Limited Expected Loss at a Policy Limit

Why a Policy Limit Changes Expected Loss

A ground-up claim may be larger than the amount an insurer pays under a policy limit. The limited expected loss measures the insurer’s average payment per loss when payment is capped at that limit. It is therefore different from the mean ground-up severity, which includes the full amount of every loss, and from expected excess, which measures the portion above the cap.

These distinctions matter when analyzing claim costs at different limits. A calculation must apply the policy limit to each loss before taking the expectation. The focus here is the limited expected loss for a single ground-up loss with no deductible or other coverage adjustment. The treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: Let $X$ be a nonnegative ground-up loss with finite mean, and let $u\geq 0$ be a per-loss policy limit. If the policy pays the loss up to that limit and has no deductible or other payment adjustment, the payment per loss is $Y^L=\min(X,u)$. The limited expected loss at limit $u$ is $E[\min(X,u)]$. The expected excess above the limit is $E[(X-u)^+]$, where $(z)^+=\max(z,0)$.

The limit caps each loss individually: a loss below $u$ is paid in full, while a loss above $u$ produces a payment of $u$. The expected excess is not the expected full loss conditional on exceeding the limit. It is the unconditional expected amount by which a loss exceeds the limit, counting zero excess whenever $X\leq u$.

The Relationship Between Limited Loss and Expected Excess

For each realized loss, the payment and the amount above the limit add back to the ground-up loss. This pointwise identity gives a useful check on any calculation.

X=\min(X,u)+(X-u)^+

To verify it, consider the two possible cases. If $X\leq u$, then $\min(X,u)=X$ and $(X-u)^+=0$, so the right-hand side is $X$. If $X>u$, then $\min(X,u)=u$ and $(X-u)^+=X-u$, so the right-hand side is again $X$. Taking expectations on both sides, which is valid because $X$ has finite mean, gives

E[X]=E[\min(X,u)]+E[(X-u)^+].

Thus the limited expected loss equals the uncapped mean less the expected excess. This identity does not require a continuous distribution; it also holds for discrete and mixed severity distributions, including distributions with probability mass exactly at the limit. It requires a nonnegative loss with finite mean and the payment structure just defined. A deductible, coinsurance, an aggregate limit, or another coverage term changes the payment function, so the limited-loss formula by itself would no longer describe the insurer’s payment.

Key distinction: $E[(X-u)^+]$ is expected excess per ground-up loss, including zero for losses at or below $u$. By contrast, $E[X-u\mid X>u]$ is the average excess conditional on a loss exceeding the limit. These are different quantities: the unconditional expected excess also reflects how often the limit is exceeded.

Worked Calculations

Worked Example: A Discrete Severity Distribution

A hypothetical insurer models ground-up severity for one type of claim as follows. The loss is $0 with probability 0.10, $20,000 with probability 0.30, $50,000 with probability 0.40, and $100,000 with probability 0.20. Find the limited expected loss at a $40,000 per-loss limit, the uncapped mean, and the expected excess above the limit.

Method and assumptions. The probabilities sum to $0.10+0.30+0.40+0.20=1.00$. With no deductible or other adjustment, the insurer’s payment for each possible loss is the smaller of the loss and $40,000.

Ground-up loss	Probability	Payment $\min(X,40{,}000)$	Excess $(X-40{,}000)^+$
$0	0.10	$0	$0
$20,000	0.30	$20,000	$0
$50,000	0.40	$40,000	$10,000
$100,000	0.20	$40,000	$60,000

Uncapped mean. Weight each ground-up loss by its probability:

E[X]=0.10(0)+0.30(20{,}000)+0.40(50{,}000)+0.20(100{,}000) =0+6{,}000+20{,}000+20{,}000 =\$46{,}000.

Limited expected loss. Weight each capped payment by its probability:

E[\min(X,40{,}000)] =0.10(0)+0.30(20{,}000)+0.40(40{,}000)+0.20(40{,}000) =0+6{,}000+16{,}000+8{,}000 =\$30{,}000.

Expected excess. Weight each excess amount by its probability:

E[(X-40{,}000)^+] =0.10(0)+0.30(0)+0.40(10{,}000)+0.20(60{,}000) =4{,}000+12{,}000 =\$16{,}000.

Check. The two limited components add to the uncapped mean: $\$30{,}000+\$16{,}000=\$46{,}000$. A common mistake is to cap only the $100,000 outcome but leave the $50,000 outcome unchanged. Both losses exceed the $40,000 limit, so both payments must be capped at $40,000.

Worked Example: An Exponential Severity Model

Suppose hypothetical ground-up severity $X$ has an exponential distribution with mean $80,000, and the per-loss limit is $160,000. Find the limited expected loss and expected excess above the limit. Report money to the nearest cent.

Method and assumptions. The exponential model has rate $\lambda=1/80{,}000$ per dollar, so $E[X]=1/\lambda=\$80{,}000$. Its memoryless property states that the conditional distribution of $X-u$, given $X>u$, is again exponential with mean $80,000. The probability of exceeding the limit is

P(X>160{,}000)=e^{-\lambda(160{,}000)} =e^{-160{,}000/80{,}000} =e^{-2} \approx 0.1353352832.

Expected excess. Conditional on exceeding $160,000, the expected amount above the limit is $80,000 by memorylessness. The unconditional expected excess is therefore the exceedance probability multiplied by the conditional expected excess:

E[(X-160{,}000)^+] =P(X>160{,}000)E[X-160{,}000\mid X>160{,}000] =e^{-2}(80{,}000) \approx \$10{,}826.82.

Limited expected loss. Use the pointwise decomposition of the ground-up loss:

E[\min(X,160{,}000)] =E[X]-E[(X-160{,}000)^+] =80{,}000-80{,}000e^{-2} \approx \$69{,}173.18.

The unrounded limited expected loss is $80{,}000(1-e^{-2})=\$69{,}173.177\ldots$, and the unrounded expected excess is $80{,}000e^{-2}=\$10{,}826.822\ldots$. Both amounts are rounded to the nearest cent only at the final step.

Check and common mistake. Their rounded values add to $80,000. Do not calculate the excess as $P(X>u)E[X\mid X>u]$: that is the expected full loss on the event of exceeding the limit, not the expected amount above the limit. Here $E[X\mid X>160{,}000]=\$240{,}000$, so that incorrect calculation gives $e^{-2}(240{,}000)\approx \$32{,}480.48$. The excess calculation must subtract the $160,000 limit from the conditional severity before multiplying by the exceedance probability.

Worked Example: A Uniform Severity Distribution

A hypothetical model assumes that ground-up severity is uniformly distributed from $0 to $120,000. A policy pays up to $90,000 per loss. Calculate the uncapped mean, limited expected loss, and expected excess above the limit.

Method and assumptions. For $X$ uniform on $[0,120{,}000]$, the density is $1/120{,}000$, and the mean is the midpoint of the interval. The limit is below the maximum possible loss, so losses from $90,000 through $120,000 are capped at $90,000.

E[X]=\frac{0+120{,}000}{2}=\$60{,}000.

For the limited expected loss, integrate the payment over the range of possible severities. Losses below $90,000 are paid in full; losses above $90,000 are paid $90,000:

\begin{aligned} E[\min(X,90{,}000)] &=\frac{1}{120{,}000} \left(\int_0^{90{,}000}x\,dx+\int_{90{,}000}^{120{,}000}90{,}000\,dx\right)\\ &=\frac{1}{120{,}000} \left(\frac{90{,}000^2}{2}+90{,}000(30{,}000)\right)\\ &=\frac{4{,}050{,}000{,}000+2{,}700{,}000{,}000}{120{,}000} =\$56{,}250. \end{aligned}

For the expected excess, only losses above $90,000 contribute. Over that interval, the excess rises from $0 to $30,000:

\begin{aligned} E[(X-90{,}000)^+] &=\frac{1}{120{,}000}\int_{90{,}000}^{120{,}000}(x-90{,}000)\,dx\\ &=\frac{\tfrac12(30{,}000)(30{,}000)}{120{,}000} =\$3{,}750. \end{aligned}

Check. The results reconcile: $\$56{,}250+\$3{,}750=\$60{,}000$, the uncapped mean. The limited expected loss is less than the mean because the cap removes some payment on losses above $90,000.

Interpretation and Common Traps

Limited expected loss is an average payment per ground-up loss under a specified cap. It is not an average conditional on a claim exceeding the limit, and it is not the expected full severity among losses that exceed the limit. Keeping the conditioning event and the amount being averaged explicit prevents common errors.

Using the uncapped mean as the capped mean. Unless $P(X>u)=0$, at least some losses are reduced by the cap, so $E[\min(X,u)]\leq E[X]$.
Subtracting the limit from every loss. Losses at or below the limit have zero excess. The positive-part expression $(X-u)^+$ enforces this.
Confusing unconditional and conditional excess. $E[(X-u)^+]$ includes the probability of exceeding the limit. A conditional excess mean, given $X>u$, does not.
Applying the cap to the wrong loss basis. The definition here starts with ground-up severity and a limit-only payment. If a deductible or another coverage term applies, first specify the actual payment function; do not silently reuse $\min(X,u)$.

As a basic reasonableness check, for nonnegative $X$ and $u\geq0$, the limited expected loss must be between zero and both $u$ and $E[X]$. The expected excess must be nonnegative, and their sum must equal $E[X]$. These checks can reveal a mistaken cap, an omitted probability weight, or a confusion between excess and full severity.

Key takeaway: For a ground-up loss capped at $u$, the limited expected loss is $E[\min(X,u)]$. It measures expected payment under the cap, while $E[(X-u)^+]$ measures expected excess; together they equal the uncapped mean.

Check Your Understanding

For each problem, identify the capped payment and distinguish it from expected excess. Answers are rounded to the nearest cent where needed.

A loss is $0, $10,000, $40,000, or $80,000 with respective probabilities 0.20, 0.30, 0.30, and 0.20. At a $30,000 limit, find the uncapped mean, limited expected loss, and expected excess. (Answer: uncapped mean $31,000; limited expected loss $18,000; expected excess $13,000.)
Severity is exponential with mean $50,000, and the limit is $100,000. Find the limited expected loss and expected excess. (Answer: limited expected loss $43,233.24; expected excess $6,766.76.)
Severity is uniform from $0 to $100,000, and the limit is $60,000. Find the uncapped mean, limited expected loss, and expected excess. (Answer: uncapped mean $50,000; limited expected loss $42,000; expected excess $8,000.)
A loss is $90,000 with probability 0.25, $150,000 with probability 0.15, $300,000 with probability 0.10, and $0 otherwise. At a $120,000 limit, find the uncapped mean, limited expected loss, and unconditional expected excess. (Answer: uncapped mean $75,000; limited expected loss $52,500; expected excess $22,500.)
A loss is $20,000 with probability 0.40, $100,000 with probability 0.35, and $200,000 with probability 0.25. The limit is $100,000. Find the limited expected loss, expected excess, and expected excess conditional on exceeding the limit. (Answer: limited expected loss $68,000; unconditional expected excess $25,000; conditional expected excess $100,000.)

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