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Derivation of Time Value of Money Formulas

Peter F. Colwell

Anyone who has studied business has at least a passing familiarity with “time value of money” formulas. Yet while many people can comprehend explanations of loan repayment schedules or retirement savings accounts, or even use financial calculators, their level of time value understanding may not include any sense of where the time value formulas come from, or how the various time value applications are related. This brief article presents a derivation of all of the standard (and some less well-known) time value of money formulas from the future value of one dollar factor. Basic future value applications are easily understood at an intuitive level. Suppose you have $1,000 in an account today. If this amount earns a rate of return of 10% for a year, the balance grows to $1,100 (the original $1,000 plus $100 in earnings). Symbolically, this time value adjustment can be represented as (1) PV0 (1 + i) = FV1 , an equation in which PV0 is the $1,000 in the present, i = .1 or 10%, and FV1 is the $1,100 future amount one year hence. This process can be repeated, in the sense that we can now view the $1,100 as the initial balance in a second future value analysis. If this amount grows at 10% for one year, the account balance grows to $1,210 by the end of that second period. Symbolically, (2) FV1 (1 + i) = FV 2 . Substituting equation (1)’s representation of FV1 into equation (2) reveals how the initial present value can be transformed into a future value two years later:

PV0 (1 + i)(1 + i) = FV 2 or

2

In other words, we multiply an initial dollar amount – a present value – by the appropriate future value factor to find a value that will not exist until n periods into the future, but is equivalent to our present value in a time-value adjusted sense. (This equivalence was explored in an earlier article on financial mechanics.) The second of the time value factors is obtained very simply from...