Corporate Finance
Lecture Five – 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.
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:
m 
APR 

EAR  1 

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
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.
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.
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;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
5.2 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.
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
FV = $2,000,000;
N = 12*40=480;
I/Y = APR = 12%;
PV = 0;
C/Y = 12
P/Y = 12
PMT = $169.99
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.
5.3 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.
5.3 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; 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.
5.3 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
5.4 Nominal and Real Interest Rates
• The nominal risk-free rate is the rate of interest earned on a riskfree 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)
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?
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)
Total cost of 11 t-shirts = $10.40*11
PV = $100; n=1; I/Y = (FV/PV) -1
= $10.40
= $114.40 = FV
= (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%
5.5 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 T-bond yield
– Due to the increased length of time and the higher default risk on the
corporate bond investment.
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.
5.6 A Brief History of Interest Rates and Inflation in
the United States (continued)
Figure 5.5 Interest rates for the three-month
treasury bill, 1950–1999.
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
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).
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?
Additional Problems with Answers
Problem 1 (Answer)
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
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?
Additional Problems with Answers
Problem 2 (Answer)
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%
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?
Additional Problems with Answers
Problem 3 (Answer)
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
Additional Problems with Answers
Problem 3 (Answer) (continued)
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.
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?
Additional Problems with Answers
Problem 4 (Answer)
Calculate the current monthly payment under the 30-year, 6% terms:
n=360; i/y = 6%/12; PV = 300000; 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 = 300000; 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
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?
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
Table 5.1 Periodic Interest Rates
Table 5.2 $500 CD with 5% APR,
Compounded Quarterly at 1.25%
TABLE 5.3 Abbreviated Monthly Amortization Schedule for
$25,000 Loan, Six Years at 8% Annual Percentage Rate
TABLE 5.4 Advertised Borrowing and Investing Rates at a
Credit Union, January 22, 2012
Table 5.6 Yields on Treasury Bills, Treasury Bonds,
and AAA Corporate
Bonds, 2000–2010
FIGURE 5.1 Interest rate
dimensions.
Figure 5.2 Upward-sloping yield
curve.
Figure 5.3 Downward-sloping yield
curve.