# Managing Your Money. UNIT 4D Loan Payments, Credit Cards, and Mortgages: We calculate monthly payments and explore loan issues.

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17 4B The Power of Compoudig 231 Cotiuig the patter, we fid that the balace after N years is the origial pricipal times 1.04 raised to the Nth power. For example, the balace after N 5 10 years is \$224 3 A1.04B 10 5 \$ We ca geeralize this result by lookig carefully at the last equatio above. Notice that the \$224 is the origial pricipal that we bega with. The 1.04 is 1 plus the iterest rate of 4%, or The expoet 10 is the umber of times that the iterest has bee compouded. Let s write the equatio agai, addig these idetifiers ad turig it aroud to put the result o the left: \$ (''')'''* accumulated balace, A 5 \$224 ('')''* origial pricipal, P d umber of compoudig periods ('')''* 11 iterest rate 3 A1.04B 10 Whe iterest is compouded just oce a year, as it is i this case, the iterest rate is called the aual percetage rate, or APR. The umber of compoudig periods is the simply the umber of years, which we call Y, over which the pricipal ears iterest. We therefore obtai the followig geeral formula for iterest compouded oce a year. where A 5 P 3 A1 1 APRB Y A 5 accumulated balace after Y years P 5 startig pricipal APR 5 aual percetage rate Aas a decimalb Y 5 umber of years Be sure to ote that the aual iterest rate, APR, should always be expressed as a decimal rather tha as a percetage. I the New College case, the aual iterest rate is APR 5 4% , ad iterest is paid over a total of 535 years. Thus, the accumulated balace after Y years would be A 5 P 3 A1 1 APRB Y 5 \$224 3 A B \$224 3 A1.04B \$ ,296,691,085 < \$ \$290 billio As the admiistrator claimed, a 4% iterest rate meas the Quee owes about \$290 billio. Techical Note For the more geeral case i which the iterest rate is ot ecessarily set o a aual (APR) basis, you may see the compoud iterest formula writte A 5 P 3 A1 1 i B N where i is the iterest rate ad N is the total umber of compoudig periods. USING YOUR CALCULATOR Most calculators have a key for raisig to powers, labeled y x or. For example, calculate by pressig 1.04 y x 535 or EXAMPLE 2 Simple ad Compoud Iterest You ivest \$100 i two accouts that each pay a iterest rate of 10% per year. However, oe accout pays simple iterest ad oe accout pays compoud iterest. Make a table that shows the growth of each accout over a 5-year period. Use the compoud iterest formula to verify the result i the table for the compoud iterest case. SOLUTION The simple iterest is the same absolute amout each year. The compoud iterest grows from year to year, because it is paid o the accumulated iterest as well as o the startig balace. Table 4.2 summarizes the calculatios.

18 232 CHAPTER 4 Maagig Your Moey TABLE 4.2 Calculatios for Example 2 SIMPLE INTEREST ACCOUNT COMPOUND INTEREST ACCOUNT Ed of Old balace 1 iterest Old balace 1 iterest year Iterest paid 5 ew balace Iterest paid 5 ew balace 1 10% 3 \$100 5 \$10 \$100 1 \$10 5 \$110 10% 3 \$100 5 \$10 \$100 1 \$10 5 \$ % 3 \$100 5 \$10 \$110 1 \$10 5 \$120 10% 3 \$110 5 \$11 \$110 1 \$11 5 \$ % 3 \$100 5 \$10 \$120 1 \$10 5 \$130 10% 3 \$121 5 \$12.10 \$121 1 \$ \$ % 3 \$100 5 \$10 \$130 1 \$10 5 \$140 10% 3 \$ \$13.31 \$ \$ \$ % 3 \$100 5 \$10 \$140 1 \$10 5 \$150 10% 3 \$ \$14.64 \$ \$ \$ To verify the fial etry i the table with the compoud iterest formula, we use a startig pricipal P 5 \$100 ad a aual iterest rate APR 5 10% with iterest paid for Y 5 5 years. The accumulated balace A is A 5 P 3 A1 1 APRB Y 5 \$100 3 A B 5 5 \$ \$ \$ This result agrees with the oe i the table. Overall, the accout payig compoud iterest builds to \$ while the simple iterest accout reaches oly \$150, eve though both pay at the same 10% rate. This is a sigificat differece, especially whe you cosider that the differece will cotiue to grow with time. Clearly, compoud iterest is much better for a ivestor tha simple iterest at the same rate. Now try Exercises Accumulated value \$15,000 \$10,000 \$5000 \$ Years FIGURE 4.2 The value of the debt i the New College case durig the first 100 years, at a iterest rate of 4% per year. Note that the value rises much more rapidly i later years tha i earlier years a hallmark of expoetial growth. Compoud Iterest as Expoetial Growth The New College case demostrates the remarkable way i which moey ca grow with compoud iterest. Figure 4.2 shows how the value of the New College debt rises durig the first 100 years, assumig a startig value of \$224 ad a iterest rate of 4% per year. Note that while the value rises slowly at first, it rapidly accelerates, so i later years the value grows by much more each year tha it did durig earlier years. This rapid growth is a hallmark of what we geerally call expoetial growth. You ca see how expoetial growth gets its ame by lookig agai at the geeral compoud iterest formula: A 5 P 3 A1 1 APRB Y Because the pricipal P ad the iterest rate APR have fixed values for ay particular compoud iterest calculatio, the growth of the accumulated value A depeds oly o Y (the umber of times iterest has bee paid), which appears i the expoet of the calculatio.

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