### Slide 1

```Understanding Interest Rates
Chapter 4
1
Chapter Definitions
Present Value
 Yield to maturity
 Simple Loan
 Fixed payment loan
 Coupon bond
 Discount bond
 Rate of return and interest rate risk
 Real vs nominal interest rates

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Yield to Maturity
Yield to maturity is the most accurate
measure of interest rates.
 Sometimes it is called internal rate of
return.
 It is the interest rate that relates all the
future returns to the present value of a
debt/investment.

3
Future Value


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Suppose you put a \$1000 in the bank at 5%
interest rate. What would be the future value in
three years?
1000 + 1000*(0.05) = 1000 (1.05) at the end of the
first year: \$1050.
1000 (1.05) + 1000 (1.05)(0.05) = 1000
(1.05)(1+.05) = 1000(1.05)(1.05) at the end of the
second year: \$1102.50.
1000(1.05)(1.05) + 1000(1.05)(1.05)(0.05) =
1000(1.05)(1.05)(1+.05)= 1000(1.05)(1.05)(1.05)
at the end of the third year: \$1157.625.
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Present Value
If you expect to be paid \$1157.625 three
years from today, the present value of
this future payment is \$1000 if the funds
earn 5% interest.
 Would the present value of this future
payment be greater or less than \$1000
if the funds were to earn 10% interest?
 FV=1000(1+i)3
 PV=1000/(1+i)3

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Present Value

If you only want to earn 5% interest for
the next two years on the \$900 you have,
and someone offers you \$1102.50 in two
years, would you take the offer?
PV
r
n
FV
900
0.05
2
992.25
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Debt Types

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Simple loan: you borrow today and pay the
principal plus the interest in the future.
Fixed payment loan: you borrow today but
make equal payments per time period until
Coupon bond: firm sells the bond (borrows
the funds) pays interest per period and at the
maturity date pays the principal, too.
Discount bond: firm sells the bond at less
than face value pays the face value at the
maturity date.
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Yield to Maturity: Simple Loan

If you borrow \$1000 today and asked to
pay \$1100 a year from now, what is the
interest rate?
1000 = 1100/(1+i)
1+i = 1100/1000
i = 1100/1000 - 1
i = 1.1 - 1
i = .1 = 10%
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Yield to Maturity: Fixed
Payment Loan
Suppose you got a \$50,000 mortgage. The
bank wants you to pay \$550.60 per month
for 20 years. What is the interest rate you
are charged?
50,000 = [550.60/(1+i)] +[550.60/(1+i)2] +
[550.60/(1+i)3] + … + [550.60/(1+i)240]
i = .01 or 1% per month or 12% per year.
 If the interest rate were lower than 12%,
would your monthly payments increase or
decrease?

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Yield to Maturity: Coupon Bond
You buy a ten-year corporate bond with a
face value of \$1000, coupon rate of 10%
for \$885.30. What is the interest rate you
are earning?
885.30 = [100/(1+i)] + [100/(1+i)2 + …
+1100/(1+i)10
i = 0.12 or 12%.
 What is the interest rate you are earning if
you bought it for \$1000?

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Yield to Maturity: Consol
You bought a consol for \$1250. It pays
\$100 per year. What is the interest rate
you are getting?
i = 100/1250
i = 0.08 or 8%.
 What if you bought the consol for \$1000;
what would be the interest rate?

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Yield to Maturity: Discount Bond
You bought a discount bond for \$800 which
will pay \$1000 a year from now. What is
the interest rate?
i = (1000 - 800)/800
i = 200/800
i = 0.25 or 25%.
 Would the interest rate be higher or lower
than 25% if you had bought the bond for
\$900?

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Other Measures of Interest Rates

Current Yield: calculating the interest rate
on a coupon bond as if it were a consol.


Current yield approaches yield to maturity when
the price of the bond is close to the face value
and the maturity date is far away.
Yield on a discount basis: the price of
Treasury Bills is quoted in this fashion. It is
always less than the yield to maturity.
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An Example of Treasury Bill

A T-Bill that has 91 days to maturity
sells for 5.4% discount. Find the price
and yield to maturity.
.054 (91/360) = [10,000 - P]/10,000
P = \$9863.50
i = [(10,000 - 9863.50)/9863.50][360/91]
i = 0.0547 or 5.47%.
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Debt Types
SIMPLE LOAN:
Principal Interest Rate # of prd End Payment Check
1000
0.0800
5
\$1,469.33
FIXED PAYMENT LOAN:
Principal Interest Rate # of prd Payment
Check
9700
0.00333
60
(\$178.62)
DISCOUNT BOND < 1 year
Price
Face Value Days Interest Rate Check
995
1000
91
0.0202
DISCOUNT BOND > 1 year
Price
Face Value Months Interest Rate Check
950
1000
18
0.0348
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Interest Rate vs. Rate of Return
Rate of return on a bond includes the
capital gains/losses.
 Capital gain is the increase in the price
of the bond. Capital loss is the
decrease in the price of the bond.
 If you bought a bond for \$1000, held it
for a year, received \$100 interest
payment and sold it for \$1100, the
return is
R = [100 + 1100]/1000 = .20 or 20%.

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Why Bonds Are Risky
If you hold the bond until maturity, return
and interest rate will be the same.
 If interest rates rise while you own a bond,
the price of the bond falls yielding capital
losses.
 The farther the maturity date the larger is
the capital loss (or capital gain when
interest rates fall).
 Prices and returns for long-term bonds are
more volatile than those for shorter term
bonds.

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Real and Nominal Interest Rates
If people expect inflation, they will not
lend money at low rates. They would
want to preserve the value of their
money by including the inflation rate in
the nominal interest rate.
 Suppose you lend \$1. You want to
receive a real interest rate of r and want
to compensate the erosion of the value
of the money by the expected inflation
rate, p.

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Real and Nominal Interest
Rates
1+i = (1+r)(1+p)
= 1 + r + p + rp
i = r + p + rp
 Nominal interest rate is equal to real
interest rate plus the expected inflation
plus the product of the two.
This is called the Fisher equation.
 For low values of real interest rates and
expected inflation the last term can be
ignored.

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