• No results found

Past and present trends in aggregate claims analysis

N/A
N/A
Protected

Academic year: 2021

Share "Past and present trends in aggregate claims analysis"

Copied!
38
0
0

Loading.... (view fulltext now)

Full text

(1)

Past and present trends

in aggregate claims analysis

Gordon E. Willmot

Munich Re Professor of Insurance

Department of Statistics and Actuarial Science University of Waterloo

1st Quebec-Ontario Workshop on Insurance Mathematics January 28, 2011 Montreal, Quebec

(2)

• goal of talk is a discussion of modelling and analysis of aggregate claims on a portfolio of business

– historical perspective; techniques and complexity of models has changed over time

– models interdisciplinary

∗ credit risk

∗ operational risk

∗ profit analysis

(3)

• modelling incorporates two basic components

– random number of “events” of interest (frequency)

– each event generates a random quantity of interest (severity)

• main goal is to aggregate these quantities

– fixed period of time; aggregate claims analysis

– tracking of behaviour over time; surplus analysis

• complexity of models increased greatly recently

– attempt to realistically model quantities

(4)

• historical description of aggregate model

– interested in the evaluation of aggregate claims distribution function (df) G(y) = 1 − G(y) where

S = Y1 + Y2 + . . . + YN

N = number of claims, Yi = amount of i-th claim

– traditionally, {Y1, Y2, . . .} assumed to be an iid sequence, inde-pendent of N

– also of interest is stop-loss random variable

(S − y)+ = max(S − y, 0)

∗ stop-loss premium R1(y) = E{(S − y)+} = Ry∞ G(t)dt

(5)

• original approaches to evaluation involved parametric approxima-tions applied to G(y) directly

– easy to use, typically requires simple quantities such as moments

– questionable accuracy, particularly in right tail

– difficult to incorporate changes in individual policy characteris-tics such as deductibles and maximums

(6)

• commonly used approximations

– normal-based

∗ normal approximations give light right tail

∗ normal power, Haldane’s, and Wilson-Hilferty all assume h(S) is normally distributed for some h(·)

– gamma-based

∗ Beekman-Bowers, translated gamma

(7)

– exponential approximations

∗ motivated by ruin theory (compound geometric)

∗ includes Cramer-Lundberg asymptotic formula, Tijms, De Vylder’s method

∗ light right tail

– subexponential approximations

∗ heavy right tail

∗ often based on extreme value arguments

(8)

– Esscher’s method

∗ surprisingly good accuracy

∗ gave rise to Esscher transform (applied probability and math-ematical finance; change of measure)

∗ adopted by statistical community for approximating distribu-tion of sample statistics which involve sums of independent random variables)

∗ often referred to as saddlepoint approximations, exponential tilting

(9)

• numerical procedures

– simulation

∗ used in 1970’s

∗ advantage in that complicated models may be used

∗ disadvantage in that right tail may be inaccurate unless many values used

∗ disadvantage in that difficult to modify assumptions at indi-vidual claim level

(10)

– transform inversion techniques

∗ FFT in discrete case

∗ complex inversion based on aggregate pgf

∗ “black box”

∗ may require discretization of claim size distribution

– continuous inversion approaches

∗ Heckman-Meyers (characteristic function, piecewise constant density)

∗ Laplace transform inversion (much recent progress in queueing community)

(11)

– recursions

∗ computation of (discretized) probability mass function of S recursively, beginning with Pr(S = 0)

∗ let pn = a + nb pn−1 for n = 2, 3, . . . ∗ Panjer-type recursion Pr(S = y) = {p1 − (a + b)p0} Pr(Y = y) + y X x=0 a + bx y ! Pr(Y = x) Pr(S = y − x)

(12)

– includes most of basic compound models, e.g., Poisson, Bino-mial, negative binoBino-mial, logarithmic series

– extensions to other models as well

– simple to understand and use

– compound Poisson due to Euler, Adelson in queueing context, Panjer (1981) in actuarial context

(13)

• individual policy modifications

– deductibles, maximums, and coinsurance on each claim

– easy to incorporate with numerical procedures such as recursions

– statistically, deductibles involve “left truncation” on loss sizes and “thinning” of loss numbers

– maximums involve “right censoring”; coinsurance results in scale changes, both on loss sizes

(14)

• trends in aggregate loss modelling in last quarter century

– removal of independent and/or identically distributed assump-tions

∗ possible due to mathematical and computational advances

∗ claim count dependencies

· time series models

· dependence through latent variables as in mixed Poisson and MAP models

∗ claim size dependencies

· MAP models, mixtures as in credibility

(15)

· strong dependency concepts such as comonotonicity

∗ dependence between claim sizes and numbers in dependent Sparre Andersen (renewal) risk models

∗ removal of identically distributed assumption

· discounted aggregate claims incorporating inflation

· claim sizes independent but depend on time of occurrence

· (mixed) Poisson process allows reduction to iid case

(16)

– stronger inter-disciplinary influences

∗ phase-type assumptions borrowed from queueing theory

· greater flexibility even for simple models

· very useful for complex models (fluid flow techniques)

∗ Gerber-Shiu techniques for option pricing and Esscher trans-form analysis in mathematical finance

∗ wide variety of probabilistic, statistical, and applied mathe-matical tools used in risk analysis

(17)

– use of ‘semiparametric’ distributional assumptions

∗ phase-type distributions, combinations of exponentials, mix-ture of Erlangs

∗ all are dense in class of distributions in R+, flexible

∗ use of these models involves a hybrid of analytic and numerical approaches

∗ semi-parametric nature makes estimation nontrivial

∗ can be numerical root-finding difficulties with phase-type dis-tributions (location of eigenvalues for calculation of matrix-exponentials) and combinations of exponentials (partial frac-tion expansions on Laplace transforms)

(18)

∗ phase-type distributions

· absorption time in time-homogeneous Markov chain

· particularly useful for fairly complex stochastic models (ad-vantage over other two classes) as well as for simple models

· calculations of most quantities of interest straightforward

· disadvantages

1) knowledge of matrix calculus needed

2) often necessary to assume that all components of model are of phase-type

(19)

• mixed Erlang distributions

– huge class of distributions

∗ includes class of phase-type distributions

∗ includes many distributions whose membership in class is not obvious from definition

(20)

– extremely useful for simple risk models

∗ model for claim sizes

∗ all quantities of interest computed easily using infinite series (even finite time ruin probabilities)

∗ no root finding needed

∗ only requires use of simple algebra

– present discussion mainly from Willmot and Woo (2007) and Willmot and Lin (2011)

(21)

– mathematical introduction ∗ Erlang-j pdf for j = 1, 2, . . . , β > 0 ej(y) = β (βy) j−1 e−βy (j − 1)! , y > 0 ∗ mixed Erlang pdf f (y) = ∞ X j=1 qjej(y), y > 0

where {q1, q2, . . .} is a discrete counting measure

∗ includes Erlang-j as special case qj = 1, and exponential as special case q1 = 1; for many class members, {qj; j = 1, 2, 3, . . .} is most easily expressed through its probability gen-erating function (pgf)

(22)

· let Q(z) = P∞j=1qjzj be the pgf, then the mixed Erlang Laplace transform (LT) is ˜ f (s) = Z 0 e −syf (y)dy = Q β β + s ! ,

implying that f (y) is itself a compound pdf with an expo-nential secondary pdf, or expoexpo-nential “phases” in queueing terminology

· if the LT may be put in this form then the distribution is a mixed Erlang

(23)

– loss model properties

∗ tail ¯F (y) = Ry∞ f (x)dx is given by ¯ F (y) = e−βy ∞ X k=0 ¯ Qk(βy) k k! = ∞ X k=1 ¯ Qk−1 β ek(y) where ¯Qk = ∞P j=k+1 qj

∗ for value at risk (VaR) or quantiles; at level p, VaR = vp where ¯

F (vp) = 1 − p, and vp is easily obtained numerically

∗ asymptotic Lundberg type formula available for ¯F (x) via com-pound distribution representation

(24)

∗ moments ∞ Z 0 ykf (y)dy = β−k ∞ X j=1 qj(k + j − 1)! (j − 1)!

∗ excess loss (residual lifetime) pdf (payment per payment basis with a deductible for x) still of mixed Erlang form

fx(y) = f (x + y)¯ F (x) = ∞ X j=1 qj,xej(y) with qj,x = ∞ P i=j qi (βx)i−j (i−j)! ∞ P j=1 ¯ Qm(βx) m m!

(25)

– force of mortality (failure or hazard rate) µ(y) = f (y)/ ¯F (y) satisfies µ(0) = βq1, µ(∞) = β 1 − z0−1 where z0 is the ra-dius of convergence of Q(z) (µ(∞) = β for finite mixtures), and µ(y) ≤ β (dominates exponential in failure rate and hence stochastic order)

– equilibrium distribution (useful in tail classification and ruin the-ory) still of mixed Erlang form

fe(y) = ¯ F (y) R 0 F (x)dx¯ = ∞ X j=1 qj∗ej(y) with qj∗ = Q¯j−1 P k=1 kqk

(26)

– mean excess loss (mean residual lifetime) is r(y) = R0∞ yfx(y)dy (also reciprocal of equilibrium failure rate)

r(y) = 1 µe(y) = ∞ P j=0 ¯ Q∗j(βy)j! j β ∞P j=0q ∗ j+1 (βy)j j! with ¯Q∗j = ∞P k=j+1 qk∗ ∗ also, r(0) = R0∞ yf (y)dy = ∞P j=1jqj/β is the mean, r(∞) = z0/ {β (z0 − 1)} and r(∞) = 1/β if z0 = ∞, and r(y) ≥ 1/β

(27)

– aggregate claims with mixed Erlang claim sizes

∗ let {c0, c1, c2, . . .} have the compound pgf

C(z) = ∞

X

n=0

cnzn = P {Q(z)}

where P (z) = E(zN) = P∞n=0pnzn, and {c0, c1, c2, . . .} is it-self a compound distribution which may often be computed recursively ∗ aggregate claims LT Z 0 e −sydG(y) = P {f (s)} = Pe ( Q β β + s !) = C β β + s !

(28)

∗ for stop-loss moments (k = 1 ⇒ stop-loss premium) Rk(y) = e−βy ∞ X n=0 rn,k(βy) n n! = ∞ X n=1 rn−1,k β en(y) where rn,k = β−k ∞ X j=1 cn+jΓ(k + j) Γ(j)

(valid for all k ≥ 0, and R0(y) = ¯G(y))

∗ for TVaR, E(S|S > x) = x + P j=0 C ∗ j(βx) j j! β P∞j=0 c∗j+1(βx)j! j where c∗j = Cj−1/P∞k=1 kck and C∗j = P∞k=j+1 c∗k

also simpler asymptotic formulas (as x → ∞) for VaR and TVaR using Lundberg light-tailed approach

(29)

– nontrivial examples of Erlang mixtures

∗ many distributions of mixed Erlang form, after changing the scale parameter

∗ identity for Laplace transforms

β1 β1 + s = β β + s      β1 β 1 − 1 − β1 β  β β+s     

∗ for β1 < β this expresses the well known result that a zero-truncated geometric sum of exponential random variables is again exponential

(30)

– Example 1 (mixture of two exponentials)

∗ suppose that (without loss generality) β1 < β2, 0 < p < 1, and f (y) = pβ1e−β1y + (1 − p)β2e−β2y, y > 0

∗ then f (s) = pe β1

β1+s + (1 − p)β2β+s2 , and using the identity with β replaced by β2, it follows that

e f (s) = β2 β2 + s     (1 − p) + p β1 β2 1 − 1 − β1 β2  β 2 β2+s      ∗ that is, f (s) = Qe ββ2 2+s  where Q(z) = z     (1 − p) + p β1 β2 1 − 1 − β1 β2  z      i.e., q1= (1−p)+pβ1 β2  , and qj= p β1 β2   1−β1 β2 j−1 for j = 2, 3, . . .

(31)

– Example 2 (countable mixture of Erlangs) ∗ suppose that f (y) = n X i=1 ∞ X k=1 pikβi(βiy) k−1e−βiy (k − 1)!

∗ assuming that βi < βn for i < n, the identity may be used with β1 replaced by βi and β by βn for each i = 1, 2, . . . , n, to express the Laplace transform

e f (s) = n X i=1 ∞ X k=1 pik βi βi + s !k in the form f (s) = Qe ββn n+s  and qj = n X i=1 j X k=1 pikj − 1 k − 1  βi βn !k 1 − βi βn !j−k , j = 1, 2, . . .

(32)

in the following example, the distribution is not necessarily of phase-type or a combination of exponentials, and there is no simple representation for the qj’s in general, but they may be obtained numerically in a straightforward manner

– Example 3 (a sum of gammas)

∗ consider the Laplace transform of f (y) given by

e f (s) = n Y i=1 βi βi + s !αi ,

corresponding to the distribution of the sum X1+X2+· · ·+Xn, with the Xi’s being independent random variables, and Xi has the gamma pdf βiiy)αi−1e−βiy/Γ(αi)

∗ we assume that the αi’s are positive (not necessarily integers), but the sum m = Pni=1 αi is assumed to be a positive integer

(33)

∗ assuming that βi < βn for i < n, it follows that e f (s) = Qββn n+s  where Q(z) = zm n−1Y i=1      βi βn 1 − 1 − βi βn  z      αi

∗ the probabilities {q1, q2, . . .} correspond to convolutions of neg-ative binomial probabilities, shifted to the right by m

∗ simple analytic formulas for {q1, q2, . . .} may be derived in some cases, such as when αi = 1 for all i or when n = 2

∗ in general, however, it follows that qj= 0 for j < m,

qm = Qn−1i=1i/βn)αi, and {qm+1, qm+2, . . .} may be computed using the Panjer-type recursion

qj = 1 j − m j−mX k=1    n−1X i=1 αi 1 − βi βn !k qj−k, j = m+1, m+2, . . .

(34)

– applications in ruin and surplus analysis

∗ Sparre Andersen (renewal) risk model

∗ mixed Erlang claim sizes

∗ let hδ(x) be the ‘discounted’ (with parameter δ ≥ 0) density of the surplus immediately prior to ruin with zero initial surplus

∗ geometric parameter φδ = ∞R 0 hδ(x)dx, 0 < φδ < 1 ∗ ladder height pdf bδ(y) = ∞ Z 0 fx(y) ( hδ(x) φδ ) dx

(35)

∗ Laplace transform of the time of ruin (ruin probability is spe-cial case δ = 0) is the compound geometric tail

¯ Gδ(x) = ∞ X n=1 (1 − φδ) φnδδ∗n(x), x ≥ 0

∗ in mixed Erlang claim size case with f (y) = ∞P

j=1qjej(y), fx(y) is also mixed Erlang, in turn implying that the ladder height pdf bδ(y) = ∞ X j=1 qj(δ)ej(y)

is still mixed Erlang, with LT ˜bδ(s) = Qδ β+sβ  where

Qδ(z) = ∞

X

j=1

(36)

∗ hence, define the discrete compound geometric pgf Cδ(z) = ∞ X n=0 cn(δ)zn = 1 − φδ 1 − φδQδ(z), and the previous results imply that

¯ Gδ(x) = e−βx ∞ X j=0 ¯ Cj(δ)(βx) j j! = ∞ X j=1 ¯ Cj−1(δ) β ej(x) where ¯Cj(δ) = ∞P n=j+1 cn(δ)

∗ explicit expression for mixed Erlang mixing weights are avail-able for some interclaim time distributions (e.g. Coxian), in which case recursive numerical evaluation is straightforward

(37)

∗ deficit at ruin given initial surplus x (relevant quantity for risk management decisions), denoted by |UT|, has mixed Erlang pdf (given that ruin occurs)

hx(y) =

X

m=1

pm,xem(y)

where the distribution np1,x, p2,x, . . .o is given by

pm,x = ∞ P j=mqj(0)τj−m (βx) ∞ P j=1qj (0) j−1P i=0τi (βx) , and τn(x) = ∞ X i=0 ci(0) x i+n (i + n)!

(38)

∗ more generally, it can be shown that E ne−δTw (|UT|) I (T < ∞) |U0 = xo = ∞ X m=1 Rm,δem(x)

References

Related documents

The Glass Factory, art adviser and invitation of Alex Mirutziu to The lab Konstfack University College of Arts, crafts and Design, ceramic and glass, lecture and.

The major types of data needed for each SGM region include: economic input-output tables, energy balance tables, supplemental data on energy consumption, national income accounts, and

As an extension of the six elements of the current system described above, the six advanced scenarios are as follows: (1) RFID readers are installed everywhere in the whole town,

Briefly, semiconfluent monolayers of HeLa cells grown without antibiotics in 12-well plates were infected with bacterial suspensions (100 bacteria per cell) in the early exponen-

Raccoon Pass is located between East Timbalier Island to the west and West Belle Pass Barrier Headland to the east, both of which are located downdrift of Belle Pass, west of

For “L” type G+4 storey building, it was observed, when compared to frame with expansion joint to frame without expansion joints, there was an decrease in percentage of

(23) Comp lete excision was bas ed on histolog ical mar gin status of the BLES sample 19/43 (44.2) j 6/10 (60) i 2/3 (66.7 ) 0/2 (0) 27/5 8 (46.6 ) Values as presented in the

“Distinguishing between heterogeneity and inefficiency: Stochastic frontier analysis of the World Health Organization’s panel data on national health care systems”, Health Economics