88 CHAPTER 2 Time Value of Moneyof 10 percent with a 360-day year. How much will be in your account after nine
months? The answer is $107.79:^13
Periodic rate iPER0.10/360 0.00027778 per day.
Number of days 0.75(360) 270.
Ending amount $100(1.00027778)^270 $107.79.
Now suppose you borrow $100 from a bank that charges 10 percent per year “sim-
ple interest,” which means annual rather than daily compounding, but you borrow the
$100 for only 270 days. How much interest would you have to pay for the use of $100
for 270 days? Here we would calculate a daily interest rate, iPER, as above, but multi-
ply by 270 rather than use it as an exponent:
Interest owed $100(0.00027778)(270) $7.50 interest charged.
You would owe the bank a total of $107.50 after 270 days. This is the procedure most
banks actually use to calculate interest on loans, except that they generally require you
to pay the interest on a monthly basis rather than after 270 days.
Finally, let’s consider a somewhat different situation. Say an Internet access firm had
100 customers at the end of 2002, and its customer base is expected to grow steadily at
the rate of 10 percent per year. What is the estimated customer base nine months into
the new year? This problem would be set up exactly like the bank account with daily
compounding, and the estimate would be 107.79 customers, rounded to 108.
Themostimportantthinginproblemslikethese,asinalltimevalueproblems,isto
becareful!Thinkaboutwhatisinvolvedinalogical,systematicmanner,drawatimeline
ifitwouldhelpyouvisualizethesituation,andthenapplytheappropriateequations.Amortized Loans
One of the most important applications of compound interest involves loans that are
paid off in installments over time. Included are automobile loans, home mortgage
loans, student loans, and most business loans other than very short-term loans and
long-term bonds. If a loan is to be repaid in equal periodic amounts (monthly, quar-
terly, or annually), it is said to be an amortized loan.^14
Table 2-2 illustrates the amortization process. A firm borrows $1,000, and the loan
is to be repaid in three equal payments at the end of each of the next three years. (In
this case, there is only one payment per year, so years periods and the stated rate
periodic rate.) The lender charges a 6 percent interest rate on the loan balance that is
outstanding at the beginning of each year. The first task is to determine the amount
the firm must repay each year, or the constant annual payment. To find this amount,
recognize that the $1,000 represents the present value of an annuity of PMT dollars
per year for three years, discounted at 6 percent:(^13) Here we assumed a 360-day year, and we also assumed that the nine months all have 30 days. This con-
vention is often used. However, some contracts specify that actual days be used. Computers (and many fi-
nancial calculators) have a built-in calendar, and if you input the beginning and ending dates, the computer
or calculator would tell you the exact number of days, taking account of 30-day months, 31-day months, and
28- or 29-day months.
(^14) The word amortizedcomes from the Latin mors,meaning “death,” so an amortized loan is one that is
“killed off” over time.