Understanding Interest Rates

Four Types of Credit Market

Instruments

Simple loan

Discount bond

Coupon bond

Fixed-payment loan

Time Lines for Credit Market

Instruments

Measuring Interest Rates

Present Value:

A dollar paid to you one year from now is less valuable than a dollar paid to you today

Why?

A dollar deposited today can earn interest and become

$1 x (1+i) one year from today.

Let i = .10

In one year $100 X (1+ 0.10) = $110

In two years $110 X (1 + 0.10) = $121 or 100 X (1 + 0.10)

2

In three years $121 X (1 + 0.10) = $133 or 100 X (1 + 0.10)3

In n years

$100 X (1 + i ) n

Discounting the future

What is the value today of a future return?

PV of $133 received in 3 years is $133/(1+0.10)3

Generally:

Time Line

One cannot directly compare payments scheduled in different points in the time line

Year

PV

$100

$100

$100

$100

0

1

2

n

100

100/(1+i)

100/(1+i)2

100/(1+i)n

$110

$121

Yield to Maturity

We know the price of a debt instrument and the future payment schedule

The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today.

Yield to Maturity

Question: did I make a sound investment?

If I pay a price P today for a set of future payments, what is the interest rate at which I could invest P and get the same set of future payments?

Simple Loan

PV = amount borrowed = $100

CF = cash flow in one year = $110 n = number of years = 1

$110

$100 =

(1 + i )1

(1 + i ) $100 = $110

$110

(1 + i ) =

$100

i = 0.10 = 10%

For simple loans, the simple interest rate equals the yield to maturity

Fixed Payment Loan

The same cash flow payment every period throughout the life of the loan

LV = loan value

FP = fixed yearly payment n = number of years until maturity

FP

FP

FP

FP

LV =

...+

2

3

1 + i (1 + i ) (1 + i)

(1 + i) n

Coupon Bond

Using the same strategy used for the fixed-payment loan:

P = price of coupon bond

C = yearly coupon payment

F = face value of the bond n = years to maturity date

C

C

C

C

F

P=

. . . +

2

3

n

1+i (1+i ) (1+i )

(1+i) (1+i ) n

Relationship Between Price and Yield to

Maturity

• When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate

• The price of a coupon bond and the yield to maturity are negatively related

• The yield to maturity is greater than the coupon rate when the bond price is below its face value

Consol or Perpetuity

• A bond with no maturity date that does not repay principal but pays fixed coupon payments forever.

i =

C

-------P

• For coupon bonds, this equation gives the current yield, and easy way to calculate approximation to the yield to maturity. Discount Bond (P = $900, F=

$1000)

Yield on a Discount Basis idb =

(F – P)

F

x

360

(number of days to maturity)

One year bill, P = $900, F = $1000

idb =

$1000 – $900

$1000

x

360

365

=0.099 = 9.9%

Two Characteristics

1.…