119
Chapter 5
Interest Rates
LEARNING OBJECTIVES (Slides 5-2 to 5-3)
1. Discuss how financial institutions quote interest rates and compute the effective
annual rate on a loan or investment.
2. Apply the time value of money equation by accounting for the compounding periods
per year.
3. Set up monthly amortization tables for consumer loans and illustrate the payment
changes as the compounding or annuity period changes.
4. Explain the real rate of interest and the effect of inflation on nominal interest rates.
5. Summarize the two major premiums that differentiate interest rates: the default
premium and the maturity premium.
6. Understand the implications of the yield curves.
7. Amaze your family and friends with your knowledge of interest rate history.
IN A NUTSHELL…
Interest rates, which represent the cost of or rent charged on loanable funds, can often be
a cause of concern and confusion for many students, due to the various ways in which
they can be computed, calculated, interpreted, and quoted by financial institutions. In the
six sections within this chapter, the author explains the difference between an annual
percentage rate (APR) and an effective annual rate (EAR); illustrates the proper interest
rate inputs to be used when solving time value of money problems using an equation,
financial calculator, or spreadsheet; emphasizes the impact that frequency of
120 Brooks ◼ Financial Management: Core Concepts, 4e
LECTURE OUTLINE
5.1 How Financial Institutions Quote Interest Rates:
Annual and Periodic Interest Rates (Slides 5-4 to 5-5)
The annual percentage rate (APR) is the most commonly quoted interest rate on most
loans and fixed-income instruments. However, unless interest is charged only once per
year, the APR will understate the true cost of interest to the borrower. Very often, lenders
charge interest more often than once a year, for example, once a month. In such a case,
the APR is divided by the number of compounding periods per year (C/Y or “m”) to
calculate the periodic interest rate. Thus if the APR is 12%, compounded monthly, i.e.,
m = 12, the periodic interest rate is 1%, i.e., 12%/12 = 1%. An increase in the frequency
of compounding leads to a reduction in the periodic interest rate but to an increase in the
Example 1: Calculating EAR or APY
The First Common Bank has advertised one of its loan offerings as follows:
“We will lend you $100,000 for up to three years at an APR of 8.5% (interest
compounded monthly.” If you borrow $100,000 for one year, how much interest will you
have paid and what is the bank’s APY?
Chapter 5 ◼ Interest Rates 121
5.2 Effect of Compounding Periods
on the Time Value of Money Equations (Slides 5-6 to 5-10)
When calculating payments to be made on a loan, the TVM equations require the
periodic rate (r%) and the number of periods (n) to be entered as inputs. Paying off a loan
more frequently than once a year results in the total payments per year being less than the
single annual payment. The reason for this is that as each periodic payment is received, a
smaller amount of the loan remains outstanding, resulting in less interest being charged
overall and per year.
Example 2: Effect of payment frequency on total payment
Jim needs to borrow $50,000 for a business expansion project. His bank agrees to lend him
the money over a five-year term at an APR of 9% and will accept either annual, quarterly,
or monthly payments with no change in the quoted APR. Calculate the periodic payment
under each alternative and compare the total amount paid each year under each option.
Loan amount = $50,000
Loan period = 5 years
APR = 9%
With annual payments, the periodic rate = 9% and the number of periods = 5:
( )
5
$50,000 $50,000
3.88965
1
11 0.09
0.09
−
+
With quarterly payments, the periodic rate = 9%/4 = 2.25% or 0.0225 and n = 20;
( )
20
$50,000 $50,000
15.96371
1
11 0.0225
0.0225
−
+
Thus, the total amount paid per year = 4*$3132.1 = $12,528.41
122 Brooks ◼ Financial Management: Core Concepts, 4e
It is important to remind students that there are multiple ways of entering the TVM inputs
into their calculators, depending on the type of calculator being used. The key thing to
remember is that the interest rate entered should be consistent with the frequency of
compounding and the number of payments involved. For example, if the periodic rate
(APR/m) is entered, the number of periods (i.e., m*n) should be accordingly entered in as
“N,” and the C/Y, and P/Y inputs should be set to 1. On the other hand, if the APR is
Example 3: Comparing annual and monthly deposits
Joshua, who is currently twenty-five years old, wants to invest money into a retirement
fund so as to have $2,000,000 saved up when he retires at age 65. If he can earn 12% per
year in an equity fund, calculate the amount of money he would have to invest in equal
annual amounts and, alternatively, in equal monthly amounts starting at the end of the
current year or month respectively.
With annual deposits: With monthly deposits:
Chapter 5 ◼ Interest Rates 123
5.3 Consumer Loans and
Amortization Schedules (Slides 5-11 to 5-14)
In this section, the author emphasizes the point that interest is charged only on the
outstanding balance of the loan. Therefore, if a borrower increases the frequency and
amount of the payment, he or she can pay off the loan much quicker than the original
term and save on interest expense. The minimum loan payments are calculated based on
the agreed upon terms of the loan, and any additional payments made have to be applied
to the principal balance, thereby reducing the outstanding balance and resulting interest
charged over the next period.
Example 4: Paying off a loan early!
Kay has just taken out a $200,000, thirty-year, 5%, mortgage. She has heard from friends
that if she increases the size of her monthly payment by one-twelfth of the monthly
payment, she will be able to pay off the loan much earlier and save a bundle on interest
costs. She is not convinced. Use the necessary calculations to help convince her that this
is in fact true.
We first solve for the required minimum monthly payment:
Next, we calculate the number of payments required to pay off the loan, if the
With minimum monthly payments:
124 Brooks ◼ Financial Management: Core Concepts, 4e
5.4 Nominal and Real Interest Rates (Slides 5-15 to 5-17)
The rate of interest earned on a risk-free investment such as a bank CD or a treasury
security is essentially compensation paid to the investor for giving up current
consumption and is known as a nominal interest rate. However, inflation can erode the
purchasing power of money, and therefore, the true reward for waiting is often expressed
as the real rate of interest earned and is the rate that remains after inflation has been
adjusted for. The Fisher Effect shown below is the equation that shows the relationship
between the real rate (r*), the inflation rate (h), and the nominal interest rate (r):
(1 + r) = (1 + r*) × (1 + h)
➔ r = (1 + r*) × (1 + h) – 1
➔ r = r* + h + (r* × h)
For example: If the real rate = 3% and the inflation rate = 4%,
the nominal rate = 3% + 4% + (3%*4%) = 7.12%.
Typically, the nominal rate is calculated by adding the real rate and the inflation rate, i.e.,
3% + 4% = 7%, because the cross product of r*and h is very small.
Example 5: Calculating nominal and real interest rates
Jill has $100 and is tempted to buy ten t-shirts, with each one costing $10. However, she
realizes that if she saves the money in a bank account, she should be able to buy eleven
t-shirts. If the cost of the t-shirt increases by the rate of inflation (i.e., 4%), how much
would her nominal and real rates of return have to be?
128 Brooks ◼ Financial Management: Core Concepts, 4e
Problems
1. Periodic interest rates. In the following table, fill in the periodic rates and the
effective annual rates.
Period
APR
Compounding
Per Year
Periodic
Rate
Effective
Annual Rate
Semi-Annual
8%
2
Quarterly
9%
4
Monthly
7.5%
12
Daily
4.25%
365
ANSWER
Period
APR
Compounding
Per Year
Periodic
Rate
Effective
Annual Rate
Semi-Annual
8%
2
4.0%
8.16%
Quarterly
9%
4
2.25%
9.31%
Monthly
7.5%
12
0.625%
7.76%
Daily
4.25%
365
0.01164%
4.34%
Periodic Rate = APR / (C/Y) = 0.08 / 2 = 0.04 = 4.0%
Periodic Rate = APR / (C/Y) = 0.09 / 4 = 0.0225 = 2.25%
Chapter 5 ◼ Interest Rates 133
Cell B1 is the Monthly Payment.
Then for cells B2 through E2 copy the formulas down from the row above.
A
B
C
D
E
1
$18,000.00
$400.40
$180.00
$220.40
$17,779.60
2
$17,779.60
$400.40
$177.80
$222.60
$17,557.00
3
$17,557.00
$400.40
$175.57
$224.83
$17,332.17
…
…
57
$1,562.35
$400.40
$15.62
$384.78
$1,177.57
58
$1,177.57
$400.40
$11.78
$388.62
$788.94
59
$788.94
$400.40
$7.90
$392.50
$396.44
60
$396.44
$400.40
$3.96
$396.44
$0.00
Repeat this for the sixty months.
12. Amortization schedule with periodic payments. Moulton Motors is advertising the
following deal on a used Honda Accord: “Monthly Payments of $245.00 for the next
48 months and this beauty can be yours!” The sticker price of the car is $9,845.00. If
you bought the car, what interest rate would you be paying in both APR and EAR
terms? What is the amortization schedule of these forty-eight payments?
ANSWER
The periodic or monthly interest rate, r, is the solution to the equation.
134 Brooks ◼ Financial Management: Core Concepts, 4e
or.
Amortization Schedule (Can be done effectively on a spreadsheet).
Repeat this for the forty-eight months.
A
B
C
D
E
1
$9,845.00
$245.00
$73.85
$171.15
$9,673.85
2
$9,673.85
$245.00
$72.57
$172.43
$9,501.41
3
$9,501.41
$245.00
$71.27
$173.73
$9,327.68
…
…
45
$961.89
$245.00
$7.22
$237.78
$724.11
46
$724.11
$245.00
$5.43
$239.57
$484.54
47
$484.54
$245.00
$3.63
$241.37
$243.18
48
$243.18
$245.00
$1.82
$243.18
$0.00