Cat Bond Design And Evaluation Workshop

From Exposure to Bond Terms

A catastrophe bond transfers a specified layer of catastrophe risk to investors. Designing one requires more than choosing a trigger label: the contract must identify what event measure is used, when the bond begins to lose principal, when principal is fully exhausted, what assets support repayment, and when cash flows occur. Evaluation then asks two distinct questions: how much protection does the sponsor receive, and how much principal can investors lose?

The design below is hypothetical. A regional property insurer wants protection against severe wind-catastrophe losses. Its desired protection layer is large enough that a catastrophe bond could provide meaningful risk transfer, but the insurer also wants a trigger that can be measured independently and settled on a stated timetable. An industry-loss trigger is selected: the bond responds to a defined estimate of total insured industry losses from a qualifying event, not directly to the sponsor’s own losses. This can make verification more independent, but it creates basis risk if the industry index and sponsor losses differ.

Definition: A catastrophe bond design specifies a covered event or trigger measure, a rule mapping that measure to principal impairment, collateral supporting the transaction, and a schedule for coupons and principal payments. The attachment point is the trigger level at which principal impairment begins. The exhaustion point is the trigger level at which the entire principal is impaired.

For a proportional industry-loss trigger, let $I$ be the measured industry loss in billions of dollars, $A$ the attachment point, $E$ the exhaustion point, and $P_0$ the initial bond principal. Assume $E>A$, and assume the contract uses linear principal impairment between these points. The fraction of principal impaired is

g(I)= \begin{cases} 0, & I\le A,\\ \dfrac{I-A}{E-A}, & A<I<E,\\ 1, & I\ge E. \end{cases}

The dollar principal impairment is $P_0g(I)$, and the principal remaining after the event is $P_0[1-g(I)]$. This rule applies only to the stated proportional trigger structure. A contract with stepped, binary, or otherwise non-linear terms requires its own mapping rule.

Set the Trigger and Layer

Suppose the insurer issues a two-year bond with $P_0=\$40$ million. The selected industry-loss attachment point is $A=\$50$ billion and exhaustion point is $E=\$90$ billion. Thus, the covered index layer has width $E-A=\$40$ billion. If the industry loss is at or below $50 billion, the bond principal is unchanged. If it is at or above $90 billion, all $40 million is impaired. Between those values, impairment increases in direct proportion to the index loss above attachment.

Choosing these values is a risk-design decision, not a mechanical calculation. The attachment should reflect how severe an event must be before the sponsor wants bond protection to begin. The exhaustion point determines the maximum index level at which the bond can still provide increasing protection. A higher attachment can reduce the frequency of principal impairment but leaves more severe sponsor losses below the bond layer. A lower exhaustion point makes full impairment occur at a smaller index loss. The calibration should therefore be reviewed against the sponsor’s exposure and the consequences of index-to-sponsor mismatch.

Worked Example: Converting Industry Loss Into Principal Impairment

The hypothetical bond has $40 million of principal, a $50 billion attachment point, and a $90 billion exhaustion point. Calculate principal impairment and principal remaining for industry losses of $42 billion, $70 billion, and $98 billion.

Method. Apply the piecewise proportional trigger rule separately to each index value. The layer width is $90-50=40$ billion dollars.

At $I=\$42$ billion, the index is below attachment, so $g(42)=0$. Impairment is $\$40\text{ million}\times0=\$0$, leaving $\$40$ million.

At $I=\$70$ billion, the index is between attachment and exhaustion. The impaired fraction is $g(70)=(70-50)/(90-50)=20/40=0.5$. Impairment is $\$40\text{ million}\times0.5=\$20$ million, leaving $\$40-\$20=\$20$ million.

At $I=\$98$ billion, the index is above exhaustion, so $g(98)=1$. Impairment is $\$40$ million and principal remaining is $\$0$.

Check. The impairment amounts are nondecreasing as the industry loss rises, never exceed the original $40 million principal, and are zero below attachment and complete above exhaustion. The common mistake is to divide the excess index loss by the exhaustion point itself; the correct denominator is the layer width, $E-A$.

Why the Proportional Rule Uses the Layer Width

The linear rule is the unique straight-line mapping that is zero at attachment and one at exhaustion. To see this, suppose the impaired fraction between the points is a linear function $g(I)=b(I-A)$. Requiring full impairment at $I=E$ gives $1=b(E-A)$, so $b=1/(E-A)$. Therefore,

g(I)=\frac{I-A}{E-A},\qquad A<I<E.

This also explains why the formula uses the amount of index loss within the covered layer, $I-A$, divided by the full width of that layer, $E-A$. Values below or above the layer must be capped at zero or one; extending the straight line without those caps could imply negative impairment or impairment greater than the original principal.

Collateral and Payment Schedule

In this hypothetical transaction, a special purpose vehicle issues the notes and holds the $40 million proceeds in a segregated account invested only in short-term, high-quality government securities. The collateral is intended to support repayment of principal, subject to the bond’s event-loss terms. It reduces exposure to the operating credit of an insurer or another party, but it does not eliminate investment, custody, legal, or operational risks. The contract would need to state eligible investments, valuation and account controls, and what happens if an investment fails to meet its requirements.

Assume the bond pays quarterly coupons at a fixed annual rate of 8%, or 2% per quarter, on principal outstanding at the start of each quarter. This is a simplified contractual coupon assumption, not a market quote. Assume coupons are paid at each quarter-end, an event’s principal impairment is settled after independent verification, and the impairment takes effect after that quarter’s coupon is calculated. Remaining principal is repaid at the end of the eighth quarter. A real transaction must specify its reference rate, spread, calculation conventions, event definition, verification procedure, and settlement deadlines.

Worked Example: Investor Cash Flows After an Event

Use the two-year, $40 million bond and its 8% annual coupon, paid quarterly on opening-quarter principal. An industry-loss event produces an index loss of $70 billion during quarter 2. It is verified and settled after the quarter 2 coupon is calculated. Find the investor’s coupons, principal impairment, and maturity principal repayment.

Method. First determine the event impairment using the trigger mapping, then calculate each coupon using the outstanding principal at that quarter’s start.

The quarterly coupon rate is $8\%/4=2\%$. In quarters 1 and 2, opening principal is $40 million, so each coupon is $\$40\text{ million}\times0.02=\$0.8$ million. The two coupons total $\$1.6$ million.

As in the preceding calculation, the $70 billion index loss impairs 50% of principal, or $\$40\text{ million}\times0.5=\$20$ million. After settlement, principal outstanding is $\$40-\$20=\$20$ million. In each of quarters 3 through 8, the coupon is $\$20\text{ million}\times0.02=\$0.4$ million. There are six such quarters, so these coupons total $6\times\$0.4=\$2.4$ million.

At maturity, the investor receives the remaining $20 million of principal. Total coupons over the term are $\$1.6+\$2.4=\$4.0$ million; principal impairment is $20 million; and maturity principal repayment is $20 million.

Check. The $20 million impairment plus $20 million repaid equals the original $40 million principal. The event occurs after the quarter 2 coupon calculation by assumption, so using $20 million to calculate that quarter’s coupon would be a timing error.

Evaluate Sponsor Protection and Basis Risk

An industry-loss trigger does not guarantee that bond recovery equals the sponsor’s own covered loss. The sponsor may experience a large loss in an event that produces an industry index below attachment, or the index may be high while the sponsor’s own loss is comparatively modest. These differences are basis risk. A scenario evaluation should therefore display both the index value and the sponsor’s covered loss, rather than infer sponsor recovery from the index alone.

For this illustration, assume bond proceeds are available to the sponsor dollar for dollar as principal is impaired, up to the bond’s $40 million limit. Define the sponsor’s net retained covered loss in a scenario as its covered loss less the bond recovery, with a floor of zero. This simplified comparison does not include premiums, transaction costs, taxes, or other risk transfers.

Worked Example: Comparing Sponsor Loss With Bond Recovery

For the $40 million bond with $50 billion attachment and $90 billion exhaustion, consider four hypothetical catastrophe outcomes. The sponsor’s covered losses are $30 million, $24 million, $48 million, and $60 million, respectively; the corresponding industry index losses are $45 billion, $70 billion, $85 billion, and $100 billion. Calculate bond recovery and sponsor net retained loss in each case.

Method. Map each industry index loss to principal impairment. Treat that impairment as sponsor recovery, then subtract recovery from the sponsor’s covered loss, subject to a zero floor.

For the $45 billion index, the value is below attachment. Recovery is $0, so retained loss is $\$30-\$0=\$30$ million.

For the $70 billion index, the impaired fraction is $(70-50)/40=0.5$. Recovery is $0.5\times\$40=\$20$ million, so retained loss is $\$24-\$20=\$4$ million.

For the $85 billion index, the impaired fraction is $(85-50)/40=35/40=0.875$. Recovery is $0.875\times\$40=\$35$ million, so retained loss is $\$48-\$35=\$13$ million.

For the $100 billion index, the index is above exhaustion. Recovery is the full $40 million, so retained loss is $\$60-\$40=\$20$ million.

Check. Each recovery is between $0 and $40 million, and no scenario gives more recovery than the bond limit. The $45 billion scenario illustrates a substantial basis-risk gap: the sponsor has a $30 million covered loss but receives nothing because the index does not reach attachment. A bond recovery is determined by the contract’s trigger, not by the sponsor’s loss amount.

Common Design and Evaluation Traps

Using the wrong denominator. The proportional impairment denominator is the index layer width, $E-A$, not the exhaustion point $E$.
Confusing trigger loss with sponsor loss. An industry index determines bond impairment under the stated terms; it does not directly measure the sponsor’s covered loss.
Ignoring caps. Impairment cannot be less than zero or greater than initial principal. Apply the below-attachment and above-exhaustion cases explicitly.
Overlooking settlement timing. Whether an event affects a coupon, the next coupon, or maturity principal depends on the contract’s schedule and calculation conventions.
Treating collateral as eliminating all risk. Collateral can support repayment, but eligible-investment, custody, legal, and operational provisions still matter.

Key takeaway: A useful cat bond evaluation links four things explicitly: the sponsor’s exposure, the trigger measurement, the rule converting trigger results into principal impairment, and the payment schedule. Scenario comparisons reveal both investor principal loss and sponsor basis risk. This treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each hypothetical problem, apply the stated trigger and payment terms. Assume no costs or other risk transfers unless stated.

A $30 million bond has a proportional industry-loss trigger with attachment at $24 billion and exhaustion at $54 billion. An event produces a $39 billion index loss. Find the impaired fraction, dollar impairment, and principal remaining. (Answer: 50%; $15 million impaired; $15 million remaining.)
A $25 million bond attaches at $40 billion and exhausts at $65 billion. The index loss is $31 billion, while the sponsor’s covered loss is $18 million. Find recovery and net retained loss. (Answer: $0 recovery; $18 million retained.)
A $60 million bond has attachment at $70 billion and exhaustion at $110 billion. An index loss of $95 billion occurs. The sponsor’s covered loss is $29 million. Find the impairment fraction, recovery, and net retained loss. (Answer: 62.5%; $37.5 million recovery; $0 retained.)
A $20 million, one-year bond pays quarterly coupons at 6% annually on opening-quarter principal. An event is settled after the quarter 1 coupon and causes 25% principal impairment. Calculate the quarter 1 coupon, each later quarterly coupon, and principal repaid at year-end. (Answer: $0.3 million; $0.225 million per quarter for quarters 2–4; $15 million at year-end.)
A $50 million bond attaches at $100 billion and exhausts at $150 billion. In one scenario, the industry index is $140 billion and the sponsor’s covered loss is $42 million. Calculate recovery and retained loss. In a second scenario, the index is $95 billion and the sponsor’s covered loss is $35 million. Calculate recovery and retained loss, and identify the basis-risk issue. (Answer: First scenario, $40 million recovery and $2 million retained; second scenario, $0 recovery and $35 million retained, illustrating a below-attachment protection gap.)

What You'll Learn