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CAS Exam 5 · Premium Measurement and Components · Tutorial 43 of 989

Deposit Premiums and Final Audits

Use estimated and audited premium calculations to reconcile a deposit with the final policy charge and identify any balance due or return.

Beginner 9 min read

What You'll Learn

  • Explain why an initial premium may be based on estimated exposure.
  • Calculate an estimated premium and a deposit from stated rates and assumptions.
  • Recalculate the final premium using audited exposure values.
  • Determine whether the audit results in additional premium or a return.
  • Distinguish the change in premium from the balance remaining after the deposit.

Why a Policy Starts With a Deposit Premium

For some hypothetical policies, the amount of exposure during the policy period is not known with certainty when coverage begins. The insurer and policyholder therefore use an estimate to calculate an initial premium. That initial amount is provisional: when the exposure is later audited, the premium is recalculated using the audited values. The initial amount credited toward the premium is the deposit premium.

An audit does not mean that the original rate is recalculated. In the basic setting here, the rate and rating rules are fixed; the audit supplies updated exposure values for the final calculation. If the audited premium exceeds the credited deposit, additional premium is due. If it is less, the policy may be eligible for a return, subject to its terms.

Definition: Let \(P_{\text{est}}\) be the premium calculated from estimated exposure, \(P_{\text{aud}}\) the premium recalculated from audited exposure, and \(D\) the deposit credited to the policy. The audit balance is $$ B=P_{\text{aud}}-D. $$ If \(B>0\), \(B\) is additional premium due. If \(B<0\), \(-B\) is a return premium, assuming the policy terms allow a return. If \(B=0\), there is no audit balance.

The calculation concerns the premium charge, not necessarily the timing or amount of cash received. For the examples, assume the stated deposit has been credited to the policy, and ignore taxes, installment charges, minimum premiums, and other adjustments unless stated. There is no interest accrual (effective annual rate \(i=0\) for settlement purposes); mortality assumptions and probability distributions do not apply to these deterministic premium calculations.

From Estimated Premium to Final Premium

For a policy with one or more rating categories, calculate the premium for each category using the applicable rate and exposure, then add the category premiums. If the deposit is a stated proportion \(\alpha\) of the estimated premium, then \(D=\alpha P_{\text{est}}\). The audit balance is not generally the difference between the audited and estimated premiums: that difference omits the part of the estimated premium that was never deposited.

Worked Example: Two Categories and an 80% Deposit

A hypothetical policy charges $25 per unit in Category X and $40 per unit in Category Y. The estimated exposures are 200 units in X and 150 units in Y. The deposit is 80% of the estimated premium. The audit establishes 220 units in X and 140 units in Y. Find the final premium and the audit balance.

Assume the rates do not change, the deposit is fully credited, and no interest, fees, taxes, minimum premium, mortality model, or probability distribution applies. The method is to calculate the estimated premium, calculate the credited deposit, calculate the audited premium, and subtract the deposit from the audited premium.

1. Estimated premium. Category X contributes \(200\times \$25=\$5{,}000\). Category Y contributes \(150\times \$40=\$6{,}000\). Thus:

$$ P_{\text{est}}=\$5{,}000+\$6{,}000=\$11{,}000. $$

2. Deposit credited. The deposit is 80% of the estimated premium:

$$ D=0.80\times \$11{,}000=\$8{,}800. $$

3. Audited premium. Category X contributes \(220\times \$25=\$5{,}500\). Category Y contributes \(140\times \$40=\$5{,}600\). Therefore:

$$ P_{\text{aud}}=\$5{,}500+\$5{,}600=\$11{,}100. $$

4. Audit balance.

$$ B=\$11{,}100-\$8{,}800=\$2{,}300. $$

The final premium is $11,100, and $2,300 of additional premium is due. As a check, the audited premium is $100 greater than the estimated premium, while $2,200 of the estimated premium was not included in the 80% deposit. The balance is therefore $2,200 plus the $100 increase, or $2,300.

Common mistake avoided: Subtracting the estimated premium from the audited premium gives \(\$11{,}100-\$11{,}000=\$100\). That is the change in total premium, not the balance due after crediting the deposit. The balance is \(P_{\text{aud}}-D\), so the correct calculation is \(\$11{,}100-\$8{,}800=\$2{,}300\).

Why the Deposit Percentage Matters

The relationship between the audit balance, the deposit percentage, and the change in premium makes the calculation easier to interpret. If the deposit is a proportion \(\alpha\) of the estimated premium, substitute \(D=\alpha P_{\text{est}}\) into the balance formula:

$$ B=P_{\text{aud}}-\alpha P_{\text{est}}. $$

Add and subtract \(P_{\text{est}}\) to separate the unpaid portion of the estimate from the change found at audit:

$$ \begin{aligned} B &=P_{\text{aud}}-P_{\text{est}}+P_{\text{est}}-\alpha P_{\text{est}}\\ &=(P_{\text{aud}}-P_{\text{est}})+(1-\alpha)P_{\text{est}}. \end{aligned} $$

This identity follows directly from algebra and applies when the deposit is exactly the stated proportion of the estimated premium. The first term is the change from estimated to audited premium. The second is the portion of the estimated premium not covered by the deposit. Both terms are needed to determine the balance. When audited exposure is lower, the first term may be negative; a return occurs only if the full calculation makes \(B\) negative.

Key result: When \(D=\alpha P_{\text{est}}\), the balance is \(B=(P_{\text{aud}}-P_{\text{est}})+(1-\alpha)P_{\text{est}}\). Do not treat the audit change \(P_{\text{aud}}-P_{\text{est}}\) as the balance due unless the deposit equals the entire estimated premium.

Fixed Charges and Consistent Premium Bases

A policy may have a fixed charge in addition to a premium that varies with exposure. The calculation must follow the contract’s stated deposit basis: if the deposit percentage applies to the total estimated premium, include the fixed charge in that total before applying the percentage. Include the same charge in the final premium when it remains part of the audited policy charge. If a charge is not adjustable, do not treat a change in exposure as changing that charge.

Worked Example: Deposit Percentage Applied to a Total Including a Fixed Charge

A hypothetical policy charges $18 per unit and has a $120 fixed policy charge. Estimated exposure is 360 units. The deposit is 75% of the total estimated premium, including the fixed charge. The audit establishes 390 units. Find the final premium and audit balance.

Assume the $120 charge remains unchanged and is included in both the estimated and final premium. Assume no interest, tax, other adjustment, mortality assumption, or probability distribution applies. The method is to add the fixed charge to each exposure-based premium before calculating or comparing totals.

1. Estimated premium. The exposure-based portion is \(360\times \$18=\$6{,}480\). Including the fixed charge:

$$ P_{\text{est}}=\$6{,}480+\$120=\$6{,}600. $$

2. Deposit credited. Since the 75% applies to the total estimated premium:

$$ D=0.75\times \$6{,}600=\$4{,}950. $$

3. Audited premium. The exposure-based portion is \(390\times \$18=\$7{,}020\). Add the unchanged fixed charge:

$$ P_{\text{aud}}=\$7{,}020+\$120=\$7{,}140. $$

4. Audit balance.

$$ B=\$7{,}140-\$4{,}950=\$2{,}190. $$

The final premium is $7,140, and $2,190 is due. Check the result by decomposition: the unpaid part of the estimate is \(0.25\times \$6{,}600=\$1{,}650\), and the premium increase is \(\$7{,}140-\$6{,}600=\$540\). Their sum is \(\$1{,}650+\$540=\$2{,}190\).

Common mistake avoided: Applying 75% only to the $6,480 exposure-based portion would give a deposit of $4,860, which contradicts the stated deposit basis. The problem says the percentage applies to the total estimated premium, so the $120 fixed charge must be included first.

When the Audit Produces a Return

A lower audited premium does not automatically mean that a return is due. The credited deposit may still be below the audited premium. Calculate the full balance before deciding which direction the settlement takes. If the balance is negative, its absolute value is the potential return; the policy’s terms determine whether that amount is refundable.

Worked Example: Lower Audited Exposure and a Return

A hypothetical policy charges $14 per unit in Category A and $20 per unit in Category B. Estimated exposures are 300 units in A and 250 units in B. The deposit is 95% of estimated premium. The audit establishes 260 units in A and 230 units in B. Find the final premium and determine the settlement.

Assume the rates stay fixed, the deposit is credited, and any negative audit balance is refundable under the policy terms. No interest, fees, taxes, mortality assumption, or probability distribution applies. The method is to calculate both category totals at the estimated and audited exposures, then compare the audited premium with the deposit.

1. Estimated premium. Category A contributes \(300\times \$14=\$4{,}200\). Category B contributes \(250\times \$20=\$5{,}000\). Thus:

$$ P_{\text{est}}=\$4{,}200+\$5{,}000=\$9{,}200. $$

2. Deposit credited.

$$ D=0.95\times \$9{,}200=\$8{,}740. $$

3. Audited premium. Category A contributes \(260\times \$14=\$3{,}640\). Category B contributes \(230\times \$20=\$4{,}600\). Therefore:

$$ P_{\text{aud}}=\$3{,}640+\$4{,}600=\$8{,}240. $$

4. Audit balance.

$$ B=\$8{,}240-\$8{,}740=-\$500. $$

The final premium is $8,240. The negative balance means a $500 return under the stated assumption. Check by decomposition: the audited premium decreased by \(\$8{,}240-\$9{,}200=-\$960\), while 5% of the estimate, or \(0.05\times \$9{,}200=\$460\), was not deposited. The balance is \(-\$960+\$460=-\$500\), consistent with the direct calculation.

Common mistake avoided: Reporting the full $960 decrease as the return ignores the $460 of estimated premium that was never deposited. Only the amount by which the credited deposit exceeds the audited premium can be returned.

Interpreting and Checking the Settlement

A useful final check is to compare three amounts: the estimated premium, the credited deposit, and the audited premium. The deposit must not be mistaken for the estimated premium unless the problem says it is 100% of that estimate. A deposit fraction also should not be applied to the audited premium unless the contract specifically defines the deposit that way.

  • Confirm the basis. Use the estimated exposure for the initial calculation and the audited exposure for the final calculation.
  • Follow the stated deposit rule. A percentage deposit is applied to the premium base specified in the problem, which may include fixed charges.
  • Subtract the credited deposit. The audit balance is final premium minus deposit, not final premium minus estimated premium.
  • Interpret the sign. A positive balance is due; a negative balance indicates a potential return.
  • Keep premium separate from cash timing. A balance describes the premium adjustment. Whether it has been billed or collected is a separate question.

This distinction matters when premium records are reviewed or summarized. The deposit is an initial amount credited toward the policy charge, while the final audited premium reflects the exposure established after the audit. Recording only the difference between estimated and audited premium can omit the unpaid portion of a partial deposit. Recording the full audited premium as a new charge without crediting the deposit can count part of the premium twice. A consistent calculation makes the final charge and any settlement direction clear.

Key takeaway: Recalculate the premium using audited exposure, then subtract the deposit actually credited. The result—not merely the change from estimated premium—determines whether additional premium is due or a return may be made.

Check Your Understanding

For each hypothetical policy, assume the stated rates remain unchanged, the deposit is credited, and no interest, taxes, fees, minimum premiums, or other adjustments apply unless specified. Calculate the final premium and audit balance.

  1. A policy charges $12 per unit. Estimated exposure is 500 units, and the deposit is 70% of estimated premium. The audit establishes 540 units. (Answer: Final premium $6,480; $2,280 additional premium due.)
  2. A policy charges $45 per unit in Category A and $30 per unit in Category B. Estimated exposures are 80 units in A and 120 units in B. The deposit is 85% of estimated premium. Audited exposures are 90 units in A and 110 units in B. (Answer: Final premium $7,350; $1,230 additional premium due.)
  3. A policy charges $16 per unit and has a $90 fixed charge included in both estimated and final premium. Estimated exposure is 250 units; the deposit is 80% of the estimated total, including the fixed charge. Audited exposure is 230 units. (Answer: Final premium $3,770; $498 additional premium due.)
  4. A policy charges $22 per unit. Estimated exposure is 400 units, and the deposit is 90% of estimated premium. Audited exposure is 350 units. A negative balance is refundable under the policy terms. (Answer: Final premium $7,700; $220 return.)
  5. A policy charges $28 per unit in Category A and $36 per unit in Category B. Estimated exposures are 140 units in A and 100 units in B. The deposit is 75% of estimated premium. Audited exposures are 150 units in A and 90 units in B. (Answer: Final premium $7,440; $1,800 additional premium due.)

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