W4 Question Set Bond-Solution PDF

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Tutorial 4: Bonds – Bills (Chapter 6) A. Bond Pricing 6-3.

The following table summarizes prices of various default-free, zero-coupon bonds

(expressed as a percentage of face value):

a.

Compute the yield to maturity for each bond.

b.

Plot the zero-coupon yield curve (for the first five years).

c.

Is the yield curve upward sloping, downward sloping, or flat?

a.

Use the following equation with FV = $100, P = price per $100 face value, and n =

maturity (years) from the table, 1/ n

 FV  1  YTM n   n   P 

We get Price (P)

Maturity (n)

YTM

95.77

1

4.42%

91.46

2

4.56%

86.93

3

4.78%

82.25

4

5.01%

77.09

5

5.34%

b.

The yield curve is as shown below.

c.

The yield curve is upward sloping.

6-10. Suppose a seven-year, $1000 bond with a 10.46% coupon rate and semiannual coupons is trading with a yield to maturity of 8.78%. 1

a.

Is this bond currently trading at a discount, at par, or at a premium? Explain.

b.

If the yield to maturity of the bond rises to 9.54% (APR with semiannual

compounding), at what price will the bond trade? a.

Because the yield to maturity is less than the coupon rate, the bond is trading at a

premium. P

52.3 52.3   2  0.0878   0.0878   1   1  2    2 

52.3  1000 14

 0.0878   1  2  

 $1, 086.49

b. P

52.3 52.3   2  0.0954   0.0954   1   1  2    2 

52.3  1000 14

0.0954   1  2  

 $1, 046.21

6-11. Suppose that Ally Financial Inc. issued a bond with 10 years until maturity, a face value of $1000, and a coupon rate of 11% (annual payments). The yield to maturity on this bond when it was issued was 5%. a.

What was the price of this bond when it was issued?

b.

Assuming the yield to maturity remains constant, what is the price of the bond

immediately before it makes its first coupon payment? c.

Assuming the yield to maturity remains constant, what is the price of the bond

immediately after it makes its first coupon payment? a. P

b.

When it was issued, the price of the bond was 110 110   (1  0.05) (1  0.05)2

Before the first coupon payment, the price of the bond is

P  110 

c. P

110  1, 000  $1, 463.30. (1  0.05)10

110 110   (1  0.05) (1  0.05)2

110 1, 000  $1,536.47. (1  0.05)9

After the first coupon payment, the price of the bond will be 110 110  2  (1  0.05) (1 0.05)

110 1, 000 $1, 426.47. 9  (1 0.05)

2

6-13. Consider the following bonds:

a.

What is the percentage change in the price of each bond if its yield to maturity

fallsfrom 7% to 6%? b.

Which of the bonds A–D is most sensitive to a 1% drop in interest rates from 7%

to 6% and why? Which bond is least sensitive? Provide an intuitive explanation for your answer. a.

We can compute the price of each zero coupon bond at each YTM using Eq. 6.2. For

example, with a 7% YTM, the price of bond A per $100 face value is P(bond A, 7% YTM) 

100  $33.87. 1.0716

The price of bond D, using Eq 6.5 is P(bond C, 7% YTM)  2

1  1  100 1   $52.77.  0.07  1.0716  1.0716

One can also use the Excel formula to compute the price: –PV(YTM, NPER, PMT, FV). Once we compute the price of each bond for each YTM, we can compute the % price change as

 Price at 6% YTM   Price at 7% YTM .  Price at 7% YTM  Percent change = The results are shown in the table below. Coupon

b.

Price

at

Price

at

Bond

Rate

Maturity

7%

6%

Change

A

0%

16

$33.87

$39.36

16.2%

B

0%

12

$44.40

$49.70

11.9%

C

2%

16

$52.77

$59.58

12.9%

D

7%

12

$100.00

$108.38

8.4%

Bond A is most sensitive, because it has the longest maturity and no coupons, so its

future cash flows have the highest discount factors. Bond D is the least sensitive. Intuitively, higher coupon rates and a shorter maturity mean that relatively more of the bond’s cash flows happen early and thus cannot be as greatly affected by changes in interest rates as bonds with low coupon rates and longer maturities.

3

6-14. Suppose you purchase a 30-year, zero-coupon bond with a yield to maturity of 4%. You hold the bond for five years before selling it. a.

If the bond’s yield to maturity is 4% when you sell it, what is the internal rate of

return of your investment? b.

If the bond’s yield to maturity is 5% when you sell it, what is the internal rate of

return of your investment? c.

If the bond’s yield to maturity is 3% when you sell it, what is the internal rate of

return of your investment? d.

Even if a bond has no chance of default, is your investment risk free if you plan to

sell it before it matures? Explain. a.

Purchase price = 100 / 1.0430 = 30.83. Sale price = 100 / 1.0425 = 37.51. Return = (37.51

/ 30.83)1/5 – 1 = 4.00%. I.e., since YTM is the same at purchase and sale, IRR = YTM. b.

Purchase price = 100 / 1.0430 = 30.83. Sale price = 100 / 1.0525 = 29.53. Return = (29.53

/ 30.83)1/5 – 1 = -0.86%. I.e., since YTM rises, IRR < initial YTM. c.

Purchase price = 100 / 1.0430 = 30.83. Sale price = 100 / 1.0325 = 47.76. Return = (47.76

/ 30.83)1/5 – 1 = 9.15%. I.e., since YTM falls, IRR > initial YTM. d.

Even without default, if you sell prior to maturity, you are exposed to the risk that the

YTM may change.

6-16.* Suppose the current yield on a one-year, zero coupon bond is 4%, while the yield on a five-year, zero coupon bond is 5%. Neither bond has any risk of default. Suppose you plan to invest for one year. You will earn more over the year by investing in the five-year bond as long as its yield does not rise above what level? The return from investing in the one-year is the yield. The return for investing in the five-year p1 1 bond for initial price p0 and selling after one year at price p1 is p0 . We have

1 , (1.05)5 1 . p1  (1  y )4 p0 

So you break even when

4

1 p1 (1  y) 4  1  1  0.04 1 p0 (1.05)5

.04 



1  y 

4

y

(1.05)5 1 (1 y )4 

(1.05)5 (1.04)

(1.05) 5/4 1  5.25%. (1.04) 1/4

5

For Problems 21–22, assume zero-coupon yields on default-free securities are as summarized in the following table:

6-21. Consider a four-year, default-free security with annual coupon payments and a face value of $1000 that is issued at par. What is the coupon rate of this bond? Solve the following equation: 1 1 1 1  1000  CPN   2  3  4     (1 0.046) (1 0.05) (1 0.054) (1 0.058)  CPN  $57.44

1000   (1 0.058) 4 

Therefore, the par coupon rate is 5.744%.

6-22. Consider a five-year, default-free bond with annual coupons of 5% and a face value of $1000. a.

Without doing any calculations, determine whether this bond is trading at a

premium or at a discount. Explain. b.

What is the yield to maturity on this bond?

c.

If the yield to maturity on this bond decreased to 5.2%, what would the new price

be? a.

The bond is trading at a discount. The yield to maturity is a weighted average of the

yields of the zero-coupon bonds. Looking at the yields for the 5 years, we can see the weighted average will be higher than 5% coupon and therefore will sell at a discount. b.

To compute the yield, first compute the price.

P 

CPN CPN CPN  FV   ...  1  YTM1 (1  YTM2 ) 2 (1  YTMN ) N

50 50 50 50 50  1000      $956.69 (1 0.046) (1 0.050)2 (1 0.054)3 (1 0.058)4 (1 0.061) 5

The yield to maturity is: CPN CPN CPN  FV   ... 1  YTM (1  YTM) 2 (1  YTM) N 50 50  1000  ...   YTM  6.029%. 956.69  (1  YTM ) (1 YTM )N P

c.

If the yield decreased to 5.2%, the new price would be:

6

CPN CPN CPN  FV   ... 1  YTM (1  YTM) 2 (1  YTM ) N 50 50  1000   ...   $991.39. (1  .052) (1 .052) N

P

6-26. Explain why the expected return of a corporate bond does not equal its yield to maturity. The yield to maturity of a corporate bond is based on the promised payments of the bond. But there is some chance the corporation will default and pay less. Thus, the bond’s expected return is typically less than its YTM. Corporate bonds have credit risk, which is the risk that the borrower will default and not pay all specified payments. As a result, investors pay less for bonds with credit risk than they would for an otherwise identical default-free bond. Because the YTM for a bond is calculated using the promised cash flows, the yields of bonds with credit risk will be higher than that of otherwise identical default-free bonds. However, the YTM of a defaultable bond is always higher than the expected return of investing in the bond because it is calculated using the promised cash flows rather than the expected cash flows.

B. Bank Bill On 15 October 2002, BabyCo issues a 180-day bank bill with a face value of $400,000. The bill was accepted and discounted by BankFive on that date (be sure you can explain what this means). BankFive held the bill until 13 January 2003, when it sold the bill to ProfitBank. On 12 February 2003, ProfitBank sold the bill to YetAnotherBank. The quoted interest rates on bank bills on the dates in question were: Date

60-day bills 90-day bills 180-day bills

15/10/02 4.57%

4.68%

4.90%

13/1/03

4.89%

4.96%

5.02%

12/2/03

4.48%

4.55%

4.82%

Answer the following: a. How much did BabyCo receive when the bill was issued? b. What did BankFive sell the bill for on 13 January 2003? c. What did ProfitBank sell the bill for on 12 February 2003? d. What was ProfitBank’s effective annual yield for the period they held the bill? e. Why isn’t ProfitBank’s effective yield the same as the bill’s quoted rate when purchased?

7

a. BabyCo issued the bank bill on 15 October. The relevant interest rate was 4.90%. The value of the bill was:

B

400,000  390,562.30 1  0.0490  180 365 

b. BankFive sold the bill on 13 January 2003. At that point the bill had 90 days remaining, so the relevant interest rate was 4.96%. The value of the bill was:

B

400,000  395,167.05 90  1  0.0496  365 c. ProfitBank sold the bill on 12 February 2003. At that point the bill had 60 days remaining, so the relevant interest rate was 4.48%. The value of the bill was:

B

400,000  397, 075.78 60 1  0.0448  365 

d. ProfitBank held the bill for 30 days. The original cost of the bill to ProfitBank was $395,167.05, and the sales price after 30 days was $397,075.78. Therefore, ProfitBank’s effective annual yield was:

 397, 075.78  EAR     395,167.05 

365

30

 1  6.04%

e. ProfitBank’s effective yield was not the same as the quoted yield for two reasons: 

Due to market conventions, the quoted yield will never be the same as the annual effective yield on a bill. The quoted yield is a simple interest rate to be applied for the remaining life of the bill, while the annual effective yield is an annually compounded interest rate.

ProfitBank did not hold the bill to maturity, but sold the bill with 60 days remaining. This meant that ProfitBank benefited from the decrease in interest rates.

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