The Statistical Distribution of Incurred Losses and Its Evolution Over Time

Casualty Actuarial Society1

The statistical distribution of incurred losses
and its evolution over time

III: dynamic models

Greg Taylor

November 2000

Casualty Actuarial Society1

Table of Contents

1.Introduction

2.The parametric framework

3.Kalman filter

4.Evolutionary model with unknown second moments

5.Estimation of second moments

6.Prediction and prediction error

7.Numerical example

8.Acknowledgements

9.References

Appendices

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1.Introduction

This paper is written at the request of, and is partly funded by, the CasualtyActuarial Society’s Committee on Theory of Risk. It is the third of a trio of papers whose purpose is to answer the following question, posed by the Committee:

Assume you know the aggregate loss distribution at policy inception and you have expected patterns of claims reporting, losses emerging and losses paid and other pertinent information, how do you modify the distribution as the policy matures and more information becomes available? Actuaries have historically dealt with the problem of modifying the expectation conditional on emerged information. This expands the problem to continuously modifying the whole distribution from inception until it decays to a point. One might expect that there are at least two separate states that are important. There is the exposure state. It is during this period that claims can attach to the policy. Once this period is over no new claims can attach. The second state is the discovery or development state. In this state claims that already attached to the policy can become known and their value can begin developing. These two states may have to be treated separately.

In general terms, this brief requires the extension of conventional point estimation of incurred losses to their companion distributions. Specifically, the evolution of this distribution over time is required as the relevant period of origin matures.

Expressed in this way, the problem takes on a natural Bayesian form. For any particular year of origin (the generic name for an accident year, underwriting year, etc), one begins with a prior distribution of incurred losses which applies in advance of data collection. As the period of origin develops, loss data accumulate, and may be used for progressive Bayesian revision of the prior.

When the period of origin is fully mature, the amount of incurred losses is known with certainty. The Bayesian revision of the prior is then a single point distribution. The present paper addresses the question of how the Bayesian revision of the prior evolves over time from the prior itself to the final degenerate distribution.

This evolution can take two distinct forms. On the one hand, one may impose no restrictions on the posterior distributions arising from the Bayesian revisions. These posterior distributions will depend on the empirical distributions of certain observations. Such models are non-parametric.

Alternatively, the posterior distributions may be assumed to come from some defined family. For example, it may be assumed that the posterior-to-data distribution of incurred losses, as assessed at a particular point of development of the period of origin, is log normal. Any estimation questions must relate to the parameters which define the distribution within the chosen family.

These are parametric models. They are, in certain respects, more flexible than non-parametric, but lead to quite different estimation procedures.

The first paper (Taylor 1999a) dealt with non-parametric models only. The second (Taylor 1999b) dealt with parametric models. The present paper addresses the case of dynamic models, in which parameters are allowed to evolve from one period of origin to the next.

Familiarity with the earlier papers will be assumed here. In particular, the Bayesian background introduced and described there will be assumed.

As far as possible, the notation used here will be common with the earlier papers.

2.The parametric framework

2.1Basic framework

Let X(i,j) denote some real valued stochastic variable that is indexed by year of origin i and development year j, i 0, 0 jJ for fixed J > 0.

Note that i + j labels a particular year of calendar time (experience year), and the data for the year k = i + j constitute the vector:

(2.1)

which will be denoted X(k).

For brevity, the whole discussion here is formulated in terms of years. However, any other period, consistently replacing years, would be equally satisfactory.

Let denote the data triangle:

(2.2)

Generally, the use of the prime, as in (2.2), will indicate the incorporation of all past experience years in the associated quantity. Equation (2.2) shows that the data triangle grows over time by the addition of diagonals.

Denote

(2.3)

and suppose the form a mutually stochastically independent set.

In this set-up, corresponds to one component of the vector , and to one component of the vector , in the second of the earlier papers.

Just as the were represented in diagonals (see (2.1)), so may the be represented. Thus

(2.4)

2.2Parameters linking development years

Section 8 of the paper on parametric models represented parameters ( depended only j there) in terms of basis functions, weighted by a smaller set of underlying parameters, as follows:

(2.5)

where were the pre-defined basis functions, and the constants requiring estimation.

More precisely, in the earlier paper was a vector, each component having the representation (2.5).

This representation is extended in the present paper to the which depend on both i and j:

(2.6)

Note that the basis functions have not been changed, and still depend only on j. However, their coefficients now vary with i, creating variation in with i.

It is convenient to express (2.6) and related formulas in matrix form:

(2.7)

where matrix and vector dimensions are optionally written below their associated symbols, and

(2.8)

(2.9)

It is now possible to represent the vector in the form:

(2.10)

where

(2.11)

(2.12)

2.3Evolution of parameters

In the previous two papers, all parameters such as have been assumed constant over time. For the dynamic models that form the subject of the present paper, these parameters are assumed to evolve from one year of origin to the next, as follows:

(2.13)

where is a deterministic Q x Q matrix and is a stochastic Q-vector with

(2.14)

(2.15)

From (2.12) and (2.13), evolves over k as follows:

(2.16)

with

(2.17)

each submatrix here of order Q x Q and

(2.18)

Substitution of (2.14) and (2.15) into (2.18) yields

(2.19)

(2.20)

denoted .

Now recall (2.10), but expressed in the form:

(2.21)

with

(2.22)

In addition, denote

(2.23)

The and are assumed mutually stochastically independent.

The structure described by (2.16) and (2.21) may be recognised as the Kalman filter structure (Kalman, 1960).

3.Kalman filter

Consider the forecast of a new diagonal from data . Let denote the forecast, and suppose that it must satisfy two conditions:

(i)it must be linear in the data; and

(ii)it must be least squares in the sense that

is a minimum within the admissible forecasts, where the expectation is unconditional.

These two conditions correspond to (4.8) and (4.9) in the previous paper (Taylor 1999b).

The forecasts , k = 0, 1, 2, etc may be calculated according to the Kalman filter recursion (Kalman, 1960), as follows:

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

The recursion is initiated at h = -1 in (3.3) and (3.4) with given values and , which are the prior-to-data estimates of and its uncertainty . The recursion then proceeds through (3.3) – (3.7), and then cycles through (3.1) – (3.7) with increasing h.

Throughout this recursive scheme, a symbol of the form represents an estimate of quantity at time on the basis of data up to and including time . Thus, (3.3) with h = k is the estimator sought in the present case, and (3.4) its covariance matrix.

Equation (3.6) indicates how the parameter estimate is affected by new data X. It is seen that the previous estimate is adjusted by a multiple (in fact, a matrix multiple) of its associated prediction error.

Thus, as data emerge and reveal errors in earlier predictions, the filter corrects for them in its subsequent predictions. The matrix K is called the Kalman gain matrix.

A very useful dissertation on the Kalman filter is given by Neuhaus (1989), where it is noted that

(3.8)

(3.9)

for .

Note also that, by (3.1),

(3.10)

if is an unbiased estimator of . Then, by (2.14) and (2.16),

(3.11)

Thus, unbiasedness of as an estimator of implies unbiasedness of as an estimator of .

Similarly, by (3.3), one can show that unbiasedness of implies unbiasedness of as an estimator of . Induction on h leads to the following result.

Proposition 3.1

Suppose that is an unbiased estimator of . Then, for each h = 0, 1, etc,

is an unbiased estimator of

is an unbiased estimator of

is an unbiased estimator of .

4.Evolutionary model with unknown second moments

Section 2.3 described a model which incorporated parameter evolution. It introduced the two sets of second order quantities V(i) and W(k).

These were regarded as known quantities. Consider, however, the case where they are unknown. In the simplest such case, they will involve just one unknown parameter, thus:

(4.1)

(4.2)

where is the unknown scalar parameter, and and are known matrices.

Generally, a starred symbol will represent the corresponding unstarred symbol after removal of the multiplier .

Suppose in addition that the initial value of the Kalman filter (Section 3) is subject to the uncertainty:

(4.3)

where is a known matrix, independent of .

Proposition 4.1

When (4.1) – (4.3) hold, the quantities and in the Kalman filter are proportional to . Moreover, the Kalman gain matrices K(h) are independent of .

This result is simply proved by reference to (3.2), (3.4), (3.5) and (3.7), and induction on h.

As a consequence of the proposition, one may write

(4.4)

(4.5)

where the starred quantities here may be obtained from a “starred recursion” corresponding to (3.1) – (3.7), but with (3.2), (3.4) and (3.7) replaced as follows:

(3.2a)

(3.4a)

(3.7a)

where

(2.20a)

This recursion is initiated with given values of and

By (3.8) and (4.5),

(4.6)

where is a known matrix given by (3.4a).

Write (4.6) as

(4.7)

with denoting the prediction error

(4.8)

Then

(4.9)

Recall from Proposition 3.1 that

(4.10)

Hence is an unbiased estimator of . Then, by (4.9), an unbiased estimator of is

(4.11)

5.Estimation of second moments

5.1Unbiased estimation

When data are available, (4.11) provides h + 2 unbiased estimators of , viz . Then any convex combination of these estimators is itself an unbiased estimator.

Proposition 5.1. Let

(5.1)

and

(5.2)

Consider estimators of of the form

(5.3)

where a(h + 1) is a non-stochastic (h + 2)-vector. The MVU estimator is given by

(5.4)

where 1 denotes a vector with all entries equal to unity.

Moreover, for this a ( h + 1),

(5.5)

Proof

The proof is an elementary and well known exercise in constrained optimism.

Application of (5.3) and (5.4) requires a knowledge of R(h + 1). One requires quantities of the form

(5.6)

This last quantity involves covariances of second moments, ie involves fourth moments, of the observations. A practical estimate requires parametric assumptions which enable the expression of fourth moments in terms of second.

Appendices A and B calculate the covariance in (5.6) on the assumption that the observations X(i,j) are normally distributed. The relevant results obtained from there are as follows:

[from (B.18)](5.7)

where, by (A.25) and the comment at the end of Appendix A,

(5.8)

and and are calculated by means of the following recursions (see (A.20) and (A.14)):

(5.9)

(5.10)

with

[by (A.11)].(5.11)

The recursion for Q is initiated (see A.18)) by:

(5.12)

Also

(5.13)

The recursion for N* is initiated by , which is itself obtained from another recursion (see A.22) – (A.24)):

(5.14)

(5.15)

[by (A.30)].(5.16)

By definition of M(f,g) in Appendix B,

and so

(5.17)

Substitute (5.7) and (5.17) into (5.6):

(5.18)

for f, g = 0, 1, 2, …, h + 1.

By definition (5.2), R(h + 1) has elements given by (5.18). They each involve the unknown factor , but note that this cancels in (5.4). Therefore, (5.4) is equivalent to:

(5.19)

with having (f,g) element

(5.20)

Then the estimator is defined by (5.3) with a (h + 1) given by (5.19). By (5.5), (5.18) and the definition of ,

(5.21)

5.2Credibility estimation

Now assume that the parameter is a latent parameter, ie it is a sampling from some prior distribution with d.f. , mean and variance .

According to the credibility of the earlier papers, specifically Taylor (1999b, Section 4), the least squares estimate of that is linear in is

(5.22)

with

(5.23)

(5.24)

By (5.21),

(5.25)

Since is an unbiased estimator of ,

(5.26)

Substitute (5.25) and (5.26) in (5.24):

(5.27)

In the case h = -1 (no data), is given by its prior:

(5.28)

5.3Summary of algorithm

The complete algorithm incorporating the Kalman filter of Section 3, as modified by Section 4, and the present section’s estimation of second moments, is as follows.

(3.1)

(3.2a)

(4.4)

(3.3)

(3.4a)

(4.11)

At this point, there follows an inner recursion to calculate the matrix R(h + 1) (see below)

(5.19)

(5.1)

(5.3)

(5.27)

(5.23)

(5.22)

(4.5)

(3.5)

(3.6)

(3.7a)

(4.4)

The recursion is initiated, as in Section 4, at h = -1 with given values of and and with

(5.28)

It begins at (3.3), runs through to (4.4) with h = -1, then cycles through (3.1) – (4.4) with h = 0, 1, etc.

The inner recursion, between steps (4.11) and (5.19), is as follows, where throughout

.(5.11)

For each value of h in the outer recursion, calculate:

(5.10)

(5.12)

(5.13)

(5.9)

where in the case g = h + 1, is given by

(5.14)

g = 0, 1, …, h + 1(5.8)

Calculate R* (h + 1) as the symmetric matrix with (f, g) element

(5.20)

This inner recursion calculates values of in terms of , all of which are calculated at previous iterations of the outer recursion.

The first loop of the outer recursion, h = -1, requires the value of R(0), hence and . These are given by

(5.13)

(5.16)

The second loop of the outer recursion, h = 0, requires the value of R(1), hence . This is given by

[by (A.6) and (A.28)]

(5.29)

by (5.15).

Note that the inner recursion calculating and for
g = 0, 1, …, h + 1 calls on just and for g = 0, 1, …, h.

Once the and have been calculated, and are ready for use in the next iteration of recursion, the and may be discarded.

Note also that, at each iteration of the outer recursion, R(h + 1) is constructed by the augmentation of R(h) with an additional row and column.

6.Prediction and prediction error

6.1Prediction

Predictors are given by Neuhaus (1989, Section 2.3). Define

(6.1)

whence

(6.2)

with

(6.3)

Also define

(6.4)

Then is the Kalman filter’s unbiased estimate of X(h) based on data up to and including experience year k.

6.2Prediction error

The same section of Neuhaus (1989) gives:

(6.5)

where denotes

This provides a recursion by which , etc can be calculated starting from given by (4.4).

Also

(6.6)

where denotes .

Equation (6.6) gives MSEP for each future experience year h. It is also possible to obtain prediction covariances across different future years from Neuhaus (1989, Section 2.2). Define

(6.7)

(6.8)

Note that

(6.9)

(6.10)

From Neuhaus,

(6.11)

(6.12)

for .

Note that and are given for the case by

(6.13)

(6.14)

7.Numerical example

The present section will illustrate the results of Section 5 by reference to the same real data set as used in the earlier papers. The data appeared in the form of incurred losses, adjusted to constant dollar values for inflation, in Table 7.1 of Paper II. They were converted to logged age-to-age factors in Table 7.2 of the same paper.

These tables are repeated as Tables 7.1 and 7.2 below.

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Table 7.2Logged incurred loss age to age factors

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As in the example of Section 8.3 of Paper II (see (8.38)), it is assumed that (2.6) takes the form:

(7.1)

ie in (2.11),

(7.2)

Consistently with (8.39) in Paper II, it is assumed that

(7.3)

ie in (2.23)

(7.4)

ie

(7.5)

The evolution of parameters (see (2.13), (2.16) and (2.20)) will be assumed subject to the following

(7.6)

(7.7)

ie

(7.8)

The following parameters are assumed for :

.(7.9)

These parameters, together with (6.4), set the prior for W(k) as representing half the parameter variances assumed in the example of Paper II (see just after (8.55)). This allows for the parameter variance of the earlier paper to be only partly attributable to uncertainty in in the present case, and also partly attributable to evolutionary variation.

In common with the same example in the earlier paper, the Kalman filter is initiated with

(7.10)

It is assumed that

(7.11)

Table 7.3 gives estimates from (3.3). These are used to develop estimates , hk, from (6.4). Table 7.4 gives the associated standard errors , where the subscript indicates the element of the covariance matrix.

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Table 7.3

Filtered logged age to age factors

Table 7.4

Standard errors of filtered logged age to age factors

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Figure 7.1 illustrates the credibility smoothing of estimated , displaying the raw estimates from (5.3) and their smoothed versions according to (5.22).

Figure 7.1

Credibility smoothing of theta

Let Y(i) denote the vector of quantities X(i,j) relating to accident year i, ie

(7.12)

Define an unbiased estimator of based on data :

(7.13)

where denotes the j-th component of vector , j = 0, 1, etc.

Note that in (7.13) the first k – i + 2 components are obtained from the Kalman filter, whereas the last J – k + i – 1 are obtained from (6.2) and (6.4). The first components are not used in the following.

Adopt the convention in (7.13) that

Write for the (J + 1)-vector:

(7.14)

and define

(7.15)

In the present example, this is the logged age-to-ultimate factor for accident year i when data are available up to and including experience year k.

The values of R(i|k) are displayed in Table 7.5.

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Table 7.5

Filtered logged age to ultimate factors

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The MSEP of the R(i|k) may also be estimated. By (7.15),

.(7.16)

Because of the positioning of the zeros in , only the bottom right sub-matrix of is required. A typical element of this sub-matrix is

(7.17)

which is equal to the element of from (6.8). That is,

(7.18)

for = k + 1, …, i + J.

Equivalently,

(7.19)

for

After substitution of (7.19), equation (7.16) generates a triangle of quantities , whose square roots are set out in Table 7.6.

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Table 7.6

MSEP of filtered logged age to ultimate factors

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Tables 7.5 and 7.6 lead to estimates of (unlogged) ultimate incurred losses through the log normal formula:

incurred losses of accident year i as at end of experience year k

x

(7.20)

These estimates appear in Table 7.7.

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Table 7.7

Estimates of ultimate claims incurred

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Figure 7.2 gives a plot of the evolving distribution of estimated ultimate incurred losses for accident year 1980. The illustrated distributions are based on the parameters in Tables 7.5 and 7.6. The figure corresponds to Tables 7.1 and 8.1 in Taylor (1999b).

Figure 7.2

The different distributions can be identified by their increasing concentration (and therefore increasing peak height) with increasing development. Thus, the distribution at the end of 1995, with only two years of development remaining, is highly concentrated.

Figure 7.2 differs little visually from Figure 8.1 of the previous paper. However, Figure 7.3 provides a little more detail of the comparison of results derived from the Kalman filter (present paper) and from the credibility approach (previous paper).