4.4 Dynamics of the CAT Bond Price

In this section, we present prices for different CAT bonds. We illustrate them while focusing on the influence of the choice of the loss amount distribution and the claim arrival process on the bond price. We analyse cases using the Burr distribution with parameters $ \alpha= 0.4801$, $ \lambda= 3.9495\cdot 10^{16}$ and $ \tau= 2.1524$, and the log-normal distribution with parameters $ \mu= 18.3806$ and $ \sigma= 1.1052$. We also analyse the homogeneous Poisson process with an annual intensity $ m=34.2$ (HP) and the non-homogeneous Poisson processes with the rate functions $ m_s^1 = 35.32 + 2.32\cdot 2\pi\sin\left\{2 \pi (s - 0.20)\right\}$ (NHP1) and $ m_s^2= 35.22+0.224\sin^2\left\{2\pi(s-0.16)\right\}$ (NHP2).

Consider a zero-coupon CAT bond defined by the payment of an amount $ Z$ at maturity $ T$, contingent on a threshold time $ \tau>T$. Define the process $ Z_s = \mathop{\textrm{E}}(Z\vert\mathcal{F}_s)$. We require that $ Z_s$ is a predictable process. This can be interpreted as the independence of payment at maturity from the occurrence and timing of the threshold. The amount $ Z$ can be the principal plus interest, usually defined as a fixed percentage over the London Inter-Bank Offer Rate (LIBOR).

The no-arbitrage price of the zero-coupon CAT bond associated with a threshold $ D$, catastrophic flow $ M_s$, a distribution function of incurred losses $ F$, and paying $ Z$ at maturity is given by Burnecki and Kukla (2003):

$\displaystyle V_t^1$ $\displaystyle =$ $\displaystyle \mathop{\textrm{E}}\left[Z \exp\left\{-R(t,T)\right\}(1-N_T)\vert\mathcal{F}{}_t\right]$  
  $\displaystyle =$ $\displaystyle \mathop{\textrm{E}}\Bigg[Z\exp\left\{-R(t,T)\right\}$  
  $\displaystyle \cdot$ $\displaystyle \left\{1-\int\limits_t^T m_s\left\{1-F(D-L_s)\right\}I(L_s<D) ds\right\}\vert\mathcal{F}{}_t\Bigg].$ (4.2)

We evaluate this CAT bond price at $ t=0$, and apply appropriate Monte Carlo simulations. We assume for the purposes of illustration that the annual continuously-compounded discount rate $ r=\ln (1.025)$ is constant and corresponds to LIBOR, $ T\in [1/4,\; 2]$ years, $ D\in [2.54,\;30]$ billion USD (quarterly - 3*annual average loss).

Furthermore, in the case of the zero-coupon CAT bond we assume that $ Z=1.06$ USD. Hence, the bond is priced at $ 3.5\%$ over LIBOR when $ T=1$ year. Figure 4.4 illustrates the zero-coupon CAT bond values (4.2) with respect to the threshold level and time to expiry in the Burr and NHP1 case. We can see that as the time to expiry increases, the price of the CAT bond decreases. Increasing the threshold level leads to higher bond prices. When $ T$ is a quarter and $ D=30$ billion USD the CAT bond price approaches the value $ 1.06\exp\left\{-\ln(1.025)/4\right\}\approx$ $ 1.05$ USD. This is equivalent to the situation when the threshold time exceeds the maturity ( $ \tau\geqslant T$) with probability one.

Figure 4.4: The zero-coupon CAT bond price with respect to the threshold level (left axis) and time to expiry (right axis) in the Burr and NHP1 case.
\includegraphics[width=1.6\defpicwidth]{STFcat04.ps}

Consider now a CAT bond which has only coupon payments $ C_t$, which terminate at the threshold time $ \tau $. The no-arbitrage price of the CAT bond associated with a threshold $ D$, catastrophic flow $ M_s$, a distribution function of incurred losses $ F$, with coupon payments $ C_s$ which terminate at time $ \tau $ is given by Burnecki and Kukla (2003):

$\displaystyle V_t^2$ $\displaystyle =$ $\displaystyle \mathop{\textrm{E}}\left[\int_t^T \exp\left\{-R(t,s)\right\}C_s(1-N_s)ds\vert\mathcal{F}{}_t\right]$  
  $\displaystyle =$ $\displaystyle \mathop{\textrm{E}}\Bigg[\int_t^T\exp\left\{-R(t,s)\right\}C_s$  
  $\displaystyle \cdot$ $\displaystyle \Bigg\{1-\int\limits_t^s m_\xi\left\{1-F(D-L_\xi)\right\}I(L_\xi<D) d\xi\Bigg\}ds\vert\mathcal{F}{}_t\Bigg].$ (4.3)

We evaluate this CAT bond price at $ t=0$ and assume that $ C_t \equiv 0.06$. The value of $ V_0^2$ as a function of time to maturity (expiry) and threshold level in the Burr and NHP1 case is illustrated by Figure 4.5. We clearly see that the situation is different to that of the zero-coupon case. The price increases with both time to expiry and threshold level. When $ D=30$ USD billion and $ T=2$ years the CAT bond price approaches the value $ 0.06\int^{2}_0 \exp\left\{-\ln(1.025)t\right\}dt\approx$ $ 0.12$ USD. This is equivalent to the situation when the threshold time exceeds the maturity ( $ \tau\geqslant T$) with probability one.

Figure 4.5: The CAT bond price, for the bond paying only coupons, with respect to the threshold level (left axis) and time to expiry (right axis) in the Burr and NHP1 case.
\includegraphics[width=1.6\defpicwidth]{STFcat05.ps}

Finally, we consider the case of the coupon-bearing CAT bond. Fashioned as floating rate notes, such bonds pay a fixed spread over LIBOR. Loosely speaking, the fixed spread may be analogous to the premium paid for the underlying insured event, and the floating rate, LIBOR, is the payment for having invested cash in the bond to provide payment against the insured event, should a payment to the insured be necessary. We combine (4.2) with $ Z$ equal to par value (PV) and (4.3) to obtain the price for the coupon-bearing CAT bond.

The no-arbitrage price of the CAT bond associated with a threshold $ D$, catastrophic flow $ M_s$, a distribution function of incurred losses $ F$, paying $ PV$ at maturity, and coupon payments $ C_s$ which cease at the threshold time $ \tau $ is given by Burnecki and Kukla (2003):

$\displaystyle V_t^3$ $\displaystyle =$ $\displaystyle \mathop{\textrm{E}}\Bigg[PV \exp\left\{-R(t,T)\right\}(1-N_T)$  
  $\displaystyle +$ $\displaystyle \int_t^T \exp\left\{-R(t,s)\right\}C_s(1-N_s)ds\vert\mathcal{F}{}_t\Bigg]$  
  $\displaystyle =$ $\displaystyle \mathop{\textrm{E}}\Bigg[PV\exp\{-R(t,T)\}$  
  $\displaystyle +$ $\displaystyle \int\limits_t^T \exp\{-R(t,s)\}\Bigg \{C_s\Bigg(1-\int\limits_t^s
m_\xi\left\{1-F(D-L_\xi)\right\}I(L_\xi<D) d\xi\Bigg)$  
  $\displaystyle -$ $\displaystyle PV\exp\left\{-R(s,T)\right\} m_s\left\{1-F(D-L_s)\right\}I(L_s<D)\Bigg\}ds\vert\mathcal{F}{}_t\Bigg].$ (4.4)

We evaluate this CAT bond price at $ t=0$ and assume that $ PV=1$ USD, and again $ C_t \equiv 0.06$. Figure 4.6 illustrates this CAT bond price in the Burr and NHP1 case. The influence of the threshold level $ D$ on the bond value is clear but the effect of increasing the time to expiry is not immediately clear. As $ T$ increases, the possibility of receiving more coupons increases but so does the possibility of losing the principal of the bond. In this example (see Figure 4.6) the price decreases with respect to the time to expiry but this is not always true. We also notice that the bond prices in Figure 4.6 are lower than the corresponding ones in Figure 4.4. However, we recall that in the former case $ PV=1.06$ USD and here $ PV=1$ USD.

Figure 4.6: The coupon-bearing CAT bond price with respect to the threshold level (left axis) and time to expiry (right axis) in the Burr and NHP1 case.
\includegraphics[width=1.6\defpicwidth]{STFcat06.ps}

The choice of the fitted loss distribution affects the price of the bond. Figure 4.7 illustrates the difference between the zero-coupon CAT bond prices calculated under the two assumptions of Burr and log-normal loss sizes in the NHP1 case. It is clear that taking into account heavier tails (the Burr distribution), which can be more appropriate when considering catastrophic losses, leads to higher prices (the maximum difference in this example reaches 50% of the principal).

Figure 4.7: The difference between zero-coupon CAT bond prices in the Burr and log-normal cases with respect to the threshold level (left axis) and time to expiry (right axis) under the NHP1 assumption.
\includegraphics[width=1.6\defpicwidth]{STFcat07.ps}

Figures 4.8 and 4.9 show how the choice of the fitted Poisson point process influences the CAT bond value. Figure 4.8 illustrates the difference between the zero-coupon CAT bond prices calculated in the NHP1 and HP cases under the assumption of the Burr loss distribution. We see that the differences vary from $ -14\%$ to $ 3\%$ of the principal. Finally, Figure 4.9 illustrates the difference between the zero-coupon CAT bond prices calculated in the NHP1 and NHP2 cases under the assumption of the Burr loss distribution. The difference is always below 12%.

Figure 4.8: The difference between zero-coupon CAT bond prices in the NHP1 and HP cases with respect to the threshold level (left axis) and time to expiry (right axis) under the Burr assumption.
\includegraphics[width=1.6\defpicwidth]{STFcat08.ps}

Figure 4.9: The difference between zero-coupon CAT bond prices in the NHP1 and NHP2 cases with respect to the threshold level (left axis) and time to expiry (right axis) under the Burr assumption.
\includegraphics[width=1.6\defpicwidth]{STFcat09.ps}

In our examples, equations (4.2) and (4.4), we have assumed that in the case of a trigger event the bond principal is completely lost. However, if we would like to incorporate a partial loss in the contract it is sufficient to multiply $ PV$ by the appropriate constant.