Parallelogram Method

Study guide: Exam 5

The Parallelogram Method (or geometric method) is the on-leveling method that restates a period's aggregate earned premium to the current rate level with a single factor: it uses geometry — and the assumption that policies are written evenly through time — to find what share of the period's premium was earned at each historical rate level.

$\text{OLF}_Y = \frac{I_{\text{current}}}{\sum_k A_{Y,k}\, I_k}$

$A_{\text{new}} = \tfrac{1}{2}(1 - p)^2$

$I_k$ is the cumulative rate level index of rate level group $k$ (each index is the previous one times $1 +$ that change), $I_{\text{current}}$ the latest index — even if the change came after the experience period — and $A_{Y,k}$ the share (area) of year $Y$ 's earned premium written at level $k$ . The second formula is the area earned in the same calendar year at a new level effective a fraction $p$ of the way through it, for annual policies.
The picture. A calendar year is a unit square: time across, fraction of the policy term earned up. A rate change for policies written on or after a date is a diagonal line whose slope is set by the term (steeper for six-month policies). A change forced by law onto policies already in force is a vertical line. Areas come from triangles ( $\tfrac{1}{2}bh$ ), parallelograms ( $bh$ ) and trapezoids.
Werner's six steps: date the changes and form rate level groups; find each group's area in each year; build cumulative indices; take the area-weighted average index; divide it into the current index; apply the factor to the year's earned premium.
Policy-year and written premium are grouped by write date, so the rate level boundary cuts them cleanly: a change effective a fraction $p$ into the year affects the share $1 - p$ . In the policy-year picture the regions are parallelograms rather than triangles.
Two weaknesses. It assumes uniform writing — false for seasonal business such as boat owners, whose policies are bought ahead of the season; refining to quarters or months, or weighting by the actual distribution of writings, helps. And it applies the overall average change, so if changes varied by class or territory the segment premiums are not on-level, which makes the result unacceptable for classification work — hence the move to extension of exposures.

Three Rate Changes Across Three Calendar Years Annual policies, written uniformly. Rate changes apply to policies written on or after: $+4.0\%$ on $7/1/2021$ , $+7.5\%$ on $4/1/2022$ , $+3.0\%$ on $10/1/2023$ . Calendar year earned premium was $\$9{,}500{,}000$ ( $2022$ ), $\$10{,}400{,}000$ ( $2023$ ) and $\$11{,}000{,}000$ ( $2024$ ).

Compute each year's on-level factor and on-level earned premium.
Answer Indices: $1.0000$ ; $1.0400$ ; $1.04 \times 1.075 = 1.1180$ ; $1.1180 \times 1.03 = 1.15154$ (current).

CY 2022. Policies written before $7/1/2021$ still earning: a triangle with base and height $0.5$ , area $\tfrac{1}{2}(0.5)^2 = 0.125$ . Written on or after $4/1/2022$ : $\tfrac{1}{2}(0.75)^2 = 0.28125$ . The $1.0400$ level holds the rest, $0.59375$ .

CY 2023. Still earning at $1.0400$ (written $1/1$ – $3/31/2022$ ): $\tfrac{1}{2}(0.25)^2 = 0.03125$ . At $1.15154$ (written from $10/1/2023$ ): $\tfrac{1}{2}(0.25)^2 = 0.03125$ . At $1.1180$ : $0.9375$ .

CY 2024. At $1.1180$ (written before $10/1/2023$ ): $\tfrac{1}{2}(0.75)^2 = 0.28125$ . At $1.15154$ : $0.71875$ .
 $\begin{align*} \bar{I}_{2022} &= 0.125(1.0000) + 0.59375(1.0400) \\ &\quad + 0.28125(1.1180) \\ &= 1.05694 \\ \bar{I}_{2023} &= 0.03125(1.0400) + 0.9375(1.1180) \\ &\quad + 0.03125(1.15154) \\ &= 1.11661 \\ \bar{I}_{2024} &= 0.28125(1.1180) + 0.71875(1.15154) \\ &= 1.14211 \end{align*}$ 
 $\begin{align*} \text{OLF}_{2022} &= 1.15154 / 1.05694 = 1.0895 \\ \text{OLF}_{2023} &= 1.15154 / 1.11661 = 1.0313 \\ \text{OLF}_{2024} &= 1.15154 / 1.14211 = 1.0083 \end{align*}$ 
On-level earned premium: $\$9{,}500{,}000 \times 1.0895 = \$10{,}350{,}250$ ; $\$10{,}400{,}000 \times 1.0313 = \$10{,}725{,}520$ ; $\$11{,}000{,}000 \times 1.0083 = \$11{,}091{,}300$ . Even $2024$ needs a factor: policies written before $10/1/2023$ , at the old level, still earn $28\%$ of its premium.

Six-Month Policies Earn Faster A personal auto book writes six-month policies uniformly. Rates rose $10\%$ for policies written on or after $4/1/2024$ , with no other change. Compute the CY 2024 on-level factor, and the error from using annual-policy geometry.
Answer With a six-month term the diagonal runs from $4/1$ at the bottom of the square to $10/1$ at the top. The region to its right is a trapezoid with bases $0.75$ and $0.25$ and height $1$ :
 $\begin{align*} A_{\text{new}} &= \tfrac{1}{2}(0.75 + 0.25)(1) = 0.50 \\ \bar{I} &= 0.50(1.00) + 0.50(1.10) = 1.050 \\ \text{OLF} &= 1.10 / 1.050 = 1.0476 \end{align*}$ 
Annual-policy geometry would give $A_{\text{new}} = \tfrac{1}{2}(0.75)^2 = 0.28125$ , $\bar{I} = 1.0281$ and an OLF of $1.0699$ — overstating CY 2024 on-level premium by $2.1\%$ . Six-month policies take on a change faster, so they need a smaller adjustment; and CY 2025 needs none at all, since every policy written before $4/1/2024$ expired by $10/1/2024$ .