One equation, five unknowns
Every time value of money problem is the same balance equation. The value of today's amount, grown at the interest rate, plus the value of the stream of payments, plus the final amount, must net to zero. That is why PV and FV need opposite signs when there are no payments: you pay in one and receive the other.
PV x (1+i)^N + PMT x (1 + i x t) x [(1+i)^N - 1] / i + FV = 0. Here i is the rate per period (I/Y divided by periods per year), and t is 0 for payments at the end of each period or 1 at the beginning. At i = 0 the middle term becomes PMT x N. This is the standard annuity formula from corporate finance textbooks such as Berk and DeMarzo. N, PV, PMT and FV have closed-form solutions. I/Y is found numerically.
Worked example: how much will a monthly saving grow to?
You save $400 at the end of every month for 15 years at 6.5% a year, compounded monthly, starting from nothing. Monthly periods mean N = 15 x 12 = 180 and the rate per period is 6.5% / 12.
| Field | Entry |
|---|---|
| N | 180 |
| I/Y | 6.5 |
| PV | 0 |
| PMT | -400 (money you pay in is negative) |
| FV | Solve: $121,417.91 |
You pay in $72,000 and the account reaches $121,417.91, so interest supplies $49,417.91. A second example runs the other direction. Borrow $20,000 at 6% for 60 monthly payments: enter PV = 20,000, N = 60, I/Y = 6, FV = 0 and solve for PMT. The answer is -$386.66, negative because it is money you pay.
End or beginning of the period
An ordinary annuity pays at the end of each period, as most loans do. An annuity due pays at the start, as rent usually does. Each payment in an annuity due earns one extra period of interest, so the future value is the ordinary figure multiplied by (1 + i). Ten yearly payments of $100 at 5% grow to $1,257.79 when each is made at year end and to $1,320.68 when each is made at the start of the year.
On a BA II Plus the calculator must be in the matching mode. The keystrokes under the result show how to switch, and they include setting payments per year with P/Y. If you want to practice the keystrokes on a handheld, the guides below walk through them with full problems.
Mistakes that cost points
- Giving PV and FV the same sign when there are no payments. Money you pay in and money you receive need opposite signs, otherwise the equation has no solution.
- Entering N in years when payments are monthly. N counts payment periods, so 10 years of monthly payments is 120.
- Typing the rate as 0.065. Enter 6.5, the same as the I/Y key expects.
- Leaving a calculator in BGN mode from the last problem.
- Rounding N or the payment partway and then reusing the rounded value. Keep full precision until the end.
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