topic 8. forwards and futures in risk management

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Topic 8. Forwards and futures in risk management

8.1 Introduction of forward and futures contract

8.2 Hedging interest rate risk

8.3 Hedging foreign exchange risk

8.4 Hedging credit risk

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Forward contract A forward contract is a contractual agreement made

directly between two parties, says A and B,

Party A (Long the forward contract/Long position) (Buyer of the forward contract):

He agrees to buy the underlying asset at certain future time (maturity date) for a agreed contractual price (forward price) (delivery price).

Party B (Short the forward contract/Short position) (Seller of the forward contract):

He agrees to sell the underlying asset at maturity date for the forward price.

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t = 0 t = T months (maturity)

Price agreed between A (forward buyer) and B (forward seller)

A pays the agreed price to B to buy the underlying asset from B

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8.1 Introduction of forward and futures contract The underlying asset of the forward contact can be

commodities such as live cattle, oil and gold, and financial assets like bonds, currencies and stock indices.

Forward contract is traded in the over-the-counter (OTC) market.

The payoff ( 收益 ) of the forward contract at the maturity date T is given by:

Long position: ST – K

Short position: K – ST

where K is the delivery price and ST is the underlying asset price at T.

Zero cost to enter the forward contract for both parties.

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Long position

Payoff - Forward contract

STK0

Payoff - Forward contract

ST

K0

Short position

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8.1 Introduction of forward and futures contract Example 8.1

Forward price of the 3-month forward contract on Gold is $100 for 1 ounce. One forward contract corresponds to 200 ounces of gold.

ABC company long (buys) one of this forward contract. At the maturity of the contract, ABC has to pay 200×100 = $20,000 to buy 200 ounces of gold from the party whom in the short position of this forward contract.

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If the price of gold at the maturity of the contract is $105 ($95) per ounce, ABC company will gain (loss) $1000. The corresponding party in the short position will loss (gain) $1000.

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8.1 Introduction of forward and futures contractFutures contract Like a forward contract, a futures contract is an

agreement between two parties to buy or sell an asset at the maturity of the futures contract for an agreed price (futures price).

Differences between a futures and forward contract:• Traded over the exchange market (e.g. Hong Kong

Exchange (HKEx), Chicago Board of Trade (CBT) and Chicago Mercantile Exchange (CME)). So, futures contract is more liquid than the forward contract.

• Standardized contract

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8.1 Introduction of forward and futures contract Differences between a futures and forward contract

(cont.):• With less credit risk than forward contract.• Subject to margin requirement – initial deposit by the

investor.• Marking to market.

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8.1 Introduction of forward and futures contract Specification of a futures contract:

• The quality of the delivered asset especially for the commodities e.g. the grade of corn in corn futures.

• Contract size e.g. the number of barrel in oil futures and the face value of the bond in bond futures.

• Maturity date

• Settlement procedure: Physical delivery or cash settlement

• Delivery arrangement such as location and method

• Delivery months

• Delivery price

• Price limit: the daily movement limit

• Position limit: the maximum number of contract can hold

Chicago Mercantile Exchange: http://www.cmegroup.com

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Daily Settlement and margins To reduce the credit (default) risk of the both parties in the

futures contract, the exchange requires the broker, who represents the investor to perform trading in the exchange, to deposit an initial margin in the margin account.

At the end of trading day, the margin account is adjusted to reflect the investor’s gain or loss. This practice is referred to as marking to market.

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If the account balance falls below certain specified limit - maintenance margin (< initial margin), the investor receives margin call to alert him to top up the balance of margin account to the level of initial margin.

The top up balance (Variation margin)

= Initial margin current margin account balance. If the investor can not do that, his broker will close out the

position.

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Example 8.2• An investor contacts his broker to buy (long) two December

gold futures contract. • Suppose the contract size of each contract is 100 ounces and the

current futures price is $1,250/ounce. • Initial margin = $6,000 (per contract) × 2 = $12,000• Maintenance margin = $4,500 (per contract) × 2 = $9,000• Assume no interest is paid to margin account (in realistic the

interest is payable to the margin account).• The future position is closed out on Day 16 by selling (short) 2

futures contract. So, the daily gain (loss) from the original and closing position are offset with each other beyond Day 16.

14Sources: Table 2.1 (Options, Futures and other derivatives, 8th Ed., J. Hull)

At the end of the day

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Since the futures is traded on exchanges, both parties can close out their futures position anytime before the maturity by entering the opposite position of the same contract. This effectively ends any net cash flow implication from futures positions beyond this date.

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8.1 Introduction of forward and futures contract Since the delivery of the underlying asset is only

occurred in a future time period - at the maturity of the futures and forward contract, the underlying asset does not appear on the balance sheet, which records only current and past transactions. Thus, forward and futures contract are examples of the off-balance-item.

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Single bond Neglect the convexity adjustment, from Eq. (2.10),

the change of bond price, P = P(R+ R) – P(R), is given by

where

P(R) is the price of the bond when the bond yield is R (P(R) > 0 : Long position; P(R) < 0 : Short position);

MD is the modified duration of the bond;

R is the changes of the bond yield.

(8.1) )( RRPMDP

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8.2 Hedging interest rate risk Suppose there is a futures contract with the

underlying is an interest rate sensitive financial asset (e.g. bond) in the market, the sensitivity of the price of a futures contract with respect to the changing of interest rate can be well approximated by

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where

F = Price of one future contract

F = change in price of one future contract

MDF = Modified duration of the underlying bond

RF = Annual yield of the underlying bond

RF = change in annual yield of the underlying bond

(8.2) FF RFMDF

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8.2 Hedging interest rate risk By considering a hedged portfolio (Bond + N futures

contracts), the value of the hedged portfolio is given by

The required number of futures contract to hedge the interest rate risk of the bond is a number N which makes QH = 0. The corresponding N is denoted as NF.

FNPQH

(8.3) )(

0

FFF

F

RFMD

RRPMDN

FNP

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8.2 Hedging interest rate risk NF < 0 the hedger should sell (short) NF futures

contract.

NF > 0 the hedger should buy (long) NF futures contract.

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8.2 Hedging interest rate risk Rounding Rule in Futures Market:

In realistic, the number of the futures contract should buy or sell must be an integer. In practice, the actual number of the futures contract is obtained by rounding down the value of NF to the nearest whole number.

Round down a given number to the nearest whole number is defined as discarding all the decimal places of that number irrelevant to whether it is a positive or negative number.

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8.2 Hedging interest rate riskIllustration of the rounding rule

Use the function “Rounddown” in Excel to do more practice of this rounding rule.

NF: Number of Futures Contracts Required for hedging

Before rounding After rounding down

15.89 15

20.25 20

40.79 40

50.12 50

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8.2 Hedging interest rate risk Example 8.3

Suppose an investment manager in a FI longs a bond with the face value of $100 million and intends to sell the bond at the end of one year. The price and the modified duration of the bond are $97 million and 17.5 respectively.

He predicts that the bond’s annual yield will increase by 1.5% over the next year.

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8.2 Hedging interest rate riskHe intends to use 1-year futures contract on a bond with the modified duration of 18.5 to hedge the interest rate risk of his bond at the end of one year.

The current futures price quote is $95 per $100 face value of the underlying bond. The contract size is $100,000 in face value. The annual yield of the underlying is expected to increase by 1.1% over the next year.

Determine the number of the futures contracts that the investment manager should buy or sell in order to hedge the interest rate risk of his bond.

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8.2 Hedging interest rate riskSolution:

From Eq. (8.3), we have

Therefore, the number of futures contracts should be sold by the investment manager is 1,317.

number) olenearest wh thedown to (round 317,1

%1.1100095$5.18

%5.1million 97$5.17

FN

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Balance sheet Neglect the convexity adjustment and assume

parallel yield shift (R), the impact of interest rate on the equity value for a FI is given

(8.4) whereA

LkRAMDkMDE LA

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8.2 Hedging interest rate risk Similar to Eq. (8.3), the required number of futures

contract to hedge the interest rate risk on the equity value of a FI is given by

By substituting Eqs. (8.4) and (8.2) into Eq. (8.5),

So,

(8.5) 0 FNE F

0 FFFLA RFMDNRAMDkMD

(8.6)

FF

LAF RFMD

RAMDkMDN

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8.2 Hedging interest rate risk Example 8.4

XYZ Bank has the following market value balance sheet.

Assets ($ millions)

Liabilities ($ millions)

Cash 202-year zero 50 coupon bond(Yield = 1.5% p.a.)

2-year zero 20 coupon bond 4-year zero 60 coupon bond (Yield = 3.5% p.a.)

3-year zero 30 coupon bond(Yield = 2.3% p.a.)

Equity 20

100 100

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Suppose XYZ Bank intends to hedge the interest rate risk on its equity value by using a 1-year futures contract with the underlying asset of 5-year zero coupon bond. The annual yield of the underlying asset is 4%. The current future price quote is $97 per $100 face value of the underlying bond. The contract size of a futures contract is $100,000 face value.

8.0100

80

39.3%5.31

460

%5.11

220

80

1

86.1%3.21

330

%5.11

250020

100

1

A

Lk

MD

MD

L

A

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Assume R = RF. From Eq. (8.6),

Hence, the number of futures contract should be purchased (long) by XYZ Bank is 182.

000,97$000,197

81.4%41

5

85.039.38.086.1

gapduration Modified

F

MD

MDkMD

F

LA

integer)nearest thedown to (round 182

000,9781.4

000,000,10085.0

FN

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R = RF = + 0.5% R = RF = –0.5%

E

(Eq. (8.4))

( 0.85)100 million

(0.005)

= $425,000

(0.85)100 million

(0.005)

= $425,000

F

(Eq. (8.2))

4.8197000(0.005)

= $2,332.85

4.8197000( 0.005)

= $2,332.85

E + NF∙F $421.3 $421.3

From the last row of the table, it can observed that E + NF∙F is not exactly equal to 0 since NF has been rounded.

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With the convexity adjustment If R is not small, the convexity adjustment has to be

considered. With the convexity adjustment, Eqs (8.3) and (8.6) will be modified as follows:

Single Bond:

where CXF is the convexity of the underlying asset in the futures contract.

(8.7)

21

)(21

)(

2

2

FRCXRFMD

RPRCXRRPMDN

FFFF

F

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Balance sheet:

where CXF is the convexity of the underlying bond in the futures contract.

(8.8)

2121

2

2

FRCXRFMD

ARCXkCXRAMDkMDN

FFFF

LALA

F

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8.3 Hedging foreign exchange risk Hedging of foreign exchange (FX) risk is similar to

hedge of interest rate risk. The forward price of a FX forward is quoted in terms of

exchange rate ($/foreign currency). Suppose the forward price of a FX forward is f and the

contract size is PF (in foreign currency). At the maturity of the forward, the forward buyer will pay f PF to the forward seller to buy PF amount of foreign currency under one FX forward contract.

The FX futures is similar to FX forward. Different from forward contract, the futures is traded on an exchange.

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8.3 Hedging foreign exchange risk Example 8.5

Suppose on today, ABC Bank long two 6-month FX forward contracts. The today’s quote of the forward is $2.0483/₤ and the size of each forward is ₤500,000.

At the end of 6 months, ABC Bank pays

2500,0002.0483=$2,048,300

to the FX forward seller to buy ₤1,000,000 under the 2 FX forward contracts.

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8.3 Hedging foreign exchange risk Suppose a FI has a foreign asset which is worth QF (in

foreign currency) and intends to hedge the FX risk by the futures contract on exchange rate ($/foreign currency).

(QF > 0: Long position; QF < 0: Short position)

Let Q be the value of the asset in $.

Let S be the spot exchange rate ($/foreign currency).

Let f be the price of the futures contract on exchange rate ($/foreign currency).

Let PF be the size of the futures contract (in foreign currency).

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Let S be the predicted change of S over a given time horizon.

Let f be the predicted change of f over a given time horizon.

If the asset is only exposed to FX risk, then

Q=QF S (8.9)

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The NF (no. of futures contract) can be obtained by solving

(8.10)

0

0

fP

SQN

fPNSQ

fPNQ

F

FF

FFF

FF

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8.3 Hedging foreign exchange risk Example 8.6

Suppose a US based FI has a foreign asset which is worth ₤100 million. The FI intends to hedge the FX risk of this foreign asset by using futures contracts. Suppose the current spot exchange rate is $1.9483/₤. The current futures price of a 1-year futures contract on the exchange rate $/₤ is $1.9468/₤.

The size of each British pound futures contract is ₤62,500.

Suppose the predicted changes on spot and futures price over the next year are $0.005/₤ and $0.003/₤ respectively.

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By Eq. (8.10),

So, the FI should sell (short) 2,666 British pound futures contract.

Ex. Redo Example 8.6 by changing the long position on the foreign asset to short position.

integer)nearest thedown to (round 666,2

003.0500,62

005.0000,000,100

FN

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Let CS be the credit spread of a defaultable bond.

Note:

In this course, we treat the following terms are equivalent to each other.

Credit spread Credit risk yield spread Bond yield spread Credit risk premium

The yield of the bond (R) = Rf (risk-free yield) + CS

With R, the price of the defaultable bond P(R) is given by Eq. (2.7) of Topic 2.

Suppose Rf = 0.

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Use the Taylor expansion with 1st order on Rf and CS , we have

We have neglected the convexity adjustment in Eq. (8.11).

If CS > 0 (< 0), then the party who long the bond will incur a loss (gain) from the bond.

(8.11) 0) be toassumed is (since )(

)(

)()()(

f

f

ff

RCSRPMD

CSRRPMD

CSCS

RPR

R

RP

RPRRPRP

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Credit spread forward A credit spread forward is an agreement that written

on a defaultable loan/bond. It is used to hedge against an increase in default risk (a decline in the credit quality of a borrower) of the defaultable bond.

A contractual credit spread CSF of the credit spread forward is usually set equal to the credit spread of the intended hedged bond at the commencing date of the forward contract.

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8.4 Hedging credit risk Suppose A is the contractual amount of the credit

spread forward. Settlement procedure: Cash settlement. Define CSM as actual credit spread of the bond at the

maturity of the credit spread forward.

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Credit Spread at the maturity of credit forward

Long Position

(Credit spread forward buyer)

Short Position

(credit spread forward seller)

CSM > CSF Pays

(CSM – CSF)A

Receives

(CSM – CSF)A

CSM < CSF Receives

(CSF – CSM)A

Pays

(CSF – CSM)A

Cash settlement at the maturity of the credit spread forward

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8.4 Hedging credit risk The credit spread forward seller hedges itself against

an increase in the borrower’s default risk. The credit spread forward buyer bears the risk of an

increase in default risk of the underlying loan.

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8.3 Hedging foreign exchange riskThe number of credit spread forward contract to hedge the credit spread risk (NF ) can be obtained by solving

From Eq. (8.11), we have

0

forward CS ofposition long thefrom receivedAmount

)(

FN

RP

(8.12) )(

0)(

A

RPMDN

CSCSANCSCSRPMD

F

MFFFM

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8.4 Hedging credit risk Example 8.7

A bank longs a $5 million bond with credit rating A. The corresponding credit spread is 2%. Suppose the bond is priced at par.

MD of the bond = 4.5 years

An one-year credit spread forward contract with the contractual amount of $1 million and contractual credit spread of 2% p.a. (CSF = 2%).

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Suppose at the maturity of the forward contract, the credit rating of the bond drops to “BB” and the corresponding credit spread is 5% (CSM = 5%).

Suppose Rf = 0, the change in the market value of the bond is given by:

Hence, the bank will incur a loss of $675,000 when the credit rating of the bond moves down to “BB”.

000,675$03.05.4000,000,5

)()(

FMFf CSCSMDCSRPRP

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From Eq. (8.12), the required number of credit spread forward contracts to hedge the increasing of the credit spread (or credit downgrade).

The bank should short 22 credit spread forward to hedge the credit spread movement.

integer)nearest thedown to (round 22

5.22million 1$

million 5$5.4

FN

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From the credit spread forward, the bank receives, from the credit spread forward buyer,

22 $1,000,000 (5% 2%) = $660,000.

Thus, the loss in the value of the bond due to a drop in the credit rating is approximately offset with the gain from the credit spread forward contract.

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8.4 Hedging credit risk The credit spread forward is used to hedge against the

spread risk of a loan. To hedge against the default risk, credit default swap (CDS) can be used.

Structure of CDS:

In credit default swap, the protection seller (swap seller) receives fixed periodic payments (swap premium or CDS spread) from the protection buyer (swap buyer) in return for making a single contingent payment covering losses on a reference asset following a default.

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Protection seller

Protection buyer

Long a defaultable

asset

Swap premium/CDS spread

(periodic)

default payment

(the default of the reference asset)

topic 8. forwards and futures in risk management

Documents

month forward contract

forward contact

contract size

standardized contract

bond futures

oil futures

corn futures

futures contract8