Gross Premiums for Life Annuity Contracts

Why Life Annuity Premiums Need Careful Timing

A life annuity exchanges a premium for payments made while a designated life survives. Its gross premium must fund both the promised annuity payments and the expenses attached to issuing and administering the contract. The timing matters: an expense paid whenever an annuity installment is paid has the same survival contingency as that installment, while an expense paid at issue is certain at time zero.

The main task is to put each expected cash flow on a consistent basis. For a single premium, compare the premium at issue with the present value of benefits and expenses. For level premiums, also value the stream of premiums and any expenses that arise when those premiums are collected. The premium-payment period need not be the same as the annuity-benefit period.

Definition: For a life aged $x$, a discrete life annuity-due pays one unit at time $k$ if the life is then alive, including a payment at time zero. Its actuarial present value is $\ddot{a}_x=\sum_{k=0}^{\infty}v^k\,{}_kp_x$. A life annuity-immediate pays at the end of each year if the life is alive at that time, with value $a_x=\sum_{k=1}^{\infty}v^k\,{}_kp_x$. Here $v=1/(1+i)$, and ${}_kp_x$ is the probability of surviving $k$ years from age $x$.

These formulas describe annual payments. They assume the stated payment is made only when the life satisfies the contract’s survival condition at its payment date. A due annuity has a payment at issue; an immediate annuity does not. If the contract has a finite payment period, include only payment dates within that period. For example, a three-year temporary annuity-immediate has value $\sum_{k=1}^{3}v^k\,{}_kp_x$.

Gross Premium Equations for Annuity Contracts

A useful setup is to value each cash flow at its actual time and under its own contingency. Let $B$ be the annuity installment. If an expense of $e$ dollars is incurred with every installment, its present value is $e$ times the same annuity factor as the benefit. If an expense is a fraction $c$ of each installment, the combined benefit and payment expense per installment is $B(1+c)+e$. A fixed issue expense is instead added once at time zero.

Suppose a gross single premium is $G$, a fraction $\alpha$ of that premium is spent at issue, and the annuity is a life annuity-due. With a fixed issue expense $C$, a fixed payment expense $e$, and an expense fraction $c$ of each benefit installment, the balance equation is

G=\alpha G+C+\bigl(B(1+c)+e\bigr)\ddot{a}_x.

The equation can be rearranged to isolate the premium:

G=\frac{C+\bigl(B(1+c)+e\bigr)\ddot{a}_x}{1-\alpha}.

This version applies when the annuity payments continue for life and the listed expenses occur at the stated times. A temporary annuity uses its temporary annuity factor instead. An expense that is not contingent on an annuity payment must not automatically be multiplied by the annuity factor.

For level premiums, let $P$ be payable annually in advance while the life is alive for $m$ years. Define the value of this premium stream as

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

If $C$ is an issue expense, $f$ is a fixed expense at each premium date, and $\alpha P$ is an expense proportional to each premium, the present value of premium income net of premium-related expenses is $(1-\alpha)P H_m-fH_m$. If a life annuity-due of amount $B$ continues for life and has payment expenses $e$ and $cB$ per installment, the balance equation becomes

(1-\alpha)P H_m = C+fH_m+\bigl(B(1+c)+e\bigr)\ddot{a}_x.

This equation assumes the fixed and percentage premium expenses occur at each premium date, including time zero, and that premiums are payable only while the life is alive. If the contract specifies a different premium timing or an expense that continues after premium payments stop, the corresponding present value must be changed.

Why the Present-Value Equations Work

The balance equations follow from the expected present value principle: set the expected present value of gross premium income equal to the expected present value of benefits and expenses. The annuity factor is appropriate for a cash flow paid only when an annuity installment is due because each such cash flow has the same payment dates and survival condition as the installment itself.

List each cash flow.
Separate benefits, issue expenses, expenses per annuity payment, premiums, and expenses per premium.

Match timing and survival.
Use a factor for payments at the beginning of each year for an annuity-due stream, or at year-end for an annuity-immediate stream. Use a temporary factor if payments stop after a fixed period.

Equate present values.
Place premium income on one side and benefits plus expenses on the other. Keep expenses proportional to the unknown premium on the premium side until solving.

For instance, the single-premium equation follows directly by writing the premium inflow as $G$, the issue expense as $\alpha G+C$, and the expected present value of the payment stream as $\bigl(B(1+c)+e\bigr)\ddot{a}_x$. Moving the term $\alpha G$ to the left gives $(1-\alpha)G=C+\bigl(B(1+c)+e\bigr)\ddot{a}_x$, which yields the stated formula when $1-\alpha\ne0$. The level-premium equation follows by applying the same balance to each premium date and summing its survival-weighted present value, $H_m$.

Worked Premium Calculations

Worked Example: Gross Single Premium for a Life Annuity-Due

A hypothetical contract pays \$10,000 at the beginning of each year while the annuitant is alive, including immediately at issue. The insurer pays a \$25 expense with each annuity installment, incurs a fixed \$500 issue expense, and spends 2% of the gross single premium on premium-related expense. The annual effective interest rate is $i=4\%$. Assume a discrete mortality model in which the conditional probability of surviving each year is $p=0.96$, independent of duration. Find the gross single premium.

Method. Value the annuity and its payment expense with the life annuity-due factor, add the certain issue expense, and gross up for the percentage-of-premium expense. This avoids treating the installment at issue as if it were paid one year later.

The one-year discount-survival factor is $vp=0.96/1.04=12/13$. Thus

\ddot{a}_x=\sum_{k=0}^{\infty}(vp)^k =\frac{1}{1-12/13}=13.

The expected present value of annuity installments is $10{,}000(13)=\$130{,}000$. The payment expense has the same timing, so its present value is $25(13)=\$325$. Add the \$500 issue expense to obtain total outgo before the percentage premium expense:

130{,}000+325+500=\$130{,}825.

If $G$ is the gross premium, 98% of it remains after the 2% expense. Therefore $0.98G=130{,}825$, giving $G=130{,}825/0.98=\$133{,}494.8979\ldots$. Rounded to the nearest cent, the gross single premium is \$133,494.90. Check: 2% of this premium is approximately \$2,669.90, leaving \$130,825.00 to fund the valued benefits and other expenses.

Worked Example: Level Premiums for a Lifetime Annuity

A hypothetical contract pays a life annuity-due of \$8,000 per year. Premiums are payable annually in advance for at most three years, and only while the annuitant is alive. The insurer incurs a \$400 issue expense, a \$30 fixed expense at each premium date, a \$20 administration expense at each premium date, and a premium-related expense equal to 5% of each gross premium. Assume $i=4\%$ and conditional annual survival probability $p=0.96$ at every duration. Determine the level gross annual premium.

Method. Use the lifetime annuity factor for benefits and the three-date temporary annuity-due factor for premiums and expenses incurred at premium dates. The two factors differ because premiums stop after the third date, while annuity benefits continue for life.

As in the previous model, $vp=12/13$, so $\ddot{a}_x=13$. The premium dates are time zero, one, and two, if the life is alive. Their factor is

H_3=1+\frac{12}{13}+\left(\frac{12}{13}\right)^2 =\frac{169+156+144}{169} =\frac{469}{169}.

The benefit present value is $8{,}000(13)=\$104{,}000$. The fixed and administration expenses total \$50 per premium date, with present value $50(469/169)=\$138.7574\ldots$. The issue expense is \$400. If $P$ is the gross annual premium, premium income is $PH_3$, and the 5% premium expense is $0.05PH_3$. The balance equation is

PH_3=104{,}000+400+50H_3+0.05PH_3,

P= \frac{104{,}000+400+50(469/169)} {0.95(469/169)} = \frac{17{,}667{,}050}{8{,}911} =\$39{,}652.2277\ldots.

The level gross premium is \$39,652.23 per year, rounded to the nearest cent. Substitution into the balance equation gives equal present values on both sides. The key modeling point is that only three premium dates are valued in $H_3$, but the benefit is valued over the full lifetime.

Worked Example: Single Premium for a Temporary Annuity-Immediate

A hypothetical contract pays \$6,000 at the end of each year for at most three years, provided the annuitant is alive at the payment date. There is a \$20 expense per installment and an additional expense equal to 1% of each \$6,000 benefit. The insurer incurs a \$250 issue expense and spends 2% of the gross single premium on premium-related expense. Assume $i=4\%$ and a constant conditional annual survival probability $p=0.96$. Find the gross single premium.

Method. Since payments are at year-end, use a temporary annuity-immediate factor. Include both payment-related expenses at each installment date, then account for the issue and premium expenses. This avoids incorrectly using a due-annuity factor that includes a payment at issue.

The discount-survival factor remains $vp=12/13$. The three possible installment dates therefore have factor

a_{x:\overline{3}|} =\frac{12}{13}+\left(\frac{12}{13}\right)^2+\left(\frac{12}{13}\right)^3 =\frac{2{,}028+1{,}872+1{,}728}{2{,}197} =\frac{5{,}628}{2{,}197}.

The percentage-of-benefit expense per installment is $0.01(6{,}000)=\$60$, so total payment-related expense is \$80 per installment. The combined benefit and payment expense is \$6,080 per installment. Its present value is

6{,}080\left(\frac{5{,}628}{2{,}197}\right) =\frac{34{,}218{,}240}{2{,}197} =\$15{,}574.9841\ldots.

Adding the \$250 issue expense gives \$15,824.9841$\ldots$ before the premium-related expense. If $G$ is the gross premium, $0.98G=15{,}824.9841\ldots$, so

G=\frac{15{,}824.9841\ldots}{0.98} =\$16{,}147.9430\ldots.

The required gross single premium is \$16,147.94, rounded to the nearest cent. Multiplying it by 0.98 recovers the present value of benefits and the issue expense, subject to rounding.

Common Timing and Loading Traps

In the first example, a common incorrect calculation is to value the \$25 payment expense with $a_x=12$, giving $25(12)=\$300$, while valuing the annuity benefit with $\ddot{a}_x=13$. That would produce $(130{,}000+300+500)/0.98=\$133{,}469.39$. It is wrong because the contract incurs the \$25 expense at the time-zero annuity installment too. The expense and benefit have identical payment timing, so both use $\ddot{a}_x$; the correct premium is \$133,494.90.

Using the wrong annuity timing. An annuity-due includes a payment at issue; an annuity-immediate begins at the end of the first year. Match the factor to the contract’s stated payment dates.
Using the benefit period for premiums. If level premiums stop before annuity benefits, value premiums with their own temporary survival-weighted factor.
Forgetting that premium expenses depend on the premium. A percentage expense is part of the unknown-premium equation; it is not a fixed dollar amount known before solving.
Applying a payment expense at the wrong dates. An expense incurred only when an installment is paid follows that installment’s timing and survival condition. An issue expense does not.

These checks are useful whenever a contract has multiple expense bases. First identify which cash flow triggers each expense. Then assign its payment date and survival condition before choosing an annuity factor. This keeps a gross premium calculation aligned with the contract rather than with a convenient but inaccurate shortcut.

Check Your Understanding

For each problem, use the stated annual interest and mortality assumptions, and round dollar premiums to the nearest cent.

A hypothetical life annuity-due pays \$5,000 annually for life. At each installment, a \$10 expense is incurred. The issue expense is \$200, and 4% of the gross single premium is spent at issue. If $i=4\%$ and the conditional annual survival probability is $0.96$ at every duration, find the gross single premium. (Answer: \$68,052.08)
A contract pays a lifetime annuity-due of \$4,000 annually. Gross premiums are due at times zero and one, while the life is alive. The issue expense is \$100, the fixed expense per premium date is \$25, and 10% of each gross premium is spent on premium-related expenses. With $i=4\%$ and conditional annual survival probability $0.96$, find the level gross premium. (Answer: \$30,130.00)
A hypothetical three-year temporary annuity-immediate pays \$3,000 at each year-end if the annuitant is alive. Each payment triggers a \$10 expense; the issue expense is \$100, and 2% of the gross single premium is spent on premium-related expenses. With $i=4\%$ and conditional annual survival probability $0.96$, find the gross single premium. (Answer: \$7,970.04)
A life annuity-due pays \$2,000 annually for life. Gross premiums are due for at most two years, in advance while the life is alive. The issue expense is \$250, the fixed expense per premium date is \$20, and 5% of each gross premium is spent on premium-related expenses. With $i=4\%$ and conditional annual survival probability $0.96$, find the level gross premium. (Answer: \$14,389.47)

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