## Continuous Compounding

As has been seen, compounding can be done yearly, quarterly, monthly, or daily. The periods can be made even smaller, as small as desired; the main disadvantage in having very small periods is having to do more calculations. If the period is made infinite simally small, we say that interest is compounded continuously. There are situations in which very frequent compounding makes sense. For instance, an improvement in materials handling may reduce downtime on machinery. There will be benefits in the form of increased output that mav be used immediately. If there are several additional runs a dav, there will be benefits several times a day. Another example is trading on the stock market. Personal and corporate investments are often in the form of mutual funds. .Mutual funds represent a changing set of stocks and bonds, in which transactions occur very frequently, often mam" times a dav.

A formula for continuous compounding can be developed from Equation (2.3) by allowing the number of compounding periods per year to become infinitely large:

By noting from a definition of the natural exponential function, e, that limf 1 + l-Y" = e-

we set

EXAMPLE 2.7

Cash flow at the Arctic Oil Company is continuously reinvested. An investment in a new data logging system is expected to return a nominal interest of 40%, compounded continuously. \\ nat is the effective interest rate earned by this investment?

The nominal interest rate is given as r = 0.40. From Equation (2.5), i, = e" A - l

The effective interest rate earned on this investment is about 49.2%.B

Although continuous compounding makes sense in some circumstances, it is rarely used. As with effective interest and nominal interest, in the days before calculators and computers, calculations involving continuous compounding were difficult to do. Consequently, discrete compounding is, by convention, the norm. As illustrated in Figure 2.3, the difference between continuous compounding and discrete compounding is relatively insignificant, even at a fairly high interest rate.  