The Annuity-Due and Annuity-Immediate Factor Identity

Why the Factor Identity Matters

An annuity-due and an annuity-immediate can have the same number of equal payments and the same payment spacing, yet different values. The distinction is timing: an annuity-due pays at the beginning of each period, while an annuity-immediate pays at the end. Moving each payment one period earlier changes its value by the accumulation factor for one period, $1+i$.

This relationship gives a useful way to move between matching present-value factors or matching accumulated-value factors. It also provides a check on calculations: if the rate per payment period and the payment count are unchanged, the due factor must be exactly $1+i$ times the immediate factor.

Definition: Let $i$ be the effective interest rate per payment period, let $n$ be a positive integer, and let $v=1/(1+i)$. A level annuity-immediate has payments at times $1,2,\ldots,n$; a level annuity-due has payments at times $0,1,\ldots,n-1$. The present-value factors are $a_{\overline{n}|}$ and $\ddot{a}_{\overline{n}|}$, respectively. When each payment is accumulated to time $n$, the corresponding accumulated-value factors are $s_{\overline{n}|}$ and $\ddot{s}_{\overline{n}|}$.

The factors describe the value of a unit payment at each specified time. For a level payment of $R$, the present value or accumulated value is the corresponding factor multiplied by $R$. The rate $i$ must match the interval between payments. For example, quarterly payments require the effective quarterly rate, not automatically the effective annual rate.

The Present-Value Identity

For an annuity-immediate, the payment at time $k$ is discounted $k$ periods to time 0. For an annuity-due, the payment at time $k-1$ is discounted $k-1$ periods to time 0. With the same $n$ and $i$, the due payment is one period earlier in every matching pair.

a_{\overline{n}|}=\sum_{k=1}^{n}v^k, \qquad \ddot{a}_{\overline{n}|}=\sum_{k=1}^{n}v^{k-1}.

Since $v=1/(1+i)$, multiplying each immediate present-value term by $1+i$ reduces its discount exponent by one:

(1+i)a_{\overline{n}|} =(1+i)\sum_{k=1}^{n}v^k =\sum_{k=1}^{n}v^{k-1} =\ddot{a}_{\overline{n}|}.

Theorem: For level payments at equal intervals, with the same positive integer $n$ and effective rate $i$ per payment period, $$ \ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}. $$

This derivation depends on matching the number of payments and the payment interval. It does not say that any two annuities with different payment patterns or different rates have values in this ratio. At $i=0$, the identity still holds: both factors equal $n$, and $1+i=1$.

The Accumulated-Value Identity

Now accumulate each payment to time $n$. The annuity-immediate payments occur at times $1,\ldots,n$, so their accumulation periods are $n-1,\ldots,0$. The annuity-due payments occur at times $0,\ldots,n-1$, so their accumulation periods are $n,\ldots,1$. Each due payment therefore accumulates for one additional period compared with its matching immediate payment.

s_{\overline{n}|}=\sum_{k=0}^{n-1}(1+i)^k, \qquad \ddot{s}_{\overline{n}|}=\sum_{k=1}^{n}(1+i)^k.

Factoring $1+i$ from every term in the due sum leaves exactly the terms in the immediate sum:

\ddot{s}_{\overline{n}|} =\sum_{k=1}^{n}(1+i)^k =(1+i)\sum_{k=0}^{n-1}(1+i)^k =(1+i)s_{\overline{n}|}.

Theorem: For the same $n$, payment interval, and effective rate $i$, $$ \ddot{s}_{\overline{n}|}=(1+i)s_{\overline{n}|}. $$

The multiplier has the same timing interpretation in both identities: each due payment is one period earlier. For present value, that means one fewer period of discounting; for accumulated value, it means one more period of interest earned. In either case, the corresponding value is multiplied by $1+i$.

Using the Identities in Calculations

A reliable method is to identify the value being calculated, confirm that the rate is per payment period, and then use the matching immediate factor. Apply $1+i$ once to the factor, not once for every payment. The payment amount, if any, is multiplied by the resulting due factor afterward.

Match the conditions.
Check that the due and immediate annuities have the same $n$, level payments, payment interval, and effective rate per interval.

Select the value type.
Use the present-value pair $a_{\overline{n}|}$, $\ddot{a}_{\overline{n}|}$, or the accumulated-value pair $s_{\overline{n}|}$, $\ddot{s}_{\overline{n}|}$.

Shift the factor.
Multiply the immediate factor by $1+i$ to obtain its due counterpart.

Apply the payment and check the timing.
Multiply by the level payment, then verify with the due payment dates or the relevant sum of discount or accumulation terms.

Worked Example: Present Value of Three Payments at the Beginning of Each Year

A hypothetical contract pays \$1,000 at the beginning of each year for three years. The effective annual interest rate is 10%. Find the present value. Use the corresponding annuity-immediate factor, and check the result from the due payment dates.

Method. The payments are annual, so the effective rate per payment period is $i=0.10$. There are $n=3$ payments. First calculate the matching immediate factor:

a_{\overline{3}|} =v+v^2+v^3 =\frac{1}{1.10}+\frac{1}{1.10^2}+\frac{1}{1.10^3} =0.9090909091+0.8264462810+0.7513148009 =2.4868519910.

Multiply by $1+i=1.10$ to get the due factor:

\ddot{a}_{\overline{3}|} =1.10(2.4868519910) =2.7355371901.

The present value is therefore $1{,}000(2.7355371901)=\$2{,}735.5371901$, which rounds to \$2,735.54 to the nearest cent.

Check. The due payments occur at times 0, 1, and 2. Their present value is $1{,}000(1+v+v^2)=1{,}000(1+0.9090909091+0.8264462810)=\$2{,}735.5371901$, giving the same rounded value. The method avoids the mistake of discounting the first due payment by one year.

Worked Example: Accumulated Value of Quarterly Deposits

A hypothetical savings plan makes three deposits of \$600 at the beginning of each quarter. The effective interest rate is 5% per quarter. Find the accumulated value at the end of the third quarter using the corresponding annuity-immediate factor, then check it by accumulating the deposits directly.

Method. Here $n=3$ and $i=0.05$ per quarter. The matching immediate accumulated-value factor is:

s_{\overline{3}|} =1+1.05+1.05^2 =1+1.05+1.1025 =3.1525.

The due accumulated-value factor is:

\ddot{s}_{\overline{3}|} =1.05(3.1525) =3.310125.

Thus the accumulated value is $600(3.310125)=\$1{,}986.075$, which rounds to \$1,986.08 to the nearest cent.

Check. The deposits at times 0, 1, and 2 accumulate to time 3 for three, two, and one quarters, respectively: $600(1.05^3+1.05^2+1.05)=600(1.157625+1.1025+1.05)=\$1{,}986.075$. As an optional rate-unit check, a 5% effective quarterly rate corresponds to an effective annual rate of $1.05^4-1=0.21550625$, or 21.550625%. The calculation uses 5% per quarter because the deposits are quarterly.

Worked Example: Recovering the Rate and Finding a Due Present Value

For a two-payment level annuity, the present-value factor for payments at times 0 and 1 is 1.15 times the factor for payments at times 1 and 2. The annuity-due pays \$640 each time. Find the effective rate per payment period and the due present value.

Method. The matching factor identity is $\ddot{a}_{\overline{2}|}=(1+i)a_{\overline{2}|}$. The stated ratio gives $1+i=1.15$, so $i=0.15$, or 15% per payment period.

The immediate factor for two payments at this rate is:

a_{\overline{2}|} =\frac{1}{1.15}+\frac{1}{1.15^2} =0.8695652174+0.7561436673 =1.6257088847.

The matching due factor is $1.15(1.6257088847)=1.8695652174$. The due present value is $640(1.8695652174)=\$1{,}196.521739$, which rounds to \$1,196.52 to the nearest cent.

Check. The due payments occur at times 0 and 1, so their present value is $640+640/1.15=\$1{,}196.521739$. This agrees. The common error of multiplying the payment by the rate itself, 0.15, is avoided: the timing multiplier is $1+i=1.15$, not $i$.

Common Mistakes and Checks

The identity is simple, but it is easy to use the wrong rate, factor, or payment timing. In particular, “one extra period” refers to the shift between matching payments, not to adding a payment or changing $n$.

Using an annual rate for nonannual payments. Use the effective rate for one payment interval. If payments are quarterly, apply the quarterly effective rate in the identity.
Changing the payment count. The due and immediate factors in an identity must have the same $n$. A timing shift does not create an additional payment.
Multiplying by $i$ instead of $1+i$. A one-period shift changes value by the full accumulation factor $1+i$, which includes return of principal as well as interest.
Applying the multiplier repeatedly. Each corresponding payment is shifted one period, so the whole matching factor is multiplied by $1+i$ once. Repeated multiplication would represent a different timing change.
Using an accumulated-value factor for present value, or vice versa. Choose the factor pair that matches the requested focal date and direction of valuation.

Key takeaway: With equal level payments, the same positive integer $n$, and the same effective rate $i$ per payment period, $\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}$ and $\ddot{s}_{\overline{n}|}=(1+i)s_{\overline{n}|}$. In both cases, $1+i$ represents the one-period shift of every due payment earlier than its matching immediate payment.

Check Your Understanding

For each problem, identify the matching factor and keep the rate on the same time scale as the payment interval.

A plan pays \$300 at the beginning of each year for four years. The effective annual rate is 6%, and the matching annuity-immediate present-value factor is 3.465106. Find the due present value, rounding to the nearest cent. (Answer: due factor $3.672012$; present value \$1,101.60.)
A fund receives three \$450 deposits at the beginning of each quarter. The effective quarterly rate is 4%. Find the accumulated value at the end of the third quarter, rounding to the nearest cent. (Answer: $s_{\overline{3}|}=3.1216$, $\ddot{s}_{\overline{3}|}=3.246464$; accumulated value \$1,460.91.)
For the same payment count and interval, a due present-value factor is 1.15 times its matching immediate factor. What is the effective rate per payment period? (Answer: $i=15\%$.)
A two-payment annuity-immediate has an accumulated-value factor $s_{\overline{2}|}=2.04$ at an effective rate of 4% per period. Find the matching due accumulated-value factor. (Answer: $\ddot{s}_{\overline{2}|}=2.1216$.)
A hypothetical plan pays \$800 at the beginning of each year for two years. The effective annual rate is 8%. Find the due present value using the immediate factor, then verify by valuing the payment dates directly. Round to the nearest cent. (Answer: $a_{\overline{2}|}=1.783264746$, $\ddot{a}_{\overline{2}|}=1.925925926$; present value \$1,540.74.)

What You'll Learn