- Email: [email protected]
Some alternatives for the individual model
ELSEVIER
Insurance: Mathematics and Economics 1.5(1994) 127-132
Some alternatives for the individual model R. Kaas ay1,H.U. Gerber b a University of Amsterdam, Amsterdam, Netherlands b Ecole des HEC, Lausanne, Switzerland
Received March 1994; revised July 1994
Abstract Three approximations for the total claims of a risk portfolio (individual model) are compared: the traditional collective (compound Poisson) model, a compound binomial model, and a compound pseudo-binomial model (nearly homogeneous portfolio). Of these, only the collective model is a prudent choice in terms of stop-loss order. The binomial model has a larger variance, but the nearly homogeneous approximation might have a lower one. The binomial model also leads to an exponentially larger distribution.
1. Introduction
The model of choice for the total claims distribution of a portfolio is without doubt the individual model, which assumes only independence between different policies in a portfolio. To accelerate computations, it is often approximated by more structured models such as a compound Poisson distribution (collective model). The usual choice for this compound Poisson distribution is the one where each policy is replaced by a compound Poisson distribution with expected number of terms equal to 1 and terms with the same distribution as the original policy. This compound Poisson distribution is equivalent to one which
’ Paper written on the occasion of a visit of the first author to the University of Lausanne. Work performed under contract SPES-ClY91-0089.
has an expected number of positive claims A equal to the original portfolio’s, and an individual claim amount distribution equal to a weighted average of the original claims distributions, conditional upon the fact that there is a claim. Since this leads to overestimation of the stop-loss premiums, and also of exponential premiums, we look at two other approximations. In one, the Poisson(h) distributed number of claims of the collective model used is replaced by a binomiahn, p) random variable, with n equal to the number of contracts in the portfolio and p = A/n. In the other, the actual distribution of the number of positive claims is used, which is a sum of II Bernoulli distributions with average probability of success p. The latter model is known as a nearly homogeneous portfolio. Approximations are often appraised by means of an upper bound on the difference in probabilities of sets (variational distance) or of intervals C-00, xl. In Kaas (1993) it is argued, however,
0167-6687/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0167-6687(94)00028-X
128
R. Kaas, H. U. Gerber /Insurance:
Mathematics and Economics 15 (1994) 127-132
that for actuarial purposes, it is better to look at (stop-loss) premiums. The maximal absolute difference between stop-loss premiums of two random variables is called their stop-loss distance. See Gerber (1984) for a derivation of bounds for the stop-loss and variational distances between the collective model and the individual model, and De Pril and Dhaene (1992) 2 for an improved stop-loss distance. For stop-loss ordered random variables with the same mean, the integrated error in the stop-loss premiums equals one half the difference in the variances, see again Kaas (1993). It seems helpful to have bounds for each single error in a stop-loss premium, but the decisive advantage of using the variance difference is that it enables one to actually compute the aggregated error, while other methods merely give an upper bound, which might or might not be unduly pessimistic, for the maximal value of each single error. All approximations studied in this paper have the same expected value as the original claims total, which means that there will be no stochastic order (i.e. uniformly ordered distribution functions). So to compare these approximations we look at weaker order concepts, namely stop-loss order (uniformly ordered stop-loss premiums), exponential order (ordered exponential premiums, which is tantamount to moment generating functions uniformly ordered on (0, a)), as well as the resulting variances of the random variables approximating the total claims. The nearly homogeneous portfolio is stop-loss smaller than the binomial model, which in turn is smaller than the collective model. The individual and binomial models cannot always be ordered by this criterion, but the individual model is exponentially smaller than the binomial approximation, so it also has a smaller variance. The nearly homogeneous model has smaller or larger variance than the individual model, depending on the variability of the mean claims and of the probabilities of non-zero claims.
2. Stop-loss ances
an anonymous referee for supplying us with references and also suggesting other improve-
order and vari-
From the textbook Kaas et al. (1994) on ordering of risks, we recall that a non-negative random variable X is smaller than Y in (net) stop-loss order, written XI,, Y, if for the stop-loss premiums TX(d) = E[(Xd),] and r,(d) we have d) 5 rTTy(d)
rx(
for all d 2 0.
There are some more lines of reasoning leading to exactly this order relation between non-negative random variables. One is that X ssl Y if and only if all risk-averse decision makers prefer losing X to losing Y. This is why stop-loss order should be a fundamental tool for comparing risks. Looking at the common preferences of the subgroup of decision makers with exponential utility functions only, one gets the exponential order, written X5, Y, and defined by XI,
Y if E[eaX]
5 E[e*‘]
for all (Y2 0.
Note that this condition on the moment generating functions mx(a) = E[eaXl and m,(a) is equivalent to X needing smaller premiums in case of exponential utility with arbitrary parameter LY,since such premiums are equal to (l/a) log m,(a). Exponential order is weaker than stoploss order. If X& Y and E[X] = E[Y], then Var[ X] 5 Var[Y] holds (we also write X I,Z Y for this last inequality). This property can easily be checked by looking at the coefficient of t2 in a Taylor expansion near 0 of the moment generating functions of X and Y. Kaas (1993) argues that instead of looking at stop-loss premiums, it is often sufficient to look only at variances. His reasoning is based on the following observations. If X and Y are two nonnegative random variables with E[Xl = E[Y], then
=
xthank some extra ments.
order, exponential
$(Var[Y]
- Var[X]),
(1)
The left-hand side is the integrated (signed) error in the stop-loss premiums. If moreover XI,, Y, then it also equals the integrated absolute error
R. Kaas, H. U. Gerber/Insurance:
in the stop-loss premiums. ality between the stop-loss one has approximately 7~~(t)/r~(t)
Mathematics
Relying on proportionpremiums of X and Y,
=Var[Y]/Var[X]
for t2~.
From (1) we see right away that if X I,[ Y and E[ X] = E[ Y], then X I,Z Y. Also, if additionally Var[X] = Var[Y], in view of (1) their stop-loss premiums are equal for all t, which can only happen if X and Y have the same distribution. Note that it is possible to construct X and Y with equal mean and variance, but X
3. Order relations
...
+x,,
where y1 is the number of policies in the portfolio, and Xi, j = 1,. . . , n is the claim on policy j. The random variables Xi are assumed to be nonnegative and independent, but not necessarily identically distributed. The homogeneous portfolio approximation to S arises by replacing each contract with an ‘average’ contract: U=Y,
+ *** +y,,
where the 5 are iid random as Y, with
variables
distributed
,kp[xj’yl’
p[Y
,=l The collective model given by v=y,+
...
+YN,
15 (1994) 127-132
approximation
to S is
129
where Yr, Y2,. . . are as above, and N, independent of Y,, Y2,. . . , has a Poisson (n) distribution. Other collective model approximations are conceivable, but this is the customary choice. See also Biihlmann et al. (1977). that policy If pj, j= l,..., IZ is the probability j has a non-zero claim, one may show that I/ as defined above has the same cdf as the one resulting from the following more common definition of the standard collective model. The compound Poisson approximation for S is given by I/‘=Z,+ ... +z,, where A4 is Poisson(h) distributed, - Z iid, with Z and Z,, Z,,... Y]Y>O, so
of the models
In this section we introduce the different portfolio models to be used and state results about order between the models, as regards stop-loss premiums, variances, as well as exponential premiums. The individual model for the total claims of a portfolio is simply s=x,+
and Economics
Replacing A4 by number of claims equivalent to the for S. The compound U’=Z,+
...
with h = Cpj, distributed as
a binomial (n, p> distributed L with p = h/n gives a model homogeneous approximation U binomial approximation to S is
+z,,
L has a binomial (n, h/n) distribution and is as above, independent of L. The Z,, Z,,... resulting approximation is called the ‘natural approximation’ by Sundt (1985). The actual number of non-zero claims of S is not binomial, but only pseudo-binomial, in the sense that it equals the number of successes in n experiments with possibly different probabilities pj of success, j = 1,. . . , n. This leads us to a potentially better approximation. The nearly homogeneous portfolio approximation for S is T=Z, + ... +Z,, where
where K has a distribution which is the convolution of Bernoulli(pj) distributions, j = 1,. . . , n, and Z,, Z,, . . . is as above, independent of K. It is easy to see that all random variables S, T, U and V have the same mean. From the paper of Biihlmann et al. (1977) we know that the individual model S I,, V. From Example 111.3.3.1 of Kaas et al. (1994) we know that K ssr L ssl M.
130
R. Kaas, H. U. Gerber/Insurance:
Mathematics
So T ssr U I,, V follows immediately since stoploss order is preserved under compounding, see e.g. Kaas et al. (19941, Theorem 111.3.2.4, and we have:
and Economics
By the usual properties of compound disProof. tributions, we have Var[V]=E[N]E[Y2]=n~~2~~&(~) x
Theorem 1. The following stop-loss order relations between S, T, l-Jand V are valid: S ss, V;
T ss, U <,, V.
= C
Cx2fj(
i
x);
Var[S] = iVL[Xj]
No further stop-loss order relations can be established, since examples exist with rrs(d) < r,(d) for some d, but m-s(d) > z-,(d) for other d. Such examples are easy to find: in view of the final remarks of the previous section it is sufficient to construct an example with both E[S] = E[ U] and Var[ S] = Var[ U]. This is accomplished for instance by taking X, = 2 and P[X, = 11= P[ X, = 3]= l/2, so Var[ X, +X,1 = 2Var[ Y] = 1. The results of Theorem 1 also hold if the order ssr is replaced by $2 . There was no stop-loss order between S and U, but as will see, S $2 U does hold, so the stop-loss premiums of U tend to be higher. Additional notation: without critical loss of generality, assume that theXj have a discrete density, with
fi(~)=P[X~=x],
15 (1994) 127-132
x20,
j=l,...,
n.
Write again pj = 1 -J(O) for the probability of a positive claim on the jth contract, let p = l/nCpi. and let pj denote the mean claim on contract j,
=
C Cx2fjtx) - C (CxfjCx,)zJ
i
i
x
Var[U] =nVar[Y] Var[T]
x
=nE[Y2]
=E[K]Var[Z]
-n(E[Y])2;
+ (E[Z])2Var[K]
= E[ Z2]E[ K] + (E[ Z])” x(Var[K] =nE[Y2]
- E[K]) -
since EIZkl = (l/p)EIYkl, and Var[K] = Cpj(l -pj).
k = 1, 2, E[K] = Cpj 0
As an immediate consequence lemma, we have: Lemma 2.
of the previous
Var[S] 5 Var[ T] holds if and only if
(2)
SO
Lemma 1. The variances of S, T, U and V are Var[Vl=
C Cx2fj(x), i x
Var[ S] = Var[ V] - C~_L;,
Var[T] =Var[V]
- ( cpj)‘$$, i
Var[U] =Var[V]
- i(
xpi)2. i
Inequality (2) is equivalent to the means of the claims on each contract having a greater variation than the probabilities of claim, in the sense that if W, and W, are random variables with P[ WI = pj] = P[ W, = pIl = (l/n>, j = 1,. . . , n, (2) expresses that the variation coefficient of W, is at least the one of W,. Since Var[U] - Var[S] is just nVar[W2], we can also deduce that Var[U] 2 Var[ S]. Write pi = cjpj for all j, so cj is the conditional mean of contract j, given there is a non-zero claim. Then equality in (2) holds for instance if all cj are equal. This gives us a whole class of instances of U and S which are not stop-loss ordered. For relation (2) not to hold, it can be seen
R. Kaas, H.U. Gerber/Insurance:
Mathematics and Economics I5 (1994) 127-132
that large values of cj must in general correspond to small values of pj and vice versa. One would like to simplify (2) for instance the condition that the sequences cj and pj are ‘positively correlated’ (in the sense that Ccjpj 2 (l/n)CcjCpj) and/or that c,! and p,? are positively correlated, but it turned out that these conditions are neither necessary nor sufficient for (2). Since I,, is a sufficient condition for I,Z , by Theorem 1 we have Theorem 2. The following variance relations between S, T, U and V hold: S <,z U;
T 1,s U 5,~ V.
The individual model S not only has a smaller variance than the compound Binomial model U, but it is also smaller in exponential order: for the moment generating functions we have m,(t) 5 m,(t) for all t 2 0. This means that loss S is preferable over loss U for those decision makers who have an exponential utility function. Indeed, we have Theorem 3. The following exponential order relations between S, T, U and V hold: S_<,U;
T<,Us,V.
Proof. The second half of the theorem follows again from Theorem 1. For the first inequality, consider the logarithms of the moment generating functions (i.e., the cumulant generating functions) of S and U. We have
klogmx.jt),
logms(t)=log~mx~t)= j=l
j=l
and log mU( t) = log(E[e’Y])n
= n
logi
,? ]=l
mx,w
so m,(t) I m,(t) for all t 2 0 follows by concavity of the log-function. But this is equivalent to S I, U. See Section V.3 of Kaas et al. (1994). 0
4. Concluding
remarks
The basic reason for using approximating models is that calculations can be done by fast recur-
131
sive techniques. But when using an approximation, it shows good business sense to base one’s decisions only on a conservative approximation, i.e., a more dangerous random variable. That is why, in view of Theorem 1, we recommend the use of the compound Poisson model, even though it leads to premiums which are too high. If greater accuracy is desired, even at the risk of obtaining stop-loss premiums or exponential premiums which are insufficient, the nearly homogeneous portfolio approximation can be used. This will in general be safe if the variability of the mean claims is larger than that of the probabilities of non-zero claims, see Lemma 2. Note that there are several ways to improve on the performance of the models discussed here. For instance Jewel1 and Sundt (1981) use a compound binomial distribution with number of trials parameter possibly different from the number of policies n, to match both mean and variance of the individual model. We also mention the paper by Kuon et al. (19931, who by the way quite convincingly present the case for using Central Limit Theorem based approximations for large portfolios, instead of recursive algorithms. In a recent paper, Weba (1993) derives an upper bound for the variational distance between the nearly homogeneous model and the collective model, and concludes that the binomial model leads to an approximation of the nearly homogeneous model which is closer than the one provided by the collective model. In our opinion, it makes no sense to study how good the binomial and collective model approximate the nearly homogeneous model, since this model has no meaning per se; its only reason for existence consists of being an approximation to the individual model. It turns out that also the variational distance between nearly homogeneous model T and collective model V is large when the probabilities of non-zero claims exhibit a large variation. Kaas (1993) also investigated the precision achievable by the individual model, considering the presence of other sources of inaccuracy such as cumulation and uncertainty about both the claim probabilities and the claims distribution, and the fact that algorithms generally require the losses Xj constituting S to be not only integer-
132
R. Kaas, H. U. Gerber/Insurance:
Mathematics and Economics 15 (1994) 127-132
valued, but at the same time small, so a process
of rounding and scaling must take place. This means that the exactness of the results is already lost before any approximation has taken place.
References Biihlmann, H., B. Gagliardi, H.U. Gerber and E. Straub (1977). Some inequalities for stop-loss premiums Astin Bulletin 9, 169-177. De Pril, N. and J. Dhaene (19921 Error bounds for compound Poisson approximations of the individual risk model Astin Bulletin 22, 135-148. Gerber, H.U. (1984). Error bounds for the compound Poisson approximation. Insurance: Mathematics and Economics 3, 191-194.
Jewell, W.S. and B. Sundt (1981). Improved approximations for the distribution of a heterogeneous portfolio. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, 221-240. Kaas, R. (1993). How to (and how not to) compute stop-loss premiums in practice. Insurance: Mathematics and Economics 13, 241-254. Kaas, R., A.E. van Heerwaarden, M.J. Goovaerts (1994). Ordering of actuarial risks. Education Series 1, CAIRE, Brussels. Kuon, S., M. Radtke and A. Reich (1993). An appropriate way to switch from the individual model to the collective model. Astin Bulletin 23, 23-54. Sundt, B. (1985). On approximations for the distribution of a heterogeneous risk portfolio. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, 189-203. Weba, M. (1993). Approximating the aggregate claims distribution of a nearly homogeneous portfolio by means of a compound binomial distribution. Technical report, Universitlt Hamburg.
Insurance: Mathematics and Economics 1.5(1994) 127-132
Some alternatives for the individual model R. Kaas ay1,H.U. Gerber b a University of Amsterdam, Amsterdam, Netherlands b Ecole des HEC, Lausanne, Switzerland
Received March 1994; revised July 1994
Abstract Three approximations for the total claims of a risk portfolio (individual model) are compared: the traditional collective (compound Poisson) model, a compound binomial model, and a compound pseudo-binomial model (nearly homogeneous portfolio). Of these, only the collective model is a prudent choice in terms of stop-loss order. The binomial model has a larger variance, but the nearly homogeneous approximation might have a lower one. The binomial model also leads to an exponentially larger distribution.
1. Introduction
The model of choice for the total claims distribution of a portfolio is without doubt the individual model, which assumes only independence between different policies in a portfolio. To accelerate computations, it is often approximated by more structured models such as a compound Poisson distribution (collective model). The usual choice for this compound Poisson distribution is the one where each policy is replaced by a compound Poisson distribution with expected number of terms equal to 1 and terms with the same distribution as the original policy. This compound Poisson distribution is equivalent to one which
’ Paper written on the occasion of a visit of the first author to the University of Lausanne. Work performed under contract SPES-ClY91-0089.
has an expected number of positive claims A equal to the original portfolio’s, and an individual claim amount distribution equal to a weighted average of the original claims distributions, conditional upon the fact that there is a claim. Since this leads to overestimation of the stop-loss premiums, and also of exponential premiums, we look at two other approximations. In one, the Poisson(h) distributed number of claims of the collective model used is replaced by a binomiahn, p) random variable, with n equal to the number of contracts in the portfolio and p = A/n. In the other, the actual distribution of the number of positive claims is used, which is a sum of II Bernoulli distributions with average probability of success p. The latter model is known as a nearly homogeneous portfolio. Approximations are often appraised by means of an upper bound on the difference in probabilities of sets (variational distance) or of intervals C-00, xl. In Kaas (1993) it is argued, however,
0167-6687/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0167-6687(94)00028-X
128
R. Kaas, H. U. Gerber /Insurance:
Mathematics and Economics 15 (1994) 127-132
that for actuarial purposes, it is better to look at (stop-loss) premiums. The maximal absolute difference between stop-loss premiums of two random variables is called their stop-loss distance. See Gerber (1984) for a derivation of bounds for the stop-loss and variational distances between the collective model and the individual model, and De Pril and Dhaene (1992) 2 for an improved stop-loss distance. For stop-loss ordered random variables with the same mean, the integrated error in the stop-loss premiums equals one half the difference in the variances, see again Kaas (1993). It seems helpful to have bounds for each single error in a stop-loss premium, but the decisive advantage of using the variance difference is that it enables one to actually compute the aggregated error, while other methods merely give an upper bound, which might or might not be unduly pessimistic, for the maximal value of each single error. All approximations studied in this paper have the same expected value as the original claims total, which means that there will be no stochastic order (i.e. uniformly ordered distribution functions). So to compare these approximations we look at weaker order concepts, namely stop-loss order (uniformly ordered stop-loss premiums), exponential order (ordered exponential premiums, which is tantamount to moment generating functions uniformly ordered on (0, a)), as well as the resulting variances of the random variables approximating the total claims. The nearly homogeneous portfolio is stop-loss smaller than the binomial model, which in turn is smaller than the collective model. The individual and binomial models cannot always be ordered by this criterion, but the individual model is exponentially smaller than the binomial approximation, so it also has a smaller variance. The nearly homogeneous model has smaller or larger variance than the individual model, depending on the variability of the mean claims and of the probabilities of non-zero claims.
2. Stop-loss ances
an anonymous referee for supplying us with references and also suggesting other improve-
order and vari-
From the textbook Kaas et al. (1994) on ordering of risks, we recall that a non-negative random variable X is smaller than Y in (net) stop-loss order, written XI,, Y, if for the stop-loss premiums TX(d) = E[(Xd),] and r,(d) we have d) 5 rTTy(d)
rx(
for all d 2 0.
There are some more lines of reasoning leading to exactly this order relation between non-negative random variables. One is that X ssl Y if and only if all risk-averse decision makers prefer losing X to losing Y. This is why stop-loss order should be a fundamental tool for comparing risks. Looking at the common preferences of the subgroup of decision makers with exponential utility functions only, one gets the exponential order, written X5, Y, and defined by XI,
Y if E[eaX]
5 E[e*‘]
for all (Y2 0.
Note that this condition on the moment generating functions mx(a) = E[eaXl and m,(a) is equivalent to X needing smaller premiums in case of exponential utility with arbitrary parameter LY,since such premiums are equal to (l/a) log m,(a). Exponential order is weaker than stoploss order. If X& Y and E[X] = E[Y], then Var[ X] 5 Var[Y] holds (we also write X I,Z Y for this last inequality). This property can easily be checked by looking at the coefficient of t2 in a Taylor expansion near 0 of the moment generating functions of X and Y. Kaas (1993) argues that instead of looking at stop-loss premiums, it is often sufficient to look only at variances. His reasoning is based on the following observations. If X and Y are two nonnegative random variables with E[Xl = E[Y], then
=
xthank some extra ments.
order, exponential
$(Var[Y]
- Var[X]),
(1)
The left-hand side is the integrated (signed) error in the stop-loss premiums. If moreover XI,, Y, then it also equals the integrated absolute error
R. Kaas, H. U. Gerber/Insurance:
in the stop-loss premiums. ality between the stop-loss one has approximately 7~~(t)/r~(t)
Mathematics
Relying on proportionpremiums of X and Y,
=Var[Y]/Var[X]
for t2~.
From (1) we see right away that if X I,[ Y and E[ X] = E[ Y], then X I,Z Y. Also, if additionally Var[X] = Var[Y], in view of (1) their stop-loss premiums are equal for all t, which can only happen if X and Y have the same distribution. Note that it is possible to construct X and Y with equal mean and variance, but X
3. Order relations
...
+x,,
where y1 is the number of policies in the portfolio, and Xi, j = 1,. . . , n is the claim on policy j. The random variables Xi are assumed to be nonnegative and independent, but not necessarily identically distributed. The homogeneous portfolio approximation to S arises by replacing each contract with an ‘average’ contract: U=Y,
+ *** +y,,
where the 5 are iid random as Y, with
variables
distributed
,kp[xj’yl’
p[Y
,=l The collective model given by v=y,+
...
+YN,
15 (1994) 127-132
approximation
to S is
129
where Yr, Y2,. . . are as above, and N, independent of Y,, Y2,. . . , has a Poisson (n) distribution. Other collective model approximations are conceivable, but this is the customary choice. See also Biihlmann et al. (1977). that policy If pj, j= l,..., IZ is the probability j has a non-zero claim, one may show that I/ as defined above has the same cdf as the one resulting from the following more common definition of the standard collective model. The compound Poisson approximation for S is given by I/‘=Z,+ ... +z,, where A4 is Poisson(h) distributed, - Z iid, with Z and Z,, Z,,... Y]Y>O, so
of the models
In this section we introduce the different portfolio models to be used and state results about order between the models, as regards stop-loss premiums, variances, as well as exponential premiums. The individual model for the total claims of a portfolio is simply s=x,+
and Economics
Replacing A4 by number of claims equivalent to the for S. The compound U’=Z,+
...
with h = Cpj, distributed as
a binomial (n, p> distributed L with p = h/n gives a model homogeneous approximation U binomial approximation to S is
+z,,
L has a binomial (n, h/n) distribution and is as above, independent of L. The Z,, Z,,... resulting approximation is called the ‘natural approximation’ by Sundt (1985). The actual number of non-zero claims of S is not binomial, but only pseudo-binomial, in the sense that it equals the number of successes in n experiments with possibly different probabilities pj of success, j = 1,. . . , n. This leads us to a potentially better approximation. The nearly homogeneous portfolio approximation for S is T=Z, + ... +Z,, where
where K has a distribution which is the convolution of Bernoulli(pj) distributions, j = 1,. . . , n, and Z,, Z,, . . . is as above, independent of K. It is easy to see that all random variables S, T, U and V have the same mean. From the paper of Biihlmann et al. (1977) we know that the individual model S I,, V. From Example 111.3.3.1 of Kaas et al. (1994) we know that K ssr L ssl M.
130
R. Kaas, H. U. Gerber/Insurance:
Mathematics
So T ssr U I,, V follows immediately since stoploss order is preserved under compounding, see e.g. Kaas et al. (19941, Theorem 111.3.2.4, and we have:
and Economics
By the usual properties of compound disProof. tributions, we have Var[V]=E[N]E[Y2]=n~~2~~&(~) x
Theorem 1. The following stop-loss order relations between S, T, l-Jand V are valid: S ss, V;
T ss, U <,, V.
= C
Cx2fj(
i
x);
Var[S] = iVL[Xj]
No further stop-loss order relations can be established, since examples exist with rrs(d) < r,(d) for some d, but m-s(d) > z-,(d) for other d. Such examples are easy to find: in view of the final remarks of the previous section it is sufficient to construct an example with both E[S] = E[ U] and Var[ S] = Var[ U]. This is accomplished for instance by taking X, = 2 and P[X, = 11= P[ X, = 3]= l/2, so Var[ X, +X,1 = 2Var[ Y] = 1. The results of Theorem 1 also hold if the order ssr is replaced by $2 . There was no stop-loss order between S and U, but as will see, S $2 U does hold, so the stop-loss premiums of U tend to be higher. Additional notation: without critical loss of generality, assume that theXj have a discrete density, with
fi(~)=P[X~=x],
15 (1994) 127-132
x20,
j=l,...,
n.
Write again pj = 1 -J(O) for the probability of a positive claim on the jth contract, let p = l/nCpi. and let pj denote the mean claim on contract j,
=
C Cx2fjtx) - C (CxfjCx,)zJ
i
i
x
Var[U] =nVar[Y] Var[T]
x
=nE[Y2]
=E[K]Var[Z]
-n(E[Y])2;
+ (E[Z])2Var[K]
= E[ Z2]E[ K] + (E[ Z])” x(Var[K] =nE[Y2]
- E[K]) -
since EIZkl = (l/p)EIYkl, and Var[K] = Cpj(l -pj).
k = 1, 2, E[K] = Cpj 0
As an immediate consequence lemma, we have: Lemma 2.
of the previous
Var[S] 5 Var[ T] holds if and only if
(2)
SO
Lemma 1. The variances of S, T, U and V are Var[Vl=
C Cx2fj(x), i x
Var[ S] = Var[ V] - C~_L;,
Var[T] =Var[V]
- ( cpj)‘$$, i
Var[U] =Var[V]
- i(
xpi)2. i
Inequality (2) is equivalent to the means of the claims on each contract having a greater variation than the probabilities of claim, in the sense that if W, and W, are random variables with P[ WI = pj] = P[ W, = pIl = (l/n>, j = 1,. . . , n, (2) expresses that the variation coefficient of W, is at least the one of W,. Since Var[U] - Var[S] is just nVar[W2], we can also deduce that Var[U] 2 Var[ S]. Write pi = cjpj for all j, so cj is the conditional mean of contract j, given there is a non-zero claim. Then equality in (2) holds for instance if all cj are equal. This gives us a whole class of instances of U and S which are not stop-loss ordered. For relation (2) not to hold, it can be seen
R. Kaas, H.U. Gerber/Insurance:
Mathematics and Economics I5 (1994) 127-132
that large values of cj must in general correspond to small values of pj and vice versa. One would like to simplify (2) for instance the condition that the sequences cj and pj are ‘positively correlated’ (in the sense that Ccjpj 2 (l/n)CcjCpj) and/or that c,! and p,? are positively correlated, but it turned out that these conditions are neither necessary nor sufficient for (2). Since I,, is a sufficient condition for I,Z , by Theorem 1 we have Theorem 2. The following variance relations between S, T, U and V hold: S <,z U;
T 1,s U 5,~ V.
The individual model S not only has a smaller variance than the compound Binomial model U, but it is also smaller in exponential order: for the moment generating functions we have m,(t) 5 m,(t) for all t 2 0. This means that loss S is preferable over loss U for those decision makers who have an exponential utility function. Indeed, we have Theorem 3. The following exponential order relations between S, T, U and V hold: S_<,U;
T<,Us,V.
Proof. The second half of the theorem follows again from Theorem 1. For the first inequality, consider the logarithms of the moment generating functions (i.e., the cumulant generating functions) of S and U. We have
klogmx.jt),
logms(t)=log~mx~t)= j=l
j=l
and log mU( t) = log(E[e’Y])n
= n
logi
,? ]=l
mx,w
so m,(t) I m,(t) for all t 2 0 follows by concavity of the log-function. But this is equivalent to S I, U. See Section V.3 of Kaas et al. (1994). 0
4. Concluding
remarks
The basic reason for using approximating models is that calculations can be done by fast recur-
131
sive techniques. But when using an approximation, it shows good business sense to base one’s decisions only on a conservative approximation, i.e., a more dangerous random variable. That is why, in view of Theorem 1, we recommend the use of the compound Poisson model, even though it leads to premiums which are too high. If greater accuracy is desired, even at the risk of obtaining stop-loss premiums or exponential premiums which are insufficient, the nearly homogeneous portfolio approximation can be used. This will in general be safe if the variability of the mean claims is larger than that of the probabilities of non-zero claims, see Lemma 2. Note that there are several ways to improve on the performance of the models discussed here. For instance Jewel1 and Sundt (1981) use a compound binomial distribution with number of trials parameter possibly different from the number of policies n, to match both mean and variance of the individual model. We also mention the paper by Kuon et al. (19931, who by the way quite convincingly present the case for using Central Limit Theorem based approximations for large portfolios, instead of recursive algorithms. In a recent paper, Weba (1993) derives an upper bound for the variational distance between the nearly homogeneous model and the collective model, and concludes that the binomial model leads to an approximation of the nearly homogeneous model which is closer than the one provided by the collective model. In our opinion, it makes no sense to study how good the binomial and collective model approximate the nearly homogeneous model, since this model has no meaning per se; its only reason for existence consists of being an approximation to the individual model. It turns out that also the variational distance between nearly homogeneous model T and collective model V is large when the probabilities of non-zero claims exhibit a large variation. Kaas (1993) also investigated the precision achievable by the individual model, considering the presence of other sources of inaccuracy such as cumulation and uncertainty about both the claim probabilities and the claims distribution, and the fact that algorithms generally require the losses Xj constituting S to be not only integer-
132
R. Kaas, H. U. Gerber/Insurance:
Mathematics and Economics 15 (1994) 127-132
valued, but at the same time small, so a process
of rounding and scaling must take place. This means that the exactness of the results is already lost before any approximation has taken place.
References Biihlmann, H., B. Gagliardi, H.U. Gerber and E. Straub (1977). Some inequalities for stop-loss premiums Astin Bulletin 9, 169-177. De Pril, N. and J. Dhaene (19921 Error bounds for compound Poisson approximations of the individual risk model Astin Bulletin 22, 135-148. Gerber, H.U. (1984). Error bounds for the compound Poisson approximation. Insurance: Mathematics and Economics 3, 191-194.
Jewell, W.S. and B. Sundt (1981). Improved approximations for the distribution of a heterogeneous portfolio. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, 221-240. Kaas, R. (1993). How to (and how not to) compute stop-loss premiums in practice. Insurance: Mathematics and Economics 13, 241-254. Kaas, R., A.E. van Heerwaarden, M.J. Goovaerts (1994). Ordering of actuarial risks. Education Series 1, CAIRE, Brussels. Kuon, S., M. Radtke and A. Reich (1993). An appropriate way to switch from the individual model to the collective model. Astin Bulletin 23, 23-54. Sundt, B. (1985). On approximations for the distribution of a heterogeneous risk portfolio. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, 189-203. Weba, M. (1993). Approximating the aggregate claims distribution of a nearly homogeneous portfolio by means of a compound binomial distribution. Technical report, Universitlt Hamburg.