SP7 General Insurance

Reserving

Chain Ladder

How are they made

First we need to consider how policies written, accidents happening, being reported and claims being paid out are used to fill values into the various different run off triangles that we can use.

Once we have data in a triangle the mathematics of calculating the reserve is the same whatever the triangle represents

Spreadsheets

There are a number of different spreadsheets you can look at to back up the calculation in this section of the course:

Classic triangulation methods Basic Reserving Calculations. This spreadsheet contains 6 years of data to illustrate the methods more clearly.

Simplified 4 year spreadsheet (suitable for hand calcs in lecture) Basic reserving (4 year).xls

I have also included a spreadsheet that allows us to compare the accuracy of using annual and quarterly chain ladder

Chain Ladder - Calculation Method

The chain ladder requires us to follow the steps below

Gather Data

Gather our data into a run-off triangle for whatever kind of reserve we are trying to calculate

Cumulate Data

Then we sum along the rows to cumulate the data

Filling in the Blank Cells

The blank cells represent the future - that we do not yet know. The purpose of this process is to try and make as good an estimate as possible as to what is going to happen in the future

Can you guess a figure you might put in cell(2014,4)

Guess = $125 \times \frac{100}{95} = 132$

What about cell(2015,4)

We might be tempted to choose $70 \times \frac{100}{80}$, but this would not be a good guess because we have not used the accident year 2014 data that we have.

Cell(2015,3) is more intuitive. This time we have two years of data which has been developed for 3 years so we can use both of these years to guess this cell.

Guess = $70 \times \frac{95+125}{80+100} = 86$

We can now see that:

the ratio of development year 4 to development year 3 is just $\frac{100}{95}$ and

the ratio of development year 3 to development year 2 is just $\frac{95+125}{80+100}$

These numbers are called the development factors and once we have calculated them for each development year we can use them to fill in the whole triangle

The following table sets out the calculation as you will often see in a spreadsheet as a convenient way of organising the data is to sum each column and then take the last value of when calculating the following year's development factor

We often notate the development factors $f_{1,2}$ and $f_{2,3}$ etc.

We should note the relationship $f_{1,3} = f_{1,2} \times f_{2,3}$ etc and specifically:

$f_{1,n} = f_{1,2} \times f_{2,3} \times f_{3,4}... \times f_{n-1,n}$ and

$f_{2,n} = f_{2,3} \times f_{3,4}... \times f_{n-1,n}$ and so on

And so we can easily continue to finish off the run-off triangle:

For each accident year the reserve to be held is the projection to the end of the triangle MINUS the LAST piece of "hard" data for that year. In the case of IBNR - this last piece of data is the last year for which we actually have the accident reports.

The total IBNR (or whatever reserve we are calculating) is then the sum of these values for each accident year

Chain ladder - further considerations

Issues to consider when looking at development triangles are:

• What do link ratios of less than 1 mean and what do we do about them
• What happens where we have incomplete triangles

Many issues we come across are similar to issues around handling data

There are many other methods which are variations on a theme of the chain ladder method:

Berquist Sherman

Adjust historic claims values to bring them into line with up to date claims handling practices

Curve fitting

Chain ladder is in fact a special case of curve fitting in which we fit the development factors exactly. More generally we could find a curve which was a close approximation to the actual development factors to be fitted

Trend analysis

Similar to curve fitting in that data can be cut into different cohorts and then any key features and trends can be analysed before recompiling back into a set of development factors or more general relationship between different development years

Class Exercise

Calculate the IBNR claims estimate for the data given below using the basic chain ladder method

Expected Loss ratio

Reserves calculated by comparing the claims paid with the total claims expected to be paid based on an a-priori expectation of the total losses incurred on a policy

Useful when data is scant or not reliable

Often reliant on underwriters best judgement or industry data

Subject to risk from underwriting cycle or spuriousness of past practices

Again using the data from above:

but now we need to know what the expected loss ratio is and also we need to know what the earned premiums were for each year

So assume an expected loss ratio of 80% and using the following written premiums

The calculation proceeds as follows:

Notes

The earned premium for each year is half the written premium in the previous year + half the written premium in the current year

Where the total expected claims has already been exceeded then we would hold a zero rather than a negative reserve

Class Example

Assuming an expected loss ratio of 80% and the written premiums given below:

Calculate the IBNR of the same data using the expected loss ratio method

Bornheutter-Ferguson

The chain ladder can produce very volatile results especially for undeveloped years and the expected loss ratio method does not use the data from claims that have already emerged.

The Bornheutter-Ferguson method is a composite of the two in which we count the claims already reported (paid depending on triangle) but then assume the future claims will be the unreported proportion of our original expected loss ratio

The following simple example should make this clear

Bornheutter-Ferguson - Method

So Bornheutter-Ferguson is all about accepting what has happened and then predicting what will happen next on the basis of what you though was going to happen.

The difference between this and the above illustration is we do not know that we are half way through the time interval.

So for the BF calculation we need to:

• Calculate the proportion of the triangle that has already been run-off
• Multiply the remaining proportion by the original expected claims

Going back to our original data and assuming an expected loss ratio of 80%:

What proportion of the claims has been run off after 3 years?

Guess = $\frac{95}{100} = 0.95$

What proportion have not been run-off (after 3 years)

1 - 0.95 = 0.05

What was our original expected loss for year 2014? (We assume that year 2013 is fully run-off)

Expected Loss Ratio (80%) times Earned Premium (135) = 108

So our reserve (2014) = $0.05 \times 108 = 5.40$

You may notice that the proportion not run off: $1-\frac{95}{100} = 1-\frac{1}{f_{3,4}}$ where $f_{3,4}$ is the development factor from year 3 to year 4 in the chain ladder model

This logic extends simply to the other accident years so

the proportion of claims not run-off after 2 years is $1-\frac{1}{f_{2,4}} = 1-\frac{1}{f_{2,3} \times f_{3,4}}$ and

the proportion of claims not run-off after 1 year is $1-\frac{1}{f_{1,4}} = 1-\frac{1}{f_{1,2} \times f_{2,3} \times f_{3,4}}$

Thinking of this expression: $\frac{1}{f_{1,2} \times f_{2,3} \times f_{3,4}}$ as the proportion actually run off after each successive development year may be an easier way to think of this

This all makes the BF method relatively easy once we have done the chain ladder as above. The following tables illustrate how the result develop:

Cape Cod

Cape Cod is a special case of the Bornheutter Ferguson method where we use our existing claims data to calculate the expected loss ratio

We do this by dividing the total claims by the total earned premiums but given that the total claims are not yet run off we adjust by factoring the total earned premiums down by the proportion of total claims run off which we calculate from the existing claims data

Using the same example again a very simple estimate of the expected loss ratio could be made by using the fully developed year (2013)

$Expected\ loss\ ratio\ =\ \frac{total\ claims\ paid}{total\ earned\ premiums} = \frac{100}{125} = 80\%$

However this would be ignoring the subsequent years so in fact we include the paid (/reported) claims so far on the numerator and then ratio down the earned premiums on the denominator by the proportion of the claims that have been run-off like so

$Expected\ loss\ ratio\ = \frac{100 + 125 + 70 + 80}{125 + 135 \times 95\% + 140 \times 77.73\% + 145 \times 46.64\%} = 87.27\%$

We then simply redo the BF method with our new Expected loss ratio and get the following results:

Class Exercise

Now use the Cape Cod method to calculate the reserves for the data given above

Inflation Adjusted Reserving (see spreadsheet)

To perform the inflation adjusted reserving we first need to decide what our base time is for calculating present values

We must then make sure we use incremental data to adjust for inflation as we do not wish to work on data in which claims in two different periods are added together

It is also important to decide the measure of inflation we are going use to adjust for inflation. This could be a subjective decision or we could inform this by using the inflation claims data from previous years

Having adjusted our incremental claims data to a consistent point in time we can then proceed as normal with whatever reserving method we wish to use

Having projected forward our reserves we then have to adjust for inflation in the future as well by adding the projected inflation to mid years to each of the incremental claim values

Class Exercise

Assuming inflation was 3% in 2012, 2013 and 2014 and has been 2% in 2015 and 2016 and then the forecast for 2017 and future years is 4%, calculate the undiscounted reserves which should be held in respect of the above data as at 31 December 2016

Average Cost Per claim Method (see spreadsheet)

The average cost per claim method largely proceeds by common sense

It essentially involves calculating the average cost per claim as a separate triangle and then projecting this triangle and the number of claims triangle forward

Points to note:

• The number of claims triangle can be treated exactly the same as a normal chain ladder, which it effectively is, where each claim happens to be £1
• The average cost triangle must be calculated using the incremental triangle and then having produced a triangle with average costs in it we project this forward as if it is a cumulative triangle and the values in each cell are the projected forward average costs per claim which are then multiplied back to the incremental number of claims triangle to produce the final reserve.

It is quite a fiddly process, which is aided by just following it through on the spreadsheet, always remembering what you are doing here is common sense

Data Grouping

Development Period

Year, quarter or month

Homogeneity of Data

Natural trade-off between the statistical properties of large numbers and the fact that different tranches of data from different sources will have different properties

Treatment of large Losses

May be necessary to treat some losses separately due to their distorting effect

Latent claims

Triangulation may not be appropriate for such risks as asbestos, pollution and health hazards as the development will be more associated with calendar years than time after exposure

Claims Cohorts

Accident year or Reporting year (Does reporting year make sense in the case of run off?) or Underwriting year

Accident year is grouping all the data by those claims relating to accidents which occurred in a given year

Reporting year is grouping all the data by those claims reported in a given year

Underwriting year is grouping all the data by those claims relating to policies written in a given year

Data

Every area of the course can be used to generate ideas for questions on data. These are NOT repeated here

Issues to consider:

• Who are the users of data
• What consequences do poor and inadequate data have for each user and use
• How does data quality vary with its source
• What is the data used for
• What would a really good data system look like and what information would it contain

Operational Issues

• There is an infinite variety of ways of generating marks in an exam from this issue
• Administrator errors
• Wrong policy number
• Wrong claim date
• Decimal point is wrong place
• etc etc etc

Claims handling procedures

• A change of personnel could change the proportion of claims that are paid out
• Legacy systems
• Grouping of data (homogeneity etc)
• Paid claims or claims requested (what was actually recorded)
• How far does data go back
• Distribution methods
• Different brokers may record data in different ways

Reserving Bases

Reserving Bases (2)

How can bases vary?

• 1 in x year insolvency events
• Data period for setting model parameters
• Claim delay patterns
• Claim size distributions
• Future business growth
• Expenses and costs
• Stochastic vs deterministic vs ranges
• Inflation and investment return

Additional reserve for unexpired risk (AURR)

This reserve is where the unearned premium reserve is inadequate as may be the case if the initial premium was inadequate

May occur where aspects of risk have not been adequately anticipated in advance of writing business

Can be very difficult to calculate exactly and is therefore very sensitive to the setting of the basis