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 cornerstone 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 FactorsToUltimate
 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
 Factortoultimate 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: Factortoultimate 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) + LossestoDate.
 Using unpaid as proxy for unpaid or unreported
 BornhuetterFerguson estimate unpaid losses as (Ultimate Losses) x (11/f)
where f = FactortoUltimate, giving the BF estimate of ultimate losses:
Ultimate = (Expected Ultimate Losses) x (1  1/f) + LossestoDate.
 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 BornhuetterFerguson
 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
underestimate 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 "pricingreserving" feedback loop, on an accountbyaccount
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).(11/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 hardtoquantify and hardtoobtain 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 ChainLadder
method.
The Bühlmann credibility estimate is dervied by minimizing the expected
squared error loss:
To do this, differentiate with respect to a and b and set equal to zero giving
and

¶Q
¶b

= 2EL(UabL) = 0. 

These give two equations
and
E(LU) = aE(L) + bE(L^{2}). 

Multiplying the first equation by E(L) and subtracting from the second gives
E(LU)E(L)E(U) = b(E(L^{2})E(L)^{2}) 

or
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
Note that L is a known quantity. This equation is exactly the BournuetterFerguson
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
where f is the factortoultimate. Thus we can estimate E(U)E(L) as E(U)E(U)/f
= E(U)(11/f). The estimate of E(U) is generally derived as premium times expected
loss ratio. It is a prior estimate of ultimate losses.