Present Value of an Annuity-Due by Shifting Payments

Why Payment Timing Changes Present Value

Two contracts can promise the same number and amount of payments but have different present values if their payment dates differ. In an annuity-due, each payment is made at the beginning of its period. The first payment is therefore made at time zero, rather than one period after the valuation date. That earlier timing increases present value.

The method here is to start with an annuity-immediate, whose payments occur at the ends of periods, and shift every payment one period earlier. The shift gives a direct way to derive the present-value factor for an annuity-due. It also makes the timing assumption visible, which is essential when setting up an exam calculation.

Definition: An annuity-due with $n$ equal payments of amount $R$ has payments at times $0,1,\ldots,n-1$. Its present value is measured at time zero, immediately before the payment at time zero. The standard factor is $\ddot{a}_{\overline{n}|}$, so the present value is $R\ddot{a}_{\overline{n}|}$.

Assume a constant effective interest rate $i$ per payment period, with $i>0$, and let $v=1/(1+i)$. There are no fees, payment contingencies, or changes in the payment amount. The rate period must match the interval between payments. For annual payments, for example, $i$ must be an effective annual rate.

Deriving the Present-Value Factor by Shifting

An annuity-immediate with $n$ payments of $R$ has payments at times $1,2,\ldots,n$. Its present value at time zero is $R a_{\overline{n}|}$. Move each of those payments one period earlier: the payment at time 1 moves to time 0, the payment at time 2 moves to time 1, and so on, until the payment at time $n$ moves to time $n-1$. The resulting stream is an annuity-due.

For each payment, moving its date one period earlier reduces its discount exponent by one. Since $v^{k-1}=(1+i)v^k$, its present value is multiplied by $1+i$. This applies to every payment, so it applies to their sum. Equivalently, the due stream’s discount factors are $1,v,\ldots,v^{n-1}$.

\begin{aligned} \ddot{a}_{\overline{n}|} &= \sum_{k=0}^{n-1}v^k\\ &= (1+i)\sum_{k=1}^{n}v^k\\ &= (1+i)a_{\overline{n}|}. \end{aligned}

The finite geometric sum also gives a direct expression. Because $1-v=d$, where $d=i/(1+i)$ is the effective discount rate,

\ddot{a}_{\overline{n}|} =\frac{1-v^n}{1-v} =\frac{1-v^n}{d}.

The sum form is often the safest way to check timing: it displays the time-zero payment as the first term, $1$. The formula using $d$ is a compact equivalent expression. Both require the same model assumptions: level payments at regular intervals, a constant effective rate per interval, and the first payment at time zero.

Mark the payment dates.
For $n$ payments in an annuity-due, write $0,1,\ldots,n-1$. Count the payment at time zero as one of the $n$ payments.

Match the rate period.
Use the effective rate for one payment interval. Then calculate $v=1/(1+i)$.

Value the payments.
Use $R\sum_{k=0}^{n-1}v^k$, or the equivalent factor $R(1-v^n)/d$.

Check the timing.
Compare with the annuity-immediate stream. The due stream is shifted earlier, so its present value must be larger when $i>0$.

Worked Applications

Worked Example: Three Payments Made Immediately and Annually

A hypothetical service contract requires three payments of $1,000 each, with the first payment due immediately and the remaining payments due one and two years later. The effective annual interest rate is 10%. Find the present value at time zero, immediately before the first payment.

Method. The payment dates are $0,1,2$, so this is a three-payment annuity-due. The common mistake to avoid is valuing it as though the first payment were at time 1.

The discount factor is $v=1/1.10=10/11$. Thus the present value is

\begin{aligned} PV &=1000(1+v+v^2)\\ &=1000\left(1+\frac{10}{11}+\frac{100}{121}\right)\\ &=1000\left(\frac{121+110+100}{121}\right)\\ &=1000\left(\frac{331}{121}\right)\\ &=\$2,735.537190\ldots \end{aligned}

Rounded to the nearest cent, the present value is $2,735.54. As a shift check, the corresponding annuity-immediate factor is $a_{\overline{3}|}=(v+v^2+v^3)=2.486852\ldots$. Multiplying by $1.10$ gives $2.735537\ldots$, the same due factor. The result exceeds $2,000 because all three payments are included, with two discounted and one paid at time zero.

Worked Example: Five Annual Payments at a 4% Effective Rate

A hypothetical equipment lease calls for five annual payments of $2,400, the first payable immediately. The effective annual interest rate is 4%. Calculate the present value at the time the first payment is due, immediately before that payment.

Method. Use the due factor with five payments. A common mistake is to use $a_{\overline{5}|}$, which values payments at times 1 through 5 rather than 0 through 4.

Here $v=1/1.04=25/26$, and $d=0.04/1.04=1/26$. The five-payment due factor is

\begin{aligned} \ddot{a}_{\overline{5}|} &=\frac{1-v^5}{d}\\ &=26\left(1-\left(\frac{25}{26}\right)^5\right)\\ &=4.629895224\ldots \end{aligned}

Therefore,

PV=2400(4.629895224\ldots) =\$11,111.7485\ldots \approx \$11,111.75.

For comparison, if the first payment were one year later, the present value would be $PV/1.04=\$10,684.37$. The difference is approximately $427.37, which is exactly the time-zero payment less the present value of the payment at time 5: $2400(1-v^5)=\$427.37$ to the nearest cent. This comparison checks that the extra value comes from the change in timing, not from changing the number or amount of payments.

Worked Example: Discounting a Deferred Annuity-Due

A hypothetical grant will provide four annual payments of $700, beginning two years from today. Thus, the payments occur at times 2, 3, 4, and 5. The effective annual interest rate is 6%. Find the present value today.

Method. First value the four-payment annuity-due at time 2, just before its first payment. Then discount that value two years to time zero. The trap is to treat the first payment at time 2 as if it were at time 0 without discounting the whole stream.

At time 2, the annuity-due factor is

\begin{aligned} \ddot{a}_{\overline{4}|} &=1+\frac{1}{1.06}+\frac{1}{1.06^2}+\frac{1}{1.06^3}\\ &=1+0.9433962264+0.8899964400+0.8396192830\\ &=3.6730119494. \end{aligned}

The value at time 2 is $700(3.6730119494)=\$2,571.1083646$. Since time 2 is two years after today, discount by $v^2=1/1.06^2=1/1.1236$:

\begin{aligned} PV_0 &=\frac{2571.1083646}{1.1236}\\ &=700\left(\frac{1}{1.06^2}+\frac{1}{1.06^3} +\frac{1}{1.06^4}+\frac{1}{1.06^5}\right)\\ &=\$2,288.2773\ldots\\ &\approx \$2,288.28. \end{aligned}

The direct sum confirms the dates: the first discount exponent is 2, not 0. The present value today is therefore less than the annuity-due value at time 2.

Common Timing Errors

The most consequential error is using the right number of payments with the wrong first payment date. For a due annuity, the present-value sum starts with $R$, because the time-zero payment is not discounted. For an immediate annuity, the sum starts with $Rv$, because its first payment is one period away.

For example, suppose there are four payments of $500 at 5% effective annual interest, with the first payment made immediately. The correct setup is

$$ 500(1+v+v^2+v^3). $$

The incorrect immediate-annuity setup is $500(v+v^2+v^3+v^4)$. It shifts every payment one year later and understates the value. The difference between the correct and incorrect sums is $500(1-v^4)$, the time-zero payment less the present value of the payment shifted to time 4.

Do not count periods instead of payments. Payments at times 0 through 3 are four payments over three years between the first and last payment dates.
Do not omit the time-zero payment. It is part of the annuity and has present value equal to its full amount.
Do not use an annual rate with nonannual periods without conversion. The $i$ in the factor must apply to one payment interval.
Do not discount a deferred stream from the wrong date. First identify the date at which its annuity-due value is measured, then discount that value to the requested focal date.

Key takeaway: An $n$-payment annuity-due has payment dates $0,1,\ldots,n-1$ and present value $R\ddot{a}_{\overline{n}|}=R\sum_{k=0}^{n-1}v^k$. Shifting an annuity-immediate one period earlier multiplies its present value by $1+i$.

The ability to read payment dates correctly is central to valuing contracts, deposits, and withdrawals. The same amount and payment count do not determine value by themselves: the first payment date matters. This timing method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payment dates before calculating. Round monetary answers to the nearest cent.

A hypothetical membership plan requires two annual payments of $900, first payable immediately. The effective annual interest rate is 5%. Find the present value just before the first payment. (Answer: $1,757.14)
A hypothetical fund makes four annual payments of $600 at times 0, 1, 2, and 3. The effective annual interest rate is 8%. Find the present value at time zero. (Answer: $2,146.26)
A hypothetical service agreement requires four annual payments of $1,000, beginning immediately. The effective annual interest rate is 10%. Calculate its present value and state the payment dates. (Answer: $3,486.85; dates 0, 1, 2, and 3)
A hypothetical scholarship provides three annual payments of $800 at times 2, 3, and 4. The effective annual interest rate is 5%. Find the present value today by first valuing the annuity-due at time 2. (Answer: $2,074.86)
A hypothetical four-payment contract requires annual payments of $1,100 at times 0, 1, 2, and 3. At an effective annual rate of 6%, calculate its present value. What timing error would result from using an annuity-immediate factor instead? (Answer: Due-annuity present value $4,040.31; using an immediate factor gives $3,811.62 and understates the value by $228.70)

What You'll Learn