How to calculate present value, step by step
To calculate present value, take the future value, work out the discount factor, and multiply them together. The formula is PV = FV ÷ (1 + r)^n, where FV is the amount you will receive, r is the discount rate per period as a decimal, and n is the number of periods until the money arrives. The discount factor is (1 + r)^n in the denominator, or 1 ÷ (1 + r)^n as a multiplier — the two are the same number written two ways. On the worked example below, $10,000 in 5 years at 10% has a discount factor of 0.620921, and the present value is $6,209.21.
The worked example: $10,000 in 5 years at 10%
The worked example, step by step. Step 1 — write down the three inputs: the future value is $10,000, the wait is 5 years, and the discount rate is 10%, or 0.10 as a decimal. Step 2 — work out (1 + r)^n: 1.10 to the power of 5 is 1.610510. Step 3 — work out the discount factor: 1 ÷ 1.610510 is 0.620921, or 0.620921323 unrounded. Step 4 — multiply: $10,000 × 0.620921 is $6,209.21. Step 5 — check it by going forward: $6,209.21 at 10% for 5 years is $6,209.21 × 1.610510, which is $10,000.00 — the round trip closes, so the answer is right. Step 6 — read the discount: $10,000 − $6,209.21 is $3,790.79, which is what waiting five years costs at a 10% return.
What the discount factor is
The discount factor is the price of a future dollar in today's money. At 10% over 5 years it is 0.620921, which means every dollar arriving in five years is worth about 62 cents today — because 62 cents invested at 10% for five years grows into a dollar. The factor is always between 0 and 1 for a positive rate, and it falls as the wait gets longer or the rate gets higher: at 3 years it is 0.751315, so the same $10,000 is worth $7,513.15; at 10 years it is 0.385543, so the value is $3,855.43. A factor of 1 would mean the money is worth its face value, which only happens at a 0% rate or a zero wait. If you remember one number from this guide, remember the factor: it is the whole calculation in a single decimal.
What discount rate to use
The discount rate is the return you could earn on money of similar risk, and it is the number the answer turns on. The same $10,000 in 5 years is worth $7,835.26 at 5% and $4,971.77 at 15% — the rate moves the answer by thousands, so a rate you guessed is a guess, not a valuation. For a safe promise, use a safe return: a savings rate, or the yield on a government bond of the same length. For a risky promise, use a higher rate, because the risk belongs in the rate — a payment that might not arrive is worth less than a certain one, and the only honest way to show that is to discount it harder. For an inflation adjustment, use the inflation rate, and the answer is the value in today's buying power rather than today's dollars. If two people disagree about a present value, they almost always disagree about the rate, not about the arithmetic.
Present value of an annuity
An annuity is a series of equal payments, and its present value is the payments discounted one by one and added up. The shortcut is the annuity factor: (1 − (1+r)^−n) ÷ r. At 10% over 5 years the annuity factor is 3.790786769, so $2,000 a year for 5 years is worth $2,000 × 3.790786769, which is $7,581.57. Notice that the annuity is worth more than the single $10,000 payment — $7,581.57 against $6,209.21 — even though the total paid is the same $10,000. The early payments are discounted less, so they are worth more, and the annuity collects five payments instead of one. A single payment is one discounted number; an annuity is five, and the factor does the adding for you. Work it out in the present-value-calculator, which shows the annuity figure next to the single-sum figure for the same rate.
Present value of a perpetuity
A perpetuity is a payment that never stops, and its present value is the payment divided by the rate: PV = PMT ÷ r. At 10%, $1,000 a year forever is worth $1,000 ÷ 0.10, which is $10,000.00. The payments after the first few decades are worth almost nothing — a dollar in 50 years at 10% is worth less than a cent — so the infinite tail adds almost nothing to the total, and the simple division is exact. The formula only works when the rate is positive and the payment never grows.
Common mistakes
The first mistake is using the wrong rate: the discount rate is the return you could earn, not the inflation rate and not the interest rate on the loan — unless the question is about inflation or a loan. The second is mixing up periods: a 10% annual rate with monthly payments needs the monthly rate and the number of months, not 10% and 5. The third is discounting a payment that arrives at the start of the year as if it arrived at the end: a payment today needs no discounting at all, and a payment at the start of each year is worth one more year of growth than the same payment at the end. The fourth is adding present values that were discounted at different rates: every payment in a sum must use the same rate, or the total means nothing. And the fifth is treating the answer as a price: the present value is what the money is worth at your discount rate, not what anyone will pay for it — a buyer with a different rate will value the same promise differently.
A quick check you can do in your head
You can sanity-check any present value without a calculator. First, the answer must be smaller than the future value when the rate is positive: $10,000 in 5 years at 10% must be less than $10,000, and $6,209.21 is. Second, the discount factor must sit between 0 and 1: 0.620921 does, and a factor above 1 means the rate or the exponent went the wrong way. Third, the round trip must close: $6,209.21 × 1.610510 is $10,000.00, so the discounting and the compounding agree. Fourth, the annuity must be worth more than the single payment with the same total: $7,581.57 against $6,209.21. If a calculator breaks any of these checks, the number is wrong — run your numbers through the present-value-calculator and compare them with the worked example above.