Written Premium at Policy Inception

What Is Written at Policy Inception?

When an insurer issues a policy, it records a premium amount associated with the coverage as issued. For ratemaking and data analysis, identifying that amount correctly matters: it provides a starting measure of the premium attached to the original policy terms. The calculation begins with the rating information and premium charges specified for that policy, not with a later payment or a later change in coverage.

Definition: The initial written premium is the premium recorded for a policy when it is issued, based on the policy’s initial terms and the applicable rating calculation. It is measured at policy inception, before subsequent premium-changing transactions. Include a charge in the calculation only when the stated policy or data definition treats it as part of premium.

The amount is determined by the rating plan and issuance rules given in the problem. There is no single universal sequence for every policy: the instructions may specify which base amounts a factor applies to, when fixed charges are added, and whether a minimum premium is tested before or after those steps. Follow the stated sequence rather than assuming one.

Initial written premium is not necessarily the amount the policyholder pays on the issue date. A policy may be paid in installments, but the initial written premium is the amount recorded for the policy at issuance under the stated premium calculation. Nor is it the premium earned as coverage time passes. Earned premium, cash receipts, and later premium transactions are distinct measures; they are not needed to calculate the inception amount in the examples below.

Build the Amount From the Stated Inputs

A common rating calculation starts with one or more exposure or rating units. If class $j$ has $n_j$ units and a rate $r_j$ dollars per unit, its base premium is $n_jr_j$. Add the applicable class base premiums to obtain the combined base. Then apply any stated factors and add fixed premium charges in the order specified. A minimum premium, if present, is a floor applied at the stage stated in the policy rule.

Calculation framework: Let $n_j$ be the number of rating units and $r_j$ the rate per unit for class $j$. The combined base premium is $B=\sum_j n_jr_j$. If one factor $k$ applies to the combined base and a fixed premium charge $C$ is added afterward, the adjusted amount is $Q=kB+C$. If a minimum $M$ applies to $Q$, the initial written premium is $W_0=\max(M,Q)$. Use this last expression only when the stated rule makes $M$ a minimum on that resulting amount.

If different factors apply to different coverages or classes, calculate those amounts separately before adding them. In that case, the adjusted premium may be written $Q=\sum_j k_jn_jr_j+C$, where $k_j$ is the factor applicable to class $j$. Do not apply a factor to a class or charge unless the problem says it applies there.

Worked Example: Combined Base, Factor, Charge, and Minimum

A hypothetical policy has 24 rating units at $37 per unit and 16 rating units at $29 per unit. A factor of 1.05 applies to the combined base premium. A $50 premium surcharge is then added, and a $1,300 minimum applies after both adjustments. Calculate the initial written premium.

Method. Calculate each class base, combine them, apply the factor to that combined base, add the stated surcharge, and then test the minimum. The common mistake to avoid is applying the minimum to the unadjusted base or calculating the factor on only one class.

The first class base is $24\times \$37=\$888$. The second class base is $16\times \$29=\$464$. Therefore, the combined base is $\$888+\$464=\$1,352$.

Apply the factor to the combined base: $\$1,352\times1.05=\$1,419.60$. Add the surcharge: $\$1,419.60+\$50=\$1,469.60$.

The amount after adjustments exceeds the minimum because $\$1,469.60>\$1,300$. Thus, the initial written premium is $1,469.60.

Check. The factor increases the combined base by 5%, or $\$1,352\times0.05=\$67.60$. Adding that increase and the $50 surcharge to $1,352 gives $\$1,352+\$67.60+\$50=\$1,469.60$, consistent with the calculation above.

Why the Calculation Is Built This Way

The sum of class base premiums represents the rating amount before a shared policy-level adjustment. If the same factor applies to every class, multiplying the combined base by the factor gives the same result as applying it separately to each class and then summing. This follows from the distributive property:

k\left(\sum_j n_jr_j\right)=\sum_j kn_jr_j

The equality depends on the factor $k$ being the same for every term. If the factors differ by coverage or class, the combined-base shortcut does not apply; instead, use $\sum_j k_jn_jr_j$. A fixed charge is added separately because it is not a per-unit rate multiplied by the number of units. Finally, a minimum premium is a floor: when the calculated amount is below the stated floor, the recorded amount is raised to that floor; when it is above, the floor does not change the result.

Identify the rating bases.
For each class or coverage, note the unit count, rate, and any factor that applies specifically to it.

Calculate base premiums.
Multiply each stated unit count by its corresponding rate, then combine the amounts where the rating instructions call for it.

Apply adjustments in the stated order.
Apply factors to the bases they govern and add fixed premium charges at the specified stage.

Test any minimum.
Compare the amount at the stage named in the rule with the minimum. Use the larger amount only if the rule makes the minimum a floor at that stage.

State the inception amount.
Report the resulting initial written premium in dollars, then check that each stated component has been included once and only once.

Worked Example: Different Factors by Coverage

A hypothetical small-business policy has a Coverage A base premium of $820 and a Coverage B base premium of $640. A factor of 1.10 applies only to Coverage A; a factor of 0.95 applies only to Coverage B. A fixed premium charge of $27.50 is added after both coverage amounts are adjusted. No minimum is specified. Find the initial written premium.

Method. Apply each coverage’s own factor, add the adjusted coverage premiums, and then add the fixed charge. The common mistake to avoid is applying one factor to the combined $1,460 base.

For Coverage A, $\$820\times1.10=\$902$. For Coverage B, $\$640\times0.95=\$608$. The adjusted coverage total is $\$902+\$608=\$1,510$. Adding the charge gives $\$1,510+\$27.50=\$1,537.50$.

The initial written premium is $1,537.50. The separate calculation is necessary because the two factors are not equal.

Check. Coverage A increases by $\$82$, while Coverage B decreases by $\$32$. The net change to the original combined base is $\$82-\$32=\$50$; $\$1,460+\$50+\$27.50=\$1,537.50$, which agrees.

Minimum Premiums and the Specified Order

A minimum premium is not an extra charge automatically added to every policy. It sets a lower bound at the stage named in the rule. For example, if a minimum applies after factors and fixed charges, first calculate those items and then compare the result with the minimum. If the result already exceeds the floor, the initial written premium remains the calculated result.

The wording matters. A minimum applied to the base premium can produce a different result from a minimum applied after a factor or fixed charge. If a problem says a minimum applies to the final amount, represent the comparison as $\max(M,Q)$. If it specifies a different stage, perform the comparison at that stage before proceeding. Do not silently move the minimum earlier or later.

Worked Example: Minimum Applied After the Charge

A hypothetical equipment policy has a base premium of $1,097. A factor of 0.92 applies to the base. A $36.75 fixed premium charge is then added. The policy has a $1,050 minimum, which applies after the factor and charge. Calculate the initial written premium.

Method. Calculate the adjusted amount first and compare that amount with the minimum. The common mistake to avoid is comparing the unadjusted base with the minimum, which skips the stated calculation sequence.

Apply the factor: $\$1,097\times0.92=\$1,009.24$. Add the fixed charge: $\$1,009.24+\$36.75=\$1,045.99$. Compare with the minimum: $\$1,045.99<\$1,050$. The minimum therefore raises the amount to $1,050.00.

Check. The minimum increases the calculated amount by $\$1,050.00-\$1,045.99=\$4.01$. The resulting initial written premium is exactly the specified floor.

What the Inception Amount Does—and Does Not—Measure

The initial written premium describes the premium recorded for the original policy at issuance. It does not, by itself, indicate how much coverage has elapsed, how much cash has been collected, or whether the premium will remain unchanged over the policy’s full term. Those questions require different premium measures or later policy information.

For example, a down payment is a payment arrangement, not a replacement for the premium calculation. If the stated initial written premium is $1,185 and the policyholder pays 25% at issue, the payment is $\$1,185\times0.25=\$296.25$, but the initial written premium remains $1,185. Likewise, a later change to the policy is not part of the original inception calculation. Keep the time point fixed: use the policy terms at issuance and do not include later transactions.

In practice, premium data must also be interpreted consistently. If a problem supplies a charge and says it is included in premium, include it according to the stated sequence. If a fee or other amount is not defined as part of premium, do not assume it belongs in written premium. This keeps the calculation aligned with the intended measure and avoids mixing premium with other amounts billed or recorded.

Common Exam Traps

Applying a factor to the wrong base. A factor on the combined base applies to all included base components. A factor stated for one coverage applies only to that coverage.
Changing the order of operations. A minimum after adjustments is tested after those adjustments, not against the original base.
Adding a minimum instead of applying a floor. A minimum is generally a comparison, not an amount to add on top of the calculated premium.
Confusing premium with a payment. An installment or down payment can be less than the initial written premium without changing the premium recorded at issuance.
Including an unstated charge. Count a fee or surcharge only when the problem identifies it as part of the premium calculation.
Rounding too early. Keep exact intermediate amounts where possible and round the final dollar amount to cents unless the question specifies another convention.

Key takeaway: Initial written premium is the amount recorded for the policy as issued. Calculate each stated rating component, apply factors and charges to the specified bases in the specified order, and use a minimum only as directed. This distinction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each hypothetical policy, calculate the initial written premium from the stated issuance information. Keep track of the base to which each factor applies.

A policy has 9 units at $48 per unit and 14 units at $26 per unit. A factor of 1.04 applies to the combined base, followed by an $18 premium surcharge. A $800 minimum applies after the surcharge. What is the initial written premium? (Answer: $845.84)
A policy has two coverages with base premiums of $520 and $360. Factors of 1.10 and 0.90 apply to the first and second coverage, respectively. Add a $25 premium charge, then apply a $900 minimum to the resulting amount. What is the initial written premium? (Answer: $921.00)
A policy has a $1,250 base premium. A factor of 0.96 applies, followed by a $40 premium charge. A $1,250 minimum applies after both adjustments. What is the initial written premium? (Answer: $1,250.00)
A policy has 32 units at $19.50 per unit and 15 units at $28 per unit. A factor of 1.08 applies only to the first class. A $12.08 premium charge is then added, with no minimum. What is the initial written premium? (Answer: $1,106.00)
A policy has two base premiums of $680 and $420. A factor of 1.05 applies to their combined base, and a $30 premium charge is added. The policyholder pays 25% of the amount at inception. What is the initial written premium, regardless of the payment amount? (Answer: $1,185.00)

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