Deriving the Annuity-Due Adjustment

Why the Annuity-Due Adjustment Works

The distinction between an ordinary annuity and an annuity due is a distinction in payment timing. An ordinary annuity makes each payment at the end of its period. An annuity due makes each payment at the beginning of its period. With the same number and amount of payments, every annuity-due payment occurs exactly one period earlier than its ordinary-annuity counterpart.

That one-period shift explains a useful relationship: the present value or future value of an annuity due is the corresponding ordinary-annuity value multiplied by $1+i$. Here, $i$ is the effective interest rate for one payment period. If payments are annual, it is an effective annual rate. For monthly payments, use the effective rate per month, not automatically the effective annual rate.

Definition: An ordinary annuity has $n$ equal payments at times $1,2,\ldots,n$. An annuity due has $n$ equal payments at times $0,1,\ldots,n-1$. The amount of each payment is $R$, and $i$ is the effective rate per payment period, with $i>-1$. The payment count $n$ is a positive integer.

The payment dates must be clear before using an annuity formula. For the present-value comparison, value both streams at time 0. For the future-value comparison, accumulate both streams to time $n$. These common valuation dates make it possible to compare the streams payment by payment.

Deriving the Present-Value Adjustment

Use $v=1/(1+i)$, the discount factor for one payment period. The ordinary annuity has payments at times $1$ through $n$, so its present value is $R(v+v^2+\cdots+v^n)$. The annuity due has payments at times $0$ through $n-1$, so its present value is $R(1+v+\cdots+v^{n-1})$. Each due payment is one period earlier, and therefore has one fewer discount period.

Write both streams at time 0.
The ordinary-annuity value is $R(v+v^2+\cdots+v^n)$. The annuity-due value is $R(1+v+\cdots+v^{n-1})$.

Compare the terms.
Multiplying the ordinary-annuity expression by $1+i$ gives $R(1+v+\cdots+v^{n-1})$, since $(1+i)v^k=v^{k-1}$.

Apply the adjustment.
The expression in Step 2 is exactly the annuity-due present value. Thus $PV_{\text{due}}=(1+i)PV_{\text{ordinary}}$.

The familiar ordinary-annuity formula $a_{\overline{n}|}=(1-v^n)/i$ applies when $i\ne 0$. Therefore the annuity-due present value can be written as follows. The sum form also covers $i=0$.

\begin{aligned} PV_{\text{ordinary}} &= R a_{\overline{n}|},\\ PV_{\text{due}} &= R\ddot{a}_{\overline{n}|} =(1+i)R a_{\overline{n}|}. \end{aligned}

Worked Example: Present Value of an Annuity Due

A hypothetical service contract requires four annual payments of $1,200, with the first payment due immediately. The effective annual interest rate is 8%. Find the present value at time 0. Assume the rate remains 8% for all four annual periods and payments are made as scheduled.

Method. Treat the four payments as an annuity due. First calculate the corresponding ordinary-annuity present value, then multiply by $1+i=1.08$. This uses the same four payments and changes only their timing.

At 8%, $v=1/1.08=0.925925926$. The ordinary-annuity factor is $a_{\overline{4}|}=v+v^2+v^3+v^4$. The terms are $0.925925926$, $0.857338820$, $0.793832241$, and $0.735029853$, giving a factor of $3.312126840$.

The ordinary-annuity present value is $1{,}200(3.312126840)=$3{,}974.552208$. Applying the due adjustment gives $1.08($3{,}974.552208)=$4{,}292.516385$, or $4,292.52 to the nearest cent.

Check. Discount the due payments directly: $1{,}200(1+v+v^2+v^3)$. The factor is $1+0.925925926+0.857338820+0.793832241=3.577096987$, giving $1{,}200(3.577096987)=$4{,}292.516384$, which agrees to the nearest cent. The small difference in unrounded values comes from displayed intermediate precision.

Common mistake avoided. Adding the immediate payment of $1,200 to the ordinary-annuity value would give $5,174.55, which is too large. Shifting the stream earlier does not simply add a new payment: the final ordinary-annuity payment at time 4 is no longer part of the due stream. The adjustment accounts for both changes.

Deriving the Future-Value Adjustment

For future value, compare both streams at time $n$. The ordinary annuity’s payment at time $k$ earns interest for $n-k$ periods. Its future value at time $n$ is $R(1+i)^{n-1}+\cdots+R(1+i)+R$. Each payment in the annuity due occurs one period earlier than its ordinary counterpart, so each earns interest for one additional period at the same valuation date.

Consequently, every accumulated payment in the due stream is $1+i$ times its counterpart in the ordinary stream. Adding those corresponding amounts preserves the same factor for the whole stream. The ordinary-annuity future-value factor is $s_{\overline{n}|}$, so the annuity-due future value is $(1+i)R s_{\overline{n}|}$.

\begin{aligned} FV_{\text{ordinary at }n} &= R s_{\overline{n}|},\\ FV_{\text{due at }n} &= R\ddot{s}_{\overline{n}|} =(1+i)R s_{\overline{n}|}. \end{aligned}

The notation sheet uses $s_{\overline{n}|}$ for the ordinary-annuity accumulated value. The factor $\ddot{s}_{\overline{n}|}$ here denotes the corresponding annuity-due accumulated-value factor. For $i\ne0$, $s_{\overline{n}|}=((1+i)^n-1)/i$. The sum of accumulated payments gives the result directly when $i=0$.

Worked Example: Future Value at the End of the Payment Schedule

A hypothetical fund receives three beginning-of-year contributions of $2,000. The effective annual interest rate is 5%, and each contribution earns interest at that rate. Find the accumulated value at the end of year 3. Assume the contributions are made at times 0, 1, and 2, with no additional contribution at time 3.

Method. Compare with an ordinary annuity accumulated to time 3. The ordinary payments would occur at times 1, 2, and 3; each actual due payment earns one extra year of interest compared with its ordinary counterpart.

The ordinary-annuity factor is $s_{\overline{3}|}=((1.05)^3-1)/0.05=(1.157625-1)/0.05=3.1525$. Thus the ordinary-annuity future value would be $2{,}000(3.1525)=$6{,}305.00$. Apply the annuity-due adjustment: $1.05($6{,}305.00)=$6{,}620.25$.

Check. Accumulate the actual contributions to time 3: $2{,}000(1.05)^3=$2{,}315.25$, $2{,}000(1.05)^2=$2{,}205.00$, and $2{,}000(1.05)=$2{,}100.00$. Their sum is $6,620.25. The result matches. The common mistake avoided here is valuing the due stream at time 3 as if its payments occurred at times 1, 2, and 3, thereby missing one year of interest on each actual contribution.

Using the Adjustment to Find a Payment

The relationship is useful in reverse as well. If the present value of an annuity due is known, divide by $1+i$ to obtain the present value of the corresponding ordinary annuity. Then use the ordinary-annuity factor to solve for $R$. Keep the payment count fixed: the adjustment changes timing, not the number of payments.

Worked Example: Solving for a Due Payment

A hypothetical four-payment obligation is payable at the beginning of each year. Its present value is $7,446.50 at time 0, using an effective annual interest rate of 5%. Find the equal annual payment. Assume the rate applies to each annual period and the given present value is rounded to the nearest cent.

Method. The due present-value factor is $(1.05)a_{\overline{4}|}$. Compute the ordinary factor first: $v=1/1.05=0.952380952$, $v^2=0.907029478$, $v^3=0.863837598$, and $v^4=0.822702475$. Thus $a_{\overline{4}|}=v+v^2+v^3+v^4=3.545950503$.

The due factor is $1.05(3.545950503)=3.723248028$. Divide the given present value by this factor: $R=$7{,}446.50/3.723248028=$2{,}000.001\ldots$. To the nearest cent, the equal annual payment is $2,000.00.

Check. A $2,000 payment has a due present value of $2{,}000(3.723248028)=$7{,}446.496056$, which rounds to the stated $7,446.50. The small difference before rounding is consistent with the present value being given to cents. The common mistake avoided is dividing by the ordinary factor alone, which would treat the first payment as due in one year instead of immediately.

Conditions, Rate Periods, and Common Traps

The adjustment is a timing identity for level payments separated by equal periods. It is not a substitute for identifying the payment dates or the rate that applies over one such period. If the rate is stated for a different period than the payment interval, first express it as an effective rate per payment period. Applying an annual rate directly to monthly payments, for example, generally gives the wrong adjustment.

The relation also does not require a positive rate. It remains valid for $i>-1$, including zero and negative rates. At $i=0$, $1+i=1$, so ordinary and due streams with the same number and amount of payments have the same present value and the same accumulated value at their respective common dates. For a negative rate, $1+i<1$; the formula still holds, but the due value will be lower than the corresponding ordinary value at that common date.

For unequal payments, the same one-period timing principle applies to each payment, but a single level-payment annuity factor may not represent the stream. Likewise, an irregular payment schedule or a rate that changes by period requires valuation of the actual cash flows under the applicable rates. Do not use a constant-rate annuity formula unless its assumptions fit the problem.

Key takeaway: With the same number and amount of level payments, shifting every payment one period earlier multiplies the corresponding ordinary-annuity value by $1+i$, provided $i$ is the effective rate for one payment period and both values use a consistent valuation date.

Check Your Understanding

For each problem, identify the payment timing and valuation date before calculating. Use the stated effective rate per payment period.

A hypothetical two-payment annuity due pays $500 at times 0 and 1. The effective annual rate is 10%. Find its present value at time 0, using the ordinary-annuity adjustment. (Answer: $954.55)
A hypothetical fund receives four beginning-of-year deposits of $300 at times 0, 1, 2, and 3. The effective annual rate is 4%. Find the accumulated value at time 4. (Answer: $1,324.90)
A two-payment annuity due pays $1,250 at times 0 and 1. The effective annual rate is 8%. Find its present value at time 0 by first finding the ordinary-annuity value and then applying the adjustment. (Answer: $2,407.41)
A hypothetical two-payment annuity due has a future value of $4,305 at time 2. The effective annual rate is 5%. Find the equal payment. (Answer: $2,000.00)
A hypothetical five-payment stream pays $700 at the beginning of each annual period, starting at time 0. The effective annual rate is 0%. Find its present value at time 0 and compare it with the present value of five $700 ordinary-annuity payments. (Answer: Both present values are $3,500.00; the adjustment factor is 1.)

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