Net Single Premiums for Life Annuities

What a Net Single Premium for an Annuity Measures

A life annuity pays only when its survival condition is met. Its value at issue is therefore not the present value of a certain stream of payments: it is the expected present value of payments that may stop when the annuitant dies. The calculation requires both the interest rate and the probabilities of survival to each payment date.

A net single premium is the expected present value of the contractual benefits, with no allowance for expenses, profit, or other loading. For a life annuity, this is the annuity benefit amount multiplied by the expected present value factor when the benefit is level. We use annual effective interest $i$ and $v=1/(1+i)$. Unless stated otherwise, payments are annual and the life is age $x$ at issue.

Definition: An annuity-immediate makes its first possible payment at the end of the first year, provided the annuitant is then alive. An annuity-due makes its first payment immediately at issue, provided the annuitant is alive at issue. A whole-life annuity continues for as long as the annuitant is alive; a temporary life annuity makes payments only during a specified finite period.

The whole-life annuity-immediate factor is $a_x$, and the whole-life annuity-due factor is $\ddot{a}_x$. For a temporary annuity of $n$ years, we use $a_{x:\overline{n}|}$ for the annuity-immediate factor and $\ddot{a}_{x:\overline{n}|}$ for the annuity-due factor. The temporary factor counts at most $n$ payments; it does not guarantee that all $n$ are made.

Building the Expected Present Value

Let ${}_kp_x$ be the probability that a life aged $x$ survives for at least $k$ years, with ${}_0p_x=1$. At time $k$, a payment can be made only if the life has survived to that payment time. Its expected present value is the payment amount multiplied by the discount factor and the relevant survival probability.

\begin{aligned} a_{x:\overline{n}|} &=\sum_{k=1}^{n}v^k\,{}_kp_x,\\ \ddot{a}_{x:\overline{n}|} &=\sum_{k=0}^{n-1}v^k\,{}_kp_x,\\ a_x &=\sum_{k=1}^{\infty}v^k\,{}_kp_x,\\ \ddot{a}_x &=\sum_{k=0}^{\infty}v^k\,{}_kp_x. \end{aligned}

The immediate annuity uses survival probabilities from year 1 onward because its possible payments occur at the ends of years 1 through $n$. The due annuity includes a payment at time 0, so its first term is $1$; its last possible payment is at time $n-1$. The whole-life sums extend over all possible annual payment times. An infinite sum represents a finite value when the series converges.

Net single premium: For a level annual payment $B$, the net single premium is $B$ times the appropriate annuity factor. Use $a$ for payments in arrears and $\ddot{a}$ for payments in advance, and use the temporary factor if payments stop after the stated term.

Why Each Term Has This Form

Consider a temporary annuity-immediate with level payment $B$ and term $n$. Define $I_k$ to be 1 if the annuitant survives to the end of year $k$, and 0 otherwise. The present value of the benefits is $B\sum_{k=1}^{n}v^k I_k$. Taking expectations and using the fact that $E[I_k]=P(\text{survive to }k)={}_kp_x$ gives the annuity-immediate formula.

Write the payment present values.
At the end of year $k$, the discounted payment, if made, is $Bv^k I_k$.

Add the possible payments.
The total present value is $B\sum_{k=1}^{n}v^k I_k$.

Take the expectation.
By linearity of expectation, the expected present value is $B\sum_{k=1}^{n}v^k E[I_k]=B\sum_{k=1}^{n}v^k{}_kp_x$.

The same argument gives the due-annuity formula, except that payments occur at times $k=0,\ldots,n-1$, and survival to time 0 has probability 1. For a whole-life annuity, apply the argument through a finite number of payment dates and extend the sum as the number of dates increases. This extension requires the resulting expected present-value series to converge.

Worked Calculations

Worked Example: Comparing Two Temporary Payment Timings

A hypothetical contract pays $1,000 annually for at most four years while the annuitant is alive. The annual effective interest rate is 5%, and the probability of surviving each year, conditional on being alive at the start of that year, is 0.90. Calculate the net single premium for (a) an annuity-due and (b) an annuity-immediate.

Method. The conditional annual survival probability is constant, so ${}_kp_x=0.90^k$. The discount factor is $v=1/1.05=20/21$. Thus $v^k{}_kp_x=(v\cdot0.90)^k=(6/7)^k$.

(a) The annuity-due can pay at times 0, 1, 2, and 3. Its factor is

\ddot{a}_{x:\overline{4}|} =1+\frac67+\left(\frac67\right)^2+\left(\frac67\right)^3 =\frac{343+294+252+216}{343} =\frac{1105}{343} =3.221574\ldots

The net single premium is $1{,}000(1105/343)=\$3{,}221.57$, rounded to the nearest cent.

(b) The annuity-immediate can pay at times 1, 2, 3, and 4. Its factor is

a_{x:\overline{4}|} =\frac67+\left(\frac67\right)^2+\left(\frac67\right)^3+\left(\frac67\right)^4 =\frac{2058+1764+1512+1296}{2401} =\frac{6630}{2401} =2.761349\ldots

The net single premium is $1{,}000(6630/2401)=\$2{,}761.35$, rounded to the nearest cent. The due annuity costs more because it has an immediate payment and, for the remaining corresponding payments, less discounting.

Check and mistake avoided. The due factor has four terms beginning with 1; the immediate factor has four terms beginning with $6/7$. Treating both as payments at the same dates would give the wrong premium.

Worked Example: Net Single Premium for a Whole-Life Annuity

A hypothetical whole-life annuity-immediate pays $200 at each year-end while the annuitant is alive. Assume an annual effective interest rate of 5% and a constant conditional probability of death of 0.05 in every year. Calculate the net single premium.

Method. The conditional probability of survival for one year is $p=1-0.05=0.95$, so ${}_kp_x=0.95^k$. The present-value factor for a possible payment at time $k$ is $v^k{}_kp_x=(vp)^k$, where $v=1/1.05=20/21$. Hence $vp=(20/21)(0.95)=19/21$.

The whole-life annuity-immediate factor is a geometric series. Since $19/21<1$, it converges:

a_x=\sum_{k=1}^{\infty}\left(\frac{19}{21}\right)^k =\frac{19/21}{1-19/21} =\frac{19}{2} =9.5.

The net single premium is $200(9.5)=\$1{,}900.00$, rounded to the nearest cent. This calculation uses the specified constant conditional mortality assumption for every future year; a one-year survival probability alone would not justify this infinite-series calculation unless the assumption about later years were also supplied.

Check and mistake avoided. The first payment is at the end of year 1, so the series starts with $vp=19/21$, not with 1. Starting with 1 would value an annuity-due instead.

Worked Example: Using a Survival Table for a Temporary Annuity

A hypothetical three-year annuity pays $1,200 annually while the annuitant is alive. Payments are annual, interest is 4% effective, and the survival probabilities are ${}_1p_x=0.88$, ${}_2p_x=0.74$, and ${}_3p_x=0.55$. Find the net single premium if payments are made (a) in advance and (b) in arrears.

Method. The survival probabilities are given directly, so no constant mortality assumption is needed. Here $v=1/1.04=25/26$.

(a) For payments in advance, the possible payment times are 0, 1, and 2. The factor is

\ddot{a}_{x:\overline{3}|} =1+v\,{}_1p_x+v^2\,{}_2p_x =1+\frac{11}{13}+\frac{925}{1352} =\frac{3421}{1352} =2.530325\ldots

Therefore the net single premium is $1{,}200(3421/1352)=\$3{,}036.39$, rounded to the nearest cent.

(b) For payments in arrears, the possible payment times are 1, 2, and 3. The factor is

\begin{aligned} a_{x:\overline{3}|} &=v\,{}_1p_x+v^2\,{}_2p_x+v^3\,{}_3p_x\\ &=\frac{11}{13}+\frac{925}{1352}+\frac{34375}{70304}\\ &=\frac{141963}{70304} =2.019273\ldots \end{aligned}

The net single premium is $1{,}200(141963/70304)=\$2{,}423.13$, rounded to the nearest cent. The third payment uses ${}_3p_x$, not ${}_2p_x$, because it is payable at time 3 only if the annuitant survives to time 3.

Check and mistake avoided. The advance factor includes a payment at time 0; the arrears factor does not. Both calculations use the same stated survival probabilities and differ because the payment dates differ.

Common Timing and Model Traps

A frequent error is to use a due-annuity formula for an immediate annuity, or vice versa. For example, in the first worked example, using the due factor $3.221574\ldots$ for the annuity-immediate would give a premium of $3,221.57 instead of $2,761.35. That incorrect calculation includes a payment at issue and places the later payments one year too early. The correction is to list the payment dates first: an immediate annuity pays at $1,\ldots,n$, while a due annuity pays at $0,\ldots,n-1$.

Other checks help prevent mistakes:

Match survival to the payment date. A payment at time $k$ requires survival to time $k$, so use ${}_kp_x$.
Do not treat a temporary annuity as guaranteed. The term limits the number of possible payments; it does not remove the survival condition.
State the mortality assumption for a whole-life sum. A finite set of survival probabilities is not, by itself, a complete model for all future years.
Keep the premium and factor distinct. The annuity factor values a unit payment; multiply it by the actual level payment to obtain the net single premium.

These calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The key discipline is to identify each possible payment date, attach its survival probability, and discount it to the valuation date before summing.

Check Your Understanding

For each problem, identify the payment timing before applying a formula. Amounts are hypothetical; round money to the nearest cent and factors to four decimal places where needed.

A three-year temporary annuity-due has annual payments of $1, with $i=5\%$, ${}_1p_x=0.80$, and ${}_2p_x=0.64$. Find its factor. (Answer: 2.3424)
A three-year temporary annuity-immediate has annual payments of $1, $i=4\%$, and survival probabilities ${}_1p_x=0.90$, ${}_2p_x=0.81$, and ${}_3p_x=0.729$. Find its factor. (Answer: 2.2624)
A whole-life annuity-due has annual payments of $1. The annual effective interest rate is 6%, and the conditional probability of death each year is 0.04. Find its factor. (Answer: 10.6000)
A two-year deferred temporary annuity-due makes three annual payments of $1 at times 2, 3, and 4, provided the annuitant is alive at each payment date. Assume $i=5\%$ and a constant conditional annual survival probability of 0.90. Find its expected present value factor at issue. (Answer: 1.9042)
A two-year temporary annuity-immediate pays $250 annually while the annuitant is alive. Let $i=5\%$, ${}_1p_x=0.95$, and ${}_2p_x=0.85$. Find the net single premium. (Answer: $418.93)

What You'll Learn