A dollar today is worth more than a dollar promised in ten years, because today’s dollar can be invested and earn interest in the meantime. Present value turns that idea into a number. It is the second building block of the site, after life tables, in our guides to how the maths works: every annuity and pension value is a sum of present values.
The idea in plain English
Suppose you can earn interest at a rate Interest rate iThe interest rate per period, used to move money between dates.How it works: Present value a year. Money invested today grows by a factor of each year: that is compounding, and the end result is the future value. Run the same calculation backwards and you get discounting: how much you would need to put aside today to have a given amount later. That amount is the present value.
So a present value is not a guess about the future. It is an exact answer to “what is this future payment worth today, if money earns rate ?”. Change the rate and the answer changes, which is why every valuation on this site states its interest rate.
The notation
- Interest rate iThe interest rate per period, used to move money between dates.How it works: Present value: the interest rate per period, usually per year.
- Discount factor vWhat 1 paid one period from now is worth today: 1/(1 + i).How it works: Present value: the discount factor, the value today of 1 paid one period from now: .
- Discount factor for t yearsWhat 1 paid t years from now is worth today at interest rate i: 1/(1 + i) multiplied by itself t times.How it works: Present value: the value today of 1 paid periods from now.
- Present valueWhat a set of future payments is worth today, at a given interest rate.How it works: Present value and Future valueWhat a set of payments grows to by the end, at a given interest rate.How it works: Present value: present value and future value.
- Rate of discount dInterest expressed as a share of the amount at the end of the period: i/(1 + i), or 1 − v.How it works: Present value: the rate of discount, interest measured against the end amount rather than the start.
- Force of interest δThe interest rate compounded continuously that matches i over a year: the natural log of (1 + i).How it works: Present value: the force of interest, the equivalent continuously compounded rate.
The formulas
Compounding forward and discounting back are the same formula read in opposite directions.
Actuaries also use two equivalent ways of expressing the same rate. The rate of discount is the interest paid in advance, and the force of interest is the continuous rate that gives the same growth over a year:
All three describe the same growth; they just measure it differently.
When there are several payments, each is discounted by its own time and the results are added: , where is the amount paid at time . A stream of equal payments has a closed form, covered in annuity formulas.
Worked example: $10,000 due in 10 years
At 5% a year:
- The one-year discount factor is = 1 ÷ (1 + 0.05) = 0.952381.
- Over 10 years, = 0.613913.
- The present value of $10,000 due in 10 years is $10,000 × 0.613913 = $6,139.
Going the other way, $10,000 invested today at 5% grows by a factor of = 1.628895 to $16,289 after 10 years. At this rate money doubles in about 14.2 years, and a payment 14.2 years away is worth about half its face value today.
The equivalent rate of discount is = 0.047619, and the force of interest is = ln(1 + 0.05) = 0.048790.
Try it: slide the rate and the years
Move the sliders to see how the discount factor falls with time, and how much faster it falls at higher rates. The bars show for every whole year on the slider.
Discounting: what a future amount is worth today
- One-year factor,
- 0.9523811 ÷ (1 + i)
- Over t years,
- 0.613913v multiplied 10 times
- Present value,
- $6,139of $10,000 due in 10 years
- Future value,
- $16,289of $10,000 today after 10 years
At 5% a year, $10,000 due in 10 years is worth $10,000 × 0.613913 = $6,139 today.
Nominal and real rates
A nominal rate discounts amounts in future dollars. A real rate, roughly the nominal rate minus inflation, discounts amounts in today’s dollars. Mixing them gives wrong answers: an income that rises with prices should be valued in today’s dollars at a real rate, or projected in future dollars and discounted at a nominal rate. The calculators on this site say which they use.
Present value also does not include risk on its own. A promised payment from a safe borrower and an uncertain one have the same present value at the same rate; the difference shows up in the rate chosen, or, for life-contingent payments, in the probability of the payment being made. Weighting each payment by the chance of being alive to receive it is exactly what turns these formulas into the value of a life annuity.
Where this is used
Questions
What is the formula for present value?
Present value equals the future amount divided by (1 + i) raised to the number of years, where i is the interest rate per year. Written with the discount factor v = 1/(1 + i), it is PV = FV × vᵗ. It tells you how much you would need to invest today at rate i to have the future amount after t years.
How do you calculate present value?
Work out the discount factor for one year, 1 ÷ (1 + i), raise it to the power of the number of years, and multiply the future amount by the result. With several future amounts, discount each one by its own number of years and add them up.
What is the difference between present value and future value?
They are the same calculation in opposite directions. Future value grows an amount today forward at interest, multiplying by (1 + i) each year. Present value brings a future amount back to today, dividing by (1 + i) each year. At the same rate, one exactly undoes the other.
What interest rate should be used to discount?
There is no single right rate; it is an assumption, and the answer depends on it. A rate on safe bonds of the right term values money that is certain to be paid; a higher rate allows for risk. Nominal amounts need a nominal rate and inflation-adjusted amounts a real rate.