### Chapter 5

```Chapter 5
Interest Rates
Learning Objectives
1. Discuss how interest rates are quoted, and compute the effective
annual rate (EAR) on a loan or investment.
2. Apply the TVM equations 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:
6. Amaze your family and friends with your knowledge of interest rate
history.
5-2
5.1 How Interest Rates Are Quoted:
Annual and Periodic Interest Rates
The most commonly quoted rate is the annual percentage rate (APR).
It is the annual rate based on interest being computed once a year.
Lenders often charge interest on a non-annual basis.
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.
For example: APR = 12%; m=12; i%=12%/12= 1%
The EAR is the true rate of return to the lender and the true cost of
borrowing to the borrower.
An EAR, also known as the annual percentage yield (APY) on an
investment, is calculated from a given APR and frequency of
compounding (m) by using the following equation:
APR 

EAR   1 


m 
m
1
5-3
5.1 How Interest Rates Are Quoted:
Annual and Periodic Interest Rates
(continued)
Example 1: Calculating an 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 3 years at an APR of 8.5%
(interest compounded monthly).” If you borrow \$100,000 for 1
year, how much interest will you have paid and what is the bank’s
APY?
Nominal annual rate = APR = 8.5%
Frequency of compounding = C/Y = m = 12
Periodic interest rate = APR/m = 8.5%/12 = 0.70833% =
.0070833
APY or EAR = (1.0070833)12 - 1 = 1.08839 - 1 = 8.839%
Total interest paid after 1 year = .08839*\$100,000 = \$8,839.05
5-4
5.2 Effect of Compounding Periods
on the Time Value of Money
Equations
TVM equations require the periodic rate (r%) and the
number of periods (n) to be entered as inputs.
The greater the frequency of payments made per year, the
lower the total amount paid.
More money goes to principal and less interest is
charged.
The interest rate entered should be consistent with the
frequency of compounding and the number of payments
involved.
5-5
5.2 Effect of Compounding Periods
on the Time Value of Money
Equations
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 5year term at an APR of 9% and will accept 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.
5-6
5.2 Effect of Compounding Periods
on the Time Value of Money
Loan amount = \$50,000
Loan period = 5 years
APR = 9%
Annual payments: PV = 50000;n=5;i = 9; FV=0;
P/Y=1;C/Y=1; CPT PMT = \$12,854.62
Quarterly payments: PV = 50000;n=20;i = 9; FV=0;
P/Y=4;C/Y=4; CPT PMT = \$3132.10
Total annual payment = \$3132.1*4 = \$12,528.41
Monthly payments: PV = 50000;n=60;i = 9; FV=0;
P/Y=12;C/Y=12; CPT PMT = \$1037.92
Total annual payment = \$1037.92*12 = \$12,455.04
5-7
5.2 Effect of Compounding Periods
on the Time Value of Money
Equations
Example 3: Comparing Annual and Monthly Deposits
Joshua, who is currently 25 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.
5-8
5.2 Effect of Compounding Periods
on the Time Value of Money
With annual deposits:
With monthly
deposits:
(Using the APR as the interest rate)
FV = \$2,000,000;
N = 40 years;
I/Y = APR = 12%;
PV = 0;
C/Y=1;
P/Y=1;
PMT = \$2,607.25
FV = \$2,000,000;
N = 12*40=480;
I/Y = APR = 12%;
PV = 0;
C/Y = 12
P/Y = 12
PMT = \$169.99
5-9
5.3 Consumer Loans and
Amortization Schedules
Interest is charged only on the outstanding
balance of a typical consumer loan.
Increases in frequency and size of payments
result in reduced interest charges and quicker
payoff due to more being applied to loan balance.
Amortization schedules help in planning and
analysis of consumer loans.
5-10
5.3 Consumer Loans and
Amortization Schedules (continued)
Example 4: Paying Off a Loan Early!
Kay has just taken out a \$200,000, 30-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.
5-11
5.3 Consumer Loans and
Amortization Schedules (continued)
We first solve for the required minimum monthly
payment:
PV = \$200,000; I/Y=5; N=30*12=360; FV=0; C/Y=12;
P/Y=12; PMT = ?\$1073.64
Next, we calculate the number of payments required
to pay off the loan, if the monthly payment is
increased by 1/12*\$1073.64 i.e. by \$89.47
PMT = 1163.11; PV=\$200,000; FV=0; I/Y=5; C/Y=12;
P/Y=12; N = ?N= 303.13 months, or 303.13/12 =
25.26 years.
5-12
5.3 Consumer Loans and Amortization
Schedules (continued)
With minimum monthly payments:
Total paid = 360*\$1073.64 = \$386, 510.4
Amount borrowed
= \$200,000.0
Interest paid
= \$186,510.4
With higher monthly payments:
Total paid = 303.13*\$1163.11 = \$353,573.53
Amount borrowed
= \$200,000.00
Interest paid
= \$153,573.53
Interest saved=\$186,510.4-\$153,573.53 = \$32,936.87
5-13
5.4 Nominal and Real Interest Rates
• The nominal risk-free rate is the rate of interest earned
on a risk-free investment such as a bank CD or a
treasury security.
• It is essentially a compensation paid for the giving up of
current consumption by the investor.
• The real rate of interest adjusts for the erosion of
• 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*) x (1 + h)
 r = (1 + r*) x (1 + h) – 1
 r = r* + h + (r* x h)
5-14
5.4 Nominal and Real Interest
Rates (continued)
Example 5: Calculating Nominal and Real
Interest Rates
Jill has \$100 and is tempted to buy 10
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 11
t-shirts. If the cost of the t-shirt increases by
the rate of inflation, i.e., by 4%, how much
would her nominal and real rates of return
have to be?
5-15
5.4 Nominal and Real Interest Rates
(continued)
Real rate of return = (FV/PV)1/n -1
= (11shirts/10shirts)1/1-1
= 10%
Price of t-shirt next year = \$10(1.04) = \$10.40
Total cost of 11 t-shirts = \$10.40*11 = \$114.40 = FV
PV = \$100; n=1; I/Y = (FV/PV) -1
= (114.4/100)-1
= 14.4%
Nominal rate of return = 14.4%
= Real rate + Inflation rate + (real rate*inflation rate)
= 10% + 4% + (10%*4%) = 14.4%
5-16
• The nominal risk-free rate of interest such as the
rate of return on a Treasury bill includes the real
rate of interest and the inflation premium.
• The rate of return on all other riskier investments
(r) would have to include a default risk premium
(dp)and a maturity risk premium (mp): i.e.
r = r* + inf + dp + mp.
• 30-year corporate bond yield > 30-year T-bond
yield
– Due to the increased length of time and the higher
default risk on the corporate bond investment.
5-17
5.6 A Brief History of Interest Rates
FIGURE 5.2 Inflation Rates in the United States, 1950–1999
5-18
5.6 A Brief History of Interest Rates
(continued)
FIGURE 5.3 Interest Rates for the Three-Month Treasury Bill, 1950–
1999
5-19
5.6 A Brief History of Interest Rates
(continued)
TABLE 5.5 Yields on Treasury Bills, Treasury Bonds, and AAA
Corporate Bonds, 1953–1999
5-20
5.6 A Brief History of Interest Rates
(continued)
• A fifty year analysis (1950-1999) of the historical
distribution of interest rates on various types of investments
in the United States shows:
• Inflation at 4.05%
• Real rate at 1.18%
• Default premium of 0.49% (for AAA-rated over government
bonds)
• Maturity premium at 1.28% (for twenty-year maturity
differences)
5-21
Problem 1
Calculating APY or EAR.
The First Federal Bank has advertised one of
its loan offerings as follows:
“We will lend you \$100,000 for up to 5 years
at an APR of 9.5% (interest compounded
monthly.)”
If you borrow \$100,000 for 1 year and pay it
off in one lump sum at the end of the year,
how much interest will you have paid and
what is the bank’s APY?
5-22
Nominal annual rate = APR = 9.5%
Frequency of compounding = C/Y = m = 12
Periodic interest rate = APR/m = 9.5%/12 = 0.79167% = .0079167
APR 

EAR   1


m 
m
1
APY or EAR = (1.0079167)12 - 1 = 1.099247 - 1 9.92%
Payment at the end of the year = 1.099247*100,000
 \$109,924.70
Amount of interest paid = \$109, 924.7 - \$100,000
 \$9,924.7
5-23
Problem 2
EAR with Monthly Compounding
If First Federal offers to structure the
9.5%, \$100,000, 1 year loan on a monthly
payment and the amount of interest paid
at the end of the year. What is your EAR?
5-24
Calculate monthly payment:
N i/y
PV
PMT FV
12 9.5/12 100,000 -8,768.35 0
Total interest paid after 1 year = 12*\$8,768.35 - \$100,000
= \$105,220.20 -\$100,000
= \$5,220.20
EAR is still 9.92%, since the APR and m are the same as #1
above,
APY or EAR = (1.0079167)12 - 1 = 1.099247 - 1 =9.92%
5-25
Problem 3
Monthly versus Quarterly Payments:
Patrick needs to borrow \$70,000 to start a business
expansion project. His bank agrees to lend him the
money over a 5-year term at an APR of 9.25% and
will accept either monthly or quarterly 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.
Which payment term should Patrick accept and why?
5-26
Calculate monthly payment:
n=60; i/y = 9.25%/12; PV = 70000; FV=0; PMT= -1,461.59
Calculate quarterly payment:
n=20; i/y = 9.25%/4; PV = 70000; FV=0; PMT= -4,411.15
Total amount paid per year under each payment type:
With monthly payments = 12* \$1,461.59 = \$17,539.08
With quarterly payments = 4*\$4,411.15 = \$17,644.60
5-27
Total interest paid under monthly compounding:
Total paid - Amount borrowed
= 60*\$1,461.59 - \$70,000
= \$87,695.4 - \$70,000
= \$17,695.4
Total interest paid under quarterly compounding:
 20 *\$4,411.15 -\$70,000
= \$88,223 - \$70,000
= \$18,223
Since less interest is paid over the 5 years with the
monthly payment terms, Patrick should accept monthly
rather than quarterly payment terms.
5-28
Problem 4
Computing Payment for an Early
Payoff:
You have just taken on a 30-year, 6%,
\$300,000 mortgage and would like to pay
it off in 20 years. By how much will your
monthly payment have to change to
accomplish this objective?
5-29
Calculate the current monthly payment under the
30-year, 6% terms:
n=360; i/y = 6%/12; PV = 300,000; FV=0;
CPT PMT1,798.65
Next, calculate the payment required to pay off the
loan in 15 years or 180 payments:
n=180; i/y = 6%/12; PV = 300,000; FV=0;
CPT PMT2,531.57
The increase in monthly payment required to pay off the loan
in
20 years = \$2,531.57 - \$1,798.65 = \$732.92
5-30
Problem 5
You just turned 30 and decide that you
would like to save up enough money so as
to be able to withdraw \$75,000 per year
for 20 years after you retire at age 65,
with the first withdrawal starting on your
66th birthday. How much money will you
have to deposit each month into an
account earning 8% per year (interest
compounded monthly), starting one month
from today, to accomplish this goal?
5-31
Calculate the amount of money needed to be
accumulated at age 65 to provide an annuity of
\$75,000 for 20 years with the account earning 8%
per year (interest compounded monthly)
n=20; i/y = 8%; FV=0; PMT=75,000; P/Y = 1;
C/Y=12
CPT PV720,210.86
Next, calculate the monthly deposit necessary to
accumulate a FV of \$720,210.86 over 35 years or
12*35 = 420 months:
n=420; i/y = 8%; FV=720,210.86; P/Y = 12; C/Y=12
CPT PMT313.97
5-32
TABLE 5.1 Periodic Interest Rates
5-33
TABLE 5.2 \$500 CD with 5% APR,
Compounded Quarterly at 1.25%
5-34
TABLE 5.3 Abbreviated Monthly
Amortization Schedule for \$25,000 Loan,
Six Years at 8% Annual Percentage Rate