Integrated Premium Calculation Workshop

Putting a Premium Calculation Together

A premium calculation is an expected-present-value balance: the present value of premiums must fund the promised benefits and, for a gross premium, the expenses specified by the contract. The difficult part is often not a single formula but keeping the cash-flow timing, survival conditions, and expense bases consistent throughout the calculation.

The integrated method in this tutorial applies to discrete contracts with annual cash flows. The same structure extends to other payment frequencies by using the corresponding interest and payment timing. The examples use hypothetical contracts and mortality assumptions; the method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Under the equivalence principle, a premium is chosen so that the expected present value of premiums equals the expected present value of the outgo being funded. A net premium funds benefits only. A gross premium also funds the contract's specified expenses.

Set Up the Cash Flows Before Calculating

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. For a life aged $x$, ${}_kp_x$ is the probability of surviving $k$ years, and ${}_kq_x$ is the probability of dying within $k$ years. In an annual discrete model, a death benefit is commonly paid at the end of the year of death, while an annual premium due at the start of a year is paid only if the insured is alive then.

For example, if premiums are due at times $0,1,\ldots,n-1$ while the insured is alive, their expected present-value factor is the temporary life annuity-due value $\ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k{}_kp_x$. If a benefit $B_k$ is paid at time $k$ only when a specified event occurs, its contribution to expected present value is the payment amount multiplied by the event probability and $v^k$. List each cash flow separately before combining them.

Specify the contract and timing.
Record when each benefit, premium, and expense is due, and the survival or claim condition for each payment.

Discount expected outgo.
For each benefit and expense, multiply its amount by its payment probability and discount factor.

Value the premium stream.
Use the probability that each premium is actually paid, not the probability that a benefit is paid.

Equate the sides and solve.
For gross premiums, separate fixed expenses from expenses that are a percentage of premium.

Why the Equivalence Equation Works

The equivalence principle balances expected present values at issue. Suppose a level annual premium $P$ is due at times $0$ through $n-1$ while the insured survives to each payment date. The payment at time $k$ has present value $Pv^k$ if it is made, and probability ${}_kp_x$ of being made. Its expected present value is therefore $Pv^k{}_kp_x$. Adding across payment dates gives $P\ddot{a}_{x:\overline{n}|}$. If the benefit present value is $C$, the net premium equation is $P\ddot{a}_{x:\overline{n}|}=C$, so $P=C/\ddot{a}_{x:\overline{n}|}$. The argument depends on valuing each payment at its actual date and applying the correct payment condition.

P\ddot{a}_{x:\overline{n}|}=\operatorname{EPV}(\text{benefits}), \qquad G\ddot{a}_{x:\overline{n}|}=\operatorname{EPV}(\text{benefits and expenses})

For a gross premium $G$, an expense that is a proportion $\alpha$ of each premium leaves only $(1-\alpha)G$ available toward benefits and other expenses. A fixed expense per premium, by contrast, is included as a separate outgo. If $C$ is benefit and claim-related expense present value, $F$ is the present value of fixed expenses not tied to premium dates, $c$ is a fixed expense per premium, and $a$ is the expected present-value factor for premiums, then

(1-\alpha)Ga=C+F+ca, \qquad G=\frac{C+F+ca}{(1-\alpha)a}.

This form assumes the stated percentage expense applies to each premium and the fixed per-premium expense is incurred exactly when that premium is paid. If the contract uses a different expense basis or timing, the equation must be adjusted accordingly.

Worked Example: Net Premium for a Two-Year Term Benefit

A hypothetical two-year term contract pays \$10,000 at the end of the year of death. The annual effective interest rate is $5\%$. Conditional death probabilities are $q_0=0.02$ in year 1 and $q_1=0.03$ in year 2, so the corresponding survival probabilities are $p_0=0.98$ and $p_1=0.97$. Two level premiums are due at times 0 and 1 while the insured is alive. Find the annual net premium.

Method. Calculate the expected present value of the death benefit, then divide by the expected present value of the premium stream. There are no expenses in a net premium calculation.

The discount factor is $v=1/1.05=0.95238095$. The year 1 death benefit has probability $0.02$, so its expected present value is $10{,}000(0.02)v=\$190.47619$. The probability of surviving year 1 and dying in year 2 is $0.98(0.03)=0.0294$. That benefit's expected present value is $10{,}000(0.0294)v^2=\$266.66667$.

Thus the total benefit present value is $\$190.47619+\$266.66667=\$457.14286$. The time 0 premium is certain and has present-value factor 1. The time 1 premium is paid with probability $0.98$, so the premium factor is $1+0.98v=1+0.93333333=1.93333333$.

The net premium is $\$457.14286/1.93333333=\$236.4532$, or \$236.45 per year rounded to the nearest cent. As a check, $\$236.4532(1.93333333)$ is approximately $\$457.14286$. The common mistake avoided here is counting the second premium with certainty: it is conditional on survival to time 1.

Gross Premiums: Keep Each Expense on Its Own Basis

Expenses can depend on the policy, the premium, or the claim. Acquisition and renewal expenses are often fixed dollar amounts at specified times. A fixed per-premium expense is incurred whenever a premium is collected. A percentage-of-premium expense is proportional to the premium amount. Claim-related expenses are incurred when a claim occurs and should be valued using the same claim probability and timing as the claim, unless the contract specifies otherwise.

A reliable calculation does not combine these categories prematurely. First value each expense stream on its own, then add the present values to the benefit outgo. This prevents, for example, applying a claim expense to every policy year or applying a renewal expense to a policy that has already terminated.

Worked Example: Term Insurance with Multiple Expense Bases

A hypothetical two-year term policy pays \$20,000 at the end of the year of death. The annual effective interest rate is $5\%$, with conditional death probabilities $q_0=0.04$ and $q_1=0.05$; hence $p_0=0.96$. Premiums are due at times 0 and 1 while the insured is alive. Expenses are: \$150 at issue, \$25 at time 1 if the insured is alive, \$12 for each premium collected, a $5\%$ expense on each premium, and \$100 for each death claim. Find the level annual gross premium.

Method. Value claims and claim expenses together, then value each fixed expense stream. Apply the percentage expense by allowing only $95\%$ of each gross premium to fund the other outgo.

Here $v=20/21$. The probability-weighted discount factor for a claim is $0.04v+(0.96)(0.05)v^2=4/49=0.0816326531$. Therefore, the benefit present value is $\$20{,}000(4/49)=\$1{,}632.6531$, and claim expenses have present value $\$100(4/49)=\$8.1633$. Combined claim and claim-expense present value is $\$1{,}640.8163$.

The premium annuity-due factor is $1+0.96v=1+32/35=67/35=1.91428571$. The \$150 acquisition expense is paid at issue, so its present value is \$150. The expected present value of the renewal expense is $\$25(0.96v)=\$25(32/35)=\$22.8571$. The \$12 per-premium expense has present value $\$12(67/35)=\$22.9714$.

The gross premium equation is therefore \[ 0.95G\left(\frac{67}{35}\right) = 1{,}640.8163+150+22.8571+22.9714 = 1{,}836.6448. \] Using unrounded values, the coefficient of $G$ is $0.95(67/35)=1273/700=1.81857143$, giving $G=\$1{,}836.6449/1.81857143=\$1{,}009.9383$. The gross premium is \$1,009.94 per year, rounded to the nearest cent.

As a check, the premium stream has present value $G(67/35)$; $5\%$ of that is the percentage expense, while the remaining $95\%$ contributes about \$1,836.64 toward the other outgo. The common mistake avoided is to use the full premium stream as available funding even though $5\%$ is explicitly spent on premium-based expenses.

Common exam trap: If a renewal expense is paid only when the insured survives to the renewal date, discounting the amount alone is not enough. Its expected present value includes the probability of being alive at that date. Likewise, a premium due at that date is not automatically certain.

Integrating a Life Annuity Benefit and Expenses

For an annuity benefit paid only while the annuitant is alive, the survival probabilities determine the expected benefit payments. Premiums may have a different timing or duration, so they need their own payment factor. The benefit and premium factors can happen to use the same survival probabilities while still using different discount exponents.

Worked Example: Gross Premium for a Temporary Life Annuity

A hypothetical contract pays \$2,000 at the end of each of years 1 and 2 if the annuitant is alive at that payment date. The annual effective interest rate is $25\%$. The probability of surviving year 1 is $0.90$, and the conditional probability of surviving year 2, given survival to year 1, is $0.80$. Level premiums are due at times 0 and 1 while the annuitant is alive. At issue there is a \$50 expense; at time 1 there is a \$20 renewal expense if the annuitant is alive. There is also an \$8 expense per premium and a $10\%$ expense on each premium. Find the gross premium.

Method. Value each annuity payment at its payment date and survival probability. Then calculate the premium and expense factors separately. At time 1, the premium is collected immediately before the annuity payment; both are conditional on survival to that date.

The discount factor is $v=1/1.25=0.8$. The probability of surviving to the end of year 2 is $0.90(0.80)=0.72$. The benefit present value is $\$2{,}000[0.90(0.8)+0.72(0.8^2)]$ $=\$2{,}000(0.72+0.4608)=\$2{,}361.60$.

The premium factor is $1+0.90(0.8)=1.72$. The issue expense is \$50. The renewal expense present value is $\$20(0.90)(0.8)=\$14.40$. The per-premium expense present value is $\$8(1.72)=\$13.76$. Thus the total outgo other than the percentage-of-premium expense is $\$2{,}361.60+\$50+\$14.40+\$13.76=\$2{,}439.76$.

The $10\%$ expense leaves $90\%$ of the premiums available, so $0.90G(1.72)=\$2{,}439.76$. The coefficient is $1.548$, and $G=\$2{,}439.76/1.548=\$1{,}576.0724$. The gross premium is \$1,576.07, rounded to the nearest cent. Substitution gives $0.90(1.72)(\$1{,}576.0724)=\$2{,}439.76$, confirming the balance. The common mistake avoided is valuing the year 2 benefit using only the one-year survival probability instead of the two-year probability $0.72$.

Checks That Catch Integrated-Calculation Errors

Before accepting a premium, check the model and the arithmetic. Confirm the interest rate is effective for the payment interval; confirm each survival probability is conditional on the correct starting age or duration; and check that premiums, benefits, and expenses are discounted to the same valuation date. Then inspect the expense equation: percentage expenses multiply premium, fixed per-premium expenses use the premium payment factor, and claim expenses follow claim timing.

A useful final check is substitution. Insert the calculated premium into the original expected-present-value equation, not a rearranged equation with a potentially omitted term. The two sides should agree before rounding. A discrepancy can reveal a missing expense, an incorrect survival condition, or an inconsistent payment date.

Wrong premium duration: Do not count premiums after the contract ends or after death if the premium is conditional on survival.
Wrong payment timing: An annuity-due premium at time 0 is not discounted; a payment at time 1 is discounted once.
Wrong expense base: A fixed dollar expense is not a percentage of premium, and a claim expense is not incurred in every policy year.
Premature rounding: Retain unrounded factors and intermediate amounts; round the final dollar premium to cents.

Check Your Understanding

For each hypothetical contract, identify the relevant present-value factors before solving. Answers are rounded to the nearest cent where premiums are in dollars.

A two-year term policy pays \$1,000 at the end of the year of death. There is no interest. Conditional death probabilities are $0.10$ in year 1 and $0.20$ in year 2. Annual premiums are due at times 0 and 1 while alive. Find the net premium. (Answer: \$147.37 per year)
A two-year term policy has the same mortality as in question 1 and pays \$5,000 at death. There is no interest. Premiums are annual and due while alive. Expenses are \$40 at issue, \$10 at time 1 if alive, and $10\%$ of each premium; there are no other expenses. Find the gross premium. (Answer: \$847.37 per year)
A single-premium contract pays \$1,000 at time 2 if the insured survives two years. The annual effective interest rate is $25\%$; survival probabilities are $0.80$ for year 1 and $0.75$ for year 2 conditional on surviving year 1. The contract has a \$30 issue expense and a $5\%$ expense on the single premium. Find the gross single premium. (Answer: \$435.79)
A two-year temporary life annuity pays \$600 at the end of each year if alive. The annual effective interest rate is $25\%$; conditional survival probabilities are $0.90$ in year 1 and $0.80$ in year 2. Annual premiums are due at times 0 and 1 while alive, with no expenses. Find the net annual premium. (Answer: \$411.91 per year)
A two-year term policy pays \$10,000 at death. The annual effective interest rate is $5\%$, with conditional death probabilities $0.02$ and $0.03$. Annual premiums are due at times 0 and 1 while alive. Expenses are \$100 at issue, \$20 at time 1 if alive, a claim expense equal to $2\%$ of the death benefit, and $5\%$ of each premium. Find the gross annual premium. (Answer: \$318.49 per year)

What You'll Learn