ACTUARIAL METHODS AND REINSURANCE COMPANY RESULTS

Stephen Mildenhall

Summary

Actuarial methods, and problems with applying actuarial methods, help explain various aspects of insurance company management
The Actuary is in a unique position to understand these methods and their pitfalls, and therefore has an important role to play in insurance company management
Actuarial methods, which may seem dry and abstract, have direct and tangible impact upon insurance company management and results
Need to apply actuarial methods correctly and strive to get best possible data
Need for continuous improvement in actuarial methods

Reinsurance and Reinsurance Company Results

Property / Casualty covers (AL, GL, WC, Property, HO, ... )
Reinsurance: insurance of insurance companies. Mostly done through a reinsurance treaty, insurance of an underlying book of business. See Cowan's class for more information.
Results often measured using combined ratio: losses plus expenses divided by premium. Example.
Operating result: includes investment income from holding insurance premiums. Combined ratio still regarded as good measure of underwriting effectiveness of an insurance company.
Accident year vs. calendar year
Slide 1: Historical results on a calendar year basis
Why have results all deteriorated so badly?
Need to look at the two main actuarial functions: pricing and reserving, combined with credibility theory, a corner-stone of Actuarial Science
Techniques below are more pricing and reserving principles than specific to reinsurance

Reserving

Reserving: determining ultimates losses on a book of business
Example of development of an individual claim, stress AY / CY difference
Simple LDF with three years (Slide 2)
How to estimate the lower half of the triangle?
Chain Ladder Method and Factors-To-Ultimate
Loss development triangle (Slide 3); describe example (fixed $80/unit loss cost, no trend, stable exposure base, changing premium adequacy)
Chain Ladder method (Slide 4); simple example
Factor-to-ultimate selection (Slide 5)
Chain Ladder Projections (Slide 6)

Credibility Theory

Credibility: a measure of the predictive power of data
Credibility theory tries to balance lower variance, biased estimators with higher variance, unbiased estimators
Shooting at targets analogy
High credibility if you are an accurate shot relative to the distances between your targets (i.e. you hit the target you are shooting at!)
Accuarcy of your shooting: expected value of process variance (EVPV)
Dispersion of targets: variance of hypothetical means (VHM)
Bühlmann credibility: Z = n/(n+K), K = EVPV / VHM, n = number of years experience

Credibility and Reserving using the Chain Ladder Method

Target: ultimate loss amount (unknown)
Shot: Factor-to-ultimate times observed losses
Accuracy of shot: variability in observed losses and uncertainty in selecting FTU
Highly variable FTUs (in first year, one very large observation; in second year 99% confidence interval about 2 to 9) lowers credibility of CL method results
We can use a Bühlmann credibility method to estimate ultimate losses (see Appendix). We get

Ultimate = (Expected Unpaid Losses) + Losses-to-Date.

Using unpaid as proxy for unpaid or unreported
Bornhuetter-Ferguson estimate unpaid losses as (Ultimate Losses) x (1-1/f) where f = Factor-to-Ultimate, giving the BF estimate of ultimate losses:

Ultimate = (Expected Ultimate Losses) x (1 - 1/f) + Losses-to-Date.

The Prior Expected Ultimate Losses are a key component of the estimate.

Pricing

Pricing is the process of determining a prospective rate for an insured
Unlike reserving, nothing is known about events during the exposure period when premium is determined
Determining ultimate historical losses is a key component (average of last five years losses)
Must adjust for changes in exposure, inflation, changing policy terms and conditions
Result of pricing: Premium = Loss Component + Expense Component + Profit Provision
Pricing determines a prior expected ultimate loss, as needed in BF estimate

Applying Bornhuetter-Ferguson

Slide 7 shows an application of BF to the current example (CL method unchanged)
Assumes constant pricing to an 80% loss ratio
Assumes Pricing and Reserving departments not in communication!
Results more stable that CL method; as expected, credibility has reduced the variance of the estimated Ultimate losses
What does the reserving actuary select?

Chicken and Egg

Here is the problem: if pricing and reserving do not communicate then an under-estimate of ultimate losses by reserving (caused by ignorance of some key factors) leads pricing to believe a line is more profitable than it actually is
Pricing is then more aggressive in its pricing, exacerbating the problem for the coming year
Adjusting historical premiums to current rate level (i.e. to a true measure of exposure) is a key component of all rate making, and is always attempted in a revision of manual rates
Even for manually rated classes with little / no underwriting judgement, this can be difficult
For individually rated large accounts, becomes more difficult
For reinsurers, even more difficult: pricing on policies in a treaty unknown when treaty terms are set; can be hard to get necessary information from ceding companies.
Requires a close connection between pricing and reserving actuaries and some kind of "pricing-reserving" feedback loop, on an account-by-account basis

Adjustment for Changes in Rate Adequacy

Slide 8 shows Slide 7, with premium replaced by an objective measure of exposure (e.g. number of trucks, receipts, payroll etc.)
Prem/Unit column shows premium adequacy has been decreasing
Information was missing from Slide 7 and so the key prior estimate of ultimate losses was understated
BF method now nearly 20 points higher for 1999; 15 points higher in 1998

Other Methods of Loss Development

Slide 9 shows an additive approach to estimating unpaid or unreported losses
Replace E(U).(1-1/f) with the average incremental losses at each development point
Estimating 1/f is fraught with hazards!
Slide 10 shows results of this method
Est Ult columns use estimates of reserves from the triangle; these would be available to the actuary
Actual Ult columns use the (generally unknown) actual means for each development period; if available would be the best estimate of Ultimate losses for this model
Expected loss ratio = known expected loss of $80/unit divided by actual average premium per unit. Ignores information about observed losses to date

Comparison of Methods

Slide 11 shows all six methods discussed so far, together with the high/low estimate
In practice only the first four methods would be available to an actuary
Exposure based methods, with reliable rate change information are harder to get, particularly in reinsurance
Extreme variability of CL method evident in early years; typically leads to its rejection and reliance on a BF type method
BF with no adjustment for premium adequacy a poor performer in 1998 and 1999
BF with Units and Additive BF (Est IBNR) methods both much closer to Actual IBNR column

Summary and Conclusions

Reinsurance pricing is one step further removed from account pricing than primary company pricing
Projecting further into future, treaty terms are set at the start of the year for all policies written during the year
Pricing relies on hard-to-quantify and hard-to-obtain information about primary company rate levels
Pricing relies on ultimate losses that can take many years to know with certainty
Leads to a reliance on pricing and reserving methods that use a prior estimate of ultimate losses
Can show such a formula is a Bühlmann credibility estimator and is actuarially sound
but...need best estimate of prior ultimate losses
Requires that pricing and reserving communicate and both have realistic assessments of current price adequacy

Appendix: Bühlman Credibility Estimate of Ultimate Losses

Let U be ultimate losses and L be losses observed at some evaluation (e.g. after 12 months, after 24 months, etc.) Bühmlann credibility is defined as the best linear approximation to the Bayesian estimate of U. A linear estimator means one of the form a+bL. If a = 0 and b = FTU this reduces to the usual Chain-Ladder method.

The Bühlmann credibility estimate is dervied by minimizing the expected squared error loss:

Q = E(U-a-bL)².

To do this, differentiate with respect to a and b and set equal to zero giving

¶Q

¶a

= -2E(U-a-bL) = 0

and

¶Q

¶b

= -2EL(U-a-bL) = 0.

These give two equations

E(U) = a + bE(L)

and

E(LU) = aE(L) + bE(L²).

Multiplying the first equation by E(L) and subtracting from the second gives

E(LU)-E(L)E(U) = b(E(L²)-E(L)²)

Cov(L,U) = bVar(L).

Now, U = L+B where B is the (unknown) reserve, and we can assume L and B are independent. Thus

Cov(L,U) = Cov(L,L+B) = Var(L)

and so b = 1. Substituting into the first equation above we get a = E(U)-E(L) = E(B), expected unpaid or unreported losses. Thus we have shown that the Bühlmann credibility estimate of Ultimate losses is given by

E(U)-E(L) + L.

Note that L is a known quantity. This equation is exactly the Bournuetter-Ferguson estimate of ultimate losses, derived using a different technique in the 1970's.

We can estimate E(U)-E(L) using loss development factors, as follows. From the definition of link ratios we know

E(U) = fE(L)

where f is the factor-to-ultimate. Thus we can estimate E(U)-E(L) as E(U)-E(U)/f = E(U)(1-1/f). The estimate of E(U) is generally derived as premium times expected loss ratio. It is a prior estimate of ultimate losses.