Interest does not always compound annually, as assumed in the problems already presented in this chapter. Sometimes it compounds quarterly, monthly, daily, even continuously. The more frequent the compounding period, the more valuable the bond or other instrument, all else constant. The mathematics remains the same (though a little more difficult when compounding is continuous), but you must be careful about what you plug into the equation for i and n. For example, $1,000 invested at 12 percent for a year compounded annually would be worth $1,000 × (1.12)1 = $1,120.00. But that same sum invested for the same term at the same rate of interest but compounded monthly would grow to $1,000 × (1.01)12 = $1,126.83 because the interest paid each month is capitalized, earning interest at 12 percent. Note that we represent i as the interest paid per period (.12 interest/12 months in a year = .01) and n as the number of periods (12 in a year; 12 × 1 = 12), rather than the number of years. That same sum, and so forth with interest compounded quarterly (4 times a year) would grow to $1,000 × (1.03)4 = $1,125.51. The differences among annual, monthly, and quarterly compounding here are fairly trivial, amounting to less than $7 all told, but are important for bigger sums, higher interest rates, more frequent compounding periods, and longer terms. One million dollars at 4 percent for a year compounded annually comes to $1,000,000 × (1.04) = $1,040,000, while on the same terms compounded quarterly, it produces $1,000,000 × (1.01)4 = $1,040,604.01. (I’ll take the latter sum over the former any day and “invest” the surplus in a very nice dinner and concert tickets.) Likewise, $100 at 300 percent interest for 5 years compounded annually becomes 100 × (4)5 = $102,400. Compounded quarterly, that $100 grows to $100 × (1.75)20 = $7,257,064.34! A mere $1 at 6 percent compounded annually for 100 years will be worth $1 × (1.06)100 = $339.30. The same buck at the same interest compounded monthly swells in a century to $1 × (1.005)1200 = $397.44. This all makes good sense because interest is being received sooner than the end of the year and hence is more valuable because, as we know, money now is better than money later.
Do a few exercises now to make sure you get it.
For all questions in this set, interest rates are stated in annual terms, but the interest compounds quarterly (four times a year). Also, assume there are no transaction fees, defaults, etc.
