Align Annuity-Due and Annuity-Immediate Premium Cash Flows

Why Payment Timing Must Be Aligned

A level premium payable at the beginning of each year does not have the same present value as one payable at the end of each year. For a life-contingent contract, the difference is not only a matter of discounting: a payment is made only if the insured is alive at its payment date. Changing the date therefore changes both the discount factor and, for later payments, the survival probability attached to the payment.

The key is to write down the actual payment dates before choosing an annuity factor. In particular, a finite series of $n$ premiums paid in advance has payments at times $0,1,\ldots,n-1$, while a finite series of $n$ premiums paid in arrears has payments at times $1,2,\ldots,n$. The two schedules have the same number of payments, but they do not have the same first or last payment date.

Definition: Under annual effective interest rate $i$, let $v=1/(1+i)$. For an individual aged $x$ at issue, an $n$-payment life annuity-due pays at times $0,1,\ldots,n-1$, provided the individual is alive at each payment time. Its unit present value is $\ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k\,{}_kp_x$. An $n$-payment life annuity-immediate pays at times $1,2,\ldots,n$, subject to the same alive-at-payment condition. Its unit present value is $a_{x:\overline{n}|}=\sum_{k=1}^{n}v^k\,{}_kp_x$.

Here ${}_0p_x=1$, so the first due payment at time 0 is certain for a person alive at issue. A payment at time $k$, for $k\geq 1$, is conditional on survival to time $k$, represented by ${}_kp_x$. These formulas assume annual payments, payments made only while the insured is alive, and a constant annual effective interest rate. Different payment rules, such as premiums continuing after death or a payment triggered by death, require a different cash-flow model.

The Finite-Term Conversion

A useful conversion can be derived directly from the payment dates. The due stream includes a certain payment at time 0 and payments at times 1 through $n-1$. The immediate stream includes payments at times 1 through $n$. The payments at times 1 through $n-1$ are identical in both streams: they have the same amounts, discount factors, and survival probabilities. What differs is the first payment of the due stream and the last payment of the immediate stream.

\ddot{a}_{x:\overline{n}|}-a_{x:\overline{n}|} =1-v^n\,{}_np_x

The adjustment is therefore the time-0 payment less the discounted expected payment at time $n$. It is generally not correct to multiply one finite-term factor by $1+i$ to obtain the other. That deterministic-annuity conversion would shift payment dates without accounting for the probability of survival or the lost/added boundary payment.

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

For a whole-life annuity, the same comparison gives $\ddot{a}_x=1+a_x$, provided the discounted survival term tends to zero as the payment time tends to infinity. For a finite term, the last immediate payment remains and must be included in the boundary adjustment.

Key result: To value a finite premium schedule, count the payments and label their dates. For $n$ level payments of amount $P$, the present values are $P\ddot{a}_{x:\overline{n}|}$ when paid at times $0,\ldots,n-1$, and $Pa_{x:\overline{n}|}$ when paid at times $1,\ldots,n$. The factor must match both the payment dates and the survival condition.

Worked Applications

Worked Example: Converting a Three-Payment Premium Schedule

A hypothetical policyholder is alive at issue. The annual probability of surviving each policy year is $p=0.8$, independently of duration, and the effective annual interest rate is $i=25\%$. A premium of $500 is payable at the beginning of each of the first three policy years if the policyholder is alive. Calculate its expected present value. Then calculate the present value if the same three premiums are instead payable at each year-end, conditional on survival to that date.

Method. Use the due factor for payments at times 0, 1, and 2, and the immediate factor for payments at times 1, 2, and 3. This avoids treating a timing change as a change in the number of premiums.

The discount factor is $v=1/(1.25)=0.8$. Each year’s discounted survival factor is $vp=(0.8)(0.8)=0.64$. Thus the due payments have present-value weights $1$, $0.64$, and $0.64^2=0.4096$. The due factor and premium present value are

\ddot{a}_{x:\overline{3}|}=1+0.64+0.4096=2.0496, \qquad 500(2.0496)=\$1{,}024.80.

For the immediate schedule, the weights are $0.64$, $0.4096$, and $0.64^3=0.262144$. Therefore

a_{x:\overline{3}|}=0.64+0.4096+0.262144=1.311744, \qquad 500(1.311744)=\$655.872.

Rounded to the nearest cent, the immediate-premium present value is $655.87. As a check, the due-minus-immediate difference is $500[1-0.262144]=500(0.737856)=\$368.928$, or $368.93 to the nearest cent. This agrees with the difference between the two unrounded present values. The common mistake is to shift the three due payments one year later but still stop at time 2; that would omit the third immediate payment at time 3.

Worked Example: Using the Boundary Adjustment for a Finite Term

A hypothetical insured has a 10% probability of dying in each policy year, independent of duration. The effective annual interest rate is 5%. A level premium of $800 is payable for three years, either at the beginning of each year while alive or at the end of each year if alive. Find both present values and verify their difference using the finite-term conversion.

Method. First form the one-year discounted survival factor. Then use the appropriate three payment dates for each schedule. The mistake to avoid is converting between the factors using interest alone; the last immediate payment also depends on survival to year 3.

The annual survival probability is $p=0.9$, and $v=1/1.05=20/21$. Hence $vp=(20/21)(0.9)=6/7$. The due factor is

\ddot{a}_{x:\overline{3}|} =1+\frac{6}{7}+\left(\frac{6}{7}\right)^2 =1+\frac{42}{49}+\frac{36}{49} =\frac{127}{49}.

The due premium present value is $\$800(127/49)=\$101{,}600/49=\$2{,}073.469\ldots$, or $2,073.47 to the nearest cent.

The immediate factor has payments at times 1, 2, and 3:

a_{x:\overline{3}|} =\frac{6}{7}+\frac{36}{49}+\frac{216}{343} =\frac{294+252+216}{343} =\frac{762}{343}.

The immediate premium present value is $\$800(762/343)=\$609{,}600/343=\$1{,}777.259\ldots$, or $1,777.26 to the nearest cent.

For the check, the boundary adjustment is

800\left(1-v^3\,{}_3p_x\right) =800\left[1-\left(\frac{6}{7}\right)^3\right] =800\left(\frac{127}{343}\right) =\$296.209\ldots.

The difference between the unrounded present values is also $\$2{,}073.469\ldots-\$1{,}777.259\ldots=\$296.209\ldots$, giving $296.21 to the nearest cent. This confirms both calculations.

Worked Example: Keeping Survival Conditions Attached to Payment Dates

A hypothetical two-year contract has annual survival probabilities of 0.9 for year 1 and 0.8 for year 2, conditional on being alive at the start of each year. The effective annual interest rate is 5%. Compare the present values of two $600 premium schedules: one payable at times 0 and 1 while alive, and one payable at times 1 and 2 while alive. Separately, value a $4,000 benefit payable at time 2 only if the insured survives both years.

Method. Calculate survival to each payment date before applying the discount factor. Keep the benefit’s survival condition separate from the premium timing. A common error is to attach the two-year survival probability to a time-1 payment.

Here $v=20/21$, ${}_1p_x=0.9$, and ${}_2p_x=(0.9)(0.8)=0.72$. At time 1, the discounted survival weight is $v{}_1p_x=(20/21)(0.9)=6/7$. At time 2, it is $v^2{}_2p_x=(400/441)(0.72)=288/441=32/49$.

The due premiums are paid at time 0 and at time 1. Their expected present value is

600\left(1+\frac{6}{7}\right) =600\left(\frac{13}{7}\right) =\$1{,}114.2857\ldots =\boxed{\$1{,}114.29}.

The immediate premiums are paid at time 1 and time 2. Their expected present value is

600\left(\frac{6}{7}+\frac{32}{49}\right) =600\left(\frac{74}{49}\right) =\$906.1224\ldots =\boxed{\$906.12}.

The benefit is payable only at time 2 if the insured survives both years, so its present value is

4{,}000v^2{}_2p_x =4{,}000\left(\frac{32}{49}\right) =\$2{,}612.2448\ldots =\boxed{\$2{,}612.24}.

The benefit value is the same whichever premium schedule is chosen: changing premium timing does not change the benefit’s payment date or survival condition. This is a check on the cash-flow separation, not a comparison of contract premiums with benefits.

A Reliable Alignment Procedure

When a problem describes premiums in words, translate the wording into a timeline before calculating. “At the beginning of each year” ordinarily means a due payment at times $0,1,\ldots$; “at the end of each year” means an immediate payment at times $1,2,\ldots$. For life-contingent payments, attach survival to the date of each payment, not to the start or end of the premium-paying period by habit.

List the payment dates.
Write each date as a time from issue. Confirm the number of payments and the first and last payment dates.

Attach the survival condition.
A payment at time $k$ made only while alive has probability weight ${}_kp_x$; at time 0 this weight is 1.

Discount each payment.
Multiply its amount by $v^k$ and by the applicable survival probability, then sum the expected present values.

Check the boundary payments.
For equal finite payment counts, verify that a due-versus-immediate difference is the time-0 payment less the discounted, survival-weighted payment at time $n$.

This alignment is useful whenever premium timing is specified separately from benefit timing. It prevents a one-year shift from changing the wrong quantity—for example, discounting a payment to a new date but leaving its survival probability unchanged. It also makes later equivalence calculations more reliable: the premium present value must reflect the actual contractual dates before it is compared with a benefit value.

Exam trap: “Three premiums payable annually” does not by itself identify an annuity factor. Determine whether the dates are $0,1,2$ or $1,2,3$, and whether each payment requires survival to its date. The finite-term factors differ at both ends of the schedule, not just by an interest conversion.

Check Your Understanding

For each problem, identify the payment dates and use the stated survival and interest assumptions.

A two-payment due premium stream pays $200 at times 0 and 1 while the insured is alive. Annual survival probability is 0.75 and the effective annual interest rate is $1/3$. Find its expected present value. (Answer: $312.50)
With the same assumptions and premium amount as question 1, premiums are instead payable at times 1 and 2 if the insured is alive on each payment date. Find the expected present value. (Answer: $175.78)
Annual survival probability is 0.9 and the effective annual interest rate is 5%. Find the present value of three $1,000 premiums payable at times 0, 1, and 2 while alive. (Answer: $2,591.84)
Survival probabilities for years 1 and 2 are 0.8 and 0.9, respectively, and the effective annual interest rate is 10%. Find the present value of two $500 premiums payable at times 1 and 2, conditional on survival to each payment date. (Answer: $661.16)
Annual survival probability is 0.9 and the effective annual interest rate is 5%. For three $1,000 payments, find the due-stream present value minus the immediate-stream present value. (Answer: $370.26)

What You'll Learn