The Credibility of the Overall Rate
Indication
Paper by Joseph Boor, FCAS
Florida Office of Insurance Regulation
Presented by Glenn Meyers, FCAS, MAAA,
Background-Why is this
needed?
– Actuaries in North America and elsewhere base general insurance rates on claims data, not tables.
– Claims data often has too few claims to fully rely on it. – Actuaries combine claims data with some other data
in a credibility formula
cost estimate = Z*(claims data) + (1-Z)*(other data) – Z is called the credibility
Background-Why is this
needed?
• For making broad (i.e., not individual insured) rates,
two general approaches are regularly used
– Square root rule
[(proxy for expected claims or losses)/(some F)]1/2 • ‘F’ is ‘full credibility standard’
– Bühlmann credibility
• (expected loss proxy ‘P’)/(P+K)
• ‘K’ is based on variance structure of two data elements combined
Background-Why is this
needed?
• Problems with square root rule
– Square root rule is not optimum credibility in terms of predicting claims costs
– Merely guarantees, up to a certain probability, that the effect of random fluctuations in the claims data is contained at a certain level.
– Does not address fluctuations or uncertainty in the ‘other data’ (called complement of credibility)
Background-Why is this
needed?
• Problems with Bühlman credibility
– Not designed for the overall rate indication, suitable for making rates for a class that is part of a much larger line of business.
– Most actuaries do not know a good method to compute the constant ‘K’
What is needed
– Credibility formula for credibility producing ‘greatest accuracy’ estimate of future claims costs
– Suitable for use for a line of business as a whole
• Complement of credibility is claims costs expected in current rate, updated for inflation, etc.
– Reasonably simple formula
– Advice to practitioners on how to compute key constants
Overview of Results
2 2 2 22
1
4
1
ε
δ
ε
δ
−
+
=
Z
– Formula is
Constants in formula
• Constants are:
– δ² is the coefficient of variation(squared) of the change in claims cost levels factor each year
• Each year’s cost change is (1+T)*(1+Δ) where δ² is the variance of Δ and everything else is constant.
– ε² is the coefficient of variation(squared) of the
observed results around the real underlying loss costs each year
• Each year’s observed claims costs are (L)*(1+E) where
ε² is the variance of E, X is the true expected costs and everything else is constant.
Underlying assumptions
– Almost all credibility models involve
assumptions
–
To understand δ² and ε
² you must understand
the assumptions of this model
• The true underlying expected losses follow a geometric Brownian motion
• The data does not contain the true expected
losses from each prior year, the expected losses are imperfectly observed
Geometric Brownian motion
assumption
– Expected change in losses for each year is
increase of T
• E.g., next years losses L(y+1) have expected value L(y)*(1+T)
– Actual changes vary from expected by
multiplicative factor of (1+
Δ
)
• so L(y+1) =L(y)(1+T)*(1+ Δ)
• Δ’s for each consecutive year are independent, but
Geometric Brownian motion
assumption
– GBM requires slightly more restrictive
assumptions than those of prior page
• GBM not required for the formula
– GBM is however, a very popular financial
model that meets criteria above.
– Model should also accommodate the Levy
processes that are also popular with financial
professionals
Observation Error
– Typically, actuary’s data never quite
represents the true expected claims costs
• Law of Large Numbers never guarantees exact calculation of costs, just approximation
• More importantly much US data is loss development estimates, not true costs
Observation Error
– Model assumes each years observed data is
multiplicatively distributed around true expected
costs
– Instead of seeing true costs L(y) we see
)]
y
(
E
[
)
y
(
L
)
y
(
Lˆ
+
1
=
×
1
+
– E(y)’s are independent and identically
Estimating δ² and ε
²
– Reference item- Bühlmann credibility
significantly underused in US due to lack of
general knowledge on how to compute
Estimating δ² and ε
²
• Paper presents methods for estimation
• Methods based on subtracting squared differences of
s y Lˆ( )'
• Squared differences between terms different #years
apart are different linear combinations of δ² and ε². • Credibility that would have worked in the past
• Fitting parameters across a wider group of data
• Estimating δ² or ε² structurally (e.g estimating ε² from loss development variance and collective risk)
• Suggest use two methods and understand the error in each.
Other relevant items
– Formula is for steady-state credibility and
updating the rate with one year of data.
• Over a series of papers, the author intends to
expand the analysis to embrace non-steady state credibility, conversion from non-optimal credibility, multiple years of data, rates made less often that annually, etc.
Other relevant items
– For geometric Brownian motion, the formula is
actually slightly different
• The formula given is essentially the formula for linear (standard) Brownian motion with additive error
• It is a high quality approximation to the slightly more complex formula for geometric Brownian motion
Other relevant items
– Other papers have dealt with this issue, but
not to this level
• E.g. Gerber and Jones (Transactions of the Society of Actuaries, 1975) dealt essentially with linear –type
problems and had more elegant formulas
• This paper uses nonlinear geometric Brownian motion, more realistic for North American general insurance
• Paper’sapproach is more calculation-oriented
– Technique works in a wider variety of situations and is in itself an important aspect of the paper.
Other relevant items
– This paper employs a more
calculation-oriented approach
• The technique can be applied to a wider variety of situations and is in itself an important aspect of the paper.
• Involves computing the variance of the observed data from the detrended true future cost level
• Lengthy basic statistics and summing of
mathematical series does lead to a more robust approach
Enhancement to Bühlmann
model
• Bonus item in
paper-– Several authors, including this presenter, have proven that, when losses change from year to year as with this model, the Z= P/(P+K) model should be Z = P/[P(1+J)+K]
– An appendix in the paper shows that this is also true when the data has loss development uncertainty.
