Future Value - Compound Interest

Compound Interest

To determine future value using compound interest:

where PV is the present value, t is the number of compounding periods (not necessarily an integer), and i is the interest rate for that period. Thus the future value increases exponentially with time when i is positive. The growth rate is given by the period, and i, the interest rate for that period. Alternatively the growth rate is expressed by the interest per unit time based on continuous compounding. For example, the following all represent the same growth rate:

• 3 % per half year
• 6.09 % per year (effective annual rate, annual rate of return, the standard way of expressing the growth rate, for easy comparisons)
• 2.95588022 % per half year based on continuous compounding (because ln 1.03 = 0.0295588022)
• 5.91176045 % per year based on continuous compounding (simply twice the previous percentage)

Also the growth rate may be expressed in a percentage per period (nominal rate), with another period as compounding basis; for the same growth rate we have:

• 6% per year with half a year as compounding basis

To convert an interest rate from one compounding basis to another compounding basis (between different periodic interest rates), the following formula applies:

where i₁ is the periodic interest rate with compounding frequency n₁ and i₂ is the periodic interest rate with compounding frequency n₂.

If the compounding frequency is annual, n₂ will be 1, and to get the annual interest rate (which may be referred to as the effective interest rate, or the annual percentage rate), the formula can be simplified to:

where r is the annual rate, i the periodic rate, and n the number of compounding periods per year.

Problems become more complex as you account for more variables. For example, when accounting for annuities (annual payments), there is no simple PV to plug into the equation. Either the PV must be calculated first, or a more complex annuity equation must be used. Another complication is when the interest rate is applied multiple times per period. For example, suppose the 10% interest rate in the earlier example is compounded twice a year (semi-annually). Compounding means that each successive application of the interest rate applies to all of the previously accumulated amount, so instead of getting 0.05 each 6 months, one must figure out the true annual interest rate, which in this case would be 1.1025 (one would divide the 10% by two to get 5%, then apply it twice: 1.052.) This 1.1025 represents the original amount 1.00 plus 0.05 in 6 months to make a total of 1.05, and get the same rate of interest on that 1.05 for the remaining 6 months of the year. The second six month period returns more than the first six months because the interest rate applies to the accumulated interest as well as the original amount.

This formula gives the future value (FV) of an ordinary annuity (assuming compound interest):

where r = interest rate; n = number of periods. The simplest way to understand the above formula is to cognitively split the right side of the equation into two parts, the payment amount, and the ratio of compounding over basic interest. The ratio of compounding is composed of the aforementioned effective interest rate over the basic (nominal) interest rate. This provides a ratio that increases the payment amount in terms present value.