0 INTRODUCTION
1
Chapter 7
Ruin theory
0
Introduction
• In the last two chapters we used the collective risk
model to look at the aggregate claims S arising
during a fixed period of time.
• In this chapter we will extend this model by treating S(t) as a function of time.
• This gives us the equation
S(t) = X1 + X2 + · · · + XN (t)
where N (t) denotes the number of claims occurring before time t.
• We can use this time dependent model to describe
the cash flows of an insurer and to determine properties of the probability of ruin in both the short
term and the long term.
1 BASIC CONCEPTS
1
2
Basic concepts
1.1
Notation
• In Chapters 5 and 6 the aggregate claims generated by a portfolio of policies over a single time
period were studied.
• In the actuarial literature, the word ”risk” is often
used instead of the phrase ”portfolio of policies”.
• In this chapter both terms will be used, so that by
a ”risk” will be meant either a single policy or a
collection of policies.
• In this chapter this study will be taken a stage
further by considering the claims generated by a
portfolio over successive time periods.
1 BASIC CONCEPTS
3
• Some notation is needed.
– N (t): the number of claims generated by the
portfolio in the time interval [0, t], for all t ≥ 0
– Xi: the amount of the i-th claim, i = 1, 2, 3, . . .
– S(t): the aggregate claims in the time interval
[0, t], for all t ≥ 0.
• {Xi}∞
i=1 is a sequence of random variables.
{N (t)}t≥0 and {S(t)}t≥0 are stochastic processes.
• It can he seen that
S(t) =
N (t)
X
Xi
i=1
with the understanding that S(t) is zero if N (t) is
zero.
• The stochastic process {S(t)}t≥0 as defined above
is known as the aggregate claims process for the
risk.
• So we have just taken the idea of a compound
distribution and generalized it to cover different
time periods.
• The insurer of this portfolio will receive premiums
from the policyholders.
1 BASIC CONCEPTS
4
• It is convenient at this stage to assume, as will
be assumed throughout this chapter, that the premium income is received continuously and at a
constant rate.
• Let c denote the rate of premium income per unit
time, so that the total premium income received
in the time interval [0, t] is ct.
• It will also be assumed that c is strictly positive.
1 BASIC CONCEPTS
1.2
5
The surplus process
• Suppose that at time 0 the insurer has an amount
of money set aside for this portfolio. This amount
of money is called the initial surplus and is denoted by U .
• It will always be assumed that U ≥ 0.
• The insurer needs this initial surplus because the
future premium income on its own may not be
sufficient to cover the future claims.
• We are ignoring expenses.
• The insurer’s surplus at any future time t(> 0) is
a random variable since its value depends on the
claims experience up to time t.
• The insurer’s surplus at time t is denoted by U (t).
• Then we can write:
U (t) = U + ct − S(t)
(t ≥ 0)
where U (0) = U .
• Notice that the initial surplus and the premium
income are not random variables since they are
determined before the risk process starts.
1 BASIC CONCEPTS
6
• For a given value of t, U (t) is a random variable
because S(t) is a random variable.
• {U (t)}t≥0 is a stochastic process, which is known
as the cash flow process or surplus process.
• See Fig. 1 on page 4.
• It is assumed that claims are settled as soon as
they occur and that no interest is earned on the
insurer’s surplus.
• It is also assumed that there are no expenses associated with the process (or, equivalently, that S(t)
makes allowance for expense amounts as well as
claim amount), and that the insurer cannot vary
the premium rate c.
• We are also ignoring the possibility of reinsurance.
1 BASIC CONCEPTS
1.3
7
The probability of ruin in continuous time
• Speaking loosely for the moment, when the surplus
falls below zero the insurer has run out of money
and it is said that ruin has occurred.
• Still speaking loosely, ruin can be thought of as
meaning insolvency, although determining whether
or not an insurance company is insolvent is, in
practice, a very complex problem.
• Another way of looking at the probability of ruin
is to think of it as the probability that, at some future time, the insurance company will need to provide more capital to finance this particular portfolio.
• The following two probabilities are defined:
Ψ(U ) = P[U (t) < 0, for some t, 0 < t < ∞]
Ψ(U, t) = P[U (τ ) < 0, for some τ, 0 < τ ≤ t]
• Ψ(U ) is the probability of ultimate ruin (given initial surplus U ).
• Ψ(U, t) is the probability of ruin within time t
(given initial surplus U ).
1 BASIC CONCEPTS
8
• These probabilities are sometimes referred to as
the probability of ruin in infinite time and the
probability of ruin in finite time.
• Here are some important logical relationships between these probabilities for 0 < t1 < t2 < ∞ and
for 0 < U1 < U2:
Ψ(U2, t) ≤ Ψ(U1, t)
Ψ(U2) ≤ Ψ(U1)
Ψ(U, t1) ≤ Ψ(U, t2) ≤ Ψ(U )
lim Ψ(U, t) = Ψ(U )
t→∞
lim Ψ(U, t) = 0
U →∞
1 BASIC CONCEPTS
9
• The intuitive explanations for these relationships
ewe as follows:
– The larger the initial surplus, the less likely it
is that ruin will occur either in a finite time
period or an unlimited time period.
– For a given initial surplus U , the longer the
period considered when checking for ruin, the
more likely it is that ruin will occur.
– The probability of ultimate ruin can be approximated by the probability of ruin within finite
time t provided t is sufficiently large.
– As the amount of initial surplus increases, ruin
will become less and less likely.
1 BASIC CONCEPTS
1.4
10
The probability of ruin in discrete time
• The two probabilities of ruin considered so far
have been continuous time probabilities of ruin,
so-called because they check for ruin in continuous time.
• In practice it may be possible (or even desirable)
to check for ruin only at discrete intervals of time.
• For a given interval of time, denoted h, the following two discrete time probabilities of ruin are
defined:
Ψh(U ) = P[U (t) < 0],
for some t, t = h, 2h, 3h, . . .
Ψh(U, t) = P[U (τ ) < 0],
for some τ, τ = h, 2h, 3h, . . . , t − h, t
Note that it is assumed for convenience in the definition of Ψh(U, t) that t is an integer multiple of
h.
• See Figure 2 on page 7.
1 BASIC CONCEPTS
11
• Listed below are five relationships between different discrete time probabilities of ruin for 0 < t1 <
t2 < ∞ and for 0 < U1 < U2:
Ψh(U2, t) ≤ Ψh(U1, t)
Ψh(U2) ≤ Ψh(U1)
Ψh(U, t1) ≤ Ψh(U, t2) ≤ Ψh(U )
lim Ψh(U, t) = Ψh(U )
t→∞
Ψh(U, t) ≤ Ψ(U, t)
• Ψ(U, t) involves checking for ruin at all possible
times. Since the more often we check for ruin, the
more likely we are to find it, we would expect that
Ψ(U, t) would be greater than Ψh(U, t).
• Intuitively, it is expected that the following two
relationships are true since the probability of ruin
in continuous time could be approximated by the
probability of ruin in discrete time, with the same
initial surplus, U , and time horizon, t, provided
ruin is checked for sufficiently often, i.e. provided
h is sufficiently small.
lim Ψh(U, t) = Ψ(U, t)
h→0+
lim Ψh(U ) = Ψ(U )
h→0+
1 BASIC CONCEPTS
1.5
12
Probability of ruin in the short term
• If we know the distribution of the aggregate claims
S(t), we can often determine the probability of
ruin for the discrete model over a finite time horizon directly (without reference to the models), by
looking at the cash flows involved.
1 BASIC CONCEPTS
• Example on page 9.
The aggregate claims arising during each
year from a particular type of annual insurance policy are assumed to follow a normal
distribution with mean 0.7P and standard
deviation 2.0P , where P is the annual premium. Claims are assumed to arise independently. Insurers are required to assess
their solvency position at the end of each
year. A small insurer with an initial surplus
of £0.1m for this type of insurance expects
to sell I 100 policies at the beginning of the
coming year in respect of identical risks for
an annual premium of £5, 000. The insurer
will incur expenses of 0.2P at the time of
writing each policy. Calculate the probability that the insurer will prove to be insolvent
for this portfolio at the end of the coming
year. Ignore interest.
13
1 BASIC CONCEPTS
14
• Question 7.3 on page 10.
If the insurer expects to sell 200 policies
during the second year for the same premium and expects to incur expenses at the
same rate, calculate the probability that the
insurer will prove to be insolvent at the end
of the second year.
• In fact, the normal distribution is probably not
a very realistic distribution to use for the claim
amount distribution in most portfolios, as it is
symmetrical, whereas many claim amount portfolios will have skewed underlying distributions.
• The distribution that is perhaps used most often
in this model is the Poisson distribution for claim
numbers. This means that the number of claims
occurring in a time interval (0, t) are P oisson(λt),
where λ is the Poisson parameter representing the
mean number of claims per unit time. This is
called a Poisson process.
1 BASIC CONCEPTS
15
• If we use the Poisson distribution to model the
number of claims, then the aggregate claim distribution takes the form of a compound Poisson
distribution.
• We can then use an appropriate claim amount distribution to model the size of an individual claim.
We then say that the claim amounts come from a
Compound Poisson process. We can then find the
mean and variance of the aggregate claim amount
for any time period using the formulae from Chapter 5 for the compound Poisson distribution.
2 THE POISSON PROCESS
2
16
The Poisson process
• The Poisson process
(λt)n
P [N (t) = n] = pn(t) = exp{−λt}
n!
• The Poisson process for the number of claims will
be combined with a claim amount distribution to
give a compound Poisson process for the aggregate
claims.
S(t) = X1 + X2 + · · · + XN (t)
• The following three important assumptions are made:
1. the random variables {Xi}∞
i=1 are i.i.d.
2. the random variables {Xi}∞
i=1 are independent
of N (t) for all t ≥ 0
3. the stochastic process {N (t)}t≥0 is a Poisson
process whose parameter is denoted λ
2 THE POISSON PROCESS
17
• For a compound Poisson process S(t), the mean,
the variance and the MGF of the process are given
by:
E[S(t)] = λt E(X)
Var[S(t)] = λt E(X 2)
MS(t)(r) = exp{λt[MX (r) − 1]}
• The kth moment about zero of the Xis, if it exists,
are denoted mk , i.e.
mk = E[Xik ] for k = 1, 2, 3, . . .
2 THE POISSON PROCESS
18
• In the following sections, we will discuss:
– The adjustment coefficient and Lundberg’s inequality
– The effect of changing parameter values on finite and infinite time ruin probabilities
– Reinsurance and ruin
3 THE ADJUSTMENT COEFFICIENT AND LUNDBERG’S INEQUALITY
3
19
The adjustment coefficient and Lundberg’s inequality
3.1
Lundberg’s inequality
• Lundberg’s inequality states that:
Ψ(U ) ≤ exp[−RU ]
where U is the insurer’s initial surplus, Ψ(U ) is the
probability of ultimate ruin and R is a parameter
associated with a surplus process known as the
adjustment coefficient.
• The value of R depends on the distribution of aggregate claims and on the rate of premium income.
3 THE ADJUSTMENT COEFFICIENT AND LUNDBERG’S INEQUALITY
20
• If we can find a value for R, then Lundberg’s inequality tells us that we can find an upper bound
for the probability of ruin.
• See a figure, which shows a graph of both exp[−RU ]
and Ψ(U ) against U when claim amounts are exponentially distributed with mean 1, and when the
premium loading factor is 10%.
3 THE ADJUSTMENT COEFFICIENT AND LUNDBERG’S INEQUALITY
21
• R can be interpreted as measuring risk. The larger
the value of R, the smaller the upper bound for
Ψ(U ) will be.
i.e. Larger values of R imply smaller ruin probabilities.
• R is a function of the parameters that affect the
probability of ruin.
• See a graph of R as a function of θ (premium
loading factor), when Xi ∼ Exponential(1/10)
and Xi = 10.
• RExponential < REqual
3 THE ADJUSTMENT COEFFICIENT AND LUNDBERG’S INEQUALITY
3.2
22
The adjustment coefficient – compound Poisson processes
• When S(t) is a compound Poisson process, then
R is defined in term of λ, MX (t) and the premium
income per unit time c.
• The following assumption will also be made:
c > λm1
so that the insurer’s premium income (per unit
time) is greater than the expected claims outgo
(per unit time).
• R can be found by solving the equation:
λMX (R) = λ + cR
or
MX (R) = 1 + (1 + θ)m1R
(c = (1 + θ)λm1)
3 THE ADJUSTMENT COEFFICIENT AND LUNDBERG’S INEQUALITY
23
• It can be shown that there is indeed only one
positive root of λMX (R) = λ + cR.
• See a graph of g(r) = λMX (r) − λ − cr
• g(0) = 0
d
d
g(r) = λ MX (r) − c = λm1 − c < 0
dr
dr
r=0
r=0
d2
d2
2 rX
g(r)
=
λ
M
(r)
=
λ
(X
e )>0
E
X
dr2
dr2
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES24
4
Effect of changing parameter values on ruin probabilities
4.1
Introduction
• The effect of changing parameter values on Ψ(U, t)
and Ψ(U ) will be discussed.
• It will be assumed that the aggregate claims process is a compound Poisson process with the parameter λ = 1.
• The implication is that the unit of time has been
chosen to be such that the excepted number of
claims in a unit time is 1.
• Ψ(U, 500) is the probability of ruin (given initial
surplus U ) over a time period in which 500 claims
are expected.
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES25
• We also assume that the expected value of an individual claim is 1.
• This implication is that the monetary unit has
been chosen to be equal to the expected amount
of a single claim.
• Ψ(20, 500) is the probability of ruin (over a time
period in which 500 claims are expected) given
an initial surplus equal to 20 times the expected
amount of a single claim.
• It will be also assumed that individual claims have
an exponential distribution.
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES26
4.2
A formula for Ψ(U ) when F (x) is the exponential distribution
• Individual claims amounts are exponentially distributed with mean 1
F (x) = 1 − e−x
• The premium loading factor is θ.
• Ψ(U ) is then: (the derivative of this result is beyond the scope of our discussion.)
1
θU
Ψ(U ) =
exp −
1+θ
1+θ
• If θ = 0 then Ψ(U ) = 1 irrespective of the value
of U .
• This result is in fact true for any form of F (x).
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES27
4.3
Ψ(U, t) as a function of t
• Assume that θ = 0.1.
• See a graph of Ψ(15, t) for 0 ≤ t ≤ 500
• Ψ(15, t) for 0 ≤ t ≤ 500 can be worked out using
a numerical method.
• Ψ(15) = 0.23248 can be calculated using
0.1 × 15
1
exp −
Ψ(15) =
1 + 0.1
1 + 0.1
• The adjustment coefficient R = 0.1/1.1 and
exp{−15R} = 0.2557 can be calculated using
MX (R)
m1
θ
MX (R)
=
=
=
=
1 + (1 + θ)m1R
1
0.1
(1 − R)−1
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES28
• Ψ(15, t) is an increasing function of t.
• For small value of t, Ψ(15, t) increases very quickly.
• For large values of t, Ψ(15, t) increases less quickly
and approaches asymptotically the value of Ψ(15).
• We would expect a much higher probability of ruin
before time 50 than before time 25 since the overall performance of the fund could easily change in
such a short time period.
• If premium rates are expected to be profitable in
the long term (say, at time 400), then a significant
surplus will have built up in most cases and so the
probability of ruin at time 425 will not be that
much higher than at time 400.
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES29
4.4
Ruin probability as a function of initial surplus U
• See a graph of Ψ(U, t) for 0 ≤ t ≤ 500 where
θ = 0.1 and U = 15, 20, 25.
• Increasing the value of U decreases the value of
Ψ(U, t) for any value of t.
• In the case of exponentially distributed individual
claim amounts,
−θ
d
Ψ(U ) =
Ψ(U )
dU
1+θ
Hence Ψ(U ) is a decreasing function of U .
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES30
4.5
Ruin probability as a function of premium loading θ
• See a graph of Ψ(15, t) for 0 ≤ t ≤ 500 where
θ = 0.1, 0.2, 0.3.
• Increasing the value of θ decreases the value of
Ψ(15, t) for any value of t.
• When θ = 0.2 and 0.3, Ψ(15, t) is more or less
constant for t greater than about 150.
• For θ = 0.2 and 0.3, (and for U=15 and for this
S(t)) ruin, if it occurs at all, is far more likely to
occur before time 150 than after time 150.
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES31
• See a graph of Ψ(10) as a function of θ.
• Ψ(U ) must be a non-increasing function of θ.
• For Xi ∼ Exponential(λ),
d
Ψ(U ) = −(1 + θ)−1Ψ(U ) − U (1 + θ)−2Ψ(U )
dθ
Hence, Ψ(U ) is a decreasing function of θ.
4 EFFECT OF CHANGING PARAMETER VALUES ON RUIN PROBABILITIES32
4.6
Ruin probability as a function of the Poisson parameter λ
• See a graph of Ψ(15, 10) as a function of λ for
θ = 0.1, 0.2, 0.3.
• The value of Ψ(15, 10) when λ = 50 is the same
as the value of Ψ(15, 500) when λ = 1.
• cf. Ψ(15, 10) with λ = 50 WITH Ψ(15, t) with
λ = 1.
5 REINSURANCE AND RUIN
5
33
Reinsurance and ruin
5.1
Introduction
• One of the options open to an insurer who wishes
to reduce the variability of aggregate claims from
a risk is to effect reinsurance.
• A reduction in variability would be expected to
increase an insurer’s security, and hence reduce
the probability of ruin.
• A reinsurance arrangement could be considered
optimal if it minimizes the probability of ruin.
• As it is difficult to find explicit solutions for the
probability of ruin, the effect of reinsurance on the
adjustment coefficient will be considered instead.
5 REINSURANCE AND RUIN
34
• If a reinsurance arrangement can be found that
maximizes the value of the adjustment coefficient,
the upper bound for the probability of ultimate
ruin will be minimized.
• As the adjustment coefficient is a measure for risk,
it seems a reasonable objective to maximize its
value.
• We will consider the effect on the adjustment coefficient of proportional and of excess of loss reinsurance arrangements.
5 REINSURANCE AND RUIN
5.2
35
Maximizing the adjustment coefficient under proportional
reinsurance
• The insurer’s premium income per unit time, before payment of the reinsurance premium, is (1 +
θ)λm1.
• The reinsurance premium is assumed as (1+ξ)(1−
α)λm1.
• The insurer’s premium income, net of reinsurance,
is [(1 + θ) − (1 + ξ)(1 − α)]λm1
5 REINSURANCE AND RUIN
36
• We also assume that ξ ≥ θ.
• If this were not true, it would be possible for the
insurer to pass the entire risk on to the reinsurer
and to make a certain profit.
• This of course ignores commission, expenses and
other adjustment to the theoretical risk premium.
5 REINSURANCE AND RUIN
37
• For the insurer’s premium income, net of reinsurance, to be positive:
(1 + θ) > (1 + ξ)(1 − α)
α>
ξ−θ
1+ξ
• The insurer’s net of reinsurance premium income
per unit time must exceed the expected aggregate
claims per unit time. Otherwise ultimate ruin is
certain. i.e.
[(1 + θ) − (1 + ξ)(1 − α)]λm1 > αλm1
α>
ξ−θ
ξ
5 REINSURANCE AND RUIN
38
• When ξ ≥ θ,
ξ−θ ξ−θ
≥
α>
ξ
1+ξ
• When ξ = θ, α > 0.
• When ξ > θ, α >
ξ−θ
ξ .
5 REINSURANCE AND RUIN
5.2.1
39
Same loadings ξ = θ
• Suppose that F (x) = 1 − e−0.1x.
• The distribution of the insurer’s individual claims
net of reinsurance
Yi = αXi ∼ Exponential(0.1/α).
P [Y ≤ y] = P [X ≤ y/α] = 1 − exp{−0.1y/α}
• Hence
Z
λ + (1 + θ)λ10αR = λ
∞
eRx(0.1/α)e−0.1x/αdx
0
1
⇒ 1 + (1 + θ)10αR =
1 − 10αR
⇒ R=
θ
for 0 < α ≤ 1
(1 + θ)10α
• It can be seen that R is a decreasing function of
α.
5 REINSURANCE AND RUIN
5.2.2
40
Different loadings ξ = 0.2 > θ = 0.1
0.1
• Suppose that F (x) = 1 − e
, Mx(t) =
0.1 − t
• The insurer’s net premium income per unit time
is
−0.1x
[(1 + θ) − (1 + ξ)(1 − α)]λm1
= [1.1 − 1.2 × (1 − α)]λ × 10
= (12α − 1)λ
• The equation defining R is:
λ + (12α − 1)λR = λ
0.1/α
λ
=
0.1/α − R 1 − 10αR
2α − 1
⇒ R=
for 0.5 < α ≤ 1
10(12α2 − α)
5 REINSURANCE AND RUIN
41
• We should find the value of α that maximizes R.
dR 20(12α2 − α) − (2α − 1)10(24α − 1)
=
dα
100(12α2 − α)2
⇒ 20(12α2 − α) = (2α − 1)10(24α − 1)
⇒ 24α2 − 24α + 1 = 0
⇒ α = 0.9564, α = 0.0436
• See a graph of R as a function of α.
5 REINSURANCE AND RUIN
5.3
42
Maximizing the adjustment coefficient under excess of loss
reinsurance
• We consider the effect of excess of loss reinsurance
on the adjustment coefficient.
• We assume that
– insurer’s premium income per unit time (before
reinsurance) is (1 + θ)λm1.
– reinsurer’s premium income per unit time is
(1+ξ)λ E(Z) (ξ ≥ θ) and Z = max(0, X−M )
– insurer’s premium income per unit time (net of
reinsurance) is
c∗ = (1 + θ)λm1 − (1 + ξ)λ E(Z)
5 REINSURANCE AND RUIN
43
• Equation defining R is
"Z
#
M
λ + c∗ R = λ
eRxf (x)dx + eRM [1 − F (M )]
0
• The equation for R must be solved numerically for
given values of θ and ξ.
• See a graph of R as a function of M when θ =
ξ = 0.1.
• R is a decreasing function of M .
• R goes to ∞ as M goes to 0.
• When ξ > θ there is a minimum retention level.
5 REINSURANCE AND RUIN
44
• See a graph of R as a function of M for the following combinations of θ and ξ:
1. θ = 0.1 and ξ = 0.2
2. θ = 0.1 and ξ = 0.4
• For ξ = 0.2, it is possible for the insurer to increase
the value of the adjustment coefficient by effecting
reinsurance.
• For ξ = 0.4, the insurer should retain the entire
risk in order to maximize the value of the adjustment coefficient.