Worked Case: Premium-Based Expected Ultimate And Reserve

Purpose and Scope of a Premium-Based Estimate

A premium-based expected loss estimate starts with the amount of premium earned for a period and a selected expected loss ratio. It converts that premium into an estimate of ultimate losses. Subtracting losses already paid gives a residual estimate of total unpaid losses at the evaluation date.

This is useful when estimating ultimate losses from an a priori expectation rather than projecting the observed losses through a development pattern. The calculation is simple, but its inputs must refer to the same exposure period and loss scope. A mismatch between earned and written premium, or between loss-only and loss-plus-expense amounts, can materially distort the result. The method and calculations here are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $EP$ be earned premium for the period and $ELR$ be the selected expected loss ratio, expressed as a decimal. The premium-based expected ultimate loss estimate is $U^{\text{sel}}=EP\times ELR$. If $D$ is paid loss through the evaluation date, on the same scope as $U^{\text{sel}}$, the selected total unpaid loss estimate is $R^{\text{sel}}=U^{\text{sel}}-D$. This residual includes all losses not yet paid; it is not limited to claims that have not yet been reported.

The expected loss ratio is an assumption or selection representing expected losses per dollar of earned premium. In this calculation, it must be suitable for the period, coverage, and loss definition being estimated. For example, an expected loss ratio for losses excluding loss adjustment expense (LAE) should be paired with paid losses excluding LAE. The estimate also assumes that the premium is earned for the same business and period as the losses being estimated.

Why the Calculation Works

The ultimate-loss calculation follows directly from the meaning of the expected loss ratio. If the selected ratio is expected losses divided by earned premium, multiplying the ratio by earned premium recovers the expected losses:

ELR=\frac{\text{selected expected ultimate losses}}{EP} \quad\Longrightarrow\quad U^{\text{sel}}=EP\times ELR.

The unpaid-loss calculation follows from the accounting identity for a fixed set of losses. At an evaluation date, ultimate losses equal paid losses to date plus the remaining unpaid losses. Rearranging gives unpaid losses as ultimate losses minus paid losses:

U_{\text{actual}}=D+R_{\text{actual}} \quad\Longrightarrow\quad R_{\text{actual}}=U_{\text{actual}}-D.

This identity does not mean that subtracting observed paid losses from a premium-based ultimate estimate automatically gives the conditional expected value of actual unpaid losses. If paid losses $D$ are random, the conditional expectation identity is

E[R_{\text{actual}}\mid D] = E[U_{\text{actual}}\mid D]-D.

The premium-based calculation instead produces a selected ultimate estimate from premium and an expected loss ratio, then takes the observed paid amount as the portion already settled. It is a reserve estimate based on those selected inputs; without a model linking paid experience to ultimate losses, it should not be described as a conditional expectation given the observed paid amount.

Confirm the basis.
Identify the period, earned premium, expected loss ratio, evaluation date, and whether the loss figures include LAE or other components.

Calculate expected ultimate losses.
Multiply earned premium by the expected loss ratio, using the ratio as a decimal.

Deduct paid losses.
Subtract paid losses through the evaluation date only after confirming that they have the same scope as the expected ultimate loss estimate.

Interpret the residual.
State that the result estimates total unpaid losses on the stated basis. Do not relabel it as an estimate of unreported losses alone.

Worked Applications

Worked Example: One Period With Written and Earned Premium

A hypothetical insurer is estimating losses for a policy period. Written premium is $2,600,000, but earned premium for the period is $2,400,000. The selected expected loss ratio is 62%. Paid losses through the evaluation date are $910,000. All loss figures exclude LAE. Estimate expected ultimate losses and total unpaid losses.

Method. Use earned premium, not written premium, because the estimate is for losses associated with premium earned during the period.

First convert the ratio to decimal form:

62\%=0.62.

Calculate the expected ultimate losses:

U^{\text{sel}} =EP\times ELR =\$2{,}400{,}000\times 0.62 =\$1{,}488{,}000.

Then subtract paid losses on the same loss-only basis:

R^{\text{sel}} =U^{\text{sel}}-D =\$1{,}488{,}000-\$910{,}000 =\$578{,}000.

Check. The estimate reconciles: $910,000 paid plus $578,000 unpaid equals $1,488,000 ultimate. The $578,000 is total unpaid loss, not just unreported loss.

Common mistake avoided. Using written premium would produce $2,600,000 × 0.62 = $1,612,000 of expected ultimate losses and an apparent $702,000 unpaid estimate. That calculation uses the wrong premium measure for this stated earned-period estimate.

Worked Example: Combining Two Segments

A hypothetical portfolio has two segments, both evaluated on a loss-only basis. Segment A has earned premium of $1,200,000, an expected loss ratio of 55%, and paid losses of $370,000. Segment B has earned premium of $800,000, an expected loss ratio of 70%, and paid losses of $560,000. Calculate the portfolio expected ultimate losses and unpaid losses.

Method. Calculate expected ultimate losses within each segment, then add the segment estimates. This preserves the different expected loss ratios and avoids applying an unweighted average ratio to the portfolio.

For Segment A:

U_A^{\text{sel}} =\$1{,}200{,}000\times 0.55 =\$660{,}000.

For Segment B:

U_B^{\text{sel}} =\$800{,}000\times 0.70 =\$560{,}000.

Add the segment ultimate estimates:

U_{\text{portfolio}}^{\text{sel}} =\$660{,}000+\$560{,}000 =\$1{,}220{,}000.

Total paid losses are $370,000 + $560,000 = $930,000. Therefore:

R_{\text{portfolio}}^{\text{sel}} =\$1{,}220{,}000-\$930{,}000 =\$290{,}000.

The equivalent portfolio loss ratio is premium-weighted:

ELR_{\text{portfolio}} =\frac{\$1{,}220{,}000}{\$1{,}200{,}000+\$800{,}000} =\frac{\$1{,}220{,}000}{\$2{,}000{,}000} =0.61.

Check. Applying 61% to the $2,000,000 total earned premium gives $1,220,000, matching the sum of segment estimates. Adding $930,000 paid and $290,000 unpaid also gives $1,220,000.

Common mistake avoided. The unweighted average of 55% and 70% is 62.5%. Applying it to $2,000,000 would produce $1,250,000, overstating expected ultimate losses by $30,000. The correct aggregate ratio weights each segment by its earned premium.

Worked Example: Keeping Loss and LAE Scope Consistent

A hypothetical book has $5,600,000 of earned premium. Its selected expected loss ratio is 58% for losses excluding LAE. At the evaluation date, paid losses plus paid LAE total $2,600,000, of which $190,000 is paid LAE. Estimate expected ultimate losses and unpaid losses, excluding LAE.

Method. The expected loss ratio targets loss only, so the paid amount used in the subtraction must also exclude LAE.

Expected ultimate losses are:

U^{\text{sel}} =\$5{,}600{,}000\times 0.58 =\$3{,}248{,}000.

First isolate paid losses from the reported combined paid amount:

D_{\text{loss}} =\$2{,}600{,}000-\$190{,}000 =\$2{,}410{,}000.

Now calculate unpaid losses, excluding LAE:

R_{\text{loss}}^{\text{sel}} =\$3{,}248{,}000-\$2{,}410{,}000 =\$838{,}000.

Check. On the loss-only basis, $2,410,000 paid plus $838,000 unpaid equals $3,248,000 expected ultimate losses.

Common mistake avoided. Subtracting all $2,600,000 of combined paid loss and LAE would give $648,000. That is not a loss-only unpaid estimate: it subtracts $190,000 of LAE from an ultimate estimate that contains no LAE. The calculation also does not estimate unpaid LAE; a loss-only expected loss ratio provides no LAE estimate by itself.

Interpretation and Common Traps

The premium-based estimate is driven by two selections: earned premium and the expected loss ratio. The paid amount determines how much of the selected ultimate has already been paid, but it does not, by itself, revise the selected ultimate in this method. That distinction matters when interpreting the result: a large residual does not prove that the ultimate estimate is too high, and a small residual does not prove that the expected loss ratio is accurate.

Check	What to confirm	Typical error
Premium measure	Use earned premium for the period being estimated.	Substituting written premium without a reason.
Ratio format	Use the expected loss ratio as a decimal in multiplication.	Multiplying by 62 instead of 0.62.
Loss scope	Match loss-only or loss-plus-LAE scope in the ultimate and paid amounts.	Subtracting combined paid amounts from a loss-only ultimate.
Period and evaluation date	Ensure paid losses relate to the same period and are measured through the stated date.	Including payments from another period or after the evaluation date.
Interpretation	Describe the residual as total unpaid losses on the selected basis.	Calling all unpaid losses unreported losses.

A negative residual can occur if paid losses exceed the selected expected ultimate losses. It is a signal to check the inputs and assumptions, not an algebraic reason to silently replace the result with zero. Possible causes include unusually high observed payments, a low selected expected loss ratio, a scope mismatch, or a data error. The appropriate interpretation depends on the context and on any applicable estimation guidance.

Key takeaway: Calculate expected ultimate losses as earned premium times the expected loss ratio, then subtract paid losses measured on the same basis. Keep the selected estimate, the observed paid amount, and any interpretation of their difference conceptually distinct.

Check Your Understanding

For each problem, keep the premium period, loss scope, and evaluation date in view. Report dollar answers to the nearest dollar.

A hypothetical insurer has earned premium of $1,800,000, an expected loss ratio of 57%, and paid losses of $735,000 through the evaluation date. Estimate expected ultimate and unpaid losses. (Answer: expected ultimate losses = $1,026,000; unpaid losses = $291,000.)
A portfolio has two loss-only segments. Segment A has earned premium of $900,000, an expected loss ratio of 60%, and paid losses of $310,000. Segment B has earned premium of $1,400,000, an expected loss ratio of 50%, and paid losses of $480,000. Find total expected ultimate losses, total unpaid losses, and the premium-weighted portfolio expected loss ratio. (Answer: expected ultimate losses = $1,240,000; unpaid losses = $450,000; portfolio expected loss ratio = 53.91%.)
A hypothetical book has earned premium of $4,200,000 and a selected expected loss ratio of 72%. Paid losses through the evaluation date are $2,150,000. Calculate expected ultimate losses and the unpaid estimate, then state whether the unpaid estimate includes unreported losses only or all unpaid losses. (Answer: expected ultimate losses = $3,024,000; unpaid losses = $874,000; it estimates all unpaid losses, not only unreported losses.)
Earned premium is $2,200,000 and the expected loss ratio is 65% for losses excluding LAE. Paid losses plus paid LAE equal $1,170,000, including $120,000 of paid LAE. Estimate expected ultimate losses and unpaid losses excluding LAE. (Answer: expected ultimate losses = $1,430,000; paid losses excluding LAE = $1,050,000; unpaid losses excluding LAE = $380,000.)
A hypothetical segment has earned premium of $3,000,000, an expected loss ratio of 48%, and paid losses of $1,560,000 on the same basis. Calculate the selected ultimate and residual unpaid estimate, and identify one input or assumption to review before interpreting a negative residual as an error. (Answer: expected ultimate losses = $1,440,000; residual unpaid estimate = −$120,000; review, for example, the selected loss ratio or the period and scope of paid losses.)

What You'll Learn