Present Value of an Annuity: Discount Each Payment

Define the payment stream before discounting

Suppose you expect three payments of 5,000, arriving exactly one, two and three years from today. You want their value today under an illustrative 6% annual discount rate. The amounts are equal and the intervals are regular, so this is an ordinary annuity. Present value moves each payment backward to the same valuation date. It is different from adding the nominal payments, which total 15,000 but occur at different times. The discount rate is a supplied assumption for this example, not a universal rate for all contracts or investments.

Discount the three payments separately

The first payment has present value 5,000/1.06, approximately 4,716.981132. The second has present value 5,000/1.06², approximately 4,449.982200. The third has present value 5,000/1.06³, approximately 4,198.096415. Adding the unrounded figures gives approximately 13,365.059747, or 13,365.06. This term-by-term method shows why the later payment contributes less to today's value under a positive discount rate. It also provides an independent arithmetic check before using a compact annuity factor or a spreadsheet formula.

Use the local tool one payment at a time

The site's Present Value Calculator accepts one future value, one annual rate and one number of years. It does not accept a repeating payment schedule. Enter future value 5000, rate 6 and years 1, then repeat for years 2 and 3. The displayed values are 4,716.98, 4,449.98 and 4,198.10; their rounded sum is 13,365.06. Do not enter the 15,000 total at year 3, because that would incorrectly assume all three payments arrive at the end of the schedule rather than on their individual dates.

Check the ordinary-annuity factor

For equal payment P, periodic rate i and payment count n, present value is P×[1−(1+i)^(-n)]/i when i is nonzero. Here the factor is approximately 2.673011949, and multiplying by 5,000 reproduces 13,365.059747. The count n is the number of payments, not the number of gaps between the first and last payment. Three payments at years 1, 2 and 3 give n=3 even though only two years separate the first receipt from the last. The valuation date supplies the remaining initial interval.

Consider an immediate first payment

If the three payments instead arrive today, one year from today and two years from today, the stream is an annuity due. Its present value is 5,000+5,000/1.06+5,000/1.06², approximately 14,166.963332, or 14,166.96. This equals the ordinary result multiplied by 1.06. The immediate payment is already at the valuation date and does not need discounting. The local single-payment tool requires a positive year input, so add today's 5,000 directly instead of pretending it arrives after a positive fraction of a year.

Compare rates while keeping the same cash flows

At zero discount, the present value of the three future 5,000 payments is simply 15,000. At 10%, it is approximately 12,434.259955, or 12,434.26. A higher positive discount rate reduces the present value of this unchanged positive payment stream. That sensitivity reflects the model, not a change in the promised nominal payments. Choosing a rate involves the purpose and risk of the valuation. Do not select a rate solely because it makes the result match a preferred purchase price or settlement amount.

Distinguish the mathematical value from contract terms

A present-value calculation does not establish that payments are guaranteed, transferable, available for an early cash-out or free of tax. An insurance annuity can have benefits and conditions that the level-payment formula does not model. Unequal amounts or irregular dates require separate discounting, and monthly schedules require a consistent periodic-rate convention. Keep the schedule, rate and valuation date together so the result can be reproduced. The local tool checks the value of each specified lump sum; the complete annuity value comes from correctly combining those cash flows.

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