern for this policy may look like:

Date	(1) Dollars of loss on claims that are each less than $100k	(2) Number of claims over $100k	(3) Dollars of loss on claims over $100k	(4) Deductible	(5) Insurance payment	(6) Insured's cost so far this year
Q1	$132,500	0	0	$132,500	0	$132,500
Q2	$93,000	2	$350,000	$293,000	$150,000	$425,500
Q3	$105,000	0	0	$74,500	$30,500	$500,000
Q4	$122,500	1	$150,000	0	$272,500	$500,000

The main thing to do is to keep track of the cumulative deductible amount (6) in relation to the aggregate limit. Claims below the per-occurrence limit are covered by the insured's deductible unless the aggregate deductible is reached. Claims above the per-occurrence limit have the insurer cover the difference between the claim and the per-occurrence deductible. If a claim would cause the aggregate deductible to be exceeded, then the insured is only responsible for the amount up to the aggregate deductible.

Put differently, (4) = min( (1) + (2)*[Per-occurrence deductible] , [Aggregate deductible] - [Prior row (6)]). Then (5) = (1) + (3) - (4).

Alice: "Here's a practice problem for you to try."

Working with aggregate deductibles

Question: What are three key questions which help price an insurance contract?

Solution:

1. How much does the insured pay before the insurer is responsible?
2. How much does the insurer pay before hitting its policy limits?
3. How much is the insured responsible for above the policy limits?

Key Definitions and Notation

Let A be a random variable for the actual loss experience and [math]E=E[A][/math] be the expected losses. The entry ratio is the ratio of actual to expected losses, and is written as [math]r=\frac{A}{E}[/math]. It may equivalently be defined as the ratio of actual loss ratio to expected loss ratio.

Pop Quiz    :-o

A policy is expected to have $300,000 in losses. At the end of the policy, actual losses were $198,000. What is the entry ratio for this policy?

We may view the entry ratio as a random variable. Similarly sized policies having the same entry ratio and coverage should behave similarly. It's common practice to estimate aggregate excess losses based on the entry ratio.

The entry ratio is often used to look up various risk characteristics that have been derived from a group of similar risks. There are several tables that may be used and the correct one depends on the underlying conditions of the policy you're asked to price.

Aggregate Limit Only Policies

One particular table is Table M. This consists of entry ratios with corresponding Table M charges and Table M savings and are usually grouped by policy size and limit. Table M is used for aggregate excess policies without a per-occurrence deductible.

A Table M Charge corresponds to an entry ratio, r, and is denoted by [math]\phi(r)[/math]. It is the expected ratio of excess losses over rE to all expected losses. It may also referred to as the Aggregate Excess Loss Factor, Aggregate Excess Ratio, or Excess Pure Premium Ratio.

Mathematically, [math]\phi(r)=E\left[\max\left(0,\frac{A}{E}-r\right)\right]=\displaystyle\int_r^\infty(y-r)\mathrm{d}F(y)[/math], where [math]Y=\frac{A}{E}[/math] is the entry ratio function and [math]F(Y)[/math] is its cumulative distribution function.

Let's use a small example to understand the Table M Charge.

Question: An insurer has five similar policies, each with an expected loss of $150,000. In normal year, the actual losses on those policies are: $132,000; $141,000; $150,000; $159,000; and $168,000. Calculate [math]\phi(1), \phi(0.6), \mbox{ and } \phi(1.12)[/math].

Solution:

Since the expected loss is $150,000 for each policy, [math]\phi(1)[/math] is the ratio of excess losses over $150,000 to the total expected losses. That is [math]\phi(1) = \frac{0+0+0+9,000+18,000}{5\cdot 150,000}=0.036[/math].

Similarly, [math]\phi(0.6)[/math] requires us to aggregate the loss dollars in excess of [math]0.6E=0.6\cdot\$150,000=\$90,000[/math]. This yields [math]\begin{align} \phi(0.6)&=\frac{132,000+141,000+150,000+159,000+168,000 - 5\cdot 90,000}{5\cdot 150,000}\\ &=0.4\end{align}.[/math]

Lastly, [math]\phi(1.12)[/math] requires us to consider the loss dollars in excess of [math]1.12E=1.12\cdot\$150,000=\$168,000[/math]. Since the largest actual loss is $168,000, there are no losses over this threshold, so [math]\phi(1.12)=0[/math]. In fact, [math]\phi(r)=0 \mbox{ for } r\ge 1.12[/math].

Note that a risk contributed 0 to the Table M Charge, [math]\phi(r)[/math] if its actual loss was lower than rE.

The insurance charge is [math]\phi(r)\cdot E[/math]. That is, the expected loss multiplied [math]\phi(r)[/math]. For a retrospectively rated policy the insurance charge is the part of the premium which is fixed and pays for losses.

A Table M Savings is very similar to a Table M Charge but in the opposite direction. The Table M Savings measures the expected gap between the actual loss and rE and then divides this by the total expected loss. It is denoted by [math]\psi(r)[/math] and may also referred to as the Aggregate Minimum Loss Factor.

A risk which has its actual loss in excess of rE contributes 0 to the Table M Savings, [math]\psi(r)[/math].

Mathematically we have, [math]\psi(r)=E\left[\max\left(r-\frac{A}{E},0\right)\right]=\displaystyle\int_0^r(r-y)\mathrm{d}F(y)[/math], where Y and F(Y) are as defined above.

Let's return to our example.

Question: Using the same five risks as the previous example, compute [math]\psi(1), \psi(0.6) \mbox{ and } \psi(1.12)[/math].

Solution:

To calculate [math]\psi(1)[/math] we need to measure the gap between the actual loss and [math]1\cdot \$150,000 =\$150,000[/math]. We get [math]\psi(1)=\frac{(150,000-132,000)+(150,000-141,000)+0+0+0}{5\cdot150,000}=0.036[/math].

Similarly, [math]\psi(0.6)[/math] uses the gap between the actual loss and [math]0.6E=90,000[/math]. Since none of the actual losses are lower than $132,000 we deduce [math]\psi(0.6)=0[/math].

Lastly, [math]\psi(1.12)[/math] uses the gap between the actual loss and [math]1.12E=168,000[/math]. We get [math]\begin{align}\psi(1.12)&=\frac{5\cdot 168,000-(132,000+141,000+150,000+159,000+168,000)}{5\cdot 150,000}\\ &=0.12\end{align}.[/math]

A neat trick to associate the symbols with the terminology is the Table M Savings uses the Greek letter Psi, [math]\psi[/math].

The insurance savings is [math]\psi(r)\cdot E[/math].

Technical Point: Although in the sample calculations of [math] \phi(r) \mbox{ and } \psi(r)[/math] we said we're calculating them, technically we're estimating them empirically by averaging N similar risks.

The net insurance charge is the difference between the insurance charge and the insurance savings. The ratio of the net insurance charge to expected losses is called the Net Table M Charge or Net Aggregate Loss Factor.

Per-Occurrence and Aggregate Limit Policies

So far we've considered policies which only have an aggregate excess limit. When there is also a per-occurrence limit, D, we need to modify our approach as follows. One way is to estimate the expected loss in excess of D and then consider the function of limited losses.

Fisher uses a subscript D to indicate the quantity is related to a policy having a per-occurrence limit of D.

So A_D is the actual policy loss when each event has been limited to D. Similarly, [math]E[A_D][/math] is the expected value of the losses limited to D. Then the excess ratio is defined as [math]k=\frac{E-E[A_D]}{E}[/math], where E is the expected loss without the per-occurrence limit.

Alice: "Pay attention! This notation is very similar to Bahnemann's but it's used in a different way. Bahnemann's A_D refers to the actual loss in excess of the per-occurrence limit, where as Fisher's A_D refers to the actual loss under the per-occurrence limit."

Table M_D is a table of aggregate excess loss factors and related savings where each occurrence has been limited by a per-occurrence limit, D, before accounting for the policy aggregate limit. Table M_D is sometimes also called a Limited Table M.

The entry ratio for Table M_D is [math]r=\frac{A_D}{E[A_D]}[/math].

When working with a Table M_D we use the cumulative distribution function of limited losses, [math]F_D=\frac{A_D}{E[A_D]}[/math], which is the limited version of F(r).

Important! The Table MD insurance charge and insurance savings are both calculated as a ratio to the total expected limited loss.

Alice: "Memory trick: Table M_D is the Limited Table M so the insurance charge and insurance ratio are divided by the expected limited loss."

We'll go into greater detail about Table M_D in Fisher.LimitedTableM.

Table L

Rather than first calculating a distribution of losses limited by a per-occurrence limit and then calculating losses in excess of an aggregate limit, we estimate the covered loss on a policy with both a per-occurrence and aggregate limit in one go using Table L.

A Table L contains a single factor, known as the Table L Charge. We'll learn about this in detail in Fisher.TableL.

The Table L entry ratio, r, is defined as [math]r=\frac{A_D}{E}=\frac{\mbox{actual }{\color{red}\textbf{limited}}\mbox{ aggregate losses}}{\mbox{expected }{\color{red}\textbf{unlimited}}\mbox{ aggregate losses}}.[/math]

The cumulative distribution of losses used in Table L is denoted by F* and is the cumulative distribution of [math]\frac{A_D}{E}[/math].

Important! Notice the subtle difference between the cumulative distributions for the Limited Table M and Table L. Table L uses unlimited expected loss where as the Limited Table M uses limited expected loss.

The Table L excess ratio, k, is the same as the Limited Table M excess ratio.

The Table L insurance charge at entry ratio r is [math]\phi_D^\star(r) =\displaystyle\int_r^\infty (y-r)\mathrm{d}F^\star (y) +k[/math]. It is the average difference between the risk's actual unlimited loss and its actual limited loss, plus the risk's limited loss in excess of r times the expected unlimited loss.

The Table L insurance savings at entry ratio r is [math]\psi_D^\star(r)=\displaystyle\int_0^r (r-y)\mathrm{d}F^\star (y)[/math]. It is the average amount by which the risk's actual limited loss falls short of r times the expected unlimited loss.

They are average amounts because in the definition of F* we divided by the expected unlimited loss, E.

Lastly, Table L is sometimes called the California Table L.

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Pop Quiz Answers

The entry ratio is [math]r=\frac{A}{E}=\frac{198,000}{300,000}=0.66[/math].