$"$An insurance company in Florida is concerned about catastrophic hurricane losses. The insurer decides to issue some dollar$-$denominated bonds to U$.$S$.$ investors. The bonds will have a $$1,000$ maturity value, mature in $10$ years, and provide a $10$ percent annual interest $($coupon$)$ payment.

The bond indenture specifies that if a Category $4$ Hurricane hits Florida in any year, only $1/4$ of the annual interest will be paid, and if a Category $5$ Hurricane occurs, no interest is payable. Assume the investors agree with the meteorological estimates of the probabilities of major hurricanes, which are provided below. Further, assume that the annual probabilities are independent.

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If the investor$-$required rate of return $($YTM$)$ for these bonds is $6$ percent, what is the most an investor should be willing to pay for one of these newly issued CAT bonds?

Hurricane Category Probability Coupon Payment Expected Coupon Payment

Category $1-30\mathrm{.}91$ $$100$ $$91\mathrm{.}00$

Category $40\mathrm{.}06$ $$25$ $$1\mathrm{.}50$

Category $50\mathrm{.}03$ $$-$ $$0\mathrm{.}00$

Total $1\mathrm{.}00$ $$92\mathrm{.}50$

Step $1$: Calculate the expected coupon payment using the table above. PMT $$92\mathrm{.}50$

PMT $$(92\mathrm{.}50)$

NPER $10$

FV $$1,000$

RATE $6\mathrm{.}00\%$

Step $2$: Use the PV$()$ function to calculate the bond value. PV$()$ $$681$

Step $3$: Calculate the PV of the only coupon payments. PV of coupon payments

Step $4$: Calculate the PV of the only par value. PV of the par value

Step $5$: Show that the bond value calculated above could also be found by adding up the value of the coupon payments and par value separately: Bond value