Ch10 HW Solutions

Chapter 10 Derivatives: Risk Management with Speculation,

Hedging, and Risk Transfer

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. A Swiss portfolio manager has a significant portion of the portfolio invested in dollar-denominated assets. The money manager is worried about the political situation surrounding the next U.S. presidential election and fears a potential drop in the value of the dollar. The manager decides to sell the dollars forward against Swiss francs.

a. Give some reasons why the Swiss money manager should use futures rather than forward currency contracts?

b. Give some reasons why the Swiss money manager should use forward currency contracts rather than futures?

Solution

a. Some reasons to use futures rather than forward currency contracts:

The money manager does not require a specific maturity as there is no specific cash flow to hedge. A futures contract with an expiration date extending beyond the election date would be acceptable.

A professional money manager is well-equipped to deal with daily marking-to-market.

Futures can be cheaper than forward as they are standardized and traded on an organized market. Forwards are customized contracts, and hence, are often more expensive unless they are of a large size.

Futures are tradable at any time while forwards are not, so the hedge can easily be removed at any time, while removing a forward hedge usually requires the writing of an opposite contract.

b. Some reasons to use forward currency contracts rather than futures:

It is easy to arrange a forward through a bank for a specific amount and maturity.

Forward contracts do not require the daily cash adjustments required by the marking-to-market procedure of futures contracts.

Currency forwards are administratively less burdensome than futures contracts.

Forward contracts can be arranged for large amounts, while the liquidity of currency futures contracts could be limited. So, the cost of implementing a currency forward could be less than the cost of implementing a currency futures hedge.

Chapter 10 Derivatives: Risk Management with Speculation, Hedging, and Risk Transfer 117

2. Why are futures contracts commonly believed to be less subject to default risk than forward contracts?

Solution

Futures markets have put in place successful procedures to protect clients from the default of

a counterparty:

The counterparty is always the clearinghouse, not a private party.

A centralized margin deposit system.

Guarantees posted by all members who are collectively responsible.

Daily marking-to-market. Variation limits can make this process take place during the day if

needed.

Liquidity of an organized market for standardized contracts.

3. Let’s consider a Swiss franc futures contract traded in the United States. On February 18 (a Friday),

the March contract closed at 0.7049 dollar per Swiss franc. The size of the contract is 125,000 Swiss

francs. The initial margin is $2,600 per contract and the maintenance margin is $1,600. Assume that

you buy one March contract on February 19 at 0.7049 $/SFr and you deposit, in cash, an initial

margin of $2,600. Listed below are the futures quotations (settlement prices) observed on three

successive days:

Feb. 18 Feb. 20 Feb. 21 Feb. 22

0.7049 0.7009 0.6949 0.7089

What are the cash flows associated with the marking-to-market procedure?

Solution

Let’s review the cash flows associated with these price fluctuations:

February 20: You lose 0.0040 dollars per franc, or $500 per contract which is debited from your

initial deposit. Your margin is now equal to $2,100, which is above the maintenance margin.

You do not have to reconstitute the initial margin.

February 21: You lose an additional 0.0060 dollars per franc, or $750 per contract, which is

debited from your margin account. Your margin is now equal to $1,350, which is below the

maintenance margin. You have to reconstitute the initial margin up to $2,600 by transferring

$1,250 to your margin account.

February 22: You gain 0.0140 dollars per franc, or $1,750 per contract. You can use this $1,750

as you like, since your initial margin is intact at $2,600.

The net result on February 22, is that you have a net gain of $500 (1,750 750 500) from the day

you initially bought the contract. If you decided to sell back the contract on February 22, your margin

deposit of $2,600 would be given back to you, and the net gain of $500 would be yours.

4. A German investor holds a portfolio of British stocks. The market value of the portfolio is £20 million,

with a of 1.5 relative to the FTSE index. In November, the spot value of the FTSE index is 4,000.

The dividend yield, euro interest rates, and pound interest rates are all equal to 4% (flat yield curves).

a. The German investor fears a drop in the British stock market (but not in the British pound).

The size of FTSE stock index contracts is 10 pounds times the FTSE index. There are futures

contracts quoted with December delivery. Calculate the futures price of the index.

b. How many contracts should you buy or sell to hedge the British stock market risk?

118 Solnik/McLeavey • Global Investments, Sixth Edition

c. You believe that the capital asset pricing model (CAPM) applies to British stocks. The expected

stock market return is 10%. What is the expected return on this portfolio before and after hedging?

d. You now fear a depreciation of the British pound relative to the euro. Will the strategies above

protect you against this depreciation? (Assume that the margin on the futures contract is

deposited in euros.)

e. The forward exchange rate is equal to 1.4 € per £. How many pounds should you sell forward?

Solution

a. The arbitrage value of the futures price of a stock index contract is equal to its spot value plus the

basis. The basis is equal to the difference between the interest rate and the dividend yield, times

the spot value of the index. In a cash and carry arbitrage, the arbitrageur buys the stocks in the

index and sells the futures contract forward. In carrying the stocks, the arbitrageur has to finance

the position at the pound interest rate, but receives the dividends on the stocks (which are not

paid or received on the futures contract). The futures price should be equal to the spot price since

all yields are equal to 4%:

F S 4,000.

b. The minimum-variance hedge ratio is equal to the beta of the portfolio. You should sell N stock

index futures contracts, adjusting for the beta of the portfolio:

N portfolio value

betastock index contract size

N ? 0 million 1.5

7,500.4,000 ? 0

c. According to the CAPM, the expected return on the British portfolio (before hedging) is:

E(R) 4% 1.5 (10% 4%) 13%.

If you hedge the portfolio by selling FTSE contracts as in Question (b), the expected return

becomes the risk-free interest rate of 4%.

d. Hedging with British stock index futures does not protect you against a depreciation of the

British pound.

e. Your current exposure is £20 million and this is the amount to be sold forward.

5. You hold a portfolio made of French stocks and worth €10 million. The beta ( ) of this portfolio

relative to the CAC index is 1.5. The interest rate for the euro is 4% for all maturities and the annual

dividend yield is 2%. The spot value of the CAC index on January 1, 2000, is 5,000. A CAC contract has a size of €10 for each index point.

a. What should be the future price of the CAC contract with a three-month maturity?

b. You fear a fall in the French stock market. What should be your hedge ratio? How many

contracts do you buy/sell?

Solution

a. F 5,000 ((4% – 2%)/4) 5000 5,025.

b. h* 1.5.


and

N portfolio value

betastock index contract size

€10 million/(5,000 10) 1.5.

N 300.

6. To capitalize on your expectation of a 10% gold price appreciation, you consider buying futures or

option contracts to speculate. The spot price of gold is $400. Near-delivery futures contracts are

quoted at $410 per ounce with a margin of $1,000 per contract of 100 ounces. Call options on gold

are quoted with the same delivery date. A call with an exercise price of $400 costs $20 per ounce.

The rate of return on your speculation will be the return on your invested capital, which is the initial

margin for futures and the option premium for options.

a. Based on your expectation of a 10% rise in gold price, what is your expected return at maturity

on futures contracts?

b. Based on your expectation of a 10% rise in gold price, what is your expected return at maturity

on option contracts?

c. Simulate the return of the two investments for various movements in the price of gold.

Solution

a The expected rate of return on the futures margin deposit is equal to 300%. This is found by

observing that the margin per ounce of gold is $10($1,000 for contract of 100 ounces). With a

10% gold price appreciation of $40, the spot price of gold will rise to $440, which will also be

the futures price on delivery date. Hence, a profit of $440 – $410 $30, for an initial investment

of $10.

b. At expiration, the option is expected to be worth $40 per ounce, since the gold price is expected

to be $440 and the exercise price is $400 per ounce. This leads to a net profit of $20 and a rate of

return on the initial $20 investment of 100%.

c. Gold Price Simulation

$320 $360 $400 $440 $480

Rate of Return:

Gold Bullion 20% 10% 0% 10% 20%

Futures 900% 500% 100% 300% 700%

Option 100% 100% 100% 100% 300%

7. In Hong Kong, the size of a futures contract on the Hang Seng stock index is HK $50 times the index.

The margin (initial and maintenance) is set at HK $32,500. You predict a drop in the Hong Kong

stock market following some economic problems in China and decide to sell one June futures

contract on April 1. The current futures price is 7,200. The contract expires on the second-to-last

business day of the delivery month (expiration date: June 27). Today is April 1, and the current spot

value of the stock market index is 7,140.

a. Why is the spot value of the index lower than the futures value of the index?

b. Indicate the cash flows that affect your position if the following prices are subsequently observed:

April 1 April 2 April 3 April 4

Hang Seng Futures 7,200 7,300 7,250 6,900


Solution

a. The futures price is higher than the spot price probably because the short-term interest rate is

higher than the dividend yield (positive basis).

b.

April 1 April 2 April 3 April 4

Gain/Loss 0 5,000 2,500 17,500

Margin before Cash

Flow

0 27,500 35,000 50,000

Cash Flow 32,500 5,000 2,500 17,500

Margin after Cash Flow 32,500 32,500 32,500 32,500

8. Derive a theoretical price for each of the following futures contracts quoted in the United States and

indicate why and how the market price should deviate from this theoretical value. In each case,

consider one unit of underlying asset. The contract expires in exactly three months, and the

annualized interest rate on three-month dollar London InterBank Offered Rate (LIBOR) is 12%.

All interest rates quoted are annualized.

Contract Useful Information

a. Gold Futures: Spot gold price $300 per ounce; cost of storage

$0.50 per ounce per month

b. Currency Futures: $/€ spot exchange rate 1.10 dollars per euro;

3-month euro interest rate 4%

c. Eurodollar Futures: (3-month

$ LIBOR): 6-month $ LIBOR interest rate 10%

d. Stock Index Futures: Current value of stock index 1,200; annual

dividend yield 2%

Solution

F is the futures price and S the spot price.

a. Gold futures

F S (1 3-month interest rate) cost of storage.

F 300 (1.03) 1.5 $310.5.

This is a pure arbitrage relation. It must hold exactly for a forward contract and closely for a

futures contract. The market price of the futures could deviate from this theoretical price because

arbitrage costs on this physical asset are quite high.

b. Currency futures

€

€$1 1 12%/ 4

1.1 1.1218 $ /1 1 4%/ 4

rF S

r

where: $r is the dollar interest rate,

€r is the euro interest rate.

This is a pure arbitrage relation. It must hold exactly for a forward contract and hold closely for a

futures contract. Arbitrage costs are very small on the currency market.


c. Eurodollars futures

The futures price is equal to 100% minus the annualized forward interest rate. The forward

interest rate rF is the three-month interest rate that will be valid in three months (the delivery date

of the futures contract). By arbitrage, it should be equivalent today to buy the futures contract or

to invest for six months (rm interest rate to maturity) and simultaneously borrow for three months

(rd interest rate to delivery).

11 .

1

m

F

d

rr

r

Here, the interest rate to maturity is rm 10% 6/12 5%.

The interest rate to delivery is rd 12% 3/12 3%.

The futures interest rate is then rF 1.05/1.03 1 1.9417%.

On an annual basis, this is equal to 1.9417% 12/3 7.7670%.

Therefore, the futures price on the eurodollar contract should be equal to:

F 100 7.7670 92.233%.

d. Stock index

F S (1 rs rd)

where: rs is the short-term interest rate,

rd is the dividend yield.

F 1200 (1 12%/4 2%/4) 1230.

The market price may diverge from this theoretical value because:

The dividend yield is an annual approximation,

and arbitrage costs are quite high for the large number of stocks represented in the index.

Note: In all these applications, one must be very careful to calculate interest pro rata temporis.

Interest rates are always quoted on an annual basis. For example, the 12% rate on a three-month bill

yields a 12% 3/12 3% per quarter.

9. You wish to establish the theoretical futures price on a Euribor contract quoted on the London

International Financial Futures Exchange (LIFFE) in London. The futures contract is for a 90-day

Euribor rate at expiration of the futures contract. You look at the current term structure of Euribor

interest rates. Following the standard conventions for short-term rates, all interest rates are quoted as

annualized linear rates. In other words, the interest paid for a maturity of T days is equal to the

annualized rate quoted, divided by 360 and multiplied by T. The observed rates are as follows:

60-Day 90-Day 150-Day 180-Day

Euribor Rate 4.125% 4.250% 4.500% 4.550%

a. What should be the Euribor futures price quoted today with an expiration date in exactly

90 days?

b. What should be the Euribor futures price quoted today with an expiration date in exactly

60 days?


Solution

a. This futures contract is for a 90-day bill issued in 90 days and maturing in 180 days. The

annualized forward interest rate rF is given by:

180360

90360

1 4.55%1 1.011998

4 1 4.25%

Fr

rF 4.80%

F 100% 4.80% 95.20%.

b. This futures contract is for a 90-day bill issued in 60 days and maturing in 150 days. The annualized forward interest rate rF is given by:

150360

60360

1 4.5%1 1.011794

4 1 4.125%

Fr

rF 4.72%

F 100% 4.72% 95.28%.

10. You specialize in arbitrage between the futures and the cash market on the Paris Bourse. The CAC stock index is made up of 40 leading stocks. The futures price of the CAC contract with delivery in a month is 2,120. The size of the contract is €10 times the index. The spot value of the index is given as 2,000. Actually, there are transaction costs in the cash market; the bid–ask spread is around 40 points. You can buy a basket of stocks representing the index for 2,020 and sell the same basket for 1,980. Transaction costs on the futures contracts are assumed to be negligible. During the next month, the stocks in the index will pay dividends amounting to 5 per index. These dividends have already been announced, so there is no uncertainty about this cash flow. The current one-month interest rate in euros is 6

1/2

5/8%.

a. Do you detect any arbitrage opportunity?

b. What profit could you make per contract?

c. What is the theoretical value of the futures bid and ask prices?

Solution

a. The basis is equal to 120 per index or 6% of the spot value. This seems very large. An arbitrage would be to sell futures, buy spot, and carry the position till expiration of the futures contract. At expiration, both positions would be liquidated (futures contracts on the index are settled by cash, not by physical delivery of the shares).

b. Let’s look at the exact arbitrage per index:

Sell the futures at 2,120.

Buy a basket of stocks at 2,020.

Carry the position for a month with a financing cost at a rate of 6 5/8% and with the receipt of €5 in dividends.

Buy back the futures at expiration at the prevailing spot index value S (by definition of the contract, the futures price is equal to the spot value in expiration).

Sell the basket of stocks at S minus 20.

Note that the futures contract is settled in cash, so the basket of stocks cannot be used for

physical delivery; it increases the transaction costs.

Let’s look at the profit at the end of the month:

Profit 2120 2020 2020 5

865 ( 20)

12 100S S

73.85.


c. By arbitrage, the bid (Fbid) and ask (Fask) are given by the following equations:

0 Fbid 1,980 1,9801

265 ( 20)

12 100S S

Fbid 1,980 10.72 5 20 1,965.72.

0 Fask 2,020 2,0205

865 ( 20)

12 100S S

Fask 2,020 11.15 5 20 2,046.15.

The futures price should be between 1,965.72 and 2,046.15.

In practice, the transaction cost on a basket of shares will generally be much less than the 2%

assumed here on a return transaction.

11. A few years ago when the French franc (FF) still existed, the MATIF futures exchange in Paris had a

very active market for the French government bond contract. The underlying asset is a notional long-

term government bond with a yield of 10%. The size of the contract is FF 500,000 of nominal value.

Futures prices are quoted in percentage of the nominal value. On April 1, the French term structure of

interest rate is flat. The bond futures price for delivery in June is equal to 106.21%. The three French

government bonds that can be used for delivery have the following characteristics:

Market Price Duration Conversion Factor

Bond A 107.46% 7.00 101.1771%

Bond B 105.57% 7.90 98.1441%

Bond C 106.32% 8.80 99.3104%

a. Is the futures price consistent with the spot bond prices? (Find the bond cheapest to deliver.)

b. Estimate the interest rate sensitivity (duration) of the futures price.

c. You are an insurance company with a portfolio of French government bonds. The portfolio has a

nominal value of FF 100 million and a market value of FF 110 million. Its average duration is 3.5.

You are worried that social unrest in France could lead to an increase in French interest rates.

Rather than selling the bonds, you wish to temporarily hedge the French interest rate risk. How

many futures contracts would you sell and why?

Note to the instructor: The section on optimal hedge ratios for bond portfolios has been

removed from the 5th edition. We include a brief summary of the theoretical derivations

given at the end of the solution.

Solution

a. To answer this question we need to determine which is the cheapest-to-deliver bond.

We search for the cheapest-to-deliver bond. If we deliver bond i with price Pi and conversion

factor CFi , the net receipt will be (assuming a flat yield-curve):

F CFi Pi.


Since F 106.21%

Bond A: 0

Bond B: 1.33%

Bond C: 0.84%

Bond A is the cheapest-to-deliver bond and its price should drive the futures price:

F PA/CFA.

Since PA/CFA 106.21%, spot bond prices are consistent with the futures price.

b. The duration of the futures should be equal to that of Bond A, or:

DF 7.00

since .AA

A

dPdFD dr

F P

c. A naive hedge would be to sell an equal nominal quantity of futures contract, that is, a nominal

value of FF 100 million or 200 contracts. However, the futures price is twice as sensitive to

interest rate movements as the portfolio (durations of 7 and 3.5, respectively). So, you should sell

only 100 contracts.

More precisely the optimal hedge ratio is (see Appendix):

h*

110 3.50.518.

106.21 7

You should sell N contracts:

N 0.518 100 millions

103.6.0.5 million

Appendix: Theoretical Derivations

Theoretical value of the futures price:

The theoretical value of the futures price is derived by arbitrage between the futures and the cheapest-

to-deliver bond. Assume that the futures price is “too high.” Then an arbitrageur could buy a

deliverable bond “B” at a price PB on the cash market and simultaneously sell the futures at F. Bond B

has a conversion factor CFB. The carrying cost of this position is the difference between the short-

term interest rate paid to finance the purchase of the bond and the long-term interest rate (yield)

received while holding the bond. Let’s assume that the yield curve is flat, so that there is no carrying

cost in this arbitrage (basis equals zero).

At delivery the arbitrageur will make a profit equal to:

F CFB PB.


Of course, the arbitrageur will choose the bond (Bond B) that maximizes this profit (i.e., the

cheapest-to-deliver bond). By arbitrage this riskless profit must be zero (it will be negative for

deliverable bonds that are not the cheapest-to-deliver). So, the futures price should be equal to:

.B

B

PF

CF

The price of the cheapest-to-deliver bond (Bond B) drives the futures prices (the conversion factor is

a constant).

Optimal hedge ratio:

Let’s assume that we wish to hedge the interest rate risk of a portfolio with a value V (here

FF 110 million), consisting of a nominal value Q (here FF 100 million) times an average spot bond

price S % (here 110%). The duration equation for the portfolio for a small variation dr in the market

yield is:

S

dV Q dS dSD dr

V Q S S

or

.S

dV D Q S dr

The duration equation for the futures price is driven by the equation duration for the cheapest-to-

deliver bond (remember that the conversion factor is a constant):

BB

B

dPdFD dr

F P

hence

.B

dF D F dr

We hedge by selling N futures contracts with a fixed size (here FF 0.5 million). For a small variation

dr in the market yield, the futures position will generate a gain of:

Gain .B

N size dF N size D F dr

The net result on the hedged portfolio is:

( ) .S B B SD Q S dr N size D F dr N size D F D Q S dr

The optimal number of contracts that will immunize the hedged portfolio to small variations in

market yield is such that:

0B S

N size D F D Q S

or

.S

B

D SQN

Size D F

The optimal hedge ratio is * .S

B

D Sh

D F


12. An American investor wants to invest in a diversified portfolio of Japanese stocks but can invest only

a rather small sum. The investor also worries about fiscal and transaction cost considerations. Why

would futures contracts on the Nikkei index be an attractive alternative?

Solution

With little capital, an investor can only buy a few Japanese shares and will only hold a poorly

diversified portfolio. Through a stock index futures contract this same investor holds a

participation in a fully diversified Japanese stock portfolio.

Transaction costs on individual shares are higher than on a stock index futures contract.

The futures prices should be set by Japanese investors who arbitrage between the futures contracts

and the stock market. On the other hand, foreign buyers of Japanese stocks tend to lose the

withholding tax on dividends paid; this is certainly the case for U.S. pension funds. Therefore, the

futures contract allows a purchase at fair prices without losing the withholding tax on dividends.

One has to be careful about the currency exposure, which is different in a direct stock purchase and a

long position in the futures contract.

13. A money manager holds $50 million worth of top-quality international bonds denominated in dollars.

Their face value is $40 million, and most issues are highly illiquid. She fears a rise in U.S. interest

rates and decides to hedge, using U.S. Treasury bond futures. Why would it be difficult to achieve a

perfect hedge (list the various reasons)?

Solution

This is a typical example of a cross-hedge where the asset to be hedged is different from the futures

contracts. Among the factors that could make the hedge imperfect:

The maturity (and duration) of the portfolio of bonds is different from that of the notional bond.

Movements in the Treasury bond rates are not perfectly correlated with those on international

bonds, which are mainly corporate bonds.

Basis risk.

14. A manager holds a diversified portfolio of British stocks worth £5 million. He has short-term fears

about the market but feels that it is a sound long-term investment. He is a firm believer in betas, and

his portfolio’s is equal to 0.8. What are the alternatives open to temporarily reduce the risk on his

British portfolio?

Solution

One alternative is to sell all the British shares and buy them back when his fears disappear. At least

he could buy and sell shares to reduce the beta of his portfolio. This is a costly solution in terms of

transaction costs.

Another alternative is to sell Financial Times stock index futures contracts to hedge the British

market risks and remove the hedge when the fears disappear. Given the beta of his portfolio, the

investor should sell for 0.8 5 £4 million.

Another alternative would be to buy put options on individual shares in the portfolio or on the stock

index.


15. You are the treasurer of a major Japanese construction company. Today is January 15. You expect to receive €10 million at the end of March, as payment from a client on some construction work in

France. You know that you will need this sum somewhere else in Europe at the end of June. Meanwhile, you wish to invest these €10 million for three months. The current three-month interest rate in euros

is 4%, but you are worried that it will quickly drop. Listed below are Euribor futures quotations on

EUREX:

Maturity (month-end) Price

February 96.02%

March 96.08%

June 96.20%

September 96.25%

a. Knowing that Euribor contracts have a size of €1 million, what should you do to freeze a lending

rate when you will receive the money?

b. At the end of March, when you receive the money, the three-month Euribor is equal to 3%.

How much money (number of euros) have you gained by engaging in the above transaction

(as opposed to doing nothing on January 15)?

Solution

a. In order to freeze a lending rate when I will receive the money, I will buy 10 futures contracts

that expire in March and have a price of 96.08%. I am now freezing a three-month lending rate of

3.92% for the end of March.

b. At the end of March, the futures price will converge to 97%, given the 3% interest rate at that

time. Hence, I will make a profit on the futures contracts equal to:

Profit €10 million [97% 96.08%]/4 €23,000.

This profit will offset the drop in interest rate from January to March. I can then invest from March to June the €10 million received at a rate of 3%.

16. A dollar-Swiss franc swap with a maturity of five years was contracted by Papaf Inc. three years ago.

Papaf swapped $100 million for CHF 250 million. The swap payments were annual, based on market

interest rates of 8% in dollars and 4% in CHF. In other words, Papaf Inc. contracted to pay dollars

and receive CHF. The current spot exchange rate is 2 CHF/$, and the current interest rates are 6% in

CHF and 10% in $ (the term structures are flat).

a. What is the swap payment at the end of year three? Does Papaf pay or receive?

b. On the final date of the swap, the spot exchange rate is 1.5 CHF/$.

What is the final swap payment at the end of year five?

Solution

a. At the end of year three, Papaf receives the balance of:

Receipt of CHF 10 million.

Payment of $8 million.

The net cash flow is:

10 – 8 2 CHF 6 million.

Papaf has to pay CHF 6 million (or $3 million).


b. At the end of year five, Papaf receives the balance of:

Receipt of 250 10 CHF 260 million.

Payment of 100 8 $108 million.

The net cash flow is:

260 108 1.5 CHF 98 million.

Papaf receives CHF 98 million (or $65.33 million).

17. An Italian corporation enters into a two-year interest rate swap in euros on April 1, 2000. The swap is based on a principal of €100 million, and the corporation will receive 7% fixed and pay six-month Euribor. Swap payments are semiannual. The 7% fixed rate is quoted as an annual rate using the European method, so the implied semiannual coupon is 3.44% [since (1.0344)

2 1.07]. Two years

later, the swap is finally settled, and the following Euribor rates have been observed:

Apr. 1, 2000 Oct. 1, 2000 Apr. 1, 2001 Oct. 1, 2001 Apr. 1, 2002

6.5% 7.5% 8% 7.5% 6%

a. What have the swap payments or receipts for the corporation been on each swap payment date?

b. The same Italian corporation also entered another two-year interest rate swap in euros on April 1, 2000. The swap is based on a principal of €100 million, and the corporation contracted to receive

7% fixed and pay six-month Euribor. On this swap, the payments are annual. Hence, the two

successive six-month Euribor are compounded. Assuming that the Euribor rates given in the

previous problem have been observed, what have the two annual swap payments been?

Solution

a. Semiannual swap:

Swap receipts by the corporation

(based on the Euribor observed at the start of the six-month period)

Date

Net Receipt in € Million

(payment if negative)

October 1, 2000 100 (3.44 3.25)% 0.19

April 1, 2001 100 (3.44 3.75)% 0.31

October 1, 2001 100 (3.44 4.00)% 0.56

April 1, 2002 100 (3.44 3.75)% 0.31

b. On each annual payment date, the floating leg is determined by compounding the two six-month

LIBOR observed in the previous year. For example, the floating leg on April 1, 2001, is equal to:

€7,121,875

since 7.121875% (1.0325)(1.0375) 1.

The floating leg on April 1, 2002, is equal to:

€7,900,000

since 7.9% (1.04)(1.0375) – 1.


The swap receipts by the corporation are

Date

Swap Receipts in € Million

(payment if negative)

April 1, 2001 100 (7 7.1219)% 0.1219

April 1, 2002 100 (7 7.9)% 0.9

18. A swap dealer provides the following quotations for a yen/$ currency swap. The quotes are for a yen

fixed rate against the U.S. Treasury yield flat, with annual payments.

Years Fixed (ann.)

2 6.00 6.08

3 6.12

6.21

4 6.14 6.23

5 6.15 6.24

7 6.18 6.28

A client wishes to enter a five-year swap, paying yen and receiving $. The current yield on five-year

U.S. Treasury bonds is 7.20%, using the semiannual method, which amounts to 7.33%, using the

annual European method.

What will the exact terms of the swap be if the client accepts these quotations?

Solution

As stated in the table, the swap quotes use the European, or annual yield, method. A client entering a

five-year swap to pay yen and receive $ will pay an annual yield of 6.24% in yen and receive an

annual yield of 7.33% in $, with both yields using the annual European method.

19. Pouf is a rapidly growing and pleasant country in the Austral hemisphere. Its inhabitants are called

Poufans, and its currency is the pof. The bond market is fairly active with many issues by Poufan

companies, but there are no foreign investors or issuers. The current yield on pof bonds is 10%.

Poufan investors have to pay a 15% tax on interest income received. The newly elected Poufan

government wishes to internationalize its bond market and attract foreign issuers. To do so, it decides

to remove any taxation of income on bonds issued by foreign corporations in Pouf. Several changes

take place after the enactment of this tax provision:

Several well-known foreign corporations issue pof-denominated bonds in the Poufan bond market.

Several well-known Poufan corporations issue international bonds denominated in U.S. dollars.

Several dollar/pof swaps are arranged.

Try to provide a sensible explanation for this phenomenon.

Solution

Foreign corporations issue bonds in pof and swap pofs for dollars (they receive pofs and pay $).

Poufan corporations issue bonds in dollars and swap dollars for pofs (they receive $ and pay pofs).

Foreign corporations can issue bonds in pofs at a lower yield than Poufan corporations because of

the tax advantage. Income is tax-free for Poufan investors. This tax advantage (15% of the yield) is

shared between Poufan and foreign corporations through the pof/dollar swap.


20. A Dutch institutional investor has decided to bet on a drop in U.S. dollar bond yields. It engages in a

leveraged strategy, borrowing $100 million at LIBOR plus 0.25% and investing the proceeds in

attractive, newly issued, long-term dollar international bonds. Suddenly, the investor becomes worried

that bond yields have hit bottom and will rise because of inflationary pressures. The investor wishes

to keep the specific international bonds that have been selected, partly because of their attractiveness

and partly because of their lack of market liquidity. What kind of swap could be arranged to hedge

this U.S. dollar bond yield risk?

Solution

The Dutch institutional investor should swap fixed for floating in dollars. It should contract to pay

fixed and receive floating (LIBOR).

21. A small German bank has the following portfolio of loans in U.S. dollars, valued at market value:

Assets Liabilities

$50 million of a five-year FRN at

LIBOR plus 0.5%

$10 million of a five-year loan at a fixed

rate of 9%

The German bank fears a long-term depreciation of the U.S. dollar relative to the euro and believes in

stable U.S. interest rates.

a. What is its currency exposure?

b. What type of swap arrangements should it contract?

c. What should the principal of the swaps be?

Solution

a. The net exposure to the $ exchange rate is a net asset position of $40 million.

b. The bank should enter in a swap to pay $ and receive euro.

c. The swap amount that minimizes the currency exposure is $40 million. If the German bank

wanted to speculate on a dollar depreciation, it would swap more than $40 million.

The answer would be different if the German bank feared a movement in U.S. interest rates.

22. A five-year currency swap involves two AAA borrowers and has been set at current market interest

rates. The swap is for US$100 million against AUD 200 million at the current spot exchange rate of

AUD/$ 2.00. The interest rates are 10% in U.S. dollars and 7% in Australian dollars, or annual swaps

of US$10 million for AUD 14 million. A year later, the interest rates have dropped to 8% in U.S.

dollars and 6% in Australian dollars, and the exchange rate is now AUD/$ 1.9.

a. What should the market value of the swap be in the secondary market?

Assume now that the swap is instead a currency–interest rate swap whereby the dollar interest is set

at LIBOR.

b. What would the market value of the currency–interest rate swap be if these conditions prevailed a

year later?

Solution

a. The new value of the swap is derived by considering the market value of two streams of cash

flows:

P$: a bond in dollars with four years remaining, with annual cash flows of $10 million, and a

principal repayment of $100 million;

PAUD: a bond in Australian dollars with four years remaining, with annual cash flows of AUD

14 million, and a principal repayment of AUD 200 million.


The swap to receive AUD and pay dollars is worth in AUD:

Swap PAUD (6%) spot AUD/$ P$ (8%).

Swap 2 3 4 2 3 4

14 14 14 214 10 10 10 1101.9 .

1.06 1.06 1.06 1.06 1.08 1.08 1.08 1.08

Swap 206.93 1.9 (106.63) AUD 4.34 million.

The U.S. dollar value of the swap is 4.34/1.9 $ 2.28 million.

Of course, the seller of this swap who receives dollars for Australian dollars will realize a

corresponding loss.

b. Without further information, we can assume that the value of a bond with a floating rate stays

constant. Therefore, the swap value will change only because of a change in the AUD interest

rate and a change in the exchange rate. This second swap is now worth:

Swap 206.93 – 1.9 (100) AUD 16.93 million.

Swap $ 8.91 million.

23. A five-year currency swap involves two AAA borrowers and has been set at current market interest

rates. The swap is for US$100 million against AUD 200 million at the current spot exchange rate of

AUD/$ 2.00. The interest rates are 4% in U.S. dollars and 7% in Australian dollars, or annual swaps

of $4 million for AUD 14 million. A year later, the interest rates have dropped to 3% in U.S. dollars

and 6% in Australian dollars, and the exchange rate is now AUD/$ 1.9.

a. What should the market value of the swap be in the secondary market?

Assume now that the swap is instead a currency–interest rate swap whereby the dollar interest is set

at LIBOR.

b. What would the market value of the currency–interest rate swap be if these conditions prevailed a

year later?

Solution

a. The new value of the swap is derived by considering the market value of two streams of cash

flows:

P$: a bond in dollars with four years remaining, with annual cash flows of $4 million, and a

principal repayment of $100 million;

PAUD: a bond in Australian dollars with four years remaining, with annual cash flows of AUD

14 million, and a principal repayment of AUD 200 million.

The swap to receive AUD and pay U.S. dollars is worth in AUD:

Swap PAUD (6%) spot AUD/$ P$ (3%).

Swap 2 3 4 2 3 4

14 14 14 214 4 4 4 1041.9 .

1.06 1.06 1.06 1.06 1.03 1.03 1.03 1.03

Swap 206.93 1.9 (103.72) AUD 9.86 million.

The U.S. dollar value of the swap is 9.86/1.9 $5.19 million.

Of course, the seller of this swap who receives dollars for Australian dollars will realize a

corresponding loss.


b. Without further information, we can assume that the value of a bond with a floating rate stays

constant. Therefore, the swap value will change only because of a change in the AUD interest

rate and a change in the exchange rate. This second swap is now worth:

Swap 206.93 1.9 (100) AUD 16.93 million.

Swap $8.91 million.

24. Four years ago, a Swiss firm contracted a currency swap of US$100 million for 250 million Swiss

francs (SFr), with a maturity of seven years. The swap fixed rates are 8% in dollars and 4% in francs,

and swap payments are annual. The Swiss firm contracted to pay dollars and receive francs. The

market conditions are now (exactly four years later) as follows:

Spot exchange rate: 2.00 Swiss francs/U.S. dollar.

Term structure of zero swap rates:

Maturity Years U.S. Dollar % (ann.) Swiss Franc % (ann.)

1 9 5

2 9.5 5.75

3 10 6

4 10.25 6.25

5 10.75 6.5

6 11 7

7 11.5 7.5

a. What should the swap payment (receipt) be at the end of the fourth year, that is, today?

b. Right after this payment, what is the swap market value for the Swiss firm?

Solution

a. Each year, the company receives SFr and pays $. Given the interest rates as of the signature of

the swap contract (8% in $ and 4% in SF), the balance for year four is the following:

Receipt of SFr 10 (250 4%) million.

Payment of $8 (100 8%) million, or SFr 16 (8 2) million.

The net cash flow is 10 16 SFr 6 million. The company has to pay SFr 6 million on year

four.

b. Right after the fourth payment, the market value of swap, V, is:

V PSF spot SFr/$ P$

where

• PSF is a SFr 250 million three-year bond with a SFr 10 million annual coupon.

• P$ is a $100 million three-year bond with a $8 million annual coupon.

Hence, we have:

2 3 2 3

10 10 260 8 8 1082 .

1.05 (1.0575) (1.06) 1.09 (1.095) (1.1)V

V SFr 46.46 million

V $23.23 million


25. A small French bank has the following balance sheet, based on historical (nominal) values.

Assets Liabilities

Loan of 100 million: Debt of 50 million:

3 years, @ 3month Euribor ½ 5year maturity, @ 10%

Net worth: 50 million

All assets and liabilities are denominated in euros. The net worth is calculated as the difference

between the value of assets and liabilities. The current interest rate term structure in euro is flat at 8%.

The risk premium over Euribor required on the loan to a client remains at 50 basis points.

a. Value the balance sheet based on market value.

b. The bank anticipates a sharp drop in French interest rates. Would this drop be good for the bank?

The current market conditions for interest rate swaps with a maturity of three or five years are 8%

against Euribor.

c. Assume that the bank simply wishes to immunize its market value against any movements in

interest rates (drop or rise). What swap would you make to hedge this interest rate risk?

d. Assume that the bank is quite confident in its interest rate prediction (a drop). What would you

suggest?

The next day, all interest rates drop to 7%.

e. Value the balance sheet again, assuming that the floating rate debt remains at 100% and that the

bank has undertaken the swap that you recommended. How much did the bank save by

undertaking this swap?

Solution

a. The market value of the asset (A) is that of a three-year floating rate loan at Euribor 1/2. Since

the current risk premium required for this type of client is still 50 basis points, the loan should still be valued at 100%, or €100 million.

The market value of the liability L can be determined by using the current market rate of 8%.

L €5 5

1 1 505 53.99

1.08 1.08 1.08

million.

The net worth V of the bank is now:

V A L 100 53.99 €46.01 million.

b. This drop in interest rates is bad for the bank as the market value of its liability will rise, while

the market value of its assets would remain unchanged.

c. The item exposed to interest rate risk is the €50 million debt with a maturity of five years. Its

market value would rise if interest rates drop. The objective is to maintain the market value of the equity (net worth of €46.01 million). Hence, the bank would enter into a five-year swap to

receive fixed and pay floating.

The remaining question is the amount of the swap. A swap of €50 million would transform the

cash flow on the debt from a fixed 10% to a floating Euribor 2% (since the swap is 8% fixed

for Euribor). If the bank focuses on market values, it would be a better idea to contract for a swap amount of €53.99 million, the current market value of the debt.


d. If the bank is quite confident in a drop in interest rates, it will swap more than the above amount

to speculate. This can be risky if the forecast is proved wrong.

e. Let’s market value the balance sheet items and the swap (off-balance sheet):

A €100 million

L €5 5

1 1 505 56.15

1.07 1.07 1.07

million.

The net worth without the swap is V0:

V0 100 56.15 €43.85 million.

If we had swapped €50 million, the value of the swap would now be:

Swap1 €55

1 1 504 50 2.05

1.07 1.07 1.07

million.

The net worth V1 of the bank is therefore:

V1 100 2.05 56.15 €45.90 million.

The previous net worth of €46.01 million is almost preserved, but not fully.

If we had swapped €53.99 million, the value of the swap would now be:

Swap2 €5 5

1 1 53.994.32 53.99 2.21

1.07 1.07 1.07

million.

The net worth V2 of the bank is therefore:

V2 100 2.21 56.15 €46.06 million.

The hedge is better, as the result is very close to the initial market value of 46.01. The difference

between 46.01 and 46.06 comes from the fact that quoted swaps had a 8%-fixed rate, while the

debt was written with a 10%-fixed coupon. The interest sensitivities (duration) of the swap and of

the debt are not identical because they have different coupons, albeit the same maturity.

26. A small Dutch bank has the following balance sheet (in euros), based on historical or nominal values.

Assets Liabilities

Loan of 200 million: FRN borrowing of 150 million:

3 years, @ 7% @ 3-month Euribor, 5-year maturity

Net worth: 50 million

All assets and liabilities are denominated in euros. The bank borrows short-term on the Euro-currency

market. The bank and its client are AAA quality. The net worth is calculated as the difference

between the value of assets and liabilities. The current euro term structure for AAA borrowers is flat

at 6.5%.

a. Value the balance sheet based on market value.

b. Compute the interest-rate sensitivity (duration) of the asset. Infer the interest rate sensitivity of

the net worth of the bank. For example, how much would stockholders lose if euro interest rates

moved up by 0.10%? (Assume that the interest rate sensitivity of an floating-rate note (FRN) is

zero, as the coupon is reset to the market interest rate.)


c. The bank fears a rise in all euro interest rates. The current market conditions for interest rate

swaps in euros are as follows:

With a maturity of three years are: 6.5% against Euribor.

With a maturity of five years are: 6.75% against Euribor.

What would you do to hedge this interest rate risk?

d. The next day, all interest rates move up to 8%. Value again the balance sheet, assuming that the

floating-rate debt remains at 100% and that the bank has undertaken the swap that you

recommended. Is the hedge perfect? Why?

Solution

a. Let’s value the bank equity based on market values:

Assets A 14 (1/1.065 1/1.065² 1/1.0653) 200/1.065

3

A €202.65.

The market value of the FRN remains constant:

Liabilities L €150

Net Worth V AL €52.65 million.

b. The interest rate sensitivity (or duration * )VD

of the bank net worth comes from that of the asset.

We have:

, or

, or

dr 0, or .

V V

A A

L L

dVD dr dV D V dr

V

dAD dr dA D A dr

A

dLD dL D L dr

L

Note that V A L, and dV dA dL.

So we get:

dV dA – dL AD A dr

LD L dr

VD V dr

AD A

LD L

VD V

hence

A A L LV

D V D VD

A V

.

In this case, the duration of the liability L is nil. The duration of the asset can be calculated as:

DA 2 3

1 14 14 214 11 2 3

1.065 1.065 1.065 1.065 202.65

DA 2.64.

DV 202.65

2.64 10.15.52.65


If Dutch interest rates move up by 10 basis points, the market value of the bank should drop by

101.5 basis points. The stockholders should lose:

dV DV. V. dr 10.15 52.65 0.10% €0.53 million.

c. A rise in all euro interest rates would lead to a decrease in the market value of the bank’s assets,

liabilities being unchanged.

To hedge this risk, we should swap to pay fixed at 6.5% and receive floating with a maturity

of three years.

The amount of the swap should be the market value of the assets: €202.65 million.

(An adjustment could be made for the differences in sensitivities.)

d. Let’s revalue the balance sheet:

A 14 (1/1.08 1/1.08² 1/1.083) 200/1.08

3 €194.85

L €150.

Off-balance sheet, the swap has taken a positive value:

Swap Value 202.65 – (13.17/1.08 13.17/1.08² 215.82/1.083)

Swap Value €7.83.

Total market value:

V A L Swap Value 194.85 – 150 7.83 €52.68.

It is, hence, a good hedge as the market value remains very stable (52.68 instead of 52.65),

but not perfect because of usual assumptions of sensitivity.

27. A differential swap, or switch LIBOR swap, involves the LIBOR rates in two different currencies but

with both legs denominated in the same currency. A Japanese insurance company engages in a

differential swap whereby it receives the six-month Japanese yen LIBOR and pays the six-month U.S.

dollar LIBOR plus 50 bp but with both legs denominated in yen. No principal is exchanged at the end.

The current LIBOR for the yen and the dollar are 6% and 4%, respectively, and the principal is 100

million yen. Hence, the first swap payment will be based on a differential of 1.5% in yen [6% (4% 0.5%)].

The current yield pick-up is 150 bp. There is no currency risk on this swap.

Provide some intuitive explanation for the pricing of such a swap, knowing that at the time, the dollar

yield curve was very steep (long-term rates are much higher than short-term rates) and the yen yield

curve was almost flat.

Solution

This is not an easy task. Some preliminary remarks can be useful.

a. In a standard floating-for-floating currency swap, the terms would be yen LIBOR against dollar

LIBOR flat. The 50-basis-point addition on the dollar LIBOR of the differential swap comes

from the fact that all cash flows on the dollar leg (interest and principal) are denominated in yen,

not in dollars.

b. The forward exchange rates reflect the differences in the two interest rates term structures. This

impacts on the pricing of the differential swap.

The difficulty in valuing a differential swap is the valuation of the dollar LIBOR (in yen) leg. The

“fair” spread to add (here 50 basis points) is the spread that makes the market value of this leg equal

to the market value of the yen LIBOR leg. The basic pricing idea is to replicate the dollar LIBOR


(in yen) leg by using forward dollar LIBOR rates and forward ¥/$ exchange rates. This is detailed in

R. Litzenberger, “Swaps: Plain and Fanciful,” Journal of Finance, July 1992 (pages 842–844).

28. You are an investment banker working in Switzerland where yields are very low (1% for all maturities).

You are planning to offer a five-year Swiss franc/British pound bond with the following characteristics:

Issuer: Brit Ltd., a top-quality British company.

Issue amount: SFr 100 million.

Coupon in SFr: 5% (or SFr 5 million).

Reimbursed value: £40 million.

This bond qualifies as a Swiss franc bond for the portfolio of a Swiss insurance company.

The current spot exchange rate is 2.5 Swiss francs per British pound. The yield curve in British

pounds is flat at 7%. The pound/franc swap rates are 7% in pounds against 1% in francs for all

maturities.

a. Assume that Swiss insurance companies can account for their Swiss franc bond holdings

at historical costs. Give a reason why it would be attractive to invest in this bond.

b. Is the coupon rate set at fair pricing (i.e., consistent with current market conditions)?

c. The British company desires to borrow in pounds and does not wish to carry any currency risk on

its debt. The investment banker needs to design a coupon swap that would hedge the currency

risk on that dual-currency bond for Brit Ltd. The designed swap should have a zero value at time

of contracting. Give one possible design for the swap and calculate its associated swap rate.

d. What is the pound yield paid by the British company, once it has hedged its currency risk on the

dual-currency bond using the swap described above? What is the annual cost-saving in British

pounds compared to a straight pound bond?

Solution

a. The bond seems attractive to Swiss institutional investors because it offers a higher SFr coupon

than a straight bond in Swiss francs. The annual income statement will show a higher income

than if the insurance company had subscribed to a straight Swiss franc bond (yield 1%).

b. The coupon C consistent with fair pricing is such that:

5

1

40100 2.5

(1.01) (1.07)t tt

C

hence a coupon rate of 5.91%.

If we value the bond at fair market value we find that it should be priced below par at 95.57%.

c. Brit Ltd. needs to swap the stream of Swiss franc coupons of SFr 5 million cash flow into a

stream of pounds coupon cash flows. A plain-vanilla currency swap (1% in francs against 7% in

pounds) would not do it, because there are no “reimbursement” values to be swapped. We need

to do a coupon-swap where the SFr leg is a stream of SFr 5 million annual cash flow, and the

£ leg is a stream of £X million annual cash flow. As the swap has a zero value at time of

contracting, we get:

5 5

1 1

50 2.5

(1.01) (1.07)t tt t

X

hence X £2.37 million or a rate of x X/40 2.37/40 5.92%.


So this is a coupon-swap with a 1% swap rate on the SFr leg and a 5.92% swap rate on the £ leg.


d. Brit Ltd. needs to get into a five-year swap to receive the SFr coupons and pay the £ coupons.

This is a swap to receive annually SFr 5 million and pay annually £2.37 million. The sum of

issuing the dual-currency bond and entering into the swap will yield annual cash flows of

£2.37 million and a final reimbursement of £40 million. The all-in-cost on the £40 million bond

is an interest rate in pounds of 5.92%; hence, a reduction of 1.08% 7% 5.92% in borrowing

cost, or an annual cost savings of £0.432 million.

29. Assume that an AAA customer pays 8% on a five-year loan and can contract a five-year interest

rate swap (paying fixed) at 8% against LIBOR. Assume that a BBB customer pays (8 m)% on a

five-year loan and can contract a five-year interest rate swap (paying fixed) at (8 µ)% against

LIBOR. Should a customer pay the same credit-quality spread (m and µ) on a loan and on a swap?

Solution

No. The bank writing both contracts stands to lose a much greater amount on the loan than on

the swap.

On the loan, the bank stands to lose all interest payments (8 m)% plus the principal of 100%. On the

swap, the bank stands to lose only an interest rate differential (fixed minus floating) and no principal.

Furthermore, if the floating rate rose above the fixed rate, the bank would have to pay the difference

anyway, so the default of the other party does not worsen the situation.

The credit-quality markup on the swap should be much smaller than on the loan m.

30. The current market conditions for an AAA client are 8% on a one-year dollar loan, and 8% fixed U.S.

dollars for 9% fixed British pounds on a one-year dollar/pound currency swap. Let’s consider a BBB

client borrowing at (8 m)% on a one-year dollar loan. The same client can enter a dollar/pound

currency swap, paying (8 µ)% fixed dollars and receiving 9% fixed pounds. Assume that the

customer has a probability of p% to default within a year. In case of default, the bank knows that it

will recover nothing on either transaction. The probability of default p (e.g., 5%) is known and

independent of movements in interest and exchange rates. The spot exchange rate is S 0 1 $/£.

Assuming that you can observe the prices of $/£ currency options, suggest some approach to

determine the fair values of m and µ. (Assume that the bank has a large number of clients whose

probabilities of default are independent; therefore, the bank can diversify away the uncertainty of

default on this specific client.)

Solution

The bank writing both contracts (loan and swap) stands to lose much more on the loan than on the

swap in case of default.

Let’s write the dollar cash flows for the bank, in a year, for a principal of $100 assuming an initial

exchange rate S0 1 $/£. Let’s denote the spot exchange rate in a year as S1, in $/£. All rates are

expressed for 100.

Cash flows

a. Loan

if no default CF1 100 8 m

if default CF2 0.

b. Swap

if no default CF3 100 8 109 S1

if default CF4 Min {0 ; 108 109 S1}.


Let’s now discuss the valuation of the two contracts at time of contracting; that is, let’s try to

determine the fair values of m and under the stated assumptions.

a. It is useful to start by outlining the valuation principles for a client without default risk

(m 0):

Default-free loan

Let’s call V(CF), the present value of a future cash flow CF. The current one-year risk-free rates

are 8% in dollars and 9% in pounds. The dollar loan is issued at 100 such that:

100 V(108) 108/1.08.

Default-free swap

The swap is written with a zero initial market value. Hence:

0 V($108 £109).

As seen in the text, the two legs of the swap can be valued separately and the £ leg is converted at

the initial spot exchange rate S0 1 $/£.

0 V($108 £109) V($108) S0 V(£109)

0 108 109

11.08 1.09

.

b. Let’s now turn to the valuation of the loan and swap in the presence of default risk:

Risky loan

The loan is still issued at 100 with a rate of (8 m)% and a risk p of full default. Given that the

uncertainty of default can be diversified across many clients, the valuation gives:

100 (1 p) V(108 m) p 0

100 (1 p) 108

1.08

m

hence

m 108 .1

p

p (10.1)

If p 5%, m 5.68.

The underlying idea is that the rate markup of m 5.68% should compensate for the total loss on

the 5% clients who default. The extra charge on 95% of the clients offsets the principal and

interest rate loss on 5% of the clients.

Note that we discount cash flows at 8% because the uncertainty can be diversified away by the

bank across many clients.

Risky swap


We make the additional assumption that default is unrelated to the final exchange rate S1. The swap is

still issued with a zero initial value. The valuation yields:

0 (1 p) V(CF3 ) p V(CF4 ) (10.2)

with

V(CF3) V($108 $ £109) V($108 £109) V($)

V(CF3) 0 1.08

V(CF4) V(min{0; 108 109 S1})

V(CF4) V(max{0; 109 S1 (108)})

V(CF4) 109 V(max{0; S1 108

109

}).

Note that the last term is the value of a one-year pound call option with a strike of 108

109.

Remember

that the forward exchange rate quoted at the time of contracting is F 108/109, so the strike is equal to F (1 /108). The term /108 is the present value, in %, of the markup .

Hence, Equation (10.2) can be rewritten as:

0 (1 ) 109 CALL(?1.08

p p

(10.3)

where the call has a strike of 108

109

.

From Equation (10.3), we derive:

1.08 109 CALL(?1

p

p

. (10.4)

m 1.09 CALL(£). (10.5)

Equation (10.5) is somewhat misleading, as the markup enters the strike price of the call. So an

iterative procedure must be used to solve it for . To get some ballpark estimate, assume that a pound

call with a strike equal to the current forward exchange rate quotes at $0.04 (a reasonable figure).

Then a pound call written with a higher strike price, F(1 /1.08), should quote slightly less than

$0.04. If it quoted $0.04 anyway, the fair value of would be:

5.68 1.09 0.04 0.25

or 0.25% of the principal.

Actually, the fair value of should be found to be slightly less than 0.25, because the call will be

worth slightly less than $0.04. Note that the credit-quality markup on the swap is 20 times less than

on the loan (m 5.68).


On an interest rate swap in a single currency, the fair value of the swap markup would be even less

because there cannot be any difference in principal at the end of the swap. On a currency swap, the

principals on the two legs are affected by currency movements. A formula similar to Equation (10.3)

could be found for an interest rate swap, with the option value being that of a floor on the variable

interest rate. See Solnik, “Swap Pricing and Default Risk: A Note,” Journal of International

Financial Management and Accounting, Spring 1990.

31. A traditional interest rate swap has a notional capital of 100 and exchange LIBOR (the floating leg)

against 6% (fixed leg). At maturity of the swap there is no capital exchange as the same notional

capital of 100 is “exchanged” on both legs. Assume that the swap has a five-year maturity.

A company needs to create an immediate cash flow to offset an immediate loss and decides to use an

amortizing swap. Its off-balance sheet items are accounted at their book or historical values. The

floating leg is LIBOR, paid quarterly, with a notional capital of 100. The fixed leg also has a notional

capital of 100, however, there is only an initial cash flow of X on the fixed leg of the amortizing swap

and no other cash flow (zero coupons). Hence, there is no capital exchanged at maturity of the swap

(capital identical on both legs). The swap is priced (the value of X is set) so that the initial swap value

is zero.

The company enters the amortizing swap to pay floating and receive fixed. In other words, its cash

flows on the swap are as follows:

Receive X at time 0.

Pays LIBOR every quarter for five years.

No cash flow at maturity.

a. Why is the amortizing swap interesting for this company, which wants to window-dress an

immediate loss? How will it impact its future earnings?

b. The term structure is flat at 6%. What should be the “fair” value of X?

c. The company expects a loss of 10 million, what should be the notional capital of the amortizing

swap that should be contracted?

d. Assume now that the company must value all off-balance sheet items at their market value. What

would happen to the value of the swap immediately after the payment of X is received by the

company? Are amortizing swaps useful in deferring losses with this accounting convention?

Solution

a. The company will have an immediate income of X appearing in its income statement, while the

swap only appears in the off-balance sheet. This income will offset the expected loss. However, it

generates a stream of future expenses that will reduce future earnings.

b. Let’s treat the amortizing swap as the difference between a floating and the fixed bond (initial

cash flow of X, zero-coupons, and identical reimbursement value 100%). The floating leg should

have a current value of 100%. The fixed leg will be priced so that it also has an initial value

of 100. Hence, 100% X% 100%/(1 6%)5 X 25.274% of the notional capital.

c. X should be set to 10 million. Hence, the notional capital 10 million/25.274%

39.566 million.

d. Right after the payment of X 10 million , the value of the swap for the company would drop

from zero to –10 million. There is no advantage in window dressing.


32. LTCM observed the following ten-year swap spreads on July 1, 1998. You remember that U.S. dollar

interest swaps (fixed rate against LIBOR three-month) are quoted as a spread over the Treasury yield

(so the fixed rate on the swap is equal to the interest rate on Treasury bonds for the same maturity

plus the quote spread). LTCM believes that the normal spread is 40 bp (basis points). The current

spread of 80 bp is expected to converge back to normal in three months.

If you borrow securities, you have to deposit as collateral an equivalent amount of cash that is

marked-to-market. For example, if you borrow a security that is worth 100, you have to deposit 100;

if the value of the security increases to 110, you have to deposit 10 more. Swaps are also marked-to-

market and free of default risk.

a. What arbitrage using Treasury bonds and swaps could you put in place if you believe that the

spread will revert back to its normal level? Be very precise and assume you do the above

arbitrage for $100 million. How much of LTCM capital is invested in the arbitrage?

b. Suppose that the spread is still at 80 bp on October 1, but that Treasury yields have moved

up by 40 bp on October 1 (reset date for the floating leg). What is your gain/loss in dollars?

[You only need to provide a rough estimate assuming that the sensitivity (duration) on the

Treasury bond and fixed leg of the swap is equal to 10.]

c. Other scenario: How much would you gain (from July 1) if Treasury yields do not move, but

the spread reverts back to 40 bp three months later on October 1 (reset date for the floating leg)?

[You only need to provide a rough estimate assuming that the sensitivity (duration) on the

Treasury bond and fixed leg of the swap is equal to 10.]

Solution

a. I would borrow 100 million of Treasury bonds and sell them. The proceeds of the sale would be

used as a deposit. So, I am short in Treasury bonds. I would then swap for 100 million paying

LIBOR and receiving fixed (at a spread of 80 bp above the Treasury yield). There is no invested

capital by LTCM. But because of the risk, I would earmark some capital.

b. The value of the Treasury bonds decreases, so LTCM gains on its short position. I can proxy this

gain on the short Treasury bond position by a simple duration calculation. The return on Treasury

bonds is proxied by:

dP/P D dr 4 %.

Being short on bonds, this translates into a gain of 4% on the short position or a gain of

$4 million.


The value of the swap becomes negative because LTCM receives on the swap a fixed rate (old

Treasury yield plus 80 bp), which is smaller than the new swap rate (new Treasury yield plus

80 bp): the difference in yield is 40 bp. I could value the swap as the difference between a fixed-

rate bond and an FRN. The value of the FRN is still 100 on reset date. The value of the fixed leg

goes down. I can proxy the loss by a simple duration calculation:

dP/P D 0.40% 4 %. The loss is $4 million.

The net result is that there is no gain or loss on the arbitrage.

c. The value of the short treasury bonds remains unchanged. The value of the swap becomes

positive because I receive Treasury yield plus 80 bp, while current market conditions are

Treasury plus 40 bp. I could value the swap as the difference between a fixed-rate bond and an

FRN. The value of the FRN is still 100 on reset date. The value of the fixed leg goes up. I can

proxy this gain by a simple duration calculation:

dP/P D 0.40% 4 %. The gain is $4 million.

Hence, the net gain is $4 million.

33. If the average premium on gold call options declines, does this mean that they are becoming

undervalued and, therefore, should be bought? Using valuation models, give at least two possible

reasons for this decline.

Solution

Probably not.

The premium on a gold call could decline because:

The price of gold declines,

the short-term interest rate drops,

the perceived volatility of the price of gold declines,

the time to expiration declines.

In all four cases, the fair market value of the option will go down.

34. The average premium on currency calls has decreased, whereas the premium on currency puts has

increased. What explanations can you provide?

Solution

This phenomenon could have several causes:

The value of the underlying currency, that is, the spot exchange rate, has gone down. This would

be the major reason for the described phenomenon.

A change in the interest rate differential, domestic minus foreign.

A change in the volatility of the underlying spot exchange rate would affect both puts and calls in

the same direction and therefore cannot explain this phenomenon.

35. You will receive $10 million at the end of June and will invest it for three months on the Eurodollar

market. The current three-month Eurodollar rate is 6%, and you are worried that the rate will drop by

the end of June. Here are some market quotes:

Eurodollar LIBOR futures, June delivery: Price 94%.

Call eurodollar, June expiration, strike price 94%: Premium 0.4%.

Put Eurodollar, June expiration, strike price 94%: Premium 0.4%.


The contract sizes are $1 million.

a. Should you buy or sell futures to hedge your interest rate risk?

b. Should you buy (or sell) calls (or puts) to insure a minimum rate at the time you will invest your

money? What is this rate?

c. In June, the Eurodollar rate has moved to 4%. What is the result of your strategies using futures

and using options?

d. What if the rate is equal to 8% in June?

Solution

a. You should buy Eurodollar futures and thereby freeze a 6% investment rate.

b. You should buy Eurodollar calls and thereby insure a minimum investment rate of: 6.0%

0.4% 5.6%. But you might invest at a higher rate if the interest rate is higher than 6% at the

end of June.

c. If the Eurodollar rate moves down to 4%:

You will be able to invest at 6% in the strategy of buying futures,

and you will be able to invest at 5.6% in the strategy of buying calls.

d. If the Eurodollar rate moves to 8%:

You will be able to invest at 6% in the first strategy,

and you will be able to invest at 8.0% 0.4% 7.6% in the second strategy.

36. The French futures market, MATIF, trades Euribor contracts. The Euribor is the three-month interbank interest rate on euros. The contract size is €1 million, and the margin is €3,000. On

January 10, March futures trade at 90.74%. Options on the Euribor futures contract are also listed.

The premiums (in %) on March options are as follows:

Strike Price Call Put

90.40 0.30 0.06

90.80 0.17 0.18

91.00 0.09 0.34

A few days later (January 14), the futures price moves to 89.50.

a. What is the gain or loss, in euros, for someone who sold a futures contract on January 10?

b. What is the return, as a percentage of the initial investment (margin)?

c. Are all option premiums quoted on January 10 reasonable?

d. You know that you will have to borrow €10 million in March and fear a rise in interest rates.

What are the maximum borrowing rates that you can insure using the various options?

e. To cap your borrowing rate, you decide to use options with a strike price of 90.80. How many

calls (or puts) should you buy (or sell)?

On January 14, the premium on the call March 90.80 moves to 0.02, and the premium on the put

March 90.80 moves to 1.33.

f. What is the € profit (or loss) on your option position?

g. What is the rate of return on your option position?


Solution

a. The futures price moves to 89.50. The total gain or loss on a contract is equal to:

Gain (loss) (futures price variation/4) size of contract

Gain €(90.74 89.50)% 1,000,000

4

Gain €3,100.

b. It is equivalent, in percentage of initial investment, to 3,100/3,000 103.33%

c. No, the options intrinsic values are the following:

Call Put

Strike 90.40 0.34 0.00

Strike 90.80 0.00 0.06

Strike 91.00 0.00 0.26

The 90.40 call’s premium (0.30) is lower than its intrinsic value (0.34). Thus, the premium is

unreasonably low.

d. In order to insure a maximum interest rate, you have to buy a put. The maximum interest rates

you can insure by using the various options are the following:

Put 90.40: 9.60 0.06 9.66%.

Put 90.80: 9.20 0.18 9.38%.

Put 91.00: 9.00 0.34 9.34%.

e. The contracts’ size being €1 million, 10 puts must be bought in order to cap your interest rate on

a €10 million borrowing.

f. Profits (losses) in euros are the following:

€€

(1.33 0.18)% 10 1,000,000Gain 28,750.

4

g. The cost of the puts was:

€€

0.18% 10 1,000,000Cost 4,500.

4

So, the return is, in percentage of initial investment, to 28,750/4,500 638.89%

37. You are currently borrowing €10 million at three-month Euribor 75 basis points. The Euribor is

at 3%. You expect to borrow this amount for five years but are worried that Euribor will rise in the

future. You can buy a 4% cap on three-month Euribor over the next five years with an annual cost of

0.75% (paid quarterly). Describe the evolution of your borrowing costs under various interest rate

scenarios (i.e., above and below 4%).


Solution

Let’s study the net cash flow on each semiannual payment date:

If Euribor < 4%

The cap is worthless.

The annualized borrowing cost interest rate on loan premium paid for the option Euribor

0.75% 0.75% Euribor 1.5%.

The dollar cost (Euribor 1.5%)(€10,000,000).

If Euribor > 4%

The cap pays the option holder the difference between Euribor and 4%.

The annualized borrowing cost interest rate on loan premium paid for the option 4% –

Euribor Euribor 0.75% 0.75% 4% – Euribor 5.5%.

The dollar cost (5.5%)(€10,000,000) €550,000.

Thus, the interest rate cap ensures that borrowing costs are never higher than 5.5% per year.

38. You would like to protect your portfolio of British equity against a downward movement of the

British stock market.

a. What are the relative advantages of stock index futures and options?

b. Should you prefer in-the-money or out-of-the-money options?

Solution

a. Futures allow removal of the uncertainty on the future value of the stock index, both up and

down. Put options on the index allow protection of your portfolio of British equity in case of a

fall in the stock market while retaining your upside potential; there is a cost associated with this

asymmetric insurance characteristic.

b. An in-the-money put gives a better downside protection than does an out-of-the-money put but it

reduces the upside potential more.

39. The current dollar yield curve on the dollar international bond market is flat at 7% for top-quality

borrowers. A company of good standing can issue plain-vanilla straight and floating-rate dollar bonds

under the following conditions:

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

Bond B: FRN. Five-year dollar FRN with a semiannual coupon set at LIBOR plus 0.25% and

a cap of 14%. The cap means that the coupon rate is limited to 14% even if the LIBOR passes

13.75%.

An investment banker proposes to a French company to issue bull and/or bear FRNs under 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% and a cap

of 20.5%.

Coupons on all bonds cannot be negative. The investment bank also proposes a five-year floor option

at 3.5%. This floor will pay to the French company the difference between 3.5% and LIBOR, if it is

positive, or zero if LIBOR is above 3.5%. The cost of this floor is spread over the payment dates and


set at an annual 0.1%. The investment bank also proposes a five-year cap option at a strike of 13.75%.

The cost of this cap is spread over the payment dates and set at an annual 0.05%. The company can

also enter into a five-year interest rate swap at 7% fixed against LIBOR.

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

current market conditions for plain-vanilla straight bonds and FRNs.

b. Find out the borrowing cost reduction that can be achieved by issuing bull notes compared to a

fixed-coupon rate of 7.25%.

c. Find out the borrowing cost reduction that can be achieved by issuing bear notes compared to an

FRN at LIBOR plus 0.25%.

Solution

a. There are many ways to combine the bonds to illustrate the advantage of the bull and bear notes

for the issuers. Let’s assume that the company issues both notes in similar amounts. As long as

LIBOR stays between 3.50% and 13.75%, the coupon on the sum of a bear and a bull note is:

Coupon1 (13.75% – LIBOR) (2 LIBOR – 7%) LIBOR 6.75%.

This is better than issuing a straight bond plus a FRN:

Coupon2 (LIBOR 0.25%) (7.25%) LIBOR 7.5%.

If LIBOR is above 13.75%, the coupons become:

Coupon1 0 20.5% 20.5%.

Coupon2 7.25% 14.00% 21.25%.

If LIBOR is below 3.5%, the bull and bear can become more costly because of the fact that the

coupon on the bear FRN cannot be negative. To cover this risk, the issuer should buy two 3.5%

floor options from the bank, with an annual cost of only 0.2%. This is still cheaper than issuing

Bond A plus Bond B.

b. Let’s see how we could use the bull and bear notes to get into a fixed-rate borrowing at a cost

inferior to 7.25%.

The idea is to combine a bull FRN plus a swap:

Issue the Bull FRN,

and swap to pay LIBOR and receive fixed 7%.

As long as LIBOR remains below 13.75%, the net annual cost will be:

Coupon3 (13.75% – LIBOR) (LIBOR – 7%) 6.75%.

To be protected against the risk that LIBOR will go above 13.75%, the company should buy a

LIBOR cap at 13.75% from the bank. The cost of this out-of-the-money cap (current LIBOR

around 7%) is extremely low and equal to 0.05%. The net result is a cost of 6.80%, or a cost

reduction of 0.45% compared to a direct issue of a straight bond at 7.25%.

c. To replicate the plain-vanilla FRN, the company could:

Issue a Bear FRN,

swap to pay fixed 7% and receive LIBOR,

or buy two LIBOR floors at 3.5%.


The net annual cost will be:

Coupon4 (2 LIBOR – 7%) (7% – LIBOR) 0.2% LIBOR 0.2%.

This is a 0.05% cost reduction compared to Bond B. The cost reduction will be even greater if

LIBOR goes over 13.75% because of the cap on a bear FRN coupon.

40. The current yield curve on the international bond market in euro is flat at 4% for top-quality

borrowers. A French company of good standing can issue plain-vanilla straight and floating-rate

bonds at the following conditions:

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

Bond B: FRN. Five-year dollar FRN with a semiannual coupon set at 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 a

strike of 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 a strike of 7.60%. The annual

premium on the cap is 0.1%. The company can also enter in a five-year interest-rate swap 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 compared to

issuing a fixed-coupon straight bond at 4%.

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

issuing a plain-vanilla FRN at LIBOR.

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

issuing a fixed-coupon straight bond at 4%.

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

issuing a plain-vanilla FRN 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%), 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 the company should issue C D and buy two floors at 2.1% and a

cap at 7.6%, then the total cost is:

(7.60% – LIBOR 0.1%) (2 LIBOR – 4.2% 2 0.05%)/2 (3.40% LIBOR 0.2%)/2.

This is still better than A B.

b. Issue the bull, swap receive fixed pay LIBOR, and buy a cap at 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 at 7.6%.

Total annual coupon:

(7.60% – 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 at 2.1.

Total coupon:

(2 LIBOR – 4.2%) 2 (4% – LIBOR) 2 0.05% 3.9% or a 0.10% cost reduction.

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

(2 LIBOR – 4.2%) (4% – LIBOR) 2 0.05% LIBOR – 0.1% or a.10% cost reduction.

41. 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 ¼ %, 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 premium 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 at 5.30%?

d. Same question assuming that the Kingdom wanted to issue an FRN at LIBOR ¼%?


Solution

a. For the investor, this is equivalent to:

Long in three straight bonds.

Short two FRNs.

Long two caps with a strike of 7.3%.

b. To transform the very-bull bond into a straight bond, the issuer needs to do the following:

Issue the very-bull 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%, or a savings of 30 bp.

d. To transform the very-bull bond into an FRN, the issuer needs to do the following:

Issue the very-bull 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 LIBOR, or a savings of 25 bp.

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

should be:

A. A one-year currency option bond. The bond is issued in dollars with a face value of $100. The

bondholder can choose to have the coupon and principal paid in dollars or in SFr, at a specified

exchange rate of SFr/$ 2, that is, receive either $100 or SFr 200 as principal repayment, and

receive either $C or SFr 2C as interest if C is the coupon set in dollars. The coupon rate is

c C/100.

B. A two-year currency option bond. The bond is issued in dollars, with a face value of $100 and

pays an annual coupon C . The bondholder can choose to have the coupons and principal paid in

dollars or in SFr, at a specified exchange rate of SFr/$ 2, that is, receive either $100 or SFr 200

as principal repayment, and receive either $C or SFr 2C as interest, if C is the coupon set in

dollars. The coupon rate is c C /100.

Current market conditions are given below:

Interest Rates 1-Year 2-Year

Zero-coupon rates

US$ 8% 8%

SFr 4% 4%

Spot exchange rate: SFr/$ 2

Currency options:

SFr call, strike price 50 U.S. cents, expiration one year: 2 U.S. cents.

SFr call, strike price 50 U.S. cents, expiration two years: 5 U.S. cents.

a. Compute the coupon C on Bond A that would be consistent with market conditions at time of

issue.

b. Compute the coupon C' on Bond B that would be consistent with market conditions at time of

issue.


Solution

a. The one-year currency option bond can be valued as the sum of:

A zero-coupon one-year bond paying $(100 C) in one year,

and a currency option to exchange $(100 C) for SFr (200 2C) in one year.

Hence, buying Bond A is equivalent to buying a zero-coupon one-year bond paying $(100 C)

in one year, plus buying 2 (100 C) SFr calls with a strike of $0.50.

Hence

100100 2(100 ) 0.02

1.08

$3.53.

CC

C

Then the coupon rate is: c 3.53%.

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

A zero-coupon one-year bond paying $C in one year,

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

a one-year SFr call option to exchange $C for 2C SFr,

and a two-year SFr call option to exchange $(100 C ) for 2 (100 C ) SFr.

Hence

100100 0.04 0.10 (100 )

21.08 1.08

' $2.22.

C CC C

C

And a coupon rate c 2.22%.

43. Titi, a Japanese company, issued a six-year international bond in dollars convertible into shares of the

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 dollars

($/¥ 0.005, ¥/$ 200).

a. Why can the bond be issued with a yield of only 8%, below the market rate for straight dollar

bonds?

b. What would happen if:

The stock price of Titi increases?

The yen appreciates?

The market interest rate of dollar bonds drops?

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 ¥ 2,000.

The exchange rate is $/¥ 0.006.

c. 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 straight bonds issued by Titi, options or warrants on the yen value of Titi stock,

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


Solution

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

b. The effects are the following:

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 dollar bonds drop, 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 the use of the valuation of options on

options. The problem comes from the fact that the conversion value of the bond 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.

44. 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). Let’s 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.

45. In 1990, the French bank, BNP, issued exchangeable bonds denominated in French francs (FF).

These are bonds issued for FF 100 on April 1, 1990, with an annual coupon of FF 5, plus an exchange

right. The bonds can be redeemed for FF 100 on April 1, 1996. The right can be exchanged on

April 1, 1991, with payment of an additional FF 100, for another bond identical to the old bond

(annual coupon of FF 5 and redeemed for FF 100 on April 1, 1996). If you exercise your right, you

will have paid an additional FF 100 on April 1, 1991, but you will then hold two BNP bonds with

maturity in 1996.

a. Under what scenario would you exercise the exchange right (exchange the right plus FF 100 for

an additional bond) on April 1, 1991? What is the attraction of such an exchangeable bond for

investors?

b. On April 1, 1990, the yield curve is flat at 6%. You can buy a call on a five-year bond with a

coupon of 5%. The call has a strike price of 100% and expires on April 1, 1991. Its premium is

2%. Construct a replication portfolio to determine at what price the exchangeable bond can be

issued by BNP.

Solution

a. The option is exercised on April 1, 1991, if the current market price of the BNP bond is more

than FF 100. Thus, using such an option is interesting in the case of a sharp fall in interest rates.

This bond allows the investor to speculate on a drop in interest rates with only limited risk.

b. The cash flows of this exchangeable bond can be replicated by a portfolio including a six-year

5%-coupon bond and a one-year call option on this bond (strike 100). Consequently, the value

of the newly issued exchangeable bond (“P”) is equal to the value of this portfolio:

P 5 6

5 5 1052 97.082.

1.06 1.06 1.06


46. Digital options: Digital (or binary) options can only have two payoffs at maturity. If the strike

condition set in the option is met, the buyer will receive the full prespecified payoff. If not, the buyer

receives no payoff. This is different from a traditional option where there exists an infinite number of

payoffs. For example, we could have a digital option on the French CAC index, stating that the option buyer will get €200 if the CAC index is above 4,000 at expiration and zero otherwise. On this digital

option, the buyer will get exactly 200 as soon as the CAC index is above the 4,000 level at expiration,

whether it be 4,001, 4,100, or 5,000.

a. Draw the profit and loss curve at expiration as a function of the CAC index for these two options:

Traditional call on the CAC index: Exercise price: 4,000; premium: 40.

Digital call on the CAC index: Exercise price: 4,000; payoff if exercised: 200; premium: 40.

b. What are the relative advantages of the two options?

c. Assume that the volatility of the French stock market increases suddenly. Should the premium on

the digital call increase more (or less) than the premium on the traditional call?

Solution

a. Profit and loss simulation at expiration:

4 0 0 0 4 1 0 0 4 2 0 0

- 4 0

4 0

1 6 0

8 0

1 2 0

d i g i t a l

t r a d i

t i o n a l

C A C i n d e x a t m a t u r i t y

P r o f i t

b. On a digital option, the fixed 200 payoff will be paid if the CAC index is above 4,000 at maturity.

Thus, the digital option is preferable to a traditional option if a small movement

above 4,000 is expected. However, a traditional option remains preferable if a large upswing

is anticipated.

c. The holder of a traditional option has more to gain from large movements in the index. Hence,

the value of a traditional call will increase more than the value of a digital call if the volatility of

the index increases.

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. A 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. You are a young investment banker considering the issuance of a guaranteed note with stock index

participation for a client. The current yield curve is flat at 8% for all maturities. Long-term at-the-

money options on the stock market index are traded by banks. Two-year at-the-money calls trade

at 17.84% of the index value; three-year at-the-money calls trade at 20% of the index value. You

are hesitant about the terms to set in the structured note. You know that if you guarantee a higher

coupon rate, the level of participation in the stock appreciation will be less. Your boss asks you to

compute the “fair” participation rate that would be feasible for various guaranteed coupon rates and

maturities. In other words, based on the current market conditions (as described above), estimate the

participation rates that are feasible with a maturity of two or three years, and a coupon rate of: 0%,

1%, 2%, 3%, 5%, and 7%.

Solution

The guaranteed note can be decomposed as the sum of a straight bond with a coupon C plus p times a

call option on the index (p is the participation rate).

For a two-year note, we have:

2 2

100100 17.84

1.08 1.08 1.08

C Cp

hence 14.266 1.7833

.17.84

Cp


For a three-year note, we have:

2 3 3

100100 20

1.08 1.08 1.08 1.08

C C Cp

hence

20.617 2.5771.

20

Cp

The next table gives the fair participation rates for various coupon rates:

Coupon Rate p(2-Year Bond) p(3-Year Bond)

0% 80% 103%

1% 70% 90%

2% 60% 77%

3% 50% 64%

5% 30% 39%

7% 10% 13%

49. You’re a banker. A client wishes to buy a guaranteed note with a 100% indexation to the stock

index’s growth. In other words, he doesn’t 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 maturity, the price of the option, that of the

zero-coupon, and 100%-zero.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 2 4 6 8 10 12 14 16 18 20

Maturity

100-Zero

Option

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 (that is, 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.

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