The Time Value of Money
A dollar today is worth more than a dollar next year, because today's dollar can be invested and earn a return. That single idea is the foundation of finance, and the time value of money (TVM) equation expresses it formally.
Five variables describe any stream of cash flows at a constant rate:
| Symbol | Meaning |
|---|---|
| N | Number of compounding periods |
| I/Y | Interest rate per period |
| PV | Present value — the amount now |
| PMT | Payment made each period |
| FV | Future value — the amount at the end |
Know four and the fifth is determined. That is exactly what an HP 12C or a TI BA II Plus does, and what this calculator does.
The Equation
PV × (1 + i)N + PMT × (1 + i·d) × ((1 + i)N − 1) ÷ i + FV = 0
where i is the rate per period as a decimal and d is 1 for payments at the beginning of each period and 0 for the end. Solving for PV, FV or PMT is algebra. Solving for N or the rate has no closed form, so those are found numerically — the calculator narrows the answer by bisection until it converges.
Sign Convention
This is where almost everyone gets stuck. The equation sums to zero, which only works if money flowing in and money flowing out carry opposite signs.
| Cash flow | Sign | Example |
|---|---|---|
| Money you pay out | Negative | A deposit, a loan repayment, an investment |
| Money you receive | Positive | A loan you take out, a withdrawal, a maturity value |
From a borrower's perspective, a $20,000 loan is PV = +20,000 (you received it) and PMT = −400 (you pay it). From a saver's, a $10,000 deposit is PV = −10,000 and the balance at the end is FV = positive. Getting the signs wrong is the single most common cause of a nonsensical answer or a solver that fails to converge.
Ordinary Annuity vs. Annuity Due
| Ordinary annuity | Annuity due | |
|---|---|---|
| Payment timing | End of each period | Beginning of each period |
| Typical use | Loans, mortgages, bonds | Rent, leases, insurance premiums, some savings plans |
| Value | Baseline | Higher by a factor of (1 + i) |
Each payment in an annuity due earns one extra period of interest, so over long horizons the difference is meaningful — at 6% over 30 years it is about 6% more accumulated value.
Matching Rate and Period
The rate must be per period, not per year. This is a frequent error.
| If payments are | N for 10 years | I/Y for a 6% annual rate |
|---|---|---|
| Monthly | 120 | 0.5% |
| Quarterly | 40 | 1.5% |
| Semi-annually | 20 | 3% |
| Annually | 10 | 6% |
Dividing the annual rate by the number of periods gives the nominal periodic rate, which is the convention for loans. Note that this is not the same as the effective rate: 0.5% monthly compounds to 6.17% a year, not 6%.
What Each Variable Answers
| Solve for | Typical question |
|---|---|
| FV | What will my savings be worth in 20 years? |
| PV | What is a stream of future payments worth today? What lump sum funds a pension? |
| PMT | What is the monthly payment on this loan? How much must I save each month? |
| N | How long until I reach my target, or until this debt is cleared? |
| I/Y | What return does this investment imply? What rate am I actually being charged? |
Worked Examples
A loan payment
$20,000 borrowed at 6% annually over 5 years, paid monthly: N = 60, I/Y = 0.5, PV = 20,000, FV = 0. Solving for PMT gives −$386.66 — the minus sign confirming it is money leaving your pocket.
A savings goal
You want $50,000 in 10 years and can earn 7% annually, saving monthly: N = 120, I/Y = 0.5833, PV = 0, FV = 50,000. PMT solves to about −$288.
The implied rate
A dealer offers a $15,000 car for $320 a month over 60 months. N = 60, PV = 15,000, PMT = −320, FV = 0. Solving for I/Y gives about 0.86% per period, or roughly 10.3% a year — considerably more than a bank would charge a good-credit borrower.
What This Model Cannot Do
- Uneven cash flows. The TVM equation assumes an identical payment every period. Irregular flows require NPV and IRR calculations instead.
- Changing rates. Adjustable-rate loans and variable returns need period-by-period modelling.
- Taxes and fees. Neither appears in the equation; adjust the rate or the payments yourself.
- Risk. A constant rate implies certainty, which no investment offers. TVM tells you what a given rate produces, not what rate you will get.
Frequently Asked Questions
Why is my answer negative?
Because it is money flowing out. A negative PMT when solving a loan is correct — you pay it. Read the sign as direction, not as an error.
Why does the solver fail on my inputs?
Almost always because PV, PMT and FV all carry the same sign, which describes cash flowing in one direction only. No interest rate or period count can balance that; at least one value must have the opposite sign.
How does this differ from the loan calculator?
The loan calculator is a preset for one common case with friendlier inputs. This one is the general engine underneath it, and can solve for any variable in any direction.
What if the rate is zero?
The equation degenerates to simple addition: PV + PMT × N + FV = 0. The calculator handles this case separately, since dividing by zero would otherwise fail.