Financial Markets and Institutions (2-downloads)

Price of Bond ($)

Yield to Maturity (%)

1,200

7.13

1,100

8.48

1,000

10.00

900

11.75

800

13.81

www.teachmefinance.com

Access a review of the key

financial concepts: time

value of money, annuities,

perpetuities, and so on.

G O   O N L I N E

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Chapter 3 What Do Interest Rates Mean and What Is Their Role in Valuation?

45

that makes fixed coupon payments of $C forever. The formula in Equation 3 for the

price of a perpetuity, P

c

, simplifies to the following:

3

(4)

where

P

c

= price of the perpetuity (consol)

C = yearly payment

i

c

= yield to maturity of the perpetuity (consol)

One nice feature of perpetuities is that you can immediately see that as i

c

goes

up, the price of the bond falls. For example, if a perpetuity pays $100 per year for-

ever and the interest rate is 10%, its price will be $1000 = $100/0.10. If the interest

rate rises to 20%, its price will fall to $500 = $100/0.20. We can also rewrite this for-

mula as

(5)

i

c

⫽

C

P

c

P

c

⫽

C

i

c

3

The bond price formula for a perpetuity is

which can be written as

in which 

. From your high school algebra you might remember the formula for an infi-

nite sum:

for x < 1

and so

which by suitable algebraic manipulation becomes

P

c

⫽ C ¢

1

⫹ i

c

i

c

⫺

i

c

i

c

≤ ⫽

C

i

c

P

c

⫽ C ¢

1

1

⫺ x

⫺ 1≤ ⫽ C B

1

1

⫺ 1>11 ⫹ i

c

2

⫺ 1R

1

⫹ x ⫹ x

2

⫹ x

3

⫹ p ⫽

1

1

⫺ x

x

⫽ 1/11 ⫹ i2

P

c

⫽ C1x ⫹ x

2

⫹ x

3

⫹ p 2

P

c

⫽

C

1

⫹ i

c

⫹

C

11 ⫹ i

c

2

2

⫹

C

11 ⫹ i

c

2

3

⫹ p

What is the yield to maturity on a bond that has a price of $2,000 and pays $100 annu-

ally forever?

Solution

The yield to maturity would be 5%.

i

c

⫽

C

P

c

E X A M P L E   3 . 5 Perpetuity

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46

Part 2 Fundamentals of Financial Markets

The formula in Equation 5, which describes the calculation of the yield to matu-

rity for a perpetuity, also provides a useful approximation for the yield to maturity

on coupon bonds. When a coupon bond has a long term to maturity (say, 20 years

or more), it is very much like a perpetuity, which pays coupon payments forever.

This is because the cash flows more than 20 years in the future have such small 

present discounted values that the value of a long-term coupon bond is very close

to the value of a perpetuity with the same coupon rate. Thus, i

c

in Equation 5 will

be very close to the yield to maturity for any long-term bond. For this reason, i

c

,

the yearly coupon payment divided by the price of the security, has been given the

name current yield and is frequently used as an approximation to describe inter-

est rates on long-term bonds.

Discount Bond

The yield-to-maturity calculation for a discount bond is similar to

that for the simple loan. Let us consider a discount bond such as a one-year U.S.

Treasury bill, which pays a face value of $1,000 in one year’s time. If the current

purchase price of this bill is $900, then equating this price to the present value of

the $1,000 received in one year, using Equation 1, gives

and solving for i,

More generally, for any one-year discount bond, the yield to maturity can be writ-

ten as

(6)

where

F = face value of the discount bond

P = current price of the discount bond

i

⫽

F

⫺ P

P

i

⫽

$1,000

⫺  $900

$900

⫽ 0.111 ⫽ 11.1%

$900i

⫽ $1,000 ⫺ $900

$900

⫹ $900i ⫽ $1,000

11 ⫹ i2 ⫻ $900 ⫽ $1,000

$900

⫽

$1,000

1

⫹ i

where

C

=

yearly payment

= $100

P

c

=

price of perpetuity (consol)

= $2,000

Thus,

i

c

⫽ 0.05 ⫽ 5%

i

c

⫽

$100

$2,000

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Chapter 3 What Do Interest Rates Mean and What Is Their Role in Valuation?

47

In other words, the yield to maturity equals the increase in price over the year

F – P divided by the initial price P. In normal circumstances, investors earn positive

returns from holding these securities and so they sell at a discount, meaning that the

current price of the bond is below the face value. Therefore, F – P should be posi-

tive, and the yield to maturity should be positive as well. However, this is not always

the case, as extraordinary events in Japan indicated (see the Global box below).

An important feature of this equation is that it indicates that for a discount bond,

the yield to maturity is negatively related to the current bond price. This is the same

conclusion that we reached for a coupon bond. For example, Equation 6 shows that a

rise in the bond price from $900 to $950 means that the bond will have a smaller increase

in its price over its lifetime, and the yield to maturity falls from 11.1% to 5.3%. Similarly,

a fall in the yield to maturity means that the price of the discount bond has risen.

Summary

The concept of present value tells you that a dollar in the future is not

as valuable to you as a dollar today because you can earn interest on this dollar.

Specifically, a dollar received n years from now is worth only 

today. The

present value of a set of future cash flows on a debt instrument equals the sum of the

present values of each of the future cash flows. The yield to maturity for an instru-

ment is the interest rate that equates the present value of the future cash flows on

that instrument to its value today. Because the procedure for calculating the yield

to maturity is based on sound economic principles, this is the measure that finan-

cial economists think most accurately describes the interest rate.

Our calculations of the yield to maturity for a variety of bonds reveal the impor-

tant fact that current bond prices and interest rates are negatively related: