Summary of Section 9.1 (Interest) The rate at which money grows is called the interest rate, and the interest rate depends on the type of loan involved. (Generally, lower interest rates are paid for loans where the lender is more free to demand the money back at an unexpected time and when the borrower is more likely to pay back.) Simple Interest The first concept is simple interest: if you loan out an amount of money P you can expect to be paid back the same amount plus an additional amount I called interest: Let P = principal = amount of initial loan, let I = interest, and let A = amount paid back. Then A = P + I. The interest is proportional to the principal: I = k*P, where the proportionality constant k is called the *interest rate* for the investment period (the "percent interest"). That is, k=I/P, i.e., percent interest equals interest per principal. So A = P + k*P, i.e., A = P*(1+k). That is, the amount paid back is the principal P times a growth factor (1+k). Define a = growth factor = A/P = 1+k. Generally the longer you lend out money the more interest you expect to earn. Let t = time of the investment. For a simple loan the interest rate is proportional to the period of time t of the investment: k = r*t, where the proportionality constant r is the nominal interest rate per time. That is, the nominal interest rate is defined by r = k/t = I/(Pt) = interest per principal per time. So the amount of interest earned is I = P*r*t and the growth factor is a = 1+rt. In brief: P = principal (amount loaned out), t = time of loan, r = interest rate per time (interest per principal per time) = k/t = I/(P*t) k = interest rate per investement period (interest per principal) = I/P, I = interest (amount paid back in addition to the principal), a = growth factor = A/P = (1+k), and A = final amount (amount paid back). For example: consider $100 lent at 12% annual interest for 3 months. We have: P = $100, t = 3 months, r = 12% per year, k = r*t = 3%, I = P*k = $3, a = 103%, and A = P + I = $103. Compound interest: Generally when money is paid back all the money including the interest is reinvested. Each time the money is invested it gets multiplied by a growth factor. The result is that money tends to grow exponentially with time. An important instance of this is compound interest: Let t = investment period = time period between "reinvestments", let n = total number of time periods, let k = interest rate per time period let P = P_0 = principal, and let P_m = amount after m time periods (which becomes the principal of the (m+1)th period). Then P_1 = P_0 * (1+k), P_2 = P_1 * (1+k) = P_0 * (1+k)^2, P_3 = P_2 * (1+k) = P_0 * (1+k)^3, ... P_n = P_0 * (1+k)^n, which is the formula on page 415 of the text. In other words, the growth factor after n steps is the growth factor after 1 step raised to the n-th power. To make this formula clear, let T = n*t = total time of investment. Then the interest rate per period is k = r*t = r*T/n and the final amount is P_n = P * (1+r*T/n)^n. For example: consider $100 lent at 12% annual interest for 12 months compounded quarterly (every 3 months). We have: principal: P = $100, total time interval: T = 12 months, time period per compounding: t = 3 months, times compounded: n = 12/3 = 4 times, interest rate per time r = 12% per year, interest rate per period: k = r*t = 3%, growth factor per period: a = (1+k) = 103%, and investment growth factor: (1+k)^n = 1.03^4 = 1.12550881. P_0 = $100 P_1 = $103 = 103% * $100 = 1.03^1 * $100 P_2 = $106.09 = 103% * $103 = 1.03^2 * $100 P_3 = $109.2727 = 103% * $106.09 = 1.03^3 * $100 P_4 = $112.550881 = 103% * $109.2727 = 1.03^4 * $100 Annual Percentage Yield (APY) An interest rate is not fully specified unless you state how often the interest is compounded. Interest grows faster if you compound more frequently. To compare compound interest rates we compare how much the money would grow over a standard length of time: one year. The percent interest earned over one year is calle the Annual Percentage Yield (APY). Suppose we know that an investment with principal P yields an amount A at the end of a period of T years. To find the annual percentage yield we pretend that the interest is compounded annually. In this case the final amount would be A = P*(1+k)^T, where k is the interest rate for an investment period of one year. Solving this equation for k gives: k = (A/P)^(1/T)-1. The annual percentage yield is k times 100, so API = 100*((A/P)^(1/T)-1), which is the formula on page 417 of the book.