Application of Jump Diffusion Models in Insurance Claim Estimation

Abstract

We investigated if general insurance claims are normal or rare events through systematic, discontinuous or sporadic jumps of the Brownian motion approach and Poisson processes. Using firm quarterly data from March 2010 to December 2018, we hypothesized that claims with high positive (negative) slopes are more likely to have large positive (negative) jumps in the future. As such, we expected salient properties of volatile jumps on the written products/contracts. We found that insurance claims for general insurance quoted products cease to be normal. There exist at times some jumps, especially during holidays and weekends. Such jumps are not healthy to the capital structures of firms, as such they need attention. However, it should be noted that gaps or jumps (unless of specific forms) cannot be hedged by employing internal dynamic adjustments. This means that, jump risk is non-diversifiable and such jumps should be given more attention.

1. Introduction

Insurance claim jumps are irregularities of the claims frequency from the policyholder to the insurer. They are crucial and fundamental in the understanding and tackling of insurable risks. We therefore, explore insurance claim jumps in general insurance products. Specifically, we investigate whether claims are associated with systematic, sporadic or discontinuous jumps or they undergo through a normal process. Our aim was to explore if insurance claims are rare or normal events using the Brownian motion model and Poisson processes in testing diffusion and jump risk. The second aim was to explore how the identified jumps affect the company’s solvency status. We put forward that, the knowledge of claim jumps is useful in proper pricing of products and better claim reserve calculations. We hypothetically state that, persistent claim jumps lead to the ruin problem. Furthermore, we conjectured that claims with high positive (negative) slopes are more likely to have large positive (negative) jumps in the future. A mismatch between liabilities and assets is central to insurance. High frequency claims exhibit fat-tailed distributions (excess kurtosis), skewness and are in most cases clustered together. We can say, the infrequent movements of large magnitude in claim counts are attributed to sudden-jumps that we want to really explore in this study.

Literally, diffusion models are tools used to describe the movement, decay and evolution of products or items in a given environment over a specified period. The variables are normally random in nature. The general application of diffusion processes is to describe the evolution of asset/product’s behavior over time in terms of their prices or returns. In finance, we see the application of the models in explaining the evolution of asset returns [1, 2]. Randomness and persistence are two salient properties of claim jumps and volatility. Jump diffusion models are applied in the financial arena to estimate stock volatilities of both prices and returns [3, 4, 5, 6]. The statistical properties of claim amounts have long been of curiosity to insurers and actuaries in pricing and risk management. Higher order expectations are less considered as much of the information about any financial or insurance data is believed to have been carried by the standard arithmetic mean and standard deviations. However special cases like positive kurtosis implies concave, U-shaped implied Black-Scholes volatility (IV) curves. Practitioners rarely do taking higher order expectations in statistical distributions such as the Gaussian, Binomial, and Poison and so on.

Claimant distributions are Longley modeled using the compound poison and gamma distributions where the former captures the frequency and the latter captures the severity of the claims [7]. No serious attention was exerted to the jumps associated with the insurance claims and the jump effects too. In general, insurance where the frequency of claim arrival is high, jump analysis is quite necessary utmost for economical reserving and capital solvency.

In option pricing, the use of Black and Scholes-type formulae is considered to price European options on written underlying assets such as stocks, foreign currencies, commodities and interest rates. We therefore intend to apply the taste of diffusion models in modeling the evolution and behavior of insurance claims for the written non-life products. The Jump Diffusion model chosen in this study can potentially explain the evolution of claims and its behavior (frequency and severity) more accurately at the expense of making the market incomplete, since jumps in premiums cannot be hedged easily. The reason behind the existence of claim reserves such as the unearned premium reserves is a key indication of the jumps in premium payments. However, underestimation is commonly burdening insurers. Underestimation is a tendency of deriving and providing values, which are excessively low and unfavorable. In our context, underestimation negatively affects the insurer’s capital structures and reserving. Thus, jump is indeed an important aspect that should be taken into consideration at regular times. We do this using the Gaussian, Poison model and by extending Merton’s [1, 8] jump-diffusion model, which we presented in our methods section. The remainder of our paper is organized as follows. The next section generalizes our jump diffusion models to a firm level. Empirical tests for the presence of jump components in the claims are contained in Section 3. Section 4 concludes the paper.

2. Materials and tools

The study employed the model contained in [1, 5]. We interpret the model as the one which contains a finite number of insurance contracts and insurers and insured. The model is based on the following assumptions:

1. No transaction costs, no taxes, and frictionless insurance markets.

2. Competitive markets (insurers are price (premiums) takers)

3. Continuous trading at equilibrium prices (premiums)

4. There are m risky contracts whose premiums and claims satiate

$$\frac{d{C}_j}{C_j}={\alpha }_j\mathit{dt}+{\sigma }_{j}d{Z}_j+\left(-{\lambda }_j{K}_j\mathit{dt}+{\pi }_{j}d{Y}_j\right),j=1:m\dots .$$E1

where $C_j\left(t\right)$is the claim amount of a contract $j$ at time $t$; ${\alpha }_j$, ${\sigma }_j$, ${\lambda }_j$, and $K_j$ are constants where ${\alpha }_j$ and ${\sigma }_j$ are the drift and diffusion components respectively; $d{Z}_j$ is a Wiener process; $d{Y}_j$ is a Poisson process with parameter ${\lambda }_j$; ${\pi }_j$ is the jump amplitude with expected value equal to $K_j$; and $d{Z}_j$, $d{Y}_j$, and ${\pi }_j$ are independent.

1. Further, Insurers have standardized opinions over {${\alpha }_j$, ${\sigma }_j$, ${\lambda }_j$, $K_j$, j = 1 … m}

2. Insurers’ and insured’s tastes are represented by a von Neumann-Morgenstern utility functional theory which is strictly increasing and strictly concave. All our assumptions except 4 are found commonly in literature [1, 5]. Assumption number 4 is the key conjecture in our analysis.

We now rewrite assumption 4 in an equivalently alternate way that separates systematic and unsystematic risk components.

Consider the diffusion part of assumption 4,

$$d{D}_j={\alpha }_j\mathit{dt}+{\sigma }_{j}d{Z}_j;j=1;\dots ;m\dots .$$E2

Following the argument from Ross [9], expression (2) implies that there exists.

{$u_j$, $f_j$, $g_j,\mathit{dØ},d{W}_j\};$ j = 1, … m, such that

$$d{D}_j={\alpha }_j\mathit{dt}+f_{j}d\mathrm{Ø}+g_j{\mathit{Wd}}_j;j=1;\dots ;m\dots .$$E3

where ${\mathit{fj}}_2+g_j^2={\sigma }_j^2$; $d\mathrm{Ø}$, $d{W}_j$ are Wiener processes;

$$E\left[d\mathrm{Ø}d{W}_j\right]=0,j=1,\dots ,m;$$

and,

$$\sum _{j=1}^m{u}_j=1,{\left(x+a\right)}^n=\sum _{j=1}^m{u}_j\left(g_{j}d{W}_j\right)=0=\sum _{j=1}^m{u}_j{\alpha }_j>r\dots$$E4

It is always likely to decompose a restricted number of normal arbitrary variables into a common factor, $\mathit{dØ}$, and error terms, $d{W}_j$, which are normally distributed. The key property of normal claims employed is that covariance of zero implies numerical independence. This same assumption is confirmed in asset prices and returns analysis [10]. Note that $\mathit{dØ}$, $d{W}_j$ will be independent of $d{Y}_j$ and ${\pi }_j$ by assumption 4. This disintegration gives $\mathit{dØ}$ the interpretation of being the unsystematic risk factor.

Substitution of expression (3) and (4) into (1) gives assumption 6: There are m risky insurance contracts whose claims satisfy:

$$\frac{d{C}_j}{C_j}={\alpha }_j\mathit{dt}+f_{j}d\mathrm{Ø}+{\mathrm{g}}_{\mathrm{j}}d{W}_j+\left(-{\lambda }_j{K}_j\mathit{dt}+{\pi }_{j}d{Y}_j\right),j=1,\dots ,m\dots$$E5

where $C_j\left(t\right)$ is the claim of a contract $j$ at time $t$; ${\alpha }_j$, $f_j$, $g_j,{\lambda }_j$, $K_j$ are constants; $\mathit{dØ}$, $d{W}_j$ are Wiener processes; $d{Y}_j$ is a Poisson process with parameter ${\lambda }_j$; ${\pi }_j$ is the jump amplitude with expected value equal to $K_j$; and $\mathit{dØ}$, $d{W}_j$, $d{Y}_j,{\pi }_j$ are independent. The jump component in expression (5), $\left(-{\lambda }_j{K}_j\mathit{dt}+{\pi }_{j}d{Y}_j\right)$, infers that insurance claims can have discontinuous ample paths. This generalizes existing models.

3. Data and model

The section tests the written insurance contracts claims to see if they contain jumps. If no jump component is present, then this would be consistent with the proposition of the previous deduction. In addition, it implies that the claims are normal events. Thus, the satisfaction of instantaneous claim reserves calculation frameworks such as the Chain ladder method and pricing models (collective risk model). We used the written insurance contacts and the recorded claims for a period spanning from March 2010 to December 2018. We performed the following hypothesis tests:

$H_0$, jump risk is diversifiable.

$H_1$, jump risk is non-diversifiable.

From the above hypothesis, we will see whether jump risk leads to capital insolvency for insurance firms. We will survey the sample path of the claims. To advance the testing procedure, note that under expression (5) the insurance claims dynamics are given by:

$$\frac{\mathit{dC}}{C}=\sum _{j=1}^m{C}_j{\alpha }_j\mathit{dt}+\left(\sum _{j=1}^m{C}_j{f}_j\right)d\mathrm{Ø}+\sum _{j=1}^m{C}_j\left(g_{j}d{W}_j-{\lambda }_j{K}_j{\mathit{dt\pi }}_{j}d{Y}_j\right)+\mathit{log}{V}_j$$E6

Where, $C={\sum }_{j=1}^n{m}_j{C}_j$, $\mathit{log}{V}_j\sim i.i.d.N\left(\alpha ,{\sigma }^2\right),$ normally distributed and models jumps in claims. Under the null hypothesis, expression (6) reduces to:

$$\frac{\mathit{dC}}{C}=\mathit{\alpha dt}+\mathit{\sigma d}\mathrm{Ø}\dots$$E7

Where, $\alpha ={\sum }_{j=1}^n{m}_j{\alpha }_j$ and $\sigma ={\sum }_{j=1}^n{m}_j{f}_{j.}$

Under the alternative hypothesis, expression (6) reduces to:

$$\frac{\mathit{dC}}{C}={\alpha }^{\prime }\mathit{dt}+\mathit{\sigma d}\mathrm{Ø}+\mathit{dq}\dots$$E8

where $\mathit{dq}=\mathit{\pi dY}$ denotes a Poisson process with parameter $\lambda$, $\pi =$ jump amplitude with estimated value equal to $K$, and $\alpha ’=\alpha —\mathit{\lambda K}$.

Another assumption is added to (8), that is, $\left(\pi \right)$ has a lognormal distribution with parameters ($a,b^2$). We add this assumption to easy up the Maximum Likelihood Estimation procedure in estimating the parameters of Eqs. (7) and (8).

Now, we conveniently re-write the hypothesis to be tested as follows:

$H_0$, jump risk is diversifiable

$$\frac{\mathit{dC}}{C}=\mathit{\alpha dt}+\mathit{\sigma d}\mathrm{Ø}\dots .$$E9

$H_1$, jump risk is non-diversifiable

$$\frac{\mathit{dC}}{C}={\alpha }^{\prime }\mathit{dt}+\mathit{\sigma d}\mathrm{Ø}+\mathit{dq}\dots .$$E10

and $\left(\pi \right)$ is dispersed lognormal $\left(a,b^2\right)$

Now, to properly test the above stated null hypothesis, a likelihood ratio test can be used: $A=-2\left(\mathit{ln}{L}_r—\mathit{ln}{L}_u\right)$, where $L_r$ is the likelihood value for the reserved density function (i.e., the null hypothesis, Eq. (9)) and $L_u$ signifies the likelihood function for the unconstrained density function (i.e., the alternative hypothesis, Eq. (10)).

Table 1 presents estimates of parameters of the diffusion-only process for diverse observation intervals and time periods. The results suggest that the total claims frequency and severity are not constant over time. The total standard deviation of claims on the firm is measured by the total claims index over 8-year period.

Component	Statistic value
Constant drift	0.21%
Drift deviance	2%
Probability jumps	4%
Mean of jumps	3%
Jump deviance	5%

Table 1.

Diffusion process parameter estimates.

Table 1 is a summary of the parameter estimates of the diffusion model used over a time horizon for a basket of diversified observations. Having the parameter values the jump-diffusion model can be safely used to infer the likely consequences of the claim jumps towards an efficient insurance engineering. The jump probability is our spanner for dealing with ruin issues and proper reserve estimations. Jump deviance is the standard deviation of the jumps, which gives the spread of the claim jumps (positive or negative) over time for the written contracts.

4. Methodology

Throughout this paper we assume that $C_t$ to be the claim amount of each insurance contract at time $t,$ whose dynamics are given by;

$$\frac{d{C}_t}{C_t}=\left(\mu -\mathit{\lambda \kappa }\right)\mathit{dt}+\mathit{\text{σdBt}}+\left[e^J-1\right]\mathit{dNt},,,,,,$$E11

where $\mu ,$ is the instantaneous expected claim amount per unit time, and $\sigma$ is the instantaneous volatility per unit time. The stochastic process $B_t$ is a standard Wiener process under the market measure $P$. The process $N\left(t\right)$ is a Poisson process, independent of the jump-sizes $J$ and the Wiener process $\mathit{Bt},$ with arrival intensity $\lambda$ per unit time under the measure $P$, so that its increments satisfy the following:

$$d\left(\mathit{Nt}\right)=\left\{\begin{array}{c}1,,,,,,,\mathit{\text{with probability}}\left(\mathit{\lambda dt}\right)\\ 0,,\mathit{\text{with probability}}\left(1-\mathit{\lambda dt}\right)\end{array}\right\},,,,,,$$E12

The expected proportional jump size is;

$$\kappa =E\left[e^J-1\right],,,,,,.$$E13

In this study, jumps are assumed independent of each other as they arrive at different times. We then defined an information set through a filtered probability measure space $\left(\Omega ,F,\left\{\mathit{\text{Ft}}\right\},P\right)$, where the filtration $\left\{\mathit{\text{Ft}}\right\}$ is the natural filtration generated by the Wiener process $\mathit{Bt}.$ In the jump-diffusion model, the insurance claims $C_t$ are defined to follow the random process given by:

$$\frac{d{C}_t}{C_t}=\mathit{\mu dt}+\mathit{\sigma d}{W}_t+\left(J-1\right)d{N}_t,,,,,.$$E14

The first two terms are familiar from the Black Scholes [11] model: The drift rate $\mu ,$ volatility $\sigma ,$ and random walk (Wiener process), $\mathit{Wt}$. The last term represents the jumps: $J$ is the jump size and $N\left(t\right)$ is the number of jump events that have occurred up to time $t$. $N\left(t\right)$ is assumed to follow the Poisson process;

$$P\left(\left(\mathit{Nt}\right)=k\right)=\frac{{\left(\mathit{\lambda t}\right)}^k}{k!}{e}^{-\mathit{\lambda t}},,,,,$$E15

where, $\mathit{\lambda t}$ is the average number of jumps per unit time. Note that, there is no specific distribution for the jump sizes. However, a common choice is a log-normal distribution given as:

$$J\sim m{e}^{\frac{y^2}{2}+\mathit{vN}\left(0,1\right)},,,,,$$E16

where $N\left(0,1\right)$ is the standard normal distribution, $m$ is the average jump size, and $v$ is the volatility of jump size. The key parameters that characterize the jump-diffusion model are $\lambda ,m,v$.

5. Model

We use the basic excel spreadsheet to model the effects of jump-diffusion on claims and the respective reserves. Our equation is as follows:

$$r_t=\alpha +{\epsilon }_t+I_t{u}_t,,,,,$$E17

where $r_t$ is the log claim amount, $\alpha$ is the mean drift, ${\epsilon }_t$ is the diffusion which follows a normal distribution calculated as $\sigma \ast \mathit{\text{NORMSINV}}\left(\mathit{\text{RAND}}\left(\right)\right)$, where $\sigma$is the standard deviation of the jumps, $I$ is the indicator variable (0 or 1), for either absence or presence of the claim jump. The value is determined by the jump probability; $u_t$ is the value of the jump. This follows a normal distribution and is determined by

$$\mathrm{E}\left[\mathrm{u}\right]+{\mathrm{\sigma }}_{\mathrm{u}}\ast \text{NORMSINV}\left(\mathrm{RAN}\left(\right)\right),$$

where $E\left[u\right]$ and ${\sigma }_u$ are the mean and standard deviation of the jump, respectively. Space is too much

6. Results

Using Excel Visual Basic for Applications (VBA) and R scrip, we perform our analysis, using the calibrated parameters. Parameter calibration was done using the maximum likelihood approach. Concisely, we presented the model results in a nicely and user-friendly manner. The user can only enter the input values on the designed user form and click ‘run’. The inputs included are the sigma (volatility value that we normally called the implied volatility), the risk free interest rate, time component (T), the number of paths for simulations (we used 174 for our case, but can be varied). Table 2 summary is in the subsequent tables in the Appendices section. The number of jumps are then estimated and modeled within the selected paths number and period. Ones denote jumps, otherwise they are normal claim movements.

7. Discussions

We tested whether or not there are systematic jumps insurance claims or they are normal events. We found that insurance claims for general insurance quoted products cease to be normal. There exist at times some jumps, especially during holidays and weekends. Such jumps are not healthy to the capital structures of firms, as such, they need attention. However, it should be noted that gaps or jumps (unless of specific forms) cannot be hedged by employing internal dynamic adjustments. This means that, according to our hypothesis tested, jump risk is non-diversifiable. A firm can manage jump-induced risks by buying options. Option derivatives help the firm to protect it against negative jumps and its consequences on its capital status. If, however, it establishes its own reserves, it must ensure and enforce a dynamic reserve adjustment. The reserves must increase as position values fall. This is an alternative option. The insurers must bear in mind the cushion, so to speak, that is dynamic. However, by dynamically hedging its own capital account, the insurer cannot wholly protect itself. Gaps or jumps are truly difficult to hedge; we thus need an idea of option hedging.

8. Conclusion

This paper develops and tests sufficient conditions for a model when insurance claims follow a jump-diffusion process. Based on weekly claims data, our results are that the reported claims contain a jump component, with a slightly high magnitude. We measure the jump component over both short (monthly and larger intervals in time (quarterly interval) and find that the weekends and holidays tend to cover up the high jump component. The economic intuition is that jump risk is not diversifiable and hence can ruin the firm leading to capital insolvency.

Table 2.

Claim amounts statistics, paths and jump forecasts.