Earning Premium for Nonuniform Coverage

Why Premium May Not Be Earned Evenly

Straight-line earning allocates premium evenly over equal portions of a policy term. That is appropriate only when equal portions of time are a reasonable representation of how coverage or risk is provided. Some hypothetical policies instead insure activities, property, or hazards whose covered exposure varies substantially across the term. A seasonal equipment policy, for example, might provide much more protection during the operating season than during storage.

When the coverage pattern is not uniform, premium can be allocated using a stated earning pattern. The key is to identify the period weights or earning proportions specified by the problem, apply them to the relevant premium, and confirm that the allocation accounts for the full amount. The pattern is an allocation basis; it does not mean that premium should be assigned according to claims that happen to occur.

Definition: For a policy with written premium $P$ and $n$ earning periods, let $w_j$ be the nonnegative earning weight for period $j$. If the weights are not already proportions, the earning proportion for period $j$ is $p_j=w_j/\sum_{k=1}^{n}w_k$. Premium earned in that period is $P_j=Pp_j$. The weights must have a positive total. When they represent the entire policy term, the proportions sum to one and the period allocations sum to written premium.

A period can be a phase or another interval defined in the problem. The weights may be provided directly, or they may be constructed from an exposure measure multiplied by a relative risk factor. For example, if $e_j$ is the exposure expected in period $j$ and $r_j$ is that period’s relative risk index, a stated risk-based pattern might use $w_j=e_jr_j$. The interpretation depends on the problem: use the supplied basis rather than assuming that every weight has the same meaning.

These calculations are deterministic allocation exercises: no mortality assumption, interest rate, or probability distribution is required unless the problem specifically supplies one. In the worked examples below, assume an effective annual interest rate $i=0$; no mortality model or loss distribution is assumed. The premium amounts are hypothetical.

From Period Weights to Earned Premium

The normalized proportions $p_j$ are the shares of the total premium assigned to each period. If the weights already sum to one, they are themselves the proportions. If the question gives percentages, divide each percentage by 100 to obtain the corresponding proportion. For cumulative earned premium at the end of period $m$, add the allocations through that period. Unearned premium is the remainder of written premium.

p_j=\frac{w_j}{\sum_{k=1}^{n}w_k}, \qquad P_j=Pp_j, \qquad P_{\mathrm{earned}}(m)=\sum_{j=1}^{m}P_j, \qquad P_{\mathrm{unearned}}(m)=P-P_{\mathrm{earned}}(m).

If a problem states period proportions directly, use those proportions in $P_j=Pp_j$ and check that they sum to one. If they do not, first determine whether the listed values are actually weights that need normalization or whether the problem has omitted part of the policy term. Do not silently treat incomplete proportions as a complete pattern.

Identify the premium and policy periods.
Use the written premium being allocated and the periods defined by the hypothetical coverage schedule.

Establish the earning basis.
Use the stated proportions or weights. If weights are built from exposure and a relative risk index, calculate each product before summing.

Normalize and allocate.
Divide each weight by the total weight, then multiply the resulting proportion by written premium.

Accumulate and reconcile.
Sum allocations through the requested period and calculate unearned premium as the written premium less earned premium. Check the full-term allocation against written premium.

Worked Allocations

Worked Example: Unequal Weights Across Four Coverage Phases

A hypothetical policy has written premium of $1,200 and four defined coverage phases. The stated earning weights are $1,1,2,4$, reflecting a progressively more active covered operation. Find the premium allocated to each phase, earned premium through phase 3, and unearned premium at that boundary.

Method and assumptions. This is a deterministic weighted allocation. Assume $i=0$; no mortality assumption or loss distribution is used. The weights cover the entire policy term.

Step 1: Total the weights.

\sum_{j=1}^{4}w_j=1+1+2+4=8.

Step 2: Normalize and allocate. The proportions are $1/8,1/8,2/8,4/8$. Multiplying each by $1,200 gives:

P_1=1{,}200\left(\frac{1}{8}\right)=150,\quad P_2=1{,}200\left(\frac{1}{8}\right)=150,\quad P_3=1{,}200\left(\frac{2}{8}\right)=300,\quad P_4=1{,}200\left(\frac{4}{8}\right)=600.

Step 3: Calculate earned and unearned premium. Through phase 3, earned premium is $150+150+300=\$600$. Unearned premium is $1{,}200-600=\$600$.

Check. The four allocations total $150+150+300+600=\$1{,}200$, the written premium. The common mistake is to divide $1,200 equally into four allocations of $300. That would disregard the stated unequal pattern; the weighted allocation above is the correction.

Worked Example: Building Weights From Exposure and Relative Risk

A hypothetical insurer assigns $4,500 of written premium to a three-phase policy. The earning basis is defined as phase exposure multiplied by the phase’s relative risk index. Exposures are 100, 200, and 300 units; the corresponding relative risk indices are 1, 1, and 2. Calculate each phase’s weight and premium allocation, then find earned premium after phase 2.

Method and assumptions. Apply the stated exposure-times-risk-index method, then normalize its weights. Assume $i=0$; no mortality assumption or probability distribution is used, and the exposure values are hypothetical.

Step 1: Calculate the weights.

w_1=100(1)=100,\qquad w_2=200(1)=200,\qquad w_3=300(2)=600.

The total weight is $100+200+600=900$.

Step 2: Calculate each proportion and allocation. The proportions are $100/900=1/9$, $200/900=2/9$, and $600/900=2/3$. Therefore:

P_1=4{,}500\left(\frac{1}{9}\right)=500,\qquad P_2=4{,}500\left(\frac{2}{9}\right)=1{,}000,\qquad P_3=4{,}500\left(\frac{2}{3}\right)=3{,}000.

Step 3: Accumulate through phase 2. Earned premium is $500+1{,}000=\$1{,}500$; unearned premium is $4{,}500-1{,}500=\$3{,}000$.

Check. The allocations total $500+1{,}000+3{,}000=\$4{,}500$. A likely error is to use exposure alone and ignore the relative risk index, which would give different weights. The stated basis requires both factors.

Worked Example: Separate Premium Components With Different Patterns

A hypothetical policy has two premium components: a $2,400 basic component earned equally over four defined phases, and a $1,800 seasonal hazard component with weights $0,1,2,3$ over those phases. Find total premium allocated to each phase and earned and unearned premium through phase 3.

Method and assumptions. Allocate each component using its own stated pattern, then add the phase allocations. Assume $i=0$; no mortality assumption or loss distribution is used. The zero weight means the seasonal component has no allocation to phase 1 under the stipulated pattern.

Step 1: Allocate the basic component. Equal allocation over four phases gives $2{,}400/4=\$600$ per phase.

Step 2: Normalize the seasonal weights. Their total is $0+1+2+3=6$, so their proportions are $0,1/6,2/6,3/6$. The seasonal allocations are:

1{,}800(0)=0,\qquad 1{,}800\left(\frac{1}{6}\right)=300,\qquad 1{,}800\left(\frac{2}{6}\right)=600,\qquad 1{,}800\left(\frac{3}{6}\right)=900.

Step 3: Combine the component allocations. The phase totals are $600+0=\$600$, $600+300=\$900$, $600+600=\$1{,}200$, and $600+900=\$1{,}500$.

Through phase 3, earned premium is $600+900+1{,}200=\$2{,}700$. Total written premium is $2{,}400+1{,}800=\$4{,}200$, so unearned premium is $4{,}200-2{,}700=\$1{,}500$.

Check. The phase totals sum to $600+900+1{,}200+1{,}500=\$4{,}200$. Applying one pattern to the combined premium would be a mistake: the two components have different stipulated earning bases.

Why the Normalization Works

The allocation formula follows from a proportionality assumption: each period receives premium in proportion to its weight. Suppose the allocation to period $j$ is $P_j=cw_j$, where $c$ is the same constant for every period. Since the allocations must exhaust the written premium, their sum must equal $P$. Summing the proportional allocations gives $c\sum_{k=1}^{n}w_k=P$, so $c=P/\sum_{k=1}^{n}w_k$. Substituting that value into $P_j=cw_j$ yields the normalized formula.

\sum_{j=1}^{n}P_j =\sum_{j=1}^{n}cw_j =c\sum_{j=1}^{n}w_j=P \quad\Longrightarrow\quad c=\frac{P}{\sum_{k=1}^{n}w_k} \quad\Longrightarrow\quad P_j=P\frac{w_j}{\sum_{k=1}^{n}w_k}.

This justification also identifies the model’s limits. The weights must be nonnegative and have a positive total, and they must represent the full allocation basis for the premium being divided. If the pattern applies only to one component, it should not automatically be applied to another. If a problem supplies earning proportions rather than relative weights, those proportions should already describe the intended shares; check that they sum to one before treating them as complete.

Interpretation and Common Traps

A nonuniform pattern changes when premium is recognized as earned across the term; it does not change the written premium itself. A larger weight assigns a larger share of the premium to that period, but only relative to the other weights. Multiplying every weight by the same positive number leaves the proportions unchanged. For example, weights $1,2,3$ and $10,20,30$ produce the same allocation.

Assuming equal time means equal earning. Equal allocation is not appropriate when the problem explicitly provides a different coverage or risk pattern.
Forgetting to normalize weights. Weights such as $2,3,5$ are not automatically 20%, 30%, and 50% unless they are divided by their total of 10.
Confusing risk weights with actual claims. A stipulated earning pattern allocates premium according to its stated basis; a claim in one period does not, by itself, change the allocation.
Using one pattern for unlike components. Allocate components separately when the problem specifies different earning patterns, then add the results.
Failing to reconcile. Full-term earned allocations should sum to the written premium being allocated. At a boundary, earned plus unearned premium should also equal that premium.

Key takeaway: State the earning basis, convert its weights to proportions, apply those proportions to the appropriate written premium, and reconcile the resulting earned and unearned amounts. A nonuniform pattern is appropriate only when the stipulated coverage or risk basis supports it.

Check Your Understanding

For each hypothetical policy, use only the stated earning basis and check the allocation against the written premium.

A policy has written premium of $2,100 and three phase weights of $2,1,4$. Find the allocation to each phase and earned premium through phase 2. (Answer: $600, $300, and $1,200 by phase; $900 earned through phase 2.)
A $4,500 premium is allocated using three phase exposure values of 100, 200, and 300 and relative risk indices of 1, 1, and 2. Find the phase allocations. (Answer: $500, $1,000, and $3,000.)
A policy has a $2,400 basic component earned equally over four phases and a $1,800 rider with weights $0,1,2,3$. Find total premium by phase and unearned premium after phase 3. (Answer: $600, $900, $1,200, and $1,500 by phase; $1,500 unearned.)
A $8,000 premium uses exposure values 40, 80, and 120 with relative risk indices 1, 1.5, and 2. Find each weight, the allocation by phase, and earned premium through phase 2. (Answer: weights 40, 120, and 240; allocations $800, $2,400, and $4,800; $3,200 earned through phase 2.)
A $7,600 premium has stipulated period proportions of 15%, 35%, and 50%. Find the allocations and the unearned premium at the end of period 2. (Answer: $1,140, $2,660, and $3,800; $3,800 unearned.)

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