4.3 Calibration of the Pricing Model

We conducted empirical studies for the PCS data obtained from Property Claim Services. ISO's Property Claim Services unit is the internationally recognized authority on insured property losses from catastrophes in the United States, Puerto Rico and the U.S. Virgin Islands. PCS investigates reported disasters and determines the extent and type of damage, dates of occurrence and geographic areas affected (Burnecki, Kukla, and Weron; 2000). The data, see Figure 4.1, concerns the US market's loss amounts in USD, which occurred between $ 1990$ and $ 1999$ and adjusted for inflation using the Consumer Price Index provided by the U.S. Department of Labor. Only natural perils like hurricane, tropical storm, wind, flooding, hail, tornado, snow, freezing, fire, ice and earthquake were taken into consideration. We note that peaks in Figure 4.1 mark the occurrence of Hurricane Andrew (the 24th August 1992) and the Northridge Earthquake (the 17th January 1994).

In order to calibrate the pricing model we have to fit both the distribution function of the incurred losses $ F$ and the process $ M_t$ governing the flow of natural events.

The claim size distributions, especially describing property losses, are usually heavy-tailed. In the actuarial literature for describing such claims, continuous distributions are often proposed (with the domain $ \mathbb{R_+}$), see Chapter 13. The choice of the distribution is very important because it influences the bond price. In Chapter 14 the claim amount distributions were fitted to the PCS data depicted in Figure 4.1. The log-normal, exponential, gamma, Weibull, mixture of two exponentials, Pareto and Burr distributions were analysed. The parameters were estimated via the Anderson-Darling statistical minimisation procedure. The goodness-of-fit was checked with the help of Kolmogorov-Smirnov, Kuiper, Cramér-von Mises and Anderson-Darling non-parametric tests. The test statistics were compared with the critical values obtained through Monte Carlo simulations. The Burr distribution with parameters $ \alpha= 0.4801$, $ \lambda= 3.9495\cdot 10^{16}$ and $ \tau= 2.1524$ passed all tests. The log-normal distribution with parameters $ \mu= 18.3806$ and $ \sigma= 1.1052$ was the next best fit.

A doubly stochastic Poisson process governing the occurrence times of the losses was fitted by Burnecki and Kukla (2003). The simplest case with the intensity $ m{}_s$ equal to a nonnegative constant $ m$ was considered. Studies of the quarterly number of losses and the inter-occurence times of the catastrophes led to the conclusion that the flow of the events may be described by a Poisson process with an annual intensity of $ m=34.2$.

The claim arrival process is also analysed in Chapter 14. The statistical tests applied to the annual waiting times led to a renewal process. Finally, the rate function $ m_s^1 = 35.32 + 2.32\cdot 2\pi\sin\left\{2 \pi (s - 0.20)\right\}$ was fitted and the claim arrival process was treated as a non-homogeneous Poisson process. Such a choice of the intensity function allows modelling of an annual seasonality present in the natural catastrophe data.

Baryshnikov, Mayo, and Taylor (1998) proposed an intensity function of the form $ m_s^2 = a+b\sin^2\left\{2\pi(s+S)\right\}$. Using the least squares procedure (Ross; 2001), we fitted the cumulative intensity function (mean value function) given by $ \mathop{\textrm{E}}(M_s)=\int_0^s m_z dz$ to the accumulated quarterly number of PCS losses. We concluded that $ a=35.22$, $ b=0.224$, and $ S=-0.16$. This choice of the rate function allows the incorporation of both an annual cyclic component and a trend which is sometimes observed in natural catastrophe data.

It appears that both the mean squared error (MSE) and the mean absolute error (MAE) favour the rate function $ m_s^1$. In this case MSE = $ 13.68$ and MAE = $ 2.89$, whereas $ m_s^2$ yields MSE = $ 15.12$ and MAE = $ 3.22$. Finally the homogeneous Poisson process with the constant intensity gives MSE = $ 55.86$ and MAE = $ 6.1$. All three choices of the intensity function $ m_s$ are illustrated in Figure 4.3, where the accumulated quarterly number of PCS losses and the mean value functions on the interval $ [4,\;6]$ years are depicted. This interval was chosen to best illustrate the differences.

Figure 4.3: The aggregate quarterly number of PCS losses (blue solid line) together with the mean value functions $ \mathop {\textrm {E}}(M_t)$ corresponding to the HP (red dotted line), NHP1 (black dashed line) and NHP2 (green dashed-dotted line) cases.

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