ruin theory
TRANSCRIPT
Code: risk0373
Ruin Theory
X. Sheldon Lin∗ and Gordon E. Willmot†
Ruin theory examines the solvency of an insurer in a theoretical setting. The insurer’s
surplus or reserve for an insurance portfolio over time is modelled by a stochastic process.
Of central importance in the evaluation of the surplus are the time of ruin at which the
surplus becomes negative for the first time, the surplus immediately before the time of ruin
and the deficit at the time of ruin. The use of stochastic processes for modelling the surplus
of an insurer can be dated back to 1909, when F. Lundberg presented his paper “On the
theory of risk” at the Sixth International Congress of Actuaries (see [44]). In his paper,
the surplus was described by a shifted compound Poisson process. The compound Poisson
surplus process was further investigated by H. Cramer (see [10]) and is still being studied
nowadays. The compound Poisson surplus process has also been extended and generalized in
several directions. Interest rates are incorporated into surplus processes. The claim arrivals
are assumed to follow a more general counting process than a Poisson process. Further,
the fluctuation of the surplus is considered and modelled by a Brownian motion and more
generally a diffusion process. Also considered are surplus processes with dividend policies.
1. The compound Poisson surplus process
The compound Poisson surplus process can be described as follows. The Poisson process
N(t) with intensity λ represents the number of claims experienced by the insurance portfolio
∗X. Sheldon Lin, Department of Statistics, University of Toronto, Toronto, Ontario M5S 3G3, Canada,
e-mail: [email protected]†Gordon E. Willmot, Department of Statistics and Actuarial Science, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada, e-mail: [email protected]
1
over time. More precisely, the number of claims over the time period [0, t] is N(t). The
random variable Yi represents the amount of the i-th claim. Hence, the aggregate claims
arising from the insurance portfolio over the time period [0, t] is given by
S(t) = Y1 + Y2 + · · ·+ YN(t),
with the convention that S(t) = 0 if N(t) = 0. It is assumed that these individual claim
amounts Y1, Y2, · · ·, are positive, independent and identically distributed with common dis-
tribution function P (x), and they are independent of the number of claims process N(t).
Thus, the aggregate claims process {S(t)} is a compound Poisson process. As shown in
Collective Risk Models, the mean and the variance of S(t) are E{S(t)} = λp1t and
V ar{S(t)} = λp2t, where p1 and p2 are the first two moments about the origin of Y , a
representative of the claim amounts Yi’s. Suppose that the insurer has initial surplus u ≥ 0
and receives premiums continuously at rate c per unit time. The insurer’s surplus process
then is given by
U(t) = u+ ct− S(t). (1)
The surplus process (1) is referred to as the classical compound Poisson surplus process. It
is normally assumed that the insurer charges a risk premium and in this case, the premuim
rate c will exceed the expected aggregate claims per unit time, i.e., c > E{S(1)} = λp1. Let
θ = [c − E{S(1)}]/E{S(1)} > 0. Then one has c = λp1(1 + θ). The parameter θ is called
the relative security loading.
Mathematically, the time of ruin is given by T = inf{t; U(t) < 0}. The first quantity
under consideration is the probability that ruin occurs in finite time, i.e., ψ(u) = Pr{T <
∞}, the probability of ruin. With the independent increments property of the Poisson
process, it can be shown that the probability of ruin ψ(u) satisfies the defective renewal
equation
ψ(u) =1
1 + θ
∫ u
0ψ(u− x)dP1(x) +
1
1 + θ[1− P1(u)], (2)
where P1(x) is the equilibrium distribution function of P (x) and is given by P ′
1(x) = [1 −
P (x)]/p1.
2
Since the probability of ruin ψ(u) is the solution of the defective renewal equation (2),
methods for renewal equations are applicable. The Key Renewal Theorem (see [46]) implies
that
ψ(u) ∼θp1
E{XeκX} − (1 + θ)p1
e−κu, as u→∞,
where the parameter κ is the smallest positive solution of∫
∞
0eκxdP1(x) = 1 + θ, (3)
and a(u) ∼ b(u) means limu→∞ a(u)/b(u) = 1. The parameter κ is called the Lundberg
adjustment coefficient in ruin theory and it plays a central role in analysis of not only the
probability of ruin but also other important quantities in ruin theory. With the Lundberg
adjustment coefficient κ, we can also obtain a simple exponential upper bound, called the
Lundberg bound, for the probability of ruin:
ψ(u) ≤ e−κu, u ≥ 0.
The defective renewal equation (2) also implies that ψ(u) is the survival probability of a
compound geometric distribution, where the geometric distribution has the probability mass
functionθ
1 + θ
(
1
1 + θ
)n
, n = 0, 1, · · · ,
and the secondary distribution is P1(x). This compound geometric representation enables
us to obtain an analytical solution for ψ(u). It is easy to see that
ψ(u) =∞∑
n=1
θ
1 + θ
(
1
1 + θ
)n
[1− P1∗n(u)], u ≥ 0, (4)
where P1∗n(u) is the n-fold convolution of P1 with itself. The formula (4) is often referred to
as Beekman’s Convolution Formula in actuarial science (see [6]) and the use of the formula
can be found in [49] and [52]. A disadvantage of the formula is that it involves an infinite
number of convolutions, which can be difficult to compute in practice.
In some situations, a closed form solution for ψ(u) can be obtained. When the individual
claim amount distribution P (x) is a combination of exponential distributions, i.e.,
P ′(x) =m
∑
j=1
qjµje−µjx
3
where 0 < µ1 < µ2 < · · · < µm and∑m
j=1 qj = 1, then
ψ(u) =m
∑
j=1
Cje−Rju,
where Rj’s are the positive solutions of the Lundberg equation (3) and
Cj =
{
m∑
i=1
qiµi −Rj
} /{
m∑
i=1
qiµi
(µi − Rj)2
}
.
See [17] and [23].
When the individual claim amount distribution P (x) is a mixture of Erlang distributions,
i.e.,
P ′(x) =m
∑
j=1
qjµjxj−1e−µx
(j − 1)!,
where µ > 0, qj ≥ 0, j = 1, 2, · · · , m and∑m
j=1 qj = 1,
ψ(u) = e−µu∞∑
j=0
Cj
[µu]j
j!,
where the coefficients Cj’s are obtained by a recursive formula that can be found in [41]. An
analytical expression is also available for more general phase-type individual claim amount
distributions. However, its calculation is not as straightforward and requires the use of the
matrix-analytic method. See [5] for details.
The classical results discussed above may be found in many books on risk theory, including
[8], [13], [21], [33] and [45].
2. The expected discounted penalty function
Further investigation of the surplus process involves the surplus immediately before the time
of ruin U(T−) and the deficit at the time of ruin |U(T )|. Dickson in [14] studied the joint
distribution of U(T−) and |U(T )|. The expected discounted penalty function introduced
by Gerber and Shiu in the seminal paper [25] is very useful for analysis of T , U(T−), and
|U(T )| in a unified manner. The expected discounted penalty function is defined as
φ(u) = E{
e−δTw(U(T−), |U(T )|) I(T <∞)}
, (5)
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where w(x, y) is the penalty when the surplus immediately before the time of ruin is x
and the deficit at the time of ruin is y. Note that I(A) is the indicator function of event
A, and the parameter δ is the interest rate compounded continuously. Many quantities of
interest may be obtained by choosing a proper penalty function w(x, y). For example, when
w(x, y) = 1, we obtain the Laplace transform of T where δ is the variable of the transform.
If w(x, y) = I(x ≤ u, y ≤ v) for fixed u and v and δ = 0, we obtain the joint distribution
function of U(T−) and |U(T )|. Furthermore, if w(x, y) = xkyl and δ = 0, joint moments of
U(T−) and |U(T )| are obtained.
It is shown in [25] that the expected discounted penalty function φ(u) satisfies the defec-
tive renewal equation
φ(u) =∫ u
0φ(u− x)g(x)dx+
λ
ceρu
∫
∞
ue−ρx
∫
∞
xw(x, y − x)dP (y)dx,
where ρ is the unique nonnegative solution to the Lundberg equation
cξ + λp(ξ)− (λ+ δ) = 0
and the kernel density g(x) is given by
g(x) =λ
c
∫
∞
xe−ρ(y−x)dP (y).
This result is of great significance as it allows for the utilization of the theory of renewal
equations to analyze the expected discounted penalty function φ(u). For example, it can be
shown that the Laplace transform of T is a survival probability of a compound geometric
distribution. Moreover, the general expected discounted penalty function can be expressed
in terms of the Laplace transform. As a result, many analytical properties of the expected
discounted penalty function may be obtained. See [25], [41], [42] and [54].
3. The compound Poisson surplus process with interest rate
The classical compound Poisson surplus process assumes that the surplus receives no interest
over time. The compound Poisson surplus process with constant interest rate was first
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studied in [47] and the model was further considered in [7], [12], [31], [50] and [51]. They
mainly focus on the probability of ruin. It is assumed that the insurer receives interest on
the surplus at constant rate δ, compounded continuously. Thus, the surplus at time t can
be expressed as
U(t) = ueδt + ceδt − 1
δ−
∫ t
0eδ(t−s)dS(s).
Sundt and Teugels [50] show that the probability of ruin ψ(u) satisfies the integral equation
[c+ δu]ψ(u) =∫ u
0ψ(u− x)[δ + λ(1− P (x))]dx+ λ
∫
∞
u(1− P (x))dx−
θ − θ′
1 + θ′λp1, (6)
where θ′ < θ is given in [50]. The equation (6) is nonlinear and hence is in general not
solvable. However, an analytical solution exists for the exponential individual claims. See
again [50].
Recently, the expected discounted penalty function under the above model and stochastic
interest rates are considered. See [9], [34], [55], and references therein.
4. The compound Poisson surplus process with diffusion
Dufresne and Gerber [18] first considered the compound Poisson surplus process perturbed
by diffusion and assumed the surplus process is given by
U(t) = u+ ct− S(t) + σW (t), (7)
where {W (t)} is the standard Brownian motion and σ > 0. The diffusion term σW (t) may
be interpreted as the additional uncertainty of the aggregate claims, the uncertainty of the
premium income, or the fluctuation of the investment of the surplus, where σ is the volatility.
In this situation, the expected discounted penalty function needs to be modified. Let
φd(u) = E{
e−δT I(T <∞, U(T ) = 0)}
and
φs(u) = E{
e−δTw(U(T−), |U(T )|) I(T <∞, U(T ) < 0)}
.
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The expected discounted penalty function is defined as
φ(u) = w0φd(u) + φs(u).
The first term represents the penalty caused by diffusion and the penalty is w0, and the
second term represents the penalty caused by an insurance claim. It is shown in [24] that
the expected discounted penalties φd(u) and φs(u) satisfy the second-order integro-differential
equations:
Dφ′′d(u) + cφ′d(u) + λ∫ u
0φd(u− y)dP (y)− (λ+ δ)φd(u) = 0
Dφ′′s(u) + cφ′s(u) + λ∫ u
0φs(u− y)dP (y) + λ
∫
∞
uw(y, y − u)dP (y)− (λ+ δ)φs(u) = 0,
where D = σ2/2. Similar to the classical compound Poisson case, it has the Lundberg
equation
Dξ2 + cξ + λp(ξ)− (λ+ δ) = 0,
which has a non-negative root ρ. The expected discounted penalty function φ(u) satisfies a
defective renewal equation with the kernel density
g(x) =λ
D
∫ x
0e−d(x−v)
∫
∞
ve−ρ(y−v)dP (y),
and d = c/D + ρ. An interesting finding is that the probability of ruin as a special case is
a convolution of an exponential density with the tail of a compound geometric distribution.
See [18].
The compound Poisson process with diffusion (7) is a special case of Levy processes and
hence it is natural to consider a class of Levy processes for the surplus process that have
only negative jumps. A number of papers discuss the probability of ruin for this class of
surplus processes. See [32], [35] and [56]. More recently, the expected discounted penalty
function for general Levy processes is investigated in [19]. Generally, the results for Levy
surplus processes are very similar to those for the compound Poisson process with diffusion.
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5. The Sparre Andersen surplus process
The compound Poisson surplus process implicitly assumes that the inter-claim times are
independently and identically exponentially distributed. A more general model involves the
assumption that the inter-claim times are independent and identically distributed, but not
necessarily exponential. The resulting surplus process is referred to as a renewal risk process,
or as a Sparre Andersen process after its usage was proposed in [4].
Analysis of the Sparre Andersen surplus process is more difficult than the compound
Poisson special case, but some progress has been made. Because the process is regenerative
at claim instants, it can be shown that the ruin probability is still a compound geometric
tail. In fact, the expected discounted penalty function defined by (5) still satisfies a defective
renewal equation and the Laplace transform of the ruin time T is still a compound geometric
tail. The defective renewal equation structure implies that Lundberg asymptotics and bounds
hold in this more general situation. For further discussion of these issues, see [26], [38], [53],
and references therein.
The difficulty with the use of this model in general is the fact that it is difficult to identify
the geometric parameter and the ladder height (or geometric secondary) distribution. This
is true even for the special case involving the ruin probabilities. Such identification normally
requires that further assumptions be made about the inter-claim time distribution and/or
the claim size distribution. In [15], Dickson and Hipp considered Erlang(2) inter-claim times,
whereas Gerber and Shiu in [26] and Li and Garrido in [36] considered the Erlang(n) case.
Also, Li and Garrido in [38] considered the more more general case involving Coxian inter-
claims. Conversely, parametric assumptions about claim sizes rather than inter-claim times
were considered by Willmot in [53]. The situation with phase-type assumptions is discussed
in [5].
A closely related model is the delayed Sparre Andersen model where the time until the
first claim is independent but not distributed as the subsequent inter-claim times. This
model attempts to address the criticism that the Sparre Andersen model implicitly assumes
that a claim occurs at time 0. Other classes of surplus processes that do not have stationary
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and independent increments are the non-homogeneous Poission process, the mixed Poisson
process and more generally Cox processes. Similar to the renewal surplus process case,
analysis of these processes is also difficult. For further details on the Sparre Andersen and
related processes, see [5], [29], [30], [45], and references therein.
6. Dividend strategies
As mentioned earlier, an insurer normally charges a risk premium on its policies, which im-
plies that the premium incomes are greater than the expected aggregate claims. In this case,
the probability of ruin is less than one. Hence, there is a positive probability that the surplus
will grow indefinitely, an obvious shortcoming of the afore-mentioned surplus processes. To
overcome this shortcoming, De Finetti in [11] suggested that a constant dividend barrier
be imposed so that the overflow of the premium incomes is paid as dividends (the constant
dividend barrier was originally applied to a binomial surplus process in his paper). This div-
idend strategy is usually referred to as the constant barrier dividend strategy. Early studies
on this strategy can be found in [8], [20], [21], and [48]. A generalization of the constant
barrier dividend strategy is the threshold dividend strategy under which a fixed proportion
of the premiums is paid as dividends when the surplus is above the constant barrier. The
threshold dividend strategy can also be viewed as a two-step premium as described in [5].
The research on surplus processes with dividend strategies is primarily concerned with (i)
calculation of the expected discounted total dividends and identification of optimal dividend
strategies; and (ii) the expected discounted penalty function. When the surplus process is
compound Poisson, the expected discounted total dividends satisfies a homogeneous piece-
wise first-order integro-differential equation and hence is solvable. The result also holds for
the Levy surplus process except that the equation is of second-order. When the objective
is to maximize the expected discounted total dividends with no constraints, the optimal
dividend strategy for the compound Poisson surplus process is the constant barrier dividend
strategy if the individual claims follow an exponential distribution and the so-called band
strategy for an arbitrary individual claim distribution. If an upper bound is imposed for
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the dividend rate, then the threshold dividend strategy is optimal for the compound Poisson
surplus process with the exponential individual claim distribution. It is generally believed
that these results also hold for general Levy processes but they have not been proven. See [8],
[27], and references therein. Other objectives may be used for choosing an optimal constant
barrier. Dickson and Waters in [16] consider the maximization of the difference between the
expected discounted total dividends and the expected discounted deficit at the time of ruin.
It was extended to maximize the difference between the expected discounted total dividends
and the expected discounted penalty in [28].
There has been renewed interest in studying the time of ruin and related quantities for
surplus processes with dividend policies in recent years, especially since the introduction of
the expected discounted penalty function in [25]. The probability of ruin is given in [5] for the
compound Poisson surplus. It is shown in [39] and [43] that the expected discounted penalty
function satisfies a non-homogeneous piece-wise first-order integro-differential equation and
hence is solvable in many situations. Some related papers on this problem include [57] and
[58]. An interesting result among others (see [43]) is the dividends-penalty identity that states
that the increment in the expected discounted penalty due to the constant barrier strategy
is proportional to the expected discounted total dividends. This result has been extended to
the stationary Markov surplus process in [28]. Other extensions include non-constant barriers
and multiple constant barriers. See [1], [2], [3], [22], [40], and references therein. There are a
number of papers considering the renewal surplus process with a constant dividend barrier.
See [37] for example. However, a constant dividend barrier is less interesting as it can not
provide an optimal dividend strategy due to the non-stationarity of the claim arrivals.
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