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Financial and Investment Instruments

Lecture 7: Derivatives

Aims

This lecture will review the principles of pricing of futures and option contracts, and show some examples of their use in portfolio management especially for risk reduction or hedging, and to gain leverage for the purpose of speculation. After this topic you should know:

• The characteristics of derivatives

• The principal differences between forward and futures contracts

• How to construct position and minimum variance hedges using futures contracts

• How to value futures contracts in general, and use the general principle to value stock index and currency futures.

• The general factors which rationally affect option prices

• How to relate European put and call prices through put-call oarity

• How to use the Black-Scholes formula to value European call options, and the assumptions underlying it.

• The characteristics of the swaps market

Reading

• Investment Analysis and Portfolio Management, 7th edition, Frank K. Reilly and Keith C. Brown (Thomson South-Western, 2003): Chapters 21, 22, 23 (parts), 24 (parts)

• Solnik, B. and D. McLeavey (2003), International Investments, 5th ed, Addison Wesley Chapter 10

This lecture covers analytical issues in valuing and using futures and options in equity portfolio management.

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Derivatives

A derivative or derivative asset is a financial instrument whose pay-off depends on the pay-off of some other asset (the underlying asset). Derivatives are used as a technique in risk management: hedgers use derivatives to reduce risk, and need speculators to assume this risk. Pricing derivatives: Because a derivative is a function of the pay-off of the underlying, it is possible to create identical pay-offs from combinations of different financial instruments. This leads to the principle of no-arbitrage in determining derivative prices. Assets with the same pay-offs must sell for the same price, since otherwise there would be an arbitrage opportunity: for two instruments with identical pay-offs but different prices: SELL the overvalued instrument, and BUY the undervalued instrument. There are two types of derivative assets: forward commitments and contingent claims A) Forward Commitments Forward and future contracts are agreements between two counterparties that fixes the terms of an exchange today for a transaction that will take place between them at some future date. Swap contracts are equivalent to a series of forward contracts, since it is an agreement between two counterparties to exchange a series of future cash flows. Examples: Stock index futures, interest rate futures, forward contracts, swaps B) Contingent Claims Option contracts are claims to pay-offs that depend (contingent) on specific events occurring. For example a call option is the right, but not the obligation, to buy an asset at the exercise price (agreed today) at a point in the future. You would only want to exercise this right and buy this underlying asset if the price of the underlying asset was greater than the exercise price! Examples: Interest rate options, convertible bonds, callable bonds, options on stocks, warrants exotic options.

1. Futures Contracts

A futures contract commits the buyer or ’long’ to buy the commodity or to make a cash settlement at a stated period in the future. The seller or ‘short’ is committed to deliver the commodity or make a matching cash settlement to the long. Examples could be to deliver 125,00 Euros for a fixed USD price, say USD 1.01 for each EUR – this is an FX future. Other examples are given below when we work out futures prices. In principle, futures contracts are the same as forward contracts – both commit a buyer and seller to a future price and quantity. They differ in practice largely because futures contracts are traded in organised markets while forward contracts are over the counter direct transactions usually with a bank.

Key Features of Forward and Futures Contracts Forward Contracts Futures Contracts

Customised size and delivery. Standardised in terms of size and delivery date.

Direct contract between counter-parties.

Contract is between customer and clearing house.

Over-the-counter Exchange traded

Locked in – Cannot reverse. Contract can be traded in an organised market – contracts can be reversed

All cash flows are at maturity date. Contracts are marked to market daily: gains and losses are realised immediately.

Margins are set at the beginning of the contract.

An initial margin is set then this must be kept above a ‘maintenance’ level.

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Differences Between Forward and Futures Contracts

(a) Futures contracts are available only in standardised units for standardised contract dates and for a limited number of assets. Forward contracts are not standardised with respect to unit, dates, or asset. (b) Futures contracts are traded on organised exchanges, and the contract is with the exchange clearing house, not with a particular investor. Forward contracts are not traded on exchanges, and are with a particular investor. (c) Futures contracts are marked-to-market. In effect you close your position each day, settle any profits or losses, and open up a new position. Forward contracts are not marked-to-market.

Futures contracts are standardized agreements to exchange specific types of good, in specific amounts and at specific future delivery of maturity dates. For example, there might be only four contracts traded per year, with the following delivery months: March, June, September and December. In contrast details of forward contracts are negotiable. The big advantage of having a standardized contract is that it can be exchanged between counterparties very easily - liquidity Futures contracts are traded on a central regulated exchange often by open outcry. Following order execution, the order is confirmed with customers, and futures contract exists. The number of contracts outstanding at any time is known as the open interest at that time. Futures contracts eliminate the problems of illiquidity and credit risk associated with forward contracts by introducing a clearing house, a system of marking to market and margin payments, and a system of price limits. The clearing house guarantees fulfilment of all contracts by intervening in all transactions and becoming the formal counterparty to every transaction. The only credit risk is therefore with the clearing house. It is also possible to unwind a futures contract at any time by performing a reversing trade, so futures contracts are generally extremely liquid (at least for the near maturing contracts). The clearing house withstands all the credit risk involved in being the counterparty to every transaction, by using the system of daily marking to market. At the end of every day's trading, the profits or losses accruing to the counterparties as a result of that day's change in the futures price have to be received or paid. Failure to pay the daily loss results in default and the closure of the contract against the defaulting party. The credit risk to the clearing house has now disappeared because the accumulated losses are not allowed to build up. Even a single day's loss is covered by a deposit that each counterparty must make when the contract is first taken out. This deposit is known as initial margin and is set equal to the maximum daily loss that is likely to arise on the contract. As the price of the contract goes against one of the counterparties, the resulting loss is met from that counterparty's initial margin and is paid over to the other counterparty as profit. As the margin account falls below a particular threshold (the maintenance margin level), it has to be topped up with additional payments known as variation margin. (Such payments have to be made immediately.)

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2. Why Use Futures

There are two primary motivations for using futures: a) Speculation The margin requirement for a futures contract is a small fraction of the face value of the contract. The futures contract offers leverage compared with a spot position b) Hedging A position in a spot asset can be hedged, at low cost, by an opposite position in the future on the asset.

Example of Leverage in a Futures Contract

The New York Board of Trade (NYBOT) trades various stock index futures contracts. One of these is the New York Stock Exchange Composite Index future. The market quotation is in the form of the index. (Examples of Index Futures quotations are shown in Figure 1. The NYSE composite is not shown in this Table)

• The index is valued at $500 per index point.

• On December 5 the index quote was 483, giving a face value for one contract of 483*500 = $241,500.

• Initial and maintenance margins are both set at $10,000. => A 10 point rise in the index ( about 2%), triggers a gain of $5,000 for 1 long contract and a loss of $5,000 for a short contract position. This is equivalent to 50% of the maintenance margin. The contract gives a leverage multiple of 25. (Full contract details can be found at http://www.nyce.com/ and then clicking on margin requirements and contract specs in the Show Report menu)

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Figure 1 Index Future Quotations in the FT

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Hedging Example

On December 5 we have a $1,000,000 portfolio of US equities.

• We can hedge our (long) spot position by taking a short position in the futures.

• The hedge will not usually be perfect: losses on the spot position will not usually be identical to the gain on the futures position.

• Notice also that we have to decide HOW MANY futures to sell for each $ of equities.

=>A simple method is to take a short futures position equal to the spot position. This is a position hedge.

# contracts = 4500*483

000,000,1exp≅=

futuresofValueFace

osurecashofValueFace

• In this case we might short 4 NYBOT NYSE Composite contracts Face value of 4 contracts= $968000 (approx $1million).

• A 10 point fall in the index future would produce a gain of 4*10*500 = $20,000 to our SHORT futures position. If our portfolio fell by 2%, we would lose $20,000 on our long spot position. The net gain is zero – the hedge happens to be perfect!

How much to Hedge – Choosing the Hedge Ratio

• Since there is not usually a perfect correlation between spot and futures prices, it is not obvious how many futures we should go short in order to hedge along position in the spot asset.

• We can, with a few exceptions, only trade futures on stock-market indices. If our portfolio is not an index portfolio, we cannot find a future on our portfolio. We have to use an INDEX futures position to hedge our portfolio.

• The hedge ratio of a hedged portfolio is the number of futures that we need to go short for $1 spot.

• ie hedge ratio: the number of units of the hedging instrument to buy to hedge the liability of the asset.

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Methods for choosing a hedge ratio

1. Position matching

This is the simplest method: Take exactly the opposite position in the futures market as in the spot. For example, if we own $1 million of equities, we sell $1 million of equity index futures. But this takes no account of the fact that when the market moves 1 % our portfolio may well move by a different amount because it is not fully diversified and/or it more aggressive/defensive than the market.

2. Minimum Variance Hedge

Choose the hedge ratio so that the variance of the HEDGED portfolio is a

minimum.

Maths: Let RP be the return to the unhedged portfolio Let RF be the return to the futures position Let RH be the return to the hedged portfolio Let H be the hedge ratio. Then

RH = RP – H*Rf

Using the portfolio variance formula:

Var(RH) = 1*Var(RP) + H2*Var(Rf) – 2*1*H*Cov(RP,Rf )

Using calculus to minimise this w.r.t. H: We find

H = Cov(RP,Rf )/Var(Rf)

NOTE: The formula for H is the same as the formula for beta if we do a least squares regression of RP on RF i.e. we estimate

RP = alpha + beta*Rf + error Intuition: This is because least squares estimation minimises the variance of the error. If we choose H = beta, the error in the regression is also the return to the hedged portfolio, and we have chosen beta to make its variance a minimum We have a $1 million equity portfolio. Suppose we do a regression of the last 60 months returns of our portfolio on the last 60 months returns to the index future. We get the following output: RP = 0.003 + 1.2*Rf, R-squared = 0.8 => H =1.2

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Choosing a minimum variance hedge means going short $1.2 million index futures. Also we can find out how effective we expect the hedge to be:

• R-squared tells us the percentage of the variance of RP that is explained by RF,

• 1 – R-squared tells us the percentage of the variance of RP that is explained by the ERROR.

=> Since the error is the return to the hedged portfolio, 1 - R-squared tells what proportion RH is of RP, and R-squared tells us by what percentage we have reduced our original portfolio variance by hedging. Suppose the standard deviation of our portfolio return is 30% p.a. Then R-squared = 0.8

� The variance of the hedged portfolio is 20% of the variance of RP � Var(RH) = 0.2*Var(RP) � Var(RH) = 0.2*0.30

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� StDev(RH) = 2.0 *0.3 = 0.1342

The standard deviation of the hedged portfolio is NOT zero, we don’t have a perfect hedge. But the standard deviation has fallen from 30% to 13.42% because of the hedge.

Construction of Hedge Portfolio with optimal hedge ratio

# contracts = β*exp

futuresofValueFace

osurecashofValueFace

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3. Pricing Futures

The principle of futures pricing is simple:

futures price is equal to the cost of ‘cash and carry’. This can be explained by a simple commodity example: If I need a ton of wheat in 1 month, I have 2 alternatives

• Buy 1 ton 1 month wheat future. Cost = futures price, payable in 1 month

• Borrow the cost of 1 ton of wheat SPOT, store the wheat, repay the loan after 1 month. Cost = spot price + interest + storage charge

Present Value of Futures = Spot price + carry cost

The difference between the futures price and the spot price is called the basis

Pricing Examples

A. Stock index future:

Within the context of stock index futures this general expression can be written

where Ft is the futures price of the index St is the spot price of the index d are dividends rf is the risk free rate. Carry cost is the interest cost less any dividends received Suppose the S&P 500 is now 900, the 3 month interest rate is 2%p.a. (simple quote), and the S&P 500 dividend yield for the next 3 months is 1.5% p.a. Then the 3 month futures price = 900*(1 + (0.02/4) – (0.015/4)) = 901.125

B. Foreign Currency Future

Suppose Spot USD/GBP =1.5, 3 month RUS = 2%p.a., 3 month RUK = 4%p.a. (simple quotes) The carry cost is the USD interest adjusted for the foreign interest earned. Then 3 month futures price for GBP = 1.5*( 1 + 0.02/4)/(1 + 0.04/4) = $1.4926 (same as the forward price)

t

fT t

=t

T

f

F

(1+ r ) = S -

d

(1+ r )τ

ττ∑

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4. Options

Options give

• the right but not the obligation to buy(CALL) or sell (PUT) an asset

• at a fixed price (EXERCISE or STRIKE) price at (EUROPEAN) or up to (AMERICAN) a fixed date( MATURITY or EXPIRATION) . The option price is often called the option PREMIUM and the selling of an option is often referred to as WRITING an option. Options are traded in most financial centres alongside futures contracts. See http://www.im.pwr.wroc.pl/~rweron/exchlink.html for a comprehensive set of links to exchanges.

Figure 2 Euronext/LIFFE Equity Option Price Quotes(FT)

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OPTION PAYOFF DIAGRAMS Option Values at Maturity A call option is a security which gives the owner the right to buy a share on or before a given date at a fixed exercise price, K. A put option gives the holder the right to sell a share at a predetermined exercise price. A European Option can only be exercised at maturity; an American Option can be exercised at any time before maturity. Value of call option at maturity is

CT = max [0, S - X] For example let X = 100p, then call price at terminal date as a function of various values of share price at the terminal date is

ST= 90 95 99 100 101 102 105 110

CT= 0 0 0 0 1 2 5 10 So value of call at maturity will have hockey-stick shape. To examine price of call at any time before maturity we need to make an assumption about the underlying stock price process.

CT

X S

Out of the

money

In the

money

45°

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Long/Short Calls/Puts at maturity

Long Call (Bought Call) Long Put (Bought Put) Short Call (Sold/Write Call) Short Put (Sold/Write Put)

Profit From Option - Including Initial Price

CT

X S

CT

X S

CT

X S

CT

S

CT

X

S

CT

X S

CT

X S

45°

CT

260

S

Profit ↑

Loss ↓

{ 14p

274

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Combining Options and Other Securities Protecting a Stock Purchase Suppose you purchase a share, but want to “insure” this asset against a fall in value - protect it from “down-side risk” Buy 1000 ABC at 270p each Buy 1 ABC Oct 260 Put at 8p

Bull Call Spread Suppose you believe an asset ABC is likely to rise slightly in the short-term Buy 1 ABC July 200 Call at 16p Sell 1 ABC July 220 Call at 8p

CT CT

S

260 270

{ 8p

Long stock

Long

Net Position

CT CT

Long call

CT CT

S { 8p

Long call

Short

Net Position 200

220

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5. Pricing Options – Rational Relationships

The following table gives the factors affecting option prices and the direction of their effect on call and put prices. The + / - are derived by rational argument, without a model at this stage.

Factor Call Put

Underlying asset price + -

Exercise Price - +

Time to Expiration + +

Risk free Rate + -

Volatility of Underlying asset Return + +

Notice – Only Volatility is not known exactly.

• Option prices can never be negative.

• American Options are always worth at least as much as European options.

• European Put and Call prices are related by Put – Call Parity.

Explanation

Let S be the price of the underlying stock Let C & P be the prices of European put and call options with the same exercise price (X) and same time to maturity (T). To derive put call parity: Form a portfolio as follows

Asset Position Cost

Long 1 share S

Long 1 Put P

Short 1 Call -C

Total Cost S+P-C

Whatever happens, this portfolio pays the exercise price X at maturity, T. Its value is therefore PV(X). So this must be its cost.

� PV(X) = S + P – C Put-Call Parity formula

Example

Suppose we have a stock with a price of $50 and a 3-month European call option with a exercise price of $52 and the call price is $5. The risk free rate, continuously compounded, is 4% p.a. Then we can figure out the price of a 3-month European $52 put. Put-Call Parity =>

P = PV(X) – S + C

= 52*exp(-0.04*0.25) –50 +5 = $6.48

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6. Exact Option Pricing by the Black-Scholes Model

The Back Scholes model makes assumptions about the returns of the underlying asset and the markets in which options and stocks are traded. The model derives a price for a European option on a stock that pays no dividends in the period up to the expiration of the option. The assumptions are:

• Price changes are independently and normally distributed with a constant mean and variance.

• There are no transaction costs or other market frictions.

• There are no pure arbitrage profit possibilities.

Intuition

If at maturity the stock price (ST) is above the exercise price the option is said to be in the money and we will make a profit of ST – X by exercising it. If ST is below X we make nothing. So the present value of the expected profit at maturity is:

C = PV( )0)(Pr)( +−∑>

T

XT

T SobS

XS

The Black-Scholes works out C, the value of a call option, given the normal distribution assumptions i.e. Prob(ST) is the normal distribution probability function. The Black-Scholes formula is of the form:

C = K1*S – K2*PV(X)

Where K1 = N(d1)

K2 = N(d2 -σ√T)

PV(X) = exp(-rf*T)*X

rf = continuously compounded risk free rate

σ = standard deviation of the underlying asset return.

d1 = (ln(S/X) + (rf +σ 2/2)*T)/σ * T

N(d1 ) is the probability obtained from a normal table that Z, a standard normal variable, is less than d1. NORMSDIST(d1) in EXCEL. Example of Using the Black Scholes Formula, To value

• 3 month European call and put options on a stock

• with a current price of $40.

• The exercise price is $38.

• The stock’s volatility ( standard deviation) is 40% p.a.

• the 3 month continuously compounded risk free rate is 5%.

So: S= 40, X=38, σ = 0.4, rf = 0.05, T = 0.25 ( Note σ ,T and rf are all in years) Using the formula

d1 = (ln(40/38) + (0.05 +0.42/2)*0.25)/0.4* 25.0

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= 0.4190 k1 = N(0.4190) = 0.6624

k2 = N(0.4190 – 0.4 25.0 ) = 0.5867

PV(X) = exp(-0.05*0.25)*38 = 37.53

� C = $4.48 To value the put option use put-call parity P = 37.53 – 40 + 4.48

= $2.01

Financial Interpretation of the Black Scholes Formula

The value C of the call option is given by

C = K1*S – K2*PV(X) Looking at the right hand side we see that this is the value of portfolio composed of:

• K1 shares each with a price of S

• A short position in the risk free asset ( i.e. borrowing) of K2*PV(X) This means that we can REPLICATE the call option by

• A position in the stock

• Borrowing Buying a call option is just the same as a ‘leveraged’ ( through borrowing) position in the stock. In theory this means that options are ‘redundant’. In practice, since d1 and

d2 depend on T, which is changing all the time, we would have to adjust our borrowing and stock position continuously. This is costly and impracticable.

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Hedging using Options

Hedging using options can be more flexible than hedging using futures. Simplest way to hedge a cash exposure is to purchase a put option

# contracts = β*exp

IndexofValueFace

osurecashofValueFace

As before the New York Stock Exchange Composite Index is 470. Put options with an exercise price of 480 sell for 51. As before the contract is valued at $500 per index

point. You have a portfolio of $1 million with a beta of 1.2.

# contracts = 2.1*500*470

000,000,1= 5

Cost of contract = 5 * $500 * 51 = $127,500 This hedge protects against downside risk, but preserves upside potential – however it

is expensive: a cheaper alternative is to using puts and calls in combination to create a synthetic short position in the index: Buy 5 480 puts and sell 5 480 calls. If calls sell for 43, then net cost of hedge is reduced to

Net cost of hedge = 5 * (51 – 43) * 500 = $20,000 This strategy has substantially reduced the cost of the hedge, but now there is no

upside potential We could assess the effectiveness of this hedge by examining the value of the positions when the options expire

Alternative hedging startegies

Should you hedge using options or futures? - Options are more flexible

- If exposure is certain (in amount and timing): use futures - If exposure is symmetrical (up and down risks): use futures

- Compare costs of alternative hedging strategies

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Swaps

Origins

Swaps orginated when firms wanted to invest overseas but countries imposed exchange controls. Swaps are agreements (contracts) between counterparties to swap future cash flows: a UK company needs dollars to invest in the US. Similarly a US firm needs sterling to invest in the UK. So the UK firm raises a sterling loan, and the US firm raises a dollar loan, and the counterparties simply exchange the principals and repayments. These loans were called back-to-back loans.

Even after exchange controls were abolished (in 1979 in UK), provided one company has a comparative advantage in raising funds than another company then there is there is the possibility of gains from trade.

Company A (AAA rating)

Company B (BBB rating)

Advantage of A over B

Fixed-rate loan 8% 12% 4%

Floating-rate loan LIBOR+0.25% LIBOR+0.5% 0.25%

Company A has an absolute advantage in borrowing both at fixed-rates and floating-rates, but Company B has a comparative advantage in borrowing at floating-rate loans. The net differential is 3.75% (4%-0.25%), which represents the gains from trade A swap is feasible if company A prefers a floating rate loan and company B prefers a fixed rate loan. Typically a swap bank arranges the swap Possible terms of the swap:

Company A Company B

Borrows fixed at 8% Borrows floating at LIBOR+0.5%

Receives from swap bank

(8+x)% Receives from swap bank (LIBOR)

Pays to swap bank LIBOR Pays to swap bank 12-y%

Net LIBOR-x% Net 12.5-y%

So company A effectively borrows at floating rates of LIBOR-x%, and company B

effectively borrows at fixed rates of 12.5-y%. Their are gains from trade if

LIBOR+0.25% > LIBOR-x% ⇒ gains to A of 0.25+x % and

12% > 12.5-y% ⇒ gains to B of y-0.5 %

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Payments to Intermediary (Swap Bank)

Swap Bank

Pay company A (8+x%)

Receives from Company A LIBOR

Pays Company B (LIBOR)

Receives from company B 12-y%

Net 4-y-x%

Competition between intermediaries will tend to reduce the net payments to the swap bank Note total gains are gains to A 0.25+x gains to B y-0.5 gains to swap bank 4-y-x Total Net gains 3.75%

Reasons for growth in the swaps market: In general swaps are the outcome of market imperfections: a) Comparative informational advantage in home market b) Subsidised loans c) Financial arbitrage d) Regulatory arbitrage e) Hedging purposes f) Restructuring capital profile Risks inherent in swaps: a) Credit risk b) Market risk

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TYPES OF SWAPS

1. INTEREST RATE SWAPS

An interest rate swaps is a contract between two parties who agree to exchange one stream of interest rate payments for another stream of interest rate payments. Since both underlying loans are in the same currency, there is no initial exchange of principals and re-exchange of principal at maturity. The only cash flows that are swapped are the interest streams on each of the notional loans. The notional principal

is simply a reference amount against which the interest is calculated. The main usage of interest rate swaps : • To achieve funding at rates below those available in bond markets and from banks. • To obtain fixed rate financing when it is impossible to access the bond markets

directly. • To restructure a debt profile without raising new finance. •To hedge against interest rate risk.

a. Coupon swaps

Coupon swaps is the most common type of interest rate swaps between fixed and floating rates. The argument for swaps is based on the principle of comparative advantage. Even though a high credit-rated firm could borrow at a lower cost in both the fixed rate and floating rate markets, it will have a comparative advantage relative to a lower credit-rated firm in one of the markets. Under these conditions, each borrower could benefit from borrowing in the market in which it has a comparative advantage and then swapping obligations for the desired type of interest payments.

Example :

Firm A and firm B face the following interest rates for a loan of the same maturity :

Firm A Firm B Differential

Fixed 9.5% 10.2% 0.70% Floating LIBOR+0.15% LIBOR+0.30% 0.15% Net differential 0.55% Both firms could enter into a swap arrangement in order to achieve lower borrowing cost. Since firm A has a comparative advantage in fixed rate debt, firm A would issue fixed rate debt at 9.5% and B issues floating rate debt at LIBOR+0.30%. With the swap, firm A pays the intermediary bank LIBOR+0.05% in exchange for a fixed rate

of 9.8%. Firm B pays the intermediary a fixed rate of 10.05% and in return receives a floating rate of LIBOR+0.20%.

Firm A Firm B Bank

Pays 9.5% LIBOR + 0.05%

LIBOR + 0.3% 10.05%

9.8% LIBOR + 0.2%

Receives 9.8% LIBOR + 0.2% 10.05% LIBOR + 0.05%

Net LIBOR - 0.25% 10.15% Savings 0.40% 0.05% 0.10%

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Firm A has effectively issued a floating-rate debt at LIBOR-0.25%, saving 0.4% from issuing the floating rate debt directly. Firm B has effectively issued a fixed rate debt at 10.15%, saving 0.05% from issuing the fixed rate debt directly. The intermediary bank gained 0.1% for its service. The total gain from the swap to all parties is 0.55%.

b. Basis swaps

A basis interest rate swap allows a borrower to exchange a stream of payments determined by one floating interest rate for a payment stream determined by another floating interest rate.

Basis swaps can also be arranged between different reset period of the same index, for instance, one-month LIBOR against three-month LIBOR.

Example :

Barclays's US branch has made a loan to a US firm in which the firm has an option of paying LIBOR+2.5% or US prime rate+1.25%. The firm chose the later. Since Barclays's funding is in the Eurodollar market, it faces `basis' risk. So, it undertakes a basis swap with Bankers Trust (a US bank). Barclays pays Bankers Trust US prime rate - 1.2% in return for 3-month LIBOR rate from Bankers Trust.

2. CURRENCY SWAPS Currency swap is an agreement to exchange both the principal and the periodic interest payments in one currency for principal and interest payments in another currency, for a predetermined length of time and at a rate of exchange agreed at the outset. A typical currency swap agreement will require : a) Initial exchange of principal (often at the spot rate). b) Exchange of interest payment (on agreed date at a predetermined interest rate).

c) Re-exchange of principal (at a rate agreed upon before the commencement of the swap).

Currency swaps are typically used to achieve one of the following objectives : • Hedging currency exposure • Obtaining funds at lower costs • Gaining access to a restricted market • Altering the currency of payments stream or investment income

a. Fixed/fixed Currency Swaps

It is an exchange between two parties of fixed interest in one currency in return for fixed interest payment in another currency. It also allows counterparties to arbitrage their relative access to different markets, thereby introducing cost savings.

Example :

A Swiss company can borrow in the Swiss capital market at 6% fixed rate but requires Sterling finance for its investment in the UK. A UK firm requires 250m Swiss Franc (SF) and would like to borrow SF at a fixed rate. However, the firm does not have a good enough credit rating to borrow in the Swiss capital market.

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Assume the exchange rate is SF2.50/£. Under the swap agreement, the Swiss company would borrow SF250m at 6% p.a. and the British company would borrow £100m at 6.5% p.a. and enter into a swap arrangement with each other. The principal will be re-exchanged at the same exchange rate. The process : • The UK firm would exchange £100m for SF250m. • On each coupon date, the UK company will provide SF15m to the Swiss company

in exchange for £6.5m from the Swiss company. • On maturity of the swap, the Swiss company will receive back the SF250m and pay

back the £100m. In this example, the swaps have effectively been used to hedge against foreign exchange risk, and to gain access to a form of finance that would not otherwise have been available.

b. Currency Coupon Swaps

It is a combination of interest rate swap and fixed rate currency swap. Besides having the exchange of principal amounts, the fixed rate interest in one currency is exchanged for floating rate interest in another currency.

Example :

A Danish company wants to open a plant in Switzerland and to finance it with fixed rate Swiss Franc (SF) debt. It could borrow SF at a fixed rate of 5.75%. A US firm could obtain dollar funds at a rate of LIBOR+0.25%. The US firm has been advised that it can issue a SF bond at a favourable rate of 5% pa. The Danish firm could borrow cheaply in the Eurodollar market at LIBOR flat. They agreed to an exchange of interest and principal that will allow each of them to obtain funds more cheaply than in the absence of a swap. The US firm would issue SF

bond of 150 million. Suppose the exchange rate is SF1.50/US$. The Danish company would borrow $100m from the Eurodollar market. US firm Danish firm Difference

Fixed (SF) 5% 5.75% 0.75% Floating ($) LIBOR+0.25% LIBOR 0.25% • At the initial exchange of principal, the US firm swaps SF150m which it has no

need, with the Danish firm for $100m. • The swap has been carefully structured so that every year the US firm will pay the

Danish firm the US dollar LIBOR rate, in exchange for 5% in SF from the Danish firm.

• In the final year of the swap, the principal amount are re-exchanged at the spot rate of SF 1.50/US$, enabling each to repay the debt it has borrowed.

The net cost of borrowing to the US firm is the Eurodollar LIBOR rate, 0.25% cheaper than it otherwise would have been able to get directly. The Danish firm effectively gets Swiss Franc debt at a fixed rate of 5%, which is 0.75% lower than if the firm was to raise the debt directly.

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Comprehension Check:

1. What are the differences between forward and futures contracts? 2. What is the principle of futures pricing? 3. What is a hedge ratio?

4. Name 2 ways of choosing a hedge ratio?

5. What factors affect option prices? In what direction? 6. What is put-call parity? 7. What factors appear in the B-S formula?

8. What is a swap contract?

9. How are the terms of a swap arrangement determined?

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Questions

1. We have a $10 million stock portfolio, we wish to hedge against market falls. Give 2 possible hedge positions using index futures.

You are given the results of a regression of the portfolio return against the returns to a futures index: RP = 0.001 + 0.8*RF, R-squared = 0.6

If the portfolio standard deviation is 0.25, what would you expect the standard deviation of the regression based hedge portfolio to be? 2. Calculate 3 month futures prices for

i) A stock index future for a spot index , current value = 4000, dividend yield = 3.5% p.a., risk free rate 4% p.a. State any assumptions.

ii) A EUR/GBP future, given a spot rate 1.6, REUR = 2.75% p.a., RUK = 4.0% p.a.. State any assumptions.

3. Value European put and call options on a stock. The options have 2 months to

expiration, a strike price of $50. The stock price is currently $52 with a return volatility of 35% p.a.. The continuously compounded 2 month risk free rate is 6% p.a.

4. A client has $2 million to invest in an equity index and T-Bills. She specifies a

maximum loss of 10% over a 1year time horizon. The portfolio is adjusted every quarter. Assume the T-Bill rate is zero. Show the equity exposure and cash positions she would have if the following quarterly market returns

happened:

Q1 +10%, Q2 0%, Q -30%, Q4 +20%. Give answers for T-Bill and CMPI ( with a multiple of 2) methods. Comment. 5. A $1 million portfolio has a volatility of 20% p.a. Calculate the value of a 1

year put option with a strike price equal to the portfolio’s value. Assume the risk free rate is 5% p.a.

6. Use the data in 5, to calculate the positions in stock and cash which replicate the stock + put option position.