Gross Premium Equations and Expense Present Values

Why Gross Premiums Need an Expense Present Value

A net premium balances the expected present value of premiums against the expected present value of benefits. A gross premium must also provide for expenses associated with issuing and administering the contract and settling claims. The central task is to value each expected cash flow at the time it occurs, allowing for whether the policy is still in force or a claim has occurred.

The equivalence principle sets the expected present value of gross premiums equal to the expected present value of benefits plus the expected present value of expenses. This is an actuarial pricing equation under stated assumptions; it does not guarantee that actual premiums will equal actual claims and expenses for any particular policy.

Definition: The gross premium is the premium determined by equating the expected present value of premiums to the expected present value of benefits and expenses. Each expected present value accounts for the amount, payment time, and probability that the cash flow occurs.

Building the Equation from Cash Flows

Suppose a policy issued to a life aged $x$ has a level annual gross premium $G$, payable in advance at times $0,1,\ldots,m-1$, provided the policy is then in force. Let $v=1/(1+i)$, where $i$ is the effective annual interest rate. The expected present value of premiums is

G\sum_{k=0}^{m-1}v^k\,{}_kp_x.

Here ${}_kp_x$ is the probability that the life survives $k$ years from age $x$, and ${}_0p_x=1$. The time-0 premium is certain, so its term is $G$. Each later premium is discounted and multiplied by the probability that it is payable.

Write $B$ for the expected present value of benefits and $E$ for the expected present value of expenses. Then the gross premium equation is

G\sum_{k=0}^{m-1}v^k\,{}_kp_x=B+E.

If premiums vary by time, replace the level amount $G$ with the appropriate premium amount in each term. The same principle applies to benefits and expenses: value each payment at its actual time and weight it by the probability that it is payable. A payment at issue has no discount factor; a payment at the end of year $t$ is discounted by $v^t$.

Why the Equation Holds

The equation follows from adding the present values of the policy’s cash flows and taking expectations. For example, let $I_k$ equal 1 if the premium at time $k$ is payable and 0 otherwise. The present value of premiums is $G\sum_k v^k I_k$. Since $E[I_k]=P(I_k=1)={}_kp_x$, linearity of expectation gives the premium present value above.

The same reasoning applies to benefits and expenses. If an amount $C_t$ is payable at time $t$ only when an event occurs, its expected present value is $v^t E[C_t]$, where $C_t$ is zero when the event does not occur. The equivalence principle balances these expected present values. This calculation does not require the cash flows to be independent: linearity of expectation is sufficient.

List the cash flows.
Separate premiums, benefits, and each expense payment. Record the amount and time for each.

Identify the payment condition.
For each cash flow, state whether it is certain, conditional on survival, or conditional on a claim or another event.

Discount and weight.
Discount each amount from its payment time and multiply by the probability that it is payable.

Equate and solve.
Set the expected present value of premiums equal to the combined expected present value of benefits and expenses.

Worked Gross Premium Equations

Worked Example: Two-Year Term Insurance with Claim Expenses

A hypothetical two-year term policy pays $10,000 at the end of the year of death. The conditional probability of death is 0.10 in year 1 and 0.20 in year 2, given survival to that year. The effective annual interest rate is 5%. Level premiums are payable in advance at issue and one year later if the policy is in force. Expenses are $120 at issue, $30 at time 1 if the policyholder is alive, and $200 when a death benefit is paid. Find the gross premium.

Method and timing check. Calculate the probability of each death year, then value benefits, premium payments, and expenses separately. In particular, the year-2 death probability is conditional on surviving year 1.

The discount factors are $v=1/1.05=20/21$ and $v^2=400/441$. The probability of death in year 1 is 0.10; the probability of death in year 2 is $0.90(0.20)=0.18$. Thus the discounted probability-weighting factor for a payment on death is

0.10v+0.18v^2 =0.10\left(\frac{20}{21}\right)+0.18\left(\frac{400}{441}\right) =\frac{38}{147} \approx 0.2585034.

The expected present value of benefits is $10{,}000(38/147)=\$2{,}585.03$. Premiums are payable at time 0 with certainty and at time 1 with probability 0.90, so their present-value factor is

1+0.90v=1+0.90\left(\frac{20}{21}\right)=\frac{13}{7}.

The expense present values are: $120 at issue; $30(0.90)v=\$25.71$ at time 1; and $200(38/147)=\$51.70$ when a claim occurs. Therefore $E=\$120+\$25.71+\$51.70=\$197.41$. The gross premium equation gives

G\left(\frac{13}{7}\right) =10{,}000\left(\frac{38}{147}\right)+120+30(0.90)v+200\left(\frac{38}{147}\right), \qquad G=\frac{10{,}000(38/147)+120+30(0.90)v+200(38/147)}{13/7} =\$1{,}498.24.

Check. The premium present value is approximately $\$1{,}498.24(13/7)=\$2{,}782.45$, equal to the benefit present value of $\$2{,}585.03$ plus the expense present value of $\$197.41$, up to rounding. The timing error to avoid is treating the second-year death probability as 0.20 unconditionally; it is 0.18 from issue.

Worked Example: Endowment Benefit with Settlement Expenses

A hypothetical two-year endowment policy pays $10,000 at the end of year 1 if death occurs in that year, or at the end of year 2 if the life survives year 1. If death occurs in year 2, the same $10,000 is paid at the end of that year. The conditional death probability is 0.10 in each year. The effective annual interest rate is 10%. Level premiums are due at times 0 and 1 while the policy is in force. Expenses are $100 at issue, $40 at each premium date, and $50 whenever the benefit is paid. Find the gross premium.

Method and timing check. At the end of year 2, either death or survival to maturity results in a payment. Together those outcomes have probability 0.90, conditional on survival through year 1. Do not count the year-2 benefit as payable only on death.

Here $v=10/11$. The probability of a year-1 benefit is 0.10. The probability of a year-2 benefit is 0.90, since year-2 death and survival to maturity together exhaust the outcomes after survival to year 1. The benefit present-value factor per dollar is

0.10v+0.90v^2 =0.10\left(\frac{10}{11}\right)+0.90\left(\frac{100}{121}\right) =\frac{101}{121}.

The expected present value of benefits is $10{,}000(101/121)=\$8{,}347.11$. The premium present-value factor is $1+0.90v=20/11$, because the second premium is due at time 1 only if the life survives the first year. Expenses have present values of $100 at issue, $40(1+0.90v)=\$72.73$ at the two premium dates, and $50(101/121)=\$41.74$ at settlement. Thus total expected expense present value is $\$214.46$.

G\left(\frac{20}{11}\right) =10{,}000\left(\frac{101}{121}\right) +100+40\left(\frac{20}{11}\right)+50\left(\frac{101}{121}\right), \qquad G=\$4{,}708.86.

Check. The right side is $\$8{,}347.11+\$214.46=\$8{,}561.57$. Multiplying the unrounded premium by $20/11$ gives the same present value. The trap avoided is omitting maturity payments from the year-2 benefit probability.

Worked Example: Three-Year Term Insurance with Recurring Expenses

A hypothetical three-year term policy pays $5,000 at the end of the year of death. The conditional death probability is 0.20 in each year. The effective annual interest rate is 5%, and level premiums are payable in advance at times 0, 1, and 2 if the policy is in force. Expenses are $75 at issue, $18 at each of times 1 and 2 if the policyholder is then alive, and $100 at claim settlement. Find the gross premium, rounded to cents.

Method and timing check. The survival probabilities to times 1 and 2 are 0.80 and $0.80^2=0.64$. The death probabilities by year, measured from issue, are 0.20, $0.80(0.20)=0.16$, and $0.64(0.20)=0.128$.

With $v=20/21$, the discounted probability-weighting factor for a death payment is

H=0.20v+0.16v^2+0.128v^3 =\frac{4132}{9261} \approx 0.4461721.

The benefit present value is $5{,}000H=\$2{,}230.86$. The premium factor is

1+0.80v+0.64v^2 =1+\frac{16}{21}+\frac{256}{441} =\frac{1033}{441}.

The expense present value is $75+18(0.80v+0.64v^2)+100H$. In this expression, $0.80v+0.64v^2=16/21+256/441=592/441$, so the three expense components are $75.00, $24.16, and $44.62. Their total is $143.78. Therefore,

G\left(\frac{1033}{441}\right) =5{,}000H+75+18\left(\frac{592}{441}\right)+100H, \qquad G=\frac{5{,}000(4132/9261)+75+18(592/441)+100(4132/9261)}{1033/441} =\$1{,}013.76.

Check. The unrounded gross premium times $1033/441$ equals the combined benefit and expense present value, approximately $\$2{,}374.64$. The common mistake is to charge each recurring expense with certainty; the time-1 and time-2 expenses are payable only if the policyholder has survived to those times.

Common Setup Errors

A correct numerical solution starts with a correct cash-flow equation. Before calculating, write down when each amount is paid and what must be true for it to be paid. Then apply the discount factor and probability for that specific payment. A payment at issue, a payment conditional on survival, and a payment triggered by death do not share the same probability or discount factor.

Using the wrong time unit. A payment at the end of year 2 is discounted by $v^2$, not $v$. Match the discount exponent to the payment time.
Ignoring a survival condition. A renewal premium or expense payable at time $k$ only while the policy is in force must be weighted by the probability of being in force at time $k$.
Using a conditional probability as an unconditional probability. For example, a year-2 death probability from issue requires multiplying the conditional year-2 death probability by the probability of surviving year 1.
Omitting an expense or counting it twice. Give each expense its own amount, payment time, and payment condition, and include it once on the expense side of the equation.
Rounding too early. Keep intermediate values unrounded where possible and round the final premium to the requested precision.

Key takeaway: A gross premium equation is an expected-present-value balance. The premium factor reflects when premiums are payable; the benefit and expense present values reflect their own payment times and conditions. The premium is obtained by dividing total expected benefit and expense present values by the expected number of discounted premium payments.

Check Your Understanding

For each problem, identify the relevant payment conditions before forming the gross premium equation.

A one-year term policy pays $10,000 at the end of the year if death occurs. The death probability is 0.10, and the interest rate is 0%. A level premium is due at issue. Expenses are $50 at issue and $20 on a claim. Find the gross premium. (Answer: $1,052.00)
A two-year term policy pays $5,000 at the end of the year of death. The conditional death probability is 0.20 in each year, and the interest rate is 0%. Premiums are due at times 0 and 1 while in force. Expenses are $60 at issue, $25 at time 1 if alive, and $40 on a claim. Find the gross premium, rounded to cents. (Answer: $1,052.44)
A two-year endowment pays $8,000 at the end of year 1 on death, or at the end of year 2 on death or survival. The conditional death probability is 0.10 in each year, and the effective annual interest rate is 10%. Premiums are due at times 0 and 1 while in force. Expenses are $40 at issue and $30 when the benefit is paid. Find the gross premium. (Answer: $3,708.50)
A two-year term policy pays $10,000 at the end of the year of death. The year-1 death probability is 0.10, and the conditional year-2 death probability is 0.20. The interest rate is 0%. Premiums are due at times 0 and 1 while in force. Expenses are $100 at issue, $15 at time 1 if alive, and $25 on a claim. Find the gross premium, rounded to cents. (Answer: $1,537.11)

What You'll Learn