Separating Closure Effects from Closed-Claim Severity

Why Separate Closure From Severity?

A paid-loss increase does not, by itself, reveal what changed in the claims process. More claims may have closed by the valuation date, raising cumulative paid losses even if the average amount paid per closed claim is unchanged. Alternatively, the closure proportion may be stable while average paid severity rises. Open claims may also have received payments, so total cumulative paid losses can move differently from paid losses on closed claims alone.

Separating these effects is useful when interpreting paid-loss movement in a Berquist-Sherman analysis. A settlement-rate adjustment can standardize for a change in the proportion of claims closed, but it does not automatically remove changes in closed-claim severity or account for payments on open claims. The calculation must match the quantity being analyzed.

Definition: Consider a fixed cohort of $N$ claims observed at two valuation dates, indexed by $t=0,1$. Let $q_t$ be the proportion of the cohort closed by date $t$, $s_t$ the average cumulative paid amount per closed claim, and $o_t$ the average cumulative paid amount per open claim. Define closed-claim paid losses as $C_t=Nq_ts_t$, open-claim paid losses as $O_t=N(1-q_t)o_t$, and total cumulative paid losses as $P_t=C_t+O_t$. These are deterministic cohort summaries; no interest, mortality, or probability distribution is assumed. Amounts are nominal cumulative payments on a consistent valuation basis.

The separation requires consistent definitions across dates: the same cohort and claim count, the same treatment of reopened claims, the same currency and payment basis, and a clear rule for which claims count as closed. In the examples below, a claim is classified as either closed or open at each valuation date, and reopened-claim complications are excluded. The average $s_t$ is paid to date per claim closed at date $t$, not an estimate of ultimate severity. Similarly, $o_t$ is paid to date per claim still open.

Decomposing Closed-Claim Paid Losses

For closed claims alone, cumulative paid losses equal $C=Nqs$. If $q$ changes, then the number of closed claims changes; if $s$ changes, the average paid amount among those closed claims changes. The total change is

\Delta C=C_1-C_0=N(q_1s_1-q_0s_0).

There is no unique way to assign the interaction between a change in $q$ and a change in $s$ to separate effects. One exact method is a symmetric decomposition: average the closure effect calculated at the two severities, and average the severity effect calculated at the two closure proportions.

Theorem: For a fixed cohort, the exact symmetric decomposition of the change in closed-claim paid losses is $$ \Delta C= \underbrace{N(q_1-q_0)\frac{s_0+s_1}{2}}_{\text{closure contribution}} + \underbrace{N(s_1-s_0)\frac{q_0+q_1}{2}}_{\text{closed-claim severity contribution}}. $$

The closure contribution answers how much paid loss is associated with the change in the proportion closed, using the average of the two observed closed-claim severities. The severity contribution answers how much is associated with the change in average paid severity, using the average of the two observed closure proportions. These contributions sum exactly to the observed change.

Why the Symmetric Decomposition Works

The identity follows by adding and subtracting terms in two different orders, then averaging the results. Changing closure first gives $N(q_1-q_0)s_0+Nq_1(s_1-s_0)$. Changing severity first gives $Nq_0(s_1-s_0)+N(q_1-q_0)s_1$. Both expressions equal $N(q_1s_1-q_0s_0)$, because the intermediate product cancels in each telescoping sum.

Averaging the two expressions gives a closure term with the average severity and a severity term with the average closure proportion:

\begin{aligned} \Delta C &=\frac{1}{2}\left[N(q_1-q_0)s_0+Nq_1(s_1-s_0)\right]\\ &\quad+\frac{1}{2}\left[N(q_1-q_0)s_1+Nq_0(s_1-s_0)\right]\\ &=N(q_1-q_0)\frac{s_0+s_1}{2} +N(s_1-s_0)\frac{q_0+q_1}{2}. \end{aligned}

The symmetric convention avoids making the attribution depend entirely on which variable is changed first. A closure-first or severity-first decomposition is also exact, but its component amounts generally differ because each assigns the interaction differently. State the convention used whenever reporting components.

Worked Example: Symmetric Attribution for Closed Claims

A hypothetical cohort contains $1{,}200$ claims. At the first valuation date, $25\%$ are closed and the average cumulative payment per closed claim is $\$4{,}000$. At the second date, $30\%$ are closed and the corresponding average is $\$3{,}800$. Find the symmetric closure and severity contributions, and verify their sum. The cohort values are deterministic; there is no discounting or assumed probability distribution.

Method. Apply the symmetric decomposition to closed-claim paid losses, $C=Nqs$. The closure proportion changes by $0.30-0.25=0.05$, and average closed-claim paid severity changes by $\$3{,}800-\$4{,}000=-\$200$.

Closure contribution. The average of the two severities is $(\$4{,}000+\$3{,}800)/2=\$3{,}900$. Therefore,

1{,}200(0.30-0.25)(3{,}900)=\$234{,}000.

Closed-claim severity contribution. The average closure proportion is $(0.25+0.30)/2=0.275$. Therefore,

1{,}200(-\$200)(0.275)=-\$66{,}000.

Check. The contributions sum to $\$234{,}000-\$66{,}000=\$168{,}000$. Directly, $C_0=1{,}200(0.25)(\$4{,}000)=\$1{,}200{,}000$, while $C_1=1{,}200(0.30)(\$3{,}800)=\$1{,}368{,}000$. The direct change is also $\$168{,}000$. The symmetric decomposition avoids the mistake of multiplying the severity change by the sum, rather than the average, of the closure proportions.

Including Payments on Open Claims

A frequent source of error is to treat all cumulative paid losses as if they were paid only on closed claims. When open claims have received payments, total paid losses are $P=N[qs+(1-q)o]$, not merely $Nqs$. Thus, the change in $P$ can differ from the change in $C$, and a change in closure proportion affects the mix of claims represented by closed and open amounts.

For a closure-first decomposition of total paid losses, first change $q$ while holding both average payment amounts at their date-0 values. Then change the closed- and open-claim average payments using the date-1 closure proportion. This convention gives an exact identity:

\Delta P= \underbrace{N(q_1-q_0)(s_0-o_0)}_{\text{closure effect}} + \underbrace{N\left[q_1(s_1-s_0)+(1-q_1)(o_1-o_0)\right]}_{\text{payment-amount effects}}.

The closure effect depends on the difference between the baseline average paid amounts for closed and open claims. If closed claims have more paid to date on average, raising the closure proportion increases total paid under this decomposition; lowering it decreases total paid. The second component includes both the change in closed-claim average payment and the change in open-claim average payment. This is an exact closure-first attribution, not a unique one: changing the order of the steps assigns interaction terms differently.

Worked Example: Closure-First Components With Open-Claim Payments

A hypothetical fixed cohort has $250$ claims. At date 0, $q_0=0.60$, $s_0=\$5{,}000$, and $o_0=\$1{,}000$. At date 1, $q_1=0.50$, $s_1=\$5{,}600$, and $o_1=\$1{,}400$. Calculate total paid at each date and its closure-first components. Treat all amounts as cumulative, with no discounting or assumed probability distribution.

Method. Include payments on both closed and open claims in $P=N[qs+(1-q)o]$. Do not apply the closed-claim formula $Nqs$ to total paid losses.

Date-0 total. The closed-claim component is $250(0.60)(\$5{,}000)=\$750{,}000$. The open-claim component is $250(0.40)(\$1{,}000)=\$100{,}000$. Thus $P_0=\$750{,}000+\$100{,}000=\$850{,}000$.

Date-1 total. The closed-claim component is $250(0.50)(\$5{,}600)=\$700{,}000$. The open-claim component is $250(0.50)(\$1{,}400)=\$175{,}000$. Thus $P_1=\$700{,}000+\$175{,}000=\$875{,}000$, so the total increase is $\$25{,}000$.

Closure effect. The closure proportion falls by $0.10$. At the baseline averages, the closed-versus-open payment difference is $\$5{,}000-\$1{,}000=\$4{,}000$. Therefore the closure effect is $250(-0.10)(\$4{,}000)=-\$100{,}000$.

Payment-amount effects. At $q_1=0.50$, the closed-claim payment effect is $250(0.50)(\$5{,}600-\$5{,}000)=\$75{,}000$. The open-claim payment effect is $250(0.50)(\$1{,}400-\$1{,}000)=\$50{,}000$. Together, these amount effects are $\$125{,}000$.

Check. The components sum to $-\$100{,}000+\$125{,}000=\$25{,}000$, matching $P_1-P_0=\$875{,}000-\$850{,}000$. The common mistake is to report the closure effect alone as the total change; here, the payment-amount changes more than offset that negative effect.

When Settlement-Rate Standardization Misleads

A simple standardization of closed-claim paid losses to a reference closure proportion $q_r$ uses $Nq_rs_t$. Equivalently, when $q_t>0$, it scales observed closed-claim paid losses $C_t$ by $q_r/q_t$. This removes the mechanical effect of the observed closure proportion for the closed-claim component, under the cohort and measurement assumptions already stated. It does not remove a change in $s_t$, and it does not estimate total paid losses unless open-claim payments are separately handled.

The distinction matters because a closure-rate adjustment can look successful while the interpretation is wrong. A standardized closed-claim amount can still rise because average paid severity among closed claims increased. Also, the adjustment says nothing by itself about changes in average payments on claims that remain open. A one-factor adjustment to total paid losses is therefore not justified merely because the closure proportion changed.

Worked Example: Separating Standardized Closed Payments From Total Paid

A hypothetical cohort contains $400$ claims. At date 0, $q_0=0.40$, $s_0=\$6{,}000$, and $o_0=\$2{,}000$. At date 1, $q_1=0.60$, $s_1=\$7{,}000$, and $o_1=\$2{,}500$. Calculate the change in total paid, the date-1 closed-claim amount standardized to the date-0 closure proportion, and what that standardized comparison does and does not show. All amounts are deterministic cumulative payments with no discounting or assumed probability distribution.

Method. First reconcile total paid using closed and open claims. Then standardize only the closed-claim amount to $q_0$; do not label that result as standardized total paid.

Total paid at date 0. Closed claims contribute $400(0.40)(\$6{,}000)=\$960{,}000$. Open claims contribute $400(0.60)(\$2{,}000)=\$480{,}000$. Thus $P_0=\$1{,}440{,}000$.

Total paid at date 1. Closed claims contribute $400(0.60)(\$7{,}000)=\$1{,}680{,}000$. Open claims contribute $400(0.40)(\$2{,}500)=\$400{,}000$. Thus $P_1=\$2{,}080{,}000$, an increase of $\$640{,}000$.

Closure-first check. The closure effect is $400(0.60-0.40)(\$6{,}000-\$2{,}000)=\$320{,}000$. The payment-amount effects are $400[0.60(\$7{,}000-\$6{,}000)+0.40(\$2{,}500-\$2{,}000)]=\$320{,}000$. Together they equal the $\$640{,}000$ total increase.

Standardized closed-claim comparison. Scaling the date-1 closed-claim amount to the date-0 closure proportion gives $\$1{,}680{,}000(0.40/0.60)=\$1{,}120{,}000$. Date-0 closed-claim paid losses were $\$960{,}000$, so the standardized closed-claim increase is $\$160{,}000$. This agrees with $400(0.40)(\$7{,}000-\$6{,}000)=\$160{,}000$: it reflects the changed average among closed claims at the reference closure proportion.

Check and interpretation. The standardized increase of $\$160{,}000$ is not the total paid increase of $\$640{,}000$. It excludes the open-claim payment change and is not intended to represent all components of total paid. A closure-only interpretation would miss both the rise in closed-claim average paid severity and the change in open-claim average payments.

Common Exam Traps

Using the wrong average in the symmetric formula. The closure contribution uses $(s_0+s_1)/2$; the severity contribution uses $(q_0+q_1)/2$. The sums are divided by two.
Confusing closed-claim paid losses with total paid losses. $Nqs$ excludes payments on open claims. Total paid is $N[qs+(1-q)o]$ when open claims have paid amounts.
Calling a standardized closed amount “total paid.” Scaling $C_t$ by a closure-rate ratio addresses the closed-claim component under stated assumptions. It does not adjust open-claim payments automatically.
Treating a component as uniquely determined. An exact decomposition depends on the attribution convention. State whether it is symmetric, closure-first, or another clearly defined ordering.
Interpreting $s_t$ as ultimate severity. It is cumulative paid per claim closed by the valuation date. It need not equal ultimate loss per claim.

Key takeaway: First identify whether the target is paid loss on closed claims or total paid loss. For closed claims, separate the change in closure proportion from the change in average paid severity. For total paid, also account for open-claim payments, and state the attribution convention used.

Check Your Understanding

For each problem, treat the cohort data as deterministic cumulative payments, with no discounting or assumed probability distribution. Show the component calculation before checking the answer.

A fixed cohort of $500$ claims has $q_0=0.20$, $q_1=0.30$, $s_0=\$3{,}000$, and $s_1=\$3{,}200$. Use the symmetric decomposition for closed-claim paid losses. (Answer: Closure contribution $\$155{,}000$; severity contribution $\$25{,}000$; total change $\$180{,}000$.)
A fixed cohort of $100$ claims has $q_0=0.70$, $q_1=0.60$, $s_0=\$4{,}500$, $s_1=\$5{,}000$, and $o_0=o_1=\$1{,}000$. Use the closure-first decomposition for total paid. (Answer: $P_0=\$345{,}000$, $P_1=\$340{,}000$; closure effect $-\$35{,}000$; payment-amount effect $+\$30{,}000$; total change $-\$5{,}000$.)
A cohort of $200$ claims has $q_0=0.50$, $q_1=0.65$, $s_0=\$8{,}000$, and $s_1=\$7{,}600$. Find the symmetric components for closed-claim paid losses and verify the change directly. (Answer: Closure contribution $+\$234{,}000$; severity contribution $-\$46{,}000$; total change $+\$188{,}000$, from $\$800{,}000$ to $\$988{,}000$.)
A cohort of $300$ claims has $q_0=0.40$, $q_1=0.50$, $s_0=s_1=\$5{,}000$, $o_0=\$1{,}500$, and $o_1=\$2{,}500$. Use the closure-first decomposition for total paid. (Answer: $P_0=\$870{,}000$, $P_1=\$1{,}125{,}000$; closure effect $+\$105{,}000$; payment-amount effect $+\$150{,}000$; total change $+\$255{,}000$.)
A cohort of $1{,}000$ claims has closed-claim proportions $q_0=0.30$, $q_1=0.50$, and average closed-claim paid amounts $s_0=\$4{,}000$, $s_1=\$4{,}600$. Standardize the date-1 closed-claim amount to $q_0$, compare it with date 0, and identify a limitation. (Answer: Standardized date-1 closed amount $=\$1{,}380{,}000$; increase over date-0 closed paid losses of $\$1{,}200{,}000$ is $\$180{,}000$; this isolates the closed-claim severity change at $q_0$, but does not determine the total paid change without open-claim payment data.)

What You'll Learn