A Survival Model Approach to Non-life Run-off
Triangle Estimation
Casualty Loss Reserve Seminar
Washington, D.C. 18 September 2008
Brian Fannin
Agenda
1. Motivation
2. Brief Review of Survival Models
3. The Method
4. What Next?
2
Motivation
3
The Current Situation
Significant progress has been made in refining the way in which we
analyze aggregate loss triangles
 Ad hoc methods have been replaced by stochastic models
 Variance of the estimate of loss reserves now receives a great deal of
attention
 Techniques which combine triangles- particularly paid and incurredhave been developed
 Correlation between triangles or lines of business are becoming more
sophisticated.
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However …
We're still using the same aggregate data, presented in the same format in which it's
been presented for over half a century.
The target estimator in most reserving exercises (and virtually every individual size of
loss distribution) is the sum of nominal loss payments. Other quantities are of secondary
interest
 This amount is of limited use in determining the market cost to transfer the liabilities
 When a discounted value is needed, typically the nominal estimate is shoe-horned into
a payment pattern which has been derived elsewhere.
Further, although they do quite a bit to help us make better estimates, they tell us little
about the underlying processes which affect the ultimate cost of claims. What causes the
variance in the level of reserves?
Put another way, can you answer the following questions?
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Can you answer these questions?
 What is the impact to our loss reserves of a 1% rise in the rate of
medical inflation five years from now?
 What effect does a delay in claim reporting have on case reserves?
 What is the impact of changes in claims department staffing levels?
 What is the probability that claims will remain open 5 years more or
less than expected?
 How long will our current liability book remain open?
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The first step towards a different approach
Rather than looking at aggregate behavior, why not look at the life cycle
of an individual claim?
 A claim occurs (is "born")
 The claim is reported (enters a population under observation)
 Payment(s) are made
 The claim is closed ("dies")
The language in parentheses is deliberate. A pool of claims can be
regarded as being analagous to a population of lives.
We'll look at casualty claims from a survival model perspective.
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Brief Review of Survival Models
8
Survival Model Mathematics
Incredibly easy.
The function of interest S(x) measures the probability that a random
variable will be greater than or equal to some fixed quantity, x. This is
nothing more than the complement of the cumulative probabilty
distribution, or S(x) = 1 – F(x).
When used to describe age, the function describes the probability that
a life will survive to an age greater than x.
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Survival Model Mathematics Continued
Given a life at age x, the notation apx
describes the probability that a life aged x
will survive for an additional a years (or will
survive past age x+a). If a=1, the subscript
S(a  x )
a px 
S( x )
is dropped.
x 1
S(x) is related to px as follows:
We often talk about the complement of the
probability that a life will survive for an
additional time, i.e. the probability that a life
will terminate in a particular interval. We call
this quantity qx
A Survival Model Approach to Non-life Run-off Triangle Estimation
S( x )   p x p 0 p1 ...p x 1
i 0
a qx
= 1 - a px
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What is the Future Lifetime?
We use the random variable K(x) to describe the future lifetime for a
life aged x. It's expectation and variance are derived using integration
by parts, which results in the following expressions.
EK ( x )  




i  0 i 1
px
E K (x)  i0 (2i  1) i1 p x
2

Var (K(x))  i 0 (2i  1) i 1 p x 

A Survival Model Approach to Non-life Run-off Triangle Estimation


i 0 i 1

2
px
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The method
12
The Method
1. Assemble data which tabulates claim closure rates by age of claim
2. Assume a binomial for the probability that a claim will close. Use
method of moments to estimate the parameters by age
3. You're done!
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Assembling the Data
I began by looking at a database of payment transactions. Age is
defined as payment year minus accident year plus one.
I'll spare you the details, but using some SQL, I was able to arrange
data so that it showed- for each age- the number of claims open and
the number of claims which would be closed in the subsequent year.
It turns out that this can be represented via a triangle. On a personal
level this was a disappointing result, but may be of some comfort to
those who still like triangles.
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Estimation of the Probabilities
Age
N
D
qx
px
S(x)
1
985
8
0.008
0.992
0.992
Closure probabilities by age are simply
2
2270
94
0.041
0.959
0.951
given by
3
3522
309
0.088
0.912
0.867
4
4461
510
0.114
0.886
0.768
5
4636
659
0.142
0.858
0.659
6
4566
820
0.180
0.820
0.541
7
3958
699
0.177
0.823
0.445
8
3322
503
0.151
0.849
0.378
9
2927
438
0.150
0.850
0.321
10
2433
322
0.132
0.868
0.279
11
2126
242
0.114
0.886
0.247
12
1838
152
0.083
0.917
0.227
13
1715
133
0.078
0.922
0.209
14
1607
86
0.054
0.946
0.198
15
1575
119
0.076
0.924
0.183
A Survival Model Approach to Non-life Run-off Triangle Estimation
q̂ x  d x
nx
If we assume the probability of
claim closure is binomial, then the
sample estimate is an unbiased
estimator of qx
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Results
Note that the probability of survival drops for the first six
years, but then raises to a relatively constant value
Probability of Claim Survival for One Year
1.200
5000
4500
1.000
4000
3500
0.800
0.600
2500
Nx
Px
3000
Nx
Px
2000
0.400
1500
1000
0.200
500
0.000
0
0
10
20
30
40
50
60
70
80
Age
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Smoothing
Just as one can alter link ratios or tail factors via either judgement or some sort of
regression or curve-fitting, one can smooth mortality rates.
The smoothing method employed in this case was the Whitaker-Henderson technique.
This method has been on earth longer than you have and is somewhat subjective.
However, I see at least one advantage: All of the survival probabilities are adjusted at the
same time. Contrast this with the typical approach of adjusting each link-ratio
individually.
The goal of the method is to create a set of factors which strikes a balance between
smoothness and reproduction of the sample estimates. The following expression is
minimized:

max 3
x 0
(3 px ) 2   x 0 px  p̂ x 
max
2
The parameter ε controls the relative weight one places on either smoothness or the
sample values.
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Smoothed Results
1.200
1.000
1Pn
0.800
Empirical
0.600
Smoothed
0.400
0.200
0.000
0
10
20
30
40
50
60
70
80
Age
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Comparison to Other Methods
Adler & Kline and Berquist & Sherman discuss a claim closure ratio
defined as the ratio of claims closed in a particular interval to the total
number of claims. Fisher & Lange use the same sort of ratio, but
compare to number of claims reported. Unless you're working with
report year data, the total number of claims is an estimate. As the
estimate of ultimate count changes, so does the closure ratio.
Teng estimates a closure ratio equal to the number of closures at any
age relative to the total reported to date. This is 1 – S(x).
In all cases, the authors do not suggest an underlying stochastic model
for claim closure rates. Rather, the presumption is that the most recent
experience will persist in the future.
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What's Next?
20
What's Next?
Loads!
 The two biggest missing items are
 Incorporation of a payment model
 Estimation of claim emergence
 Model to forecast changing claim closure rates, i.e. change in mortality
probabilities
 More sophisticated graduation techniques
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Conclusion
If you ignore the (significant!) issue of newly reported claims, I have
answered at least two of the questions that I posed earlier.
 What is the probability that claims will remain open 5 years more or
less than expected?
 How long will our current liability book remain open?
That may not be a lot, but it's a start!
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Thank you very much for your attention.
Brian Fannin
If you have any questions, please feel free to e-mail me at
BFannin@MunichRe.com.