Getting the signs right
Almost every wrong answer from a TVM solver comes from signs rather than from arithmetic. The table shows the convention for the two most common situations.
| Situation | PV | PMT | FV |
|---|---|---|---|
| Saving into an account | Negative (you pay it in) | Negative (you pay it in) | Positive (you withdraw it) |
| Repaying a loan | Positive (you receive it) | Negative (you repay it) | Zero, or negative if a balloon remains |
| Valuing an investment | Negative (the price you pay) | Positive (income received) | Positive (sale proceeds) |
- The rate entered is a nominal annual rate divided by the number of periods per year. Where a product quotes an effective annual rate instead, convert before entering it.
- Solving for the number of periods rarely gives a whole number. A fractional result means the final period is partial.
What the time value of money means
Money available now is worth more than the same amount later, because it can earn a return in the meantime. The time value of money puts that intuition into an equation relating a lump sum today, a stream of level payments, a lump sum at the end, an interest rate and a number of periods.
The sign convention is what trips people up, and it is not arbitrary. Cash you receive is positive and cash you pay out is negative, so a deposit is negative and the balance you later withdraw is positive; a loan you receive is positive and the instalments you pay are negative. If every value carries the same sign, the equation has no sensible solution.
Payments can fall at the end of each period, which is the ordinary annuity used for most loans, or at the start, which is the annuity due used for rent and many insurance premiums. Paying at the start means every payment has one extra period to earn or accrue, so the two settings give different answers.
How to use the TVM calculator
- Choose which of the five quantities you want to find. The field for that quantity disappears — it is the answer.
- Enter the other four, observing the sign convention: money paid out is negative, money received is positive.
- Set the number of periods per year. Twelve for monthly, four for quarterly, one for annual.
- Choose whether payments fall at the end or the start of each period.
- Read the result. When solving for a rate, both the annual rate and the effective annual rate are shown.
The equation behind the solver
All five quantities satisfy one relationship. The present value is grown forward over n periods, the payment stream is valued as an annuity, and the future value is added; consistent values make the total zero.
Four of the five can be isolated algebraically. The rate cannot, so it is found by bisection — the same approach the interest rate calculator uses, and for the same reason.
Common mistakes
- Entering every value as positive. The equation needs at least one sign change to have a solution.
- Mixing an annual rate with monthly periods without setting periods per year.
- Using the ordinary-annuity setting for payments made in advance, such as rent, which are an annuity due.
- Entering an effective annual rate where a nominal rate is expected.
- Reading a fractional number of periods as an error rather than as a partial final period.
الأسئلة الشائعة
What are the five TVM variables?
Present value, future value, payment per period, number of periods and the interest rate. They are linked by one equation, so any four determine the fifth. This is the same set of variables a financial calculator labels PV, FV, PMT, N and I/Y.
Why do I have to enter negative numbers?
Because the equation balances inflows against outflows. Money you pay out is negative and money you receive is positive. Saving 100 a month towards a goal is a payment of -100; the balance you eventually withdraw is positive. If every figure has the same sign there is nothing to balance and no solution exists.
What is the difference between an ordinary annuity and an annuity due?
An ordinary annuity pays at the end of each period and is the standard for loans. An annuity due pays at the start, which is how rent and many insurance premiums work. Because each payment in an annuity due has one extra period to compound, its value is (1 + i) times that of the equivalent ordinary annuity.
How is the interest rate solved when it has no formula?
By numerical search. The rate appears both as a multiplier and inside an exponent, so it cannot be isolated. The solver brackets the answer and halves the interval repeatedly until it converges well beyond display precision.
What is the difference between the annual rate and the effective annual rate?
The annual rate here is nominal — the periodic rate multiplied by the number of periods in a year, ignoring compounding within the year. The effective annual rate, (1 + i)^m - 1, includes that compounding, and is the correct basis for comparing products with different compounding frequencies.
المراجع
- Brealey RA, Myers SC, Allen F. Principles of Corporate Finance, 13th edition. McGraw-Hill Education, 2019 — chapters on present value and valuing annuities.
- Bodie Z, Kane A, Marcus AJ. Investments, 12th edition. McGraw-Hill Education, 2021.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing, 3rd edition. Cambridge University Press, 2007 — root finding by bisection.