Investors are willing to give up liquidity of some of their money if it means a reward in the future. Therefore, a future payment is equivalent to a smaller present cash amount. A conversion from the future payment, or future value, to the present value is called “discounting.”

Compounding on the other hand has to do with adding interest earnings to the principal so it too can go on to earn interest in its own right.

## Compounding

Convert the interest rate for a compounding period into an effective rate for the whole year by exponentiating to the yearly frequency.

For example, if the rate per compounding period is 0.5 percent, and there are twelve compounding periods per year, then one year’s return equals 1.005^12 = 1.0617, or an effective annual rate of 6.17%. Here, the caret ^ indicates exponentiation. Note that the exponentiating makes sense because after one compounding period, the 0.5 percent earned during the period gets added into the principal: 1+0.5%. Then it gets multiplied by the interest factor at the end of the next compounding period. In other words, once the compounding period in which it is earned has ended, it from then on is treated as interest-earning principal.

Convert a “nominal” interest rate into an effective rate by dividing it by the number of times it is compounded and raising it to an equivalent exponent.

For example, a nominal interest rate of 3 percent compounded monthly leads to (1 + 3%/12)^12 = 1.0304, or an annual return of 3.04%. The nominal rate is really just the periodic interest rate times the number of compounding periods per year.

Interpret a nominal interest rate, r, compounded continuously as a compounding factor e^(rt) where t is in the same unit of time as r. Here, e is the base of the natural logarithm (about 2.7). For example, a $100 principal deposited for two years at a nominal rate of 3 percent to be compounded continuously gives an interest return of $100 x (e^(0.03x2)) - $100 = $6.18.

## Discounting

Convert a future payment into its present value by multiplying by a discount factor equal to 1/(1+i)^t, where i is the annual effective interest paid for the risk and loss of liquidity of the investment's present value, and t is the length of time invested.

For example, if the future value in three years is $1,000,000 and the discount factor is based on an annual accumulation rate of 3 percent, then the present value is $1,000,000 x (1/1.03^3) = $915,142.

Convert a future payment into its present value using a continuously compounded discount rate by multiplying by e^-rt, where r is the nominal rate of interest.

$100 x (1/(1+i)) + $200 x (1/(1+i)^2) for an effective annual interest rate i.

#### Tips

See the eHow articles “How to Calculate Monthly Interest and Principal Payments,” “Excel Amortization Tutorial,” and “How EMIs Are Calculated” by the current author for equations that handle a large number of interest payments or solve for unknown interest rates.

References

- The Theory of Interest; Stephen Kellison; 1996
- Experian. "What Is APR and How Does It Affect Me?" Accessed Sept. 15, 2020.
- Inc. “Why Einstein Considered Compound Interest the Most Powerful Force in the Universe.” Accessed Sept. 15, 2020.
- Corporate Finance Institute. “What is the Compound Interest Formula?” Accessed Sept. 15, 2020.
- Corporate Finance Institute. “What is the Annual Percentage Yield?” Accessed Sept. 15, 2020.
- Office of the Comptroller of the Currency. "Truth in Lending." Accessed Sept. 15, 2020.
- American Express. "APY vs. APR: The Basics About How Interest Is Calculated." Accessed Sept. 15, 2020.

Tips

- See the eHow articles “How to Calculate Monthly Interest and Principal Payments,” “Excel Amortization Tutorial,” and “How EMIs Are Calculated” by the current author for equations that handle a large number of interest payments or solve for unknown interest rates.

Writer Bio

Paul Dohrman's academic background is in physics and economics. He has professional experience as an educator, mortgage consultant, and casualty actuary. His interests include development economics, technology-based charities, and angel investing.