chapter 7 global bond investing hw solutions

Chapter 7 Global Bond Investing

Note: In the sixth edition of Global Investments, the exchange rate quotation symbols differ from previous

editions. We adopted the convention that the first currency is the quoted currency in terms of units

of the second currency.

For example, :$ 1.4 indicates that one euro is priced at 1.4 dollars. In previous editions we used

the reversed convention $/ 1.4, meaning 1.4 dollars per euro.

All problems in this test bank still use the old convention and have not been adapted to reflect the

new quotation symbols used in the 6th edition.

Questions and Problems

1. What is the difference between a foreign bond and a Eurobond?

Solution

A foreign bond is a bond issued on a national market in its local currency but by a borrower foreign

to that country. A Eurobond is placed worldwide by an international syndicate of banks. The bonds

are generally not placed in the country of the borrower. The Eurobond market has no geographic

location and therefore no supervision by a national authority.

2. List three differences between dollar Eurobonds and Yankee bonds.

Solution

The following differences exist between dollar Eurobonds and Yankee bonds:

Differences In: Eurobond Yankee

Primary Market Trading Outside of U.S. U.S.

Major Secondary Market Trading Outside of U.S. U.S.

Registration None SEC

Underwriter International Syndicate U.S. Syndicate

Issuer Any Entity Non-U.S. Entity

Coupon Usually Annually Semiannually

3. Why did U.S. commercial banks have an interest in the development of the Eurobond market?

Solution

The GlassSteagall Act of 1933 forbids U.S. commercial banks to own, underwrite, or deal in corporate stocks and bonds. This Act only applies to the United States and not to foreign activities of U.S. commercial banks subsidiaries. The Eurobonds market was a way to enter investment banking.

56 Solnik/McLeavey Global Investments, Sixth Edition

4. Give at least two reasons why Eurobonds are issued in bearer form.

Solution

Eurobonds are issued in bearer form because:

Registration by the issuing company of all international bondholders across the world would be a daunting and costly task.

The work of global clearing houses, like Euroclear or Clearstream, is greatly simplified (and cheaper) by the fact that bonds are in bearer form and therefore totally fungible.

Registration of bondholders in a centralized location would increase the risk of sudden (and retroactive) tax imposition or regulation by the country of the registrar.

International bondholders seek anonymity, possibly for tax evasion reasons.

5. To provide full protection against unexpected tax imposition, all Eurobond contracts have a covenant

stating that the issuer will increase the interest payments to make up for any tax imposed. Assume

that Paf Inc. has issued a Eurobond with a coupon of $10 per $100 bond. For some reason, Paf Inc. is

forced by its government to transfer 15% of the coupon as withholding tax, so that the net coupon

paid to the bondholder is only $8.50. What should Paf Inc. do, according to the bond covenant?

Solution

The coupon must be increased so that the net receipt by the bondholder is still $10 per $100 bond.

Hence, the new coupon rate, r, must verify:

10 100 (1 0.15)

11.7647%.

r

r

6. Lets consider the NKK dual-currency bond shown in Exhibit 7.3. It is a bond quoted in yen at 101%. What would happen to the market price if the following scenarios took place?

a. The market interest rate on (newly issued) yen bonds drops significantly.

b. The dollar drops in value relative to the yen.

c. The market interest rate on (newly issued) dollar bonds drops significantly.

d. Would you give the same answers if the same bonds were quoted in dollars?

Solution

Lets define the following notations:

P the market price of the NKK bond in yen

P$ the market price of the NKK bond in U.S. dollars

S the spot exchange rate is /$

r the Japanese interest rate

r$ the U.S. interest rate

Each note of 100 is reimbursed in ten years by $0.5524 ($110,480,000 for 20 billion).

We have:

2 10 10 ?

8 8 8 0.5524.

1 (1 ) (1 ) (1 )P S

r r r r

a. If the yen interest rate (r) drops, the market value of the NKK bond in yen should rise as is the

case with any bond with a fixed stream of cash flows denominated in yens. In other words, this

bond with a fixed 8% yen coupon becomes very attractive when compared to newly issued yen

bonds with a low coupon.

Chapter 7 Global Bond Investing 57

b. If the dollar drops in value relative to the yen (S decreases), the value of the NKK bond, which is

quoted in yen should drop. This is because part of the value of this bond is determined by the

promised repayment of the bond, which is set as a fixed amount of dollars; the present value in

yen of this fixed amount of dollars drops with the dollar exchange rate.

c. If the market interest rate on dollar bonds drops (r$), the value of the NKK bond should rise. Part

of the value of the bond is determined by the promised principal repayment in dollars. This payoff

can be seen as a zero-coupon dollar bond, and its value goes up when interest rates drop.

d. The dollar value of the bond is obtained by dividing by S:

$ 2 10 10 ?

1 8 8 8 0.5524.

1 (1 ) (1 ) (1 )P

S r r r r

If the yen interest rate (r) drops, the market value of the NKK bond in dollar should rise (see first answer). If the dollar drops in value relative to the yen (1/S increases), the value of the NKK bond in dollars should rise. If the market interest rate on dollar bonds drops (r$), the value of the NKK bond should rise (see first answer).

7. Two bond indexes of the same market tend to give the similar total return indications even if their

composition is quite different. Why?

Solution

All bond prices tend to move up or down when interest rates move down or up. This is true for bonds quoted in the same currency. It is only when there are twists in yield curves that there can be marked differences in performance among bonds of different maturity. Prices of corporate bonds are also affected by changes in the market credit risk spread. So the difference in performance of two bond indexes in the same currency might differ if their maturity and credit quality composition differ.

8. Assume that you are an international bank that has lent money to some Latin American countries.

Because of the nonpayment of interest due, you have already taken substantial reserves against these

nonperforming loans. Why would you be willing to exchange these loans for Brady bonds?

Solution

The Brady plan allows the transformation of nontradable loans into traded bonds. Even if the bank does not wish to sell these Brady bonds, the new securities are now performing and the emerging country is less likely in the future to default on public bonds than on private loans. If the loans have been properly provisioned it is likely that the market price of newly-issued Brady bonds will be higher than the book value of the loans that they replace; this is because of the collateral provided and the political importance of these bonds for the emerging country wishing to get additional funding in the future.

9. Discuss the differences between a par and a discount Brady bond.

a. Take the viewpoint of the emerging country.

b. Take the viewpoint of the bondholder.

Solution

a. In both cases, the bonds provide some debt reduction for emerging countries. The amount of debt reduction is made visible immediately in the case of a discount bond. From then on the emerging country pays a market interest rate on the reduced principal. In the case of a par bond, the debt reduction is obtained through a coupon rate reset well below the market rate, but the redemption value remains unchanged. In both cases, the bonds will pay lower coupons than the original loan. The interest payments are reduced for a discount bond because a market rate is applied to a smaller principal. The interest payments are reduced for a Par bond because a below-market rate applies to the original principal.


The question is, which interest payments are lowest? Lets assume no default until maturity and lets compare the cash flows on the two alternative forms of Brady bonds. Clearly, the final redemption value will be much greater for a Par bond than a discount bond. Hence, this will be

compensated by lower coupon payments if the two bonds are to yield the same market-required

yield-to-maturity.

b. For the bondholder, discount bonds have the advantage of clarifying the situation by immediately

revaluing the principal. In contrast, a larger part of the expected return on the Par bond comes

from the bigger principal payment at maturity. Given the long maturity on these bonds, this

makes the expected return all the more susceptible to default risk.

10. You purchase a Eurobond in euros, at a quoted price of 101.5%. The annual coupon on the bond is 6%, and we are exactly one month after the past coupon date. You buy 100,000 of nominal value of this bond. What is your total expense?

Solution

The clean price of the bond is 101.5%. Accrued interest is equal to one month of coupon, or 30/360 6%, given convention on the Eurobond market. Hence your total expense is:

(101.5% 30/360 6%) 100,000 102,000 euros.

11. What are the potential biases of the simple yield calculation? Take the example of two straight yen

Eurobonds with the same maturity of five years. Bond A has a coupon of 12% and Bond B, a coupon

of 8%. The current market yield on yen bonds is 10%. These two bonds have the same yield-to-

maturity of 10% and are correctly priced at 107.58% for Bond A and 92.42% for Bond B. What

would be the yield-to-maturity indicated by the simple yield calculation?

Solution

The simple yield calculation for Bond A gives:

100 107.58 10012 9.745%.

5 107.58Yield

For Bond B, it gives:

100 92.42 1008 10.296%.

5 92.42Yield

This method understates the true yield-to-maturity for Bond A priced over par and overstates it for

Bond B priced under par. The difference between the two yields appears to be 0.55% although they

have equal actuarial yield-to-maturity.

The bias is even more pronounced for bonds quoted at a price very different from their par value as

zero-coupon bonds.

12. Take the example of two straight yen Eurobonds with the same maturity of five years. Bond A has a

coupon of 12% and Bond B a coupon of 8%. The current market interest rate on yen bonds is 9%.

These two bonds have the same yield-to-maturity of 10% and are correctly priced at 111.67% for

Bond A and 96.11% for Bond B. What would be the yield-to-maturity indicated by the simple yield

calculation?


Solution

The simple interest calculation for Bond A gives:

Yield 8.656%.

For Bond B, it gives:

Yield 9.133%.

This method understates the true yield-to-maturity for Bond A priced over par and overstates it for

Bond B priced under par. The difference between the two yields appears to be 0.477% although they

have equal actuarial yield-to-maturity.

The bias is even more pronounced for bonds quoted at a price very different from their par value as

zero-coupon bonds.

13. A zero-coupon bond with a five-year maturity is worth 68.06% of its final reimbursement value.

a. Verify that its actuarial yield-to-maturity is equal to 8% by compounding 8% over five years.

b. What is the simple yield of this bond, and why is it so different from the actuarial yield?

Solution

a. 568.06 (1.08) 100. (1)

b. The simple yield r is given by:

100 68.06 1

9.386%.5 68.06

r

(2)

A zero-coupon bond always quotes well below par, so the simple yield systematically overstates

the actuarial yield-to-maturity. The intuitive reason is that the simple yield computes an annual

linear depreciation of the future capital gain (100 68.06) instead of compounding it as in Equation (7.1). This is seen in rewriting Equation (7.2) as:

68.06 (1 5 ) 100.r

14. What are the annual yield-to-maturity and duration for the following bonds:

a. A zero-coupon bond reimbursed at $100 in ten years and currently selling at $38.

b. A straight bond reimbursed at $100 in ten years, with an annual coupon of 10% and selling

at $110.

c. A perpetual bond with an annual coupon of $8 and currently selling at $110.

Solution

a. The yield-to-maturity r is calculated as follows:

10

10038

(1 )

10.16%.

r

r

The duration is 9.08 and the Macaulay duration is ten years.


b. The yield-to-maturity r is calculated as follows:

2 10

10 10 110110

1 (1 ) (1 )

8.48%.

r r r

r

The duration is 6.38 and the Macaulay duration is 6.92 years.

c. The yield-to-maturity is calculated as follows:

2 3

2

8 8 8110

1 (1 ) (1 )

8 1 1110 1

1 1 (1 )

8 1 8

1

7.27%.

r r r

r r r

r

r r r

r

The duration is 13.75 and the Macaulay duration is 14.75 years.

Note: Recall that: (1 q q2 q3 . . .) 1/(1 q).

15. The market price of a two-year bond is 105% of its nominal value. The annual coupon to be paid in

exactly one year is 7%. Its yield-to-maturity (European method) is 4.336%.

a. Calculate its duration.

b. Calculate its simple yield.

c. Calculate its semiannual yield (U.S. method).

Solution

We verify that:

2

7 107105

1 (1 )

with 4.336%.

r r

r

a. Macaulay duration 2

1 7 1072 1.936.

105 1 (1 )r r

Duration: D 1.936/1.04336 1.86.

b. Simple yield 1 100 105

7% % 4.286%.105 2


c. 2 4

7 107105

' '1 1

2 2

r r

or

2

1 12

rr

where r is the semiannual yield (U.S. YTM) and r is the annual yield (European YTM).

r 4.29%.

16. A bond has been issued in euros with an annual coupon rate of 10%. The previous coupon has just

been paid. This bond has a sinking fund provision: Half of the issue is reimbursed in two years and half in three years. You hold 10 million of nominal value of this bond.

a. Write the three future annual cash flows in euros, assuming that the previous coupon has just

been paid.

b. The yield curve is currently flat at 9%. What is the value of the bond, its yield-to-maturity, its

duration, and its modified duration?

c. How much do you stand to lose if the yield curve moves uniformly from 9% to 9.1% within

one day?

Solution

a. Cash flow (in French franc million):

Year 1 Year 2 Year 3

1 6 5.5

b. The market value of the bond is:

2 3

1 6 5.5

1.09 1.09 1.09

102.1452%.

V

V

The value of the portfolio is equal to

102.1452% 10 million 10.21452 million euros.

The yield-to-maturity is r 9%.

The duration is 2.13. The Macaulay duration is 2.33.

c. 2.13 0.10% 0.213%dP

P .

For a value of 10.21452 million euros, you stand to lose about:

dV 21,757 euros.

17. A straight bond with an annual coupon of 9% will be reimbursed 100% in three years. The previous

coupon has just been paid and this bond currently trades at 105.25%. Its European yield-to-maturity

is 7%.

a. What is its modified duration?

b. What is its semiannual yield-to-maturity?

c. What is its simple yield?


Solution

a. 2.58

b. 6.88%

c. 6.89%

18. You hold a bond with a duration of 17. Its yield is 6% while the cash (one-year) rate is 4%. You

expect yields to move down by 10 basis points over the year.

a. Give a rough estimate of your expected return.

b. What is the risk premium on this bond?

Solution

a. The expected return on the year is the sum of the accrued interest plus the expected capital loss:

Return 6% (17 0.1%) 7.7 %.

This is a rough estimate because the duration is going to move down over the year, as the bonds maturity shortens.

b. The risk premium is obtained by deducting the short-term interest rate:

Risk premium 3.7%.

19. A one-year bond is issued by a corporation with a 5% probability of default by year-end. In case of

default, the investor will recover nothing. The one-year yield for default-free bonds is 10%.

a. What yield should be required by investors on these corporate bonds if they are risk neutral?

b. What should the credit spread be?

Solution

a. Lets call y the yield and m the credit spread, so that y 10% m.

The bond is issued at 100%. If the bond defaults (5% probability), the investor gets nothing in a

year. In case of no default (95% probability), the investor will get (100 y)%. So the yield should be set on the bond so that its expected payoff is equal to the expected payoff on a risk-free

bond, 110%:

110 95% (100 y) 5% 0.

The yield is equal to y (110 95)/95 15.79%.

b. The credit spread is equal to m 5.79%.

The spread is above 5%, the probability of default for two reasons. First, the investor loses 110

not 100 in case of default, so the spread must offset both the lost principal and the lost interest.

Second, the spread has to be a bit larger because it is paid on bonds only 95% of the time.

An investor who is risk-neutral, or who can diversify this risk by holding a large number of

bonds issued by different corporations, would be satisfied with this 5.79% credit spread. But a

risk-averse investor would usually add a risk premium on top, because risk of defaults tend to be

correlated across firms (business cycle risk).


20. There is a 0.5% probability of default by the year-end on a one-year bond issued at par by a particular

corporation. If the corporation defaults, the investor will not get anything. Assuming that a default-

free bond exists with identical cash flows and liquidity, and the one-year yield on this bond is 4%.

a. What yield should be required by risk-neutral investors on the corporate bond?

b. What should the credit spread be?

Solution

a. Let y be the yield on the corporate bond. There are two possibilities at the year-end. One, the

corporation defaults (0.5% chance), and the investor gets nothing. Two, the corporation does not

default (99.5% chance), and the investor gets (100 y)%. Equating the expected payoff on the corporate bond to that on the identical default-free bond, we have:

104 0.005 0 0.995 (100 y).

From this equation we get that y 4.52%.

b. Let m be the credit spread on the corporate bond. Then, y 4% m.

We thus find that m 0.52%.

21. Several years ago, when the Deutsche mark and French franc still existed, the yield curves were as

follows:

Maturity US$% DM% FF%

1 month 2.10 8.00 7.00

6 months 2.50 7.75 7.15

1 year 3.00 7.00 7.30

2 years 3.50 6.90 7.50

5 years 4.00 6.80 7.60

10 years 4.25 6.75 7.70

Spot Exchange

Rate (per US$)

1.80

5.50

Calculate the implied forward exchange rates, assuming that the interest rates are international money

rates (linear convention) for maturities of less than a year and yields on zero-coupon bonds (European

convention) for maturities of more than one year.

Solution

Up to one year, interest rates are quoted in a linear way:

e.g., DM/$ 1 month 1 8%/12

1.80 1.8088.1 2.1%/12

After one year, interest rates are actuarial:

e.g., DM/$ 2 years

21 6.9%

1.80 1.9202.1 3.5%


Forward Rates

Maturity DM/$ FF/$

1 month 1.8088 5.5224

6 months 1.8467 5.6263

1 year 1.8699 5.7296

2 years 1.9202 5.9333

5 years 2.0557 6.5201

10 years 2.2813 7.6166

22. Back in 1985, when the Deutsche mark still existed, the yield curves were as follows:

Maturity

U.S.

Dollar

Deutsche

Mark

Japanese

Yen

Swiss

Franc

British

Pound

12 months 8.31 4.81 7.19 4.31 11.31

5 years 9.78 6.40 6.82 5.40 10.90

7 years 10.16 6.75 7.00 5.45 11.00

10 years 10.33 6.80 7.33 5.70 11.14

Spot Exchange

Rate (per US$)

2.50

200.00

2.10

0.70

Calculate the implied forward exchange rates, assuming that yields on zero-coupon bonds (European

convention) for maturities of more than one year.

Solution

The results of the calculation for zero-coupon yield-curves are as follows:

Implied Forward Exchange Rate and Currency Appreciation of the US$

Maturity

DM/$ Forward

(appreciation rate)

/$ Forward

(appreciation rate)

SF/$ Forward

(appreciation rate)

/$ Forward

(appreciation rate)

1 year 2.4192 197.93 2.0224 0.7194

(3.23%) (1.03%) (3.69%) (2.77%)

5 years 2.1381 174.45 1.7132 0.7364

(14.48%) (12.77%) (18.42%) ( 5.21%)

7 years 2.0061 163.14 1.5466 0.7382

(19.76%) (18.43%) (26.35%) ( 5.46%)

10 years 1.8060 151.81 1.3678 0.7531

(27.76%) (24.09%) (34.87%) ( 7.59%)


23. A young investment banker considers issuing a DM/$ currency option bond for a AAA client and

wonders about its pricing. He knows that currency options are available on the market and that they

could help set the conditions on the bond issue. As a first step, he decides to study a simple case: a

one-year bond. The current market conditions are as follows:

One-year dollar interest rate: 10%.

One-year Deutsche mark interest rate: 7%.

Spot DM/$ exchange rate: $1 DM 2.

The banker could issue a bond in dollars at 10%, a bond in DM at 7%, or a currency option bond at

an interest rate to be determined. One-year currency options are negotiated on the over-the-counter

market. A one-year currency option to exchange one dollar for two Deutsche marks is quoted at 4%,

that is, four cents per dollar. This is a European option, which can be exercised only at maturity. The

one-year forward exchange rate is:

1 7%2 .

1 10%F

a. Given these data, what should the interest rate be on a one-year DM/$ bond?

b. How would you determine how to set the interest rate on an n -year currency bond?

Solution

A currency option bond can be decomposed as a straight bond plus a currency option. Assume that the one-year currency option bond is issued with an interest rate i to be determined. The spot exchange rate was DM/$ 2. The payoff of this bond in year one will depend on the exchange rate at the time. For a $100 initial investment, we have in year one:

Payoff

$ 100(1 i) if DM/$ > 2

or

DM 200(1 i) if DM/$ < 2.

We construct a package of the straight dollar bond and the currency option (to exchange $1 for DM2), which will exactly replicate these payoffs. Then, it must be that this package must initially sell for the same value of $100.

To replicate the payoffs we buy:

1

$ 1001.10

i of the 10% dollar bond

and a currency option on $(1 i)100.

If the DM/$ exchange rate is below 2 a year from now, we will abandon the currency option and receive the payment on the bond plus a 10% interest; this would mean a payment of:

1$ 100 1.10 $100(1 ).

1.10

ii

If the DM/$ exchange rate is above 2, we exercise the option. We still receive $100(1 i) on the straight bond and use our option [which is written precisely on $100(1 i)] to convert this dollar amount to DM 2 100(1 i) 200(1 i). Therefore, the payoffs in year one are:

$100(1 i) if DM/$ > 2


or

DM 200(1 i) if DM/$ < 2.

These payoffs are exactly identical to those on our currency option bond in all states of the world, therefore the two securities (or package of securities) should have the same value, that is, $100. This implies the following:

1100 (4%(1 ) 100) 100

1.10

price paid on premium paid price of currency

straight bond on option options bond

ii

hence, i 5.36%.

The currency option should be issued with an interest rate of 5.36%. A similar reasoning would apply to a longer-term bond if currency options were traded for each cash flow date on the bond.

24. The yields on zero-coupon bonds are as follows:

US$% Yen%

1 year 3.00 5.00

2 years 3.50 6.00

A young investment banker considers issuing a $/yen dual-currency bond for 100 million. It is a

bond with interest paid in yen and principal repaid in dollars. The current spot exchange rate is

$1 100. The bond will be reimbursed for $1 million in two years. The interest is paid on year one and year two. What should the interest paid in yen be?

Solution

We value each cash flow using its appropriate discount rate and currency. Lets call x the interest rate. The final cash flow is $1 million. The present dollar value of this cash flow is converted in yen at the

spot exchange rate of 100 /$. The fair interest rate x on the bond should be found by equating the

present yen value of all cash flows to the issue value of 100 million.

We have in million yen:

2 2

100 100 100100

1.05 1.06 1.035

3.61%.

x x

x

25. A company is deciding whether to issue a one-year dual-currency bond or a one-year currency option

bond.

The dual-currency bond would be issued in CHF (Swiss francs) with a principal of 100 CHF per bond, with interest payable in CHF and principal repaid in U.S. dollars ($50). Denote x the interest

at which this bond is issued.

The currency option bond is issued in CHF (100 CHF), and the interest and principal are repaid in CHF or $ at the option of the bondholder. The principal repaid is either 100 CHF or $50, and the

interest rate is either y CHF or 1/2y dollars.


As you guessed, the current spot exchange rate is 2 CHF/$. The current one-year market interest rates

are 6% in CHF and 10% in $. One-year currency options are quoted in Chicago. A put CHF is quoted

at 1.2 U.S. cents per CHF; this option premium is for one CHF, with a strike price of 50 U.S. cents.

a. What is the fair interest rate x on the dual-currency bond?

b. What is the fair interest rate y on the currency option bond?

Solution

a. One-year dual-currency bond:

The cash flow is CHF 100x US$ 50 at the end of year one. We have to translate into CHF, hence:

100 50100 2

1.06 1.10

x .

The interest rate x at which this bond should be issued is:

x 9.64%.

Lets call C the CHF coupon on the currency-option bond. We have C 100 y. The cash flows of the currency-option bond are as follows:

Year 0 Year 1

CHF 100 CHF (C 100) or

US$ (C 100)

To replicate the cash flows, we first buy a one-year straight CHF bond repaid CHF (100 C),

then we buy the quantity (100 C) of the one-year CHF put.

This combination exactly replicates the cash flows of the currency-option bond. Hence, its total

value at time 0 should be equal to CHF 100. We write below the CHF value of this replication

portfolio. One CHF option costs 1.2 U.S. cents or 0.024 CHF at the spot exchange rate of 2 CHF/$.

Market Value Quantity Total

Bond 1/1.06 CHF 100 C (100 C)/1.06

Option (0.012 2) CHF 100 C 0.024 (100 C)

100100 0.024(100 )

1.06

3.37.

CC

C

The interest rate y on the currency option bond is:

y C/100 3.37%.


26. A young investment banker meets one of its clients, SOSO Inc. that is based in Sydney, Australia.

Current market conditions are the following:

FX Spot rate: AU$/$ 2.

Interest rate (zero-coupon):

AU$ $

1 year 10% 6%

2 year 11% 7%

Quote in U.S.$ cents for options (strike price: 50 cents per AU$).

PUT AU$ CALL AU$

1 year 2.5 2

2 year 3.5 3

For example a PUT AU$, traded in Chicago, gives the right to sell 1 Australian dollar (AU$) at 50 cents in a year. Its current price is 2.5 cents per AU$.

SOSO would like to issue a bond paying a fixed annual coupon of 6 AU$ and to be reimbursed in a year 100 AU$ or 50$ at the bearers choice.

a. Assuming that the bond is actually issued at 105 AU$, what is the implicit price of the option linked to that bond? Would you recommend that bond to an investor?

b. If the market was efficient, what is the normal issue price for such a bond?

After many thoughts, SOSO agrees to issue instead a dual-currency bond with an annual coupon in AU$ and a nominal to be reimbursed in US$ with the following characteristics:

Issue Price: 100 AU$.

Reimbursement Price: 50$

Maturity: 2 years.

Annual Coupon: C AU$.

c. Under current market conditions, at what level should Coupon C on the dual-currency

bond be set?

Solution

a. The cash flows of the bond that SOSO would like to issue can be replicated by a portfolio made

of the two following securities:

A one-year bond that pays 106 AU$.

A one-year AU$ put that enables to sell 100 AU$ for 50$.

Therefore, the value of the replicating portfolio must be equal to that of the bond at the time of

issue. Y, the implicit value of the option verifies the following equation:

106105

1.1Y

hence

Y 8.63 AU$ 4.32 $.


However, a put AU$ is traded in Chicago for 2.5 cents that gives the right to sell 1 AU$ for

50 cents in a year. The market value of the option offered by SOSO is therefore 100 2.5 cents 2.5 dollars, that is, 5 AU$. The implicit value of the option included in the bond being superior

(8.63 AU$), the bond seems to be overvalued and should not be recommended.

b. If the market is efficient, there is no arbitrage opportunity. The value of the bond (P) is equal to

that of the replicating portfolio (bond option).

106i.e., 5 101.36 AU$.

1.1P

c. Annual coupon, C, should be paid normally.

The cash flows of this dual-currency bond are C AU$ in year 1 and C 50 AU$ in year 2. Each cash flow must be assessed individually and converted in AU$. Therefore C verifies:

hence 2 250

100 21.1 1.11 1.07

7.36 AU$.

C C

C

27. Bank PAPOUF decides to issue two bonds and wonders what should be the fair interest rate on these

bonds:

Bond A: A two-year /$ dual-currency bond with interest in and principal in $. The bond is issued for 100 and pays an interest rate of i , each year for two years. The principal is reimbursed at $50.

Bond B: A two-year currency option bond. The bond is issued in $ with a face value of $ 100. The bondholder can choose to have the coupons and principal paid in dollars or in , at a

specified exchange rate of /$ 2, that is, receive either $100 or 200 as principal repayment, or receive either $C or 2C as interest if C is the coupon set in dollars.

Current market conditions are as follows:

Interest Rate 1-Year 2-Year

US$ 8% 8%

4% 4%

Currency Options 1-Year Maturity 2-Year Maturity

call 2 US cents 5 US cents per

put 1 US cent 3 US cents per

Spot exchange rate: S /$ 2.

a. What should be the coupon i set on Bond A consistent with current market conditions?

b. What should be the coupon C set on Bond B consistent with current market conditions?


Solution

a. Bond A:

The cash flows are as follows (where i is the interest rate):

Year 1 Year 2

100 i 100 i 50 US$

We value each cash flow separately and translate them into :

2 2

100 100 50100 2

1.04 1.04 1.08

0.0756.

i i

i

The interest rate of this bond should be 7.56%.

b. Bond B:

The cash flows of the currency option bond are as follows:


$100 $C $(100 C)

or 2C or 2(100 C)

To replicate the cash flows we first buy two zero-coupon bonds in US$. The first bond is a one-

year bond repaid C$ and the second bond is a two-year bond repaid (100 C)$.

Then we buy two currency options. The first one is a quantity (2C) of the one-year call and the

second one is a quantity 2(100 C) of the two-year call.

This combination of two bonds and two options exactly replicates the cash flows of the currency-

option bond. Hence its total value at time 0 should be equal to US$ 100.


1-Year Bond 1/1.08 C C/1.08

2-Year Bond 1/1.082 C 100 (100 C)/1.082

1-Year Option 0.02 2C 0.04C

2-Year Option 0.05 2(100 C) 0.1(100 C)

2

100100 0.04 0.1 (100 )

1.08 1.08

2.22.

C CC C

C

The interest rate on this currency option bond should be 2.22%.


28. The yield curves in U.S. dollars and Swiss francs are as follows:

U.S. Dollar% Swiss Franc%

1 Year 10 6

2 Years 12 7

These are yields for zero-coupon bonds of one- and two-year maturities. The spot exchange rate is

SF/$ 1.5.

a. What are the implied one-year and two-year forward exchange rates?

b. You contemplate issuing a dual-currency bond. You could issue zero-coupon bonds in both

currencies at the interest rates above. Instead, you wish to issue bonds of SF 150 with a coupon C

in Swiss francs, paid each year for two years, and reimbursed for $100 at the end of two years.

What is the interest rate c% (c C/150) on the bond that would be consistent with the yield curves above?

c. You contemplate issuing a two-year currency option bond. The bond is issued for $100 and gives

the option to receive the coupons and principal payment in either dollars or Swiss francs at a

fixed exchange rate of SF/$51.5. A bank gives you quotes on the premiums for SF calls with a

strike price of 1/1.5 0.66666 US$. The premium for a one-year call is 4 U.S. cents (per Swiss franc) and for a two-year call is 7 U.S. cents. What coupon rate should you set on your currency

option bond?

Solution

a. Implicit one-year and two-year forward rates are the following:

1

2

2 2

1.061.5 1.4454 SF/$

1.10

1.071.5 1.3691 SF/$.

1.12

F

F

b. Lets denote c% the interest rate at which the bonds should be issued. Cash flows are C Swiss francs on year one and year two, and $100 on year two.

We have:

2 2

100150 1.5

1.06 1.07 1.12

16.74

/150 11.16%.

C C

C

c C

c. Lets call x the coupon on the $100 bond. The currency-option bond cash flows are the following:


$100 $x $(100 x)

or SF 1.5x or SF 1.5(100 x)

To replicate the cash flows, we first buy two zero-coupon bonds in US$:

The first bond is a one-year bond paying $x,

and the second bond is a two-year bond paying $(100 x).


We then buy two currency options:

The first option is a one-year call SF; the amount is 1.5x,

and the second option is a two-year call SF; the amount is 1.5(100 x).

This investment strategy exactly replicates the currency-option bond cash flows. Consequently,

its total value at time t 0 should be equal to $100. We write below the dollar value of this replication portfolio.


1-Year Bond 1/1.10 x x/1.10

2-Year Bond 1/1.122 100 x (100 x)/1.122

1-Year Option 0.04 1.5x 0.06 x

2-Year Option 0.07 1.5(100 x) 0.105(100 x)

We have:

100 0.9091x 79.7194 0.7972x 0.06 x 10.5 0.105x

1.8713x 9.7806

x 5.23.

The interest rate at issuance is 5.23%.

29. Titi, a Japanese company, issued a six-year Eurobond in dollars convertible to shares of the Japanese

company. At time of issue, the long-term bond yield on straight dollar bonds was 10% for such an

issuer. Instead, Titi issued bonds at 8%. Each $1,000 par bond is convertible into 100 shares of Titi.

At time of issue, the stock price of Titi is 1,600 yen and the exchange rate is 100 yen 0.5 dollar

($/Y 0.005).

a. Why can the bond be issued with a yield of only 8%?

b. What would happen if:

The stock price of Titi increases?

The yen appreciates?

The market interest rate of dollar bonds drops?

c. A year later, the new market conditions are as follows:

The yield on straight dollar bonds of similar quality has risen from 10% to 11%.

Titi stock price has moved up to Y 2000.

The exchange rate is $/Y 0.006.

What would be a minimum price for the Titi convertible bond?

d. Could you try to assess the theoretical value of this convertible bond as a package of other

securities such as a straight bond issued by Titi, options or warrants on the yen value of Titi stock,

an futures and options on the dollar/yen exchange rate?


Solution

a. This low yield is compensated by the conversion option clause.

b. Answer for various scenarios:

If the stock price of Titi in Yen appreciates, so does the dollar price of the convertible bond.

If the yen appreciates so does the dollar price of the convertible bond.

If the market interest rate of the dollar bonds drops, the dollar price of the convertible bond goes up.

c. The valuation of such a bond is fairly complex. However, it should sell at least for its conversion

value:

100 2,000 0.006 $1,200.

d. This is a very difficult exercise. In theory it would require to use the valuation of options on

options. The problem comes from the fact that the conversion value of the bond (its price at time

of conversion) is uncertain; therefore, it is not possible to use conventional currency futures or

currency options to hedge the currency risk. The amount to hedge is variable.

30. Fuji Bank issued convertible Eurobonds in January 1989. Convertible bonds were a popular way for

Japanese banks to raise funds while the Tokyo stock market was booming in the 1980s. The lure of

capital gains from converting the bonds to equity allowed the banks to issue the securities with a very

low interest rate.

Fuji Bank Eurobond was a 500-million Swiss franc zero-coupon bond, issued at par with a maturity

of five years. A bond with a face value of 100 Swiss francs could be converted into two shares of Fuji

Bank at any time starting in 1991. At time of issue, Fujis stock was worth 3,590 yen, and a Swiss franc was worth 80 yen. The bond also had a put option that could be exercised at the start of 1991

(and only at that time). Bondholders had the option of redeeming the bond at a premium of 2.625%

over its face value. In other words, bondholders could obtain 102.625 francs for each bond. On

January 14, 1991, the Tokyo stock market and the yen dropped. A stock of Fuji Bank was worth

2,400 yen, and a Swiss franc was worth 95 yen. Yields on Swiss franc bonds were around 4%. Most

bonds were presented for early redemption.

a. Why was it advantageous for a bondholder to exercise the put option?

b. What was the total yen loss for Fuji Bank?

Solution

a. On January 14, 1991, bondholders had three alternatives:

Converting the bond.

Using the put option to redeem the bond.

Keeping the bond. This meant that the bond holding was worth its market value net of the put option that expired.

Fuji Bank stock only quoted 2,400 yen. Hence, the call option allowed for the exchange of the

bond for two stocks worth 4,800 yen (or SF 50.53), while the bond itself could be redeemed for

SF 102.625. There was no profit at all in converting the bond into stocks.

To value the bond after the expiration of the put option, one could treat it as a zero-coupon SF

bond reimbursed at 100 in three years plus two call options on Fuji stock with a strike price of

100/2 50 francs for each share, while the market price of Fuji stock was only worth 2,400/95 25.3 francs. A three-year Swiss franc zero-coupon bond was worth:

V 100/1.043 88.9.


The call option on Fuji stock with a strike price of 50 francs was deeply out-of-the-money and

worth very little. So the two call options were worth at most a couple of francs. The value of the

convertible bond was well below 100 and it was much more attractive to immediately redeem it

at 102.625.

b. In its books, Fuji Bank carried this liability at historical cost. The total accounting loss posted by

Fuji Bank is equal to the difference between the funds raised at issuance and the put options exercise cost:

500,000,000 80 (102.625%) 500,000,000 95 8.746875 billion yen.

We assumed that all bondholders exercised their put option and that no bonds had been

previously converted into stocks.

31. The current euro yield curve on the euro Eurobond market is flat at 4% for top-quality borrowers. A

French company of good standing can issue plain-vanilla straight and floating-rate dollar Eurobonds

at the following conditions:

Bond A: Straight bond. Five-year straight dollar Eurobond with a coupon of 4%.

Bond B: Floating rate note (FRN). Five-year dollar FRN with a semiannual coupon set at London InterBank Offered Rate (LIBOR).

An investment banker proposes to the French company to issue bull and/or bear FRNs at the

following conditions:

Bond C: Bull FRN. Five-year FRN with a semiannual coupon set at:

7.60% LIBOR.

Bond D: Bear FRN. Five-year FRN with a semiannual coupon set at:

2 LIBOR 4.2%.

The floor on all coupons is zero. The investment bank also proposes a five-year floor option at 2.1%.

This floor will pay to the French company the difference between 2.1% and LIBOR, if it is positive,

or zero if LIBOR is above 2.1%. The cost of this floor is spread over the payment dates and set at an

annual 0.05%. The bank also proposes a five-year cap at 7.60%. The annual premium on the cap is

0.1%. The company can also enter in a five-year interest-rate swap of 4% fixed against LIBOR.

a. Assume that the French company issues Bonds C and D in equal proportions. Is it more

advantageous than issuing Bonds A and B in equal proportion and why?

b. Find out the borrowing cost reduction that can be achieved by issuing the bull Note C compared

to issuing a fixed-coupon straight Bond A at 4%.

c. Find out the borrowing cost reduction that can be achieved by issuing the bull Note C compared

to issuing a plain-vanilla FRN B at LIBOR.

d. Find out the borrowing cost reduction that can be achieved by issuing the bear Note D compared

to issuing a fixed-coupon straight Bond A at 4%.

e. Find out the borrowing cost reduction that can be achieved by issuing the bear Note D compared

to issuing a plain-vanilla FRN B at LIBOR.

Solution

a. It is more advantageous as the average cost is (3.40% LIBOR)/2 as opposed to (4% LIBOR)/2. However one must be careful:

If LIBOR > 7.60% the average cost is (0 2 LIBOR 4.2%)/2, which can be larger than

(4% LIBOR)/2.


If LIBOR < 2.1% the average cost is (7.6% LIBOR)/2, which can be larger than

(4% LIBOR)/2.

To eliminate these risks one should issue C D and buy two floors @2.1% and a cap @7.6%. Then the total cost is:

(3.40% LIBOR 2 0.05% 0.1%)/2. This is still better than A B.

b. Issue the bull, swap the receive fixed pay LIBOR, and buy a cap @7.6%. Total annual coupon:

(7.60% LIBOR) (LIBOR 4%) 0.1% 3.70% or a 0.30% cost reduction.

c. Issue the bull, swap twice the amount to receive fixed pay LIBOR, and buy a cap @7.6%. Total

annual coupon:

7.6% LIBOR 2 (LIBOR 4%) 0.1% LIBOR 0.30% or a 0.30% cost reduction.

d. Issue the Bear note, swap twice the amount to pay fixed and receive LIBOR, buy two floors

@2.1. Total coupon:

(2 LIBOR 4.2%) 2 (4% LIBOR) 2 0.05% 3.9% or a 0.1% cost reduction.

e. Issue the bear note, swap to pay fixed and receive LIBOR, buy two floors @2.1. Total coupon:

(2 LIBOR 4.2%) (4% LIBOR) 2 0.05% LIBOR 0.1% or a 0.1% cost reduction.

32. An FRN is a bond that pays a quarterly or semiannual coupon indexed on a short-term interest rate

such as the LIBOR.

a. Why does it make sense to use a short-term interest rate as the index?

b. Why are banks heavy issuers of FRNs?

Solution

a. A semiannual FRN can be viewed as a (forced) rollover of six-month borrowings. Hence, it

makes sense to link the coupon to the alternative borrowing cost, namely six-month LIBOR.

As we see in Chapter 7, setting the coupon to be equal to the matching short-term interest rate

implies that the FRN should be priced at par on coupon date (for a borrower free of default risk).

This is the index that leads to the most stable bond price.

b. A major activity of a commercial bank is to provide short-term credits to its customers. Because

this activity is permanent, they need stable long-term resources to finance it. However, using

bonds with a fixed yield would introduce interest rate risk between the income received on

credits (floating short-term rate) and the cost paid on debt (fixed-bond rate). The bank can issue

long-term FRNs. In case of dropping interest rates the financing cost on the FRN will decrease to

match the drop in income on short-term credits.

33. A company without default risk can issue a perpetual FRN at LIBOR. The coupon is paid and reset

semiannually. It is certain that the issuer will never have default risk and will always be able to

borrow at LIBOR. The FRN is issued on November 1, 2005, when the six-month LIBOR is at 4.5%.

On May 1, 2006, the six-month LIBOR is at 5%.

a. What is the coupon paid on May 1, 2006, per $1,000 bond?

b. What is the new value of the coupon set on the bond?

c. On May 2, 2006, the six-month LIBOR has dropped to 4.9%. What is the new value of the FRN?


Solution

a. FRNs issued on the international market are fairly standardized. The rate is determined on

the first day of the coupon period and is equal to LIBOR ( margin) as of this date. Hence, the semiannual coupon paid by this perpetual Euro-dollar FRN on May 1, 2006, is equal to the

semiannualized six-month LIBOR as of November 1, 2005 (4.5%/2). This represents a

$22.5 coupon per $1,000 bond.

b. The new coupon to be paid on November 1, 2006, is set at $25.

c. Between each coupon payment, the FRN reacts as a fixed-rate bond. The new value of the FRN

on May 2, 2006, is equal to the discounted sum of the FRN value on the next coupon payment

plus the distributed coupon.

1 000 25$1,000.49

1 (4.9/2)%V

or

P 100.049%.

33. A company without default risk can issue a ten-year FRN at LIBOR. The coupon is paid and reset

semiannually. It is certain that the issuer will never have default risk and will always be able to

borrow at LIBOR. The FRN is issued on November 1, 2005, when the six-month LIBOR is at 4.5%.

Here are the dollar yield curves on two different dates:

May 1, 2006 % August 1, 2006 %

1 Month 5.00 4.00

3 Months 5.00 4.50

6 Months 5.00 5.25

12 Months 5.00 6.00

a. What should the value of the FRN be on May 1?

b. What should the value and the clean price of the FRN be August 1, 2006?

Solution

a. May 1 is a reset date. On that day, the coupon is paid and reset at an annual rate of 5%, or 2.5%

for the semester. The bond should be priced at par, or 100%, on May 1.

b. Three months later, yields have moved. We know that the bond will be worth 100% on

November 1, 2006, when it pays a semester coupon of 2.5%. The current three-month rate is

4.5%. Hence, the present value of the bond on August 1, 2006, is:

1000 25$1,000.49

1 (4.9 / 2%V

or P 100.04%.

If the bond is quoted as a clean price plus accrued interest, the accrued interest on August 1 is

equal to 1.25% (or three months of a coupon of 5%). Hence, the clean price would be:

Pc 101.36% 1.25% 100.11%.


35. A company without default risk can issue a five-year dollar FRN at LIBOR. The coupon is paid and

reset semiannually. It is certain that the issuer will never have default risk and will always be able to

borrow at LIBOR. The FRN is issued on November 1, 2005, when the six-month LIBOR is at 5%.

Here are the dollar yield curves on two different dates:

May 1, 2006 % August 1, 2006 %

1 Month 6.00 5.00

3 Months 6.00 5.50

6 Months 6.00 6.25

12 Months 6.00 7.00

a. What should the value of the FRN be on May 1?

b. What should the value and the clean price of the FRN be August 1, 2006?

Solution

a. May 1 is a reset date. On that day, the coupon is paid and reset at an annual rate of 6%, or 3% for

the semester. The bond should be priced at par, or 100%, on May 1.

b. Three months later, yields have moved. We know that the bond will be worth 100% on

November 1, 2006, when it pays a semester coupon of 3%. The current three-month rate is 5.5%.

Hence, the present value of the bond on August 1, 2006, is:

103101.6030

1 5.5%/4V

.

If the bond is quoted as a clean price plus accrued interest, the accrued interest on August 1 is

equal to 1.5% (or three months of a coupon of 6%). Hence, the clean price would be:

Pc 101.60% 1.50% 100.10%.

36. A company without default risk has issued a perpetual Eurodollar FRN at LIBOR. The coupon is paid

and reset semiannually. It is certain that the issuer will never have default risk, and will always be

able to borrow at LIBOR. The FRN is issued on March 1, 2002, when the six-month LIBOR is at 5%.

The Eurodollar yield curve on September 1, 2002, and December 1, 2002, is as follows.

September 1, 2002 % December 1, 2002 %

1 Month 4.25 4.00

3 Months 4.50 4.25

6 Months 4.75 4.50

12 Months 5.00 4.75

a. What is the coupon paid on September 1, 2002, per $1,000 FRN?

b. What is the new value of the coupon set on the FRN on September 1, 2002?

c. What is the new value and clean price of the FRN on December 1, 2002?

Solution

a. The coupon paid on September 1 is based on the rate set on March 1, which is 5%. Since the

coupon is semiannual, the coupon paid is 5% of $1,000/2 $25.

b. As per the yield curve on September 1, the six-month rate is 4.75%. Thus, the new value of the

coupon set on September 1 is 4.75% of $1,000/2 $23.75.


c. On December 1, three months have elapsed since September 1, and three months are remaining

until the next reset date of March 1, 2003. We know that on this next reset date three months later

that the bond will be worth 100%. Then it will pay a coupon of 2.375% (as set on September 1).

The three-month rate on December 1 is 4.25%. Hence, the present value of the FRN on December 1, is:

1,000 23.75$1,012.99.

1 4.25%/4V

If the bond is quoted as a clean price plus accrued interest, the accrued interest on December 1 is

equal to 1.1875% (or three months of a coupon of 4.75%), or $11.875. Hence, the clean price

would be:

Pc 1,012.9870 11.875 $1,001.11, or 100.11%.

37. A corporation rated AA issues a five-year FRN Eurobond in euros on November 1, 2005. The coupon

is paid quarterly and is equal to euro-LIBOR plus a spread of %. On November 1, the three-month

euro LIBOR is at 4%. The issuer remains rated at AA during the life of the bond.

a. Three months later (February 1, 2006), the three-month euro-LIBOR has moved to 4.5%, and the

market-required spread for AA borrowers has remained at %. What should the value of the

bond on reset date be?

b. Three months later (May 1, 2006), the three-month euro-LIBOR is still at 4.5%, but the market-

required spread for AA borrowers has increased to %. Give some estimation of the new value

of the FRN on the reset date.

Solution

a. 100%.

b. The risk premium required by the market has just moved for this type of borrower. The spread set

on the bond being constant, the bond partly reacts as a fixed-rate bond even on coupon dates.

As mentioned in the text, a common practice is to calculate an approximate value of the bond

under the freezing assumption. Freezing assumes that euro-LIBOR and credit spread will

stay constant over the life of the bond (quarterly coupon of [4.5 ]%/4 1.25%); the frozen

cash flows are then discounted at the current market-required rate ([LIBOR %]/4 1.3125%).

18

181

1.25 10099%.

(1 5.25/ 4 %) (1 5.25/ 4 %)ttP

The price dropped by 1%.

38. A corporation rated A has issued a semiannual FRN in dollars. This is a perpetual bond, which will

pay coupons indefinitely if the corporation does not default. The coupon is set at six-month LIBOR

plus a spread of %. The six-month dollar LIBOR is equal to 5%.

Six months later, the six-month dollar LIBOR has remained at 5%, but the market-required spread for

A-rated corporations on long-term FRNs has moved to 1%. Give some estimation of the new value of

the FRN on the reset date.


Solution

No exact theoretical answer can be provided. A common industry practice is to assume that the

LIBOR will remain forever at its current level (5%). Under this assumption, the bond becomes a

perpetual bond with a fixed coupon of 5.75%, while the market-required yield is 6%. The new value

of the bond should be:

2 3

2.875 2.875 2.875 2.87595.83%.

1.03 1.03 1.03 0.03P

Note that we discounted semiannual coupons of 2.875 5.75/2 at a required semiannual yield of

3% 6%/2.

39. In March 1993, the Student Loan Marketing Association (Sallie Mae) issued five-year notes with a

coupon set at 4.5% in the first year and reset quarterly subsequently. The floating quarterly coupon

rate was set to be the higher of either 4.125% or 50% of the rate on ten-year Treasury notes plus

1.25%. At time of issue, the interest rates for all maturities were well below 4%, and investors were

attracted by the high current yield (4.5%) compared to other straight bonds available.

Assume that in March 1994, interest rates have risen dramatically and that the U.S. Treasury yield

curve is now flat at 7% for all maturities.

a. What is the new coupon rate set on the Sallie Mae bond?

b. Why is the Sallie Mae bond now trading at a hefty discount?

Solution

a. The new coupon is set to 50% of the ten-year Treasury rate plus 1.25% or:

C 3.50% 1.25% 4.75%.

b. The coupon of the Sallie Mae bond is well below that of U.S. Treasuries with similar maturity

(4.75% compared to 7%). The only advantage of the Sallie Mae FRN is to provide a minimum

coupon of 4.125% if yields drop in the future. Given the current level of interest rates for top

quality notes, the value of this protective option must be very small. This Sallie Mae bond did

trade at a large discount in March 1994.

40. The current dollar yield curve on the Eurobond market is flat at 7% for top-quality borrowers. A

French company of good standing can issue plain-vanilla straight and floating-rate dollar Eurobonds

at the following conditions:

Bond A: Straight bond. Five-year straight dollar Eurobond with a coupon of 7.25%.

Bond B: FRN. Five-year dollar FRN with a semiannual coupon set at LIBOR plus % and a cap of 14%. The cap means that the coupon rate is limited at 14% even if the LIBOR passes 13.75%.

An investment banker proposes to the French company to issue bull and/or bear FRNs at the following conditions:

Bond C: Bull FRN. Five-year FRN with a semiannual coupon set at: 13.75% LIBOR.

Bond D: Bear FRN. Five-year FRN with a semiannual coupon set at: 2 LIBOR 7%.


The coupon on a bull FRN will increase when LIBOR drops. This is sometimes known as a reverse floater. The coupon on the bull FRN cannot be negative, so it has a floor of zero. The bear FRN will benefit from a rise in interest rates. The coupon on the bear FRN is set with a cap of 20.50%.

a. Explain why a bull FRN could be attractive to some investors.

b. Explain why a bear FRN could be attractive to some investors.

c. Explain why it would be attractive to the French company to issue these FRNs compared to

current market conditions for plain-vanilla straight Eurobonds and FRNs. The company assumes

that LIBOR can never be below 3.5% or above 13.75%.

Solution

a. The bull FRN is attractive to investors who believe in a future drop in interest rates. This reverse

floater can be viewed as the sum of:

A long position in two straight bonds with a coupon of 7%,

and a short position in one plain-vanilla FRN at LIBOR 0.25%.

This is a leveraged instrument in fixed-coupon bonds (invest 100 and borrow 100 to buy 200 of

fixed-coupon bond). The value of the fixed-coupon bonds will rise if long-term rates drop. In

case of an unexpected rise in LIBOR, the structure of the bull FRN caps de facto the LIBOR side

at 13.75% (as the coupon on the bull FRN cannot become negative).

b. The bear FRN is attractive to investors who believe in a rise in interest rates. It can be viewed as

the sum of:

A long position in two plain-vanilla FRNs at LIBOR flat,

and a short position in one straight bond with a fixed coupon of 7%.

If interest rates rise, the value of the FRNs should remain stable but the value of the straight bond

should drop, hence, a rise in the value of the bear FRN. In case of a dramatic drop of LIBOR, the

structure of the bear FRN provides a de facto floor on the LIBOR side at 3.5% (as the coupon on

the bear FRN cannot become negative).

c. The attractiveness of these bull and bear FRNs for the French company can be judged by

considering various issuing combinations.

Issue 2 bull FRN 1 bear FRN

The net coupon on the three bonds is:

2(13.75% LIBOR) (2 LIBOR 7%) 20.5%

or 6.8333% per bond compared to 7.25% on a straight bond.

Of course, the fact that coupons on bonds cannot become negative would affect this argument if

LIBOR dropped below 3.5% or rose above 13.75%.

Issue 1 plain FRN (Bond B) 1 bull FRN

The net coupon on the two bonds is:

(LIBOR 0.25%) (13.75% LIBOR) 14%

or 7% per bond compared to 7.25% on a straight bond.

Because the plain FRN (Bond B) is capped at 14% (LIBOR 13.75%), the cost advantage is certain for any value of LIBOR.

Issue 1 straight bond (Bond A) 1 bear FRN


The net coupon for the two bonds is:

(7.25%) (2 LIBOR 7%) 2 LIBOR 0.25%

or LIBOR 1/8% per bond compared to LIBOR 1/4% on the plain FRN.

This cost advantage will disappear if LIBOR drops below 3.5%.

41. The French luxury-goods company LVMH, Louis VuittonMot Hennesy, issued a series of perpetual floating-rate notes on the international capital market in the 1990s. These bonds have the advantage

of being quasi-equity, while benefiting from favorable tax treatment. Pioneered by state-owned

French firms that cannot sell stock to the public, and subsequently used by a number of private

European companies that were reluctant to dilute their stocks, the subordinated perpetual floating-rate

note is an instrument that remains outstanding in name only. These securities are called instantly

repackaged perpetuals, or IRPs.

After a 5-billion franc issue in 1990, LVMH sold, in March 1992, 1.5 billion francs of IRPs. The

company received 1.1 billion francs, the remaining 400 million being transferred to an offshore trust.

The trust used the proceeds to buy fifteen-year zero-coupon bonds issued by banks underwriting the

LVMH issue or by sovereign borrowers such as Denmark and Austria. The 400-million investment in

zero-coupon bonds will be redeemed for 1.5 billion in fifteen years. The IRPs have the peculiarity

that they pay interest only for the first fifteen years; the interest becomes nil thereafter. After these

fifteen years, the trust is committed to repurchase the perpetuals at their face value of 1.5 billion

francs. The trust, especially set up for this purpose, will then hold the IRPs forever, but their market

value has become zero as they are perpetuals, which pay no interest. The semiannual coupon was set

at six-month PIBOR (Paris InterBank Offer Rate) plus %.

From an accounting viewpoint, these IRPs are treated as new equity of LVMH, because they are

perpetual. From a tax viewpoint, the interest paid on the IRPs during fifteen years can be deducted as

interest expense (while dividend payments are not tax deductible).

a. Assume that you are an investment banker proposing such an IRP to a potential client. Explain in detail the advantage of such a package relative to a plain-vanilla fifteen-year FRN, or relative to a new stock issue.

b. In 1990, the French tax authorities decided to allow a write-off of interest expense for only the net amount of capital that the issuer actually takes on its books (1.1 billion for LVMH). Why does this decision reduce the attraction of issuing IRPs?

c. Following the 1992 LVMH issue, the tax authorities decide to introduce a new regulation for trusts, whereby capital gains would be taxed at the normal income tax rate. In effect, the trust would make a capital gains equal to the difference between the face value of the zero-coupon bonds and their issue price. This basically shut the market for IRPs. Why?

Solution

a. It could be useful to summarize the features of this issue. For simplicity, assume a flat yield-curve. The 1992 IRP issue has a size of FF 1.5 billion. FF 1.1 billion go to LVMH as quasi-equity. FF 0.4 billion go to an offshore trust fund and is used to purchase zero-coupon bonds whose redemption value in fifteen years will be exactly FF 1.5 billion (to repurchase the IRPs).

Hence, the package means that LVMH is basically borrowing FF 1.5 billion for fifteen years. It sets aside FF 0.4 billion earmarked for the repayment of the principal in fifteen years.


Lets focus on the tax situation until 1990. The big tax advantage is that the capital gain on the FF 0.4 billion invested by the Trust is not taxable as capital gains or income. To make the argument more transparent, it can be useful to assume that LVMH does not need the money and only uses the proceeds to invest in similar bonds to take advantage of the tax situation. LVMH could invest the FF 1.1 billion in fifteen-year FRNs. The resulting tax situation would be as follows:

All interest payments on the FF 1.5 billion of IRP are tax-deductible.

Interest received on the FF 1.1 billion FRNs is taxable.

Interest received (in the form of capital gain) on the FF 0.4 billion zero-coupons is not taxable, because of the offshore status of the trusts.

There is a tax gain on FF 0.4 billion:

Advantages relative to a plain-vanilla fifteen-year FRN

From a French accounting viewpoint, IRPs are treated as equity because they are perpetual (no need to reimburse them), so the issue helps improve financial ratios. LVMH is taking on what looks like more debt (1.1 billion) but is actually improving its debt-to-equity ratio. In that sense, IRPs are preferable to traditional FRN debt. From a tax viewpoint, IRPs are much preferable.

Advantages relative to a stock issue

IRPs do not lead to any dilution of ownership. Coupons are tax-deductible while dividends are not.

b. Under the new tax regime (19901992), the tax incentive for issuing IRPs has disappeared. The amount of capital whose interest is tax-deductible is set at FF 1.1 billion. The balance for which the tax deductibility is lost (FF 0.4 billion) corresponds exactly to the initial capital of the fund. The tax situation of IRPs is now neutral. The only advantage of IRPs is their quasi-equity status.

c. In 1992, the French government became unhappy to see so many foreign entities issuing FF zero-coupon bonds. It felt that they were tapping the French bond market and reducing its ability to raise capital to finance its budget deficit. The 1992 tax provision was a discrete manner to shut the market for IRPs. Taxing the trusts on interest received (in the form of capital gains) on their zero-coupon bond investment was a logical decision. However, having both the 1990 tax provision and this new 1992 provision meant that the IRP became at a tax disadvantage compared to a straight debt issue. The French tax authority could impose a specific tax on such offshore trust funds because they were ad-hoc entities set up for a single purpose by the borrower and with strict contractual relations.

42. Which of the following statements about the Macaulay duration of a zero-coupon bond is true? The Macaulay duration of a zero-coupon bond:

a. Is equal to the bonds maturity in years.

b. Is equal to one-half the bonds maturity in years.

c. Is equal to the bonds maturity in years divided by its yield-to-maturity.

d. Cannot be calculated because of the lack of coupons.

Solution

a. Is equal to the bonds maturity in years.


43. Which of the following bonds has the longest duration?

a. 8-year maturity; 6% coupon.

b. 8-year maturity; 11% coupon.

c. 15-year maturity; 6% coupon.

d. 15-year maturity; 11% coupon.

Solution

c. 15 year maturity; 6% coupon.

44. A bond with annual coupon payments has the following characteristics:

Coupon Rate Yield-to-Maturity Macaulay Duration

8% 10% 9

The bonds modified duration (in years) is:

a. 8.18 years.

b. 8.33 years.

c. 9.78 years.

d. 10.00 years.

Solution

a. 8.18 years.

45. A nine-year bond has a yield-to-maturity of 10% and a modified duration of 6.54 years. If the market

yield changes by 50 basis points, the bonds expected price change is:

a. 3.27%

b. 3.66%

c. 5.00%

d. 6.54%

Solution

a. 3.27%.

46. You are a U.S. investor considering purchase of one of the following securities. Assume that the

currency risk of the German government bond will be hedged, and the six-month discount on

Deutsche mark forward contracts is 0.75% versus the U.S. dollar.

Bond Maturity Coupon Price

U.S. Government June 1, 2013 6.50% 100.00

German Government June 1, 2013 7.50% 100.00

You do not expect any price change in U.S. bond prices in the next six months. Calculate the expected

price change required in the German government bond, which would result in the two bonds having

equal total returns in U.S. dollars over a six-month horizon.


Solution

To compare the German bond performance versus the U.S. bond performance, one must first

compute the return over the six-month period for each security. The return for each security is:

Return Income forward discount or premium change in bond price.

Return on German Bond 7.5/2 ( 0.75) Change in German Bond Price.

Return on U.S. Bond 6.5/2 0 0 3.25.

If Return on German Bond Return on U.S. Bond, then:

3.75 0.75 Change in German Bond Price 3.25.

Change in German Bond Price 0.25% an increase in price of 1/4 point.

47. Guaranteed note.

You are a young banker offering a client to issue a guaranteed note. The yield curve is flat at 9% for

each maturity. Options on the stock index are offered by banks. An at-the-money call with a two-year

maturity trades at 12% of the index value, whereas a three-year call is worth 15% of the index.

You wonder about the characteristics of the bond. If you offer a high coupon, the indexation will be

low. Therefore, you decide to compute the indexation levels in accordance to the current market

conditions for maturities of two and three years and coupon levels of 0%, 2%, and 5%.

Solution

The guaranteed note can be regarded as a fixed rate bond paying a coupon C and p calls on the index

(p being the indexation level).

For a two-year bond we have:

2 2

100100 15

1.09 1.09 1.09

15.832 1.759.

15

C Cp

Cp

For a three-year bond we have:

2 3 3

100100 18

1.09 1.09 1.09 1.09

22.782 2.531.

18

C C Cp

Cp

The following table shows the indexation for different coupons and maturities.

Coupon p (2 years) p (3 years)

0% 105.55% 126.57%

2% 82.09% 98.44%

5% 46.91% 56.26%


48. Youre a banker. A client wishes to buy a guaranteed note with a 100% indexation to the stock indexs growth. In other words, he does not want any coupon but requires 100% of the index growth. You wonder about the maturity of such a note. You check the prices of various index calls traded on

the market for different maturities. Their strike is the current index level and their price is expressed

as a percentage of this level. (For instance, if the CAC is worth 3,000, the strike is 3,000, and the one-

year maturity call trades at 11% of 3,000. You also check the price of a zero-coupon in percentage for

various maturities. The following graph shows, for each a maturity, the price of the option, that of the

zero-coupon and 100%-zero.

a. What is the maturity of the guaranteed note (Coupon 0%, indexation 100%)? Justify.

b. If as a banker, you want to make a profit, should you lengthen or shorten the maturity of that note?

Explain why.

c. Everything remaining constant (i.e., same volatility and interest rate), should the maturity

of the guaranteed note be shorter or longer if the index pays a low dividend rather than a

high one? Why?

Solution

a. We must take the intersection between the curves option and 100-zero, that is, about four years.

b. To make a profit, a longer maturity should be offered (sold at 100%) for the guaranteed note.

c. The buyer of index options or guaranteed note loses the dividend (compared to a spot buy). If the

dividend yield is high, the option price will be lower. Therefore, a shorter maturity can be offered.

49. The investment fund of Lemon County of California is investing $1 billion in a leveraged-bond hedge

fund. This hedge fund has the following structure:

$4 billion invested in a reverse-floater (also called bull FRN). This is a five-year bond with a coupon set at: 9% minus three-month LIBOR.

$3 billion borrowed at: three-month LIBOR.

The current yield curve is flat at 4.5%. The reverse floater is currently priced at 100%.

a. Estimate the yield-enhancement over LIBOR that the hedge fund would provide if the yield

curve drops uniformly by 10 basis points (0.1%).


Actually, the whole yield curved moved up to 7% within a few days.

b. What would be the new income (coupon rate) on this $1 billion investment made by Lemon

County?

c. Can you provide some rough estimate of the new market value of this $1 billion investment?

Solution

a. For a net investment of $1 billion, the coupon paid is equal to:

4 (9% LIBOR) 3 LIBOR 36% 7 LIBOR.

If the current yield curve drops to 4.4%, the yield-enhancement over LIBOR is:

(36% 7 4.4%) 4.5% 5.2% 4.5% 0.7%

or 80 bp from 4.4.

b. A few days later, the whole yield curved moves up to 7%. The new coupon rate is therefore:

36% 7 7% 13%.

This is an annual loss of 130 million.

c. Lets value the fund as:

$4 billion long in the reverse floater,

and, $3 billion short in a plain-vanilla floater.

Lets further assume that the market price of the reverse floater is set as the package of two bonds:

Two straight bonds with a fixed coupon of 4.5%,

a short position in one plain-vanilla floater at LIBOR flat.

Then, 100 invested in the hedged fund can be seen as 800 long in a straight fixed-coupon bond and 700 short in a plain-vanilla FRN:

PRICE (hedge fund) 8 PRICE (fixed) 7 PRICE(FRN)

The interest rate movement should leave the price of the FRN unchanged (there will be a slight

drop in value because the coupon is fixed until the next quarterly reset date), the five-year bond

with a fixed coupon of 4.5% should drop to 89.75% if rates move up to 7%.

5

51

4.5 10089.75

(1.07) (1.07)tt

hence:

PRICE (hedge fund) 8 89.75% 7 100 17.99.

For a $1 billion investment; this is a value of US$ 0.18 Bn.

The total investment is almost wiped out.


50. Strumpf Ltd. decides to issue a convertible bond with a maturity of two years. Each bond is issued

with a nominal value of 100 and an annual coupon C; of course, C has to be determined. Each bond

can be redeemed for 100 or converted into one share of Strumpf at the option of the bondholder.

The current stock price of Strumpf is 90. The yield curve for an issuer like Strumpf is flat at 6%.

Barings is ready to issue long-term options on Strumpf shares. The premiums on calls with one-year

and two-year expirations are given below:

Strike

Price

European-Type American-Type

1-Year 2-Year 1-Year 2-Year

90 11 16 12 17

100 6 8 6.5 9

a. American-type calls are more expensive than European-type calls. Is it reasonable?

b. Assume that the bond can only be converted at maturity, after payment of the second coupon.

What should be the fair coupon rate C, consistent with the above market conditions?

c. Assume that the bond is issued with the coupon rate determined above. The yield curve suddenly

moves from 6% to 6.1% and the option premiums stay the same. What should be the new market

price of the convertible bond?

d. Assume now that the bond can be converted on two dates (rather than one as above). These dates

are the first year (right after the first coupon payment) and the second year as above. It is not

possible to convert the two-year bond at any other date. Is it possible to construct an arbitrage

portfolio allowing to price the fair coupon C with the above data? Be precise in your explanation

and state what type of options you would need to price the bond.

Solution

a. American-type calls give the investor the possibility (but not the obligation) to exercise the

option at any time before maturity. This possibility, which offers additional profit opportunities,

justifies an additional price.

b. The two-year convertible bond can be valued as the sum of:

A one-year zero-coupon bond paying C at maturity,

a two-year zero-coupon bond paying (C 100),

and a two-year European-type call (strike 100).

As a result, the value of this portfolio must be equal to the value of the newly issued

convertible bond.

2

100100 8

1.06 1.06

? 1.64

C C

C

hence, the fair coupon rate, c, is equal to 1.64%.

c. The new market price of the convertible bond is:

2

1.64 101.648 99.83.

1.061 1.061P


d. The bond is now convertible either after the first coupon (one year) or after the second coupon (two years). Lets imagine that the bond is converted after one year. Thus, it no longer exists. It would, therefore, be incorrect to replicate this convertible bond with a one-year zero-coupon bond paying C, a two-year zero-coupon bond paying 100 C, a one-year European type call (strike 100), and a two-year European type call. Besides, if the bond is converted after the first coupon, the implicit strike price of this conversion right is the value of the bond (without the conversion right) after the first coupon. This value cannot be known in advance. All we know is the conversion price after the second coupon, which is 100.

51. On April 1, 2000, a corporation rated AA has issued a semiannual FRN in dollars. This is a perpetual bond, which will pay coupons indefinitely if the corporation does not default. The coupon is set at six-month LIBOR plus a spread of %. The six-month dollar LIBOR is equal to 5%.

a. Three months later (July 1, 2000), the corporation is still rated AA and the market-required credit spread for AA is still at %. We observe the following LIBOR rates:

1-Month 3-Month 6-Month

3 3/4% 4 % 5 %

Give an estimation of the total value of the bond. What should be its quoted price?

b. Three months later (October 1, 2000), the coupon has just been paid. The six-month dollar LIBOR is again at 5%, but the market-required spread for AA-rated corporations on long-term FRNs has moved to 1%. Give some estimation of the new value of the FRN on reset date.

Solution

a. On the reset date or October 1, the bond should trade at par (100%) and the coupon to be paid is @ 5%, or 2.75%, so the total value should be 102.75%. Three months before (July 1) the total value of the bond should be discounted at the three-month interest rate:

V 102.75/(1 4.5%/4) 101.607.

Accrued interest is in both cases 5.5%/4 1.375.

The quoted (clean) price is 101.607 1.375 100.232.

b. We freeze LIBOR. The value of a perpetual is given by:

V C/r 5.5/6 91.67%.

52. The Kingdom of Papou issues a very-bull bond with a coupon equal to:

14.6 2 LIBOR.

Of course, the coupon cannot be negative.

The Kingdom could have issued a FRN at LIBOR %, or a straight bond at 5.30%.

The current market conditions for swaps are 5% against LIBOR.

You could also trade in CAPS and FLOORS with different exercise prices (these are levels of interest

rates). The premiums are paid annually.

Exercise

Interest Rate

Annual Premium

Cap Floor

7.3 % 0.2% 2%

14.6 % 0.1% 10%


a. You are a buyer of this very-bull bond. Tell us what it is equivalent to, in terms of buying/selling:

FRN, straight bonds, caps or floors?

b. Assume that the Kingdom actually wanted to issue a straight bond (fixed coupon). The bank will

put in place a de-mining portfolio with swaps and options so that this very-bull bond plus the de-mining portfolio is equivalent to a straight bond. What is exactly the de-mining portfolio? [Be very precise and tell us if the Kingdom must pay fixed, receive LIBOR or vice versa, etc. ...]

c. What is the cost advantage for the Kingdom compared to issuing bonds @ 5.30 %?

d. Same question assuming that the Kingdom wanted to issue an FRN @LIBOR %?

Solution

a. For the investor, this is equivalent to:

Long in three straight bonds.

Short two FRN.

Long two caps with a strike of 7.3%.

b. To convert the very-bull bond into a fixed-coupon bond, the Kingdom will:

Issue the very-bull bond for a capital of 100.

Swap 200 to pay LIBOR and receive 5%.

Buy 200 of caps with strike of 7.3% at a cost of 0.4%.

c. The net result is a fixed cost of 5% and a saving of 30 bp.

d. To convert the very-bull bond into a FRN, the Kingdom will:

Issue the very-bull bond for a capital of 100.

Swap 300 to pay LIBOR and receive 5%.

Buy 200 of caps with strike of 7.3% at a cost of 0.4%.

The net cost is the premium on LIBOR or a savings of 25 bp.

53. Inflation indexed bonds.

Many countries, among which the United Kingdom, the United States, and France, have issued

inflation indexed bonds. Coupons and reimbursement value depend on the price index at the time of

payment. Lets assume that a bond has been issued for 100 at time 0, with a maturity n and a real

coupon equal to 0 . Let It be the price index at time t. The coupon paid at t will be:

Ct 0 It /I0.

And the reimbursement value at maturity n will be:

Rn 100 In/I0.

Since the reimbursement is also indexed on the price index, we can easily check that the actual yield

is equal to the real coupon accrued by the inflation rate.

However, the real interest rate required by the market fluctuates with time. Knowing the market price

of an inflation indexed bond and its real coupon, we can easily compute (using a discounting method)

its real yield at any time t.


If the indexed bond still has n years to maturity, we just have to use the discounting method for a real

cash-flow bond:

0 0 0

1 2

100.

(1 ) (1 ) (1 )nP

Knowing the real interest rate we can compare it with the nominal interest rate r on classic bonds.

a. In terms of risk, what is the interest of such bonds? What kind of investors is it aimed at?

b. In terms of return, assume that yield curves r and are flat and that we expect the inflation level to remain constant for the coming years. You expect an annual inflation rate of . In what case do you prefer an inflation-indexed bond to a straight bond? [Find the relation between r, and ]

Solution

a. Coupons and reimbursement increase if inflation rates increase. Indexed bonds are, therefore,

protected against the inflation risk. They are securities without any risk in real terms. A pension

fund that will pay pension indexed on wages regards these bonds as risk-free whereas straight

bonds bear an inflation risk.

b. In nominal terms, the indexed bond will pay a nominal rate P such as

R (1 ) (1 ).

It is, therefore, interesting to buy the bond if R > r or (1 ) (1 ) > (1 r).

chapter 7 global bond investing hw solutions

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