Interest Rates
Chapter 5.1-5.3
Outline
• Interest Rate Quotes and Adjustments
– The Effective Annual Rate (EAR) and the Annual
Percentage Rate (APR)
• The determinants of Interest Rates
– Real versus Nominal Interest Rates
– The Yield Curve
• Further questions
Interest Rates Quotes and
Adjustments
The Effective Annual Rate (EAR)
• The effective annual rate reflects the dollar
interest received from investing $1 for 1 year.
This is the rate we have been considering up
till now.
• This EAR is often quoted as “Annual
Percentage Rate” or “APR” when payments
are made throughout the year such as
mortgages, car loans and other investments
Effective Annual Rate
Interest rates for periods smaller than one year:
• Given a certain EAR we can calculate the interest
accumulated for a shorter period than one year.
Example: Receiving 5% for one year is equivalent to
receiving 2.47% every six months
(1+ r) = (1.05) =1.0247
0.5
0.5
At the end of the first six months we have $1.0247.
If this amount is reinvested for the rest of the year,
then $1.05 is accumulated in total
$1.0247´ (1.0247) = $1.05
Effective Annual Rate
Annual Percentage Rate (APR)
The annual percentage rate (APR) indicates the
amount of simple interest earned in one year, that
is the amount without the effect of compounding –
it does not fully take into account the time value of
money
• The APR is quoted for a given number of
compounding periods
• The compounding period can be “annual”,
“semiannual”, “quarterly”, “monthly”, or even
“daily”
Converting APR to EAR
The interest received per compounding period
for an APR with k compounding periods is:
APR
Interest Rate per Compounding Period =
k
Converting APR to EAR
k
æ APR ö
1+ EAR = ç1+
÷
è
k ø
Converting APR to EAR
Annual Percentage Rate (APR)
Annual Percentage Rate (APR)
Application: Amortizing Loans
In amortizing loans each periodic payment is the
same which means that each period you pay interest
on the loan and some part of the loan balance.
Application:
Amortizing
Loans
The Determinants of Interest Rates
Nominal and Real Interest Rates
Nominal interest rate: is the rate we have been
using till now for discounting future cash flows and
it indicates the rate at which your money will grow
if invested for a certain period
Real interest rate: is the rate of growth of your
purchasing power, after adjusting for inflation.
1+ r
rR =
-1 » r - i
1+ i
rR is the real interest rate
i is the rate of inflation
Interest Rates and Inflation
1+ r
rR =
-1 » r - i
1+ i
The Yield Curve of Interest Rates
Interest rates change depending on the horizon
of the investment:
• Banks often offer different rates on loans
depending on their term
The Yield Curve: depicts the Term Structure of
interest rates or the relation between the
investment term and the interest rate
The Term Structure of Risk-Free U.S.
Interest Rates
Present Value Calculation
We can calculate the present value of a stream
of cash flows using the term structure of interest
rates
C1
C2
CN
PV =
+
+...
2
N
1+ r1 (1+ r2 )
(1+ rN )
Present Value Calculation
C1
C2
CN
PV =
+
+...
2
N
1+ r1 (1+ r2 )
(1+ rN )
Further questions
College
Question 10 (2nd Edition): Your son has been accepted into
college. This college guarantees that your son’s tuition will not
increase for the four years he attends college. The first
$10,000 tuition payment is due in six months. After that, the
same payment is due every six months until you have made a
total of eight payments. The college offers a bank account that
allows you to withdraw money every six months and has a
fixed APR of 4% (semiannual) guaranteed to remain the same
over the next four years.
How much money must you deposit today if you intend to
make no further deposits and would like to make all the
tuition payments from this account, leaving the account
empty when the last payment is made?
College
Eight payments of $10,000 due semiannually and starting
six months from now. The six month interest rate is
4%
rsix month =
= 2%
2
The required deposit to cover all future tuition
expenses
$10, 000 æ
1 ö
PV =
= $73, 254.8
ç18÷
2% è 1.02 ø
Car Dealer Loan
Question 22 (2nd Edition): You need a new car and the dealer
has offered you a price of $20,000, with the following
payment options: (a) pay cash and receive a $2000 rebate, of
(b) pay a $5000 down payment and finance the rest with a 0%
APR loan over 30 months. But having just quit your job and
started an MBA program, you are in debt and you expect to be
in debt for at least the next 2.5 years. You plan to use credit
cards to pay your expenses; luckily you have one with a low
(fixed) rate of 15% APR (monthly).
Which payment option is best for you?
Car Dealer Loan
The car dealer is offering tow alternatives:
(a) A loan at 0% APR compounded monthly – or in other
words: an initial payment of $5,000 and 30 monthly
payments of $500 thereafter.
(b) $2,000 rebate.
The cost of the car under dealer financing is:
15%
rsix month =
=1.25%
12
ö
$500 æ
1
PV = $5, 000 +
= $17, 444.5
ç130 ÷
1.25% è 1.0125 ø