ab page 1 advanced experience ratemaking experience rating and exposure shift presented by robert...

abAdvanced Experience

Ratemaking

Experience Rating and Exposure Shift

Presented byRobert Giambo

Swiss Reinsurance AmericaSeminar on Reinsurance

June 7 & 8, 2004

Robert GiamboSwiss Reinsurance AmericaSeminar on Reinsurance 2004

abAdvanced Experience Ratemaking


The Problem: A shift in a ceding company’s limits profile over time can distort the results of experience indications for excess of loss contracts.

Distortions will persist even if:

– individual claims are trended to future levels;

– then slotted to reinsurance layer.




Extreme Example:

– Ceding company has historically written a 50%/50% mix of 500K and 1M commercial auto policy limits.

– However, for the upcoming year, company will only be writing 1M policies.

– Typical experience rating will understate the indicated 500 x 500 layer losses for the upcoming year by approximately 50% if some adjustment isn’t made.

Question: What is a simple way to make an adjustment when limits profiles are changing over time?




Statistical Model: Assume that:

– The underlying loss distribution is the same for each policy limit;

– Inflation effects all claims equally for each experience period (no differing inflation by size of claim);

– Therefore, once claims have been trended, they all arise from same loss distribution.

Standard Experience Rating Process (casualty lines):

– Trend individual claims from ground-up;

– Cap at policy limit;

– Slot to reinsurance layer.

Initial modelling and results focus on General Liability and Commercial Auto.




The expected experience indication at year t levels based on

year t - n historical experience can be modelled as

follows:

For each policy limit K:

– Ratio of expected layer losses to total limit losses based

on the underlying size-of-loss distribution at year t

levels times

– Expected total limits losses for policies at limit K based

on year t – n limits profile.

Overall indications at year t levels based on the combination

of total limits losses across policy limits.




But this is equivalent to an exposure indication (expressed as a ratio of layer to total losses) using the policy limit profile of year t – n and underlying size-of-loss distribution at year t levels.

However, the “correct” indication we want is based on the limits profile that will be in effect in year t.

Therefore, an adjustment factor for the year t – n experience period is equal to:

– Exposure rate at year t based on year t profile divided by;

– Exposure rate at year t based on year t - n profile;

Note that the size-of-loss distribution used is the same for both exposure calculations; it is the curve at year t levels.Robert Giambo

Swiss Reinsurance AmericaSeminar on Reinsurance 2004



This adjustment is “correct” if we know the underlying size-

of-loss distribution for the ceding company.

However, the calculated adjustment is not sensitive to

the distribution used; adjustments are nearly

independent of the underlying size-of-loss distribution.

Therefore, industry curves can be used in the calculation

of the adjustment factor without any loss of accuracy.

Adjustment should be done by line of business on multi-line

contracts.




Results and Examples:

– Misestimation potential increases depending on the number of years in the experience period; the more years used, the more limit profiles can shift. Examples will show overall adjustment based on number of years in experience period.

– Effects were calculated using all ISO products, prem-ops, and commercial auto curves (61 in all) and minimum and maximum effects calculated over all curves to determine the sensitivity of results to the underlying distribution.

– Also modelled was an alternative suggested by some for handling exposure shifts; trend individual claims but do not cap at policy limits.





Note the relatively tight range of results over all ISO curves. This justifies the conclusion that an approximation of

adjustment factor can be based on industry distributions.

500,000 x 500,000Trend 7.0%Prem/Ops - 2 SG CW

Real World Example 1

Expected Loss Distribution Misestimation due to Exposure ShiftYear t Year t - 9

100,000 0.0% 0.0% Experience250,000 0.0% 0.0% Period Capped Uncapped500,000 3.0% 15.0% 3 year -7.6% 2.8%750,000 15.0% 40.0% 5 year -10.1% 4.4%

1,000,000 82.0% 45.0% 7 year -12.6% 5.7%1,500,000 0.0% 0.0% Sensitivity to ISO Curve (Capped)2,000,000 0.0% 0.0% Minimum and Maximum Effect3,000,000 0.0% 0.0% 3 year 5 year 7 year5,000,000 0.0% 0.0% -7.8% -10.4% -13.0%

-7.4% -9.8% -12.3%100.0% 100.0%




Compare to previous example; Misestimation effect is not very sensitive to underlying inflation for “capped” approach. For “uncapped”

approach, results are very sensitive to underlying inflation.


Real World Example 1


100,000 0.0% 0.0% Experience250,000 0.0% 0.0% Period Capped Uncapped500,000 3.0% 15.0% 3 year -7.7% -7.7%750,000 15.0% 40.0% 5 year -10.3% -10.3%

1,000,000 82.0% 45.0% 7 year -12.8% -12.8%1,500,000 0.0% 0.0% Sensitivity to ISO Curve (Capped)2,000,000 0.0% 0.0% Minimum and Maximum Effect3,000,000 0.0% 0.0% 3 year 5 year 7 year5,000,000 0.0% 0.0% -7.9% -10.5% -13.1%

-7.6% -10.1% -12.6%100.0% 100.0%




Not capping losses appears to be a reasonable estimator in this example – but

see next example


Industry Distribution - Products Liability




-3.5% -4.6% -5.7%100.0% 100.0%

Industry limits distributions were derived from ISO increased limits

circulars for latest year and latest

year – 9. Intermediate years were

interpolated.




Not capping losses wildly overestimates this layer.

4,000,000 x 1,000,000Trend 7.0%Prem/Ops - 2 SG CW

Industry Distribution - Products Liability




-6.8% -9.1% -11.5%100.0% 100.0%





Industry Distribution - Prem Ops




-2.1% -2.8% -3.4%100.0% 100.0%





Industry Distribution - Commercial Auto




-3.2% -4.3% -5.3%100.0% 100.0%




Assumes all umbrellas written over a $1M underlying retention

4,000,000 x 1,000,000Trend 7.0%Prem/Ops - 2 SG CW

Primary Umbrella


1,000,000 40.0% 50.0% Experience2,000,000 30.0% 30.0% Period Capped Uncapped3,000,000 0.0% 0.0% 3 year -6.8% 51.0%4,000,000 0.0% 0.0% 5 year -9.1% 68.2%5,000,000 30.0% 20.0% 7 year -11.4% 85.4%6,000,000 0.0% 0.0% Sensitivity to ISO Curve (Capped)6,500,000 0.0% 0.0% Minimum and Maximum Effect

3 year 5 year 7 year100.0% 100.0% -8.2% -10.7% -13.4%

-5.4% -7.2% -9.0%



Comments on Policy Limits Profile:

– Typically we use premium by policy limit to estimate the distribution of total losses by policy limit:

- This is equivalent to assuming that the underlying loss ratio is the same for each policy limit.

- Is this approximation good enough?

– Limit Profile may be on a different time basis than experience period:

- Profile may be in-force or policy year basis;

- Experience period may be accident year or policy year;

- May need to do some interpolation to get a profile consistent with experience period.




Comments on the Policy Limits Profile (Continued):

– What to do in the absence of historical limits profile?

- Assume some shift over time consistent with industry patterns?

- Use a default adjustment based on industry shift?




Comments on On-Level Premium:

– Question: Does the On-Level Premium need to be adjusted for shift in policy limits?

– Answer: NO


Proof By ExampleLoss

Premium Loss RatioLim 1 100 60 60%Lim 2 150 90 60%

Reinsurance Layer = Layer between Lim 1 and 2Year 1: Two Policies - 1 of EachYear 2: Two Policies - 2 of Lim 2

Total Layer ExposurePremium Loss Rate

Year 1 250 30 12%Year 2 300 60 20%

Adjustment Factor 167%

The change in exposure rate combines two effects (1) the nominal increase in layer loss between year 1

and 2 and (2) the increase in premium that results from a shift in

limits profile



Comments on On-Level Premium (Continued):

– However, when inflation is introduced there is a small, second order effect because inflation effects by policy limit vary.

– Would imply that on-level factors have to vary by limit to satisfy assumption that loss ratios are the same by limit




Comments on Property:

– Same approach should work with modification;

– The limits / total sums insured in the historical profiles must be trended to year t levels first using an exposure trend, then the exposure rating done.

– Applies only to non-cat exposure.


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