Understanding the Mathematics of TVM Equations
The Time Value of Money models how compound rates change the value of money flows across a defined timeline. The standard TVM relationship equates the sum of the present value, the compound interest accumulation of recurring payments, and the final future value to zero:
Formula Framework
The general formula that binds these values is:
where:
- PV = Present Value (value at N = 0)
- FV = Future Value (value at period N)
- PMT = Periodic payment
- r = Interest rate per period (Annual Rate / Compounds per year)
- N = Total compounding periods
- d = Payment timing type (0 if payments occur at the End, 1 if payments occur at the Beginning)
Sign Convention Rules
For the algebraic equation to balance, you must distinguish cash flows:
1. **Negative (-) Values**: Represent cash outflows. Money leaving your pocket to invest, deposit, or make principal payments. (e.g. buying a bond is $PV = -10,000$).
2. **Positive (+) Values**: Represent cash inflows. Money entering your pocket from a loan disbursement, maturity payout, or withdrawal. (e.g. receiving a loan payout is $PV = 10,000$).