The Time Value of Money

Worth more now

Would you rather have $1000 today or $1000 a year from now? Today, because $1000 today can be put in an account and earn interest. At 5% a year it becomes 1000 × 1.05 = 1050 dollars in a year.

So $1000 today is worth the same as $1050 a year from now, and more than $1000 a year from now. A sum of money is worth more now than the same sum later, because it can earn interest in between. That is the time value of money.

Value now and value later

The value of a sum today is called its present value, PV. The value it grows to after N periods is its future value, FV. Write the interest rate for one period as a decimal, i, so that 5% a year is i = 0.05. Each period multiplies the sum by 1 + i, so with nothing paid in or taken out along the way, FV = PV $\times {\left(1+i\right)}^N$× (1 + i)^N.

This is the compound interest formula, with the rate written as a decimal. The single number ${\left(1+i\right)}^N$(1 + i)^N, the growth factor for N periods, links the value now to the value later.

Bringing a sum back to today

Divide both sides of FV = PV $\times {\left(1+i\right)}^N$× (1 + i)^N by ${\left(1+i\right)}^N$(1 + i)^N and the formula runs the other way: PV = FV $÷{\left(1+i\right)}^N$÷ (1 + i)^N. Finding a present value this way is called discounting: a sum due later is discounted back to what it is worth today.

It is the same move as a reverse percentage. A price that rose by 20% is undone by dividing by 1.2; N periods of growth are undone by dividing by the growth factor ${\left(1+i\right)}^N$(1 + i)^N. Taking a percentage off would not undo them.

$1000 today against a sum B due in a year, at a rate r. At r = 5% the $1000 grows to $1050 in a year, and B = 1100 due in a year is worth 1100 ÷ 1.05 = 1047.62 dollars today, which the figure rounds to 1048. So B is worth more than the $1000 on either date. Drag B down to the sum that ties with $1000 today: $1050.

A sum due in five years

$1000 is due in 5 years, and money earns 6% a year, so i = 0.06 and N = 5. The growth factor is $1.06^5=1.338226$1.06⁵ = 1.338226, to 6 decimal places, so PV = 1000 ÷ 1.338226 = 747.26 dollars, to the nearest cent.

So $747.26 today and $1000 in five years are worth the same. Put $747.26 in the account and it grows to 747.26 × 1.338226 = 1000.00 dollars, to the nearest cent, in five years.

What $1000 is worth today at 6% a year when it is due now, in 1 year, and so on up to 5 years: $1000, $943.40, $890.00, $839.62, $792.09 and $747.26. Each year further off divides by 1.06 once more.

One rule, two directions

Compounding and discounting are one relationship read in two directions. Multiplying by ${\left(1+i\right)}^N$(1 + i)^N moves a sum forward N periods; dividing by it moves a sum back. Two sums due on different dates can only be compared once they are moved to the same date, and either date gives the same verdict.

Compare $1000 today with $1100 in two years, at 5% a year. Forward: $1000\times 1.05^2=1102.50$1000 × 1.05² = 1102.50 dollars in two years, which is more than $1100. Back: $1100/1.05^2=997.73$1100 / 1.05² = 997.73 dollars today, to the nearest cent, which is less than $1000. On both dates the $1000 today is worth more.

Any four fix the fifth

A finance calculator, or the finance solver on a graphing calculator, holds five quantities: N, the number of periods; I, the interest rate; PV, the value now; PMT, a payment made every period; and FV, the value later. Enter any four and it finds the fifth. With no payments along the way, PMT = 0, and the other four are linked by FV = PV $\times {\left(1+i\right)}^N$× (1 + i)^N. Many calculators count money paid out as negative and money received as positive, so PV and FV are entered with opposite signs.

The formula can be solved for any of the four by hand too. The rate that turns $800 into $1000 in 4 years solves $800\times {\left(1+i\right)}^4=1000$800 × (1 + i)⁴ = 1000, so ${\left(1+i\right)}^4=1.25$(1 + i)⁴ = 1.25 and $1+i=1.25^{\frac{1}{4}}=1.057371$1 + i = 1.25^(1/4) = 1.057371, to 6 decimal places: a rate of 5.74% a year. The time $800 takes to reach $1000 at 5% a year solves $1.05^N=1.25$1.05^N = 1.25, so $N=\ln 1.25/\ln 1.05=4.57$N = ln 1.25 / ln 1.05 = 4.57, to 2 decimal places: the balance first passes $1000 after 5 whole years.

The usual mistakes

Growing forward when today's value is asked. 1000 × 1.338226 = 1338.23 dollars is what $1000 today becomes in 5 years. The value today of $1000 due in 5 years is smaller than $1000, not larger.

Taking the rate straight off. 5 years at 6% is not 30% off, and 1000 × 0.7 = 700 dollars is too low. Discounting divides by the growth factor, and a division is not a subtraction.

Using simple growth. Five years at 6% is a factor of $1.06^5=1.338226$1.06⁵ = 1.338226, not 1 + 5 × 0.06 = 1.3, because each year's interest is worked out on a balance that already holds the earlier interest.