Present value of an ordinary annuity versus annuity due

Payment timing changes present value. An annuity that pays at the beginning of each period is worth more today than the same series paid at the end, because each payment is received one period sooner and has less time to be discounted. CFP exam questions often test this distinction through a scenario or calculator mode rather than by naming the formula directly.

Ordinary annuity and annuity due timing

An ordinary annuity makes payments at the end of each period. A standard loan payment due after the borrower has used the funds for the month is a familiar timing pattern. An annuity due makes payments at the beginning of each period. Rent or an insurance premium due at the start of the covered period may have this structure, although the contract dates determine the actual cash-flow timing.

For five annual payments, an ordinary annuity pays at the ends of years one through five. An annuity due pays at time zero and at the beginnings of years two through five. Both have five payments; the due series simply shifts every payment one period earlier.

Present value formulas

Let PMT be the payment per period, i the discount rate per period, and n the number of payments. The present value of an ordinary annuity is PMT × [1 − (1 + i)^−n] ÷ i. The present value of an annuity due is that ordinary-annuity value multiplied by (1 + i). The formulas assume level payments at regular intervals and a consistent rate and period.

The adjustment works because every payment in the due series is one period closer to today. Moving a cash flow one period earlier multiplies its present value by one plus the periodic rate. Apply the periodic rate, not automatically an annual nominal rate, when payments are monthly or quarterly.

Worked example

Consider five annual payments of $1,000 discounted at 6% per year. For an ordinary annuity, calculate $1,000 × [1 − (1.06)^−5] ÷ 0.06, which is about $4,212. For an annuity due, multiply that result by 1.06, giving about $4,465. The due value is higher because every payment is received earlier.

Solve it on a financial calculator

1. Set the payment frequency and number of periods. Five annual payments means five periods; five years of monthly payments means sixty monthly periods.
2. Enter the periodic interest rate. Convert an annual rate to the matching periodic rate when the calculator expects a nominal annual rate and payments are more frequent.
3. Set payment timing to END for an ordinary annuity or BEGIN for an annuity due. Verify the setting before solving.
4. Enter the payment and any other known cash flows with the calculator's required sign convention.
5. Solve for present value and check whether the answer's size makes sense. The annuity due should have the larger present value when the payment and rate assumptions are otherwise identical.

Common setup errors

• Using BEGIN mode when the first payment is actually one period from now overstates present value.
• Counting a time-zero payment as both a payment today and one full period of discounting shifts the timeline incorrectly.
• Using the annual rate with monthly periods understates or overstates value depending on the setup; match rate and period units.
• Multiplying by (1 + annual rate) when the payments are monthly uses the wrong one-period adjustment. Use the periodic rate.
• Changing timing also changes future value. For an otherwise identical series, the annuity due's future value is the ordinary annuity's future value multiplied by (1 + i). Do not use the present-value answer when the prompt asks for future value.

How to map the timeline before calculating

Draw a line with time zero at the left. Place the first payment where the contract says it occurs. Then count the remaining intervals to confirm n. If the first payment is immediate and there are five payments total, there are five cash flows but only four intervals between the first and last payment dates. The standard annuity-due formula handles that timing directly; the timeline prevents accidentally entering six payments or discounting the first payment.

The same timeline method works in retirement, education, insurance, and loan questions. First identify who pays whom, how often, and whether the first cash flow is immediate. Then choose ordinary or due timing and enter the numbers.

Draw the timeline before entering a calculator

For an ordinary annuity, the first payment arrives one full period from today and the last payment arrives at the end of period n. For an annuity due, the first payment arrives today and each later payment is one period earlier than its ordinary-annuity counterpart. Label period 0 and period 1 explicitly; many wrong answers come from choosing the correct formula for the wrong timeline.

The present value of an ordinary annuity is PMT × [1 − (1+r)^−n] ÷ r. The present value of an annuity due is that ordinary-annuity value multiplied by (1+r), because each payment is received one period earlier. The periodic rate and number of periods must match the cash-flow frequency. A monthly payment discounted at an annual rate needs a monthly rate and total monthly periods.

Worked example and calculator settings

Suppose a client receives $1,000 at the end of each year for four years and the annual discount rate is 5%. Ordinary-annuity present value is $1,000 × [1 − (1.05)^−4] ÷ 0.05, about $3,546. A beginning-of-year annuity due is worth about $3,723, the ordinary value multiplied by 1.05. The difference is the interest earned by receiving every payment one period sooner.

On a financial calculator, set payment mode to END for ordinary annuity and BEGIN for annuity due. Clear old values and enter n, i/y, PMT, FV, and PV consistently. Cash paid out and cash received should use opposite signs. If the calculator remains in BGN from a previous question, it can silently produce the wrong result. Confirm the mode on the screen and reset it afterward.

Rate, growth, and irregular timing

If payments grow at a constant rate, the level-payment annuity formula does not apply unchanged. If the discount rate equals the growth rate, the standard growing-annuity formula has a special limit case. Irregular payments should be discounted individually: PV = Σ CFt ÷ (1+r)^t. When payments begin after a deferral period, first value the annuity at the date immediately before its first payment, then discount that amount back to today.

The formula assumes payments occur at exact, regular intervals and the discount rate is appropriate for the risk and timing. Taxes, inflation, default risk, and payment certainty can require different assumptions. In CFP problems, first identify whether payments are beginning or end of period, match the rate to the period, enter the number of payments, and check whether the result should be larger or smaller than the other timing convention.