Net Level Premiums for Life Annuities

Introduction

A life annuity pays while its recipient is alive. Its value therefore depends both on interest and on the probability that each scheduled payment will be made. When a level premium funds an annuity benefit, the premium must also be valued as a survival-contingent payment stream. Correct pricing requires matching the two streams’ expected present values, with their payment dates and survival conditions stated explicitly.

This tutorial develops that calculation for discrete whole-life and temporary life annuities. It uses the net equivalence principle: expected present value of premiums equals expected present value of benefits, with no expenses or profit included. The methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A net level premium is a constant premium amount determined by equating the expected present value of premiums to the expected present value of benefits. In this tutorial, both premium and benefit payments are contingent on the annuitant being alive at the scheduled payment time. A payment due at time 0 is made immediately, when the annuitant is alive by assumption.

Survival-Contingent Annuity Factors

Let the annual effective interest rate be $i$, and let $v=1/(1+i)$. For a life aged $x$, ${}_kp_x$ is the probability of surviving from age $x$ to age $x+k$, with ${}_0p_x=1$. A payment of one at time $k$, conditional on being alive then, has expected present value $v^k{}_kp_x$: it is discounted by $v^k$, and it is made with probability ${}_kp_x$.

In an annual annuity-due, payments are scheduled at times 0, 1, 2, and so on, while the annuitant is alive. In an annual annuity-immediate, payments are scheduled at times 1, 2, 3, and so on, if the annuitant is alive at those times. For whole-life annuities, their expected present values per unit of annual payment are:

\ddot{a}_x=\sum_{k=0}^{\infty}v^k{}_kp_x, \qquad a_x=\sum_{k=1}^{\infty}v^k{}_kp_x.

The first term of $\ddot{a}_x$ is 1, because the time-0 payment is certain. There is no such term in $a_x$. These expressions assume an annual payment pattern and a survival model for the life aged $x$; they do not describe continuous annuity payments.

For a temporary annuity covering $n$ annual payment dates, the due schedule uses times $0,\ldots,n-1$, while the immediate schedule uses times $1,\ldots,n$. Their factors per unit payment are, respectively, $\sum_{k=0}^{n-1}v^k{}_kp_x$ and $\sum_{k=1}^{n}v^k{}_kp_x$. The survival probability at each payment date must be used, including at the final date.

Why Survival-Contingent Present Values Have This Form

Let $I_k$ equal 1 if the annuitant is alive at time $k$, and 0 otherwise. A unit payment at time $k$, if alive, has present value $v^kI_k$. Since $E[I_k]=P(\text{alive at time }k)={}_kp_x$, its expected present value is $v^k{}_kp_x$. For finitely many scheduled payments, linearity of expectation gives the sum of these terms. For a whole-life annuity, the same sum extends over all possible payment dates when the present-value series converges.

The equivalence principle then equates the benefit sum and premium sum. If benefits have amount $B$ per payment and premiums have constant amount $P$, each sum is the relevant unit-payment factor multiplied by its payment amount. Thus the premium is the benefit present value divided by the premium payment factor. This is why the timing and survival conditions for each stream must be identified separately.

Whole-Life Net Level Premiums

For a whole-life annuity benefit of amount $B$ each year in arrears, funded by level premiums payable annually in advance for life, the benefit factor is $a_x$ and the premium factor is $\ddot{a}_x$. The net level premium is therefore:

P\ddot{a}_x=Ba_x, \qquad P=B\frac{a_x}{\ddot{a}_x}.

If both benefits and premiums use the same whole-life timing, use the same factor on both sides and the level premium equals the benefit amount. That simple result does not hold when one stream is due and the other immediate. The premium period must also match its stated conditions: a whole-life premium is paid only at scheduled dates when the annuitant is alive.

Worked Example: Whole-Life Benefit in Arrears, Premiums in Advance

A hypothetical contract pays a whole-life annuity of $1,200 at the end of each year if the recipient is alive. Level premiums are due at the start of each year while the recipient is alive. Assume a 5% annual effective interest rate and a constant annual survival probability of 0.90 at every future year. Find the net level annual premium.

Method. Calculate the two whole-life factors from the stated interest and survival assumptions, then equate their expected present values. The annual discount factor is $v=1/1.05=20/21$. Since ${}_kp_x=0.90^k$, each discounted survival term is $(v\cdot0.90)^k=(6/7)^k$.

The benefit is an annuity-immediate, so its factor is the infinite geometric sum beginning at $k=1$:

a_x=\sum_{k=1}^{\infty}\left(\frac67\right)^k =\frac{6/7}{1-6/7}=6.

The premiums are an annuity-due, so their factor includes the certain time-0 premium:

\ddot{a}_x=\sum_{k=0}^{\infty}\left(\frac67\right)^k =\frac{1}{1-6/7}=7.

The expected present value of benefits is $1{,}200(6)=\$7{,}200$. Equivalence gives $7P=\$7{,}200$, so $P=\$7{,}200/7=\$1{,}028.5714\ldots$. To the nearest cent, the premium is $1,028.57 per year. The unrounded premium gives equal present values: $7(7{,}200/7)=\$7{,}200$. The cent-rounded premium gives a small rounding difference, as expected. This method avoids using the annuity-due factor for an annuity-immediate benefit.

Temporary Annuity Benefits and Premiums

For a temporary annuity, only the specified finite number of payment dates is included. If benefits are due for $n$ years but level premiums are paid in arrears for $n$ years, the equivalence equation is:

P\sum_{k=1}^{n}v^k{}_kp_x = B\sum_{k=0}^{n-1}v^k{}_kp_x, \qquad P= B\frac{\sum_{k=0}^{n-1}v^k{}_kp_x} {\sum_{k=1}^{n}v^k{}_kp_x}.

This equation is specific to the stated schedules: annuity-due benefits and annuity-immediate premiums. If either schedule changes, its payment dates and survival probabilities must be changed as well. In particular, the last premium in arrears is contingent on survival to time $n$, whereas the last due benefit is contingent on survival to time $n-1$.

Worked Example: Three-Year Temporary Annuity with Arrears Premiums

A hypothetical contract pays $900 at times 0, 1, and 2 if the recipient is alive at each payment date. The level premium is payable at the ends of years 1, 2, and 3, if the recipient is alive then. Assume a 5% annual effective interest rate and a constant annual survival probability of 0.90. Find the net premium.

Method. Use a three-payment annuity-due factor for benefits and a three-payment annuity-immediate factor for premiums. Here $v=20/21$ and $v{}_1p_x=(20/21)(0.90)=6/7$. The discounted survival term for year $k$ is therefore $(6/7)^k$.

The benefit factor for times 0, 1, and 2 is:

1+\frac67+\left(\frac67\right)^2 =1+\frac67+\frac{36}{49} =\frac{127}{49} =\frac{889}{343}.

The premium factor for times 1, 2, and 3 is:

\frac67+\left(\frac67\right)^2+\left(\frac67\right)^3 =\frac{294+252+216}{343} =\frac{762}{343}.

Equating expected present values gives $P(762/343)=900(889/343)$. Thus $P=900(889/762)=900(7/6)=\$1{,}050$. Checking, the benefit value is $900(889/343)=\$2{,}332.653\ldots$, and the premium value is $1{,}050(762/343)=\$2{,}332.653\ldots$. The present values agree. This calculation avoids treating the arrears premiums as if they were paid at the due dates.

Different Survival Probabilities by Year

A constant annual survival probability is convenient for geometric sums, but it is not required. With a life table or a sequence of age-specific mortality assumptions, use the resulting ${}_kp_x$ separately at each payment date. The same equivalence equation applies; only the survival inputs change.

Worked Example: Whole-Life Annuity with a Finite Survival Model

A hypothetical contract pays an annuity-due benefit of $640 at times 0, 1, and 2, provided the recipient is alive. Level premiums are payable at each year-end while the recipient is alive. Assume a 10% annual effective interest rate, ${}_1p_x=0.80$, ${}_2p_x=0.40$, and ${}_3p_x=0$, with no survival beyond time 3. Find the net level annual premium.

Method. Compute the benefit factor at times 0, 1, and 2, then the premium factor at times 1, 2, and 3. The annual discount factor is $v=1/1.10=10/11$. The time-3 premium has zero expected value because ${}_3p_x=0$.

The benefit factor is:

1+v{}_1p_x+v^2{}_2p_x =1+\frac{10}{11}(0.80)+\left(\frac{10}{11}\right)^2(0.40) =1+\frac{8}{11}+\frac{40}{121} =\frac{249}{121}.

The premium factor is:

v{}_1p_x+v^2{}_2p_x+v^3{}_3p_x =\frac{8}{11}+\frac{40}{121}+0 =\frac{128}{121}.

Equivalence gives $P(128/121)=640(249/121)$, so $P=640(249/128)=5(249)=\$1{,}245$. The benefit present value is $640(249/121)=\$159{,}360/121$, and the premium present value is $1{,}245(128/121)=\$159{,}360/121$. The equality checks exactly. The zero survival probability at time 3 removes that premium term; it does not remove the benefit at time 2, when survival probability is 0.40.

Common Timing Trap

A frequent error is to use one annuity factor for both sides of the equivalence equation without checking when payments occur. For the three-year example above, using the due benefit factor $127/49$ for the premiums would incorrectly give $P=900$. That calculation assumes premiums at times 0, 1, and 2. The contract instead collects premiums at times 1, 2, and 3, so the correct factor is $762/343$, and the premium is $1,050. The correction is not an adjustment to mortality; it is the use of the actual premium payment dates.

Before calculating, write down the payment dates for each stream and ask whether survival is required at each date. A payment at time 0 is not discounted and is not reduced for mortality. An end-of-year payment at time $k$ is discounted $k$ years and weighted by ${}_kp_x$. Confusing survival to the beginning of a year with survival to its end shifts the probability index and can change the answer.

Key takeaway: A net level annuity premium is the benefit amount multiplied by its survival-contingent present-value factor, divided by the premium stream’s own survival-contingent factor. Establish payment dates first; then apply discounting and survival at those dates.

Check Your Understanding

For each hypothetical contract, set up the appropriate survival-contingent factors before solving. Amounts are annual, and premiums are level.

A five-year temporary annuity pays $400 at the end of each year if the annuitant is alive. Premiums are due at times 0 through 4 while alive. Assume a 25% annual effective interest rate and annual survival probability 0.80. Find the net premium. (Answer: $256.00)
A two-year temporary annuity pays $300 at times 0 and 1 if the recipient is alive. Premiums are paid at the ends of years 1 and 2 if alive. Assume a 25% annual effective interest rate and annual survival probability 0.75. Find the net premium. (Answer: $500.00)
A whole-life annuity pays $720 annually in advance while the recipient is alive. Level premiums are paid annually in arrears while alive. Assume a 25% annual effective interest rate and constant annual survival probability 0.75. Find the net premium. (Answer: $1,200.00)
A four-payment annuity-due benefit pays $1,000 at times 0, 1, 2, and 3 if alive. Premiums are payable at times 1, 2, 3, and 4 if alive. Assume $i=25\%$ and survival probabilities ${}_1p_x=0.90$, ${}_2p_x=0.60$, ${}_3p_x=0.30$, and ${}_4p_x=0$. Find the net premium to the nearest cent. (Answer: $1,795.17)
A three-year temporary annuity pays $600 at times 0, 1, and 2 if alive. Premiums are payable at times 1, 2, and 3 if alive. Assume $i=25\%$, ${}_1p_x=0.90$, ${}_2p_x=0.72$, and ${}_3p_x=0.36$. Find the net premium to the nearest cent. (Answer: $958.51)

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