Net Loss Distributions Under Reinsurance

Why the Net Loss Distribution Matters

Reinsurance changes more than an insurer’s average retained loss. It can also change the range and probability of net outcomes, particularly when a contract responds only above an attachment point or stops paying at a limit. To evaluate the resulting risk, the insurer needs the net loss distribution: the probabilities of the amounts it will ultimately bear, including the reinsurance premium.

This tutorial treats one defined period and one specified gross loss variable. The examples use hypothetical contracts and loss distributions. They assume that the reinsurer pays the contractual recovery in full and on time; counterparty default, expenses other than the stated premium, taxes, and timing differences are excluded.

Definition: Let $X$ be the insurer’s gross loss before reinsurance. Let $R(X)$ be the recovery due from the reinsurer under the contract, and let $\pi$ be a deterministic reinsurance premium paid for the period. The insurer’s net loss is $$ Y=X-R(X)+\pi. $$ The net loss distribution is the probability distribution of $Y$, obtained by applying this transformation to each possible value of $X$. If the premium is contingent or random, it must instead be included as a random quantity in the transformation.

Quota-Share Reinsurance

Under quota-share reinsurance, the reinsurer accepts a specified fraction $c$ of each covered loss, where $0\le c\le1$. The insurer retains the fraction $1-c$. If the premium is fixed, the recovery and net loss for a gross loss $x$ are

R(x)=cx,\qquad Y=(1-c)x+\pi.

The transformation is linear: every gross loss is scaled by the retained percentage and shifted upward by the premium. To obtain a discrete net distribution, transform every gross loss outcome and keep its probability. If several gross outcomes map to the same net amount, add their probabilities.

Worked Example: Transforming a Discrete Loss Distribution Under Quota Share

A hypothetical insurer has gross loss $X$ equal to $0 with probability 0.25, $40 with probability 0.35, $100 with probability 0.25, or $200 with probability 0.15. It cedes 30% of each loss under quota share and pays a fixed premium of $18. Find the net loss distribution and expected net loss.

Method. For each gross loss $x$, calculate the ceded recovery $0.30x$, then calculate the retained loss plus the $18 premium: $$ Y=x-0.30x+18=0.70x+18. $$

For $x=0$, the recovery is $0.30(0)=0$, so $Y=0-0+18=$18$, with probability 0.25.

For $x=40$, the recovery is $0.30(40)=$12$, so $Y=40-12+18=$46$, with probability 0.35.

For $x=100$, the recovery is $0.30(100)=$30$, so $Y=100-30+18=$88$, with probability 0.25.

For $x=200$, the recovery is $0.30(200)=$60$, so $Y=200-60+18=$158$, with probability 0.15. The resulting probabilities sum to $0.25+0.35+0.25+0.15=1.00$.

The expected net loss from the transformed distribution is $$ E[Y]=0.25(18)+0.35(46)+0.25(88)+0.15(158) =4.50+16.10+22.00+23.70=$66.30. $$

Check. The expected gross loss is $0.35(40)+0.25(100)+0.15(200)=$69.00$. The expected recovery is $0.30($69.00)=$20.70$. Therefore $E[X]-E[R(X)]+\pi=69.00-20.70+18.00=$66.30$, agreeing with the direct calculation. The check avoids forgetting that the premium is added even when the gross loss is zero.

Excess-of-Loss Reinsurance

An excess-of-loss contract pays only when a loss exceeds an attachment point. Let $a$ be the attachment point and $L$ the maximum recovery, or limit, for the covered loss. For one covered loss $x$, the recovery is

R(x)=\min\bigl((x-a)^+,L\bigr),\qquad (x-a)^+=\max(x-a,0).

Equivalently, the recovery is zero for $x\le a$, is $x-a$ for $a<x<a+L$, and is $L$ for $x\ge a+L$. The insurer’s net loss is $x-R(x)+\pi$. This formula assumes that $L$ is the maximum recovery for the loss being modeled. If the contract instead has an annual aggregate limit or covers multiple occurrences subject to separate terms, those terms must be modeled explicitly; a single-loss formula is not sufficient.

Worked Example: Applying an Attachment Point and Limit

A hypothetical insurer models one period’s gross loss as $0 with probability 0.40, $60 with probability 0.25, $140 with probability 0.20, or $300 with probability 0.15. It buys excess-of-loss reinsurance attaching at $50, with a maximum recovery of $120, for a fixed premium of $22. Find the net loss distribution and expected net loss.

Method. For each gross loss, apply $R(x)=\min(\max(x-50,0),120)$, then calculate $Y=x-R(x)+22$.

At $x=0$, the recovery is $\min(0,120)=0$, so the net loss is $0-0+22=$22$, with probability 0.40.

At $x=60$, the amount above attachment is $60-50=$10$. The recovery is $\min(10,120)=$10$, so the net loss is $60-10+22=$72$, with probability 0.25.

At $x=140$, the amount above attachment is $140-50=$90$. The recovery is $\min(90,120)=$90$, so the net loss is $140-90+22=$72$, with probability 0.20.

At $x=300$, the amount above attachment is $300-50=$250$, but the recovery is capped at $120. Thus the net loss is $300-120+22=$202$, with probability 0.15.

The two different gross losses of $60 and $140 both produce a net loss of $72. Combine their probabilities: $0.25+0.20=0.45$. The net distribution is therefore $22 with probability 0.40, $72 with probability 0.45, and $202 with probability 0.15.

The expected net loss is $$ E[Y]=0.40(22)+0.45(72)+0.15(202) =8.80+32.40+30.30=$71.50. $$

Check. The expected gross loss is $0.25(60)+0.20(140)+0.15(300)=$88.00$. Expected recovery is $0.25(10)+0.20(90)+0.15(120)=2.50+18.00+18.00=$38.50$. Hence $E[X]-E[R(X)]+\pi=88.00-38.50+22.00=$71.50$. The equality confirms that the repeated $72 outcome has been assigned the correct combined probability.

Why the Expected-Loss Check Works

For each possible gross loss $x$, the contract defines a recovery $R(x)$, and the insurer’s net loss is $x-R(x)+\pi$. Taking expectations gives

E[Y]=E[X-R(X)+\pi] =E[X]-E[R(X)]+\pi.

The final equality follows from linearity of expectation and the fact that $\pi$ is fixed. It does not require $X$ and $R(X)$ to be independent: the recovery is itself a function of $X$. For a discrete distribution, the same result follows by summing each outcome:

\sum_j p_j\bigl(x_j-R(x_j)+\pi\bigr) =\sum_j p_jx_j-\sum_j p_jR(x_j)+\pi\sum_j p_j =E[X]-E[R(X)]+\pi,

because the outcome probabilities $p_j$ sum to one. This identity is a useful arithmetic check, but it does not determine the full net distribution. Two contracts can have the same expected recovery and still produce different distributions of net losses.

Combining Quota Share and Excess of Loss

When contracts operate in sequence, identify the loss base to which each contract applies. The following example explicitly applies quota share first and then applies excess of loss to the insurer’s post-quota-share retained loss. Changing that contract order or defining the excess layer on gross loss would generally change the result.

Worked Example: Sequential Quota Share and Excess of Loss

A hypothetical insurer has gross loss $X$ equal to $0 with probability 0.20, $50 with probability 0.30, $100 with probability 0.30, or $250 with probability 0.20. It first cedes 20% of each loss under quota share, for a fixed premium of $12. It then buys excess-of-loss cover on the post-quota-share retained loss, attaching at $40 with a recovery limit of $80, for a fixed premium of $9. Find the final net loss distribution and its mean.

Method. First calculate the retained loss $Z=0.80X$. Next calculate the excess-of-loss recovery $R_2(Z)=\min(\max(Z-40,0),80)$. The final net loss is $Y=Z-R_2(Z)+12+9$.

For $X=0$, $Z=0$, $R_2=0$, and $Y=0+21=$21$, with probability 0.20.

For $X=50$, $Z=0.80(50)=$40$. The amount above the $40 attachment is zero, so $R_2=0$ and $Y=40+21=$61$, with probability 0.30.

For $X=100$, $Z=0.80(100)=$80$. The amount above attachment is $40, below the $80 limit. Thus $R_2=40$, and $Y=80-40+21=$61$, with probability 0.30.

For $X=250$, $Z=0.80(250)=$200$. The amount above attachment is $160, so the recovery is capped at $80. Thus $Y=200-80+21=$141$, with probability 0.20.

The $61 outcome arises from two gross losses, so its probability is $0.30+0.30=0.60$. The final distribution is $21 with probability 0.20, $61 with probability 0.60, and $141 with probability 0.20. Its mean is $$ E[Y]=0.20(21)+0.60(61)+0.20(141) =4.20+36.60+28.20=$69.00. $$

Check. The expected gross loss is $0.30(50)+0.30(100)+0.20(250)=$95.00$, and quota share leaves $0.80(95.00)=$76.00$ before the second treaty. The excess-of-loss recovery has expectation $0.30(0)+0.30(40)+0.20(80)=$28.00$. Adding the two premiums gives $76.00-28.00+12.00+9.00=$69.00$, as required. This check depends on applying the second treaty to $Z$, not to the original gross loss $X$.

Common Mistakes and Interpretation

A net loss distribution is built by transforming outcomes, not by subtracting an average recovery from each possible loss. The expectation identity is valid for the mean, but it cannot replace the outcome-by-outcome calculation when the full distribution is required.

Ignoring the limit. In the excess-of-loss example, an incorrect calculation at $x=$300$ would set recovery equal to $300-50=$250$, producing a net loss of $300-250+22=$72$. The contract limit is $120, so the correct recovery is $120 and the correct net loss is $300-120+22=$202$.
Forgetting the premium in low-loss outcomes. A fixed premium is paid even when the gross loss is zero. In the second example, zero gross loss produces a $22 net loss, not zero.
Failing to combine equal outcomes. If different gross losses map to the same net loss, their probabilities must be added. Leaving duplicate rows is acceptable only if their probabilities are still counted correctly in calculations.
Applying a contract to the wrong loss base. A layer written on post-quota-share retention does not automatically produce the same recovery as a layer written on gross loss. State the contract sequence and base before calculating.
Confusing a per-loss limit with an aggregate limit. A cap on recovery for one loss does not, by itself, specify a cap on total recoveries across multiple losses. Model the relevant contract terms rather than substituting one for the other.

These transformations matter because the net distribution describes both the cost of the contract and the residual loss pattern borne by the insurer. It is the necessary starting point for subsequent analysis of risk and capital effects; the distribution alone does not establish whether the reinsurance is economically worthwhile.

Key takeaway: Define the recovery on the correct loss base, transform every gross loss outcome, add the fixed premium, and combine probabilities for repeated net outcomes. Then check the mean using $E[Y]=E[X]-E[R(X)]+\pi$.

Check Your Understanding

For each problem, derive the net outcomes from the stated contract terms and include the reinsurance premium.

A hypothetical gross loss is $0 with probability 0.50, $50 with probability 0.30, or $150 with probability 0.20. The insurer cedes 40% under quota share and pays a fixed premium of $10. Give the net loss distribution and its mean. (Answer: $10 with probability 0.50, $40 with probability 0.30, and $100 with probability 0.20; mean $37.00.)
Gross loss is $20 with probability 0.20, $80 with probability 0.50, or $200 with probability 0.30. An excess-of-loss layer attaches at $60, has a recovery limit of $90, and costs a fixed premium of $15. Give the net distribution and its mean. (Answer: $35 with probability 0.20, $75 with probability 0.50, and $125 with probability 0.30; mean $82.00.)
Gross loss is $0 with probability 0.25, $100 with probability 0.50, or $300 with probability 0.25. First cede 20% under quota share. Then apply excess-of-loss cover attaching at $80 with a recovery limit of $100 to the post-quota-share retained loss. The two fixed premiums are $8 and $12. Give the final distribution and its mean. (Answer: $20 with probability 0.25, $100 with probability 0.50, and $160 with probability 0.25; mean $95.00.)
Gross loss is $40 with probability 0.20, $100 with probability 0.30, $180 with probability 0.30, or $400 with probability 0.20. An excess-of-loss layer attaches at $75, has a recovery limit of $150, and costs a fixed premium of $25. Find the net distribution after combining equal outcomes and calculate its mean. (Answer: $65 with probability 0.20, $100 with probability 0.60, and $275 with probability 0.20; mean $128.00.)
Gross loss is $50 with probability 0.40, $150 with probability 0.40, or $350 with probability 0.20. First cede 10% under quota share; then apply excess-of-loss cover attaching at $100 with a recovery limit of $100 to the retained loss. The respective fixed premiums are $14 and $6. Find the final net distribution and its mean. (Answer: $65 with probability 0.40, $120 with probability 0.40, and $235 with probability 0.20; mean $121.00.)

What You'll Learn