Worked Comparison of Indemnity and Index Covers

Why Compare Covers Scenario by Scenario?

An indemnity cover responds to the insured’s covered loss, subject to its contract terms. An index cover responds to a specified index or trigger. The two can therefore pay different amounts in the same event. Comparing only their average payouts can miss important differences: an index may pay more on average but leave a larger shortfall in the most severe loss scenario, or it may pay when the insured’s loss is relatively small.

A useful comparison starts with one set of mutually exclusive scenarios. For each scenario, record its probability, the insured loss, the indemnity recovery, and the index recovery. Then calculate each cover’s expected recovery and the residual loss after recovery. This keeps the trigger mechanics visible instead of hiding them inside a single average.

Definition: Let $L$ be the insured loss, $I$ the indemnity recovery, and $J$ the index-based recovery in a scenario. Define residual losses as $R^I=L-I$ and $R^J=L-J$. Define signed basis as $B=J-I$, so a positive $B$ means the index cover pays more than the indemnity cover in that scenario. Absolute basis is $|B|$, the size of the payout difference without regard to its direction.

For an indemnity layer with deductible $d$ and limit $u$, the payment is $I(L)=\min\{u,(L-d)_+\}$, where $x_+=\max\{x,0\}$. A simple capped index payment can be written $J(Z)=\min\{m,c(Z-k)_+\}$, where $Z$ is the index value, $k$ is its attachment threshold, $c$ is the payment per index unit above that threshold, and $m$ is the maximum payment. These formulas apply only when the stated contract terms are an ordinary deductible-and-limit layer and a linear, capped index schedule, respectively. Other contract schedules must be calculated using their own terms.

When scenarios $s=1,\ldots,n$ have probabilities $p_s$, with $\sum_s p_s=1$, the expected recovery is the probability-weighted sum of scenario recoveries. The same approach applies to expected loss, residual loss, and basis.

E[I]=\sum_{s=1}^n p_s I_s,\qquad E[J]=\sum_{s=1}^n p_s J_s,\qquad E[R^I]=\sum_{s=1}^n p_s(L_s-I_s)

Why Expected Residual Loss Follows from Expected Recovery

The comparison of expected residual losses does not require a new probability model. In each scenario, residual loss is the original loss less the recovery. Multiplying that identity by the scenario probability and summing gives the result directly:

E[R^I] =\sum_s p_s(L_s-I_s) =\sum_s p_sL_s-\sum_s p_sI_s =E[L]-E[I].

The same derivation gives $E[R^J]=E[L]-E[J]$. It follows that $E[R^J]-E[R^I]=E[I]-E[J]=-E[B]$. This is an identity for expected values, not a statement that either cover pays the same amount in each scenario. In particular, $E[|B|]$ cannot generally be found by taking the absolute value of $E[B]$.

Worked Comparisons

Worked Example: Signed and Absolute Basis Across Three Loss Scenarios

A hypothetical insured compares an indemnity layer of $50,000 excess of $50,000, with a maximum recovery of $200,000, against an index cover paying $10,000 per index unit, capped at $160,000. The following scenario amounts are in thousands of dollars. Scenario probabilities are 0.20, 0.50, and 0.30; losses are 50, 100, and 460, and corresponding index values are 1, 4, and 16. Find expected recoveries, expected residual losses, and signed and absolute basis.

Method. Apply each contract separately in each scenario, then weight the resulting amounts by their probabilities. For the indemnity layer, $I(L)=\min\{200,(L-50)_+\}$. For the index cover, $J(Z)=\min\{160,10Z\}$. All amounts below are in thousands of dollars.

Scenario 1. The indemnity pays $\min\{200,(50-50)_+\}=0$. The index pays $\min\{160,10(1)\}=10$. Thus $B=10-0=10$, and residual losses are $50-0=50$ under indemnity and $50-10=40$ under the index cover.

Scenario 2. The indemnity pays $\min\{200,100-50\}=50$. The index pays $\min\{160,10(4)\}=40$. Thus $B=40-50=-10$, and residual losses are $100-50=50$ and $100-40=60$.

Scenario 3. The indemnity pays $\min\{200,460-50\}=200$, because the layer limit applies. The index pays $\min\{160,10(16)\}=160$, because its cap applies. Thus $B=160-200=-40$, and residual losses are $460-200=260$ and $460-160=300$.

Expected indemnity recovery. $E[I]=0.20(0)+0.50(50)+0.30(200)=0+25+60=85$. Expected index recovery. $E[J]=0.20(10)+0.50(40)+0.30(160)=2+20+48=70$.

The expected loss is $E[L]=0.20(50)+0.50(100)+0.30(460)=10+50+138=198$. Therefore expected residual losses are $198-85=113$ for indemnity and $198-70=128$ for the index cover. Direct weighting confirms the first result: $0.20(50)+0.50(50)+0.30(260)=10+25+78=113$. For the index cover, $0.20(40)+0.50(60)+0.30(300)=8+30+90=128$.

The signed expected basis is $E[B]=0.20(10)+0.50(-10)+0.30(-40)=2-5-12=-15$. The expected absolute basis is $E[|B|]=0.20(10)+0.50(10)+0.30(40)=2+5+12=19$. Check: the index cover’s expected recovery is $15$ less than the indemnity recovery, matching $E[B]=-15$; its expected residual loss is $15$ greater. The common mistake is to report $|E[B]|=15$ as the average size of the scenario-level mismatch. That is incorrect: absolute values must be taken before weighting, giving $E[|B|]=19$.

Worked Example: A Trigger That Pays Mainly in Larger Events

A hypothetical portfolio has four mutually exclusive loss scenarios, with probabilities 0.25, 0.25, 0.30, and 0.20. Losses, in thousands of dollars, are 0, 40, 100, and 200. Compare an indemnity layer of $40,000 excess of $20,000 with a capped index schedule $J(Z)=\min\{100,25(Z-1)_+\}$. The corresponding index values are 0, 1, 3, and 5.

Method. First calculate each recovery using its own trigger. The indemnity recoveries are $0,20,40,40$: for example, at loss 100 the payment is $\min\{40,100-20\}=40$, and at loss 200 the layer remains capped at 40. The index recoveries are $0,0,50,100$: at $Z=3$, $25(3-1)=50$; at $Z=5$, the uncapped amount is $25(5-1)=100$.

Expected indemnity recovery is $0.25(0)+0.25(20)+0.30(40)+0.20(40)=0+5+12+8=25$. Expected index recovery is $0.25(0)+0.25(0)+0.30(50)+0.20(100)=0+0+15+20=35$.

Expected loss is $0.25(0)+0.25(40)+0.30(100)+0.20(200)=0+10+30+40=80$. Thus expected residual losses are $80-25=55$ for indemnity and $80-35=45$ for the index cover. Directly, indemnity residuals are $0,20,60,160$, giving $0+5+18+32=55$; index residuals are $0,40,50,100$, giving $0+10+15+20=45$.

The scenario bases $J-I$ are $0,-20,10,60$. Therefore $E[B]=0.25(0)+0.25(-20)+0.30(10)+0.20(60)=0-5+3+12=10$, while $E[|B|]=0+5+3+12=20$. Check: the expected payout advantage of the index cover is 10, equal to $E[B]$, and its expected residual loss is 10 less. However, at loss 40 the index pays nothing while indemnity pays 20. A higher expected payout does not mean the index cover performs better in every scenario.

Worked Example: Comparing Tail Residuals, Not Only Averages

A hypothetical insured assigns probabilities 0.40, 0.40, and 0.20 to losses of $20,000, $80,000, and $200,000. It compares an indemnity layer of $30,000 excess of $20,000 with an index schedule $J(Z)=\min\{120,60(Z-1)_+\}$. The index values are 1, 2, and 3. Calculate expected recovery and residual loss, and determine the probability that residual loss exceeds $100,000 under each cover.

Method. The indemnity recoveries are $0,30,30$: the middle loss is $80-20=60$ above the deductible, but the recovery is capped at 30; the largest loss also reaches that cap. Index recoveries are $0,60,120$, since the last scenario reaches the index cap.

Expected indemnity recovery is $0.40(0)+0.40(30)+0.20(30)=0+12+6=18$. Expected index recovery is $0.40(0)+0.40(60)+0.20(120)=0+24+24=48$. Expected loss is $0.40(20)+0.40(80)+0.20(200)=8+32+40=80$.

Expected residual losses are $80-18=62$ for indemnity and $80-48=32$ for the index cover. As a direct check, indemnity residuals are $20,50,170$, giving $0.40(20)+0.40(50)+0.20(170)=8+20+34=62$. Index residuals are $20,20,80$, giving $0.40(20)+0.40(20)+0.20(80)=8+8+16=32$.

Only the largest scenario has indemnity residual loss above 100, so that probability is 0.20. None of the index residuals exceeds 100, so that probability is 0.00. The bases are $0,30,90$, giving $E[B]=0.40(0)+0.40(30)+0.20(90)=12+18=30$; here $E[|B|]=30$ because all scenario bases are nonnegative. Check: $E[J]-E[I]=48-18=30$, consistent with the basis calculation. The index cover has lower residuals in these scenarios, but this result depends on this hypothetical scenario set and trigger schedule; it is not a general guarantee.

Interpreting the Comparison

Expected recovery summarizes average payout under the assumed scenario probabilities. Expected residual loss summarizes the corresponding average amount left after recovery. Neither quantity alone shows where the cover responds. The scenario-level residuals and basis should therefore remain part of the analysis, especially when large losses have material consequences.

Signed basis indicates direction: positive values mean the index pays more than indemnity; negative values mean it pays less.
Absolute basis measures the size of mismatch regardless of direction. It is not the absolute value of average basis.
Residual loss by scenario reveals whether a cover’s trigger leaves a shortfall in a particular event. If an index payment can exceed the insured loss, the residual $L-J$ can be negative; that represents a net payment under this arithmetic comparison, not a negative physical loss.

The probabilities used in the calculation matter. They should describe the scenario set being analyzed, and they must sum to one. A comparison based on a small or deliberately selected set of scenarios is informative about those scenarios, but does not establish the expected result for every possible outcome. In practice, contract definitions, limits, exclusions, and the reliability of the index also affect how the scenario amounts should be specified.

Key takeaway: Calculate indemnity and index payments separately in each scenario, then compare their probability-weighted recoveries and residuals. Use both signed and absolute basis: the signed mean shows the average direction of the payout difference, while the mean absolute basis preserves the average size of scenario-level differences.

Check Your Understanding

For each problem, calculate in the stated units and use the supplied scenario probabilities.

A hypothetical cover has losses of 30 and 90 thousand dollars with probabilities 0.40 and 0.60. An indemnity layer is 20 thousand excess of 10 thousand, with a 20-thousand limit. The index pays 0 and 50 thousand in the two scenarios. Find both expected recoveries, both expected residual losses, and $E[B]$ and $E[|B|]$. (Answer: Expected recoveries: indemnity $20,000, index $30,000; expected residuals: indemnity $46,000, index $36,000; $E[B]=\$10,000$; $E[|B|]=\$26,000$.)
Losses are 0, 60, and 150 thousand dollars with probabilities 0.50, 0.30, and 0.20. An indemnity layer is 50 thousand excess of 25 thousand, with a 50-thousand limit; index recoveries are 0, 20, and 80 thousand. Calculate expected loss, recoveries, residuals, and signed and absolute basis. (Answer: Expected loss $48,000; expected recoveries: indemnity $20,500, index $22,000; expected residuals: indemnity $27,500, index $26,000; $E[B]=\$1,500$; $E[|B|]=\$10,500$.)
With probabilities 0.20, 0.50, and 0.30, losses are 40, 100, and 240 thousand dollars. An indemnity layer is 50 thousand excess of 20 thousand, with a 50-thousand limit; index recoveries are 10, 40, and 100 thousand. Find expected recoveries, expected residuals, and both basis measures. (Answer: Expected recoveries: indemnity $44,000, index $52,000; expected residuals: indemnity $86,000, index $78,000; $E[B]=\$8,000$; $E[|B|]=\$22,000$.)
A hypothetical insured has losses of 20, 80, and 200 thousand dollars with probabilities 0.40, 0.40, and 0.20. Indemnity recoveries are 0, 30, and 30 thousand; index recoveries are 0, 60, and 120 thousand. Find the probability that residual loss exceeds 100 thousand under each cover, as well as expected residual loss under each. (Answer: Indemnity probability 0.2000 and expected residual $62,000; index probability 0.0000 and expected residual $32,000.)
In any scenario comparison, explain why $E[|B|]$ cannot generally be calculated as $|E[B]|$. (Answer: Absolute value must be taken separately in each scenario before probability weighting; positive and negative signed bases can cancel in $E[B]$ but cannot cancel in $E[|B|]$.)

What You'll Learn