A bond’s credit rating provides a guide to its risk. Suppose that long-term bonds rated Aa currently offer yields to maturity of 7.5%. A-rated bonds sell at yields of 7.8%. Suppose that a 10-year bond with a coupon rate of 7.6% is downgraded by Moody’s from an Aa to A rating.

A bond’s credit rating provides a guide to its risk. Suppose that long-term bonds rated Aa currently offer yields to maturity of 7.5%. A-rated bonds sell at yields of 7.8%. Suppose that a 10-year bond with a coupon rate of 7.6% is downgraded by Moody’s from an Aa to A rating.
a. Is the bond likely to sell above or below par value before the downgrade?
    Above par value
    Below par value

 Answer
Above par value

b. Is the bond likely to sell above or below par value after the downgrade?

    Above par value
    Below par value
Answer

Below par value

Explanation
 
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

The bond’s yield to maturity will increase from 7.5% to 7.8% when the perceived default risk increases.  The bond price will fall:

a.
Initially, the bond is rated Aa and by that benchmark should yield around 7.5%. The coupon is 7.6%, so the bond price will be above par to offset the excess 0.1% coupon payment.

Initial price = PV=$77×[1.082−110.082(1.082)]+$1,00010(1.082)=$966.75


b.
After, the bond is rated A and by that benchmark should yield around 7.8%. The coupon is 7.6%, so the bond price will be below par to compensate for the insufficient coupon payment.

New price= PV=$77×[1.085−110.085(1.085)]+$1,00010(1.085)=$947.51

Several years ago, Castles in the Sand Inc. issued bonds at face value of $1,000 at a yield to maturity of 7.4%.

a. Several years ago, Castles in the Sand Inc. issued bonds at face value of $1,000 at a yield to maturity of 7.4%. Now, with 8 years left until the maturity of the bonds, the company has run into hard times and the yield to maturity on the bonds has increased to 12%. What is the price of the bond now? (Assume semiannual coupon payments.)

b. Suppose that investors believe that Castles can make good on the promised coupon payments but that the company will go bankrupt when the bond matures and the principal comes due. The expectation is that investors will receive only 82% of face value at maturity. If they buy the bond today, what yield to maturity do they expect to receive?


Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

Since the bonds were issued at par, the coupon rate had to match the yield to maturity at the time of issuance. Thus, the coupon rate is 7.4%.

a.
Since the bond has semiannual payments, the coupon payment, the interest rate, and the number of periods must all be expressed in semiannual terms:

Bond price	=	PV of coupon payments + PV of face value
	=	C × ((1 / r) – {1 / [r(1 + r)t]}) + FV / (1 + r)t
	=	[(.0740 × $1,000) / 2] × [[1 / (.1200 / 2)] – (1 / {(.1200 / 2)[1 + (.1200 / 2)](8 × 2)})] + $1,000 / [1 + (.1200 / 2)](8 × 2)
	=	$767.56

b.

Bond price	=	PV of coupon payments + PV of face value
$767.56	=	C × ((1 / r) – {1 / [r(1 + r)t]}) + FV / (1 + r)t
	=	[(.0740 × $1,000) / 2] × [[1 / (r / 2)] – (1 / {(r / 2)[1 + (r / 2)](8 × 2)})] + (.8200 × $1,000) / [1 + (r / 2)](8 × 2)

To compute r, use trial-and-error, a financial calculator, or a computer. See the calculator solution below.

Calculator computations:

a.

Enter

8 × 2

12 / 2

–74 / 2

–1,000

N

I/Y

PV

PMT

FV

Solve for

767.56

b.

Enter

8 × 2

12 / 2

74 / 2

820

N

I/Y

PV

PMT

FV

Solve for

–768

YTM = 5.11% × 2 = 10.21%

A 6-year maturity bond with face value of $1,000 makes annual coupon payments of $108 and is selling at face value.

A 6-year maturity bond with face value of $1,000 makes annual coupon payments of $108 and is selling at face value. What will be the rate of return on the bond if its yield to maturity at the end of the year is: (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.)


Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

With only one year to maturity and annual coupon payments, the price formula can be simplified to:

Bond price = (C + FV) / (1 + r)^t

a.

Price	=	$1,108 / 1.06
	=	$1,045.28

Rate of return	=	[Annual interest + (Ending price – Beginning price)] / Beginning price
	=	($108 + 1,045.28 – 1,000) / $1,000
	=	.1533, or 15.33%

b.

Price	=	$1,108 / 1.11
	=	$1,000

Rate of return	=	[Annual interest + (Ending price – Beginning price)] / Beginning price
	=	($108 + 1,000 – 1,000) / $1,000
	=	0.1080, or 10.80%

c.

Price	=	$1,108 / 1.13
	=	$982.27

Rate of return	=	[Annual interest + (Ending price – Beginning price)] / Beginning price
	=	($108 + 982.27 – 1,000) / $1,000
	=	0.0903, or 9.03%

You buy a 20-year bond with a coupon rate of 9% that has a yield to maturity of 10%. (Assume a face value of $1,000 and semiannual coupon payments.) Six months later, the yield to maturity is 11%. What is your return over the 6 months?

You buy a 20-year bond with a coupon rate of 9% that has a yield to maturity of 10%. (Assume a face value of $1,000 and semiannual coupon payments.) Six months later, the yield to maturity is 11%. What is your return over the 6 months?

Answer
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

PV0 = $40× [100.050− 100.050(1.050)40]+ $1,0001.05040=$914.20
PV1=$40×[100.06−13900.06(1.05)]+$1,00039(1.05)= $840.71

Rate of return =	$40 + ($840.71 - $914.20)	= –914.2000 = –3.12%
	$914.20

A bond with a face value of $1,000 has 10 years until maturity, carries a coupon rate of 7%, and sells for $1,160. Interest is paid annually.

A bond with a face value of $1,000 has 10 years until maturity, carries a coupon rate of 7%, and sells for $1,160. Interest is paid annually.

a. If the bond has a yield to maturity of 9% 1 year from now, what will its price be at that time? (Do not round intermediate calculations.)


 b. What will be the annual rate of return on the bond? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places. Negative amount should be indicated by a minus sign.)
 

c. Now assume that interest is paid semiannually. What will be the annual rate of return on the bond?


d. If the inflation rate during the year is 3%, what is the annual real rate of return on the bond? (Assume annual interest payments.) (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places. Negative amount should be indicated by a minus sign.)

Explanation

Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

a.

Bond price	=	PV of coupon payments + PV of face value
	=	C × ((1 / r) – {1 / [r(1 + r)t]}) + FV / (1 + r)t
	=	(0.11 × $1,000) × ((1 / 0.11) – {1 / [0.11(1.11)(10 – 1)]}) + $1,000 / 1.11(10 – 1)
	=	$820

Since yield to maturity equals the coupon rate, the bond must be priced at par.

b.

Rate of return	=	[Annual interest + (Ending price – Beginning price)] / Beginning price
	=	($74 + 1,000 – 1,160) / $1,160
	=	–0.2293, or –22.93%

c.
The rate of return will be slightly higher above −22.93%, since the midyear coupon can be reinvested:

Rate of return =	$37 + $37 × (7.30).5 + ($1,000 -$1,160)	= –(expression error)%
	$1,160

d.

Rate of return	=	(1 + Nominal return) / (1 + Inflation rate) – 1
	=	[1 + (–0.2293)] / 1.03 – 1
	=	–0.2517, or –25.17%

You buy a bond for $976 that has a coupon rate of 7% and a maturity of 5-years. A year later, the bond price is $1,136.

You buy a bond for $976 that has a coupon rate of 7% and a maturity of 5-years. A year later, the bond price is $1,136. (Assume a face value of $1,000 and annual coupon payments.)
a. What is the new yield to maturity on the bond?


b. What is your rate of return over the year?



Explanation

Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

a.

Bond price	=	PV of coupon payments + PV of face value
Bond price	=	C × ((1 / r) – {1 / [r(1 + r)t]}) + FV / (1 + r)t
$1,136	=	(0.07 × $1,000) × ((1 / r) – {1 / [r(1 + r)(5 – 1)]}) + $1,000 / (1 + r)(5 – 1)
	=	0.0368, or 3.68%

To solve for r, use trial-and-error, a financial calculator, or a computer. See the calculator solution below.

b.

Rate of return	=	[Annual interest + (Ending price – Beginning price)] / Beginning price
	=	($74 + 1,136 – 976) / $976
	=	0.2398, or 23.98%

Calculator computations:

Enter

5 – 1

–1,136

74

1,000

N

I/Y

PV

PMT

FV

Solve for

3.68

Perpetual Life Corp. has issued consol bonds with coupon payments of $105. (Consols pay interest forever and never mature. They are perpetuities.)

Perpetual Life Corp. has issued consol bonds with coupon payments of $105. (Consols pay interest forever and never mature. They are perpetuities.)

a. If the required rate of return on these bonds at the time they were issued was 10%, at what price were they sold to the public?


b. If the required return today is 12%, at what price do the consols sell?


Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations. 

a.

PV	=	C / r
	=	$105 / 0.10
	=	$1,000

b.

PV	=	C / r
	=	$105 / 0.10
	=	$875

The following table shows some data for three zero-coupon bonds. The face value of each bond is $1,000.

The following table shows some data for three zero-coupon bonds. The face value of each bond is $1,000.


Bond     Price     Maturity (Years)     Yield to Maturity
A     $     310                 25                 —    
B           310                 —                 9     %
C           —                 11                 8    



a. What is the yield to maturity of bond A? (Do not round intermediate calculations. Enter your answer as a percent rounded to 3 decimal places. Assume annual compounding.)





b. What is the maturity of B? (Do not round intermediate calculations. Round your answer to 2 decimal places. Assume annual compounding.)

 



c. What is the price of C? (Do not round intermediate calculations. Round your answer to 2 decimal places. Assume annual compounding.)






Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

The general bond formula is:
Bond price = PV of coupon payments + PV of face value
With a zero-coupon bond and annual compounding, the formula is reduced to:

Bond price	=	PV of face value
Bond price	=	FV / (1 + r)t

a.

Bond price	=	FV / (1 + r)t
$310	=	$1,000 / (1 + r)25
(1 + r)25	=	$1,000 / $310
r	=	3.2258(1 / 25) – 1
r	=	.0480, or 4.796%

b.

Bond price	=	FV / (1 + r)t
$310	=	$1,000 / 1.09t
1.09t	=	$1,000 / $310
t × ln1.09	=	ln.0480
t	=	ln.0480 / ln1.09
t	=	13.59 years

c.

Bond price	=	FV / (1 + r)t
	=	$1,000 / 1.0811
	=	$428.88

Calculator computations:
a.

Enter

25

–310

1,000

N

I/Y

PV

PMT

FV

Solve for

4.796

 
b.

Enter

9

–310

1,000

N

I/Y

PV

PMT

FV

Solve for

13.59

 
c.

Enter

11

8

–1,000

N

I/Y

PV

PMT

FV

Solve for

428.88