Simulating Actuarial Models

84 Pages·2010·4 MB·English

by  Ralf Korn

Chapter 8
Simulating Actuarial Models
8.1 Introduction
In the preceding chapters we have been interested in pricing isolated fi-
nancial contracts. There, the main principle to deal with market risk, the
risk of losses due to unfavourable price moves, is to switch to an equivalent
marketmartingalemeasureandcalculatethe presentvalue ofafinancialcon-
tract. This approachis basedon the assumptionthat assets underlying these
contracts can be traded to reduce or even eliminate the inherent risks.
In insurance mathematics, we look at the risks arising from insurance con-
tracts (such as in life insurance or car insurance). However, they cannot be
traded and the arbitrage argument often plays no role in their valuation. As
these contracts are often soldin high numbers, suitable variants of the law of
large numbers suggest the expected present value of the future payments as
anindicatorforthevalueofacontract. Tobeonthesafeside,safetyloadings
are included in the premium calculations. Also, dependencies can play an
extremely important role in judging the risk arising from the whole portfo-
lio of sold contracts. We will therefore consider the two important topics of
premium principles and of dependence modelling. Further important types
of risks that we will explicitly look at are the risk of rare events in nonlife
insurance and the longevity risk in life insurance. In both cases Monte Carlo
methods are suitable tools. The concepts of copulas or quantiles that we are
dealing with in this chapter also have applications in finance.
As insurance mathematics is a classical subject, there are various mono-
graphsondifferentaspectsandtypesofinsurance. Amongthemarestandard
and recent texts such as Bu¨hlmann (2005), Gerber (1997), Mikosch (2004),
and Møller and Steffensen (2007), just to name a few.
8.2 Premium principles and risk measures
Thepremiumofaninsurancecontractisthatpartofitspricethatshould
be sufficienttocoverthe riskthattheinsurancecompanytakesoverwiththis
357
358 Monte Carlo Methods and Models in Finance and Insurance
contract. The actualprice ofthe contractalsocontainsparts thatare needed
for covering administrational expenses and further costs. This is sometimes
calledgross premium. Wewillnotconsideradministrationalcostsandonly
lookatthepremiumasdescribedabove. Tocalculatethispremiumaso-called
premium principle is used.
As premium principles are used to judge the risk inherent in an insurance
contract,itis reasonableto presentalongwiththemso-called(financial)risk
measures that are developed for judging and managing financial risks.
We will introduce both concepts, comment on the aspects of their Monte
Carlosimulation,andcommentontherelationshipbetweenriskmeasuresand
3 premium principles.
1
0
2
r
be 8.2.1 Properties and examples of premium principles
o
ct
O Tointroduceapremiumprinciple,wefirsthavetointroducethenotionofa
7 riskX asarandomvariableforwhichwealwaysassumesuitableintegrability
1
3 properties when they are needed. We further assume that the considered
1
0: insurance contract starts right away, which implies that the risk is present
0
at immediately. Wewillconsideradifferentcaseinlifeinsurancewhencontracts
y] might start in the future. For this, we then have to introduce a suitable
r
ra discounting. Formal definitions are given below.
b
Li
n
e
g DEFINITION 8.1
r
Be Let (Ω,F,P) be a probability space.
of (a) A risk X is a nonnegative random variable on (Ω,F,P).
y
sit (b) A functional p(.) on the space of risks X is called a premium principle.
r
e
v
Uni As there are many suggestions around for different premium principles to
y [ use, there are also properties in the literature by which the suitability of a
b
d premiumprincipleshouldbejudged. Westatefourofthem(seeSundt[1993])
de but refer the reader to an impressive collection of nearly 20 properties listed
a
o in Laeven and Goovaerts (2008):
nl
w
o
D
DEFINITION 8.2
Let X,Y be two risks. Then, some reasonable properties of a premium prin-
ciple p(.) are:
1. p(X+Y)≤p(X)+p(Y) subadditivity
2. p(X)≤p(X +Y)) monotonicity
3. p(X)≥E(X) nonnegative safety loading
4. p(X)≤ supX(ω) no ripoff
ω∈Ω
Simulating Actuarial Models 359
REMARK 8.3 The interpretations of the above properties are indicated
by their names:
1. Property 1 requires that it should not be profitable to split the risk
X + Y and sign two contracts, one for X and the other one for Y.
However,this property is at debate of being reasonablewithout further
assumptions on the dependence between X and Y.
2. Property 2 is a monotonicity requirement: Additional risk needs addi-
tional premium.
3 3. Property 3 is motivated by the law of large numbers: If the company
1 would charge less than the expected loss E(X) (“fair premium”), it
0
2
r would for sure go bankrupt given the number of sold contracts is large.
e
b
cto 4. Property4isreasonableasnocustomerwouldsignacontractthatcosts
O more than the highest possible claim size.
7
1
3 With the choice of presenting these properties we do not claim that they
1
0: are the most important ones. They simply serve as popular examples.
0
at
] Wepresentsomepopularpremiumprinciplesandcheckto seeifthey share
y
ar the above properties.
r
b
Li
n
e DEFINITION 8.4
g
Ber The expectation principle pexp(X) for a claim X and a constant μ>0 is
f given by
o
sity pexp(X)=(1+μ)E(X). (8.1)
r
e
v The expectation principle satisfies Properties 1, 2, and 3 of Definition 8.2.
ni
U However, it obviously violates Property 4 for constant claims or for claims
y [ with a maximum smaller than (1+μ)·E(X). However,as in both cases such
b
d a contract could not be sold, it is clear that such a value for μ would not be
e
d used in practice. A further weakness seems to be that the fluctuations of the
a
nlo risk X plays no role for its premium.
w Principlesthatexplicitlytakethefluctuationsoftheclaimsizesintoaccount
o
D are the variance and the standard deviation principles.
DEFINITION 8.5
Let μ>0 be a given constant.
1. The variance principle p (X) for a claim X is given by
var
p (X)=E(X)+μ·Var(X). (8.2)
var
2. The standard deviation principle p (X) for a claim X is given by
sd
(cid:27)
p (X)=E(X)+μ· Var(X). (8.3)
sd
360 Monte Carlo Methods and Models in Finance and Insurance
Although at first sight more sophisticated than the expectation principle,
both these premium principles might violate Property 2 of Definition 8.2
which is a serious defect. The reason for this is that expectation and vari-
ance/standarddeviationarerelatedinanonlinearandnonmonotonicway. To
demonstratethis,weassumethatwehaveaprobabilityqforaclaimX toarise
in the next period and that the height of the claim is Γ(1,500)-distributed.
Thepremiumarisingforthisconfigurationforthestandarddeviationprinciple
with μ=1 is displayed in Figure 8.1.
3
1
0 Standard deviation principle
2
r
e
b
o 0
Oct 60
7
1
3 0
1 m 0
0: u 4
0 mi
at e
y] pr 00
r 2
a
r
b
Li
n 0
e
g
er 0.0 0.2 0.4 0.6 0.8 1.0
B
f
o probability q
y
sit
r
e
v
ni
U
FIGURE 8.1: Premium from standard deviation principle.
[
y
b
d
e
d
a Notethatthe resultingpremiumforq =1,i.e.thata claimoccursforsure,
o
nl is lower than for one which only occurs with probability q = 0.856. This
w
o violates the monotonicity requirement and is not acceptable. So one should
D
take great care when using one of those premium principles. On top of that
the variance principle also violates the subadditivity property.
A premium principle that avoids these problems (see Fischer [2003]) is the
semistandarddeviationprinciple whichincorporatesonlydeviationsfromthe
mean by high claims.
DEFINITION 8.6
The semistandard deviation principle p (X) for a claim X and a con-
ssd
Simulating Actuarial Models 361
stant 0≤μ≤1 is given by
(cid:25)
(cid:30) (cid:31)
p (X)=E(X)+μ· E [max(0, X −E(X))]2 . (8.4)
ssd
All premium principles that we have considered so far were closely related
to the strong law of large numbers and the central limit theorem (which
motivates the use of standard deviation or variance as a measure for judging
the risk of deviation from the expectation of a claim) and explicitly added
somesafetyloadings. Thefollowingprinciple–whichisknownastheexpected
utility principle – incorporatesthe attitude towardsrisk of the insurer by the
3
1 introduction of a utility function.
0
2
r
e
b
o DEFINITION 8.7
Oct Let U(x) be a utility function (i.e. a concave, strictly increasing function).
7 A premium p (X) for a claim X is said to be calculated by the expected
1 eu
3 utility principle if we have
1
00: U(c)=E(U(c−X +p (X))) (8.5)
at eu
] where c is a (positive) constant, e.g. the wealth of the insurer.
y
r
a
r
b
Li REMARK 8.8 1. The premium is such that the utility from signing the
n
e new contract is equal to the utility of not signing it.
g
r
e
B 2. Property 1 of a premium principle is not fulfilled. However, this is
of desired, because higher risk should lead to overproportionallyincreasing pre-
y
sit miums. The reasoning behind this can be seen when comparing the risks
er corresponding to n customers of the same age having identical life insurance
v
ni contractswiththeriskofasinglecustomerinsuredonn-timesthesumpayable
U
[ atdeathandannuityrate. Inthesecondcase,thelongevityriskandtheearly
y
b death risk are much higher than n-times the risks in the first case, because
d
e in the first case the longevity and early death risks average out over the n
d
a customers.
o
nl
w 3. Instead of the constant c one should insert the random variable C
o
D representing the whole portfolio of claims and replace the left-hand side by
E(U(C)). Thisleadstoahighpremiumforaclaimbeinghighlycorrelatedto
C and a low premium for a claim leading to a diversificationin the portfolio.
4. Note that for the particular choice of the utility function
(cid:9) (cid:10)
1
U(x)= 1−e−αx , for a fixed α>0 (8.6)
α
the premium calculated by the expected utility principle is independent of c
and is explicitly given by
(cid:9) (cid:9) (cid:10)(cid:10)
1
p (X)= ln E eαX (8.7)
eu
α
362 Monte Carlo Methods and Models in Finance and Insurance
(see Laeven and Goovaerts [2008] for this and also for more premium princi-
ples). This principle only yields a finite premium for exponentially bounded
risks.
8.2.2 Monte Carlo simulation of premium principles
After the choice of a premium principle has been made, there remains the
task of explicitly calculating the premium of an insurance contract. If this
cannot be done explicitly, Monte Carlo simulation is a possible method of
choice. This is straightforward for the expectation principle (of course, be-
3 sides the fact that contracts with a complicated insurance payment structure
1
0
2 mightrequire methods as sophisticatedas for exotic options)andalsoforthe
er expected utility principle in case of the exponential utility function when the
b
cto explicit expression of Equation (8.7) is used.
O
However, there is a new aspect introduced by the premium principles that
7
1 includethevarianceinanyformasaningredient. Tocalculatethevariancewe
3
1 alreadyneedthe expectation. Ofcourse,forlargevalues ofN (the number of
0:
0 Monte Carlo runs) one can use the Monte Carlo estimate of the expectation,
at the arithmetic mean. As we can calculate the variance as
]
y
ar (cid:9) (cid:10)
br Var(X)=E X2 −(E(X))2 (8.8)
Li
n
ge theMonteCarloestimationofE(X)andofE(X2)canbedonesimultaneously.
Ber However, for estimating the semivariance E([max(0, X−E(X))]2) such a
of decompositionisnotavailable. Thus,onecouldperformatwo-stepprocedure:
y
versit 1. Estimate the mean E(X) by X¯N1 based on N1 simulation runs.
ni
U
y [ 2. Estimate the semivariance E([max(0, X −E(X))]2) based on N2 new
b simulation runs by
d
e
d
nloa 1 (cid:13)N2 (cid:9)max(cid:9)0, X −X¯ (cid:10)(cid:10)2.
Dow N2 i=1 N1
8.2.3 Properties and examples of risk measures
A risk measure is related to a financial position X˜ and a time horizon T.
Here,thepositionX˜ canbebothpositiveornegative. Incontrasttoinsurance
claims, X˜ >0 describes a profit.
Fo¨llmer and Schied (2002)state the requirementon a risk measure clearly:
“...a risk measure is viewed as a capital requirement: We are looking for the
minimal amount of capital which, if added to the position and invested in a
risk-free manner, makes the position acceptable.”
Simulating Actuarial Models 363
DEFINITION 8.9
A risk measure ρ is a real-valued mapping defined on the space of random
variables.
As this definition is fairly weak, we present some requirements on a risk
measure that are popular in the literature.
DEFINITION 8.10
Let X˜,Y˜ be two financial positions. Some reasonable properties of a risk
measure ρ(.) are:
3
1
0 1. ρ(X˜ +m)=ρ(X˜)−m ∀m∈R translation invariance
2
r
be 2. X˜ ≥Y˜ a.s.⇒ρ(X˜)≤ρ(Y˜) monotonicity
o
ct : ;
7 O 3. ρ λX˜ +(1−λ)Y˜ ≤λρ(X˜)+(1−λ)ρ(Y˜) for λ∈[0,1] convexity
1
3
0:1 4. ρ(λX˜)=λρ(X˜) for λ≥0 positive homogeneity
0
at
y] REMARK 8.11 The meaning of the properties of risk measures can
r
ra already be understood by their names:
b
Li
n 1. Translation invariance means that riskless money changes the risk of a
ge position by exactly the same amount. In particular, we observe ρ(X˜ +
r
Be ρ(X˜))=0,i.e.ifwe investthe risk premium ρ(X)ina risk-freemanner
of then there is no risk anymore.
y
sit 2. Monotonicity simply says that less risk requires less money set aside.
r
e
v
ni 3. Convexity of the risk measure favours diversification.
U
[
y 4. Positive homogeneity implies that risk increases linearly in the units
b
d owned of a particular risky good. This property is heavily discussed
e
ad in the literature as it totally ignores liquidity risk. From an insurance
o
nl pointofview,italsomeansthatinsuring10hightowers,inforexample
w
o San Francisco, against earthquakes bears the same risk as insuring 10
D
high towers in 10 different places. With regard to extreme risk, this is
not reasonableas an earthquake in San Francisco will likely damage all
hightowers,whereasitisratherunrealisticthatanearthquakehappens
at all the 10 places at the same time.
To normalize the range of the risk measure one can also require
ρ(0)=0 normalization
which has the reasonable interpretation that a zero position has no risk.
In the literature, mainly two types of risk measures are considered.
364 Monte Carlo Methods and Models in Finance and Insurance
DEFINITION 8.12
A risk measure is called convex if it satisfies the requirements 1 to 3 of
Definition 8.10.
A risk measure is called coherent if it satisfies the requirements 1 to 4 of
Definition 8.10.
Properties 3 and 4 imply that a coherent risk measure is also subadditive.
If it attains a finite value ρ(0) then it also has the normalization property
ρ(0)=0.
We will look at some popular risk measures. The one which is mainly used
3 in banks and has become an industry standard is the value-at-risk.
1
0
2
r
e
b DEFINITION 8.13
o
ct The value-at-risk of level α (VaR ) is the α-quantile of the loss of the
O α
7 financial position X˜ at time T:
1 (cid:30) (cid:18) (cid:28) (cid:29) (cid:31)
3 (cid:18)
1 VaR (X˜)=−inf u∈R(cid:18)P X˜ ≥u ≥1−α (8.9)
0: α
0
at where α is a high percentage such as 95% or 99%.
]
y
r
a
r Asaquantile,VaR iseasytounderstandandverypopularinapplications.
b α
Li However,itdoes notgiveus anidea aboutthe heightofthe actuallossabove
n
e that quantile. Further,it is notconvexandso it does notnecessarilysupport
g
er diversification. To see this, consider the positions X,Y with:
B ⎧
of ⎪⎨100 with probability 0.901
y
rsit X˜ =Y˜ =⎪⎩0 with probability 0.009
ve −200 with probability 0.09,
ni
U
[ then for α=90%, we obtain
y (cid:11) (cid:12)
b
ed VaR 1X˜ + 1Y˜ =50,
d 2 2
a
o
wnl 1VaR(X˜)+ 1VaR(Y˜)=−100.
o 2 2
D
DEFINITION 8.14
The conditional value-at-risk (or average value-at-risk) is defined as
(cid:22)
1
1
CVaR (X˜)= VaR (X˜)dγ. (8.10)
α 1−α γ
α
The CVaR coincides with the expected shortfall or tail conditional
α
expectation defined by
(cid:28) (cid:18) (cid:29)
(cid:18)
TCE (X˜)=−E X˜(cid:18)X˜ ≤VaR (8.11)
α α
Simulating Actuarial Models 365
iftheprobabilitydistributionofX˜ hasnoatoms. CVaR isindeedacoherent
α
risk measure (see Acerbi and Tasche [2002]).
Asforpremiumprinciplesthereisariskmeasurebasedonexpectedutility:
DEFINITION 8.15
Let U : R → R be a utility function (i.e. strictly increasing and concave).
Then, the risk measure based on utility of a financial position X˜ is given by
(cid:30) (cid:18) : ; (cid:31)
(cid:18)
ρ (X˜)=inf m∈R(cid:18)E U(X˜ +m) ≥U(0) . (8.12)
utility
3
1 Itcanbeshownthatthejustdefinedriskmeasurebasedonautilityfunction
0
2
r is a convex risk measure (see F¨ollmer and Schied [2002]).
e
b
o
ct
O 8.2.4 Connection between premium principles and risk
17 measures
3
1
0: Asbothconceptsareusedtojudgerisks,theyshouldhavemanyfeaturesin
0
at common (indeed, already in Deprez and Gerber [1985]convex premium prin-
] ciples were discussed). However, before we comment on parallels one should
y
ar also keep in mind that a premium principle is closer to a pricing principle as
r
Lib it is focused on a single contract. But the main concept behind this pricing
n approach is the strong law of large numbers and not the arbitrage principle
e
g of finance. Therefore, the classical premium principles such as the expecta-
r
e
B tion (see Definition 8.4) or the variance principle as introduced in Definition
of 8.5 are not directly related to ideas of risk measures that have a tendency to
y
sit concentrateon valuing the extreme risks. Examples for this point of view are
er the VaR, presented in Definition 8.13, or the CVaR, introduced in Definition
v
ni 8.14. However,thesemistandarddeviationpremiumprinciplefromDefinition
U
[ 8.6 also concentrates on the high claims.
y
b An approach that connects both concepts is the one based on utility, the
ed expected utility approach for premium principles and risk measures as in
d
a Definitions 8.7 and 8.15.
o
nl One can define a premium principle p out of a givenrisk measureρ via the
w
o requirement of
D
ρ(p(X)−X)=! 0 (8.13)
for each claim X. Since p(X) is riskless, this leads to the identification
p(X)=ρ(−X) (8.14)
which would also allow us to extend the definition of a premium principle to
general random variables.
Further, we can then compare the conditions imposed on premium princi-
ples and on risk measures. Given that the risk measure is convex and nor-
malized, then it follows directly that the above defined premium principle is
366 Monte Carlo Methods and Models in Finance and Insurance
monotonic and also condition 4 of a premium principle is implied. Moreover,
the subadditivity of a coherent risk measure implies the subadditivity of the
premium principle. Condition 3 of a premium principle cannot be directly
verified, as a risk measure is a priori defined without reference to a probabil-
ity measure. However, for special choices of ρ such as CVaR and expected
utility, this condition can be explicitly verified.
For convex risk measures which are not coherent,the above premium prin-
ciple will also typically fail to be subadditive.
3 8.2.5 Monte Carlo simulation of risk measures
1
0
r 2 Quantile estimation and VaR
e
b
cto Thefirstingredientofestimatingriskmeasuresisthe estimationofaquan-
O tile. The natural Monte Carlo estimator for an α-quantile q = F−1(α) of
7 α
1 a random variable X with distribution function F is obtained by generat-
3
00:1 dinisgtrNiburetiaolnizaFtNio(nxs).of X and then using the α-quantile qˆα,N of the empirical
at
]
y
r
a
br Algorithm 8.1 Crude Monte Carlo simulation of the α-quantile
n Li Let F be a given distribution function, α∈[0,1].
e
g
er 1. Simulate N independent random numbers X1,...,XN, Xi ∼F.
B
f
o 2. Compute the empirical distribution function
y
ersit 1 (cid:13)N
niv FN(x)= N 1{Xi≤x}.
U i=1
[
y
b
d 3. Estimate the quantile qα by
e
d
oa qˆ =F−1(α).
nl α,N N
w
o
D
REMARK 8.16 1. Of course, if F is explicitly known, then one would
calculate the quantile via numerically solving F(x)=α. So the crude Monte
Carlo method of Algorithm 8.1 is only used when F is not available or hard
to compute. We have already seen such examples in Sections 5.6.1 and 5.6.2
in the case of basket or Asian options, where F has been the distribution
function of sums of log-normals which is not known explicitly. The above
inversion of the empirical distribution function is of course done in a simple