Why a Date-Based Picture Helps
On-leveling requires identifying which rate level applies to premium earned in an experience period. The key dates are not interchangeable: a policy has an effective date, a term during which it provides coverage, and dates on which its premium is earned. A rate change may divide policies into different rate-level groups according to their effective dates.
The parallelogram method’s geometric model makes these relationships visible. It represents policy effective dates on one axis and dates on which coverage is earned on the other. Policy terms create diagonal boundaries; an earning period creates a horizontal band; and a rate-change date can divide the picture into regions. The geometry is useful for understanding which policies contribute earned exposure, before calculating any premium adjustment.
Building the Geometric Model
Let \(w\) denote a policy’s effective date and \(t\) a date on which that policy earns exposure. A policy with term \(L\) is in force from \(w\) to \(w+L\). Thus, its exposure is earned only where \(w \leq t \leq w+L\). In a diagram with \(w\) on the horizontal axis and \(t\) on the vertical axis, these limits are the diagonal lines \(t=w\) and \(t=w+L\).
The first diagonal marks the policy’s start: dates before it are before the policy is effective. The second marks its expiration: dates beyond it are after the policy term. The strip between the lines is the region in which a policy can earn exposure. The width \(L\) between the two diagonal boundaries represents the fixed term.
To show earning during \([a,b]\), draw horizontal boundaries at \(t=a\) and \(t=b\). Keep only the part of the in-force strip between them and between the writing dates \(c\) and \(d\). Depending on how these boundaries meet, the resulting region may contain a parallelogram, a trapezoid, or a triangle. The name “parallelogram method” refers to the characteristic date-based geometry; it does not mean every clipped region must be a complete parallelogram.
Put the writing interval \([c,d]\) on the horizontal axis. Uniform writings at rate \(q\) mean that a small effective-date interval of length \(dw\) contains \(q\,dw\) policies.
For each effective date \(w\), coverage runs from \(t=w\) through \(t=w+L\). These two parallel diagonal lines bound the in-force region.
Draw \(t=a\) and \(t=b\). The intersection with the in-force region shows which effective dates earn exposure during the selected period, and for how long.
If policies effective at or after date \(r\) receive a new rate, draw a vertical line at \(w=r\). The regions on either side correspond to policies assigned to different rate levels.
The diagram separates two questions. The first is geometric: how much exposure is earned in the selected calendar period, and which effective dates contribute it? The second is contractual: which rate applies to the policies in each region? A rate-change date must be interpreted according to the rating rule. For a renewal-effective change, the rate boundary separates policies by the effective date of the renewed term. If a change instead applies to existing policies midterm, the assignment rule differs and must be represented accordingly.
Why Area Represents Earned Exposure
For a policy effective at \(w\), the number of months it earns exposure during \([a,b]\) is the length of the overlap between its policy term \([w,w+L]\) and the earning period \([a,b]\). Because one full policy-term is one exposure unit and earning is uniform, the policy earns its overlap length divided by \(L\).
Under constant writings, the effective-date interval \(dw\) contains \(q\,dw\) policies. During a small earning-date interval \(dt\), each of those policies earns \(dt/L\) exposure units if it is in force. Their combined contribution is therefore \((q/L)\,dw\,dt\). Summing these contributions over the eligible region gives the area interpretation.
The factor \(q/L\) is essential. The area has units of time squared; \(q\) has units of policies per unit time; and \(L\) has units of time. Their combination gives exposure units. Area alone is not earned exposure. This result follows directly from adding the contribution \((q/L)\,dw\,dt\) across the region, rather than from treating every policy in the picture as fully earned.
Worked Example: A Rectangular Earning Region
A hypothetical program issues policies uniformly at \(q=3\) policies per month from effective date 0 through month 6. Every policy has a 12-month term. Find the exposure earned from month 8 through month 10. Assume no interest or mortality adjustment; the deterministic issuance density is constant at 3 policies per month, and exposure is earned uniformly over the 12-month term.
Step 1: Identify contributing effective dates. Every policy in the cohort has an effective date \(w\) between 0 and 6. Its expiration date is \(w+12\), which ranges from month 12 to month 18. Every such policy is therefore in force throughout months 8 to 10: its start is no later than month 6, and its expiration is no earlier than month 12.
Step 2: Find the diagram area. The effective-date width is \(6-0=6\) months, and the earning-period height is \(10-8=2\) months. The eligible region is a rectangle with area \(6 \times 2=12\) month-squared.
Step 3: Convert area to exposure. The scale factor is \(q/L=3/12=0.25\). Thus earned exposure is \(12 \times 0.25=3\) exposure units.
Check. There are \(3 \times 6=18\) policies in the cohort. Each earns \(2/12=1/6\) of an exposure unit in the two-month period, so total exposure is \(18 \times 1/6=3\), as found geometrically. The method avoids treating the entire cohort as fully earned in the period.
Worked Example: A Region Clipped by Policy Expirations
A hypothetical portfolio is written uniformly at \(q=2\) policies per month from effective date 0 through month 4. Policies have an 8-month term. Calculate exposure earned from month 6 through month 10. Assume no interest or mortality adjustment, a constant issuance density of 2 policies per month, and uniform earning over each 8-month term.
Step 1: Determine overlap by effective date. For policies effective from month 0 through month 2, coverage ends between months 8 and 10. The overlap with the earning period is \((w+8)-6=w+2\) months. For policies effective from month 2 through month 4, each is in force for the full four-month earning period, so its overlap is 4 months.
Step 2: Calculate the region’s area. On effective dates 0 to 2, the overlap height increases linearly from 2 to 4 months. The corresponding region is a trapezoid with area \((2+4)/2 \times 2=6\) month-squared. On effective dates 2 to 4, the overlap height is 4 months, giving a rectangle of area \(4 \times 2=8\) month-squared. Total area is \(6+8=14\) month-squared.
Step 3: Apply the exposure scale. The scale factor is \(q/L=2/8=0.25\). Earned exposure is \(14 \times 0.25=3.5\) exposure units.
Check. The total cohort is \(2 \times 4=8\) policies, so exposure of 3.5 units in a four-month earning interval is plausible: each policy can contribute at most \(4/8=0.5\) units, giving a cohort maximum of 4. The result is below that maximum. The trapezoid accounts for policies expiring during the period; counting every policy as earning all four months would overstate exposure.
Adding a Rate-Change Boundary
Suppose a rate change applies to policies with effective dates at or after \(r\), while earlier effective dates retain the old rate. The vertical line \(w=r\) divides the eligible region into an old-rate part and a new-rate part. Because the writing density and term are the same on both sides in the basic model, each area can be converted to exposure using \(q/L\).
The boundary classifies policies; it does not, by itself, change how exposure is earned within a policy term. For instance, if the change is renewal-effective, a policy already in force on \(r\) does not automatically move to the new-rate region. Its renewal must have an effective date on or after the change date. Always apply the stated policy rule before drawing the boundary.
Worked Example: Splitting Earned Exposure at a Rate Date
A hypothetical book is written uniformly at \(q=2\) policies per month from effective date 1 through month 5. Each policy has a 6-month term. A new rate applies to policies effective at or after month 3. Find old-rate and new-rate exposure earned from month 5 through month 7. The old and new full-term premiums are $600 and $720 per exposure unit, respectively. Assume no interest or mortality adjustment, constant issuance density, and uniform earning over each term.
Step 1: Establish the date regions. The rate boundary is \(w=3\). Effective dates 1 to 3 belong to the old-rate group; effective dates 3 to 5 belong to the new-rate group. Every policy in this cohort is in force throughout months 5 to 7: the earliest effective date, month 1, expires at month 7, and the latest effective date, month 5, starts at the beginning of the earning period.
Step 2: Find the area on each side. Each effective-date group has width 2 months, and the earning period has height 2 months. Each side is a rectangle of area \(2 \times 2=4\) month-squared.
Step 3: Convert each area to exposure. The scale factor is \(q/L=2/6=1/3\). Old-rate exposure is \(4 \times 1/3=4/3\) units. New-rate exposure is also \(4 \times 1/3=4/3\) units.
Step 4: Illustrate the rate assignment. At the stated full-term premiums, the old-rate earned premium represented is \((4/3)\times \$600=\$800\). The new-rate amount is \((4/3)\times \$720=\$960\). Together they represent $1,760.
Check. The whole cohort contains \(2 \times 4=8\) policies. Each earns \(2/6=1/3\) of an exposure unit during the two-month period, so total exposure is \(8 \times 1/3=8/3\). The two regions sum to \(4/3+4/3=8/3\), confirming the split. The method avoids assigning all earned exposure to the rate in effect at the midpoint of the earning period.
Assumptions and Common Traps
The geometric model is most informative when its assumptions match the data. Uniform writings allow the number of policies in an effective-date interval to be represented by \(q\) times its width. A fixed term gives parallel start and expiration boundaries. Uniform earning means that \(1/L\) of a policy’s exposure is earned per unit of time in force. If any of these conditions fails, the same diagram may still help organize dates, but the simple area scale \(q/L\) cannot be applied unchanged.
For example, strongly seasonal writings do not have a constant \(q\), and different term lengths create different expiration boundaries. A nonuniform earning pattern also changes each policy’s contribution: the fraction earned during an interval is no longer simply its overlap length divided by \(L\). In such cases, the model must reflect the actual writing pattern, terms, or earning rule rather than forcing a constant-area calculation.
A frequent exam trap is to compute the polygon’s area and report that number as exposure. In the preceding clipped-region example, that error would report 14 exposure units. The area is 14 month-squared; converting it requires multiplying by \(q/L=2/8\), giving \(14 \times 2/8=3.5\) exposure units. Another trap is to classify a policy by the date exposure is earned when the stated rate change applies by policy effective date. The rate boundary belongs on the effective-date axis in that situation.
Check Your Understanding
For each problem, assume uniform writing, fixed policy terms, uniform earning, and no interest or mortality adjustment unless stated otherwise.
- A hypothetical portfolio is written at 1.5 policies per month from effective date 0 through month 4. The term is 10 months. How much exposure is earned from month 6 through month 8? (Answer: 1.2 exposure units)
- Policies are written at 3 per month from effective date 0 through month 2, with a 6-month term. Find exposure earned from month 4 through month 8. (Answer: 3 exposure units)
- A cohort is written at 1 policy per month from effective date 2 through month 6. The term is 8 months. Find exposure earned from month 7 through month 9. (Answer: 1 exposure unit)
- A book is written at 2 policies per month from effective date 2 through month 6, with an 8-month term. A new rate applies to effective dates at or after month 4. Find exposure earned on each side of the boundary from month 7 through month 9. (Answer: 1 old-rate exposure unit and 1 new-rate exposure unit)
- A cohort is written at 2 policies per month from effective date 0 through month 5, with a 6-month term. A new rate applies at effective date 3. For earning from month 4 through month 8, find old-rate and new-rate exposure separately. (Answer: \(10/3\) old-rate exposure units and \(5/2\) new-rate exposure units)