INSURANCE RISK AND RUIN PDF
Request PDF on ResearchGate | On Feb 1, , Hanspeter Schmidli and others published Insurance Risk and Ruin. David C. M. Dickson. The focus of this book is on the two major areas of risk theory: aggregate claims distributions and ruin theory. For aggregate claims distributions. Insurance Risk and Ruin. The focus of this book is on the two major areas of risk theory: aggregate claims distributions and ruin theory. For aggregate claims.
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Cambridge Core - Finance and Accountancy - Insurance Risk and Ruin - by PDF; Export citation 1 - Probability Distributions and Insurance Applications. Finance and Insurance - Insurance Risk and Ruin - by David C. M. Dickson. PDF; Export citation 1 - Probability distributions and insurance applications. Insurance Risk and Ruin - Ebook download as PDF File .pdf), Text File .txt) or read book online.
The focus of this book is on the two major areas of risk theory: For aggregate claims distributions, detailed descriptions are given of recursive techniques that can be used in the individual and collective risk models. For the collective model, different classes of counting distribution are discussed, and recursion schemes for probability functions and moments presented. For the individual model, the three most commonly applied techniques are discussed and illustrated. Care has been taken to make the book accessible to readers who have a solid understanding of the basic tools of probability theory. Numerous worked examples are included in the text and each chapter concludes with a set of excercises for which outline solutions are provided. Panjer, Waterloo University Andrew Wilson, Watson Wyatt The International Series on Actuarial Science, published by Cambridge University Press in conjunction with the Institute of Actuaries and the Faculty of Actuaries, will contain textbooks for students taking courses in or related to actuarial science, as well as more advanced works designed for continuing professional development or for describing and synthesising research.
To see this, we note that the recursion scheme given by formula 4. Then by repeated application of formula 4. Table 4.
We conclude our discussion of the a, b, 0 class by considering the prob- ability generating function of a distribution in this class, and deriving a result Table 4. Not only is it useful in the context of aggregate claims distributions, but, as we shall see in Chapter 7, it has applications in ruin theory.
Until now we have tacitly assumed that individual claims follow some continuous distribution such as lognormal or Pareto. Indeed, we have not discussed discrete distributions as candidates to model individual claim amounts, and we defer 68 The collective risk model a discussion of this until Section 4. Since we are now assuming that individual claim amounts are distributed on the non-negative integers, it follows that S is also distributed on the non- negative integers.
It shows that g x is expressed in terms of g 0 , g 1 ,. In all practical appli- cations of this formula, a computer is requiredtoperformcalculations.
However, the advantage that the Panjer recursion formula has over formula 4. In general a recursion formula for the distribution function of S does not exist. We shall see in Chapter 7 that this is a particularly useful result. We remark that these results are consistent with results in Section 4. As the recursion formula is the same for the a, b, 1 class as for the a, b, 0 class, we can construct members of the a, b, 1 class by modifying the mass of probability at 0 in distributions in the a, b, 0 class, and there are two ways in which we can do this.
Its zero-truncated 4. There are four other members of the a, b, 1 class. When the counting distribution belongs to the a, b, 1 class and individual claim amounts are distributed on the non-negative integers, the techniques of the previous section can be used to derive a recursion formula for the probability function for aggregate claims.
Similarly, following the arguments in Section 4. Applying formula 4. Rounded values are shown in this solution. By inserting formula 4. We can now write equation 4. It is beyond our scope to discuss ranges for the parameters a, b and c in formula 4. The probability functions of S 1 and S 2 can each be calculated by the Panjer recursion formula, and values are shown in Table 4. To apply formula 4. In the above solution, we have exploited the fact that we knew that the counting variable when the risks were combined was the sum of Poisson and negative binomial random variables.
It follows from the recursive nature of formula 4. We can use this fact to set the starting value in formula 4. Care must be exercised in applying such an approach, and the obvious test to apply is to increase the upper limit of summation. In practice, however, continuous distributions such as Pareto or lognor- mal are used to model individual claim amounts. In order to apply a recursion formula in such a situation we must replace a continuous distribution by an appropriate discrete distribution on the non-negative integers, and we refer to this process as discretising a distribution.
One approachis throughmatchingprobabilities. An alternative approach is to match moments of the discrete and continuous distributions. Thus, the discretisation procedure is mean preserving. Note that although we are discretising on the integers, this technique also applies to discretising on 0, z, 2z,. To see this, suppose that X has an exponential distribution with mean , and let Y be a randomvariable whose distribution is a discretised version of this exponential distribution on 0, 1, 2,.
Thus, we can think of the quality of a discretisation process improving as the fraction of the mean on which the distribution is discretised decreases. Figures 4. In Fig. The better approximationis clearlyinFig. The quality of this approximation depends on the value of the scaling factor k. In general, the larger the value of k, the better the approximation should be. In the examples in Section 4.
This level of scaling should be appropriate for most practical purposes. This Pareto distribution has been discretised using formula 4.
Three scaling factors have been used, namely 20, 50 and We can see from this table that an increase in the scaling factor does not have a great effect on the approximation, and this is particularly the case for larger probabilities.
First, not all recursion schemes are stable. This means that a recursion formula may produce sensible answers initially, but will ultimately produce values that are clearly wrong. For example, when the claim number distribution is binomial, the Panjer recursion formula is unstable. Instability manifests itself in this case by producing values for the probability function of S that are outside the interval [0, 1].
The Panjer recursion formula is, however, stable when the claim number distribution is either Poisson or negative bino- mial. Instability does not mean that a recursion scheme is not useful. It simply means that we should be careful in analysing the output from our calculations.
This occurs when the initial value in a recursive calculation is so small that a computer stores it as zero.
One solution to the problemis to set g 0 to an arbitrary value such as 1, then proceed with the recursive calculation. However, a problem with recur- sive methods is that they can be computationally intensive, even with modern computing power.
Therefore, approximate calculation can still be very useful, particularly if it can be done quickly. We now describe two methods of approx- imating the aggregate claims distribution, each of which can be implemented easily with basic software such as a spreadsheet. As the number of variables being summed increases, we would expect the distribution of this sum to tend to a normal distribution by the Central Limit Theorem. The problem with this argument is that we are dealing with a random sum, but if the expected number of claims is large, it is not unreasonable to expect that a normal distribution would give a reasonable approximation to the true distribution of S.
Figure 4. First, the true distribution of S is positively skewed, whereas the normal distribution is symmetric. The situation is different in Fig. A common feature of Figs 4. For example, 4. In each case the normal approximation has understated the 95th percentile of the distribution. In summary, the advantages of the normal approximation are that we need little information to apply it just the mean and variance of S , it is easy to apply, 86 The collective risk model and it should give reasonable approximations if the expected number of claims is large.
In Section 1. By equation 4. Next, equation 4. Note that in Fig. As noted in the discussion following Example 4. Thus, the approximations in Example 4. The major advantage that the translated gamma approximation has over the normal approximation is that it takes account of the skewness of the distribution of S.
However, we need one more piece of information in order to apply this approximation. In Example 4. Nevertheless, the translated gamma approximation is a simple and easily implemented approach that can produce excellent approximations.
Discretisation methods are discussed by Panjer and Lutek , while the discretisation given by formula 4. Numerical aspects of recursive calculations are discussed by Panjer and Willmot and Panjer and Wang Individual claim amounts are lognormally distributed with mean 1 and variance 2.
Calculate E[S] and V[S]. Calculate the premium for this risk using the exponential premium principle with parameter 0. Aggregate claims from a risk have a compound Poisson distribution with Poisson parameter and individual claim amounts are exponentially distributed with mean Calculate the premium for this risk using the Esscher premium principle with parameter 0.
Using the notation of Section 4. The insurer of this risk effects excess of loss reinsurance with retention level with Reinsurance Company A. Aggregate claims from a risk have a compound Poisson distribution with Poisson parameter 10, and the individual claim amount distribution is exponential with mean An insurer charges a premium of to cover this risk, and arranges excess of loss reinsurance with retention level M.
Show that E [g M ] is an increasing function of M. For which values of M is E [g M ] positive? Let M be a positive integer. Aggregate claims from a risk have a compound negative binomial distribution. The distribution of the number of claims is NB 10, 0. The insurer effects excess of loss reinsurance with retention level 4. An insurer offers travel insurance policies.
The probability that a policy produces a claim is q and the amount of a claim is an exponential random variable with mean 1 The premium for such a policy is This premium has been calculated on the following assumptions: Find the value of q. An alternative approach is to consider the aggre- gate claim amount from a portfolio as the sum of the claim amounts from the individual policies that comprise the portfolio.
This approach gives rise to the individual risk model which we discuss in this chapter. In the next section we specify the model assumptions, and in subsequent sections we consider dif- ferent approaches to evaluating the aggregate claims distribution.
A numerical illustration of these methods is given in Section 5. It is important to realise that the amount paid out under an individual policy can be zero and often is in practice.
Suppose that a claim occurs under the i th policy. Note that S i has a compound binomial distribution since the distribution of the number of claims under the i th policy is B 1, q i , and so it immediately follows from formulae 4.
However, it represents perfectly the situation in life insurance. For the remainder of this chapter it is convenient to use the terminology of life insurance, so that q i is the mor- tality rate of the holder of policy i.
Further, the assumption of independence implies that there are n distinct individuals in the portfolio. We concentrate on developing formulae which can be used to cal- culate the aggregate claims distribution within this life insurance framework, which is the most important application for the individual risk model. In most practical applications n is large, and so this approach is not particularly attrac- tive.
Hence we seek alternative methods which involve fewer computations. In this section we derive the recursion formula and describe a variation of it. However, we defer application of these formulae until Section 5.
It is convenient to subdivide the portfolio according to mortality rate and sum assured. We assume that sums assured in the portfolio are integers, namely 1, 2,. Let n i j denote the number of policyholders with mortality rate q j and 5. Differentiating equation 5. Hence, we have r P.
Dickson D.C.M. Insurance Risk and Ruin
Values of q j are small in practice, and this means that for large values of k, the terms which contribute to h i, k are very small, and h i, k it- self is usually small. One way of reducing computing time is to discard small values of 5.
We use ideas similar to those in the previous section to develop this method, and our set-up is identical to that in the previous section. The reason for using absolute values is that our construction does not guarantee that each g K x is positive. The constraints on k are as follows. Note that when the upper limit is K, there is a contribution at K only if K is a divisor of x. To see this, note from equation 5.
Equation 5. As with other methods, we illustrate it numerically in Section 5. In presenting ideas, we revert to the general model described in Section 5.
Hence, we are no longer able to classify policyholders by mortality rate and sum assured as we did in those sections. Let G i be the distribution function of the amount paid out in claims under the i th policy. As noted in Section 5. There is no simple representation for the convolution of compound binomial distributions.
However, as shown in Section 4. This motivates a simple idea. Then P is a compound Poisson distribution. Note that G i and P i are both compound distributions — they have different claim number distributions, but the same individual claim amount distribution. Table 5. The message from Table 5. Further, each method produces a very good approximation to the B 1, q i distribution when q i is small.
The main result of this section is as follows: To prove this result we need the following two auxiliary results. Then n. The proof of equation 5. We are now in a position to prove equation 5. If equation 5. Thus, we have proved equation 5. Thus, for 5.
It is nevertheless true that equation 5. For this portfolio, formulae 5. The legend for this table is as follows: CP1 denotes the compound Poisson approximation when the Poisson parameter for each policy is the mortality rate; 6. N denotes the normal approximation, where the approximating normal distribution has mean We can see from Table 5.
The normal approximation is the poorest of all the approximations. However, Fig. As an illustration of the point made in Section 5. We can see in this table that for each value of i , the values of h i, k decrease in absolute value as k increases. For the compound Poisson approximations, from equation 5. We note from Table 5. We also note that a consequence of the choice of Poisson parameters in approximation CP2 is that the approximating distribution function always takes values less than the true distribution function.
The bounds in Section 5. A practical overview of the different methods presented in this chapter is given by Kuon et al. The table below shows data for a life insurance portfolio in which the lives are independent with respect to mortality. Mortality rate Sum assured Number of lives 0. Consider a portfolio of n insurance policies. In a life insurance portfolio the sums assured are 1, 2,.
Claims from policies are assumed to be independent of each other. Find expressions for b 2 x when x is even, and when x is odd. State the values of x for which these expressions are non-zero. Write computer programs to verify the values given in Table 5. In Chapter 4 we considered the aggregate amount of claims paid out in a single time period. We now consider the evolution of an insurance fund over time, taking account of the times at which claims occur, as well as their amounts.
To make our study mathematically tractable, we simplify a real life insurance operation by assuming that the insurer starts with some non-negative amount of money, collects premiums and pays claims as they occur. Our model of an insurance surplus process is thus deemed to have three components: The aim of this chapter is to provide an introduction to the ideas of ruin theory, in particular probabilistic arguments. We use a discrete time model to introduce ideas that we apply in the next two chapters where we consider a continuous time model.
Indeed, we will meet analogues of results given in this chapter in these next two chapters. We start in Section 6. For the remainder of this chapter we assume that u is a non-negative integer so that the surplus process is always at an integer value since the premium income per unit time is 1 and claim amounts are integer valued. For this surplus process, we say that ultimate ruin occurs if the surplus ever falls to 0 or below. We will see in Chapter 7 that this is simply Introduction to ruin theory a very convenient modelling assumption.
In Chapter 4 we saw that appropriate use of scaling allowed us to apply a model in which individual claims were distributed on the integers, and as we will see in Chapter 7, scaling can similarly be applied to this discrete time model.
It also follows from equation 6.
To apply the function g d it is important to note that it has an alternative interpretation. Hence equation 6. Hence, we can write equation 6. Example 6. Solution 6. If we now insert for H in equation 6. Suppose we 6. We continue in this manner, using equation 6. Since many of the probabilities used in the calculations will be very small, we can reduce the number of calculations involved by ignoring small probabilities.
Thus, we are setting values less than to be zero. First, by a suitable choice of , we can control the error in our calculation. Sec- ond, the upper limit of summation in equation 6. We illustrate an application of this algorithm in Chapter 8. Thus the function is decreasing at 0. Further, any turning point of the function is a minimum since g. Find R d. See, for example, Grimmett and Welsh However, martingale arguments are not required to prove results discussed in Chapters 7 and 8 and so will not be discussed further.
First we start with a description of the classical risk process. For sim- plicity, we assume that this distribution is continuous with density function f and, keeping the notation of Chapter 4, the kth moment of X 1 is denoted by m k. This is a technical condition which we require in Section 7. Nevertheless, this is a useful model which can give us some insight into the characteristics of an insurance operation. In our con- text, an event is the occurrence of a claim.
An important property of compound Poisson processes is that they have sta- tionary and independent increments. Thus, if a process has independent increments, the increments over non-overlapping time intervals are independent. This is exactly the same situation as at time 0. However, the opposite is not true.
In this chapter we concentrate mostly on the ultimate ruin probability. In Sections 7. It takes account of two factors in the surplus process: To see that there is a unique positive root of equation 7. To see that equation 7. In the former case, equation 7. Example 7. Find an expression for R. Solution 7. Calculate R.
Insurance Risk and Ruin | Reinsurance | Utility
We can solve numerically for R using the Newton—Raphson method with a starting value of 0. Table 7. As in the previous chapter, we can prove this result by an inductive argument. However, by eliminating the integral term, a differential equation can be created, and solved.
Hence, if we multiply equation 7. However, formula 7. Although this method of solution can be used for other forms of F, we do not pursue it Classical ruin theory further. In the next section we showhowequation 7. In Section 7. If we write c in this way in equation 7. This independence holds for any individual claim amount distribution, not just the exponential distribution.
Insurance Risk and Ruin. David C. M. Dickson
To see why this is the case, consider the following two risks: Risk I The aggregate claims process is a compound Poisson process with Pois- son parameter and individual claim amounts are exponentially distributed with mean 1. The premium income per unit time is RiskII The aggregate claims process is a compoundPoissonprocess withPois- son parameter 10 and individual claim amounts are exponentially distributed with mean 1.
If we take the unit of time for Risk II to be one month, and the unit of time for Risk I to be one year, we can see that the risks are identical. There is thus no difference in the probability of ultimate ruin for these two risks.
However, if the unit of time for Risk II were one year, there would be a difference in the time of ruin, which is discussed in Chapter 8. In this and the next chapter, we apply the following properties of Laplace transforms. Recall equation 7. From equations 7. However, it is usually a straightforward exercise to use this approach with mathematical software which has the capacity to invert Laplace transforms.
We describe each method in turn, then conclude with numerical illustrations of each method. This allows the use of the recursion formula 4. We now show that this formula is generally true. We now turn our attention to the distribution of L. Then the aggregate loss process attains a new record high at this point in time, namely l 1.
Here we are making use of the fact that the compound Poisson process has stationary and independent increments. Further, the maximum of the aggregate loss process is simply the sum of the increases in the record high of the process. Thus, we can write L as a compound geometric random variable: Figure 7. Then by applying techniques from Section 4. Further, as N has a geometric distribution, L 7. Note that the distributionof L 1 is a continuous one. Hence, if we wishtoapply formula 4.
To calculate these bounds we apply the procedure described in Section 4. This is a convenient scaling of parameters that does not affect principles. The distribution of X 1,i should be chosen such that it is a good approximation to the distribution of X i and we have seen in Chapter 4 how this can be done.
We can now approximate this by a probability of ruin in discrete time.
Insurance Risk and Ruin
Thus, in Section 7. Then we know from Section 7. To apply the method of bounds from Section 7. In Table 7. In particu- lar, the lower bounds increase and the upper bounds decrease. We also note that for each value of u, the average of the bounds gives an excellent approximation Table 7.
We remark that from Table 7. In applying this method, the scaled exponential distribution was replaced by the discrete distribution given by formula 4. We see from this table that this method also gives very good approximations, and for this individual claim amount distribution there is little difference between the methods in terms of approximations. As a second illustration we consider the situation when the individual claim amount distribution is Pa 4, 3.
We can 7. For this situation, calculation by the method of Section 7. However, the need for approximations has diminished in recent years as numerical methods such as those described in the previous section can now be implemented easily with modern computing power. The main reason for introducing this approximation will be explained in the next chapter.
Two further approximation methods are described in Exercises 9 and Then by equation 7. The recursive procedure in Section 7. Although there is no evidence in the examples presented in Section 7. An alternative, stable, algorithm is discussed in Dickson et al. De Vylder derived the approximation which bears his name. Comment on the quality of the approximation. Use formula 7. Let the individual claim amount distribution be Pa 4, 3. This result will be applied in Sections 8.
We then consider the distribution of the time to ruin, and conclude with a discussion of a problem which involves modifying the surplus process through the payment of dividends. In this chapter we use the same assumptions and notation as in Chapter 7. An alternative way of expressing this question is to ask what is the probability that ruin occurs in the presence of an absorbing barrier at b? Also, as the distribution of the time to the next claim from the time the surplus attains b is exponential, the probabilistic behaviour of the surplus process once it attains level b is inde- pendent of its behaviour prior to attaining b.
Similarly, if ruin occurs from initial surplus u, then either the surplus process does or does not attain level b prior to ruin. This can be obtained from Section 7.
Example 8. Solution 8. For the classical risk model, this occurs only in the case of an exponential distribution for individual claims, and the reason for this is the memoryless property of the exponential distribution. In Exercise 3 of Chapter 6 we saw the analogue of this result for the discrete model discussed in that chapter. Find an expression for G u, y. We explore this example further in Exercise 4. We recall from formula 7. In the notation of Section 8. In Section 8. We note that W is a defective distribution function for the same reason that G is in Section 8.
Note that if the surplus process never reaches x, then ruin must occur with a surplus prior to ruin less than x. Hence, the numerator on the right-hand side of equation 8. We can write this because ruin can occur in one of the two following ways. Inserting this expression into equation 8. As equation 8. Figure 8. In this section we consider exact and approximate calculation of the density and moments of T u. For the remainder of this section we concentrate on the special case when 8.
Then, by inserting for both f and F in equation 8. Applying the same technique as in Section 7. From equation 8. Higher moments of T u,c can be found in a similar manner, as illustrated in Exercise 6. This idea is explored further in Section 8.
Table 8. The exact values have been calculated from formula 8. This value was calculated by the method of Section 7. This distribution has mean 1 and variance A question of interest is how should the level of the barrier b be chosen to maximise the expected present value of net income to the share- holders, assuming that there is no further business after the time of ruin.
Integrating out in equation 8. Find an expression for V u, b. We can now insert the functional form 8. Continuing as in the solution to Example 8. Division of equation 8. The solution to equation 8. Strictly, we have not proved this, and we should consider the second derivative of L u, b.
As is clear from equation 8. The maximum severity of ruin is discussed in Picard , while Section 8. For an alternative approach and solution to the inversion problem, see Dickson et al. Section 8. For a discussion of other advanced 8. The concept of dual events is discussed by Feller , while tables of Laplace Transforms can be found in Schiff The insurer initially calculates its premium with a loading factor of 0.
What conclusion can be drawn from these calculations? Use equation 8. Consider the dividends problem of Section 8. First, in Section 9. Second, in Section 9. From Section 4. Thus, by formula 3. If we consider a as a function of A, we see that a is an increasing function of A. This sim- ply states that as the price of reinsurance increases, the insurer should retain a greater share of each claim. Although they are intuitively appealing, they also have limitations. As the analysis is based on an exponential utility function, the premium that the insurer receives to cover the risk does not affect the decision.
However, if we assume that the reinsurance premium is paid from the premium income that Reinsurance the insurer receives, then it seems unreasonable that the premium income does not affect the decision. This point is addressed in the next section where we consider the effect of reinsurance on a surplus process.
We remark that in Sections 9. These principles provide solutions for optimal retention levels that are expressed in simple forms in terms of the parameter of the utility function and the parameter of the reinsurance premium principle, and are thus suitable to illustrate points. However, other premium principles can equally be used for the reinsurance premium, and some of these are illustrated in the exercises at the end of this chapter.
Then we consider the situation under both proportional and excess of loss reinsurance. Our objective is to compare excess of loss reinsurance with retention level M, under which the insurer pays min X, M when a claim occurs, with any reinsurance arrangement given by a rule h.
An im- portant point to note about equation 9. Then, as shown in Fig. This assumption may not always be borne out in practice. Table 9. It is clear from Table 9. Example 9. Solution 9. In the above example, the reinsurance arrangement is effectively a risk shar- ing arrangement with the premium income and claims being shared in the same proportion by the insurer and the reinsurer.
We can see that in equation 9. Hence we must consider the net of reinsurance 9. From Example 7. Differentiation gives R. Figure 9. We also note that the same value for R a can occur for two different values of a. An obvious criterion to adopt Reinsurance 0. Figures 9. A third possible shape is that R a is an increasing function of a. In this case the cost of reinsurance outweighs the reduction in claim variability caused by reinsurance. In all our numerical illustrations so far in this section, we have considered exponential individual claimamounts, where the mean individual claimamount is 1.
For this individual claimamount distribution it follows fromformula 7. In this case the same ideas that apply to proportional reinsurance also apply. To illustrate ideas, for the remainder of this section let the individual claim amount distribution be exponential with mean 1.
Then, adapting equation 9. Dispatched from the UK in 1 business day When will my order arrive? Mark S. David C. Paul Sweeting. Edward W. Angus S. Piet De Jong. David Hindley. Roger J. Yiu-Kuen Tse. Eric Bolviken. Home Contact us Help Free delivery worldwide. Free delivery worldwide. Bestselling Series. Harry Potter.
Popular Features. New in International Series on Actuarial Science: Description The focus of this book is on the two major areas of risk theory: For aggregate claims distributions, detailed descriptions are given of recursive techniques that can be used in the individual and collective risk models.
For the collective model, the book discusses different classes of counting distribution, and presents recursion schemes for probability functions and moments. For the individual model, the book illustrates the three most commonly applied techniques.
Beyond the classical topics in ruin theory, this new edition features an expanded section covering time of ruin problems, Gerber-Shiu functions, and the application of De Vylder approximations. Suitable for a first course in insurance risk theory and extensively classroom tested, the book is accessible to readers with a solid understanding of basic probability.