A Survival Model Approach to Non-life Run-off Triangle Estimation

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A Survival Model Approach to Non-life Run-off
Triangle Estimation

• Casualty Loss Reserve Seminar
• Washington, D.C. 18 September 2008
• Brian Fannin

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Agenda

1. Motivation
2. Brief Review of Survival Models
3. The Method
4. What Next?

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Motivation
4
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• Ad hoc methods have been replaced by stochastic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 incurred- have 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
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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 xa). If a1, the subscript is dropped.
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
aqx 1 - apx
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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.

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The method
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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
2 2270 94 0.041 0.959 0.951
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
Closure probabilities by age are simply given by
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
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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

The parameter e controls the relative weight one
places on either smoothness or the sample values.
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Smoothed Results
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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?
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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_at_MunichRe.com.