The time value of money

Five numbers, one equation

Five quantities describe almost every household cash-flow problem. Present value is an amount today. Future value is an amount later. Payment is a repeating amount. The interest rate prices the wait. N counts the periods. Give a calculator any four and it can solve the fifth, because the signed cash flows have to balance.

Try it on the finance calculator. Leave N at 10, I/Y at 8, present value at 20,000, and the payment at −2,000. The future value is −14,205.38. The minus sign is not a loss. It is the balancing cash flow at the end: about $14,205 you could withdraw.

Compounding is the second dollar of interest

Simple interest pays only on the original principal. Compound interest pays on the balance, so the second year earns interest on the first year’s interest. The compound interest calculator draws both lines so the balance is not confused with the money you put in.

The rule of 72 is the version you can do in your head: divide 72 by the annual rate and you have a rough doubling time. At 8 percent that is nine years. Use the shortcut to think, and the calculator when the plan needs a number you will save toward.

Payments have a direction and a date

A loan payment is mostly interest at the beginning because the balance is large, and mostly principal at the end because the balance has shrunk. Extra principal reduces the base on which the next interest charge is calculated. A biweekly plan works mainly because twenty-six half payments equal thirteen full payments a year.

Payments at the beginning of a period, common in leases, earn or cost one extra period of interest compared with payments at the end. The finance calculator keeps that choice under settings.

Discounting runs the film backward

Present value asks what a future amount is worth today. A higher discount rate makes a future dollar worth less now. That is why a bond’s price falls when the yield investors demand rises above the coupon.

Net present value discounts a list of cash flows. Internal rate of return searches for the rate that would make that sum zero. Payback only asks when the cash comes back, and it ignores every dollar after recovery.

Inflation is the same shape, pointed at prices

Three percent inflation for twenty years multiplies. It does not add 60 percent. The inflation calculator compounds a price forward or backward. Either inflate the spending, or use a real return and keep the spending in today’s dollars. Doing both counts inflation twice.

The exact real return divides one plus the nominal return by one plus inflation. Subtracting inflation is a close mental estimate and a slightly generous one.

What the signs are protecting

Money you pay out is negative on the five-key finance calculator. Money you receive is positive. Household calculators use ordinary positive amounts and say so. When a result is negative, the sentence under the number translates it.

A mortgage adds tax and insurance. A retirement projection grows contributions forward. A loan is the payment that drives a balance to zero. The foundation does not change. Only the labels do.

A check you can do without the page

Pick a small case you can follow. One hundred dollars at 10 percent for one year is one hundred ten if interest is paid once at the end. The second year, compound interest uses one hundred ten, not the original one hundred. That single difference is the whole idea. The calculators exist so you do not have to repeat it for 360 months by hand.

When a result surprises you, shrink the term to one period and the amounts to something you can audit. If the one-period case matches your own arithmetic, the long case is the same rule repeated. If it does not match, you and the page are not using the same rate, the same timing, or the same sign.

Household pages often hide the sign convention because a mortgage payment is easier to read as a positive number you will send. The finance calculator keeps the signs visible because that is the equation. Neither page is more official. They are the same relationship with different labels. The methodology records the periodic-rate construction if you want the algebra in one place.

More from a check you can do without the page

Use this lesson before you use a long projection. If present value, future value, and the payment direction are clear, the retirement, loan, and bond pages are applications rather than new theories. If they are not clear, stay on the finance calculator and change one key at a time until the headline moves the way you expect.

The sample is worth typing yourself. N is 10. I/Y is 8. Present value is 20,000. Payment is −2,000. Payments are at the end of each period. Future value solves to −14,205.38. Change only the rate to 6 and the future value becomes −9,455.36. Those two results are the check that the sign convention and the compounding on this desk match a five-key financial calculator.

Once that check passes, the rest of the library is labeling. A mortgage adds tax and insurance on top of principal and interest. An IRA compounds a contribution and then shows a tax illustration. A bond prices coupons and face value at a yield. Bring the habit from this page with you: one change, then Calculate, then read the line that moved.

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