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**Sections 10.1, 10.2, 10.3, 10.4**

Previous discussions of “yield to maturity” and “internal rate of return” have implicitly been based on an assumption of one level interest rate throughout a term of investment. The phenomenon where interest rates differ throughout a term of investment is called the term structure of interest rates, which can be displayed in the form of a graph of the relationship between interest rates and term of investment. Such a graph is called a yield curve.

Three basic yield curves (depending on economic conditions) are as follows:

Term of Investment

**Normal Yield Curve**

Term of Investment

**Inverted Yield Curve**

Term of Investment

**Flat Yield Curve**

Recall that the net present value (NPV) of a series of future payments is

*P*

(

*i*

) =

*t n*

= 0

*v t*

*R t*

but this is based on a single rate of interest

*i*

.

An interest rate from a yield curve corresponding to a particular time is called a spot rate.

If each payment is discounted by its associated spot rate, the NPV is given by

*P*

(

*s*

) =

*t n*

= 0

(1 +

*s t*

)

*t *

*R t*

where

*P*

(

*s*

) denotes that the NPV is based on the series of spot rates

*s*

1

*s*

2

, …,

*s n*

, with

*s*

0

= 0.

,

**Chapter 10 Exercises**

1.

Find the present value of payments of $1000, $800, and $600 made respectively one, two, and three years from today with each of the given yield curves.

(a)

*s t*

= 0.04 + (0.4

0.1

*t*

)

2 where time

*t*

= 0 is today

The required spot rates are

*s*

1

0.13

*s*

2

0.08

*s*

3

The present value is 1000(1.13)

1

+ 800(1.08)

2

+ 600(1.05)

3

=

$2089.13

(b)

*s t*

= 0.15

(0.4

0.1

*t*

)

2 where time

*t*

= 0 is today

The required spot rates are

*s*

1

0.06

*s*

2

0.11

*s*

3

The present value is 1000(1.06)

1

+ 800(1.11)

2

+ 600(1.14)

3

=

$1997.68

Recall that the basic formula to find the price of a bond is

*P*

=

*Fr a *

–

*n*

**|**

*i*

+

*C*

(1 +

*i*

)

*n*

where

*i*

is a single yield to maturity. A more general formula using spot rates is

*P*

=

*Fr t n*

(1 +

*s t*

)

*t*

= 1

+

*C*

(1 +

*s n*

)

*n*

According to (modern) finance theory, the

*Law of One Price*

states that these two formulas must be equal.

If we are given values for the spot rates

*s*

1

,

*s*

2

, …,

*s n*

, then setting the two formulas equal to each other will make it possible to solve for the level yield rate

*i*

, although a SOLVER or some method from numerical analysis will generally be necessary.

If we are given a value for the level yield rate

*i*

, then there are multiple possibilities for the set of spot rates two formulas equal to each other.

*s*

1

,

*s*

2

, …,

*s n*

which will make the

If a set of coupon bond prices

*P*

1

,

*P*

2

, …,

*P n*

are given where the durations of the bonds respectively correspond to the times at which the spot rates

*s*

1

,

*s*

2

, …,

*s n*

are to be determined, then the bootstrap method can be used to obtain the spot rates using the following steps:

Step 1: Use

*P*

1 to determine

*s*

1

.

Step 2: Use

*P*

2 and

*s*

1 to determine

*s*

2

.

**Chapter 10 Exercises**

2.

Each of the prices in the table on the right is for a $1000 par value bond with annual 8% coupons. Use the bootstrap method to find the spot rates implied by these bond prices.

$1033.49 =

1 +

*s*

1

*s*

1

= 0.045

Term

1 year

2 years

3 years

Price

$1033.49

$1056.10

$1040.78

$1056.10 =

1 +

*s*

1

+

(1 +

*s*

2

)

2

=

1.045

+

(1 +

*s*

2

)

2

*s*

2

= 0.050

If a set of coupon bond prices

*P*

1

,

*P*

2

, …,

*P n*

are given where the durations of the bonds respectively correspond to the times at which the spot rates

*s*

1

,

*s*

2

, …,

*s n*

are to be determined, then the bootstrap method can be used to obtain the spot rates using the following steps:

Step 1: Use

*P*

1 to determine

*s*

1

.

Step 2: Use

*P*

2 and

*s*

1 to determine

*s*

2

.

Step 3: Use

*P*

3

,

*s*

1

, and

*s*

2 to determine

*s*

3

.

Step 4: Continue this process to generate desired spot rates for as many bond prices as are available.

**Chapter 10 Exercises**

2.

Each of the prices in the table on the right is for a $1000 par value bond with annual 8% coupons. Use the bootstrap method to find the spot rates implied by these bond prices.

$1033.49 =

1 +

*s*

1

*s*

1

= 0.045

Term

1 year

2 years

3 years

Price

$1033.49

$1056.10

$1040.78

$1056.10 =

1 +

*s*

1

+

(1 +

*s*

2

)

2

=

1.045

+

(1 +

*s*

2

)

2

*s*

2

= 0.050

$1040.78 =

1 +

*s*

1

+

(1 +

*s*

2

)

2

+

(1 +

*s*

3

)

3

=

1.045

+

(1.05)

2

+

(1 +

*s*

3

)

3

*s*

3

= 0.066

If a set of coupon bond prices

*P*

1

,

*P*

2

, …,

*P n*

are given where the durations of the bonds respectively correspond to the times at which the spot rates

*s*

1

,

*s*

2

, …,

*s n*

are to be determined, then the bootstrap method can be used to obtain the spot rates using the following steps:

Step 1: Use

*P*

1 to determine

*s*

1

.

Step 2: Use

*P*

2 and

*s*

1 to determine

*s*

2

.

Step 3: Use

*P*

3

,

*s*

1

, and

*s*

2 to determine

*s*

3

.

Step 4: Continue this process to generate desired spot rates for as many bond prices as are available.

The coupon rate need not be the same for each bond. The bootstrap method can still be used even if the bonds have different coupon rates.

The at-par yield for a bond is defined to be the yield rate which would result in a bond having a yield rate

*i*

equal to its coupon rate

*g*

. Such a bond would sell “at par”. To derive a formula for the at-par yield rate, consider a bond where

*F*

=

*C*

= 1 and

*r*

=

*g*

. Consequently, we must have

1 =

*P*

=

*Fr n*

*t*

= 1

(1 +

*s t*

)

*t*

+

*C*

(1 +

*s n*

)

*n*

from which can solve for the at-par yield rate

*r*

=

1

*n*

*t*

= 1

(1 +

*s n*

)

*n*

(1 +

*s t*

)

*t*

Note that the at-par yield rate can be denoted

*r n*

bond to which it would apply. to reflect the term of the