School Launch

Money Through Time

A dollar today is worth more than a dollar later, because today’s dollar can be put to work earning interest. Choose a rate, a number of years and how often interest compounds. Then watch savings grow, find what a future goal is worth today, or work out the monthly deposit that reaches it.

Personal Finance Lessons 7.1 and 10.1 · Banking Lesson 5.2 · Accounting Lesson 10.4 · Series 65 Lesson 3.3 · about 20 minutes

Click or tap the chart to read any year. With the chart focused, the left and right arrow keys move one year (Home and End jump to the ends).

Balance
Money you put in
Interest earned

The same thing in a spreadsheet

Rule of 72 check

What do you want to find?

Your numbers

$
$
$
%
How often interest compounds

Growing money forward

Put P dollars in an account that pays a yearly rate r, compounded n times a year. Each period the balance is multiplied by 1 + r/n, so after t years the future value is

FV = P(1 + r/n)nt

Compounding more often helps, but less and less. If interest compounded every instant (continuous compounding), the growth factor becomes ert, where e ≈ 2.71828. Compare monthly, daily and continuous above: the differences are a few dollars, not thousands. The APY (annual percentage yield) puts every choice on the same footing: it is how much one whole year adds.

Bringing money back: present value

Run the same arithmetic backwards. If you want F dollars in t years, the amount you need today is the present value

PV = F / (1 + r/n)nt

Dividing by the growth factor is called discounting. It is how businesses compare money that arrives at different times, and how a bond or a promise of future payments is priced.

Monthly deposits: an annuity

A deposit of D every month is an annuity. Here deposits are monthly, so the page first turns any compounding into the matching monthly rate i = (1 + r/n)n/12 − 1 (for continuous, i = er/12 − 1). With N = 12t deposits made at the end of each month,

FV = P(1 + i)N + D[(1 + i)N − 1] / i

Deposits made at the start of each month earn one extra month, so the deposit part is multiplied by 1 + i. Solving the same equation for D gives the deposit needed to reach a goal; that is what Deposit needed does, and what a spreadsheet’s =PMT does.

Reading the spreadsheet formula

=FV(rate, nper, pmt, pv, type), =PV(rate, nper, pmt, fv, type) and =PMT(rate, nper, pv, fv, type) work in Excel, Google Sheets and LibreOffice. The rate and the number of periods must match: a monthly rate with a count of months. Money you pay in is entered as a negative number and money you get back comes out positive, which is why the page writes −200 for a deposit and why =PV and =PMT answer with a minus sign.

The rule of 72

To guess how long money takes to double, divide 72 by the yearly rate in percent. At 6%, 72 ÷ 6 = 12 years. The exact answer for yearly compounding is ln 2 / ln(1 + r): 11.9 years. The rule is closest for rates near 8% and drifts at the ends (at 20% it says 3.6 years; the truth is 3.8). The check card above compares the rule with the exact time for the compounding you chose.

Doubling repeats. Money that doubles in 12 years is 4 times as large after 24 and 8 times after 36. The dashed red lines on the chart mark these doublings of your starting amount: this steady multiplying, not the size of the first deposit, is what makes long saving powerful.

Check yourself

  1. You put $5,000 in an account paying 6% compounded quarterly and leave it for 10 years. What is it worth?
    Answer

    5,000(1 + 0.06/4)40 = 5,000(1.015)40 ≈ $9,070.09. In a spreadsheet: =FV(6%/4, 4*10, 0, −5000).

  2. How much must you set aside today to have $20,000 in 8 years at 4% compounded yearly?
    Answer

    20,000 / 1.048 ≈ $14,613.80. =PV(4%, 8, 0, 20000) shows −14,613.80: money you pay in.

  3. Starting from nothing, what monthly deposit reaches $100,000 in 20 years at 6% compounded monthly?
    Answer

    i = 0.005 and N = 240, so D = 100,000 × 0.005 / (1.005240 − 1) ≈ $216.43 a month. You put in 240 × 216.43 ≈ $51,943; interest supplies the rest.

  4. Use the rule of 72 to estimate the doubling time at 9% compounded yearly, then find the exact time.
    Answer

    72 ÷ 9 = 8 years. Exactly, ln 2 / ln 1.09 ≈ 8.04 years. The rule is very close here.

A bit of history: Luca Pacioli stated the rule of 72 in his Summa de arithmetica (1494) as a rule merchants already used. When Benjamin Franklin died in 1790, his will left £1,000 each to Boston and Philadelphia, to be lent at 5% interest to young tradesmen for two hundred years, to show what patient compounding could do.

“He that gathereth by labour shall increase.” — Proverbs 13:11