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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 impact of inflation
on nominal rates.
5. Summarize the two major premiums that differentiate
interest rates: the default premium and the maturity
premium.
6. Amaze your family and friends with your knowledge of
interest rate history.
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5.1 How Financial Institutions Quote
Interest Rates: Annual and Periodic
Interest Rates
Most common rate quoted 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 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 

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m 
1
5.1 How Financial Institutions Quote
Interest Rates: Annual and Periodic
Interest Rates (continued)
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 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?
Answer
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
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5.2 Effect of Compounding Periods
on 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.
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5.2 Effect of Compounding Periods on
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 5-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.
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5.2 Effect of Compounding Periods on
Time Value of Money Equations
Example 2 Answer
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 (TI-BAII+: [2nd] [I/Y] Displays P/Y [4] [Enter];
C/Y=4 (TI-BAII+: [2nd] [I/Y] [↓] Display C/Y [4] [Enter];
Or Leave [P/Y], [C/Y] alone, and simply change I = 9/4 = 2.25
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
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5.2 Effect of Compounding Periods on
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.
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5.2 Effect of Compounding Periods on Time
Value of Money Equations
(Example 3—Answer)
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
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FV = $2,000,000;
N = 12*40=480;
I/Y = APR = 12%;
PV = 0;
C/Y = 12
P/Y = 12
PMT = $169.99
Total annual = 169.99 x 12
= 2039.88
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.
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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 onetwelfth 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.
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5.3 Consumer Loans and
Amortization Schedules (continued)
Example 4 Answer
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 (= 1073.64 + 89.47); 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.
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5.3 Consumer Loans and Amortization
Schedules (continued)
Example 4 (Answer) (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
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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
purchasing power caused by inflation.
• 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)
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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 tshirts. 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?
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5.4 Nominal and Real Interest
Rates (continued)
Example 5 (Answer) (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%
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5.5 Risk-Free Rate and
Premiums
• 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 Treasury bond
yield
– Due to the increased length of time and the higher default risk
on the corporate bond investment.
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5.6 A Brief History of Interest Rates
and Inflation in the United States
Figure 5.4 Inflation rates in the United
States, 1950–1999.
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5.6 A Brief History of Interest Rates
and Inflation in the United States
(continued)
Figure 5.5 Interest rates for the threemonth treasury bill, 1950–1999.
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5.6 A Brief History of Interest Rates
and Inflation in the United States
(continued)
Table 5.5 Yields on Treasury Bills, Treasury
Bonds, and AAA Corporate Bonds, 1950–1999
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5.6 A Brief History of Interest Rates
and Inflation in the United States
(continued)
• A fifty year analysis (1950-1999) of the historical
distribution of interest rates on various types of
investments in the USA shows:
• Inflation at 4.05%,
• Real rate at 1.18%,
• Default premium of 0.53% (for AAA-rated over
government bonds) and,
• Maturity premium at 1.28% (for twenty-year
maturity differences).
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Table 5.1 Periodic Interest Rates
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Table 5.2 $500 CD with 5% APR,
Compounded Quarterly at 1.25%
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TABLE 5.3 Abbreviated Monthly
Amortization Schedule for $25,000 Loan, Six
Years at 8% Annual Percentage Rate
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TABLE 5.4 Advertised Borrowing and
Investing Rates at a Credit Union, January
22, 2012
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Table 5.6 Yields on Treasury Bills,
Treasury Bonds, and AAA Corporate
Bonds, 2000–2010
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FIGURE 5.1 Interest rate
dimensions.
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Figure 5.2 Upward-sloping yield
curve.
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Figure 5.3 Downward-sloping
yield curve.
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Additional Problems with Answers
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?
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Additional Problems with Answers
Problem 2
EAR with monthly compounding
If First Federal offers to structure the 9.5%,
$100,000, 1 year loan on a monthly
payment basis, calculate your monthly
payment and the amount of interest paid at
the end of the year. What is your EAR?
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Additional Problems with Answers
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?
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Additional Problems with Answers
Problem 4
Computing payment for 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?
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Additional Problems with Answers
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?
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Additional Problems with Answers
Problem 5 (Answer)
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
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