COMBINATION Of ESTIMATES Of OUTSTANDING CLAIMS IN NON-LIFE INSURANCE

G.C.Taylor

E~S.Knight Co. Research Centre, LS.Knight Co., Consulting Actuaries, 71 York Street, Sydney, NSW, 2000, AUSTRALIA.

May 1984

Short title: Combination of estimates Keywords: Outstanding claims, estimation, combination, blending

335

SUMMARY. The paper is concerned with the estimation of outstanding claims

of a non-life insurer. Typically, the actuary carries out this estimation by

a number of different methods, and so arrives at a number of estimates. Each

method will usually provide separate estimates in respect of separate years

of origin. In addition, it is likely that physical estimates of outstandings

prepared by the insurer will be available.

The question considered is: How should the various estimates of outstanding

claims be combined to produce the actuary's final estimate.

It is suggested that three criteria must be balanced against one another:

(i) the estimates of outstanding claims should be unbiased;

(if) the different estimates arising from different methods should be

blended in such a way as to minimize the variance of total

estimated outstandings, subject to (ii) below;

(iii) the relation of the final estimates, by year of origin, to the

corresponding physical estimates should progess sufficiently ;~.-::--

smoothly over the years of origin that the credibility of the final

estimates is not strained.

Section 2 points out similarities between these criteria and those

underlying the Whittaker-Henderson method of graduating a set of decremental

(e.g. mortality) rates. Sections 3 and 4 carry out the algebra required to

solve the problem. Sections 5 and 6 add a few further comments. Section 7

provides a numerical example in detail.

336

1. I n t r oduc t i on

This paper wi l l consider the estimation of outstanding claims of a

non-life insurer. Attention w i l l not be given to the technique by which one

may proceed from the raw data to one or more estimates on the basis of

various models of the claims process. Rather w i l l i t be assumed that such

estimate(s) have been obtained, and that the issue is one of how i t (they)

should be translated into a f inal statement of the estimated mean and

variance of outstanding claims.

In most treatments of portfolios with which are associated lengthy

delays in the settlement of claims, the claims statistics are suff ic ient ly

recalcitrant that the actuary/statistician wi l l wish to apply several

different models to the data. This is the so-called "battery of methods"

discussed by Taylor and Matthews (1977). The philosophy underlying that

approach is discussed by Taylor (1984, Chapter 14), where much of what

appears here is anticipated.

The major points el ici ted there are:

( i) different methods of analysis are sensitive to different features

of the data and, in the absence of some omnibus method which can be

relied upon to capture a l l such features, the use of a collection

337

of methods enhances the likelihood that such features w i l l be

recognised;

( i i ) the ambiguity of the data w i l l usually be such that no single one

of the methods applied can be selected as absolutely preferable to

a l l the others; rather one wi l ] be inclined to give some weight to

each method.

The question to be considered here concerns the determination of those

weights.

Similar questions have been considered in the l i terature on time series

forecasting. A recent reference is Winkler and Makridakis (1983). They give

various other references, but generally the problem considered in that l i t e r -

ature is somewhat different from the present one. Section 5.2 gives a few

details of the differences.

The technique developed here is reminiscent of the Whittaker-Henderson

method of graduation of mortality rates (Mil ler, 1946), a very brief review

of which is now given.

2. WhittakerTHenderson. )rad.atton

The basic technique of this method of graduation is due to Whittaker

(1939). I t was adapted to the actuarial probIem of graduation of mortality

rates by Henderson (1924). A succinct description is given by Mi l ler (3946).

338

Very briefly, the method is as follows.

Consider an m-vector " of mortality rates. The components of the vector

represent mortality rates for an ordered sequence of ages. Now let Q denote

an "observation" of this vector of mortality rates, i.e. what is usually

referred to as the vector of crude mortality rates. This vector will be an

unbiased estimator of~. However, it would never be adopted as a vector of

mortality rates for practical use because it takes no cognizance of the fact

(or at least the universally accepted assumption) that a sequence of true

mortality rates (i.e. the~) according to age proceeds smoothly.

It is standard therefore to submit Q to a process of smoothing or

graduation. The graduation may be regarded as a mapping:

W:Q .... -~ ~ (2.1)

which satisfies the conditions:

(i) ~ is still unbiased (at least approximately) as an estimator of ~;

(ii) the components of ~. taken in order, proceed more smoothly

(according to whatever criterion of smoothness is in use) than

those of Q;

(iii) the vector \J, while adjusting Q, is nevertheless probabil-

istically compatible with Q in the sense that the latter can be

regarded as a reasonably likely random drawing from a distribution

having the former as its mean, i.e. the difference between ~ and

339

Q should not be too great.

There are various methods of graduation available, and the reader is

referred to the appropriate texts for details. However, there is one which

recognise.s the above three requirements in an explicit way, and which will be

found useful for later application to the combination of various sets of

estimates of non-life insurance outstanding claims.

This is the Whittaker-Henderson method, according to which the mapping W

of (2.1) is derived as that which minimizes a loss function:

where

L( ii :Q) = A( ii ,Q) + k S( ii), (2.2)

A( u ,Q) is some measure of the (lack of) adherence of u to Q,

possibly a Euclidean norm of the difference between the two

or some variant of it;

S( ~) is some measure of the lack of smoothness of the components of

U, possibly a Euclidean norm of the vector of differences of

r-th (say) order of the components of ii ;

k is a scalar reflecting the relative importance assigned to the

adherence and smoothness parts of the loss function.

340

As k -+ D, u tends to follow Q slavishly. As k -+ ... , u tends, in

the above example of 5(.), to a polynomial of degree r-l regardless of

whether it fits well to Q or not.

The usual forms of A(.,.) and S(.) are the squares of the Euclidean

norms mentioned above. In this case, the loss function becomes quadratic in

~, whence the resulting equation determining u is linear in that vector.

Thus, ~ is finally expressed as a linear transformation of the vector Q.

We do not give the whole of that development here. Once again, the

reader is referred to the appropriate texts should the detail be required.

The purpose of the present section has been to exhibit the basic ideas of

Whittaker-Henderson graduation for comparison with the methodology developed

in Sections 3 and 4 in connection with the estimation of outstanding claims.

Statement of the problem of combination of estimates

The problem of combination of various estimates of outstanding claims

has been described in general terms in Section 1. Before we proceed further,

it is appropriate to state it precisely.

It is usual that an estimate of outstanding claims will consist not just

of a single figure but will be the sum of a number of sub-estimates. For

example, outstanding claims may have been estimated separately for each past

341

period of occurrence. or alternatively for each period of notification.

In fact. the precise nature of the various sUb-estimates may assume many

different forms. For the sake of definiteness. we shall continually speak in

this and subsequent sections as if outstanding claims have been estimated in

respect of individual periods of origin (which could mean either occurrence

or notification). It should be remembered. however. that the methodology is

intended to be of much wider application than this.

Suppose that n methods of estimation of outstanding claims have been

applied to a particular portfolio. Each of m different periods of origin has

been dealt with separately by each of the n methods. The result will be mn

estimates.

let X • i=I •...• mj j=l •... n. denote the raw estimate in respect of ij

method j as applied to period of origin i. It is supposed that each method

is unbiased. so that

E[X ij

) = lJ

i (3. I)

where II is the mean --~lue of the random variable. amount of outstanding

claims in respect of period of origin i.

The covariance of X and X is denoted by C(X .X ). At this ij kl ij k I

stage no structural constraints are applied to the ensemble of covariances.

342

It is required that the X be used to produce a final estimate in ij

respect of each of· the m periods of origin. let Y denote this final k

estimate for period of origin k. It will be a function of the X We ij

write

Y = w(X), (3.2)

where X is the ensemble of X ,Y is the column vector of Y written out ij

m mn in order, and w:R R is a function yet to be specified.

It is assumed that w(.) is linear. This assumption can be justified on

two grounds:

(i) if the distributional properties of the X ij

are allowed to be

completely general, it is only by restricting w(.) to a certain

subset of the linear transformations that unbiasedness of Y can be

assured of following from unbiasedness of Xi

(ii) in practice, it is very likely that one or more of the methods of

estimation will have involved some anomaly in its application,

whereupon it will have been "tampered with" in a manner which

probably defies proper statistical descriptioni in this case, the

resulting X may not be completely unbiased, and the quality of

one's estimates of the C[X ,X ] may not be entirely sat is-ij k 1

factory.

343

In view of these considerations, greater refinement of w(.) than a

requirement that it be linear is almost certainly unjustified. It will be

assumed henceforth that (3.2) is a linear equation.

In order to obtain an explicit statement of the optimal combination of

estimates of outstanding claims, it is necessary to state explicit

assumptions as to the form of the loss functions to serve in the place of

A(.,.) and S(.) introduced in (2.2). For the purposes of this paper we

shall adopt specific forms for these functions. However, it is to be

emphasised that other forms may be used.

The functional forms adopted here are: m

A(Y:X) = V( t Y ]; (3.3)

i=l

m-r r 2

S(Y:X) E (~ V ] (3.4)

i=l

where V denotes the ratio of the statistical estimate Y to the corres-

ponding phySical estimate.

The problem now becomes one of selecting (linear) w(.) so as to

minimize the loss function l(Y:X) defined by (3.3), (3.4) and:

l(Y:X) = A(Y:X} + k S(Y:X). (3.5)

This minimization is subject to the side-condition that the Y selected should

be an unbiased estimator of expected outstanding claims. The effect of

A(.:.) is to force the final estimates Y toward those· which minimize the

344

variance of total oustanding claims. The effect of S(.:.) is to avoid

rather implausible situations in which The final estimates of outstandings,

relative to the corresponding physical estimates vary widely. For example,

final estimates equal to 125 percent, 70 percent, 140 percent and 75 percent

of physical estimates in respect of 4 consecutive periods of origin tend to

lack credibility for their is no apparent reason why physical estimates would

deviate from the truth in this erratic manner.

Usually, a high degree of smoothness of the V will not be required.

Moreover, the value of m is unlikely to be great. These two facts indicate

that a small value of r is appropriate. A value of r=2 or 3 could probably

be used in most practical cases.

4. Solution of the basic problem

4.1 Matrix notation

A first step is to express the loss function l(Y:X) in terms of vectors

and matrices. If a superscript T denotes transposition, then (3.3) is

simply:

T A(Y:X) V[l Y]

T 1 V[Y] 1, (4.1.1)

where I denotes the m-dimensional column vector each of whose components is

unity. The right side of this last equation can be expressed in terms of the

X Since Y=w(X) with w linear, it is possible to write: ij

n Y '" X w

ij ij j=l

= X w (4.1.Z) (i) (il

where w is the n-dimensional column vector: (il

T w = (w .w •••.• w ). (4.1.3)

(il i! iZ in

and X is the lxn matrix: ( i )

(x .X ••••• X ). (4.1.4) il iZ in

The various values of i appearing in (4.1.Z) may be included in a single

fonnu la. thus:

Y " Xw.

where X and w denote respect ive ly the mxmn matrix .and

vector:

X It X 0 0 0 (1)

0 X 0 0 (Z)

0 0 )( 0 (3)

0 0 0 0 0 0 0 0 0 0 "~().- 0 0 0 0 X

(m)

Combination of (4.1.1) and (4.1.5) gives:

T A(Y:X) V[l X w]

T V[Q w]

346

w " w

w

w

w

(4.1.5)

mn-dimensional column

(1 )

(Z) (4.1.6)

(3)

(m)

" G

'~.~

where Q=X is given (using (4.1.6)) by:

T Q = (X ,X , •.. ,X ). (4.1.7)

(1) (2) (m)

Since Q is just a vector, the last expression for A(Y:X) can be written:

T A(Y:X) = w V[Q] w

T = w C w,

where C = V[Q].

Now for the function S(Y:X). Let p

respect of period of origin i. Let

-1 -1

Then

P = diag (p 1

v = PY

, •••• p

PXw.

m

(4.1.8)

denote the physical estimates in

). (4.1.9)

(4.1.10)

Moreover, r-th differences may be expressed in matrix form by means of the

(m-r)xm matrix:

o = (-1/ (:) (:) -G) G) ° 0 ..• 0

° (:) -(:) ... (-1/-~:~ (_lJr(:)o ... o

Then (3.4) becomes:

T S(Y:X) [OV] [OV]

T [OPXw] [OPXw]

T T = w [x AX] w,

347

(4.1.11)

where

T T A = POOP

Finally, one may write (4.1.11) as:

with

T S(V:X) = w B w,

T B X AX

(4.1.12)

(4.1.13)

(4.1. 14)

Equations (4.1.8) and (4.1.13) may now be substituted into the loss

function (3.5) giving:

T L(V:X) = w [C+kB] w. (4.1.15)

4.2 The unbiasedness side-condition

As stated just after (3.5), the minimization of that loss function is

•

subject to the side-condition that the final estimate V be unbiased. It is

assumed in (3.1) that each> of the rawest imates X

given (3.1), it is required that V satisfy:

E[Vr-;; II,

where \I is the vector:

T u = (u ,\I , ••• ,Il ).

1 2 n

By (4.1.5), this is:

E[Xw] = E[X] w = u,

which, by (3.1), is:

Uw = I,

348

is unbiased. Thus, ij

(4.2.1)

(4.2.2)

where U is the mxmn matrix having the same structure as X (see (4.1.6)) but

with each X replaced by a row n-vector each of whose components is (il

un ity.

4.3 Combination of estimates - solution of the optimization problem

The problem of selecting a combination of the n sets of estimates of

outstandings has now been formulated as one of constrained optimization. The

loss function (4.1.15) must be minimized subject to the constraint (4.2.2).

This minimization may be carried out by the method of lagrange

multipliers. The objective function is defined as:

T T F = w [C+kB] w - 2. (Uw - I), (4.3.1)

where 2. is a column m-vector of lagrange multipliers.

Differentiating F with respect to wand setting the result equal to zero

gives:

T T w [C+kB] - U = 0,

-1 T i.e. w = [C+kB] U '. (4.3.2)

The constraint (4.2.2) may be combined with the result (4.3.2) to give:

-1 T -1 • ={U [C+kB] U} 1. (4.3.3)

Finally, combination of (4.3.2) and (4.3.3) yields:

-1 T -1 T -1 w = [C+kB] U ! U [C+kB] U I, (4.3.4)

or, more briefly,

-1 w = z [UZ] 1, (4.3.5)

349

where

-1 T Z [C+kBl U (4.3.6)

The final estimates of outstandings are, according to (4.1.5/,

-1 Y = Xw = XZ [UZ] 1. (4.3.71

It is evident that the final estimates Yare some type of "weighted

average" of the raw estimates X. Recall that U has the same structure as X

except that U contains a I in each position that X contains an X This ij

reinforces the concept of (4.3.7) as a ·weighted average" of the X ij

It will be seen in Section 5.2 that in a special Simple case (4.3.7)

does indeed reduce to a simple form of weighted average.

4.4 Variance of final estimates

The calculation of the variance of the vector Y of final estimates is

now quite simple. By (4.1.2),

T cry ,Y ) = w C[X ,X ) w (4.4.1)

j _/.:'- (1) (i) (j) (j)

whence n

T v[:: Y = w veX J w, (4.4.2)

where vex) denotes the mnxmn matrix whose {i,j)-block is C[X ,X (il (j)

Then substitution of (4.3.S) in (4.4.2) gives:

T -1 T -1 V[yJ = 1 [uzl z V[X) z [UZl 1. (4.4.3)

350

Because of the "weighted average" nature of Y, it is to be expected that

V[Y J would be less than VeX J. A numerical example is given in Section

7.

5. Extensions and special cases

5.1 Possible extensions

The optimization carried out in Section 4.3 was made subject to the

single (unbiasedness) constraint (4.2.2). It is possible to conceive of

other constraints which might be seen as desirable in certain cases. For

example, there may be evidence from external sources that average claim

s'izes, after correction for normal inflation, have been increasing at a

reasonably constant rate from one period of origin to the next. It may be

desired to give some weight to this collateral evidence in the formation of

final estimates of outstandings.

This objective could be achieved merely by the addition of a further

;;:;\ term to the loss function. Note that average claim size (in what follows all

'0' statistics are taken to be adjusted for "normal" inflation) for a given

-,,",( period of origin is:

past claim payments + outstanding claims number of cJalms lncurred

the denominator and first member of the numerator are known. Thus, for

period of origin, average claim size takes the form:

351

const. + const. x outstandings.

It follows that the vector of finally estimated average claim sizes is:

a + GY,

where a is column m-vector of constants, and G is an mxm diagonal matrix of

constants. The r-th differences of these average claim sizes form the

vector:

o (a + GY) = 0 (a + GXw), (5.1.1)

with 0 defined as in Section 4.1.

It is then possible to modify the loss function L(Y:X) given by (4.1.15)

by adding a further term:

T h [D(a+GXw) [D(a+GXw)), (5.1.2)

similar to S(Y~X) as exhibited in (4.1.11).

The algebra flowing from the introduction of this extra term is not

pursued here as the object of the present section is simply to hint at the

modifications to loss function (4.1.15) which might prove useful in practice.

Naturally, modifications other than the particular one discussed in the

development of (5.1.2~y-ight arise.

352

5.2 A couple of special cases

The first special case to be mentioned is just a trivial adaptation of

the general loss function (4.1.15). It deserves special mention, however,

because it corresponds to the usual case of combining estimates in the time

series literature (see the paper by Winkler and Makridakis (1983) quoted

earlier and the references given there). It is obtained by setting k=O.

In this case (4.3.6) simplifies to:

-1 T Z = e u (5.2.1)

and (4.3.7) in turn becomes:

-1 T -1 T -1 Y = XC u [ue U] 1. (5.2.2)

This formula is particularly interesting because of its resemblance to the

standard formula for Weighted least squares regression. Such a regression,

T T with X as the observation (column) vector, U as design matrix, and

-1 e as weight matrix, yields as vector of regression estimates of E[X]:

-1 T -1 -1 T [Ue U] ue X

Transposed to row form, this becomes:

T -1 T -1 T -1 X e u [ue U]

which may be compared with (5.2.2).

(5.2.3)

The second special case also involves the restriction k=O. However, it

also involves further restrictions reducing it to the case often met in the

combination of time series forecasts. Here, the forecasts· are frequently

made in respect of a scalar. The problem then becomes one of combination of

forecasts of a scalar Quantity rather than a vector as in earlier sections. 353

That is. m=l as well as k=O. Of course. since k=O. (5.2.2) is still

. applicable.

In this case it is useful to begin by noting the dimensions of the

various matrices and vectors involved in (5.2.2). For convenience, they

written out below.

Y lxn

X lxn

C nxn

U lxn

1 lxl

Because of the dimensioning of this special case, one may omit the scalar 1

T and replace U by the row vector 1 in any formulas. With this understood,

(5.2.2) reduces to:

-1 T-l-l Y XC 1 [1 e 1] (5.2.4)

where the bracketed product and the product to its left are each scalars. In

fact, expressed in terms of the individual elements of the matrices and

vectors involved, (5.2.4) is: -1 -1

Y = t (C ) X / 1: (e ) (5.2.5) jk j jk

j j,k

where all suffixes indexing period of origin have been deleted (since it is

-1 assumed that m=l), and (e )

jk

-1 denotes the (j,k)-element of e

In the case in which X •...• X are all forecasts of a particular 1 n

quantity by n different methods, (5.2.5) can be recognised as the standard

formula for the combination of forecasts which minimizes the variance of the

combined forecast (Newbold and Granger. 1974; also quoted by Winkler and

Makridakis, 1983). 354

6. Practical Difficulties

There are two major practical difficulties in the implementation of the

basic formula (4.3.7). Both of these relate to the covariance matrix C.

Firstly, for any given method (say j), it is necessary to obtain all the

covariances C between the estimates of outstandings provided by that ij,kj

method in respect of the various periods of origin. While there are a number

of methods for which the technique for obtaining these covariances is

available, it must be said that, generally, the metho~ology for estimation of

second moments of estimates of outstanding claims is still ;n a developing

stage. Taylor (1984) gives some indication of those methods in respect of

which the problem of second moments has been addressed.

Secondly, it is necessary to obtain "inter-method" covariances C ij,kl

j fl. The literature contains, to the best of my knowledge, absolutely

nothing on this subject. It is impossible, therefore, to approach this part

of the problem with the rigour it warrants. Nonetheless, as will be seen in

Section 7, the approach taken here is that a rough-and-ready but sensible

approximation of these elements of the covariance matrix is preferable to the

completely undisciplined procedures currently endemic in the estimation of

outstanding claims.

355

7. Numerical example

The portfolio chosen for use in demonstration of the theory developed

above is a small Public Liability portfolio in Australian states other than

New South Wales. Data from this portfolio were analysed and outstanding

claims projected by three different methods. These were the:

(i) payments per unit of risk (PPCI) method;

(ii) payments per claim finalized (PPCF) method;

(iii) projected physical estimates (PPE) method.

Each of these three methods is described and discussed by Taylor (1984).

As it is not the principal purpose of this paper to discuss methodology

for estimation of second moments of outstanding claims, it is not appropriate

to present in great detail the processes by the estimated second moments

associated with the above ,three analyses were obtained. Just a couple of'

basic pOints are mentioned, however.

Firstly, the PPCF method is a special case of the invariant see-saw

method (Taylor, 1983,1984). Estimation of the second moments associated with

this method is discussed by Taylor and Ashe (1983), and the technique devel-

oped there has been used in the present example.

356

I . I

j I

I 'j , I , , j 'I j ~ ~ 1

Secondly, the PPE method, being heavily based on physical estimates, can

be expected to be relatively reliable in respect of the older periods of

origin (where a good deal of information as to the nature of the claim will

have accumulated), but much less reliable in respect of the more recent

periods of origin. The precise meaning of this "reliability" will be

clarified below.

\ The results of application of the three methods of estimation were as in

I the following table, where they are compared with the insurer's physical 1-; estimates.

) -------- ------------------------------------------- --------------------- --------, , Year of PPCI me.thod PPCF method PPE method Physica 1

origin --------------------- ---------- ---------- ---------- ---------- estimate Estimated Associated Estimated Associated Estimated Associated O/S claims standard O/S claims standard O/S claims standard at 1/1/83 deviation at 1/1/83 deviation at 1/1/83 deviation

$'900 $'000 $'000 ---------- --------~-

$'000 $'000 -------1--$:000---

1982 268 85 288 171 296 207 1981 221 93 375 242 141 85 1980 176 89 328 257 308 139 1979 112 45 194 218 131 52 1978 67 59 135 211 33 12

8 2

100 20 ---------- ----------~~il~~ I .... ;; ....

10 8 17

11 45 45 ---------- ---------- ----------

201 1373 836 1017 366 TOTAL i 927 I ................. . ---------- ---------- ---------- ---------- ----------

t I

i f, 357

--------$'000

219 134 329 151 43 14

192 .-------

1082 .-------

Note that, as expected, the ratios of PPE estimates to physical

estimates progress smoothly over years of origin whereas the corresponding

PPCI and PPCF ratios are irregular.

Year of Ratio of statistical to physical estimates

origin PPCI method PPCF method PPE method

per cent per cent per cent

1982 122 132 135 1981 165 280 105 1980 53 100 94 1979 74 128 87 1978 156 314 77 1977 221 57 57

1976 and earlier 27 23 52

TOTAL 86 127 94

Some remarks are in order regarding the estimation of the

variance-covariance structures associated with the above estimates of

outstanding claims. The full covariance matrix associated with

is set out in the detailed numerical example below.

358

each method

The PPCF covariance matrix is derived formally. The PPCF method

involves the use of a regression model. Its formal structure has been used,

via the GLIM program (Bak"er and Nelder,l978), to obtain the covariance matrix

of estimated outstandings according to year of origin.

The PPCl covariance matrix is derived semi-formally. The covariance of

PPCI in respect of development years i and j has been taken as the sample

covariance. From that point the covariance matrix of estimated outstandings

according to year of origin is developed formally.

The PPE covariance matrix has been constructed on the basis of only very

rough calculations reflecting the variation observed in the underlying

statistics.

We have absolutely no information concerning the covariance structure

between the different methods. However, general reasoning suggests that:

(il because both PPCl and PPCF methods are based on past claim payment

experience, the covariances between their results will be relat­

ively high;

(ii) because the PPE method, depending on the past development of

physical estimates, is much less dependent on past claim payment

experience, the covariances between this and the other two methods

will be relatively low.

359

The detailed structure used for the grand covariance matrix C is as

follows. The mn x mn matrix C may be regarded as consisting of mxm blocks,

each block of dimension nxn. If the rows and columns of C are permuted to

yield nxn blocks each of dimension mxm, then the (i,i) block is the covar­

iance matrix associated with the i-th method. These matrices are available.

The (i,j)-block (i~ j), on the other hand, contains the covariances

between outstandings as estimated by the i-th and j-th .methods respectively.

An ad hoc rule has been used to obtain these as follows:

,(ij) (ii) (jj) 1/2 c • r [c c ] I

kl ij kl kl where

(i j) c = C[ X ,X ], kl ki lj

and r is some selecte~ constant. The 3x3 matrix of r is chosen ij ij

subject he ly as

methods.

a rough measure of the covariance between the i-th and j-th

It is to be noted that, with the grand covariance matrix C set up in

this way, there is no guarantee of its positive definiteness in general,

although this will be guaranteed if the matrix of r ij

to diagonal.

360

is sufficiently close

Numerical details of the covariance structures used in the example are

given in lower triangular form below. In the arrays displayed, higher rows

relate to more recent years of origin.

Covariances of PPCI estimates of outstandings

.719El0

.108El0 .866El0

.847E9 .111El0 .786El0

.411E9 .441E9 .495E9 .198El0

.169E9 .107E9 .247E9 .390E9 .345El0

.797E7 .890E7 .277E8 .459E8 .566E8 .101E9 -.844E7 -.942E7 -.101E8 .322E8 .664E8 .162E8 .112E9

Covariances of PPCF estimates of outstandings

.291Ell

.104Ell • 586Ell

. 114E 11 .201Ell .659Ell

.914El0 .162Ell • 185E 11 .473El1

.880Ell .156Ell .178Ell .177Ell .443E 11

.525E8 .927E8 .106E9 .106E9 .115E9 .281E9

.161E6 .285E6 .326E6 . 324E6 .352E6 .151E6 .202El0

361

Covariances of PPE estimates of outstandings

.429E 11

.701El0 .716El0

.861El0 .469El0 .192Ell

.272El0 .133El0 .291El0 .275El0

.478E9 .244E9 .479E9 .242E9 .133E9

.746E8 .406E8 .831E8 .378E8 .111E8 .576E7

.414E9 .254E9 .554E9 .262E9 . 692E8 .192E8 .400E9

As mentioned earlier in this section, covariances between the PPE and

other methods can be expected to be low. The. situation is much more

uncertain as regards covariances between the PPCI and PPCF methods. Two

alternative matrices of r terms have been used •. They reflect high and ij

low PPCI-PPCF covariances respectively.

The two 3x3 matrices of r are as shown below. Once again they are ij

given in lower triangular form. The ordering of rows from top to bottom is

PPCI, PPCF, PPE.

High covariance

1.00 0.50 1.00 0.10 0.05 1.00

~.

Low covariance

1.00 0.20 1.00 0.10 0.05 1.00

Four groups of results have been computed, with the intention of invest­

igating the effects on the computations of different parameters and covar­

iance structures. The four groups are:

(il main results - smoothing based on 3rd differences {r=31:

(al high PPCI-PPCF covariance structure;

362

" .. '

(b) low PPCI-PPCF covariance structure;

(i i) subs "id iary resu Its - smooth ing based on 2nd differences (r=2) with

high PPCI-PPCF covariance structure:

(a) based on covariance matrices for individual methods as

displayed above;

(b) based on covariance matrices for individual methods in

which the off-diagonal elements have been deleted.

For each of these four choices of parameters. computations are carried

out for various values of the "tuning constant" k. A perusal of the covar­

iance matrices displayed above suggests that. when blending of the three

methods' results has been carried out with some emphasis on variance

reduction. the resulting variance of total estimated outstanding claims

should be of the order 1.OE10.

This suggests in turn that:

(i) a cho:ce of k~1.OE10 should lead to non-negligible weight being

given to both variance reduction and smoothing in the construction

of blended estimates;

(ii) a choice of k several orders of magnitude lower than 1.0E10should

lead to a blending process which concentrates solely on minimiz-

363

ation of the variance of total estimated outstandings;

(iii) a choice of k several orders of magnitude higher than 1.0[10

should lead to a blending process which concentrates on fitting the

ratios of the blended to the physical estimates close to a poly­

nomial of degree r-l.

This reasoning appears to be validated by the numerical

results now given.

Results (i)(a)

Smoo~hing based on 3rd differences High PPCI-PPCF covariance structure

--------- ---------------.--------------------------~----------- ----Year Blended estimate of outstanding claims for k = of as a percentage of physical estimates given in parenthesis)

origin ---------- .. - -------------- -------------- ---------------0 1.OE9 1.0[10 LOE 12

--------- ------------ -------------- -------------- ------------- ... -'000 $'000 $'000 $'000

1982 268 ( 122) 268 ( 122) 267 (122) 266 (1211 1981 169 (126 ) 145 (108) 125 (93) 130 (97) 1980 187 (57) 20B (63) 231 (70) 251 (76) 1979 111 (74) 105 (70) 99 (66) 89 (59) 1978 30 (70) 32 (74) 26 (60) 19 (44) 1977 -3 (-21) 8 (57) 6 (43) 5 (36)

1976 and ~-ear 1 ier 23 (12) 25 (13 ) 30 (16) 48 (25 )

--- ... ----- ------------ -------------- -------------- ------------- ... -TOTAL 785 791 784 808

Standard deviation 163 166 173 178 ... -------- ... ----------- -------------- ... ------------- ---------------

364

" " .. ~

.: '-

, .•.. If

' ....

Results (i )(b)

Smoo'hing based on 3rd differences low PPCI-PPCF covariance structure

1982 1981 1980 1979 1978 1977

1976 and

Blended estimate of outstanding claims for k = as a percentage of physical estimates given in parenthesis) --------------------------- --------------1----------------o I 1.0E9 1.0El0. 1.0E12 ------------ -------------- ---------- .. --- ----- .. ----------'000 $'000 $'000 $'000

267 ( 122) 267 ( 122) 266 (121 ) 265 (121) 178 (133) 150 (112) 125 (93 ) 132 (99) 203 (62) 218 ' (66) 235 (71) 259 (79) 114 (75 ) 110 (73) 105 (70) 94 (62)

34 (79) 34 (79) 29 (67) 21 (49) 8 (57) 9 (64) 7 (50) 5 (36)

~;l~{~~; :~;;:::~~~~: :::~;;::::~~:: :::~;;:::~~:::l:::~;;:::~:::::: Results (ii)(a)

Smoo'hing based on 2nd differences High PPCI-PPCF covariance structure

---- ---- -------.---------------------------------------------------Year Blended estimate of outstanding claims for k c

of (as a percentage of physical est imates given in parenthesis orig in ---------------------------- -------------- ---------------

0 1.0E9 1.0ElO 1.0E12 -------- ----_ .. ----- .. - -------------- -------------- ---------------

'000 $'000 $'000 $'000

1982 268 (122) 268 (122) 267 ( 122) 266 (121 ) 1981 169 (126) 156 (116 ) 133 (99 ) 135 (101) 1980 187 (57) 198 (60) 226 (69) 263 (80) 1979 111 (74 ) 109 (72) 100 (66) 91 (60) 1978 30 (70) 29 (67) 25 (58) 18 (42 ) 1977 -3 ( -21) 6 (43) 5 (36) 3 (21)

1976 and earlier 23 (12 ) 25 (13 ) 32 ( 17) 48 (25)

--------- ------------ -------------- ------- ---.---- ---------------TOTAL 785 791 788 824

Standard deviation 163 164 170 179 --------- ------------ -------------- -------------- ------------ .. --

365

Results '( ii )(b)

Smoo~hing based on 2nd differences High PPC!-PPCF covariance structure Off-diagonal elements of covariance matrices for individual

methods set to zero

Year of

origin

1982 1981 1980 1979 1978 1977

1976 and . earlier

Blended estimate of outstanding claims for k = (as a percentage of physical estimates given in parenthesis) --------------------------- --------------j----------------o r 1.0E9 1.0El0 1.0E12 ------------ -------------- -------------- ---------------'000 $'000 $'000 $'000

275 (126) I 275 (126) 275 (l26) 274 (125) 190 (142) i 174 (130) 143 (l07) 142 (l06) 223 (68) I 229 (70) 246 (75) 288 (88) 122 (81) 121 (80) 116 (77) 107 (71) 35 (81) 33 (77) 30 (70) 23 (53) 9 (64) 8 (57) 8 (57) 5 (36)

62 (32) 64 (33) 72 (38) 41 (21)

TOTAL 916 904 890 880 Standard deviation 123 125 135 155

It may be noted that:. as expected, the standard deviation of the

estimated total outstandings increases as the parameter k (the weight given

to smoothing at the expense of smaller variation) increases. Probably the

most appropriate manner in which to apply this method in practice is to

366

compute blended outstandings for a range of values of k, as in the above

example, and then choose the minimum k which provides smoothness which is

regarded as acceptable.

The following observations may be made on the above numerical results:

(i) the choice of 2nd or 3rd differences as the criterion according

to which smoothing of the final estimates is carried out has

virtually negligible effect on the results;

(ii) the choice between high and low "between-method" covariance

structures has substantially less effect on the final results

than does the a 110wance or otherwise for the "with in-method"

covariances.

In!uitively, it would seem that these conclusions are of reasonably

general application, though care is always necessary in such extrapolation

of results to unseen examples. The following pOints may be noted, however:

(i) the "within-method" covariances, if themselves of significant

magnitude (as they are in the above example), will clearly affect

the variance of total estimated outstandings even in the case

k=O, and so must affect cases where k assumes an appreciable

value;

367

(ii) the situation as regards the "between-method" covariances is

much less clear due to the fact that the high variances assoc­

iated with the PPCF method lead to the assignment of little

weight to that method in the blending process; hence, changes in

PPC!-PPCF covariance will have little effect on the weight

assigned to the latter of these methods, and so little effect on

the blended estima1es.

The following table illustrates the last point "by displaying the

weights assigned to the PPCI, PPCF and PPE estimates in the formation of

the blended estima'"es given in the set of Results (i)(a) with ~ = 1.0E10.

Year Weight. assigned to estimat.e obtained by means of

-~~~~ ~- ::~~~;~j~(F~~:;~j~~~:: :~~~;~~~~:::::: 1982 104, -0 -4 1981 41 I -21 80 1980 60: -9 31 1979 124 ~ -13 -11 1978 -22 1 121 1977 -7 i 10 97

~;f~~!~~ ____ :_~~~ ______ l ______ :~ __________ :~~ _________ _

368

I I I I

These weights are reasonably in accord with intuition when reference

is made to the earlier table in this section in which the standard devi-

ations associated with the various estimates were given. They are not

thoroughly intuitive, of course, because of:

(i) the effects of covariances;

(ii) the requirement of smooth results.

The most obvious feature of the weights displayed is their assumption

of negative values in some places. This is perhaps unusual to those

accustomed to dealing with blending concepts. However, there appears to be

nothing theoretically objectionable involved. In any event, most of the

negative weights are reasonably small.

Nevertheless, if objection were taken to them, it would be possible to

duplicate the above theory but subject to the additional constraint that

the weight vector w contain only non-negative components. Such a

constraint would, however, be a non-linear one, and consequently the new

problem would not possess a neat matrix solution corresponding to (4.3.7).

The new solution would not, in fact, represent a global minimum of L(Y:X)

as does (4.3.7) but would lie on the boundary of the admissible set of w

(i.e. one or more components of w would be 0).

369

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371