Present value calculator

What does a present value calculator tell you?

Present value (PV) is the amount today that is equal in value to money you will receive later, once you allow for the return you could earn in the meantime. A higher discount rate or a longer wait makes a future amount worth less now. The calculator handles the four cash flow shapes that cover most textbook and exam questions: one future sum, a level annuity, a perpetuity, and a perpetuity that grows.

The default case is $10,000 due in 5 years at 10% a year. Its present value is $6,209.21. Read from a 3-decimal table, the factor is 0.621 and the answer is $6,210. The calculator shows both, so you can match a written answer and still see the precise figure.

For a project with different cash flows each year and an investment at the start, the NPV calculator discounts each year separately and nets off the outlay. This page values a single sum or one regular stream.

What is the present value formula for each case?

• Single sum: PV = F / (1 + i)^n, or F x the discount factor (1 + i)^-n.
• Annuity, payments at the end of each period: PV = PMT x (1 - (1 + i)^-n) / i. The part after PMT is the annuity factor.
• Annuity due, payments at the start: the annuity value x (1 + i).
• Perpetuity: PV = PMT / r, with the first payment one year from now.
• Growing perpetuity: PV = C1 / (r - g), where C1 is the payment one year from now and g its yearly growth. It only works when r is above g.

In the first three, i is the rate per period and n the number of periods. With yearly periods, i is simply the annual rate. With monthly periods, a 10% annual rate is 0.8333% a month, and 5 years is 60 months; $10,000 due in 5 years discounted monthly that way is worth $6,077.89.

How do you discount a single future sum, step by step?

Find the discount factor for the rate and the number of years, round it to three decimals as a printed table does, and multiply. Here is $10,000 due at a later date under five combinations:

The gap between the two columns is never more than half a thousandth of the sum, because that is the most a 3-decimal factor can be out by. Waiting twice as long at 10% cuts the value from $6,209.21 to $3,855.43, and waiting 20 years leaves less than a sixth of the original amount.

How do you calculate the present value of an annuity?

Multiply the level payment by one annuity factor instead of discounting each payment. The factor for 5 years at 10% is 3.790787, or 3.791 to three decimals, so $1,000 a year for 5 years is worth $3,791 by hand and $3,790.79 exactly. The payments add up to $5,000; waiting for them costs $1,209.21 of value.

An example in euros: €2,500 at the end of each year for 12 years at 7%. The factor is 7.943, giving €19,857.50 by hand, while the exact present value is €19,856.72. Another in dirhams: AED 20,000 a year for 10 years at 6% has a factor of 7.360 and is worth AED 147,201.74.

An annuity factor is the sum of the single discount factors for years 1 to n. That is why the annuity factor for 5 years at 10% equals 0.909 + 0.826 + 0.751 + 0.683 + 0.621 to within rounding.

What if the first payment is today? The annuity due

When payments arrive at the start of each period, the first one is not discounted at all. The value is the ordinary annuity value times (1 + i): $3,790.79 x 1.10 = $4,169.87 for five payments of $1,000 at 10%, or $379.08 more than when each payment comes a year later.

By hand there are two equivalent routes. Multiply the 5-year factor by 1.10: 3.791 x 1.10 gives $4,170.10. Or take 1 for the payment today plus the 4-year factor for the other four: 1 + 3.170 = 4.170, giving $4,170. Both land within 25 cents of the exact $4,169.87. Leases and rents paid in advance are typical annuity due cases.

How do you value a perpetuity or a growing perpetuity?

A perpetuity pays the same amount every year with no end. Its value is the payment divided by the rate, so $1,000 a year at 10% is worth $10,000. At 5% the same stream is worth $20,000, and at 12% it is worth $8,333.33. If the first payment is today rather than in a year, add that payment: $11,000.

Far-off payments count for little. The first 30 payments of the $1,000 perpetuity at 10% are worth $9,426.91, which is 94.3% of the whole value; everything after year 30 adds the remaining 5.7%.

A growing perpetuity raises its payment by g each year. With $1,000 next year, 3% growth and a 10% rate, PV = 1,000 / (0.10 - 0.03) = $14,285.71. Growth makes the far future matter more: here the first 30 payments supply 86.1% of the value. At 5% growth the value rises to $20,000, and at 7% to $33,333.33, because the denominator shrinks. The same formula values a share from its next dividend: a dividend of 2.50 expected next year, growing 4% a year, at a 9% required return, gives 2.50 / 0.05 = 50.00 a share.

How do you calculate present value in Excel or Google Sheets?

Microsoft gives the syntax as =PV(rate, nper, pmt, [fv], [type]): the rate per period, the number of periods, the level payment, an optional future amount, and type 0 for end-of-period payments or 1 for the start. Cash paid out is negative and cash received positive, so a stream you receive comes back as a negative present value, the amount you would pay for it.

• Single sum: =PV(10%, 5, 0, -10000) returns 6,209.21.
• Annuity: =PV(10%, 5, -1000) returns 3,790.79; add a fifth argument of 1 for the annuity due value of 4,169.87.
• Microsoft's own example is $500 a month for 20 years at 8%, =PV(A3/12, 12*A4, A2, , 0) in its cells, or =PV(8%/12, 12*20, 500, , 0) with the values typed in, with a result of -$59,777.15. Choose the annuity mode, monthly periods, $500, 20 years and 8% here and the calculator returns $59,777.15.

No spreadsheet function values a perpetuity; type =1000/10% for a level one, or =1000/(10%-3%) for a growing one. Google Sheets names the PV arguments rate, number_of_periods, payment_amount, future_value and end_or_beginning, in the same order.

How do you work out present value on a BA II Plus?

The TVM worksheet in the Texas Instruments guidebook uses the same five values: N, I/Y, PV, PMT and FV, with I/Y entered as the yearly rate. P/Y and C/Y start at 1, so for yearly periods there is nothing to set. Reset the TVM variables before each new problem; the guidebook gives the key sequence.

For the single sum, enter 5 for N, 10 for I/Y, 0 for PMT and 10000 for FV, then compute PV: the display shows -6,209.21, negative because it is what you would pay today to receive $10,000 later. For the annuity, enter 5 for N, 10 for I/Y, 1000 for PMT and 0 for FV, and compute PV for -3,790.79. To value the annuity due, change the payment setting from END to BGN and compute PV again. A perpetuity has no TVM variable of its own; divide the payment by the rate.

Common present value mistakes

• Counting the first payment of an annuity due as a year away. It is today, with a factor of 1.000.
• Using an annual rate with monthly periods. Divide the rate by 12 and count months, or the value will be far too low.
• Applying the growing perpetuity formula to this year's payment. C1 is next year's amount. If you know today's figure, multiply it by (1 + g) first: 2.40 growing 4% becomes 2.496, and 2.496 / 0.05 = 49.92.
• Growth at or above the discount rate. The formula breaks down, so the calculator asks for a higher rate instead of printing a meaningless number.
• Treating a 3-decimal answer as exact. It matches a written answer, not a spreadsheet; the gap grows with the size of the sum.

What this calculator leaves out, and a short glossary

It discounts at one constant rate and assumes every payment arrives on time and in full. It does not handle tax, inflation, payments that change irregularly, or a rate that differs by year, and the discount rate you choose is itself a judgment about risk and alternatives. For a full list of yearly cash flows and an outlay, use the NPV calculator; to find the rate itself from cost of debt and equity, see the WACC calculator.

Discount rate: the yearly return used to bring future money back to today. Discount factor: (1 + r)^-n, the value today of 1 due in n years. Annuity factor: the value today of 1 a period for n periods. Perpetuity: a level payment with no end date. Growing perpetuity: a payment that rises by a fixed rate each year forever. Annuity due: a stream whose first payment is today.

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