Introduction to Finance

• Institutions and Markets 
Depository institutions off er savings 
accounts and certifi cates of deposit (CDs) to individual savers. To 
entice individuals to save with them, these fi nancial institutions 
often state annual percentage rates but compound the interest more 
frequently than once a year. The result is that the eff ective annual 
rate (EAR) is higher than the stated annual percentage rate (APR). 
Of course, since most fi nancial institutions depend on the spread 
between their cost of obtaining funds and their lending rates, they 
must balance the eff ective annual rates at which they borrow and 
lend. Credit card loans typically provide for monthly compounding 
so that the EAR is higher than the APR.
• Investments 
Most fi nancial decisions are based on the rate at 
which an investment is compounded or a future value is discounted. 
Savers are interested in growing, or compounding, their savings over 
time and know that the longer an investment can compound the more 
rapidly it will grow in value at a specifi ed interest rate. Investors make 
plans, based on the compound rates of return they expect to earn on 
their investments, about when they can buy a home, when they can 
send their children to college, and when they can retire. The ability 
to compound interest more frequently than once a year means that 
investors can reach their goals sooner, or at lower interest rates.
• Financial Management
Financial managers borrow from banks 
and issue debt to raise funds to maintain and grow their fi rms. While 
some business loans are simple interest loans, others take the form 
of fully amortized loans, whereby annuity payments are composed 
of a declining interest portion and a rising principal repayment por-
tion over the life of the loan. Investors, of course, expect to earn 
compound rates of return on their debt and equity investments held 
for more than a year. Financial managers accordingly must invest 
funds in capital projects that will generate excess cash fl ows suf-
fi cient in amount to provide investors with their expected rates of 
return.
Summary
LO 9.1 
The time value of money is the math of fi nance whereby a 
fi nancial return is earned over time by saving or investing money. 
Simple interest is interest earned only on an investment’s principal.
LO 9.2 
Compounding is an arithmetic process whereby an initial 
value increases or grows at a compound interest rate over time to reach 
a value in the future. Compound interest involves earning interest 
on interest in addition to interest on the principal, or initial invest-
ment. Four approaches, or methods, are available for fi nding a future 
value when compounding is involved. They include an equation-
based solution, a fi nancial calculator solution, a spreadsheet solution, 
and a table-based solution.
LO 9.3 
Discounting to determine present values is an arithmetic pro-
cess whereby a future value decreases at a compound interest rate over 
time to reach a present value. Solving an equation, using a fi nancial 
calculator, developing a spreadsheet program, and using pre-calculated 
tables are methods for fi nding present values.
LO 9.4 
Four variables—present value, future value, interest rate, 
and number of periods—are included in the future value and present 
value equations. As long as the values for any three of these variables 
are known, we can solve for the fourth, or unknown, variable. For 
example, if the present value and future value amounts are known 
along with the number of periods, the interest rate can be found. 
Alternatively, if we knew the interest rate in addition to the present 
and future values, we could solve for the number of periods.
LO 9.5 
An annuity is a series of equal payments (or receipts) that occur 
over a number of time periods. An ordinary annuity is when equal pay-
ments (or receipts) occur at the end of each time period. Necessary 
inputs include an interest rate, the number of periods, and a periodic 
payments variable. We then can solve for the future value of the ordinary 
annuity using one or more of the previously described solution methods. 
LO 9.6 
The present value of an ordinary annuity can be calculated 
using equations, fi nancial calculators, spreadsheets, or tables. The 

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Exercises
245
necessary inputs are the amount of the periodic payments, an interest 
rate, and the number of periods. We then can solve for the present 
value of the ordinary annuity.
LO 9.7 
The interest rate for an ordinary annuity can be found by enter-
ing the future value, the periodic payments amount, and the number 
of periods, and then solving for the interest rate. The number of time 
periods can be found in the same way, except the interest rate is one of 
the inputs and the number of periods is the unknown variable. 
LO 9.8 
The periodic payments amount for ordinary annuities can be 
determined using the same approach as previously described for fi nd-
ing the interest rate or number of periods. Amortized loans (such as 
real estate mortgage loans) are examples of ordinary annuities. After 
deciding the amount of the loan (present value), the interest rate on 
the loan, and the life of the loan, we can calculate the amount of each 
periodic payment necessary to pay off the loan in full at maturity. 
LO 9.9 
When more frequent than annual compounding (or discount-
ing) occurs, the stated annual interest rate must be divided by the 
number of compounding periods within one year to get the rate per 
period, and the total number of periods calculated by multiplying the 
number of years by the number of compounding periods within a 
year. Then, future values (or present values) can be found using the 
same process employed when compounding occurred annually. The 
Truth in Lending law requires that the cost of consumer credit be 
stated as an annual percentage rate. However, when payments are 
made more frequently than annually, the eff ective annual rate meas-
ures the compounding impact on the cost of borrowing.
Key Terms
amortized loan
annual percentage rate (APR)
annuity
annuity due
compounding
compound interest
discounting
eff ective annual rate (EAR)
future value
loan amortization schedule
ordinary annuity
present value
Rule of 72
simple interest
time value of money
usury
Review Questions