1. Introduction
Typical risk considerations in the area of insurance and finance are concerned with the Risk Process
(1.1)
where
represents the capital available at time
, given the initial capital
, after paying claims
which occurred at random times during the interval
according to a Poisson process Nt . The premium income stream ct is deterministic with premium rate c per unit of time.
is known as the Crámer-Lundberg model and represents the risk reserve of a company at time t. The main objective is to calculate the odds that the company reserve will ever become negative, referred to as the probability of ultimate ruin.
Except a few special cases with closed form solutions, the analysis of this process is usually carried out by numerical inversion of the associated Laplace Transform to solve a renewal equation involving the probability of ruin in infinite time. Since the joint work of Gerber and Shiu [1] [2] in the late 1990’s it has been customary to analyze the process in terms of an expected discounted penalty function.
Various attempts have also been made to add a Lévy Process component to the model in [3] [4] and [5] among many others. Over the years, discrete time versions of the model have been studied, see for example [6] for recent work along these lines. Lately, a stochastic premium income component has been added, see for example [7]. Much of the theory and applications are elucidated in [8] and [9].
We remark, that while it may be reasonable for an insurance company to conveniently collect premiums according to deterministic formula ct, given customers contractual obligation to pay premiums to receive coverage for their claims, it certainly is not a reasonable assumption for most models of business income, as the future number of customers and their respective premiums cannot be guaranteed. Furthermore, while it may be true “on average” that an insurance company receives premiums as a continuous stream, it is still possible that the total premiums collected by time t may be substantially smaller than ct, at some future times t.
To remedy this drawback, we propose a model in which ct is replaced by a stochastic component leading to a shifted discrete time zero-mean random walk representation of the Risk Process that can be analyzed by various tools from probability theory.
The paper is organized as follows. Section 2 introduces our new model. In sections 3 - 5 we derive estimates for the probability of ruin by Kolmogorov’s Maximal Inequality, Stopping a Martingale and Large Deviation Principle. Section 6 includes summary conclusions and directions for future research.
2. Derivation of the Model
An extension of the Crámer-Lundberg model to random premiums, by Boikov [7], is as follows
(2.1)
where
are independent Poison processes and
are independent sequences of i.i.d. representing the premiums and claims respectively.
Our objective is to propose a new model that can be considered a discrete time counterpart to continuous model (2.1) which provides considerable reduction in random complexity through replacing random sums
by
. The significance of such model is that it reflects the actual real-world practice. Namely, “Ruin” is naturally defined as having a negative balance at the end of the day. Likewise, “ruin” has not occurred if the balance at the end of the day is not negative. This is irrespective of whether or not the balance may have been negative at some point before the end of the day.
Dickson and Waters [10] and Dickson [11] studied a discrete model with deterministic premiums
(2.2)
For our model we discretize time in (2.1) whereby generalize (2.2) to random premiums with simultaneous extension of the range of Xi from non-negative integers to non-negative reals as follows
(2.3)
and can be viewed as a random walk started at initial capital u at time 0.
Recall that the safety loading requires the expected value of the Risk Process gain =
to be positive, for otherwise the probability of eventual ruin is one. Therefore,
(2.4)
where
is a safety loading factor.
representation below will play a key role in establishing several estimates for the probability of ruin. Namely, thanks to (2.1) we have
(2.5)
which is a zero-mean random walk
with linear drift
started at u.
3. Probability of Ruin by Kolmogorov’s Maximal Inequality
The results in this section provide an upper estimate on the probability of ultimate ruin in relation to the initial capital. Furthermore, it is shown how to select the initial capital to achieve a low probability of ruin in the finite time interval [0, T].
Theorem 3.1. Let
, where
, and
arei.i.d. with
,
.
Then for every positive integer N there exists an initial capital u0 such that
Proof. For any subsequence of integers
, and an integer
we have
where the last two inequalities follow from the lower bound on
depicted in our graph below. (See Figure 1)
Choosing
and setting
we obtain
To complete the proof it suffices to show that for suitably chosen subsequence (
)the series
converges, and consequently there exists an integer l = l(N) such that
.
To this end, by Kolmogorov’s maximal inequality with
hence
as needed.
Corollary 3.1. For every positive integer N there exist an initial capital u0 and a finite time T such that
Proof. Choose
and
. Then by Kolmogorov’s maximal inequality
Corollary 3.2. Starting with capital
the probability of no ruin by time N is at least
.
4. Probability of Ruin by Stopping a Martingale
We show how martingale method can be applied to calculate the probability of ultimate ruin for our model. Recall that by (2.3) - (2.4) the risk process reads
Figure 1. Linear income trend and its lower bound.
(4.1)
where
are nonnegative independent random variables.
Theorem 4.1. Suppose
such that
.
Then
and
. (4.2)
Proof. Let
so
and
. By Jensen’s inequality for
we have
and therefore r must be positive. For any
and
we have
and
Define a stopping time N by
with
in the case no such n exists.
Then
is a martingale as
where the conditional expectation becomes expectation, due to indepedence of (
) and equals 1 by assumption. It is standard to check that
( [12] ) whence
. Now
giving
(4.3)
By taking
and setting
it follows that
is increasing in k. Consequently, by continuity of
for monotone sequences
Consequently by (4.3)
Finally by (4.3) with a = u
Example 4.1 (Exponential case). Let the claim size X ~ exponential with mean
and premium size Y~ exponential with mean
. Then the condition
(4.4)
in terms of the moment generating function is as follows
.
Solving for r we get
(4.5)
Some comments regarding (4.5) are in order and we collect them in the following.
Remark. For solutions r to be well defined and positive, some conditions must be satisfied as follows.
1)
,
which is always satisfied for
, whereas
,
.
2)
is always a positive solution, however
can also be solution if
and
with
.
Consequently
if
and
if
.
Example 4.2 (Binomial case). For claim and premium
,
with
,
. Then condition (4.4)
(4.6)
in terms of the moment generating function reads
and (4.2) has the form
Notice that the assumption
gives
.
For example,
gives
whence
Namely, every extra dollar of initial capital halves the probability of the Ultimate Ruin!
5. Probability of Ruin via Large Deviation Principle
This section is concerned with the derivation of the upper bound for the probability of ruin on the interval
, which we will refer to as Tail Ruin probability. Our arguments are based on the rate function—a key ingredient of the Large Deviation Principle, so for the sake of completeness we recall some relevant facts.
Large deviation results show that probabilities of atypical events An, away from typical events, all off to zero at an exponential rate. That is,
for large n where the constant
is directly computable. One of the first and important rates is concerned with the Law of Large Numbers and states
, whenever
.
A large deviation result we need is attributed to Crámer and stated below without proof.
Large Deviation (Th. I.4, [13] ).
Let
be i.i.d. with the moment generating function
,
, and
. Then for any
and rate function
.
Remark 5.1. The above result has a straightforward extension
for t from some subset of
. Key properties of the rate function
are as follows:
,
is convex,
,
may assume
,
for
,
for
.
is a convex conjugate or Legendre Transform of the convex function
.
Lemma 5.1 (upper bound). Assume
,
. Then for a
(5.1)
Proof. By Markov’s inequality
. (5.2)
Since t is arbitrary one can optimize this upper bound by maximizing the function
over t. We have
,
and therefore
in some vicinity of t = 0, because
. This in turn, since
is concave down, shows that
has a unique strictly positive maxim um, which can be readily obtained by solving
for some
, whence
.
Remark 5.2 We would like to point out and emphasize the often overlooked draw-back of the probability of ultimate ruin, which stems from that fact that it does not provide any information as to when the actual ruin occur during the time interval
. For this very reason, our theorem below fills this gap and sheds some light on the time window where the ruin is most likely to occur.
Theorem 5.1 Let
, where
, and
arei.i.d. with
. Then we have the following upper bounds for the probability of ruin
(5.3)
Proof. We have
In the example below we will illustrate how the upper bounds (5.3) can be used to estimate the probability of ruin in a
, when an upper bound for the probability of ultimate ruin is available.
Example 5.1 Consider our previous Example 4.1 where the claim size X is exponential with mean
and the premium size Y is exponential with mean
.
Then
,
. (5.4)
Furthermore,
and for
as in (5.2)
Therefore,
and gives
,
. (5.5)
Let us choose
. Then by (5.5)
or
of 1%.
On the other hand
which is 100 fold smaller than the probability of Ultimate Ruin, thus negligible in comparison!
A word about why
is in order. Given
was obtained numerically. We include our graph of concave down h(t) below. (See Figure 2)
Similarly,
is negligibly small and can be dropped. By comparing the order of smallness of the respective probabilities we infer that, if the ruin occurs, it will most likely happen within the first five years.
Figure 2. The graph of
.
6. Conclusions
We have introduced a discrete time risk model that features a convenient way of maintaining end of the day net balance of company’s capital reserve, resulting from the random size premiums income minus the incoming random size claims on the daily basis. Three different methods of estimating the probability of ruin (i.e., negative capital reserve) were presented and illustrated by examples. The key innovation is a reduction of complexity associated with randomness of the Risk Process by modeling random premiums and random claims arriving at discrete deterministic times in our model, as opposed to random claims arriving at random times according to Poisson process in the Crámer-Lundberg model studied in the literature.
Future research will focus on extending the model to allowing investment of the collected premiums into stock market equities.