The Loss Ratio Rate Indication Method

Purpose of the Loss Ratio Method

A rate indication asks how the current rate level should change to support an expected level of losses while meeting the target represented by the permissible loss ratio. The loss ratio method makes this comparison using two quantities: losses projected to the prospective period and earned premium adjusted to the current rate level. The resulting indication is a percentage change to that current level.

The method is especially useful when an actuary has a selected permissible loss ratio and a credible aggregate premium base. It does not itself determine whether the selected loss trend, development factors, or permissible loss ratio are appropriate. Those are inputs that must be chosen on a consistent basis.

Definition: The permissible loss ratio (PLR) is the proportion of premium available to cover the loss measure being used in the indication, at the target rate level. The loss measure and PLR must have matching scope. For example, if the losses being projected exclude a particular cost component, the PLR must be defined for that same loss basis.

Let $L$ denote projected losses for the experience being analyzed, $P$ the earned premium on-levelled to the current rate level, and $q$ the permissible loss ratio. The indicated rate change is

\text{Indicated rate change} = \frac{L}{qP}-1.

The quantity $qP$ is the amount of loss that the current-level premium can support under the selected PLR. If $L$ exceeds $qP$, the indication is positive; if $L$ is less than $qP$, it is negative. The indicated new rate level is $1+\text{indicated rate change}$ times the current rate level.

Why the Formula Holds

The formula follows from equating the losses to be supported with the permissible share of premium after the rate change. Suppose $r$ is the multiplier applied to the current rate level. If earned premium is proportional to the rate level, applying $r$ to on-level premium $P$ gives premium $rP$. At the permissible loss ratio, that premium supports losses of $qrP$. Set this amount equal to projected losses $L$:

qrP=L \quad\Longrightarrow\quad r=\frac{L}{qP}.

A multiplier of $r=1$ means no change. Thus the percentage change is $r-1$, giving $L/(qP)-1$. This justification assumes that the exposure and mix underlying the premium base are suitable for the projected losses, that the premium scales with the rate change, and that the selected PLR applies to the same basis as $L$ and $P$. If these assumptions do not hold, the simple formula may not give an appropriate indication.

Preparing the Inputs

The formula is compact, but its inputs often require careful preparation. For each experience period, losses may need to be developed to ultimate and trended to the prospective period. Earned premiums are adjusted to the current rate level. Once prepared, sum the appropriate losses and premiums across the periods and apply the selected PLR to the total on-level premium.

Establish the loss basis.
Identify the losses included in the indication and use development factors to estimate ultimate losses where necessary.

Project losses to the prospective period.
Apply the selected trend factors to the developed losses, using the appropriate time intervals and direction of projection.

Determine on-level premium.
Use earned premium adjusted to the current rate level. If factors are supplied, apply each to the premium for the period to which it belongs.

Calculate and interpret the indication.
Divide total projected losses by permissible losses at the current level, $qP$, and subtract one. State the result as a percentage change.

Development and trend adjustments are applied to the loss amounts; on-level factors are applied to premium. The factors should reflect the intended projection dates and rate level, not be applied mechanically to every number in the calculation. When the problem supplies adjusted losses or premiums, do not apply the same adjustment again.

Worked Example: One Experience Period With Development and Trend

A hypothetical insurer has incurred losses of $1,000.00 thousand for an experience period. The selected development factor to ultimate is 1.10, and the trend factor from the period’s loss level to the prospective period is 1.10. Earned premium is already on-level at $2,000.00 thousand. The permissible loss ratio is 0.65. Calculate and interpret the indication.

Method. Develop and trend the losses, then compare them with the losses supportable by the on-level premium. The premium is already on-level, so no additional premium adjustment is made.

Step 1: Develop losses.

1{,}000.00 \times 1.10 = 1{,}100.00 \quad\text{thousand dollars}.

Step 2: Trend developed losses.

1{,}100.00 \times 1.10 = 1{,}210.00 \quad\text{thousand dollars}.

Thus, $L=1{,}210.00$ thousand dollars. The permissible loss amount at the current rate level is

qP=0.65 \times 2{,}000.00=1{,}300.00 \quad\text{thousand dollars}.

The indicated rate change is

\frac{1{,}210.00}{1{,}300.00}-1 =0.930769\ldots-1 =-0.069231\ldots.

Rounded to two decimal places, the indication is a 6.92% decrease. The projected losses are below the $1,300.00 thousand supportable at the current premium level and PLR. As a check, the indicated rate-level multiplier is $1-0.069231\ldots=0.930769\ldots$; multiplying $2,000.00 thousand by this multiplier and then by 0.65 gives approximately $1,210.00 thousand.

Common mistake avoided: Applying the trend factor to the original incurred losses and then treating the resulting amount as developed losses would omit development. The correct sequence here is to develop first, then trend.

Worked Example: Combining Two Experience Periods

For a hypothetical line, two periods have the following data, in thousands of dollars. Period 1 has incurred losses of $600.00, a development factor of 1.05, a trend factor of 1.08, and on-level earned premium of $1,200.00. Period 2 has incurred losses of $800.00, a development factor of 1.00, a trend factor of 1.04, and on-level earned premium of $1,500.00. The permissible loss ratio is 0.60. Calculate the overall indication.

Method. Project each period’s losses to the prospective period, add those amounts, and compare the total with the PLR applied to total on-level premium.

For Period 1, the developed losses are

600.00 \times 1.05=630.00, \qquad 630.00 \times 1.08=680.40 \quad\text{thousand dollars}.

For Period 2, the developed and trended losses are

800.00 \times 1.00=800.00, \qquad 800.00 \times 1.04=832.00 \quad\text{thousand dollars}.

Total projected losses are $680.40+832.00=1{,}512.40$ thousand dollars. Total on-level premium is $1{,}200.00+1{,}500.00=2{,}700.00$ thousand dollars, so permissible losses at the current level are

0.60 \times 2{,}700.00=1{,}620.00 \quad\text{thousand dollars}.

Therefore,

\frac{1{,}512.40}{1{,}620.00}-1 =0.933580\ldots-1 =-0.066420\ldots.

Rounded to two decimal places, the overall indication is a 6.64% decrease. The check is that the projected losses are $107.60 thousand below permissible losses at the current level: $1{,}620.00-1{,}512.40=107.60$. Dividing by $1,620.00 gives approximately 6.64%.

Common mistake avoided: Do not calculate a separate percentage change for each period and take an unweighted arithmetic average. Periods have different premium volumes, so combine projected losses and on-level premiums first.

Worked Example: Applying Premium On-Level Factors

A hypothetical insurer has two experience periods, with amounts in thousands of dollars. Period 1 has incurred losses of $550.00, a development factor of 1.10, a trend factor of 1.08, earned premium of $950.00, and a premium on-level factor of 1.10. Period 2 has incurred losses of $700.00, a development factor of 1.00, a trend factor of 1.04, earned premium of $1,300.00, and an on-level factor of 0.95. The permissible loss ratio is 0.62. Find the indication, rounding the final percentage to two decimal places.

Method. Adjust losses to the prospective period and premiums to the current rate level, then aggregate each basis separately.

For Period 1, projected losses are

550.00 \times 1.10=605.00, \qquad 605.00 \times 1.08=653.40 \quad\text{thousand dollars}.

For Period 2, projected losses are

700.00 \times 1.00=700.00, \qquad 700.00 \times 1.04=728.00 \quad\text{thousand dollars}.

Total projected losses are $653.40+728.00=1{,}381.40$ thousand dollars. On-level premiums are calculated separately:

950.00 \times 1.10=1{,}045.00, \qquad 1{,}300.00 \times 0.95=1{,}235.00 \quad\text{thousand dollars}.

Total on-level premium is $1{,}045.00+1{,}235.00=2{,}280.00$ thousand dollars. Permissible losses at the current level are

0.62 \times 2{,}280.00=1{,}413.60 \quad\text{thousand dollars}.

The indicated change is

\frac{1{,}381.40}{1{,}413.60}-1 =0.977228\ldots-1 =-0.022772\ldots.

Rounded to two decimal places, this is a 2.28% decrease. The indication is modest because projected losses are only $32.20 thousand below the $1,413.60 thousand permissible loss amount. As a check, $32.20/1{,}413.60=0.022772\ldots$, consistent with the indicated decrease.

Common mistake avoided: The 1.10 and 0.95 factors adjust earned premium, not losses. Applying them to losses as well would mix the bases and distort the indication.

Interpreting and Checking an Indication

An indication is a comparison with the current rate level under stated assumptions, not a guarantee of the outcome that will emerge. A positive result means the projected losses exceed the amount supportable at the current level and PLR. A negative result means they fall short. A result of zero means $L=qP$, so no overall change is indicated by this method.

A useful check is to compare projected losses directly with permissible losses $qP$. The rate-level multiplier $L/(qP)$ should be above one when projected losses exceed permissible losses and below one when they do not. If the computed direction contradicts that comparison, check the division, the subtraction of one, and the sign of the reported percentage.

The calculation also depends on the alignment of its inputs. Development and trend assumptions should produce losses for the intended prospective period. Premium should be on the stated current rate level. The PLR must apply to the same loss measure, and the experience should be suitable for the projection. If the exposure basis or mix changes materially, a single aggregate indication may not represent the prospective portfolio; the formula alone does not correct such differences.

Key takeaway: Calculate the loss ratio indication as projected losses divided by permissible losses at the current rate level, minus one. Develop and trend losses, on-level premiums, aggregate on each basis, and interpret the resulting multiplier before reporting the percentage change.

Check Your Understanding

For each problem, calculate the overall indicated rate change and state whether it is an increase or a decrease. Amounts are in thousands of dollars unless stated otherwise.

A hypothetical portfolio has incurred losses of $420.00, a development factor of 1.05, and a trend factor of 1.04. Earned premium is already on-level at $700.00, and the permissible loss ratio is 0.70. What is the indicated change? (Answer: 6.40% decrease)
A hypothetical policy group has incurred losses of $300.00, a development factor of 1.10, and a trend factor of 1.05. On-level earned premium is $500.00 and the permissible loss ratio is 0.65. What is the indicated change, rounded to two decimal places? (Answer: 6.62% increase)
For Period 1, incurred losses are $400.00, the development factor is 1.10, the trend factor is 1.10, and on-level premium is $800.00. For Period 2, incurred losses are $600.00, the development factor is 1.00, the trend factor is 1.05, and on-level premium is $1,200.00. If the permissible loss ratio is 0.62, what is the combined indication, rounded to two decimal places? (Answer: 10.16% decrease)
Projected losses for a hypothetical portfolio are $1,260.00. On-level earned premium is $2,000.00 and the permissible loss ratio is 0.70. Find the indicated rate change and the implied premium at the indicated level. (Answer: 10.00% decrease; $1,800.00 thousand)
A hypothetical portfolio has incurred losses of $500.00, a development factor of 1.20, and a trend factor of 1.10. Earned premium of $900.00 is already on-level, and the permissible loss ratio is 0.60. Calculate the indicated change, rounding to two decimal places. (Answer: 22.22% increase)

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