11/2/2015 Nattawoot Koowattanatianchai 1 Derivatives Analysis - - PowerPoint PPT Presentation

11/2/2015 Nattawoot Koowattanatianchai 1

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Derivatives Analysis

Nattawoot Koowattanatianchai

11/2/2015 Nattawoot Koowattanatianchai 3

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Lecture 1

Forward contracts

Discussion topics

 Forward contracts

 Nature of a forward contract  Types of forwards  Pricing and valuation of equity

forwards

 Pricing and valuation of bond

and interest rate forwards

 Pricing and valuation of currency

forwards

 Credit risk in forwards

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Readings

 CFA Program Curriculum 2015 -

Level II – Volume 6: Derivatives and Portfolio Management.

 Reading 47

 Don M. Chance and Robert

Brooks, An Introduction to Derivatives and Risk Management, 9th Edition, 2013, Thomson.

 Chapters 8-9

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Derivative contracts

 Definition

 A derivative contract derives its value from the

performance of an underlying entity (e.g., asset, index, or interest rate).

 Derivative contracts covered in this course

 Forwards  Futures  Options  Swaps

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Forward contracts

 Definition

 A forward contract is an

agreement between two parties in which one party, the buyer or the long, agrees to buy from the other party, the seller or the short, an underlying asset or other derivative, at a future date at a price established at the start of the contract.

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Forward contracts

 Example

 A pension manager, anticipating the receipt of

cash at a future date, enters into a commitment to purchase a stock portfolio at a later date at a price agreed on today.

 The manager is hedged against an increase in stock

prices until the cash is received and invested.

 The manager is also hedged against any decrease in

stock prices.

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Forward contracts

 Important features

 Transaction price and quantity

are agreed today.

 Neither long or short pays any

money at the start, although some collateral may be required to minimize the default risk.

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Delivery and settlement

 At expiration

 A deliverable forward contract stipulates that the

long will pay the agreed-upon price to the short, who in turn will deliver the underlying asset to the long.

 Alternatively, a cash settlement forward contract

permits the long and short to pay the net cash value of the position on the delivery date.

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Delivery and settlement

 Example

 Two parties agree to a forward contract to trade a

zero-coupon at a price of $98 per $100 par.

 At expiration, the underlying zero-coupon bond is

selling at a price of $98.25.

 Delivery: The long is due to receive from the short an

asset worth $98.25, for which a payment to the short of $98.00 is required.

 Cash settlement: The long is due to receive $0.25 from

the short.

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Default risk

 Only the party owing the

greater amount can default.

 Delivery: If the short is obligated

to deliver a zero-coupon bond selling for more than $98, then the long would not be obligated to make payment unless the short makes delivery.

 Cash settlement: Since the

short is owing the greater amount, he/she may default.

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Termination of a forward contract

 Assuming that the contract calls for a delivery

at expiration, and that the long decides that he/she no longer wishes to buy the asset at expiration.

 The long can re-enter the market and create a

new forward contract expiring at the same time as the original contract, taking the position of the short instead. Doing so would mean that the long has no further exposure to the price of the asset.

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Termination of a forward contract

 Example

 Amy is originally long to buy at $40 and later short

to deliver $42.

 At expiration, the short of the original contract delivers

the asset to Amy, and she pays him $40.

 Amy then delivers the asset to the long of the

subsequent contract and receive $42.

 Amy nets $2.

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Termination of a forward contract

 Example

 Amy is exposed to a possibility of default from her

counterparty (credit risk).

 The counterparty of her original contract defaults, she

has to buy the asset in the market and could suffer a significant loss.

 If the counter party of her subsequent contract defaults,

she would then not deliver the asset but would be exposed to the risk of changes in the asset price.

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Termination of a forward contract

 Example

 Amy can avoid the credit risk by

contacting the original counterparty with whom she engaged in the long forward contract and go short the forward this time.

 If both agree to cancel both

contracts, the counterparty would pay Amy the present value of $2.

 The credit risk is eliminated.

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Termination of a forward contract

 Example

 Amy could choose to deal

with the other counterparty and leave the credit risk in the picture.

 She might receive a better price

from another counterparty.

 She might perceive the credit

risk to be too high.

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Types of forward contracts

Types of forward contracts Equity forwards Bond and interest rate forward contracts Currency forward contracts Other types

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Equity forwards

 Definition

 Equity forward is a contract

calling for the purchase of an individual stock, a stock portfolio, or a stock index at a later date.

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Equity forwards: Individual stocks

 Example

 A portfolio manager is responsible for the portfolio

• f a high-net-worth individual. This individual is

heavily invested in the stock called Gregorian Industries, Inc. (GII). The client notifies the manager of her need for $2 million in cash in six

• months. This cash could be raised by selling

16,000 shares at the current price of $125 per GII

• share. For whatever reason, it is considered best

not to sell the stock any earlier than necessary.

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Equity forwards: Individual stocks

 Example

 Portfolio manager’s actions:

 Contacting a forward contract dealer and obtaining a

quote of $128.13 as the price at which a deliverable forward contract to sell the stock in six months. Signing the contract for the sale of 15,600 shares at $128.13 which will raise $1,998,828.

 Possible consequences at expiration:

 The client loess if the GII stock rises to a price above

$128.13 during the six-month period.

 The client wins if the price falls below $128.13.

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Equity forwards: Stock portfolios

 Example

 A pension fund manager needs to sell about $20

million of stocks to make payments to retirees in three months. The manager has analyzed the portfolio and determined the precise identities of the stocks and number of shares of each that he would like to sell.

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Equity forwards: Stock portfolios

 Example

 Pension fund manager’s

possible options:

 Option 1: Entering into a forward

contract on each stock that he wants to sell and incurring administrative costs for each contract.

 Option 2: Entering into a forward

contract on the overall portfolio and incurring only one set of costs.

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Equity forwards: Stock portfolios

 Example

 Pension fund manager’s actions:

 Choosing Option 2 and providing a list of the stocks and

number of shares of each he wishes to sell to the dealer.

 Obtaining a quote of $20,200,000 from the dealer.

 If the stock is worth $20,500,000 at expiration.

 Deliverable: The manager will transfer the stock to the

dealer and receive $20,200,000. This means that the client effectively takes an opportunity loss of $300,000.

 Cash settlement: The client will pay the dealer $300,000

and sell the stock in the market, receiving $20,500,000.

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Equity forwards: Stock indices

 Example

 A UK asset manager wants to protect the value of

her Financial Times Stock Exchange 100 (FTSE 100) index fund. The manager wants to sell a certain amount of UK blue chip shares at the later date, but is unsure which stocks she will still be holding then. She simply knows that FTSE 100 is representative of the stock that she will sell. The manager is also concerned with the systematic risk associated with the UK stock market.

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Equity forwards: Stock indices

 Example

 Asset manager’s possible

• ptions:

 Option 1: Taking a specific

portfolio of stocks to a forward contract dealer and obtaining a forward contract on that portfolio.

 Option 2: Obtaining a forward

contract on the FTSE 100 that has a better price quote because the dealer can more easily hedge the risk with other transactions.

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Equity forwards: Stock indices

 Example

 Asset manager’s actions:

 Assume that the manager

decides to protect £15,000,000 of stocks. She chooses Option 2 and

• btains a quote price of

£6,000 on a short forward contract covering £15,000,000 from the dealer. The index contract is nearly always cash settled.

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Equity forwards: Stock indices

 Example

 At expiration, the index is at £5,925.

 The index declines by 1.25%. Thus, the manager should

receive 0.0125 × £15,000,000 = £187,500 from the dealer.

 If the portfolio perfectly matches a FTSE 100 index fund,

then it could be viewed that the loss of 1.25% of the portfolio initially worth £15,000,000 (loss of £187,500) is covered by the forward contract.

 In reality, the portfolio is not an index fund and such a

hedge is not perfect.

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Equity forwards: Dividends

 Equity forwards typically have payoffs based

• nly on the price of the equity, value of the

portfolio, or level of the index. They do not

• rdinarily pay off any dividends paid by the

component stocks.

 An exception is some equity forwards on stock

indices are based on total return indices, e.g., S&P 500 Total Return Index.

 The variablity of prices is much more important

than the variability of dividends.

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Forward contracts on bonds

 Definition

 Bond forward is a contract calling for the purchase

• f an individual bond, a bond portfolio, or a bond

index at a later date.

 Similarities between equity forwards and

bond forwards

 A bond may pay a coupon, which corresponds

somewhat to the dividend that a stock may pay.

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Forward contracts on bonds

 Differences between equity forwards and bond

forwards

 A bond matures. Thus, a forward contract on a bond must

expire prior to the bond’s maturity date.

 Bonds often have many special features such as calls and

convertibility.

 A bond carries the risk of default. So, a forward contract

written on a bond must contain a provision to recognize how default is defined, what it means for the bond to default, and how default would affect the parties to the contract.

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Forward contracts on bonds

 Example

 Consider a forward contract on a default-free

zero-coupon bond (or a Treasury bill or T-bill) in which one party agrees to buy the T-bill at a later date, prior to the bill’s maturity, at a price agreed

• n today.

 Suppose the underlying is a 180-day T-bill, which is

selling at a discount of 4%.

 Its price per $1 par will be 1- 0.04(180/360) = $0.98.  The bill will therefore sell for $0.98.  If purchased and held to maturity, it will pay off $1.

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Forward contracts on bonds

 Example

 Consider a forward contract that calls for delivery

• f a 90-day T-bill in 60 days.

 Suppose the contract sells for $0.9895. This implies a

discount rate of 4.2%.

 $1 – 0.042(90/360) = $0.9895

 T-bills are typically sold at a discount from par

value, a procedure called “discount interest”, and the price is quoted in terms of the discount rate.

 The use of 360 days is the convention in

calculating the discount.

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Forward contracts on bonds

 Example

 Consider a forward contract on a default-free

coupon-bearing bond, or a Treasury bond or T- Bond.

 The T-bond pays interest, typically in semiannual

installments, and can sell for more (less) than par value if the yield is lower (higher) than the coupon rate.

 Prices of T-bonds are typically quoted without the

interest that has accrued since the last coupon date. But we shall always work with the full price – that is, the price including accrued interest.

 Prices are often quoted by stating the yield.

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Interest rate forward contracts

 Eurodollar

 A dollar deposited outside the US.  Banks borrow dollars from other banks by issuing

Eurodollar time deposits, which are essentially short-term unsecured loans.

 The rate on such dollar loans is call the London

Interbank Rate.

 The lending rate, called the London interbank offered

rate (Libor), is more commonly used in derivative contracts.

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Interest rate forward contracts

 Libor

 Libor is the rate at which

London banks lend dollars to other London banks.

 Libor is considered to be

the best representative rate on a dollar borrowed by a private, i.e., non governmental high-quality borrower.

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Interest rate forward contracts

 Example: Eurodollar time deposit

 Suppose a London bank such as NatWest needs

to borrow $10 million for 30 days. It obtains a quote from the Royal Bank of Scotland for a rate

• f 5.25%.

 30-day Libor is 5.25%.  If NatWest takes the deal, it will owe $10,043,750 in 30

days.

 $10,000,000 x [1 + 0.0525(30/360)] = $10,043,750

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Interest rate forward contracts

 Example: Eurodollar time deposit

 Unlike the T-bill market, the interest is not

deducted from the principal. Rather, it is added on to the face value, a procedure appropriately called “add-on interest”.

 The rates on Eurodollar time deposits are

assembled by a central organization and quoted in financial newspapers.

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Interest rate forward contracts

 Example: Other time deposit instruments

 Eurosterling trades in Tokyo.  Euroyen trades in London.  A euro-denominated loan

 One bank borrows euros from another.  There are two rates on such euro deposits.  EuroLibor is complied in London by the British Bankers

Association.

 Euribor is complied in Frankfurt and published by the

European Central Bank.

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Interest rate forward contracts

 Forward rate agreement

(FRA)

 FRA is a contract in which the

underlying is neither a bond nor a Eurodollar or Euribor deposit but simply an interest payment made in dollars, Euribor, or any other currency at a rate appropriate for that currency.

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Interest rate forward contracts

 Example: FRA

 Consider an FRA expiring in 90 days for which the

underlying is 180-day Libor. Suppose the dealer quotes this instrument at a rate of 5.5%. Suppose the end user goes long and the dealer goes short. The contract covers a given notional principal of $10 million.

 The end user will benefit if rates increase.  In contrast, the dealer will benefit if rates decrease.

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Interest rate forward contracts

 Example: FRA

 At expiration in 90 days, the rate on 180-day Libor

is 6%.

 The 6% interest will be paid 180 days later.  The present value of a Eurodollar time deposit at that

point in time would be:

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Interest rate forward contracts

 Example: FRA

 At expiration in 90 days, the rate on 180-day Libor

is 6%.

 The end user, the party going long the FRA, receives

the following payment from the dealer, which is the party going short:

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Interest rate forward contracts

 Example: FRA

 At expiration in 90 days, the rate on 180-day Libor

is 6%.

 The numerator indicates that the contract is paying the

difference between the actual rate that exists in the market on the contract expiration date and the agreed- upon rate, adjusted for the fact that the rate applies to a 180-day instrument.

 The divisor appears because it is necessary to adjust

the FRA payoff to reflect the fact that the rate implies a payment that would occur 180 days later on a standard Eurodollar deposit.

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Interest rate forward contracts

 FRA payoff formula (from the perspective of

the party going long)

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Interest rate forward contracts

 FRA descriptive notation and interpretation

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Notati ation

• n

Contract ract expire res s in Underl rlyin ying rate 1×3 1 month 60-day Libor 1×4 1 month 90-day Libor 1×7 1 month 180-day Libor 3×6 3 moths 90-day Libor 3×9 3 months 180-day Libor 6×12 6 months 180-day Libor 12×18 12 months 180-day Libor

Currency forward contracts

 Definition

 A currency forward contract enables one party to

lock in a pre-agreed upon exchange rate at which it will sell one currency and buy another currency at a later date.

 Use

 Currency forwards are widely used by banks and

corporations to manage foreign exchange rate risk.

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Currency forward contracts

 Example

 Suppose Microsoft has a European subsidiary

that expects to send it €12 million in three months. When Microsoft receives the euros, it will then convert them to dollars.

 Microsoft is essentially long euros since it will have to

sell euros, i.e., Microsoft has euro-nominated assets that exceed in value its euro-nominated liabilities.

 Microsoft is essentially short dollars because it will have

to buy dollars.

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Currency forward contracts

 Example

 Microsoft’s actions

 Obtaining a quote on a currency forward for €12 million

in three months.

 JP Morgan Chase quotes a rate of $0.925.  Under this contract, Microsoft would know it could convert

its €12 million to 12,000,000 × $0.925 = $11,100,000 in three months.

 To hedge its position, Microsoft has to go short the

forward contract, i.e., goes short the euro and long the dollar.

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Currency forward contracts

 Example

 At expiration, the spot rate for euros is $0.920.

 Delivery: Microsoft is pleased that it is locked in a rate of

$0.925. It simply delivers the euros and receives $11,100,000 at an exchange rate of $0.925.

 Cash settlement: The dealer pays Microsoft 12,000,000

× ($0.925 - $0.920) = $60,000. Microsoft has to convert the euros to dollars at the current spot exchange rate of $0.920, receiving 12,000,000 × $0.920 = $11,040,000.

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Other types of forward contracts

 Examples

 Commodity forwards

 The underlying asset is oil, a

precious metal, various sources of energy (electricity, gas, etc.), or some other commodity.

 Forward contracts on whether

 The underlying is a measure of the

temperature or the amount of disaster damage from hurricanes, earthquakes, or tornados.

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Pricing and valuation

 In most markets, price of an

asset is believed to always equal its value or the price would quickly converge to the value.

 Price of an asset is what it will

sell for.

 Value of an asset is what it is

worth.

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Pricing and valuation

 With respect to certain

derivatives, however, value and price take on slightly different meanings.

 Value is what you can sell

something for or what you must pay to acquire something.

 Accordingly, valuation is the

process of determining the value of an asset or service

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Pricing and valuation

 Pricing a forward contract

 A forward price is the fixed

price or rate at which the transaction scheduled to

• ccur at expiration will take

place.

 Pricing means to determine the

forward price or forward rate that both parties agree on the contract initiation date.

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Pricing and valuation

 Valuing a forward contract

 Determining the amount of money that one would

need to pay or would expect to receive to engage in the transaction.

 Alternatively, if one already held a position,

valuation would mean to determine the amount of money one would either have to pay or expect to receive in order to get out of the position.

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Pricing and valuation

 Notation

 S0 = the price of the underlying asset in the spot market

at time 0 (St at time t, and ST at time T).

 F(0,T) = the price of a forward contract initiated at time 0

and expiring at time T.

 V0(0,T) is the value at time 0 of the forward contract

initiated at time 0 and expiring at time T (Vt(0,T) is the value at time t, and VT(0,T) is the value at expiration)

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today

t

expiration T

Pricing and valuation

 Value at expiration

 VT(0,T) = ST – F(0,T)

 If this equation does not hold, an arbitrage profit can be

easily made.

 Example: Suppose F(0,T) = $20 and ST = $23.

 VT(0,T) must be $3.  If VT(0,T) > $3, the long would be able to sell the

contract to someone for more than $3 – someone would be paying the long more than $3 to obtain the obligation

• f buying a $23 asset for $20. Obviously, no one would

do that.

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Pricing and valuation

 Value at expiration

 Example: Suppose F(0,T)

= $20 and ST = $23.

 If VT(0,T) < $3, the long

would have to be willing to sell for less than $3 the

• bligation of buying a $23

asset for $20. Obviously, the long would not do that.

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Pricing and valuation

 Value today

 Example 1: Consider a contract that expires in

• ne year. Suppose S0 is $100 and that F(0,T) is

$108. Assume that the interest rate is 5%.

 We borrow $100 to buy the asset and sell the forward

contract for $108. We hold the position until expiration.

 We lose $5 in interest on the $100 tied up in the asset.  At expiration, we receive $108 for the asset regardless

• f ST. This would result in an arbitrage profit of $3.

 V0(0,1) = $100 - $108/1.05 = -$2.8571 (the short pays

$2.8571 to the long). Note that $2.8571×1.05 = $3.

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Pricing and valuation

 Value today

 Example 2: Consider the same situation but now

F(0,T) is set to $103.

 If the asset were a financial asset, we could short sell

the asset for $100.

 We invest that $100 at the 5% rate and simultaneously

buy a forward contract.

 At expiration, we would take delivery of the asset paying

$103 and then deliver it to the party from whom we borrowed it (for short sale).

 Arbitrage profit = $105 - $103 = $2.

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Pricing and valuation

 Value today

 Example 2: Consider the same situation but now

F(0,T) is set to $103.

 If short selling is not permitted, too difficult, or too costly,

a person who already owns the asset could sell it at the spot market, invest $100 at 5%, and buy a forward contract at the same time.

 At expiration, that person would pay $103 to buy the

asset back. This means an arbitrage profit of $2.

 V0(0,1) = $100 - $103/1.05 = $1.9048 (the long pays

$1.9048 to the short).

 $1.9048 × 1.05 = $2

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Pricing and valuation

 Value today formula

 V0(0,T) = S0 – F(0,T)/(1+r)T

 r represents the interest rate.

 V0(0,T) must equal 0, otherwise one party is

required to make a payment to the other upfront to eliminate an arbitrage opportunity.

 Pricing formula

 Since V0(0,T) = S0 – F(0,T)/(1+r)T and no money

changes hands at the start with a forward contract, V0(0,T) = 0 giving F(0,T) = S0(1+r)T.

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Pricing and valuation

 Value at a point during the life of the contract

(from the perspective of the party going long)

 The long will have to pay F(0,T) at T  The long will receive the underlying asset worth

ST at T

 Vt(0,T) = St – F(0,T)/(1+r)(T-t)

 St represents the present value of the asset’s future

value.

 F(0,T)/(1+r)(T-t) represents the present value of the

payment of F(0,T) to be made at T.

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Pricing and valuation

 Advanced problem:

 An investor holds title to an asset worth $125.72.

To raise money for an unrelated purpose, the investor plans to sell the asset in nine months. The investor is concerned about uncertainty in the price of the asset at that time. The investor learns about the advantages of using forward contracts to manage this risk and enters into such a contract to sell the asset in nine months. The risk- free interest rate is 5.625%.

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Pricing and valuation

 Advanced problem:

 A. Determine the appropriate price the investor

could receive in nine months by means of the forward contract.

 Solution: $130.99

 B. Suppose the counterparty to the forward

contract is willing to engage in such a contract at a forward price of $140. Explain what type of transaction the investor could execute to take advantage of the situation.

 Solution: Overpriced contract should be sold.

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Pricing and valuation

 Advanced problem:

 C. Calculate the rate of return (annualized), and

explain why the transaction in B is attractive.

 Solution: Annualized ROR = 15.43%

 D. Suppose the forward contract is entered into at

the price in A. Two months later, the price of the asset is $118.875. The investor would like to evaluate her position with respect to any gain or loss accrued on the forward contract.

 Solution: V2/12(0,9/12) = -$8.0

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Pricing and valuation

 Advanced problem:

 E. Determine the value of the forward contract at

expiration assuming the contract is entered into at the price in A and ST is $123.50.

 Solution: V9/12(0,9/12) = -$7.49

 F. Explain how the investor did on the overall

position of both the asset and the forward contract in terms of the rate of return at expiration.

 Solution: The ROR in 9 months is 4.19%. This means

that the annualized ROR is 5.625%

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Pricing equity forwards

 The effects of dividends must be incorporated

into the pricing process.

 Given a series of these dividends of D1, D2, … Dn,

whose values are known, that occur at times, t1, t2, … tn, the present value will be computed as:

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Pricing equity forwards

 The effects of dividends must be incorporated

into the pricing process.

 The future value will be computed as:

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Pricing equity forwards

 Recall that the forward price is the spot price

compounded at the risk-free interest rate.

 In the presence of dividends, the general pricing

formula is adjusted to:

 Holders of long positions in forward contracts do not

benefit from dividends in comparison to holders of long positions in the underlying stock.

 Alternatively,

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Pricing equity forwards

 Example

 The risk-free rate is 4%. The forward contract

expires in 300 days and is on a stock currently priced at $5, which pays quarterly dividends according to the following schedule:

11/2/2015 Nattawoot Koowattanatianchai 72

Days to Ex-di dividend dend date Dividend dend ($) 10 0.45 102 0.45 193 0.45 283 0.45

Pricing equity forwards

 Example

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Pricing equity forwards

 Continuous compounding

 Assume that the stock, portfolio, or index pays

dividends continuously at a rate of δc and the continuously compounded equivalent of the discrete risk-free rate r is rc = ln(1+r).

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Pricing equity forwards

 Continuous compounding

 Example: Consider a forward contract on France’s

CAC 40 Index. The index is at 5475, δc is 1.5%, and rc is 4.625%. The contract life is 2 years.

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Valuing equity forwards

 Recall that the value of a forward contract is

the asset price minus the forward price discounted back from the expiration date.

 For discrete compounding:  For continuous compounding:

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Valuing equity forwards

 Recall that the value of a forward contract is

the asset price minus the forward price discounted back from the expiration date.

 V0(0,T) is set to zero because no cash exchanges

hands at the contract initial date.

 At expiration, no dividends remain, so the

valuation formula reduces to VT(0,T) = ST – F(0,T)

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Valuing equity forwards

 Advanced example:

 An asset manager anticipates the receipt of funds

in 200 days, which he will use to purchase a particular stock. The stock he has in mind is currently selling for $62.50 and will pay a $0.75 dividend in 50 days and another $0.75 dividend in 140 days. Assume that r is 4.2%. The manager decides to commit to a future purchase of the stock by going long a forward contract on the stock.

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Valuing equity forwards

 Advanced example:

 A. At what price would the manager commit to

purchase the stock in 200 days through a forward contract.

 Solution: PV(D,0,T) = $1.48 and F(0,T) = $62.41

 B. Suppose the manager enters into the contract

at price in A. Now 75 days later, the stock price is $55.75. Determine Vt(0,T) at this point.

 Solution: The second dividend only remains and will be

paid in 65 days. PV(D,t,T) = $0.74 and Vt(0,T) = -$6.53.

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Pricing and valuing bond forwards

 Notation

 T = the expiration date of the forward contract  Y = the remaining maturity of the bond on the

forward contract expiration

 Bc = a coupon bond  Bt

c(T+Y) = the bond price at time t

 CI = the coupon interest over a specified period of

time

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Pricing and valuing bond forwards

 Pricing and valuation formulae

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Pricing and valuing bond forwards

 Example

 Consider a bond with semiannual coupons. The

bond has a current maturity of 583 days and pays the next four semiannual coupons in 37 days, 219 days, 401 days, and 583 days, at which time the principal is repaid. Suppose that the bond price, which includes accrued interest, is $984.45 for a $1,000 par, 4% coupon bond. Assume that:

 r = 5.75%  T = 310 days  T+Y = 583 days, implying Y = 273 days

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Pricing and valuing bond forwards

 Example

 Only the first two coupons occur during the life of

the forward contract.

 PV(CI,0,T) = $20/1.057537/365 + $20/1.0575219/365 = $39.23  F(0,T) = ($984.45 - $39.23)(1.0575)310/365 = $991.18

 15 days later, the new bond price is $973.14. Let

the risk free rate now be 6.75%.

 PV(CI,t,T) = $20/1.067522/365 + $20/1.0675204/365 = $39.20  Vt(0,T) = ($973.14 - $39.20) - $991.18/(1.0675)295/365 =

• $6.28

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Pricing and valuing FRAs

 Notation

 h = the day on which the FRA expires  g = an arbitrary day prior to the expiration  h + m = days from today until the maturity date of

the Eurodollar instrument on which the FRA rate is based.

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g

expiration h

maturity m h  today

Pricing and valuing FRAs

 Notation

 Li(j) = the rate on a j-day Libor deposit on an

arbitrary day i

 The bank that borrows $1 on day i for j days will pay back

the following amount in j days.

 Lh(m) is the rate for m-day Libor on day h  FRA(0,h,m) = the rate on an FRA established on

day 0, expiring on day h, and based on m-day Libor.

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Pricing and valuing FRAs

 For a $1 notional principal, the FRA payoff at

expiration is:

 This formula is the same as the previous one:

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Pricing and valuing FRAs

 FRA rate is given by the following formula:

 The numerator is the future value of a Eurodollar

deposit of h+m days.

 The denominator is the future value of a shorter-

term Eurodollar deposit of h days.

 Multiplying by (360/m) annualizes the rate.

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Pricing and valuing FRAs

 The value of an FRA during its life

 Vg(0,h,m) = the value of an FRA on day g, which

was established on day 0, expires on day h, and is based on m-day Libor

 The first term is the present value of $1 received at day h.  The second term is the present value of 1 plus the FRA

rate to be received on day h+m.

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Pricing and valuing FRAs

 Example

 Consider a 3×9 FRA. This instrument expires in 90

days and is based on 180-day Libor. Thus, the Eurodollar deposit on which the underlying rate is based begins in 90 days and matures in 270 days.

 h = 90  m = 180  h+m = 270

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Pricing and valuing FRAs

 Example

 Let the current rate be:

 L0(h) = 90-day Libor = 5.6%  L0(h+m) = 270-day Libor = 6%

 Determine the rate that will be quoted on the 3×9

FRA.

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Pricing and valuing FRAs

 Example

 Assume that we go long the FRA, and it is 25 days

later (g = 25). Interest rates have moved significantly upward to the following:

 Lg(h-g) = L25(65) = 65-day Libor = 5.9%  Lg(h+m-g) = L25(245) = 245-day Libor = 5.9%

 Determine the market value oft the FRA for a $1

notional principal.

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Pricing currency forwards

 Exchange rates

 We shall always quote exchange rates in terms of

units of the domestic currency per unit of the foreign currency.

 Example

 From the perspective of a US investor, a stock that sells

for $50 is quoted in units of the domestic currency per unit (share) of stock.

 Likewise, if the euro exchange rate is quoted as $0.90,

then the euro sells for $0.90 per unit, which is one euro.

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Pricing currency forwards

 Notation

 S0 = current exchange rate

(units of the domestic currency per one unit of the foreign currency)

 rf = the foreign interest rate  r = the domestic interest rate  F(0,T) = exchange rate in a T-

year forward contract

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Pricing currency forwards

 Consider the following transactions executed

today (time 0), assuming a contract expiration date of T:

 Take S0/(1+ rf)T units of the domestic currency and

convert it to 1/(1+ rf)T units of the foreign currency.

 Sell a forward contract to deliver one unit of the

foreign currency at the rate F(0,T).

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Pricing currency forwards

 At expiration

 The 1/(1+ rf)T units of foreign currency will accrue

interest at the rate rf and grow to one unit of the foreign currency at T.

 Deliver one unit of the foreign currency to the

holder of the long forward contract, who pays F(0,T) units of the domestic currency.

 This amount was known at the start of the transaction.

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Pricing currency forwards

 Pricing formula

 Since the payment of F(0,T) is riskless. The

present value of F(0,T) must equal the initial outlay

• f S0/(1+ rf)T. Otherwise, an arbitrage opportunity

exists.

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Pricing currency forwards

 Arbitrage

 If F(0,T) is too high

 Sell the forward contract at the market rate.  Borrow S0/(1+ rf)T from a domestic bank at the rate r to

buy 1/(1+ rf)T units of the foreign currency.

 Hold the position, earning interest on the foreign

currency.

 At maturity of the forward contract, deliver one unit of the

foreign currency and be paid the forward rate.

 Return S0/(1+ rf)T (1+ r)T to the local bank.

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Pricing currency forwards

 Arbitrage

 If F(0,T) is too low

 Buy the forward contract at the market rate.  Sell 1/(1+ rf)T units of the foreign currency, receive S0/(1+

rf)T units of domestic currency and deposit this amount in a domestic bank at the rate r .

 Hold the position, earning interest on the domestic

currency.

 At maturity, buy back 1/(1+ rf)T units of the foreign

currency and pay the forward rate.

 Withdraw S0/(1+ rf)T (1+ r)T from the local bank.

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Pricing currency forwards

 Continuous compounding

 Notation

 rfc = ln(1+rf) = the continuously compounded foreign

interest rate

 rc = ln(1+r) = the continuously compounded domestic

interest rate

 Pricing formula

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Pricing currency forwards

 Example

 Suppose a domestic currency is the US dollar and

the foreign currency is the Swiss franc. Let the spot exchange rate be $0.5987, the US interest rate be 5.5%, and the Swiss interest rate be 4.75%. We assume that these interest rates are fixed and will not change over the life of the forward contract. We also assume that these rates are based on annual compounding and are not quoted as Libor-type

• rates. Thus, we compound using formulas like

(1+r)T.

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Pricing currency forwards

 Example

 Assuming the forward contract has a maturity of

180 days, find the forward price.

 This means that entering into a forward contract requires

the long party to purchase one Swiss franc in 180 days at a price of $0.6008.

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Valuing currency forwards

 Recall that the value of an

equity forward is the stock price minus the present value of the dividends over the remaining life of the contract minus the present value of the forward price

• ver the remaining life of the

contract.

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Valuing currency forwards

 Discrete compounding  Continuous compounding

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Valuing currency forwards

 Example

 Suppose we go long the previous forward contract,

with F(0,180/365) = $0.6008. It is now 40 days later, or 140 days until expiration. The spot rate is now $0.65. As assumed above, the interest rates are fixed. Determine the value of the contract at this point.

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Credit risk in forwards

 Example

 Consider a currency forward

contract that expires in 180 days in which the long will pay a forward rate of $0.6008 for each Swiss franc to be received at

• expiration. Assume that the

contract covers 10 million Swiss francs.

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Credit risk in forwards

 Example

 At expiration, suppose the spot rate for Swiss

francs is $0.62.

 The long is due to receive 10 million Swiss francs and

pay $0.6008 per Swiss franc, or $6,008,000 in total.

 Suppose that because of bankruptcy or insolvency, the

short cannot come up with the $6,200,000 that it would take to purchase the Swiss francs on the open market at the prevailing spot rate.

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Credit risk in forwards

 Example

 At expiration, suppose the spot rate for Swiss

francs is $0.62.

 The long needs to buy Swiss francs in the open market.

Doing so would incur an additional cost of $6,200,000 - $6,008,000 = $192,000, which can be viewed as the credit risk at the point of expiration when the spot rate is $0.62.

 This risk is an immediate risk faced by the long at expiration.  The short faces no credit risk.

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Credit risk in forwards

 Example

 Prior to expiration, the long faces a potential risk

that the short will default. In the example we used in which the long is now 40 days into the life of the contract, the market value to the long is $0.0499 per Swiss franc.

 The long exposure would be 10,000,000($0.0499) =

$499,000.

 If the probability that the short will default is 10%, the

expected credit loss from the transaction is $49,000.

 The short faces no credit risk. It will do if Vt(0,T) < 0.

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Credit risk in forwards

 Example

 Managing credit risk: Marking to market

 Suppose the market value to the long is $0.0499 per

Swiss franc after 40 days into the contract.

 Suppose that two parties had agreed when they entered

into the transaction that in 40 days, the party owing the greater amount to the other would pay the amount owed and the contract would be repriced at the new forward rate.

 The short would pay the long $499,000.  Using the pricing formula, the currency forward contract

would be repriced to $0.6518.

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Forward markets

 Important characteristics

 Forward contracts are private transactions,

permitting the ultimate customization.

 Forward markets have only a light degree of

• regulation. Default rates are high as a result.

 Forward markets serve a specialized clientele,

specifically large corporations and institutions with specific target dates, underlying assets, and risks.

 Transactions in forward contracts typically are

conducted over the phone.

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