7. derivatives part1 pdf

Derivatives – Part 1(LOs 27.x – 31.x)

Forwards (LO 27.x)1

4 Commodity Futures (LO 30.x)

Interest Rate Futures (LO 29.x)

2 Hedging Strategies (LO 28.x)

3

Swaps (LO 31.x)5

Forwards/Futures

Time (T)

0 0 0( ) rTTF E S F S e

S0

F0

ST-1

FT-1 F0ST

ST=F

T

Forwards/Futures

Time (T)

(Commodity with High

“Convenience Yield” or

High-dividend Financial Asset)

F0 = E(S

T)

S0

F0

ST-1

FT-1 F0ST

ST=F

T

Cost-of-carry modelLO 27.1: State & explain cost-of-carry model for forward prices with & without interim cash flows

Time (T)

Income/

Dividend

(q)

Storage

Cost (U, u)

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

Convenience

(y)

Cost-of-carry: Question

A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk-free rate is 5% for all maturities.

What the price of a (long) forward contract (F0) to purchase the stock in one year?




0 0 0

( 0.05)(6/12) (.05)(1)

( )

($50 [($1) ])

$51.538

rTF S I e F

e e

DerivativesLO 27.2: Compute the forward price given both the price of the underlying and the appropriate carrying costs of the underlying.

( )0 0

r u q y TF S e

( )0 0

r q TF S e

Commodity

Financial asset (e.g., stock index)

Cost-of-Carry ModelCost of carry = interest to finance asset (r)+ storage cost (u) - income earned (q)

( )0 0( ) r y TF S U I e

( )0 0

r u q y TF S e

Present values

Commodity

constant rates as %

u = storage costs

q = income (dividend)

y = convenience yield

U = Present value, storage costs

I = Present value, income


0 0( ) rTF S I e( )0 0

r q TF S e

Present values


constant rates as %

q = income (dividend) I = Present value, income

Cost-of-Carry: Question

The spot price of corn today is 230 cents per bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum.

What is the forward price in four (4) months?




( ) (6%/12 1.5%)(4)0 0

(.02)(4)

(230)

230 249.16

r u TF S e e

e

Derivatives

LO 27.3: Calculate the value of a forward contract.

0( ) rTf F K e

Value of a forward contractA long forward contract on a non dividend-paying stock has three months left to maturity.

The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%.

What is the value of the forward contract?




(5%)(0.25)0 0 10 $10.126rTF S e e

(5%)(0.25)0( ) (10.126 8) $2.153rTf F K e e

DerivativesLO 27.4: Describe the differences between forward and futures contracts.

Forward vs. Futures Contracts

Forward Futures

Trade over-the-counter Trade on an exchange

Not standardized Standardized contracts

One specified delivery date Range of delivery dates

Settled at contract’s end Settled daily

Delivery or final cash

settlement usually occurs

Contract usually closed

out prior to maturity

Derivatives

A long-futures position agrees to buy in the future

A short-futures position agrees to sell in the future.

Price mechanism maintains a balance between buyers and

sellers.(market equilibrium)

Most futures contracts do not lead to delivery, because

most trades ―close out‖ their positions before delivery.

Closing out a position means entering into the opposite

type of trade from the original.

LO 27.5: Distinguish between a long futures position and a short futures position.

Derivatives

An (underlying) asset

A Treasury bond futures contract is on underlying

U.S. Treasury with maturity of at least 15 years and not

callable within 15 years (15 years ≤ T bond).

A Treasury note futures contract is on the underlying

U.S. Treasury with maturity of at least 6.5 years but not

greater than 10 years (6.5 ≤ T note ≤ 10 years).

When the asset is a commodity (e.g., cotton, orange

juice), the exchange specifies a grade (quality).

LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled.

Derivatives

Contract size varies by type of futures contract

Treasury bond futures: contract size is a face value of $100,000

S&P 500 futures contract is index $250 (multiplier of 250X)

NASDAQ futures contract is index $100 (multiplier of 100X)

Recently, ―mini contracts‖ have been introduced:

S&P 500 ―mini‖ = $50 x S&P Index

NASDAQ ―mini‖ = $20 x NASDQ

(each contract is one-fifth the price, to attract smaller investors)


Derivatives

Delivery Arrangements

The exchange specifies delivery location.

Delivery Months

The exchange must specify the delivery month; this can

be the entire month or a sub-period of the month.


Derivatives

Margin account: Broker requires deposit.

Initial margin: Must be deposited when contract is

initiated.

Mark-to-market: At the end of each trading day, margin

account is adjusted to reflect gains or losses.

LO 27.7: Describe the marking-to-market procedure, the

initial margin, and the maintenance margin.

Derivatives

Maintenance margin: Investor can withdraw funds in the margin account in excess of the initial margin. A maintenance margin guarantees that the balance in the margin account never gets negative (the maintenance margin is lower than the initial margin).

Margin call: When the balance in the margin account falls below the maintenance margin, broker executes a margin call. The next day, the investor needs to ―top up‖ the margin account back to the initial margin level.

Variation margin: Extra funds deposited by the investor after receiving a margin call.



Derivatives

There is only a variation margin if and when there is a

margin call.

Variation margin = initial margin – margin

account balance

The maintenance margin is a trigger level—once

triggered, the investor must ―top up‖ to the initial

margin, which is greater than the maintenance level.

LO 27.8: Compute the variation margin.

Derivatives

The exchange clearinghouse is a division of the exchange

(e.g., the CME Clearing House is a division of the

Chicago Mercantile Exchange) or an independent

company. The clearinghouse serves as a

guarantor, ensuring that the obligations of all trades are

met.

LO 27.9: Explain the role of the clearinghouse.

Derivatives

Market order: Execute the trade immediately at the

best price available.

Limit order: This order specifies a price (e.g., buy at $30

or less)—but with no guarantee of execution.

Stop order: (aka., stop-loss order) An order to execute

a buy/sell when a specified price is reached.


Derivatives

Stop-limit: Requires two specified prices, a stop and a

limit price. Once the stop-limit price is reached, it

becomes a limit order at the limit price.

Market-if-touched: Becomes a market order once

specified price is achieved.

Discretionary (aka., market-not-held order): A

market order, but the broker is given the discretion to

delay the order in an attempt to get a better price.


Derivatives

A short forward (or futures) hedge is an agreement to sell in the future and is appropriate when the hedger already owns the asset.

Classic example is farmer who wants to lock in a sales price: protects against a price decline.

A long forward (or futures) hedge is an agreement to buy in the future and is appropriate when the hedger does not currently own the asset but expects to purchase in the future.

Example is an airline which depends on jet fuel and enters into a forward or futures contract (a long hedge) in order to protect itself from exposure to high oil prices.

LO 28.1: Differentiate between a short hedge and a long hedge, and identify situations where each is appropriate.

Derivatives

Basis = Spot Price Hedged Asset –

Futures Price Futures Contract = S0 – F0

Basis =

Futures Price Futures Contract –

Spot Price Hedged Asset = F0 – S0

LO 28.2: Define and calculate the basis.

Hull says first is correct but second is

common for financial assets (either is okay)

Basis risk

Time (T)

$2.50

$2.20

$2.00

$1.90

T0

$0.30$0.10

T1

No hedge

Spot = -$0.50

SPOT

Forward

Short 1.67 F

Spot = -$0.50

Future = $0.30 (1.67)

Net = 0

Basis

Weakening of the basis = Futures price

increases more than spot

Basis risk

$2.50

$2.20

$2.00

$1.90

T0

$0.30$0.30

T1

No hedge

Spot = -$0.50

SPOT

Forward

Short 1.67 F

Spot = -$0.50

Future = $0.50 (1.67)

Net = +33.5

Basis

$1.70

Time (T)

Basis unchanged.

But unexpected strengthening= Hedger improved!

Basis risk

$2.50

$2.20

$2.60

$1.90

T0

$0.30

T1

SPOT

Forward

Short 1.67 F

Spot = +$0.10

Future = -$0.40 (1.67)

Net = -56.8

Basis

$2.60

Time (T)

Basis declines

Unexpected weakening= Hedger worse!

$0.0

Derivatives

Spot price increases by more than the futures price

basis increases. This is a ―strengthening of the basis‖

When unexpected, strengthening is favorable for a short

hedge and unfavorable for a long hedge

Futures price increases by more than the spot price

basis declines. This is a ―weakening of the basis‖

When unexpected, weakening is favorable for a long hedge

and unfavorable for a short hedge

LO 28.3: Define the types of basis risk and explain how they arise in futures hedging.

Derivatives

But basis risk arises because often the characteristics

of the futures contract differ from the underlying

position.

Contract ≠ Commodity

Contract is standardized (e.g., WTI oil futures)

Commodities are not exactly commodities (e.g., hedger has a

position in different grade of oil)


Basis risk higher with cross-hedging

Derivatives



position.

Contract ≠ Commodity.





Trade-offLiquidity(exchange)

Basis risk

Derivatives

The optimal hedge ratio (a.k.a., minimum variance hedge

ratio) is the ratio of futures position relative to the spot

position that minimizes the variance of the position.

Where is the correlation and is the standard

deviation, the optimal hedge ratio is given by:

LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio.

* S

F

h

Derivatives

For example, if the volatility of the spot price is 20%, the

volatility of the futures price is 10%, and their correlation

is 0.4, then:


* S

F

h20%

* (0.4) 0.810%

h

Derivatives



is 0.4, then:


* S

F

h20%

* (0.4) 0.810%

h

** A

F

h NN

QNumber of

contracts

Derivatives

Given a portfolio beta ( ), the current value of the

portfolio (P), and the value of stocks underlying one

futures contract (A), the number of stock index futures

contracts (i.e., which minimizes the portfolio variance) is

given by:

LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.

PN

A

Derivatives

By extension, when the goal is to shift portfolio beta from

( ) to a target beta ( *), the number of contracts

required is given by:


( * )P

NA

Futures: QuestionLO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.

Assume:

Value of S&P 500 Index is 1240

Value of portfolio is $10 million

Portfolio beta ( ) is 1.5

How do we change the portfolio beta to 1.2?

PN ( * )

A

Hint: Contract = ($250 Index)

and # of futures is given by:


Assume:

• Value of S&P 500 Index is 1240

• Value of portfolio is $1 million

• Portfolio beta ( ) is 1.5

We short about 10 contracts. (-) indicates short, (+) long…

( * )

$10,000,000(1.2 1.5) 9.7

(1240)(250)

PN

A

Derivatives

When the delivery date of the futures contract occurs

prior to the expiration date of the hedge, the hedger can

roll forward the hedge: close out a futures contract and

take the same position on a new futures contract with a

later delivery date.

Exposed to:

Basis risk (original hedge)

Basis risk (each new hedge) = ―rollover basis risk‖

LO 28.6: Identify situations when a rolling hedge is appropriate, and discuss the risks of such a strategy.

DerivativesLO 29.1: Identify and apply the three most common day count conventions.

Actual/actual U.S. Treasuries

30/360 U.S. corporate and

municipal bonds

Actual/360 U.S. Treasury bills and

other money market

instruments

Derivatives

The Treasury bond futures contract allows the party with

the short position to deliver any bond with a maturity of

more than 15 years and that is not callable within 15

years. When the chosen bond is delivered, the conversion

factor defines the price received by the party with the

short position:

LO 29.2: Explain the U.S. Treasury bond (T-bond) futures contract conversion factor.

Cash Received = Quoted futures price Conversion

factor + Accrued interest

= (QFP CF) + AI

Derivatives

The convexity adjustment assumes continuous

compounding. Given that ( ) is the standard deviation of

the change in the short-term interest rate in one year, t1is the time to maturity of the futures contract and t2 is

the time to maturity of the rate underlying the futures

contract:

LO 29.3: Calculate the Eurodollar futures contract convexity adjustment.

21 2

1Forward = Futures

2t t

Derivatives

The number of contracts required to hedge against an

uncertain change in the yield, given by y, is given by:

FC = contract price for the interest rate futures contract.

DF = duration of asset underlying futures contract at maturity.

P = forward value of the portfolio being hedged at the maturity

of the hedge (typically assumed to be today’s portfolio value).

DP = duration of portfolio at maturity of the hedge

LO 29.4: Formulate a duration-based hedging strategy using interest rate futures.

* P

C F

PDN

F D

DerivativesLO 29.4: Formulate a duration-based hedging strategy using interest rate futures.

Assume a portfolio value of $10 million.

The fund manager will hedge with T-bond futures (each

contract is for delivery of $100,000) with a current futures

price of 98.

She thinks the duration of the portfolio at hedge maturity will

be 6.0 and the duration of futures contract with be 5.0.

How many futures contracts should be shorted?





price of 98.




($10 million)(6)* 122

(98,000)(5)P

C F

PDN

F D

Derivatives

Portfolio immunization or duration matching is when a

bank or fund matches the average duration of assets with

the average duration of liabilities.

Duration matching protects or ―immunizes‖ against

small, parallel shifts in the yield (interest rate) curve. The

limitation is that it does not protect against nonparallel

shifts. The two most common nonparallel shifts are:

A twist in the slope of the yield curve, or

A change in curvature

LO 29.5: Identify the limitations of using a duration-based hedging strategy.

Derivatives

LO 30.1: Explain the derivation of the basic equilibrium

formula for pricing commodity forwards and futures.

Time (T)

Discount

rate ( )

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

( )0, 0( ) r TT TF E S e

Derivatives



( )0, 0( ) r TT TF E S e

0

0,T

( ) Spot price of S at time T, as expected at time 0F Forward pricer Risk-free rate

Discount rate for commodity S

TE S

Derivatives

Lease rate = commodity discount rate – growth rate

Lease rate dividend yield

LO 30.2: Define lease rates, and discuss the importance of lease rates for determining no-arbitrage values for commodity futures and forwards.

( )0, 0

r TTF S e

( )0 0

r q TF S e

Financial asset

Derivatives

Contango refers to an upward-sloping forward curve

which must be the case if the lease rate is less than

the risk-free rate. Backwardation refers to a downward-

sloping forward curve which must be the case if the

lease rate is greater than the risk-free rate.

LO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.

DerivativesLO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.

Time (T)

Research says normal backwardation is “normal:” speculators

want compensation (risk premium) for buying the futures contract

Spot

(S0)E(ST)

Forward

(F0)

Forward

(F0)

DerivativesLO 30.4: Explain how storage costs impact commodity forward prices, and calculate the forward price of a commodity with storage costs.

Time (T)

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

Storage Cost ( )

negative dividend

Convenience (y)

dividend

Lease rate ( )

dividend


Time (T)

Lease

rate ( )

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

( )0 0

r c TF S e

Storage

Cost ( )

Convenience

(y)

DerivativesLO 30.5: Explain how a convenience yield impacts commodity forward prices, and determine the no-arbitrage bounds for the forward price of a commodity when the commodity has a convenience yield.

( ) ( )0 0 0

r c T r TS e F S e

Risk-free

Rate

Storage

Cost

Risk-free

Rate

Storage

Cost

Convenience

Yield

Commodity Futures

1460148015001520154015601580160016201640

S&P 500 Index

Rational forward curve rises by cost

of capital (risk free + premium) less

dividends

Commodity FuturesLO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures.

500

550

600

650

700

750

800

850

900

Jul-07 Nov-08 Mar-10 Aug-11 Dec-12

Gold futures

Durable, (relatively) cheap to store.

Forward curve is “uninteresting”


200

250

300

350

400

450

Sep

-07

Dec-0

7

Mar-

08

Jun

-08

Sep

-08

Dec-0

8

Mar-

09

Jun

-09

Sep

-09

Dec-0

9

Mar-

10

Jun

-10

Sep

-10

Dec-1

0

Corn


5

6

7

8

9

10

Au

g-0

7

Nov-0

7

Feb

-08

May-0

8

Au

g-0

8

Nov-0

8

Feb

-09

May-0

9

Au

g-0

9

Nov-0

9

Feb

-10

May-1

0

Au

g-1

0

Nov-1

0

Natural Gas

Costly to transport. Costly to store (storage costs). Highly seasonal


68

69

70

71

72

73

74

75

76

Sep

-07

Nov-0

7

Jan

-08

Mar-

08

May-0

8

Jul-

08

Sep

-08

Nov-0

8

Jan

-09

Mar-

09

May-0

9

Jul-

09

Sep

-09

Nov-0

9

Jan

-10

Mar-

10

May-1

0

Jul-

10

Sep

-10

Nov-1

0

Crude oil

Compared to natural gas, easier to store and transport. Global market. Long-run forward price less (<) volatile than short-run forward.

Commodity Futures

If we can take a long position on one commodity that is an input (e.g., oil) into another commodity that is an output (e.g., gas or heating oil), then we can take a short position in the output commodity and the difference is the commodity spread.

Assume oil is $2 per gallon, gasoline is $2.10 per gallon and heating oil is $2.50 per gallon.

If we take a long position in 2 gallons of gasoline and one gallon of heating oil, plus a short position in three gallons of oil, the commodity spread =

(2 long gasoline $2.10) + (1 long heating oil $2.50) – (3 oil $2) = +$0.70

LO 30.7: Describe and calculate a commodity spread.

Commodity Futures

The basis is the difference between the price of the futures contract and the spot price of the underlying asset.

Basis risk is the risk (to the hedger) created by the uncertainty in the basis.

The futures contract often does not track exactly with the underlying commodity; i.e., the correlation is imperfect. Factors that can give rise to basis risk include:

Mismatch between grade of underlying and contract

Storage costs

Transportation costs

LO 30.8: Define basis risk, and explain how basis risk can occur when hedging commodity price exposure.

LO 30.9: Differentiate between a

strip hedge and a stack hedge.

10 10 10 10 10 10 10 10 10 10 10 10

Jan Feb Mar

<120 <110 <100Jan Feb Mar

Commodity Futures

Oil producer to deliver

10K barrels per month

Strip hedge: contract for

each obligation

Stack hedge: Single maturity,

―stack and roll‖

Commodity Futures

A strip hedge is when we hedge a stream of obligations by offsetting each individual obligation with a futures contract that matches the maturity and quantity of the obligation. For example, if a producer must deliver X number of commodities per month, then the strip hedge entails entering into a futures contract for X commodities, to be delivered in one month; plus a futures contract for X commodities to be delivered in two months. The strip hedger matches a series of futures to the obligations.

A stack hedge is front-loaded: the hedger enters into a large future with a single maturity. In this case, our hedger would take a long position in a near-term futures contract for 12X commodities (i.e., a year’s worth). The stack hedge may have lower transaction costs but it entails speculation (implicit or deliberate) on the forward curve: if the forward curve gets steeper, the stack hedger may lose. On the other hand, if the forward curve flattens, then the stack hedger gains because he/she has locked in the commodity at a relatively lower price.

LO 30.9: Differentiate between a strip hedge and a stack hedge.

Swaps

A swap is an agreement to exchange future cash

flows

• “Plain vanilla” swap: company pays fixed rate on

notional principal and receives floating rate (pay

fixed receive floating)

• Interest rate swap: principal not exchanged

(i.e., that’s why it is called notional)

• Currency swap: principal is (typically) exchanged

at beginning (inception) and end (maturity)

LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.

SwapsLO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.

Add Your

Text here

Pay

Fixed

Receive

LIBOR

Add Your

Text here

Add Your

Text here

Add Your

Text here

Receive Fixed

Receive

LIBORReceive

LIBOR

Pay

FixedPay

Fixed

Pay LIBOR“Plain-vanilla”

Counterparty

Swaps

Notional principal: $100 million (notional principal is not exchanged)

Swap agreement: Pay fixed rate of 5% and receive LIBOR

Term: 3 years with payments every six months


End of

Period

(6 months)

LIBOR at the

Start of

Period

Pay Fixed

Cash Flow

Receive

Floating

Cash Flow

Net Cash

Flow

1 5.0% -2.5 +2.5 0.0

2 (Year 1) 5.2% -2.5 +2.6 +0.1

3 5.4% -2.5 +2.7 +0.2

4 (Year 2) 5.0% -2.5 +2.5 0.0

5 4.8% -2.5 +2.4 -0.1

6 (Year 3) 4.6% -2.5 +2.3 -0.2

Swaps

Intel borrowing fixed-rate @ 5.2%

MSFT borrowing floating-rate @ LIBOR + 10 bps

LO 31.2: Explain how an interest rate swap can be combined with an existing asset or liability to transform the interest rate risk.

SwapsLO 31.3: Explain the advantages and disadvantages of the comparative advantage argument often used for the existence of the swap market.

Fixed Floating

BetterCreditCorp 4% LIBOR + 1%

WorseCreditCorp 6% LIBOR + 2%

Swaps

LIBOR/swap zero given: six-month = 3%, 1 year = 3.5%, 1.5 year = 4%.

The 2 year swap rate is 5% which implies that a $100 face value bond with

a 5% coupon will sell exactly at par (why? Because the 5% coupons are

discounted at 5%)

We can solve for the two year zero rate (R) because it is the unknown

LO 31.4: Explain how the discount rates in a swap are computed.

Period Cash flowLIBOR/swap

zero rates

PresentValue of

Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R

Total PV $100.00

SwapsLO 31.4: Explain how the discount rates in a swap are computed.

Period Cash flow

LIBOR/swap zero rates

PresentValue of

Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R

Total PV $100.00

( .5)(3%) ( 1)(3.5%) ( 1.5)(4%) 2

2

( 2 )

2.5 2.5 2.5 102.5 100

2.46 2.41 2.35 102.5 100

0.90506

4.99%

R

R

R

e e e e

e

e

R

Swaps

If two companies enter into an interest rate swap arrangement, then one of the companies has a swap position that is equivalent to a long position in floating-rate bond and a short position in a fixed-rate bond.

VSWAP = BFL - BFIX

The counterparty to the same swap has the equivalent of a long position in a fixed-rate bond and a short position in a floating-rate bond:

VSWAP Counterparty = BFIX -BFL

LO 31.5: Explain how a swap can be interpreted as two simultaneous bond positions or as a sequence of forward rate agreements (FRAs).

Swaps

LO 31.6: Calculate the value of an interest rate swap.

Add Your

Text here

Receive

½ of 7%

Pay

½ LIBOR

Add Your

Text here

Add Your

Text here

Time

0.25Time

0.75Time

1.25

Receive

½ of 7%

Pay

½ LIBOR

Receive

½ of 7%

Pay

½ LIBOR

Assumptions

Notional 100

Receive Fixed 7.0%

LIBOR Rates

3 Months (0.25) 5.0%

6 Months (0.5) 5.5%

9 Months (0.75) 6.0%

12 Months (1.0) 6.5%

SwapsLO 31.6: Calculate the value of an

interest rate swap.

Assumptions

Notional 100

Receive Fixed 7.0%

LIBOR Rates

3 Months (0.25) 5.0%

6 Months (0.5) 5.5%

9 Months (0.75) 6.0%

12 Months (1.0) 6.5%

Fixed Floating

LIBOR Disc. Cash Flows Cash Flows

Time Rates Factor FV PV FV PV

0.25 5.0% 0.988 $3.5 $3.46 $102.75 $101.47

0.75 6.0% 0.956 $3.5 $3.35

1.25 6.5% 0.922 $103.5 $95.42

Total $102.23 $101.47

Value (swap) = $102.23 - $101.47 = $0.75

SwapsLO 31.7: Explain the mechanics and calculate the value of a currency swap.

Assumptions

Principal, Dollars ($MM) 10

Principal, Yen (MM) Y 1,000

FX rate 120

US rate 5.0%

Japanese rate 2.0%

SWAP:

PAY dollars @ 5%

RECEIVE yen @ 9%

SwapsLO 31.7: Explain the mechanics and

calculate the value of a currency swap.

Assumptions



FX rate 120

US rate 5.0%

Japanese rate 2.0%

SWAP:

PAY dollars @ 5%

RECEIVE yen @ 9%

Dollars (MM) Yen (MM)

Time FV PV FV PV

1 0.5 $0.48 90 Y 88

2 0.5 $0.45 90 Y 86

3 0.5 $0.43 90 Y 85

3 10 $8.61 1000 Y 942

$9.97 Y 1,201

Yen bond Y 1,201

Yen bond in US dollars $10.01

Dollar bond $9.97

Swap, yen bond - dollar bond $0.04

Swaps

Because a swap involves offsetting choir position, there is

no credit risk when the swap has negative value. Credit

risk only exists when the swap has positive value.

Further, because principal is not exchanged at the end of

the life of an interest rate swap, the potential default

losses are much less than those on an equivalent loan. On

the other hand, in a currency swap, the risk is greater

because currencies are exchanged at the end of the swap.

LO 31.8: Explain the role of credit risk inherent in an

existing swap position.

1

Derivatives – Part 1(LOs 27.x – 31.x)

Forwards (LO 27.x)1

4 Commodity Futures (LO 30.x)

Interest Rate Futures (LO 29.x)

2 Hedging Strategies (LO 28.x)

3

Swaps (LO 31.x)5

Forwards/Futures

Time (T)

0 0 0( ) rTTF E S F S e

S0

F0

ST-1

FT-1 F0ST

ST=F

T

Forwards/Futures

Time (T)

(Commodity with High

“Convenience Yield” or

High-dividend Financial Asset)

F0 = E(S

T)

S0

F0

ST-1

FT-1 F0ST

ST=F

T

Cost-of-carry modelLO 27.1: State & explain cost-of-carry model for forward prices with & without interim cash flows

Time (T)

Income/

Dividend

(q)

Storage

Cost (U, u)

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

Convenience

(y)

2







0 0 0

( 0.05)(6/12) (.05)(1)

( )

($50 [($1) ])

$51.538

rTF S I e F

e e

DerivativesLO 27.2: Compute the forward price given both the price of the underlying and the appropriate carrying costs of the underlying.

( )0 0

r u q y TF S e

( )0 0

r q TF S e

Commodity



( )0 0( ) r y TF S U I e

( )0 0

r u q y TF S e

Present values

Commodity

constant rates as %

u = storage costs

q = income (dividend)

y = convenience yield

U = Present value, storage costs

I = Present value, income

3


0 0( ) rTF S I e( )0 0

r q TF S e

Present values


constant rates as %

q = income (dividend) I = Present value, income







( ) (6%/12 1.5%)(4)0 0

(.02)(4)

(230)

230 249.16

r u TF S e e

e

Derivatives

LO 27.3: Calculate the value of a forward contract.

0( ) rTf F K e

4







(5%)(0.25)0 0 10 $10.126rTF S e e

(5%)(0.25)0( ) (10.126 8) $2.153rTf F K e e

DerivativesLO 27.4: Describe the differences between forward and futures contracts.

Forward vs. Futures Contracts

Forward Futures

Trade over-the-counter Trade on an exchange

Not standardized Standardized contracts

One specified delivery date Range of delivery dates

Settled at contract’s end Settled daily

Delivery or final cash

settlement usually occurs

Contract usually closed

out prior to maturity

Derivatives

A long-futures position agrees to buy in the future

A short-futures position agrees to sell in the future.

Price mechanism maintains a balance between buyers and

sellers.(market equilibrium)

Most futures contracts do not lead to delivery, because

most trades ―close out‖ their positions before delivery.

Closing out a position means entering into the opposite

type of trade from the original.

LO 27.5: Distinguish between a long futures position and a short futures position.

5

Derivatives

An (underlying) asset

A Treasury bond futures contract is on underlying

U.S. Treasury with maturity of at least 15 years and not

callable within 15 years (15 years ≤ T bond).

A Treasury note futures contract is on the underlying

U.S. Treasury with maturity of at least 6.5 years but not

greater than 10 years (6.5 ≤ T note ≤ 10 years).

When the asset is a commodity (e.g., cotton, orange

juice), the exchange specifies a grade (quality).


Derivatives

Contract size varies by type of futures contract

Treasury bond futures: contract size is a face value of $100,000

S&P 500 futures contract is index $250 (multiplier of 250X)

NASDAQ futures contract is index $100 (multiplier of 100X)

Recently, ―mini contracts‖ have been introduced:

S&P 500 ―mini‖ = $50 x S&P Index

NASDAQ ―mini‖ = $20 x NASDQ

(each contract is one-fifth the price, to attract smaller investors)


Derivatives

Delivery Arrangements

The exchange specifies delivery location.

Delivery Months

The exchange must specify the delivery month; this can

be the entire month or a sub-period of the month.


Derivatives

Margin account: Broker requires deposit.

Initial margin: Must be deposited when contract is

initiated.

Mark-to-market: At the end of each trading day, margin

account is adjusted to reflect gains or losses.



6

Derivatives

Maintenance margin: Investor can withdraw funds in the margin account in excess of the initial margin. A maintenance margin guarantees that the balance in the margin account never gets negative (the maintenance margin is lower than the initial margin).

Margin call: When the balance in the margin account falls below the maintenance margin, broker executes a margin call. The next day, the investor needs to ―top up‖ the margin account back to the initial margin level.

Variation margin: Extra funds deposited by the investor after receiving a margin call.



Derivatives

There is only a variation margin if and when there is a

margin call.

Variation margin = initial margin – margin

account balance

The maintenance margin is a trigger level—once

triggered, the investor must ―top up‖ to the initial

margin, which is greater than the maintenance level.

LO 27.8: Compute the variation margin.

Derivatives

The exchange clearinghouse is a division of the exchange

(e.g., the CME Clearing House is a division of the

Chicago Mercantile Exchange) or an independent

company. The clearinghouse serves as a

guarantor, ensuring that the obligations of all trades are

met.


Derivatives

Market order: Execute the trade immediately at the

best price available.

Limit order: This order specifies a price (e.g., buy at $30

or less)—but with no guarantee of execution.

Stop order: (aka., stop-loss order) An order to execute

a buy/sell when a specified price is reached.


7

Derivatives

Stop-limit: Requires two specified prices, a stop and a

limit price. Once the stop-limit price is reached, it

becomes a limit order at the limit price.

Market-if-touched: Becomes a market order once

specified price is achieved.

Discretionary (aka., market-not-held order): A

market order, but the broker is given the discretion to

delay the order in an attempt to get a better price.


Derivatives

A short forward (or futures) hedge is an agreement to sell in the future and is appropriate when the hedger already owns the asset.

Classic example is farmer who wants to lock in a sales price: protects against a price decline.

A long forward (or futures) hedge is an agreement to buy in the future and is appropriate when the hedger does not currently own the asset but expects to purchase in the future.

Example is an airline which depends on jet fuel and enters into a forward or futures contract (a long hedge) in order to protect itself from exposure to high oil prices.

LO 28.1: Differentiate between a short hedge and a long hedge, and identify situations where each is appropriate.

Derivatives

Basis = Spot Price Hedged Asset –

Futures Price Futures Contract = S0 – F0

Basis =

Futures Price Futures Contract –

Spot Price Hedged Asset = F0 – S0

LO 28.2: Define and calculate the basis.

Hull says first is correct but second is

common for financial assets (either is okay)

Basis risk

Time (T)

$2.50

$2.20

$2.00

$1.90

T0

$0.30$0.10

T1

No hedge

Spot = -$0.50

SPOT

Forward

Short 1.67 F

Spot = -$0.50

Future = $0.30 (1.67)

Net = 0

Basis

Weakening of the basis = Futures price

increases more than spot

8

Basis risk

$2.50

$2.20

$2.00

$1.90

T0

$0.30$0.30

T1

No hedge

Spot = -$0.50

SPOT

Forward

Short 1.67 F

Spot = -$0.50

Future = $0.50 (1.67)

Net = +33.5

Basis

$1.70

Time (T)

Basis unchanged.

But unexpected strengthening= Hedger improved!

Basis risk

$2.50

$2.20

$2.60

$1.90

T0

$0.30

T1

SPOT

Forward

Short 1.67 F

Spot = +$0.10

Future = -$0.40 (1.67)

Net = -56.8

Basis

$2.60

Time (T)

Basis declines

Unexpected weakening= Hedger worse!

$0.0

Derivatives

Spot price increases by more than the futures price

basis increases. This is a ―strengthening of the basis‖

When unexpected, strengthening is favorable for a short

hedge and unfavorable for a long hedge

Futures price increases by more than the spot price

basis declines. This is a ―weakening of the basis‖

When unexpected, weakening is favorable for a long hedge

and unfavorable for a short hedge


Derivatives



position.

Contract ≠ Commodity





Basis risk higher with cross-hedging

9

Derivatives



position.

Contract ≠ Commodity.





Trade-offLiquidity(exchange)

Basis risk

Derivatives

The optimal hedge ratio (a.k.a., minimum variance hedge

ratio) is the ratio of futures position relative to the spot

position that minimizes the variance of the position.

Where is the correlation and is the standard

deviation, the optimal hedge ratio is given by:


* S

F

h

Derivatives



is 0.4, then:


* S

F

h20%

* (0.4) 0.810%

h

Derivatives



is 0.4, then:


* S

F

h20%

* (0.4) 0.810%

h

** A

F

h NN

QNumber of

contracts

10

Derivatives

Given a portfolio beta ( ), the current value of the

portfolio (P), and the value of stocks underlying one

futures contract (A), the number of stock index futures

contracts (i.e., which minimizes the portfolio variance) is

given by:


PN

A

Derivatives

By extension, when the goal is to shift portfolio beta from

( ) to a target beta ( *), the number of contracts

required is given by:


( * )P

NA


Assume:

Value of S&P 500 Index is 1240

Value of portfolio is $10 million

Portfolio beta ( ) is 1.5

How do we change the portfolio beta to 1.2?

PN ( * )

A

Hint: Contract = ($250 Index)

and # of futures is given by:


Assume:

• Value of S&P 500 Index is 1240

• Value of portfolio is $1 million

• Portfolio beta ( ) is 1.5

We short about 10 contracts. (-) indicates short, (+) long…

( * )

$10,000,000(1.2 1.5) 9.7

(1240)(250)

PN

A

11

Derivatives

When the delivery date of the futures contract occurs

prior to the expiration date of the hedge, the hedger can

roll forward the hedge: close out a futures contract and

take the same position on a new futures contract with a

later delivery date.

Exposed to:

Basis risk (original hedge)

Basis risk (each new hedge) = ―rollover basis risk‖

LO 28.6: Identify situations when a rolling hedge is appropriate, and discuss the risks of such a strategy.

DerivativesLO 29.1: Identify and apply the three most common day count conventions.

Actual/actual U.S. Treasuries

30/360 U.S. corporate and

municipal bonds

Actual/360 U.S. Treasury bills and

other money market

instruments

Derivatives

The Treasury bond futures contract allows the party with

the short position to deliver any bond with a maturity of

more than 15 years and that is not callable within 15

years. When the chosen bond is delivered, the conversion

factor defines the price received by the party with the

short position:

LO 29.2: Explain the U.S. Treasury bond (T-bond) futures contract conversion factor.

Cash Received = Quoted futures price Conversion

factor + Accrued interest

= (QFP CF) + AI

Derivatives

The convexity adjustment assumes continuous

compounding. Given that ( ) is the standard deviation of

the change in the short-term interest rate in one year, t1is the time to maturity of the futures contract and t2 is

the time to maturity of the rate underlying the futures

contract:

LO 29.3: Calculate the Eurodollar futures contract convexity adjustment.

21 2

1Forward = Futures

2t t

12

Derivatives

The number of contracts required to hedge against an

uncertain change in the yield, given by y, is given by:

FC = contract price for the interest rate futures contract.

DF = duration of asset underlying futures contract at maturity.

P = forward value of the portfolio being hedged at the maturity

of the hedge (typically assumed to be today’s portfolio value).

DP = duration of portfolio at maturity of the hedge

LO 29.4: Formulate a duration-based hedging strategy using interest rate futures.

* P

C F

PDN

F D





price of 98.








price of 98.




($10 million)(6)* 122

(98,000)(5)P

C F

PDN

F D

Derivatives

Portfolio immunization or duration matching is when a

bank or fund matches the average duration of assets with

the average duration of liabilities.

Duration matching protects or ―immunizes‖ against

small, parallel shifts in the yield (interest rate) curve. The

limitation is that it does not protect against nonparallel

shifts. The two most common nonparallel shifts are:

A twist in the slope of the yield curve, or

A change in curvature

LO 29.5: Identify the limitations of using a duration-based hedging strategy.

13

Derivatives



Time (T)

Discount

rate ( )

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

( )0, 0( ) r TT TF E S e

Derivatives



( )0, 0( ) r TT TF E S e

0

0,T

( ) Spot price of S at time T, as expected at time 0F Forward pricer Risk-free rate

Discount rate for commodity S

TE S

Derivatives

Lease rate = commodity discount rate – growth rate

Lease rate dividend yield

LO 30.2: Define lease rates, and discuss the importance of lease rates for determining no-arbitrage values for commodity futures and forwards.

( )0, 0

r TTF S e

( )0 0

r q TF S e

Financial asset

Derivatives

Contango refers to an upward-sloping forward curve

which must be the case if the lease rate is less than

the risk-free rate. Backwardation refers to a downward-

sloping forward curve which must be the case if the

lease rate is greater than the risk-free rate.

LO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.

14

DerivativesLO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation.

Time (T)

Research says normal backwardation is “normal:” speculators

want compensation (risk premium) for buying the futures contract

Spot

(S0)E(ST)

Forward

(F0)

Forward

(F0)


Time (T)

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

Storage Cost ( )

negative dividend

Convenience (y)

dividend

Lease rate ( )

dividend


Time (T)

Lease

rate ( )

Risk-free

Rate (r)

Spot

(S0)

Forward

(F0)

( )0 0

r c TF S e

Storage

Cost ( )

Convenience

(y)

DerivativesLO 30.5: Explain how a convenience yield impacts commodity forward prices, and determine the no-arbitrage bounds for the forward price of a commodity when the commodity has a convenience yield.

( ) ( )0 0 0

r c T r TS e F S e

Risk-free

Rate

Storage

Cost

Risk-free

Rate

Storage

Cost

Convenience

Yield

15

Commodity Futures

1460148015001520154015601580160016201640

S&P 500 Index

Rational forward curve rises by cost

of capital (risk free + premium) less

dividends


500

550

600

650

700

750

800

850

900

Jul-07 Nov-08 Mar-10 Aug-11 Dec-12

Gold futures

Durable, (relatively) cheap to store.

Forward curve is “uninteresting”


200

250

300

350

400

450

Sep

-07

Dec-0

7

Mar-

08

Jun

-08

Sep

-08

Dec-0

8

Mar-

09

Jun

-09

Sep

-09

Dec-0

9

Mar-

10

Jun

-10

Sep

-10

Dec-1

0

Corn


5

6

7

8

9

10

Au

g-0

7

Nov-0

7

Feb

-08

May-0

8

Au

g-0

8

Nov-0

8

Feb

-09

May-0

9

Au

g-0

9

Nov-0

9

Feb

-10

May-1

0

Au

g-1

0

Nov-1

0Natural Gas

Costly to transport. Costly to store (storage costs). Highly seasonal

16


68

69

70

71

72

73

74

75

76

Sep

-07

No

v-0

7

Jan

-08

Mar-

08

May-0

8

Jul-

08

Sep

-08

No

v-0

8

Jan

-09

Mar-

09

May-0

9

Jul-

09

Sep

-09

No

v-0

9

Jan

-10

Mar-

10

May-1

0

Jul-

10

Sep

-10

No

v-1

0

Crude oil

Compared to natural gas, easier to store and transport. Global market. Long-run forward price less (<) volatile than short-run forward.

Commodity Futures

If we can take a long position on one commodity that is an input (e.g., oil) into another commodity that is an output (e.g., gas or heating oil), then we can take a short position in the output commodity and the difference is the commodity spread.

Assume oil is $2 per gallon, gasoline is $2.10 per gallon and heating oil is $2.50 per gallon.

If we take a long position in 2 gallons of gasoline and one gallon of heating oil, plus a short position in three gallons of oil, the commodity spread =

(2 long gasoline $2.10) + (1 long heating oil $2.50) – (3 oil $2) = +$0.70

LO 30.7: Describe and calculate a commodity spread.

Commodity Futures

The basis is the difference between the price of the futures contract and the spot price of the underlying asset.

Basis risk is the risk (to the hedger) created by the uncertainty in the basis.

The futures contract often does not track exactly with the underlying commodity; i.e., the correlation is imperfect. Factors that can give rise to basis risk include:

Mismatch between grade of underlying and contract

Storage costs

Transportation costs

LO 30.8: Define basis risk, and explain how basis risk can occur when hedging commodity price exposure.

LO 30.9: Differentiate between a

strip hedge and a stack hedge.

10 10 10 10 10 10 10 10 10 10 10 10

Jan Feb Mar

<120 <110 <100Jan Feb Mar

Commodity Futures

Oil producer to deliver

10K barrels per month

Strip hedge: contract for

each obligation

Stack hedge: Single maturity,

―stack and roll‖

17

Commodity Futures

A strip hedge is when we hedge a stream of obligations by offsetting each individual obligation with a futures contract that matches the maturity and quantity of the obligation. For example, if a producer must deliver X number of commodities per month, then the strip hedge entails entering into a futures contract for X commodities, to be delivered in one month; plus a futures contract for X commodities to be delivered in two months. The strip hedger matches a series of futures to the obligations.

A stack hedge is front-loaded: the hedger enters into a large future with a single maturity. In this case, our hedger would take a long position in a near-term futures contract for 12X commodities (i.e., a year’s worth). The stack hedge may have lower transaction costs but it entails speculation (implicit or deliberate) on the forward curve: if the forward curve gets steeper, the stack hedger may lose. On the other hand, if the forward curve flattens, then the stack hedger gains because he/she has locked in the commodity at a relatively lower price.

LO 30.9: Differentiate between a strip hedge and a stack hedge.

Swaps

A swap is an agreement to exchange future cash

flows

• “Plain vanilla” swap: company pays fixed rate on

notional principal and receives floating rate (pay

fixed receive floating)

• Interest rate swap: principal not exchanged

(i.e., that’s why it is called notional)

• Currency swap: principal is (typically) exchanged

at beginning (inception) and end (maturity)


SwapsLO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap.

Add Your

Text here

Pay

Fixed

Receive

LIBOR

Add Your

Text here

Add Your

Text here

Add Your

Text here

Receive Fixed

Receive

LIBORReceive

LIBOR

Pay

FixedPay

Fixed

Pay LIBOR“Plain-vanilla”

Counterparty

Swaps

Notional principal: $100 million (notional principal is not exchanged)

Swap agreement: Pay fixed rate of 5% and receive LIBOR

Term: 3 years with payments every six months


End of

Period

(6 months)

LIBOR at the

Start of

Period

Pay Fixed

Cash Flow

Receive

Floating

Cash Flow

Net Cash

Flow

1 5.0% -2.5 +2.5 0.0

2 (Year 1) 5.2% -2.5 +2.6 +0.1

3 5.4% -2.5 +2.7 +0.2

4 (Year 2) 5.0% -2.5 +2.5 0.0

5 4.8% -2.5 +2.4 -0.1

6 (Year 3) 4.6% -2.5 +2.3 -0.2

18

Swaps

Intel borrowing fixed-rate @ 5.2%

MSFT borrowing floating-rate @ LIBOR + 10 bps

LO 31.2: Explain how an interest rate swap can be combined with an existing asset or liability to transform the interest rate risk.

SwapsLO 31.3: Explain the advantages and disadvantages of the comparative advantage argument often used for the existence of the swap market.

Fixed Floating

BetterCreditCorp 4% LIBOR + 1%

WorseCreditCorp 6% LIBOR + 2%

Swaps

LIBOR/swap zero given: six-month = 3%, 1 year = 3.5%, 1.5 year = 4%.

The 2 year swap rate is 5% which implies that a $100 face value bond with

a 5% coupon will sell exactly at par (why? Because the 5% coupons are

discounted at 5%)

We can solve for the two year zero rate (R) because it is the unknown

LO 31.4: Explain how the discount rates in a swap are computed.

Period Cash flowLIBOR/swap

zero rates

PresentValue of

Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R

Total PV $100.00

SwapsLO 31.4: Explain how the discount rates in a swap are computed.

Period Cash flow

LIBOR/swap zero rates

PresentValue of

Cash Flow0.5 $2.5 3.0% $2.461.0 $2.5 3.5% $2.411.5 $2.5 4.0% $2.352.0 $102.50 X? 102.5e2R

Total PV $100.00

( .5)(3%) ( 1)(3.5%) ( 1.5)(4%) 2

2

( 2 )

2.5 2.5 2.5 102.5 100

2.46 2.41 2.35 102.5 100

0.90506

4.99%

R

R

R

e e e e

e

e

R

19

Swaps

If two companies enter into an interest rate swap arrangement, then one of the companies has a swap position that is equivalent to a long position in floating-rate bond and a short position in a fixed-rate bond.

VSWAP = BFL - BFIX

The counterparty to the same swap has the equivalent of a long position in a fixed-rate bond and a short position in a floating-rate bond:

VSWAP Counterparty = BFIX -BFL

LO 31.5: Explain how a swap can be interpreted as two simultaneous bond positions or as a sequence of forward rate agreements (FRAs).

Swaps

LO 31.6: Calculate the value of an interest rate swap.

Add Your

Text here

Receive

½ of 7%

Pay

½ LIBOR

Add Your

Text here

Add Your

Text here

Time

0.25Time

0.75Time

1.25

Receive

½ of 7%

Pay

½ LIBOR

Receive

½ of 7%

Pay

½ LIBOR

Assumptions

Notional 100

Receive Fixed 7.0%

LIBOR Rates

3 Months (0.25) 5.0%

6 Months (0.5) 5.5%

9 Months (0.75) 6.0%

12 Months (1.0) 6.5%

SwapsLO 31.6: Calculate the value of an

interest rate swap.

Assumptions

Notional 100

Receive Fixed 7.0%

LIBOR Rates

3 Months (0.25) 5.0%

6 Months (0.5) 5.5%

9 Months (0.75) 6.0%

12 Months (1.0) 6.5%

Fixed Floating

LIBOR Disc. Cash Flows Cash Flows

Time Rates Factor FV PV FV PV

0.25 5.0% 0.988 $3.5 $3.46 $102.75 $101.47

0.75 6.0% 0.956 $3.5 $3.35

1.25 6.5% 0.922 $103.5 $95.42

Total $102.23 $101.47

Value (swap) = $102.23 - $101.47 = $0.75

SwapsLO 31.7: Explain the mechanics and calculate the value of a currency swap.

Assumptions



FX rate 120

US rate 5.0%

Japanese rate 2.0%

SWAP:

PAY dollars @ 5%

RECEIVE yen @ 9%

20

SwapsLO 31.7: Explain the mechanics and

calculate the value of a currency swap.

Assumptions



FX rate 120

US rate 5.0%

Japanese rate 2.0%

SWAP:

PAY dollars @ 5%

RECEIVE yen @ 9%

Dollars (MM) Yen (MM)

Time FV PV FV PV

1 0.5 $0.48 90 Y 88

2 0.5 $0.45 90 Y 86

3 0.5 $0.43 90 Y 85

3 10 $8.61 1000 Y 942

$9.97 Y 1,201

Yen bond Y 1,201

Yen bond in US dollars $10.01

Dollar bond $9.97

Swap, yen bond - dollar bond $0.04

Swaps

Because a swap involves offsetting choir position, there is

no credit risk when the swap has negative value. Credit

risk only exists when the swap has positive value.

Further, because principal is not exchanged at the end of

the life of an interest rate swap, the potential default

losses are much less than those on an equivalent loan. On

the other hand, in a currency swap, the risk is greater

because currencies are exchanged at the end of the swap.

LO 31.8: Explain the role of credit risk inherent in an

existing swap position.

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