Comparing Basis Risk Across Trigger Types

Why Trigger Basis Risk Matters

A catastrophe bond transfers risk according to its contract trigger, but the sponsor’s loss and the bond’s recovery need not move together. An indemnity trigger refers to specified sponsor losses; an industry-loss trigger refers to an external loss index; and a parametric trigger refers to a physical measurement. When the trigger quantity differs from the sponsor’s loss experience, the bond may pay too little or too much relative to the protection the sponsor intended.

A fair comparison needs a consistent target. The sponsor generally does not expect a bond to reimburse every dollar of gross loss: an attachment, limit, and defined coverage scope may be intentional. Basis risk is therefore measured against the intended covered-layer amount, not automatically against the sponsor’s full event loss.

Definition: Let $L$ be the sponsor’s event loss in the specified coverage scope, and let the sponsor’s intended covered layer have attachment $A_s$ and limit $C$. The target amount is $$ T(L)=\min\{\max(L-A_s,0),C\}. $$ Let $R$ be the bond recovery for that event. Define signed basis mismatch as $B=R-T(L)$, underpayment as $U=\max(T(L)-R,0)$, and overpayment as $O=\max(R-T(L),0)$. Thus $B=O-U$. These measures compare event-level dollars with the intended covered layer; they are not probabilities or measures of long-run hedge performance.

A positive $B$ means the bond recovery exceeds the target amount for that event; a negative $B$ means a shortfall. “Overpayment” here describes the comparison with the stated target, not a judgment that the contract paid incorrectly. The bond may be performing exactly as written while producing an excess relative to the sponsor’s loss.

Unless stated otherwise, the examples use hypothetical event losses and trigger readings in millions of dollars, no discounting, and no probability distribution or event frequency. They compare recoveries at the event level. In each example, contract terms and scenario inputs are the complete data needed for the calculation.

Common Payout Framework

For a trigger quantity $z$, suppose the bond has maximum recovery $C$, attaches at trigger level $A$, and is exhausted at trigger level $E$, where $E>A$. For a linear trigger, the recovery is

R(z)=C\min\left\{\max\left(\frac{z-A}{E-A},0\right),1\right\}.

The quantity $z$ must be interpreted according to the trigger wording: it could be a verified sponsor loss, an industry-loss index, or a physical measurement. Its units must match $A$ and $E$. The formula assumes the contract specifies a linear payout between attachment and exhaustion, zero below attachment, and the full limit at or above exhaustion. It does not apply unchanged to a nonlinear payout schedule or a contract with different boundary rules.

Indemnity recovery: If the bond uses the same verified loss scope and the same attachment and limit as the sponsor’s target layer, then $R=T(L)$. An indemnity trigger can still differ from the target if its covered perils, event definition, loss adjustments, or limits do not match the sponsor’s intended protection.

For a linear index payout, the fraction of the bond limit paid is the fraction of the trigger interval traversed. The denominator is the width $E-A$, not the attachment level $A$. This matters because the trigger interval measures how far the index must move between zero payout and full payout.

Calculate the sponsor target.
Apply the sponsor’s intended attachment and limit to $L$, giving $T(L)$. This removes the portion the sponsor deliberately retains and respects the intended cap.

Calculate the bond recovery.
Use the trigger-specific quantity and contract terms to find $R$. Keep trigger units consistent with attachment and exhaustion levels.

Compare like with like.
Compute $B=R-T(L)$. A negative result is an underpayment; a positive result is an overpayment relative to the target.

Worked Comparisons

Worked Example: An Industry Index Underpays and Overpays

A hypothetical sponsor intends to transfer the layer attaching at $20 million and capped at $40 million. The bond has an industry-loss trigger with maximum recovery $C=\$40$ million, attachment $A=\$100$ million of industry loss, and exhaustion $E=\$180$ million. For two separate hypothetical events, calculate the target, industry-trigger recovery, and mismatch. No distribution, interest rate, or discounting is assumed.

Event 1: Underpayment. The sponsor’s loss is $L=\$50$ million and the industry index is $I=\$140$ million. The sponsor target is

T(50)=\min\{\max(50-20,0),40\} =\min(30,40)=\$30\text{ million}.

The industry index lies $140-100=\$40$ million above attachment in a trigger interval of $180-100=\$80$ million. Its recovery is

R(140)=40\left(\frac{140-100}{180-100}\right) =40\left(\frac{40}{80}\right) =40(0.5)=\$20\text{ million}.

Therefore $B=20-30=-\$10$ million, $U=\$10$ million, and $O=0$. The bond pays $10 million less than the sponsor’s intended layer.

Event 2: Overpayment. Now let $L=\$35$ million and $I=\$150$ million. The target is

T(35)=\min\{\max(35-20,0),40\} =\min(15,40)=\$15\text{ million}.

The index is $150-100=\$50$ million above attachment, so

R(150)=40\left(\frac{150-100}{180-100}\right) =40\left(\frac{50}{80}\right) =40(0.625)=\$25\text{ million}.

Thus $B=25-15=+\$10$ million, $O=\$10$ million, and $U=0$. The same trigger can underpay in one event and overpay in another because the relationship between sponsor loss and industry loss is not fixed.

Check: Both recoveries are between zero and the $40 million limit, and both are consistent with an index strictly between $100 million and $180 million.

Worked Example: A Parametric Trigger Can Miss a Severe Sponsor Loss

A hypothetical bond pays linearly from zero to its $30 million limit as a wind-speed measurement $q$ rises from 120 to 160 miles per hour. The sponsor’s intended layer attaches at $A_s=\$15$ million and has limit $C=\$30$ million. For two separate events, calculate target and recovery. Assume the parametric trigger formula applies, and no distribution, interest rate, or discounting is assumed.

Event 1: Underpayment. The sponsor reports $L=\$45$ million, while the measured wind speed is $q=130$ miles per hour. The target is

T(45)=\min\{\max(45-15,0),30\} =\min(30,30)=\$30\text{ million}.

Wind speed is $130-120=10$ miles per hour above attachment, over a full trigger interval of $160-120=40$ miles per hour. Therefore

R(130)=30\left(\frac{130-120}{160-120}\right) =30\left(\frac{10}{40}\right) =30(0.25)=\$7.50\text{ million}.

The signed mismatch is $B=7.50-30=-\$22.50$ million. The underpayment is $22.50 million and the overpayment is zero.

Event 2: Overpayment. Suppose instead $L=\$25$ million and $q=150$ miles per hour. The target is

T(25)=\min\{\max(25-15,0),30\} =\min(10,30)=\$10\text{ million}.

The measurement is $150-120=30$ miles per hour above attachment. Hence

R(150)=30\left(\frac{150-120}{160-120}\right) =30\left(\frac{30}{40}\right) =30(0.75)=\$22.50\text{ million}.

Now $B=22.50-10=+\$12.50$ million, so the bond has a $12.50 million overpayment relative to the target. A physical measurement can be informative about hazard severity without being a direct measurement of the sponsor’s insured loss.

Check: Both wind speeds are inside the 120-to-160 interval, and the calculated recoveries, $7.50 million and $22.50 million, are within the $30 million limit.

Worked Example: Compare Three Triggers Across Multiple Events

A hypothetical sponsor has an intended layer attaching at $20 million and capped at $40 million. Three hypothetical bonds each have a $40 million limit. The indemnity bond uses the sponsor’s stated loss and exactly matches the target layer. The industry bond attaches at an index of $100 million and exhausts at $180 million. The parametric bond attaches at 120 miles per hour and exhausts at 160 miles per hour. For the four events below, calculate target amounts and recoveries. All dollar amounts are in millions; no distribution, interest rate, or discounting is assumed.

Event	Sponsor loss $L$	Industry index $I$	Wind speed $q$
A	35	130	140
B	50	140	130
C	75	170	160
D	15	120	125

First calculate targets: $T_A=\min(35-20,40)=15$, $T_B=\min(50-20,40)=30$, $T_C=\min(75-20,40)=40$, and $T_D=\max(15-20,0)=0$. The indemnity recoveries equal those targets: 15, 30, 40, and 0.

For the industry trigger, use $R(I)=40\min\{\max((I-100)/80,0),1\}$. The recoveries are $R(130)=40(30/80)=15$, $R(140)=40(40/80)=20$, $R(170)=40(70/80)=35$, and $R(120)=40(20/80)=10$.

For the parametric trigger, use $R(q)=40\min\{\max((q-120)/40,0),1\}$. The recoveries are $R(140)=40(20/40)=20$, $R(130)=40(10/40)=10$, $R(160)=40(40/40)=40$, and $R(125)=40(5/40)=5$.

Event	Target	Indemnity recovery	Industry recovery	Industry signed mismatch	Parametric recovery	Parametric signed mismatch
A	15	15	15	0	20	+5
B	30	30	20	-10	10	-20
C	40	40	35	-5	40	0
D	0	0	10	+10	5	+5

Across the four events, total target is $15+30+40+0=\$85$ million. Total industry recovery is $15+20+35+10=\$80$ million, a net signed mismatch of $-\$5$ million. Total parametric recovery is $20+10+40+5=\$75$ million, a net signed mismatch of $-\$10$ million. The indemnity recovery totals $85 million and has no mismatch under the stated exact-match assumption.

Net mismatch can conceal offsetting event errors. For the industry trigger, the absolute mismatches sum to $0+10+5+10=\$25$ million, even though the net shortfall is only $5 million. For the parametric trigger, they sum to $5+20+0+5=\$30$ million, versus its $10 million net shortfall. This comparison is descriptive of these scenarios; it does not establish a probability-weighted or statistical measure of hedge performance.

Check: Every recovery is between zero and $40 million. Event C reaches the parametric exhaustion level and pays its full limit; event D has no sponsor target but positive index and parametric recoveries, producing overpayments relative to the target.

Why the Linear Payout Formula Holds

Between attachment and exhaustion, a linear trigger must increase from zero recovery at $z=A$ to full recovery $C$ at $z=E$. The straight line through these two points has slope equal to change in recovery divided by change in trigger quantity, namely $C/(E-A)$. Starting from zero at attachment gives the intermediate recovery below.

R(z)=\frac{C}{E-A}(z-A), \qquad A<z<E.

If $z\leq A$, the contract has not reached attachment, so recovery is zero. If $z\geq E$, the recovery cannot exceed the limit, so it is $C$. Combining these three regions gives the capped formula. This derivation depends on a linear payout between the two trigger points; a different contractual schedule requires its own recovery function.

Common Exam Trap and Interpretation

A frequent error is to compare bond recovery with gross sponsor loss rather than with the intended covered layer. In the first worked example’s Event 1, a mistaken calculation might call the shortfall $50-20=\$30$ million, subtracting the bond recovery from the full $50 million loss. But the sponsor intentionally retains the first $20 million, so its target is $T(50)=\$30$ million. The correct underpayment is $30-20=\$10$ million, not $30 million.

Another common error is to divide the index increase by the attachment level. For an index of $140 million, the correct payout fraction is $(140-100)/(180-100)=40/80=0.5$, not $40/100=0.4$. The latter would produce a mistaken $16 million recovery rather than the correct $20 million. Always use the width of the trigger interval in the denominator.

Basis comparisons are useful for evaluating whether a trigger’s event-level payments align with a sponsor’s stated protection objective. They also show why the direction of mismatch matters: underpayment leaves more event loss with the sponsor than intended, while overpayment can result when the bond’s index or measurement is severe relative to the sponsor’s target loss. Neither direction can be ruled out merely because a trigger is objective or contractually well-defined.

Key takeaway: Compare a bond’s recovery with the sponsor’s intended covered-layer amount, not automatically with gross event loss. Calculate the trigger payout from its own units and terms, then report signed mismatch and distinguish underpayment from overpayment.

Check Your Understanding

For each hypothetical scenario, calculate the requested recovery or mismatch using the stated linear payout terms. Dollar amounts are in millions.

A sponsor’s intended layer attaches at $12 million and has a $25 million limit. Sponsor loss is $31 million, and an exactly matching indemnity bond pays the target layer. What is the target and recovery? (Answer: $19 million target; $19 million recovery.)
An industry trigger has limit $30 million, attachment $50 million, and exhaustion $110 million. The index is $80 million and the sponsor’s intended target is $21 million. Find recovery and signed mismatch. (Answer: $15 million recovery; $-\$6$ million signed mismatch, a $6 million underpayment.)
A parametric bond has a $24 million limit, attaches at 90 units, and exhausts at 130 units. The measurement is 115 units. The sponsor has loss $28 million and an intended layer attaching at $10 million with a $24 million limit. Find recovery and mismatch. (Answer: $15 million recovery; $18 million target; $-\$3$ million signed mismatch, a $3 million underpayment.)
An industry bond has a $30 million limit, attaches at an index of 100, and exhausts at 160. Sponsor loss is $44 million, and its intended layer attaches at $14 million with a $30 million limit. If the index is 150, find the target and recovery. (Answer: $30 million target; $25 million recovery; $5 million underpayment.)
For two events, sponsor losses are $25 million and $45 million. The intended layer attaches at $10 million and has a $30 million limit. An industry trigger has a $30 million limit, attaches at 80, and exhausts at 140; its indices are 100 and 140. Calculate the net signed mismatch and the sum of absolute event mismatches. (Answer: targets are $15 million and $30 million; recoveries are $10 million and $30 million; net signed mismatch is $-\$5$ million and the sum of absolute mismatches is $5 million.)

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