Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Shot-Noise Cox Process

In this paper, we introduce tail dependene measures for collateral losses from catastrophic events. To calculate these measures, we use bivariate compound process where a Cox process with shot noise intensity is used to count collateral losses. A homogeneous Poisson process is also examined as its counterpart for the case where the catastrophic loss frequency rate is deterministic. Joint Laplace transform of the distribution of the aggregate collateral losses is derived and joint Fast Fourier transform is used to obtain the joint distributions of aggregate collateral losses. For numerical illustrations, a member of Farlie-Gumbel-Morgenstern copula with exponential margins is used. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are also provided.


Introduction
Over the recent years, numerous papers have looked at the modelling of dependence within an insurance portfolio or between insurance portfolios [1][2][3][4][5].Also in the field of financial risk management, a range of papers on dependence modelling within credit risk and operational risk can be noticed [6][7][8].Besides the construction of specific multivariate models, considerable attraction is given to the use of copulas.In particular, within the theory of Lévy processes, Lévy copulas have proven to be useful [9].
Our paper is very much based on insurance applications where two lines of business are hit by a common external event, hence the word "collateral losses" in the title.These joint losses may, for instance be triggered by events such as flood, windstorm, hail, bushfire, earthquake and terrorism.Particular examples concern collateral losses due to 2011 Great Eastern Japan Earthquake, 2010-2011 Queensland floods, 2009 Victorian Bushfires [10], 2005 Hurricane Katrina [11] and 2001 September 11 attack [12].
For the purpose of this paper, we concentrate on a very specific model and show how, within this model several explicit calculations for relevant risk quantities can be performed.The bivariate model we consider has the following structure: where is the total loss arising from risk type and t is the number of collateral losses up to time .The random variables i N t X and , 1,2, denote the individual loss amounts.In this model, the dependence between two random variables and comes from the common arrival process t , together with the dependence between the individual losses i X and i .The latter is modelled using the notion of copula [13].To be more specific, we assume the loss random variable i Y X and i Y are independent identically distributed with continuous distribution function X F and Y F respectively.The joint distribution of the vector   , X Y  is assumed to be of the form with a given copula .The uniqueness of this two stage construction goes back to Sklar's Theorem.F .Proof.See Schweizer and Sklar [14] or Nelsen ([13], p. 18).
To deal with stochastic nature of catastrophic loss arrival in practice, we use a Cox process for t .The Cox process provides flexibility by letting the intensity not only depend on time but also allowing it to be a stochastic process.Therefore the Cox process can be viewed as a two step randomisation procedure.A process t N  is used to generate another process t by acting its intensity.That is, t is a Poisson process conditional on N N t  which itself is a stochastic process.
Losses arising from a catastrophe depend on its intensity.One of the processes that can be used to measure the impact of catastrophic events is the shot noise process.Previous works of insurance application using shot noise process and a Cox process with shot noise intensity can be found in [15][16][17][18][19]. Reference [20] also used a Cox process with shot noise intensity to model operational risk.The shot noise process is particularly useful to loss arrival process as it measures the frequency, magnitude and time period needed to determine the effect of catastrophic events.As time passes, the shot noise process decreases as more and more losses are settled.This decrease continues until another event occurs which will result in a positive jump in the shot noise process.Therefore the shot noise process can be used as the parameter of a Cox process to measure the number of catastrophic losses, i.e. we will use it as an intensity function to generate a Cox process.We will adopt the shot noise process used by Cox & Isham [21]:  that is carried on from catastrophic events incurred previously; Z   is a sequence of independent and identically distributed random variables with distribution function and (i.e.magnitude of contribution of catastrophic event to intensity); S   is the sequence representing the event times of a Poisson process t M with constant intensity  ; and   is the rate of exponential decay.
Catastrophic events may take long to materialise so the decay rate may not be exponential.It is assumed to be of this form for a matter of convenience, i.e. closed-form expressions of final results are easily derived.We also make the additional assumption that a Poisson process t M and the sequences   1,2, A Pois ate son process with loss frequency r  is also st alculate the fo udied for t N , that may be considered when catastrophic loss f uency rate is deterministic.
With the above model specifications, we c req llowing relevant risk measures: , , Here and respectively.The quantities ) are asymptotic upper tail dependence measures and the quantities (4) and (6) are conditional tail expectations.The motivation for calculating these quantities that measure extremal dependence in the upper tail of a bivariate distribution is that insurance industry is more concerned with dependence between extreme losses.For a discussion on the coefficient of tail dependence parameters, see McNeil et al. [22].
In order to evaluate above risk m known as tain the joint distribution of the aggregate collateral losses   L  , which are required to improve t racy of the distributions of the aggregate collateral losses inverting joint Fast Fourier transforms.We also provide the figures of the joint dis-tribution of the aggregate collateral losses and their contours.In Section 5, we illustrate the calculations of relevant risk measures (3)-(6) using joint Fast Fourier transforms.For numerical illustrations, an exponential distribution for the jump sizes of catastrophic event, a member of Farlie-Gumbel-Morgenstern copula with exponential margins are used throughout the paper.Section 6 shows the sensitivity analysis on the parameters of a shot-noise Cox process from a Poisson process using the risk measure of (4).Concluding remarks are in Section 7.

Joint Laplace Transform of the llateral
where . Now let us find a suitable martingale in order to derive joint Laplace transform of the dis-Lemma 2.1.Considering constants tribution of the aggregate collateral losses   martingale, where is a , , e solution is s (10) ich the result follows.ined in Lemma 2.1 and sett-and th by wh we can easily obtain the general form int Laplace transform of the distribution of the aggregate collateral losses   where the conditional expectation is based on the ex E E probability space   , ,P   , and t information set he with th e filtration then it is given by roughout the paper, we firstly assume that jump Th .If loss X sizes of catastrophic event follow an exponential distribution is stationary [16], (12) i given by .Then using Theorem 2.6 in s occurs by arriva noise in ity Secondly, as a specific example for , we use the Fa as 4) where ing expression for the above astly, to make t e sume that joint Laplace transform of the distribution of the aggregate collateral losses as it can be easily obtained using the joint distribution function   , F x y driven by (14) It will be of interest to examine the joint Laplace transfo rm of the distribution of the aggregate collateral losses   and using (14) we have We omit the corresponding expressions for the Laplace transform of the distribution of , w ar    and Differentiating (12) w.r.t. and  and set 0   and we can derive the joint e The higher moments of and at time t can be obtained by differentiating it further, i.e.
The covariance between and at time is given by (25) and the linear correlation coefficient between and at time is given by Let us now illustrate the calculations of the covariance nd linear correlation between and at time and using (14) we have and the linear correlation coefficient between   1 and The parameter values used to calculate the covariance and linear correlation using (25) and (26) are

Homogeneous Poisson Process
The parameter values used to calculate the covariance and linear correlation using (28)

Collateral Losses via Bivariate Fast Fourier Transform
In order to calculate the risk measures of (3)-(6), we inver obtained in Section er transfo calculatio we present the ex-2.For details on how to use bivariate Fast Fouri rm, we refer to [26][27][28].Before we show the ns of risk measures in Section pressions for the joint probabilities of the aggregate collateral losses and their densities at  

Shot-Noise Cox Process
If we let p    and    in (13), w e the on for the joint probability of aggregate collateral losses at   Regardless of loss size distributions, we have the same joint probability of aggregate collateral losses at   the expression for the joint density of aggregate collateral losses at ss sizes (14), i.e Based on (35), we can easily obtain its expression for exponential lo using .  , which means loss X and Y move in the same direction.On the other hand, compared to when 1 Regardless of loss size distributions, we have the same ability of agg egate co sses at   expression for the joint density of aggregate collateral sses at and is given by 2 e , 2 2 Based on (38), we can easily obtain its expression for exponential loss sizes using (14), i.e.To reduce the error of used algorithm for inverting bivariate Fast Fourier transforms using Matlab, the following methods have been used: e

Now using the parameter values in Examples
Matlab, let us illustrate the c  The sampling points have been taken as many as possible, i.e. we have used 8192 × 4096 points in th calculation, which was the maximum points we could sampled in 32-bit Matlab. If the sampling interval gap is too small, there would be a large amount of truncation error.Also if the sampling interval is too big, there would be a large .amount of sampling error.Hence we have chosen carefully the right sampling interval, so that the total errors caused by inverting bivariate Fast Fourier transforms could be negligible  For the figures in this paper, we have used the 2D  low-pass/moving-average filter to filter the jitter noise out.Then we have used the linear interpolation to smooth the borders of the data.The pattern matrix used for this filter was

Example 3
Using the different at and the calculations of th Tables 5-8.
Tables 5-8 show that each quantile values are higher with respect to a shot-noise Cox process than their counterparts.They also show that the risk measures of (3) with respect to a shot-noise Cox process are higher than e aggregate collateral losses with re--noise Cox process have heavier tail than th respect Poisson pro % q VaR e risk % 95% q  measure of (3) are shown i 99% , n their counterparts.These justify that the marginal/joint distributions of th spect to a shot eir counterparts with to a cess.

Example 4
Secondly, based on the % q VaR at 95% and 99% in Example 3, the calculations of the risk measure (4), i.e.
are shown in Tables 9 and 10.Tables 9 and 10 show that the risk measures of (4) with respect to a shot-noise Cox process are higher than their counterparts regardless of the critical values.Regardless of the loss arrival process t N , the risk measures of ng bigger as the critical value goes to 99%.(4) are getti Th rences 9 Example 5 ey also show that the diffe between the values in Table 9 and their counterparts in Table 10 are getting higher as the critical value goes to 9 %.

Showing the calculations of the quantiles of the
and the joint probability be- for each cases in 4, the calculations of the ris Tables 11 14.79 29.58 0.01 .
14.79, 29.58 Table 13.Table 16.Ta 12 e go th shot-noise thei ard a t the critical value goes to 99.9%.y also show that the dif between the values in Table 13 and their counterparts in Table 14 are getting lower as the critical value goes to 99.9%.
Tables 15 and 16 show that the risk measures of (5) with respect to a shot-noise Cox process are higher than their counterparts regardless of the critical values.Regardless of the loss arrival process N , the risk m N t easures (5) are getting smaller as the critical value goes to 99.9%.
Lastly, based on the quantile values and the joint probabilities in Example 5, the calculations of the risk measure (6), i.e.
shown in Tables 17 and 18. Tables 17 and 18 show that the risk measures of (6) with respect to a shot-noise Cox process are higher than their counterparts regardless of the quantile values.Regardless of the loss arrival process t N , the risk measures of (6) are getting bigger as the critical value goes to 99.9% .They also show that the differences between the values in Table 17 and their counterparts in

Sensitivity Analysis
In this section, we examine the effec on the risk measure of (4), i.e.          19 shows  , which is the parameter in exponential jump size distribution of catastrophic event, is the most sensitive parameter in terms of changes in the value the risk measure of (4).

Conclusions
We have used bivariate compound process to model aggregate collateral losses arising from catastrophic er of collateral losses, a Cox process was sed to accommodate the stochastic nature of their frequency rate in practice.The shot noise s used as the intensity of a Cox process as the number of collateral lo ses rising from ents depends on the frequency and magn mary events and the time period needed to determine the effect of these events.o examined a Poiss process for the number of collateral losses as its counterpart.With the common collateral loss arrival process in the bivariate odel, the dependence between the individual losses om d not f copula where for numerical illustrations, a m collateral tween surance premiums.We have presented the expressions for joint probabilities of the aggregate collateral losses and their densities at   1 0 t L  and , which were used to improve the of joi utions of the aggregate collateral losses inverting the Fast Fourier transforms.We have compared the simulated risk measure values obtained using a compound Poisson and a compound shot-noise Cox model, respectively.The s ess also provided using the isk measure of (4).
Other counting processes and other copulas with different margins can be considered in the proposed bivariate model, that we leave for further research.We hope that what we have presented in this paper provides practitioners with feasible models to quantify collateral losses that would occur more often due to global warming, climate change and terrorism.
These are required to im rove the accuracy of the joint distributions of the aggregate collateral losses inverting bivariate Fast Fourier transforms.

1 0 1 
gure 4 (or Figure3) shows that joint probabilities of aggregate collateral losses at the bottom left corner and the top right corner moves to its diagonal left and right, respectively when   which means loss X and Y move in the opposite direction.
llateral losses and their contours at each value of  with respect to a Poisson process for are omitted, for which see the early version of this t N paper in http://ssrn.com/author=383758.1-2 and alculations of the risk measures of (3)-(6) with respect to a shot-noise Cox process and a Poisson process.They are shown in Tables 5-18.

Figure 4 .
Figure 4.The contour of the joint distribution of collateral losses with θ = −1.

Figure 5 .
Figure 5.The joint distribution of collateral losses with θ = 1 for a Poisson process.

Figure 6 .
Figure 6.The contour of the joint distribution of collateral losses with θ = 1 for a Poisson process.

Figure 12 .
Figure 12.The contour of the joint distribution of collateral losses with θ = 1 and γ = 0.5.

. Shot-Noise Cox Process process follows a Distribution of the Aggregate Co Losses
)

Table 1 .
Covariance between   t N tiating (17) w.r.t. and  and set t L and   2 L at time t , i.e.L at time t are own in Table 3 and Table 4 respectively.  1   t L 2 .t L 2 .

Table 18 are getting
higher With respect as the critical value goes to 99.9%.to the FGM copula correlation parameter,  , the risk measures of (3)-(6) are increasing (decreasing)as it changes to  1   Figures 7-12 are the joint distributions of aggregate collateral losses and their con- changes in the values of the parameters of a ot-noise Cox process from a Poisson process.To do so, N .