Parallelogram On-Leveling for Loss Ratio Indications

Why the Earning Pattern Matters

An earned premium total for a calendar year may combine policies charged at different rate levels. This happens because policies generally earn over their terms: after a rate change, policies already in force may continue earning at the old level until renewal. Treating all premium in the change year as if it were earned at either the old or the new level can therefore distort a loss ratio indication.

The parallelogram method estimates how earned exposure is distributed between rate levels when policy effective or renewal dates are spread uniformly through the year and policies earn pro rata. Once the share at each level is estimated, the premium can be adjusted to a selected target level. For a loss ratio indication, the adjustment changes the premium denominator; it does not change the losses.

Definition: Let $e$ be the fraction of the calendar year elapsed when a rate change takes effect, so $0 \le e \le 1$. Let $q$ be the share of a calendar year's earned exposure, valued at a common base rate, that is at the new rate level. For one rate increase of $r$, the new rate is $1+r$ times the old rate. The on-level premium factor to the new level is the new rate level divided by the average rate level represented in the earned exposure.

Assumptions and the Two Parallelogram Shares

The standard pattern assumes annual policies, a uniform distribution of effective or renewal dates throughout the year, pro rata earning, and a rate change that applies to policies effective or renewed on or after its effective date. It also assumes the exposure mix is otherwise stable. Under these assumptions, the proportion of the in-force exposure at the new level changes linearly over time, producing the parallelogram shape.

In the calendar year containing the rate change, no policies are at the new level before the effective date. After that date, the new-level in-force share rises from zero to $1-e$ by year-end. The earned-exposure share is the area under this rising line, divided by the full year's exposure area:

q_{\text{change year}} = \int_e^1 (t-e)\,dt = \frac{(1-e)^2}{2}.

In the following calendar year, the new-level share at the start is $1-e$, reflecting renewals that moved to the new level after the effective date in the prior year. During the first fraction $e$ of the year, the remaining old-level policies renew; after that, the entire portfolio is at the new level. Thus:

q_{\text{following year}} = \int_0^e (1-e+t)\,dt + \int_e^1 1\,dt = 1-\frac{e^2}{2}.

These $q$ values are shares of earned exposure (equivalently, earned premium valued at a common base rate). They are not generally the shares of actual premium dollars, because a unit of exposure at the new rate produces a different premium from a unit at the old rate.

Locate the effective date.
Express the elapsed part of the calendar year as $e$, using the relevant fraction of a year.

Choose the calendar-year pattern.
Use $(1-e)^2/2$ in the year containing the change, or $1-e^2/2$ in the following year.

Calculate the average rate level.
For an increase $r$, the average rate level relative to the old level is $1+qr$.

Adjust premium to the target level.
For the new level, multiply earned premium by $(1+r)/(1+qr)$, then divide losses by the adjusted premium to obtain the on-level loss ratio.

Deriving the On-Level Factor

Take the old rate level as 1. A share $1-q$ of earned exposure is at that level, while share $q$ is at level $1+r$. The average rate level represented by the earned exposure is therefore:

$$ (1-q)(1)+q(1+r)=1+qr. $$

To put the premium at the new level, scale the average level $1+qr$ to $1+r$. This gives the on-level factor:

F_{\text{on-level}}=\frac{1+r}{1+qr}.

This ratio has a direct premium interpretation. For each unit of base-rate exposure, the observed earned premium is proportional to $1+qr$, while premium at the new level would be proportional to $1+r$. Their ratio is the factor that converts observed premium to new-level premium. Consequently, if $P$ is earned premium and $L$ is the corresponding loss amount, then:

P_{\text{on-level}}=P F_{\text{on-level}}, \qquad LR_{\text{on-level}}=\frac{L}{P F_{\text{on-level}}} =\frac{L/P}{F_{\text{on-level}}}.

The factor is for moving premium to the stated target level. If a different target level is selected, the numerator of the factor must represent that target, and the experience premium must be expressed relative to the corresponding average level.

Worked Example: Rate Change During the Calendar Year

A hypothetical insurer introduces an 8% rate increase effective April 1. Annual policies renew uniformly through the year and earn pro rata. For the calendar year containing the change, the earned premium is $1,022,500 and losses are $540,000. Calculate the on-level premium factor to the new rate level and the adjusted loss ratio.

Method. April 1 is three months into the year, so $e=3/12=0.25$. Since this is the year containing the change, the new-level earned-exposure share is:

q=\frac{(1-0.25)^2}{2} =\frac{0.75^2}{2} =\frac{0.5625}{2} =0.28125.

The average rate level is $1+qr$, with $r=0.08$:

$$ 1+(0.28125)(0.08)=1+0.0225=1.0225. $$

The new rate level is 1.08 relative to the old level. Therefore:

F_{\text{on-level}}=\frac{1.08}{1.0225} =\frac{432}{409} \approx 1.0562.

Apply the factor to the earned premium:

P_{\text{on-level}} =\$1{,}022{,}500\left(\frac{432}{409}\right) =\$1{,}080{,}000.

The raw loss ratio is $\$540{,}000/\$1{,}022{,}500\approx 52.81\%$. The adjusted loss ratio is:

LR_{\text{on-level}} =\frac{\$540{,}000}{\$1{,}080{,}000} =0.50=50.00\%.

Check. The new-level premium is consistent with $1,000,000 of base-rate earned exposure multiplied by 1.08. The method avoids treating all change-year premium as if it were already at the new rate level.

Worked Example: The Calendar Year After the Change

A hypothetical commercial insurer made a 20% increase effective October 1. For the next calendar year, earned premium is $1,143,750 and losses are $720,000. Assume annual policies, uniformly distributed renewal dates, and pro rata earning. Find the on-level premium factor to the increased rate level and the adjusted loss ratio.

Method. October 1 is nine months into the year of the change, so $e=9/12=0.75$. The experience year in this problem is the following calendar year, so use:

q=1-\frac{e^2}{2} =1-\frac{0.75^2}{2} =1-\frac{0.5625}{2} =1-0.28125 =0.71875.

With $r=0.20$, the average rate level represented in the earned exposure is:

$$ 1+qr=1+(0.71875)(0.20)=1.14375. $$

The factor to the new level is:

F_{\text{on-level}}=\frac{1.20}{1.14375} =\frac{64}{61} \approx 1.0492.

The adjusted premium is:

\$1{,}143{,}750\left(\frac{64}{61}\right) =\$1{,}200{,}000.

The raw loss ratio is $\$720{,}000/\$1{,}143{,}750\approx 62.95\%$. After adjustment:

LR_{\text{on-level}} =\frac{\$720{,}000}{\$1{,}200{,}000} =60.00\%.

Check. The adjusted premium is 1.20 times $1,000,000 of base-rate exposure, as required at the new level. The calculation uses the following-year pattern, not the triangle for the year in which the change took effect.

Worked Example: A Change Effective at the Start of the Year

A hypothetical personal property program increases rates by 10% effective January 1. Uniform annual renewals and pro rata earning are assumed. For the calendar year of the change, earned premium is $1,050,000 and losses are $630,000. Calculate the on-level factor and adjusted loss ratio.

Method. The change is effective at the start of the year, so $e=0$. The change-year formula still applies:

q=\frac{(1-0)^2}{2}=0.5.

This does not mean all earned premium is at the new level. Policies that were in force on January 1 can continue earning at the old level until renewal, so the average exposure share at the new level over the year is one-half.

With $r=0.10$, the average level and factor are:

1+qr=1+(0.5)(0.10)=1.05, \qquad F_{\text{on-level}}=\frac{1.10}{1.05}=\frac{22}{21}\approx 1.0476.

On-level premium is:

\$1{,}050{,}000\left(\frac{22}{21}\right) =\$1{,}100{,}000.

Thus the adjusted loss ratio is:

LR_{\text{on-level}} =\frac{\$630{,}000}{\$1{,}100{,}000} \approx 0.5727=57.27\%.

Check. The raw premium corresponds to a 1.05 average rate level on $1,000,000 of base-rate exposure; the on-level premium corresponds to 1.10. The calculation avoids assuming that a January 1 change immediately puts every policy's earned premium at the new level.

Common Trap: Reversing the Loss-Ratio Adjustment

The on-level factor in these examples multiplies premium. Because the loss ratio is losses divided by premium, the same factor divides the raw loss ratio. In the first example, the raw loss ratio is approximately 52.81% and the factor is approximately 1.0562. The incorrect calculation $52.81\%\times1.0562\approx55.78\%$ moves the loss ratio in the wrong direction. The correct calculation is $52.81\%/1.0562\approx50.00\%$.

Another frequent error is to use the wrong year’s parallelogram share. A change-year share is triangular, $(1-e)^2/2$; the following-year share is $1-e^2/2$. The two formulas describe different earning patterns. Verify whether the experience period contains the effective date or follows it before calculating $q$.

The parallelogram estimate is not appropriate without qualification when renewal dates are materially nonuniform, policies have substantially different terms, earning is not pro rata, or the rate change is applied differently from the stated renewal rule. In those circumstances, the actual exposure and rate-level distribution should be modeled using the relevant policy and earning details rather than assumed to follow the standard pattern.

Key takeaway: First identify the experience year's position relative to the effective date. Estimate the new-level exposure share with the corresponding parallelogram formula, calculate the premium factor to the target level, and divide the raw loss ratio by that factor.

Check Your Understanding

For each problem, assume annual policies, uniformly distributed effective or renewal dates, and pro rata earning unless stated otherwise. Round factors to four decimals and loss ratios to two decimal places.

A 9% increase takes effect May 1. Find the new-level exposure share and the factor to the new level for the calendar year containing the change. (Answer: $q=2/9=0.2222$; factor $=1.0686$.)
A 15% increase takes effect September 1. For the following calendar year, find the new-level exposure share and the factor to the new level. (Answer: $q=7/9=0.7778$; factor $=69/67\approx1.0299$.)
A 6% increase takes effect July 1. In the calendar year of the change, the raw loss ratio is 48.00%. Find the on-level loss ratio. (Answer: $q=0.1250$, factor $=1.06/1.0075\approx1.0521$, adjusted loss ratio $\approx45.62\%$.)
A 12% increase takes effect November 1. For the following calendar year, calculate the factor to the new level. (Answer: $q=47/72\approx0.6528$; factor $=672/647\approx1.0386$.)
An 8% increase takes effect March 1. In the year of the change, earned premium is $370,000 and losses are $155,520. Find the on-level premium and adjusted loss ratio. (Answer: $q=25/72\approx0.3472$; factor $=972/925\approx1.0508$; on-level premium $388,800; adjusted loss ratio 40.00%.)

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