	The Time Value of Money

	Concepts

	Future Value

	Present Value

	Cash Flow Streams

	Annuities

	Other Compounding Periods

	Equations

	Tools & Problems

	TVM Calculator

	Cash Flow Calculator

	TVM Exercise

	Uneven Cash Flow Stream Exercise

	Time Value of Money Quiz

Future Value

The Future Value of a cash flow represents the amount, at some time in the future, that an investment made today will grow to if it is invested to earn a specific interest rate. For example, if you were to deposit $100 today in a bank account to earn an interest rate of 10% compounded annually, this investment will grow to $110 in one year. This can be shown as follows:

Year 1

$100(1 + 0.10) = $110

At the end of two years, the initial investment will have grown to $121. Notice that the investment earned $11 in interest during the second year, whereas, it only earned $10 in interest during the first year. Thus, in the second year, interest was earned not only on the initial investment of $100 but also on the $10 in interest that was paid at the end of the first year. This occurs because the interest rate in the example is a compound interest rate.

Compound Interest

Under compound interest, interest is earned not only on the initial principal but also on the accumulated interest. Interest begins to be earned on the accumulated interest as soon as it is paid, which occurs at the end of each compounding period. This is in contrast to simple interest, under which interest is only earned on the initial principal.

Valuations should generally be based on compound interest because, after the interest has been paid, the full amount, i.e., the initial principal plus interest, could be withdrawn and reinvested elsewhere. Thus, interest on the new investment would be earned on the full amount.

The interest rate in the example is 10% compounded annually. This implies that interest is paid annually. Thus the balance in the account was $110 at the end of the first year. Thus, in the second year the account pays 10% on the initial principal of $100 and the $10 of interest earned in the first year. Thus, the $121 balance in the account after two years can be computed as follows:

Year 2

$110(1+0.10) = $121 or

$100(1+0.10)(1+0.10) = $121 or

$100(1+0.10)² = $121

If the money was left in the account for one more year, interest would be earned on $121, i.e., the initial principal of $100, the $10 in interest paid at the end of year 1, and the $11 in interest paid at the end of year 2. Thus the balance in the account at the end of year three is $133.10. This can be computed as follows:

Year 3

$121(1+0.10) = $133.10 or

$100(1+0.10) (1+0.10) (1+0.10) = $133.10 or

$100 (1+0.10)³ = $133.10

A pattern should be becoming apparent. The Future Value of an initial investment at a given interest rate compounded annually at any point in the future can be found using the following equation:

where

FV_t = the Future Value at the end of year t,
CF₀ = the initial investment,
r = the annually compounded interest rate, and
t = the number of years.

Future Value Example

Find the Future Value at the end of 4 years of $100 invested today at an interest rate 10%.

Solution: