Short-Term Policies And Nonannual Terms

Why Policy Term Length Matters

A policy count does not, by itself, tell you how much time was insured. One six-month policy and one eighteen-month policy each count as one policy, but they provide different durations of coverage. If an analysis uses a time-based exposure unit, treating both as one annual exposure would make the denominator inconsistent with the coverage provided.

This tutorial converts short-term and nonannual policy terms into exposure units under an explicitly stated convention. The central rule is simple: define what one exposure unit represents, then express every policy term on that same basis. This is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: A policy-year equivalent (PYE) is a time-based exposure unit representing one policy exposed for one year, when the exposure basis is one policy. Under a month-based convention, a policy term contributes its number of months divided by 12 PYEs. Under a day-based convention with a 365-day year, it contributes its covered days divided by 365 PYEs. A policy count is the number of policies; it is not automatically the same as the number of PYEs.

The convention must be stated rather than assumed. For example, a calculation may use a month basis in which each month is one-twelfth of a year, or a day basis in which a year is fixed at 365 days. If one calculation uses months and another uses actual days, their results may differ slightly. That difference is not necessarily an error, but mixing conventions within the same calculation is.

A time-based PYE is also not the only possible exposure basis. Some analyses use a count of policies or another exposure base selected for the coverage. Those units answer different questions. The conversion rules here apply when the stated convention is time-based and one policy is the underlying exposure base.

Converting a Policy Term to Annualized Exposure

For a policy covering a stated number of months, the conversion is the term length divided by 12. For a policy with a stated number of covered days under a 365-day convention, divide the term length by 365. For a portfolio, calculate each policy’s contribution and add the contributions.

\text{PYE for one policy} = \begin{cases} \frac{\text{term in months}}{12}, & \text{month-based convention},\\[6pt] \frac{\text{term in days}}{365}, & \text{365-day convention}. \end{cases}

\text{Total PYE}=\sum_{j=1}^{n}\text{PYE}_j

The sum can be greater than the number of policies or less than it. A term longer than one year contributes more than one PYE; a term shorter than one year contributes less than one PYE. This is a consequence of measuring duration, not a claim that the number of contracts changed.

Why the conversion works: Under a 365-day convention, one PYE is defined as 365 covered days. Thus one covered day represents $1/365$ of a PYE. With $D$ covered days, the total is $D$ contributions of $1/365$, or $D/365$ PYE. The same reasoning applies to months: each month contributes $1/12$ PYE, so $M$ months contribute $M/12$ PYE. Because the contribution is proportional to time, separate policy terms can be added after conversion to the same unit.

Written and Earned Exposure for Nonannual Terms

For a policy recorded at inception, written exposure under a time-based convention reflects the full scheduled policy term. If a policy is written for nine months, its written exposure is $9/12=0.75$ PYE on a month basis. A policy written for eighteen months contributes $18/12=1.5$ PYEs.

Earned exposure reflects the portion of the term that has elapsed by the valuation date. The same conversion applies, but the duration used is the elapsed covered time rather than the full scheduled term. For example, if 120 days of a policy’s term have elapsed under a 365-day convention, its earned exposure is $120/365$ PYE. The remaining term is not earned exposure at that date.

This distinction is especially important for multi-year terms: the full term may contribute more than one written PYE, while only the elapsed portion contributes to earned exposure as of a particular date. Apply one convention consistently to both the duration being measured and the portfolio total.

Worked Example: Combining Several Short and Long Terms

A hypothetical insurer records three groups of policies for one coverage. There are 84 six-month terms, 36 fifteen-month terms, and 20 three-month terms. Use a month-based convention. Find the total policy count and total written exposure in PYEs. No interest, mortality, or loss distribution is assumed; this is deterministic exposure arithmetic.

Method. Convert each term to PYEs by dividing its months by 12, multiply by the number of policies in the group, and then add the groups.

For the 84 six-month policies, each contributes $6/12=0.5$ PYE:

84 \times \frac{6}{12}=84 \times 0.5=42\text{ PYE}.

For the 36 fifteen-month policies:

36 \times \frac{15}{12}=36 \times 1.25=45\text{ PYE}.

For the 20 three-month policies:

20 \times \frac{3}{12}=20 \times 0.25=5\text{ PYE}.

The policy count is $84+36+20=140$. Total written exposure is $42+45+5=92$ PYEs. As a check, the combined policy-months are $84(6)+36(15)+20(3)=504+540+60=1{,}104$ months, and $1{,}104/12=92$ PYEs.

Common mistake avoided: Counting all 140 policies as 140 annual exposures ignores the term lengths. The correct result is 140 policies but 92 PYEs; these are different measures.

Worked Example: Written and Earned Exposure on a Day Basis

A hypothetical portfolio contains three policy terms with durations of 120, 270, and 540 covered days. At a valuation date, 60, 135, and 365 days, respectively, have elapsed. Use a fixed 365-day year for every conversion. Find total written exposure, total earned exposure, and total unearned exposure. Treat the elapsed-day values as given; no calendar-date or leap-year adjustment is needed. No interest, mortality, or loss distribution is assumed.

Method. Divide each duration by 365. Use full term days for written exposure, elapsed days for earned exposure, and remaining days for unearned exposure.

The full terms total $120+270+540=930$ days. Written exposure is:

\frac{930}{365}=2.547945\ldots\text{ PYE}.

Elapsed days total $60+135+365=560$. Earned exposure is:

\frac{560}{365}=1.534246\ldots\text{ PYE}.

The remaining days are $120-60=60$, $270-135=135$, and $540-365=175$, for a total of $60+135+175=370$ days. Unearned exposure is:

\frac{370}{365}=1.013699\ldots\text{ PYE}.

Rounded to four decimal places, the results are 2.5479 written PYE, 1.5342 earned PYE, and 1.0137 unearned PYE. The check is $560+370=930$ days, so earned plus unearned exposure equals written exposure before rounding.

Common mistake avoided: Dividing elapsed days by each policy’s full term would produce a proportion of each policy earned, not the amount of earned exposure in PYEs. The day-based PYE denominator is 365 for every policy in this example.

Keeping the Exposure Unit Consistent

A calculated rate or average inherits the unit of its denominator. For example, dividing a dollar amount by PYEs produces dollars per PYE; dividing by policy count produces dollars per policy. The two results may both be arithmetically valid, but they are not the same measure and should not be described as though they were.

The following example uses a stipulated loss amount only to show the consequence of choosing the wrong denominator. It does not require a loss model or a claim probability assumption.

Worked Example: Matching a Loss Total to the Exposure Denominator

A hypothetical group of policies consists of 32 six-month terms and 12 twelve-month terms. The losses for the same experience group are stipulated to total $38,400. Under a month-based convention, calculate the exposure and the loss amount per PYE. Then show what goes wrong if the policy count is used as the denominator while reporting a per-PYE result. No interest, mortality, or loss distribution is assumed.

Method. Convert both term groups into PYEs, add them, and divide the stipulated losses by that exposure.

The 32 six-month terms contribute $32(6/12)=16$ PYE. The 12 twelve-month terms contribute $12(12/12)=12$ PYE. Total exposure is $16+12=28$ PYE. Therefore:

\frac{\$38{,}400}{28\text{ PYE}}=\$1{,}371.428571\ldots\text{ per PYE}.

Rounded to the nearest cent, the result is $1,371.43 per PYE. The policy count is $32+12=44$. Dividing by that count gives $\$38{,}400/44=\$872.727272\ldots$, or $872.73 per policy—not dollars per PYE. The computation using 44 is not a substitute for the PYE calculation because each denominator measures something different.

Common mistake avoided: Using a policy count merely because the data are organized as a list of policies. Choose the denominator from the stated exposure convention, not from the format of the data.

Exam Traps and Practical Checks

Short and nonannual terms create errors most often when a calculation silently switches units. Use these checks before accepting an exposure total:

Check the convention. Confirm whether the problem specifies months, days, or another basis. Do not substitute a 365-day year for a month convention midway through a calculation.
Check the term being measured. Use the scheduled full term for written exposure and the elapsed portion for earned exposure. Do not use the full term when asked for exposure earned to date.
Check for terms over one year. A 15-month policy contributes 1.25 PYEs under a month convention; capping it at one PYE discards five months of coverage.
Keep the unit in the result. “92 policies” and “92 PYEs” are not interchangeable statements, even when their numerical values happen to match.
Reconcile durations. When written, earned, and unearned exposure are measured on the same basis and dates, earned duration plus unearned duration should equal the full term duration.

These checks are useful whenever policy terms differ across a portfolio. A consistent exposure convention makes unlike durations comparable without confusing contract count with insured time.

Key takeaway: Convert every short-term or nonannual policy using the stated convention before aggregating. Under a month basis, divide months by 12; under a fixed 365-day basis, divide covered days by 365. Keep written and earned durations distinct, and label the resulting unit.

Check Your Understanding

Use the convention stated in each problem. Give exposure totals to four decimal places where the result does not terminate.

A hypothetical portfolio contains 54 four-month terms and 18 ten-month terms. Under a month-based convention, find the policy count and total written PYE. (Answer: 72 policies and 33.0000 PYE)
Three policies have terms of 75, 180, and 400 covered days. Under a fixed 365-day convention, find total written exposure. (Answer: $655/365=1.7945$ PYE)
A group contains 40 six-month terms, 16 eighteen-month terms, and 14 three-month terms. Under a month-based convention, calculate total written exposure. (Answer: $20+24+3.5=47.5000$ PYE)
A policy has a 600-day term, of which 210 days have elapsed at the valuation date. Under a fixed 365-day convention, find earned and unearned exposure. (Answer: $210/365=0.5753$ earned PYE and $390/365=1.0685$ unearned PYE)
A hypothetical group has $38,400 in stipulated losses and consists of 32 six-month terms and 12 twelve-month terms. Under a month-based convention, calculate the loss amount per PYE to the nearest cent. (Answer: $1,371.43 per PYE)

What You'll Learn