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
,
and
, and the log-normal distribution with parameters
and
. We also
analyse the homogeneous Poisson process with an annual intensity
(HP) and the non-homogeneous Poisson processes with the rate functions
(NHP1) and
(NHP2).
Consider a zero-coupon CAT bond defined by the payment of an amount at maturity
, contingent on a threshold time
. Define the
process
. We require that
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
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 , catastrophic flow
, a distribution function of incurred
losses
, and paying
at maturity is given by Burnecki and Kukla (2003):
We evaluate this CAT bond price at , and apply appropriate Monte Carlo simulations. We assume for the purposes of illustration that the annual
continuously-compounded discount rate
is constant and corresponds to LIBOR,
years,
billion USD
(quarterly - 3*annual average loss).
Furthermore, in the case of the zero-coupon CAT bond we assume that USD. Hence, the bond is priced at
over LIBOR when
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
is a quarter and
billion USD the CAT bond price approaches the value
USD. This is equivalent to the situation when the threshold time exceeds the maturity (
) with probability one.
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Consider now a CAT bond which has only coupon payments , which terminate at the threshold time
. The no-arbitrage price of the CAT bond associated with a threshold
, catastrophic flow
, a distribution function of incurred losses
, with coupon payments
which terminate at time
is given by Burnecki and Kukla (2003):
We evaluate this CAT bond price at and assume that
. The value of
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
USD billion and
years the CAT bond price
approaches the value
USD. This is equivalent to the situation when the threshold
time exceeds the maturity (
) with probability one.
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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 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 , catastrophic flow
, a distribution function of incurred losses
, paying
at maturity, and coupon payments
which cease at the threshold time
is given by Burnecki and Kukla (2003):
We evaluate this CAT bond price at and assume that
USD, and again
. Figure 4.6 illustrates this CAT
bond price in the Burr and NHP1 case. The influence of the threshold level
on the bond value is clear but the effect of increasing the time to
expiry is not immediately clear. As
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
USD and here
USD.
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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).
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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 to
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%.
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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 by the appropriate constant.