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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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**8.1 Introduction of forward and futures contract**

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. 2

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**8.1 Introduction of forward and futures contract**

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 3

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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. 4

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**8.1 Introduction of forward and futures contract**

Payoff - Forward contract ST K Short position Long position 5

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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. 6

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**8.1 Introduction of forward and futures contract**

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. 7

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**8.1 Introduction of forward and futures 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 8

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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. 9

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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: 10

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**8.1 Introduction of forward and futures contract**

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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**8.1 Introduction of forward and futures contract**

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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**8.1 Introduction of forward and futures contract**

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.

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At the end of the day Sources: Table 2.1 (Options, Futures and other derivatives, 8th Ed., J. Hull)

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**8.1 Introduction of forward and futures contract**

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. 15

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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. 16

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**8.2 Hedging interest rate risk**

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. 17

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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 18

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**8.2 Hedging interest rate risk**

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 19

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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. 20

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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. 21

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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. 22

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**8.2 Hedging interest rate risk**

Illustration 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 23

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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. 24

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**8.2 Hedging interest rate risk**

He 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. 25

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**8.2 Hedging interest rate risk**

Solution: From Eq. (8.3), we have Therefore, the number of futures contracts should be sold by the investment manager is 1,317. 26

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**8.2 Hedging interest rate risk**

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 27

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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, 28

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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 2-year zero coupon bond (Yield = 1.5% p.a.) 2-year zero coupon bond 4-year zero (Yield = 3.5% p.a.) 3-year zero coupon bond (Yield = 2.3% p.a.) Equity 100 29

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**8.2 Hedging interest rate risk**

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. 30

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**8.2 Hedging interest rate risk**

Assume R = RF. From Eq. (8.6), Hence, the number of futures contract should be purchased (long) by XYZ Bank is 182. 31

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**8.2 Hedging interest rate risk**

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. 32

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**8.2 Hedging interest rate risk**

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. 33

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**8.2 Hedging interest rate risk**

Balance sheet: where CXF is the convexity of the underlying bond in the futures contract. 34

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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. 35

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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. 36

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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). 37

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**8.3 Hedging foreign exchange risk**

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) 38

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**8.3 Hedging foreign exchange risk**

The NF (no. of futures contract) can be obtained by solving 39

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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. 40

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**8.3 Hedging foreign exchange risk**

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. 41

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8.4 Hedging credit risk 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. 42

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8.4 Hedging credit risk 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. 43

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**8.4 Hedging credit risk 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. 44

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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. 45

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8.4 Hedging credit risk Cash settlement at the maturity of the credit spread forward 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 (CSF – CSM)A 46

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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. 47

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**8.3 Hedging foreign exchange risk**

The number of credit spread forward contract to hedge the credit spread risk (NF ) can be obtained by solving From Eq. (8.11), we have 48

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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%). 49

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8.4 Hedging credit risk 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”. 50

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8.4 Hedging credit risk 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. 51

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8.4 Hedging credit risk 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. 52

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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. 53

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**8.4 Hedging credit risk Protection seller Protection buyer**

Long a defaultable asset Swap premium/CDS spread (periodic) default payment (the default of the reference asset) 54

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