futures - angelfireand a generic 20-year, 8% coupon bond that is used as the basis for quoting...

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Futures CFA Level III © Gillsie June, 1999

FUTURES 1. Commodity Futures

a) The development of forward & futures contracts resulted from a need to reduce the risk inherent in the commodity markets, especially those dealing with agricultural and mining products. Producers wanted to shift the risk that prices would fall between the time they started growing/mining the product and the time it was ready for market to speculators and Users wanted to pass the risk of prices for the raw product increasing at a faster rate than they could pass on to their retail customers in order to keep margins relatively stable.

b) There are some fundamental relationships in these markets (that are usually marked-to-market each day) DAILY SETTLEMENTS on FUTURES CONTRACTS Long Position’s Daily Gain = Daily Payment from Seller to Buyer = KSize(Ft–Ft-1) GAIN/LOSS on a FUTURES CONTRACT NOT HELD ‘til SETTLEMENT Long Gain = Short Loss = KSize(Ft - F0) GAIN/LOSS on FUTURES CONTRACT HELD ‘til SETTLEMENT Long Gain = Short Loss = KSize(Ss - F0) For Example: Suppose a Buyer and Seller enter into a Future Contract for Gold (KSize= 100 oz.). F0 = $300. Each must put up $1,000 Initial Margin t Ft Long Gain(100(Ft-F0) Buyer Seller 0 300 0 1000 1000 1 305 500 1500 500 2 303 (200) 1300 700 3 294 (900) 400 1600 4 306 1200 1600 400 5 304 (200) 1400 600

a) Stock Index Forward & Futures Contracts

��These Contracts are SETTLED in CASH (never Delivery) ��Futures Contracts Exist for Several Stock Market Indexes, each of which has

its own contract dollar multiplier Contract Dollar Multiplier S&P 500 $250 per index point (in old text, was $500) DJIA $ 10 NASDAQ $100 Nikkei $ 5 DAILY SETTLEMENT on Stock Index Futures Contracts Long Position Gain = Short Position Loss = K$Multiplier (Ft - Ft-1) GAIN/LOSS on Stock Index Future NOT Held ‘til Settlement Long Gain = Short Loss = K$Multiplier (Ft – F0) GAIN/LOSS on Stock Index Future HELD ‘til Settlement Long Gain = Short Loss = K$Multiplier (SS – F0)

��Main Difference between a Stock Index Futures and Forward Contract is that in the Futures Market, a settlement is made daily via the marked-to-market method, whereas a Forward contract is settled only once

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b) Treasury Bond Futures Contracts ��Treasury Bond Futures Contracts SETTLE with PHYSICAL DELIVERY ��To satisfy the settlement requirements, the holder of the short must deliver

100 treasury bonds (100,000 par value) that mature in (or at least cannot be Called for) at least 15 years

��Thus, there can be SEVERAL Deliverable bonds, each with very different characteristics, thus making pricing more difficult In FORWARD CONTRACTS Invoice PriceDB = (F0)(Conversion FactorDB) + Accrued InterestDB The Conversion Factor is a SCALER that makes the adjustment for the difference in price between whatever the actual treasury bond that is delivered for settlement by the holder of the short happens to be (the delivered bond) and a GENERIC 20-year, 8% coupon bond that is used as the basis for quoting prices in the Treasury Bond Futures Market For Example: The Contract Price (F0) for a Treasury Bond Forward Contract for Settlement in 1 month is 95.5. What is the Invoice Price that must be PAID by the Party who BUYS this Contract and holds it until expiration, if the bond that is delivered at settlement is 7 5/8% coupon, 22-year bond with a conversion factor of .966? Answer: Invoice PriceDB = (F0)(Conversion FactorDB) + Accrued InterestDB = (95.5)(.966) + Accrued InterestDB

= 92.253 + Accrued InterestDB ��Thus, the settlement occurs with the SELLER of the Contract delivering 100

deliverable bonds and the BUYER Paying the INVOICE Price applicable to the Deliverable Bonds (for FORWAD Contracts)

��In FUTURES Contracts, the same Delivery is Required at SETTLEMENT (100 Deliverable Bonds that cannot mature or be called for at least 15 years)

��However, the DELIVERED BOND is not Determined until DELIVERY is Made, the marked-to-market daily settlement method is BASED o a CONTRACT SIZE of 100 Bonds ($100,000) par value, generic, 8% coupon, 20-year bonds

��The formulas used to compute the various values are similar to other futures contracts, except for when the contract is HELD until SETTLEMENT and then one must consider the Conversion Factor as it Relates to the MOST DELIVERABLE BOND DAILY SETTLEMENT on Treasury Bond Futures Contracts Long Gain = Short Loss = ($1,000)(Ft – Ft-1) GAIN/LOSS on Treasury BOND Futures K NOT held thru settlement Long Gain = Short Loss = ($1,000)(Ft – F0) GAIN/LOSS on Treasury Bond Futures K HELD thru Settlement Long Gain = Short Loss = ($1,000)(FS – F0)(Conversion FactorDB)

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c) Treasury Bill and Eurodollar (Interest Rate) Forward & Futures Contracts ��TREASURY BILL Forward and Futures Contracts require the holder of the

short position to deliver $1,000,000 FACE VALUE (the contract size) of 3-month treasury bills on Settlement Day. These are quoted differently from Bond prices, in the IMM Index, the T-Bill Future is quoted based on the following relationship F0 = 100(1 – tmf90 discount) Where tmf90 discount = the ANNUALIZED percentage discount from par at which 3 month (90 day) treasury bills are EXPECTED to sell at the time the contract settles Note, this quote is NOT a % of par (like T Bond Futures) It looks ahead to 3 months POST Settlement

��The INVOICE PRICE, measured as a Percentage of $1,000,000 is determined Invoice Price (%) = 100 – (100 – F0)(90/360) For Example: Suppose a Treasury Bill Futures Contract is QUOTED at 95. What is the Invoice Price that the Buyer of the Contract is Agreeing to Pay for the DELIVERED 90-day Treasury Bills at the Time of Settlement Answer: Unlike the T-Bond contract, a futures contract price for a T-Bill does not mean that, when the $1,000,000 face value of 3-month T-bills are delivered to settle the contract, the invoice price to be paid by the holder is to be $950,000. Rather, in the T-Bill market, a quote of 95 means that the price to be paid on settlement day will correspond to an ANNUALIZED DISCOUNT of 5% (100-95 = 5) at THE TIME OF SETTLEMENT Invoice Price = 100 – (100 –95)(90/360) = 98.75% � will pay $987,500 at settlement

��For Every FULL POINT Change in a T-Bill future price corresponds to a $2,500 change in the Dollar Value of the Contract’s Invoice Price DAILY SETTLEMENTS on T-BILL FUTURES Long Gain = Short Loss = ($2,500)(Ft – Ft-1) GAIN/LOSS on T-BILL FUTURE NOT HELD ‘til SETTLEMENT Long Gain = Short Loss = ($2,500)(Ft – F0) GAIN/LOSS on T-BILL FUTURE HELD ‘til SETTLEMENT Long Gain = Short Loss = ($2,500)(SS – F0) Where SS is the spot IMM index that prevails at the time of settlement (100 – annualized discount yield of a 3-month treasury bill that prevails in the market at time of settlement)

��Hence, one may speculate on expected short-term interest rate movements by buying/selling T-Bill Futures For Example: assuming a normal positively sloped curve (tm) rtm tmf90 discount (F0) ($) K Maturity Spot tm Forward Discount K Futures Price K Invoice Price Discount on 90-day Bills 90 days 5.0% 5.87% 94.13 $985,325 180 5.4 6.17 93.83 $984,575 270 5.6 6.68 93.32 $983,300 360 5.8 NOTE: To Derive the tmf90 discount one must BOOT STRAP

� (1 – rtm + 90 Discount)[(tm+90)/360] � tmf90 discount = �1 - --------------------------------------- � (360/90)

� (1 – rtm discount) (tm / 360) � Hence, 90 dayf90 discount = {1 – [1-.054(90+90/360)]/[1-(.05)(90/360)]} (360/90) = .05873

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��CHARACTERISTICS o When the Forward Discount Curve for T-Bills is POSITIVELY

SLOPED, the Futures Contracts Prices (and Contract Invoice Prices) DECLINE as Maturity Increases

o When the Forward Discount Curve for T-Bills is NEGATIVELY SLOPED (INVERTED), the Futures Contracts Prices (and Contract Invoice Prices) INCREASE as Maturity Increases

o Also, the Futures Contract Price is BASED on what the MARKET EXPECTS the 90-day T-Bill’s Percentage from Par will be AT THE TIME THE CONTRACT EXPIRES

For Example: assuming change from the data above to (tm) rtm tmf90 discount (F0) ($) K Maturity Spot tm Forward Discount K Futures Price K Invoice Price Discount on 90-day Bills 90 days 5.1% 6.18% 93.82 $984,550 180 5.6 6.69 93.31 $983,275 270 5.9 7.01 92.99 $982,475 360 6.1 Speculators who purchased the 180-day contract earlier would suffer a .52 point loss. When marked-to-market Long Gain = Short Loss = ($2,500)(Ft – Ft-1) = (2,500)(93.31-93.83) = -1,300

��When the Market’s Expectation of what the 90 Day Treasury Bill’s Annualized discount from par will be, at the time the contract expires, INCREASES, the Price of the FUTURES FALLS, and those who are LONG Suffer a LOSS.

��When the Market’s Expectation of what the 90 Day Treasury Bill’s Annualized discount from par will be, at the time the contract expires, DECREASES, the Price of the FUTURES RISES, and those who are Long Realize a GAIN

��Thus, SPECULATORS who Believe that 90-Day TREASURY Bill Discounts will rise LESS than the Market Expects should BUY T-BILL FUTURES, and those who believe it will rise MORE than the Market expects should SELL T-BILL FUTURES

��When the 90-day T-Bill Discount Rate Rises or Falls in accordance with the market expectations, NO GAIN or LOSS will Accrue to participants in the futures market (only can win if the rate changes are different from what the market expects, and you’re on the right side – thus, even if you are correct in assuming which way rates will move, if you’re prediction is the same as the market’s, you won’t win)

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For Example: Consider the following Market (tm) rtm tmf90 discount (F0) ($) K Maturity Spot tm Forward Discount K Futures Price K Invoice Price Discount on 90-day Bills 90 days 5.0% 5.87% 94.13 $985,325 180 5.4 6.17 93.83 $984,575 270 5.6 6.68 93.32 $983,300 360 5.8 the current 90-day T-bill discount rate from par is 5.0%. IF the Speculator believes that the Discount rate on 3-month T-bills will RISE over the next 6 months. To profit from this, the Speculator decides to SELL a 180-day T-Bill Futures Contract SHORT at 93.83. What would his Profit/Loss be if the 90-day T-bill’s discount rate in 180 days rises from the Current 5.0% to

a.) 6.17% b.) 5.50% c.) 6.50%

a) When the 90-day T-Bill’s discount rate from par is 6.17 on Settlement Day, the Gain/Loss would be FS = 100 (1 – 0f90 discount) = 100(1-.0617) = 93.83 Long Gain = Short Loss =(2,500)(FS – F0) = (2,500)(93.83 – 93.83) = 0 Since this is exactly what the market expected, the Speculator made nothing

b) When the 90-day T-Bill’s discount rate from par is 5.50% on Settlement Day, then the Gain/Loss would be FS = 100 (1 - 0f90 discount) = 100(1-.055) = 94.5 Long Gain = Short Loss = (2500)(FS – F0) = (2500((94.5 – 93.83) = 1,675 Thus, as the Speculator was short, he lost $1,675

c) When the 90-day T-Bill’s Discount rate from par is 6.50% on Settlement Day, then the Gain/Loss would be FS = 100 (1-0f90 discount) = 100(1-.065) = 93.5 Long Gain = Short Loss = (2500)(FS – F0) = (2500)(93.5 – 93.83) = -825 Thus, the Speculator, being short, gained $825

��EURODOLLAR Deposit Futures are similar to Treasury Bill Futures, but they Differ in a Few Key Respects

1) The Contract Price (F0) of the Eurodollar Contract is based on the 3-month LIBOR rate that is expected tm days forward when the contract matures, rather than on the T-Bill discount rate F0 = 100 – tmf90 LIBOR

2) Eurodollar deposits are priced like CDs, i.e., they are sold at PAR and carry a COUPON interest rate, rather than being sold at a Discount PEurodollar = $1[1+c(tm/360)] / [1+rt LIBOR(tm/360)]

3) The LIBOR rate is an ADD-ON Rate that is determined for a Discounted Money Market Instrument as follows Add-on Yield = (100-P) / P For a Coupon-paying money market instrument it is Add-on Yield = c / P This is different from the discount from par for a money market instrument Discount rate (r) = [(100-P)/100] [360/tm]

4) The Eurodollar Contract, whose K Size is $1,000,000, settles in CASH, rather than with the delivery of an actual 90-day Eurodollar Bank Deposit itself. The Invoice Price is determined using the same formulas used for T-Bill Contracts DAILY SETTLEMENTS on Eurodollar FUTURES Long Gain = Short Loss = (2500)(Ft – Ft-1) GAIN/LOSS on Eurodollar Future NOT held ‘til Settlement

Long Gain = Short Loss = (2500)(Ft – F0) GAIN/LOSS on Eurodollar Future HELD ‘til Settlement Long Gain = Short Loss = (2500)(SS – F0)

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For Example: Consider the Yield Curve for the Short-Term Eurodollar Maturities indicated below (tm) Spot Libor tm Forward Rate (F0) Invoice Price (rtm) on 90 Day LIBOR tmf90 LIBOR 90 5.0% 5.73% (bootstrap) 94.27 985,675 180 5.4 5.84% 94.16 985,400 270 5.6 6.14% 93.86 984,650 360 5.8 NOTE: �1 + (rtm + 90 LIBOR)[(tm+90)/360] � tmf90 LIBOR = �------------------------------------- - 1 � [90/360] � 1 + (rtm LIBOR)(tm/360) � Suppose, after one day, the contract prices change to the following: tm) Spot Libor tm Forward Rate (F0) Invoice Price (rtm) on 90 Day LIBOR tmf90 LIBOR 90 5.1% 6.02% (bootstrap) 93.98 984,950 180 5.6 6.32% 93.68 984,200 270 5.9 6.42% 93.58 983,950 360 6.1 Thus, Speculators who bought the 180-day contract at 94.16 would lose $1,200 Long Gain = Short Loss = (2500)(Ft - Ft-1) = (2500)(93.68-94.16) = -1,200 d) Currency Forward and Futures Contracts

��Most currency trading takes place between international banks. When making forward commitments in currencies, these banks prefer to use forward contracts that can be specifically designed to meet their particular needs and do not require daily settlements. The futures market for currencies have been developed for the convenience of non-bank customers in hedging currency risks

��Since currencies are commodities, they use the same general formulas: DAILY SETTLEMENT on CURRENCY FUTURES

Long Gain = Short Loss = K (x/y) (Ft (x/y) – Ft-1(x/y)) GAIN/LOSS on CURRENCY FUTURES NOT Held ‘til Settlement Long Gain = Short Loss = K(x/y) (Ft (x/y) – F0 (x/y)) GAIN/LOSS on CURRENCY FUTURES HELD ‘til Settlement Long Gain = Short Loss = K(x/y) (SS(x/y) – F0 (x/y))

2. Hedging with Forward and Futures Contracts a) How to Hedge

��Employing a Strategy designed to reduce the PRICE RISK inherent in Owning a Commodity is called HEDGING

��A SHORT HEDGE is Accomplished by owning a commodity and simultaneously taking a SHORT Position in that commodity’s futures

��A LONG HEDGE is Accomplished by BUYING LONG a FUTURES Contract in a Commodity that one expects to PURCHASE at a FUTURE TIME For Example: for a SHORT HEDGE, consider a JEWELER who Holds an INVENTORY of 2,345 oz of Gold which is spot trading at $400/oz. To reduce exposure to price fluctuations, the Jeweler decides to SELL 20, 6-month gold futures contracts (Ksize = 100 oz.) The Gold Futures contracts are trading at $412/oz. Four Months Later, the jeweler decides to lift the hedge by closing out the futures position. To do this, he would buy back the 20 futures contracts which he sold short. When this is done, the Spot price of Gold is $395/oz. And the futures price is $399.10 (for the contract which he bought and is then to settle in 2 months). Gain/Loss on Long Position in Gold (Spot-held) � (2345)(395-400) -$11,725 Gain/Loss on Short Sale of Futures � (20)(100)(412-399.10) $25,800 Net Gain/Loss on Hedge $14,075 Initial Value of Gold (2345*400) $938,000 Net Worth on Date Hedge Lifted $952,075 Effective Price of Gold Holdings on Date Hedge Lifted � [952,075/2345] = $406.00/oz.

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��Hedges seldom work out perfectly. While they reduce price risk, they do not eliminate it, especially when the hedge is lifted before the contract settles. Also, the effective price that a hedge produces for a commodity USUALLY does not equal the spot price of the commodity at the time the hedge is initiated

��In PLANNING A HEDGING STRATEGY, Several Decisions must be made 1) DETERMINE what the HEDGING VEHICLE will be. The Most

Appropriate Hedging Vehicle should be a Security Meeting the Following Criteria ��The Price movements in the Hedging Vehicle should CORRELATE

highly with the price movements of the commodity being hedged. Usually, the best hedging vehicle is a forward/futures contract whose underlying asset is the same commodity as the one being hedged

��If there is no forward/futures contract whose underlying asset is the same commodity as the one being hedged, may have to perform a CROSS-HEDGE

��Choose a Futures contract whose CONCTRACT SIZE divides as evenly as possible into the quantity of the commodity being hedged

��Choose a Futures contract that Settles as Close to the HEDGING HORIZON as possible. For on Settlement day, the basis of a forward or futures contract is known to be ZERO, but if the hedge is lifted before that day, basis will exist and that basis is unknown and must be ESTIMATED and therefore, BASIS RISK will exist.

��Choose a Hedging Vehicle that is LIQUID so that it can be closed out whenever it becomes desirable to lift the hedge

2) DETERMINE the proper HEDGE RATIO. 3) DETERMINE the TARGET PRICE that is Likely to be achieved by Using

the Hedge. Some difficulties in determining this involve; ��The price of a Forward/Futures contract will CONVERGE toward the

Spot Price of the underlying commodity on Settlement Day. Thus HEDGES that are HELD until Settlement will LOCK in the Price that is Implied by the contract price, rather than the spot price of the commodity when the hedge is initiated

��When one lifts the hedge BEFORE SETTLEMENT, there will be Basis Risk and one won’t know ahead of time the Target Price (Vtarget) Vtarget (tl < tm) = F0 + (SL – FL) = F0 + BasisL The target price of a straight hedged, measured on the day the hedge is lifted is always equal to the current price of the forward/futures _ the BASIS that will exist on the day the hedge is lifted (which is an unknown). Therefore, BASIS RISK exists

��When one lifts the Hedge ON SETTLEMENT, one knows the BASIS will be Zero, and thus the Value of the contract on settlement is the same as the Price of the Futures on the day the hedge was initiated Vtarget (tl =tm) = F0 + (SS=L – FS=L) = F0 + (SS – SS) = F0

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When Lift on Settlement, the TARGET Value of a straight hedge equals the price at which the futures was sold short to effectuate the hedge

��When the Hedge is Lifted before settlement, the price that will actually be realized by the hedger may be different from that which is implied by the current futures or that which exists in the current spot market. When the basis at the time the hedged is to be lifted is uncertain, the Target Price can ONLY be ESTIMATED VTarget (tl<tm) = S0 + (BasisL – Basis0) = S0 + ∆Basis Hedging a commodity with a futures contract does NOT guarantee that the hedger will receive the Spot price of the commodity that exists when the hedge is initiated. Rather, the target price of a straight-hedge equals the price of the Commodity on the day the hedge is initiated, plus the amount by which the basis of the contract will Change between the time the hedge is initiated and when it is lifted. Hence, lifting a hedge before settlement is RISKY because in order to determine the target value of the hedge, the hedger must then speculate on how much the basis will change during the time the hedge is in place. BUT, when one lifts the Hedge on SETTLEMENT Day, one guarantees a Vtarget = F0

4) DETERMINE the Probable EFFECTIVENESS of the Hedge (i.e., how likely the target price will be achieved)

b) Determining the Number of Contracts Required to Perform a Total Hedge

��The Typical SHORT Hedge is a PORTFOLIO comprised of a Long Position in the Commodity plus a SHORT position in the Futures contract (hedging vehicle)

��If one wants a TOTAL Hedge, the Value of the Portfolio does NOT Change between the time the contract is initiated through the time the Hedge is Lifted. Hence: ∆Vcommodity +∆Vfutures = 0

��To determine the Number of contracts needed to complete a Hedge, one must solve the following NF = - HR[Qcommodity to be hedged / KSize]

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c) Determining the Hedge Ratio ��Hedge ratios can be formulated in several different ways, depending

on the information available The THEORETICAL Hedge Ratio on a FORWARD K to be lifted at time tL is HRforward = [∆Pcommodity/∆Punderlying commodity] / [1+rF((tm-tL)/360)] = [Price β / [1+rF((tm-tL)/360)] Since FUTURES Contracts are marked to market daily, the Theoretical Hedge Ratio for a futures is the same as above, except tL = 0 (meaning the hedge is reformulated every day) HRFutures = [∆Pcommodity/∆Punderlying commodity] / [1 + (rFtm/360)] = [Price β / [1+ (rFtm/360)] ] Eventually, ∆F = ∆Punderlying commodity [ 1 + (rFtm/360)] And, the Theoretical Hedge will be HRFutures = [∆Pcommodity/∆F] = βF So, whereas hedging with a forward is a set it & forget it strategy, hedging with a futures is not, as it must be rebalanced every day. With the Tailing of the Hedge, must constantly adjust the Hedge Ratio to adjust Nf so that one has a proper hedge.

��Note, the above exercise (from Appendix DF-5) is the basis. A Simpler method may be to use the OPTIMUM or MINIMUM VARIANCE HEDGE RATIO

HR = [COVP(C,F) / σ2F] = [ρP(C,F) σP(c) / σF] = βF

��This equation implies that the futures contract that should be used to hedge the price risk in a commodity is one whose price movements are HIGHLY CORRELATED with those of the commodity being hedged. It also implies that the effectiveness of the hedge can be measured by the R2 of the regression relating ∆PC to ∆F and that (1-R2) is a Measure of the EXPECTED BASIS RISK in the hedge

For Example: A Jeweler has an inventory of 2,000 oz. of gold at spot price of $290/oz. He wants to hedge this position against price risk for 4 months. A 6-month gold futures contract is trading at 297.50/oz. The jeweler performs a regression relating changes in the price of gold to changes in the price of the 6-month gold futures contract. Result: ∆PGold = .05 + 1.03∆F � (R2 = .965)

a.) How many gold futures contracts should the jeweler sell in order to obtain a minimum variance hedge of the 2,000 gold inventory?

Answer: NF = - HR (Qcommodity/ KSize) NF = - HR (2,000 / 100) NF = - HR (20) HR = βF = 1.03 NF = - (1.03)(20) = -20.6 The Jeweler should sell 21 contracts to perform the hedge

b.) If the Spot Price of Gold is $280/oz. and the futures contract is trading at 285.5 when the hedge is lifted, 4 months hence, what effective price will the jeweler obtain for the gold?

Initial position in Gold (2,000 @ $290) $580,000 Gain on Long in Gold (21)(100)(280-290) (20,000) Gain on Short in Gold Future (2000)(297.50-285.50) 25,200 $585,200 Effective Price (585,200/2000) $ 292.60

c.) If the spot price of gold is $240/oz. and the futures contract is trading at 244.74 when the hedge is lifted 4 months hence, what effective price will the jeweler obtain for the gold?

Initial Position in Gold (2,000 @ 290) $580,000 Gain on Long in Gold (2000)(240-290) (100,000) Gain on Short in Gold Futures (21)(100)(297.5-244.75) 110,775 $590,775 Effective Price (590,775/2000) $ 295.39

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��IMPLICATIONS to be DRAWN from the Example o The TARGET PRICE produced by a hedged does NOT equal

the Spot Price of the underlying commodity at the time the hedge was initiated; rather, the target price of a hedge is a Price that is IMPLIED by the price of the futures contract at the time the hedge is initiated

o The MINIMUM VARIANCE HEDGE is NOT Perfect: the Target price produced by the hedge is NOT the SAME for all possible spot prices at the time the hedge is lifted: REASONS�

��The Number of Contracts Sold Short must be ROUNDED to the Nearest Whole Number, negating the Ability to produce a THEORETICALY Perfect Hedge

��If the Hedge is NOT lifted at the time of Settlement, the Size of the BASIS is NOT predictable. Therefore, there is BASIS RISK

��The EX ANTE (expected) Futures β that was used as a hedge ratio may not equal the EX POST (Actual) futures β that existed during the time the hedge was in place. In theory the Ex Post Futures β should be used, but this is impossible without PERFECT Foresight by the Analyst

��The Hedge was NOT REBALANCED during the Hedging Horizon to account for changes in the number of contracts that should have been should have been sold short as the futures β changed during the life of the hedge. This is a Form of TAILING THE HEDGE

o Despite these shortcomings, the hedge worked fairly well and the effective price did not fluctuate nearly as much as the spot price of the commodity itself during the hedging period. Thus, one can effectively stabilize the effective price, though not perfectly

d) Cross-Hedging ��When the Commodity underlying the Hedging Vehicle is different from the

Commodity being hedged, the resulting hedge is a Cross-Hedge. If it can be avoided, cross-hedging should not be used because it is so risky For Example: Assume the jeweler has 2,000 oz. of gold at $290/oz. Suppose there is NO Gold Forward/Futures contract. But, there is a Silver Futures Contract with a Contract Size of 5000 oz. Therefore, the jeweler decides to sell 6-month Silver Futures Contracts short to hedge his gold inventory for 4 months. Assuming the Spot price of Silver is $3.65/oz. and the Futures Contract for silver is trading at $3.80/oz. The Regression Analysis of Gold on Silver Futures reveals: ∆PGold = $90 + 80∆FSilver How many 6-month Silver Futures contracts with a KSize of 5,000 oz. should the jeweler enter into as the holder of the short position to effectuate this hedge? When the hedge is lifted in 4 months and the spot price of gold is $240/oz. and the silver futures contract is at $3.15/oz. what will be the jeweler’s effective price of gold at the end of the Hedging Horizon? NF = - HR (QCommodity to be hedged / KSize) = - HR (2000/5000) HR = βF = 80 NF = -80(2000/5000) = -32 The Jeweler should sell 32 Silver Futures Contracts Short

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The Effective Price of Gold when using this hedge will be: Initial Position in Gold (2000 @ $290/oz.) $580,000 Gain on Long Gold (2000)(240 – 290) (100,000) Gain on Short Silver Futures (32)(5000)(3.8-3.15) 104,000 584,000 Effective Price of Gold (584,000/2000) 292.00

��Cross-hedges are more risky because there are VARIOUS Sources of ERROR o The Relationship between the Commodity to be Hedged and the

Commodity used in the Hedging Vehicle may be MISSPECIFIED or ERRATIC over the Time Horizon of the Hedge

o There can be SUBSTANTIAL BASIS RISK ��Since Basis Risk is Measured by (1-ρ2

(C,X)), if the prices of the Commodity to be Hedged (C) and the Commodity used in the Hedging Vehicle (X) are NOT highly correlated, Basis Risk can be Substantial

��Unless there is a High R2 between FX and SC, there will be sufficient Basis Risk to mean the Hedge should be ineffective

e) Hedging Equity Portfolios with Stock Index Futures Contracts ��Stock Index Futures were developed in order to have an effective way of

hedging equity portfolios against Adverse Market Fluctuations ��Usually, Short Hedges are employed, wherein a long position in an equity

portfolio is hedged against a suspected downturn in the general stock market. This is usually cheaper, when the downturn is expected to be temporary, that selling shares outright and repurchasing them later.

��The General Principles of Commodity hedging can be applied to Stock Index Futures Hedging, although some modifications are required, as Stock Index Futures are Settled in Cash (not in delivery of the commodity) NF = [-βP / [1+(rFtm/360)]] * [Value of Portfolio / ($multiplier*Spot Value of Index)] NF = (-βP)*[Value of Portfolio / ($multiplier*F0)] For Example: A Pension Fund has $400,000,000 invested in a portfolio of common stocks with a β of .92 relative to the S&P 500. The S&P Index is at 968.25. How many 6-month futures contracts trading at 992.45 are needed to hedged this Portfolio? If one month later the hedge is lifted when the S&P is at 952.45 and the S&P Futures is at 972.30, what is the overall gain/loss from the hedged portfolio compared to the gain/loss that would have occurred without the Hedge? Answer NF = (-βP)(VP/($multiplierF0) = (-.92)[400,000,000/(250)(992.45)] = -1,483 contracts Ending Value of Portfolio with the Hedge Initial Value $400,000,000 Gain from Long (.92)(400,000,000)[(952.45-968.25)/968.25] (6,005,061) Gain from Short Futures (250)(1483)(992.45-972.30) 7,470,613 Value of Portfolio when hedge is lifted $401,465,552 Ending Value of Portfolio without the Hedge Initial Value $400,000,000 Gain from Long (.92)(400,000,000)[(952.45-968.25)/968.25] (6,005,061) Value of Portfolio without the Hedge $393,994,939

��For These SHORT PORTFOLIO HEDGES to be Effective, 2 Conditions MUST Prevail

1.) The PORTFOLIO must perform in line with the market as adjusted by its β

2.) The Stock Index Futures Contract Price Must Move in Tandem with the Spot Price of the Stock Index over the course of the Hedge

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��Also, in theory, the Hedge should be REFORMULATED DAILY because futures contracts require Daily financial settlements. (the Tailing the Hedge Problem). But usually, just use a Set It & Forget It Approach

��Long Hedges can also be accomplished using Stock Index Futures Contracts For Example: The S&P 500 is at 995.50 and a 3-month S&P Futures is at 1007.95. A portfolio manager will be receiving $2,000,000 in dividends in 5 days. Believing the market may rise significantly before those funds can be invested, the manager decides to hedge against a market rise by buying S&P 500 futures contracts. How many contracts should the manager purchase in order to effectuate this long hedge? Answer NF = -βP[VP/($multiplier*F0) = -1[-2,000,000/(250)(1007.95)] = 8 Contracts

��Note, here we assumed βP was 1 because nothing was given in the facts. Do the same on the exam.

��Since the $2,000,000 represents a future outlay, it too is negative in the formula

If the S&P 500 increases from 995.50 to 1015.80 over the next 5 days, and the futures contract increased from 1007.95 to 1028.50, the manager profits Profit from the long Hedge = (8)(250)(1028.50-1007.95) = 41,100 Without the long hedge, the manager could buy fewer stocks when the $2,000,000 dividend was received. But with the hedge, the manager can use both the $2,000,000 and the $41,100 profit from the hedge to acquire approximately the same number of shares as he could when the S&P 500 is at 1015.80 as when it was at 995.50 (the market rose by 2% and $41,100 is about 2% of $2,000,000)

��A Refinement o A CRITICAL Assumption made in formulating the hedging

methodology is that the VOLATILITY of the Futures Contract is the same as the Volatility of the Stock Index that underlies it. Often, Stock Index Futures are MORE VOLATILE than the underlying index. In that case, the β of the portfolio being hedged should NOT be used as the hedge ratio (as regressed against the index). Instead, the hedge ratio should be determined by regressing the returns on the portfolio against the returns on the futures contract that will be used as the hedging vehicle.

o Using this Model NF = [-βRM/(1+(rftm/360))] * [VP/($multiplier*IndexSpot)] NF = (-βRM)[VP/($multiplier*F0)]

o Usually, βRM (regressing the Portfolio on the Hedging Vehicle) is approximately equal to βP (regressing the Portfolio on to the Stock Market Index). But, there are some technical differences between the 2

��βP is the CAPM β; however, the true market portfolio is unobservable and the S&P 500 is only a Proxy for it

��βRM is the β that is needed to perform a MINIMUM-VARIANCE hedge because it is a regression of the portfolio against the hedging vehicle. βP is NOT the minimum variance hedge ratio because it is not a regression against the hedging vehicle.

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3. Other Applications of Futures Contracts a) Creating a Synthetic Treasury Bill from an Equity Portfolio Using Stock

Index Futures Contracts ��A TOTALLY Hedged asset, in theory, will generate a return equal to the

COST-of-CARRY imbedded in the contract. Thus, if a large, well-diversified portfolio that can be expected to behave relative to the stock market index as if it had only systematic risk is totally hedged, the hedged portfolio should generate a return that is APPROXIMATELY equal to the Risk Free Rate. As T-Bills are good proxies for the risk-free asset, a totally hedged portfolio should produce a constant (risk-free) return similar to that of a T-Bill.

For Example: A portfolio manager has a $50,000,000 well-diversified portfolio whose β is 1.1. The manager expects the stock market to decline over the next six months. Currently, the S&P 500 is at 975.61 and the 6-month S&P 500 Futures is at 1,000.00. The Risk-free rate is 5%. To protect against the expected decline in the stock market, the manager could sell the portfolio with the expectation of repurchasing the stocks later. But this is too expensive. Rather, he decides to totally hedge the Portfolio for 6 months. NF = (-HR)[VP / ($multiplier*S0)] NF = - [βP / (1+(rFtm/360)]*[VP/$multiplierS0] NF = -βP[VP/($multiplierF0)] � NOTE: this is the most often used formulation NF = (-1.1)[50,000,000/(250)(1,000)] = -220 Over the 6-month life of the hedge, based on the following values of the S&P indexes, the following results will accrue

850.00 950.00 1050.00 Ending Portfolio Value (50,000,000) + (50,000,000)(1.1)(St-S0/S0) 42,918,738 48,556,237 54,193735 Gain on Futures [(220)(250)(1000-St)] 8,250,000 2,750,000 (2,750,000) Ending Value of Hedge 51,168,738 51,306,237 51,443,735

��NOTE: The Ending Value of the Hedge is Approximately the same regardless of the ending value of the manager’s portfolio (the slippage is due to the Tailing the Hedge problem)

��As all the ending values of the hedge are greater than $50,000,000, this increase in value is APPROXIMATELY the RISK FREE RETURN

��Thus, by hedging the equity portfolio, the manager has effectively turned it into a money market instrument for the life of the hedge

b) Creating Synthetic Equity Portfolios with Stock Index Forward & Futures Contracts

��Having a Long (or Short) position in a stock index forward or futures contract is equivalent to owning or (selling short) the stock market index itself.

��If the contract position is fully covered with funds invested at (or borrowed at, if a short position is being replicated) the risk-free rate, this is like having a 100% position in a stock market index fun.

��Not having the forward or futures position fully covered is like having a leveraged long, or short, position in an index fund

��To Synthetically create a portfolio that will replicate the performance of the S&P 500: 1. Buy that number of S&P 500 Futures contracts that represents the

current exposure to equities that is desired. One may use the basic hedging relationship for stock index futures contracts can be used to determine the number of contracts that is necessary to do this NF = [-βP/(1+(rFtm/360))]*[VP/($multiplier*S0)] NF = -βP[VP/($multiplier*F0)] If desire to replicate the Stock Index, βp = 1

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2. If the S&P 500 index futures contracts are purchased on MARGIN, one would be in a LEVERAGED position in the S&P 500 index. To have an UNLEVERAGED synthetic position in the stock market index, it is necessary to have sufficient cash on hand to settle the contracts at the contracted price (and to make required mark-to-market daily settlement) For Example: The S&P 500 is at 1,200. The S&P 500 6-month futures is at 1,042.95. The Risk Free Rate is 4.5%. If a portfolio manager wants to create a synthetic $102,000,000 investment in the S&P 500 index, the following steps can be taken.

a. Buy Enough S&P 500 Forward Contracts to represent a $102,000,000 exposure to stocks NF = -βP [VP/($multiplier*S0)] NF = -1{-102,000,000/(250)(1020)} = 400 contracts

b. Invest Cash is an amount equal to the present value of the amount that will have to be spent to buy the stocks via the futures contract at settlement in 6 months Amt. Required to Settle = (400 contracts)(250)(1042.95) = 104,295,000 Amt. Of Cash Invested Now to have sum at Settlement = (104,295,000)/(1+.5(.045)) = 102,000,000

c. The 2 Steps shown above will produce a portfolio whose performance will replicate that of the S&P 500 index itself, excluding the effects of dividends. If dividends are to be taken into consideration, so as to produce a portfolio that will replicate the total return of the S&P 500 index, invest additional cash in an amount to equal the present value of the expected dividends to be received on a $102,000,000 investment in the S&P 500 index over the next six months

A Combination of 400 S&P 500 Index forward contracts and a money market investment of $102,000,000 in cash will produce a portfolio that will produce the same return as a $102,000,000 equity position in the S&P 500 index (excluding dividends) over the life of the contracts One Difficulty with using this approach is that a synthetic portfolio will not behave precisely like a real portfolio of a given size unless the portfolio value, divided by the equity exposure represented by one stock index futures contract, generates a whole number of contracts. However, forward contracts can be customized to give any exposure to the index desired by a portfolio manager

4. “Minimizing Cash Drag with S&P 500 Index Tools” by Hill & Cheong

a) Holding EXCESS CASH reserves in an equity portfolio during periods of rising stock prices can ADVERSELY Impact portfolio performance.

b) If stocks are returning 12%, and the money market return is 5%, a 6% Cash Reserve Balance can reduce performance by 42 basis points Adverse ImpactPortfolio = (requity – rmoney market)(%cash reserve balance)

c) A Portfolio Manager can replicate the return on any stock market index for which there is a stock index future contract by simply buying the futures contract and holding cash in reserve

d) If the manager wants to replicate the return on the S&P 500 Index, there is an Alternative: BUY 500 DEPOSITORY RECEIPTS � these are shares in a UNIT INVESTMENT TRUST that owns the stocks that comprise the S&P 500 Index

e) SPDRs are listed on the ASE where they trade just like any other stock SPDRs v. Replicating the S&P 500 Index with S&P 500 Futures Contracts

f) Whether it is more advantageous to use SPDRs or Futures depends on the facts and circumstances. One must look at the strengths & weaknesses of each

i. Futures Contracts are MORE LIQUID than SPDRs ii. Transactions involving Futures may be Prohibited for some Portfolios

by Policy or Legal Restrictions. SPDRs are permitted in those cases because they are unit trust shares that represent ownership of securities. (some restrictions on SPDRs, like can’t have more than 5%

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of assets in SPDRs or can’t own more than 5% of any individual SPDR)

iii. Futures contracts EXPIRE, SPDRs do NOT. Ergo, no need to ROLL OVER the SPDR (save some expenses). But cost per transaction is lower for futures than SPDRs. (1-way trading costs for futures are 1.1 basis point, for SPDR, it’s 7-10 basis points). Thus, unless there are many rollovers, Futures are cheaper

iv. A MANAGEMENT FEE is charged on SPDRs (with no management fee for Futures). Usually, it’s 18.45 basis points per year

v. SPDRs may pay quarterly dividends, which are not automatically re-invested, and in a rising market, this can create CASH DRAG

vi. Historically, SPDRs have UNDERPERFORMED the S&P 500 Index more than FUTURES

For Example: Compute the Effect of CASH DRAG, for a portfolio that is 95% invested in the equity market, earning a return equal to the S&P 500 index, and 5% cash, based on the following facts: S&P Return 12.00% Cash Return 4.50% Futures Return 11.85% SPDR Return 11.67% Answer Portfolio Return with Cash = (.95)(.12)+(.05)(.045) = 11.6250% Portfolio using Futures = (.95)(.12)+(.05)(.1185) = 11.9925% Portfolio using SPDRs = (.95)(.12)+(.05)(.1167) = 11.9835%

5. Adjusting the Risk Level (ββββ) of Equity Portfolios with Stock Index Forward & Futures

Contracts ��The β of a Portfolio is the Weighted Average of the β of the Individual Assets that

comprise that portfolio, with the weights being the percentage of the total portfolio that is invested in each asset. VPβ*

P = VSβS + VFβF Where VP is the Value of the Portfolio whose β is to be changed VS is the Value of the Owned EQUITY portfolio

VF is the Value of the Current Equity Exposure represented by the Futures Contracts needed to produce the desired portfolio β

β*P is the DESIRED β of the Portfolio

βS is the β of the Equity Portfolio Currently held βF is the β of the index that underlies the Stock index futures contract

��The Number of Futures Contracts Needed to Produce the Desired Portfolio β is NF (to ∆β portfolio) = [VF/(1+(rFtm/360))] / [$multiplierS0 of Market Index] NF (to ∆β portfolio) = [VF / ($multiplierF0 of Market Index Futures)] For Example: An investment manager has a $16,000,000 equity portfolio with a β of .95 (relative to S&P 500) and $800,000 in cash. The manager wishes to increase the β of the Total Portfolio to 1.10. The S&P 500 is at 963.15 and the S&P 500 Futures for 6-month is at 987.23. How can the money manager increase the β of the Portfolio without actually selling some stocks and buying others when the risk free rate is 5% Answer: The manager can use S&P 500 futures contracts to change the beta of the portfolio VPβ*

P = VSβS + VFβF = (16,800,000)(1.10) = (16,000,000)(.95) + VF(1) � assume βF is 1 VF = 3,280,000 NF (to ∆β portfolio) = [VF/(1+rFtm/360))] / [$multiplierS0 of Market Index] = [3,280,000/( 1 + ((.05)(180)/360))] / (250)(963.15) = 13.29 NF (to ∆ β Portfolio)= [VF/($multiplierF0)] = [3.280,000/(250)(987.23)] = 13.29 Thus, the Manager should take a LONG Position in 13 (maybe 14) Contracts

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For Example: If the manager wanted to reduce the Portfolio β to .75? Answer: VPβ*

P = VSβS + VFβF = (16,800,000)(.75) = (16,000,000)(.95) + VF(1) VF = -2,600,000 NF (to ∆β portfolio) = [VF/(1+rFtm/360))] / [$multiplierS0 of Market Index] = [-2,600,000/(1+(.05)(180)/360))] / (250)(963.15) = -10.53 NF (to ∆ β Portfolio)= [VF/($multiplierF0)] = [-2,600,000 / (250)(987.23)] = -10.53 Thus, the Manager should SHORT 10 or 11 Contracts

��To INCREASE the β of an EQUITY Portfolio, BUY (LONG) FUTURES ��To DECREASE the β of an EQUITY Portfolio, SELL (SHORT) FUTURES ��NOTE: A Perfectly Hedged Portfolio has a β = ZERO. Thus, do the same exercise

and try to decrease the desired Portfolio β to 0 ��Will wind up with the same result as one would if using the Hedging Formula

NF = (-βS)[VPortfolio to be hedged/($multiplierF0)] 6. Using Stock Index Forward and Futures Contracts to Change the Weightings of

Equity Portfolios (while keeping the Equity Portfolio in tact) For Example: Suppose a portfolio manager has a $50,000,000 portfolio that is invested 70% in US equities, and 30% in Japanese equities. The manager wishes to change the mix to 60% US and 40% Japanese. The S&P 500 futures is at 975.35 and the Nikkei 225 futures is at 18,300. Futures can be used to alter this exposure. To accomplish the change in weightings, the manager must reduce the US Exposure by $5,000,000 and increase the Japanese exposure by $5,000,000. NF = (VP) / ($multiplier F0) NF (US) = (-5,000,000) / (250)/(975.35) = -20.51 (SELL SHORT 21 Futures) NF (Jap) = (5,000,000) / (5)(18,300) = 54.64 (BUY LONG 55 Futures)

7. Hedging Bond Portfolios with Treasury Bond Forward and Futures Contracts ��Treasury Bond futures contracts can be used to lock in a bond portfolio’s price (The

TARGET PRICE of the Hedge) and, thus, the YIELD associated with that price (TARGET RATE)

��A SHORT Hedge, holding a short position in the Treasury Bond futures contracts, is used to offset the risk of holding a long position in bonds, or the risk that interest rates may rise if one must borrow funds in the future

��A LONG Hedge is the holding of a long position in treasury bond futures, it is used to offset the risk associated by having a net short position in bonds or to offset the risk that interest rates might fall before funds that are to be invested in the future can be put in the bond market

o For Example: A bond dealer with an inventory of 100,000 bonds ($100,000,000 par value) when interest rates are 8%. The dealer does not want to speculate, but rather wants to earn the spread between the bid and ask. Thus, the Dealer hedges any LONG bond position held in inventory by shorting Treasury bond futures contracts against the long position.

o For Example: A money manager has received $10,000,000 to invest in bonds. It will take a few days to invest the funds. If rates fall before the bonds are purchased, there is a lost opportunity. To hedge against this possibility, the manager may buy bond futures contracts (long hedge). If rates fall, the profit on the future contracts will make up for the opportunity profit lost by not having the cash invested. If rates rise, there will be a loss on the futures, but can buy bonds at a lower rate.

o For Example: A real estate developer must borrow $10,000,000 in one year. The developer is afraid that rates might rise in the interim. To hedge against this possibility, the developer could short treasury bond futures contracts (short hedge) If rates do rise, the profit on the futures will reduce the amount of funds that will have to borrow at the higher rate; if rates fall, more money will need to be borrowed, but can do it at a lower rate.

��To totally hedge a Bond Portfolio with Treasury Bond Futures contracts, the same principles are used as when hedging any commodity NF = - HR (QBonds to be Hedged / KSize)

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��The Hedge Ratio can be Formulated in One of FOUR Ways a) The Face Value Naïve (FVN) Model

o The Face Value Naïve model is an EQUAL-DOLLAR model in which the number of bond futures contracts required to effectuate a hedge is SIMPLY the Ratio of the PAR VALUE of the Bonds being hedged to the dollar value of a contract size of the bond futures contract that is being used as the hedging vehicle. Here, the Hedge Ratio is 1. NF (FVN) = -1 (#Bonds to Hedge / 100 Bonds) = -1 (Par Value of Hedged bonds/ $100,000)

o While Simple, this Model has Distinct DISADVANTAGES ��The Model ignores the difference between the MARKET Value of the

bonds being hedged and the MARKET Value of the Futures contracts ��Any differences between the DURATIONS of the bonds to be hedged

and the futures contract that is used as the hedging vehicle are ignored (i.e., this model assumes the futures contract and the bonds being hedged have the same durations)

o These drawbacks render this hedge ratio model INEFFECTIVE most of the time. Generally used only to hedge SHORT-TERM BONDS with T-Bill or Eurodollar futures contracts as such instruments have comparable Durations and comparable prices

b) Market Value Naïve (MVN) Model ��MVN defines hedge ratio as the ratio of the price of the bond being hedged to

the price of the futures contract that is used as the Hedging Vehicle HR(MVN) = (PB/F)

��Therefore, the number of futures contracts required to effectuate this hedge would be NF(MVN) = - (PB/F) (#Bonds Hedged/100 bonds) = -(PB/F)(Par Value Bonds/$100,000)

��Disadvantages o Ignores the difference between the DURATIONS of the bonds to be

hedged and the hedging vehicle o This will ONLY work when the Durations of the 2 are equal

��Thus, it is not normally an appropriate model c) Regression Model

��The Regression model is the Same as the MINIMUM-VARIANCE or Statistical Hedge model in which the Hedge Ratio is defined as HR = (COVB,F/σ2

F) = ∆PB/∆F = βF ��The Futures β is the Regression Coefficient of a regression relating changes in

the price of the bond being hedged to changes in the price of the futures contract that is used as the hedging vehicle

��Thus the number of contracts required to hedge bonds when the hedge ratio is determined with this formula is NF(min-var) = -βF(#Bonds to Hedge / 100 bonds) = -βF (Par Value / 100,000)

��While useful, there are some DISADVANTAGES o Not possible to use this when hedging NEWLY Issued bonds whose

price history is non-existent o Model only IMPLICITY considers differences between the duration of

the bonds to be hedged and the futures contract

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o Model only Implicitly captures the tendency of bond prices to drive toward Par over time

o Model attempts to minimize tracking error, but it does not focus on minimizing changes in the value of the hedged portfolio over time

d) Price Sensitivity Models ��Price Sensitivity Models are used MOST of the TIME for Hedging Bond

Portfolios. There are 2 Models which are equivalent to each other (and 4 mathematically equivalent formulas) HR = (PVBPB/PVBPMD)(Yield β)(Conversion FactorMD) = (PVBPB/PVBPF)(Yield β) HR = (PB/PMD)(D*

B/D*MD)(Yield β)(Conversion FactorMD) = (PB/F)(D*

B/D*MD)(Yield β)

��The Yield β, or Relative Volatility, used in this formulation is the regression coefficient found by regressing yields of the bond being hedged (YB) against yields on the Treasury Bond that underlies the contract used as the hedging vehicle (YMD or YT). YB = α +βYT (or MD) Yield β = (∆YB / ∆YT (or MD)) The Yield β will be 1.0 if the bond that underlies the futures contract is the same type of bond as that which is to be hedged; when the 2 bonds are different (i.e., corporate v treasury) the yield beta may not be 1.0

��All of the parameters used in the Hedge ratio formulations given above should be measured as of the DATE when the HEDGE is LIFTED, because the goal of hedging is to lock in a target price on that date. Since that is not possible, this introduces some RISK into the hedging process

For Example: A Bond dealer holds $10,000,000 face value of corporate bonds, trading at 94.35 whose modified duration is 10.5. The dealer wants to hedge this portfolio with treasury bond futures contracts trading at 108.38. The Most deliverable bond in the treasury market has the following characteristics. Yield Modified Duration Conversion Factor Price 7.50% 10.75 .9550 103.50 A Regression of the yields on corporate bonds like those comprising the portfolio (YC) against the yields on treasuries (YT) produces the following result YC = 1.55 + .95YT --> Thus, the Yield β is .95 How many treasury bond futures contracts should be sold short in order to hedge the bond dealer’s inventory? Answer NF = -HR (Qbonds/Ksize) = -HR (10,000 bonds / 100 bonds/K) HR = (PB/PMD)(D*

B/D*MD)(Yield β)(Conversion FactorMD) = (PB/F0)(D*

B/D*MD)(Yield β)

HR = (94.35/103.50)(10.50/10.75)(.95)(.9550) = (94.35/108.38)(10.50/10.75)(.95) HR = .8078 NF = -HR(Qbonds/Ksize) = -.8038(10,000/100) = -80.78 The Bond Dealer should sell 81 Treasury Bond Futures Contracts Short

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Using Treasury Bond Futures to Change the DURATION of a Bond Portfolio ��The principle that is employed to change the duration of a bond portfolio

using bond futures contracts is that the duration of a portfolio is the weighted average of the duration of the assets in the portfolio.

��Thus, one must go through a 2 step process i. Find D*

PVP = DBVB + DMDVMD where D*

P is the desired Duration of the Bond Portfolio VP is the Value of the Portfolio DB is the Duration of the Bonds held VB is the Value of the Bonds held

DMD is the duration of the Most Deliverable bond underlying a futures contract measured at settlement day VMD is the value of the most deliverable bond

ii. Then, Solve for VMD, and Figure out NF = (VMD/F0Ksize) = [(VMDConversion FactorMD) / (PMD*Ksize)]

For Example: A manager owns a $150,000,000 portfolio consisting of $137,790,000 worth of 8% coupon, 15-year bonds that are trading at 91.86, as well as $12,210,000 in cash. The YTM on the bonds is 9% and their duration is 9.1 years. The investor would like to reduce the duration of the portfolio to 3.0 years, using 9-month Treasury bond futures contracts. The Treasury bonds futures contract settling in 9 months is currently quoted at 94. The Most deliverable Treasury bond is a 12% treasury bond, priced at 126.90 whose duration is 10 years and whose conversion factor is 1.3500. How many contracts are needed to accomplish this? Answer: D*

PVP = DBVB + DMDVMD (3.0)(150,000,000) = (9.1)(137,790,000) + (10.0)(VMD) VMD = -$80,388,900 NF = (VMDCFMD/PMDKsize) = (VMD / F0Ksize) NF = (-80,388,900)(1.35) / (1,269)(100) = (-80,388,900)/(940)(100) = -855 Thus, when 855 Treasury Bond contracts are Sold short, the resulting portfolio will have a duration of 3 years. In effect, this creates a SYNTHETIC 3-YEAR ZERO-COUPON Portfolio

��To INCREASE DURATION, BUY LONG (futures) ��To REDUCE DURATION, SELL SHORT (futures)

8. Hedging with Interest Rate Futures Contracts ��Interest rate futures contracts can be used to hedge investments in short-term money

market securities such as T-bills, CDs, Eurodollar deposits, Banker’s Acceptances, etc.

��As both T-Bill & Eurodollar Futures have a Ksize of $1,000,000, the Basic Hedging formula used to determine the number of contracts required to produce a total hedge is: NF = -HR (Value being Hedged / $1,000,000) --> $1,000,000 is the Par Value of a T-Bill contract

��As usual, determining the Hedge Ratio is the most difficult part ��The Hedge Ratio depends upon which type of hedge is being attempted.

o When an Interest Rate Futures contract is being used to hedge a short-term money market instrument that is Sold at DISCOUNT from PAR (like a T-Bill), the Hedge ratio is construed differently than when it is used to hedge an interest-bearing security such as a CD F0 = 100(1-tmf90) for T-Bill & Eurodollar Gain on Long = ($2,500)(FS-F0)

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Thus, a hedge ratio for a discount security with an interest rate futures contract should be related to:

1. the ratio of the durations of the 2 instruments 2. a Yield β relating how interest rates in the hedged security’s market

change relative to changes in the interest rate that underlies the hedging vehicle HRdiscount instrument = [(tB/360)/.25](Yield β) --> Where tB/360 is essentially duration .25 as it is based on a 90 day bill Usually, Yield β is assumed to be 1.0

For Example: A portfolio manager expects to be investing $40,000,000 in 180-day T-bills. Fearing interest rates will FALL before the funds can be invested, the investor decides to hedge against the interest rate risk by using T-Bill futures. How many contracts are needed to hedge the risk? Answer: This is a LONG HEDGE NF = - HR (Amt. To be hedged / 1,000,000) HRdiscount instrument = [(tB/360)/.25] (Yield β) = [180/360/ .25](1) = 2 NF = -2 (-40,000,000/1,000,000) = 80 contracts The Manager should BUY 80 T-Bill Futures Contracts, representing an underlying contract value that is twice as large as the prospective investment because the bill being bought is twice as volatile as the futures contract per-basis-point change in interest rate expectations. Thus, twice as many contracts with a duration of .25 are needed to produce a dollar gain/loss equal to the increase/decrease in the price of the treasury bill, whose duration is .5 for a given change of interest rates If the instrument being hedged is MONEY MARKET instrument paying a fixed coupon interest, the hedge ratio is more complex

HRcoupon instrument = {[1+C(tB/360)][tB/360] / .25[1+rtB(tB/360)]2} * (Yield β) Where C is the Coupon interest rate on the money market instrument RtB is the Yield on the security being hedged For Example: How many Eurodollar futures contracts are required to hedge a $40,000,000 position in a 5% coupon 35-day money market security that is selling at par? Answer NF = -HR (40,000,000/1,000,000) HRcoupon = {[1+.05(35/360)]/.25[1+.05(35/360)]2}(1.0) = .387 NF = -.387(40,000,000/1,000,000) = -15 contracts Thus, for this SHORT Hedge 15 futures contracts are needed

��Another common use of interest rate futures is to ENABLE Lenders & Borrowers to HEDGE against the possibility that expectations imbedded in the forward interest rate structure will NOT materialize

��NOTE: when an interest rate futures contract is used to Hedge interest rates, the TARGET RATE that is LOCKED IN is NOT the Current Spot rate, but instead it is 90-DAY T-BILL (LIBOR) DISCOUNT that the MARKET EXPECTS to PREVAIL at the TIME the HEDGE is LIFTED (if lift hedge before settlement, will have an unknown BASIS RISK)

��When Lift at SETTLEMENT, F0 = 100 (1-tmf90); tmf90 = (100-F0)/100 When use a T-Bill hedge, lock in implied forward rate (not today’s rate, but rate existing at time of maturity)

��In the Case of a Cross-Hedge, the Target Rate of the Hedge can be ESTIMATED if it is assumed that the spread between the interest rate being hedged and the rate that underlies the hedging vehicle is known and the hedge is lifted on the day the contract settles Target RatetL=tM = [(100-F0)/100] + Spread

��Cross-hedges using interest rate futures contracts will protect against the possibility that market expectations about the 90-day treasury bill discount

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may not be realized, but it will not protect against the possibility that a change may occur in the spread.

For Example: A corporation has $100,000,000 of commercial paper debt outstanding. The commercial paper was issued at 5.6% discount from par, and will be maturing in 90 days. Currently, the 90-day T-Bill rate is 4.8% and 90-day commercial paper can be issued at a discount of 5.8%. The T-Bill futures contract maturing in 3 months is priced at 95. The Corporation’s treasurer has decided to roll over the commercial paper debt when it matures by issuing $100,00,000 of new 90-day commercial paper. But, he is concerned that interest rates might be higher than current levels at that time. Consequently, he decides to hedge the interest rate risk by using T-Bill futures. How should this hedge be construed? What is the target interest rate that the treasurer can ‘lock in’ for an additional 90 days after the current borrowings come due by using the hedge? Answer: Since the treasurer fears that interest rates may be higher than the market currently expects 3 months from now, he should take a position in the futures market that would produce a profit if the rates increase. This would offset the additional borrowing costs. Since interest rate futures fall when rates rise, the hedge requires the treasurer to take a short position in the contracts. NF = -HR(Value to be Hedged/KSize) = -HR(100,000,000/1,000,000) HRdiscount instrument = [(tB/360)/.25](Yield β) = [(90/360)/.25](1.0) = 1.0 NF = -(1)(100,000,000/1,000,000) = - 100 The Treasurer should SELL SHORT 100 contracts The Target rate locked in by using this hedge (assuming the current spread between commercial paper and T-bill discounts remains at 100 basis points) would be tmf90 = (100-F0/100) + Spread = (100-95/100) + (.01) = 6% �� Note: by using this hedge, the treasurer maintains neither the original 5.6%

discount rate nor the 5.8 rate in the current market. The effective rate he locks in will be the 6% rate the market currently expects to prevail in the commercial paper market at the time the futures contracts expire 90 days in the future.

�� INTEREST RATE futures hedging protects the Hedger against UNANTICIPATED CHANGES in interest rates; it does NOT protect against interest rate changes that are already anticipated, and therefore, are already imbedded in the slope of the yield curve and futures contracts prices.

9. Using Interest Rate Futures Contracts to Shorten Maturities

��The Best way to REDUCE the Maturity of a Money Market Security is to SELL Interest Rate Futures contract whose Maturity equals the difference between the maturity of the security held and 90 days tM = tB - 90

where tM is the time until maturity of the interest rate FUTURES contract used as the hedging vehicle (measured in DAYS)

tB is the time until maturity of the security whose maturity is being SHORTENED (measured in DAYS)

��The formula for determining the number of contracts that should be sold to reduce the maturity of a security is the conventional hedging relationship NF = -HR (Par Value of Securities / 1,000,000)

��The HEDGE RATIO takes into account any differences between the duration of the security whose maturity is being reduced and the duration of the security which underlies the futures (which is .25 for interest rate futures contracts) For securities whose maturities being shorted which are DISCOUNT instruments, the Hedge Ratio is HRShorten Maturity = {[(tB-tD)/360]/.25}(Yield β)

Where tB is the time ‘til maturity of the bond whose maturity is being shortened, measured in days

tD is the Maturity that is DESIRED for the security, in Days Normally assume a Yield β of 1.0

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For Example: A firm owns $10,000,000 face value of a 270-day treasury bill. The bill is currently yielding 5% (discount basis). The Yield curve is flat at 5%, so quoted prices for t-bill futures contracts of various maturities are 95 (flat yield curve implies that all tm-forward rates are equal to the current level of interest rates 3-month T-Bill Futures 95.00 6-month T-Bill Futures 95.00 9-month T-Bill Futures 95.00 The firm would like to sell the bill in 90 days. But, it fears that rates might rise, causing the bill to be sold at a disadvantageous price. Suggest a strategy Answer: The 9-month bill can be effectively shortened to a 90-day bill, with the current 90-day forward rate locked in by using the following contracts: NF = - HR (Par Value/1,000,000)

The bill currently owned matures in 270 days. Therefore, the maturity of the T-Bill futures contract whose yield β is most likely to be equal to 1.0 is

tM = tB – 90 = 270 –90 = 180 days The firm should use a T-Bill futures that matures in 180 days. In effect, this means that the firm will sell the 270-day bill when it has 180 days to mature at a guaranteed yield. This yield will be the 180-dayd forward rate on a 90-day T-Bill that is implied by the 6-month forward contract quoted price Target Rate = f90 = (100-F0)/100 = (100-95/100) = 5% The desired maturity of the 270-day bill is in 90 days, therefore, the Hedge Ratio should be HRshorten maturity={[(tB-tD)/360]/.25}(Yield β) = {[(270-90)/.360]/.25}(1.0) = 2.0 NF = -HR (PV/1,000,000) = -2(10,000,000/1,000,000) = -20 Thus, to shorten the 270 day bill to a 90 day bill, SELL SHORT 20 Futures Contracts (180 days)

��Using Interest Rate Futures to Lengthen Maturities ��BUYING interest rate futures contracts can LENGTHEN Maturities of Investments at

a LOCKED IN rate that is equal to the CURRENT FORWARD rate imbedded in the interest rate futures market For Example: An investor holds $50,000,000 face value of treasury bills which mature in 60 days. The yield curve is flat. The investor plans to roll over these bills in 60 days into another 90-day T-Bill. But, it is feared that rates might decline between now and then. The investor would prefer to lock in the market’s current interest rate expectations rather than lower expectations that he fears will exist in 60 days. How can he do this? Answer: The Investor should BUY T-Bill Futures contracts 60-days forward in order to lengthen the current 60-day bill to 150 days. Since the yield curve is flat, the 60-day forward rate on a 90-day bill should also be 6%. This means, a T-Bill futures contract maturing in 60 days should be quoted at 94 F0=100(1-60f90) = 100(1-.06) = 94 The investor will be receiving $50,000,000 in 60 days when the currently held bills mature. To lengthen the maturity to 150 days, the number of futures contracts to buy NF = -HR(PV/1,000,000) HR = {[(tB-tD)/360]/.25}(Yield β) = {[(60-150)/360]/.25}(1.0) = -1.0 NF = - (-1.0)(50,000,000/1,000,000) = 50 Contracts The Investor should buy 50 contracts to lock in a target rate equal to the 60-day forward rate on 90-day T-Bills Target Rate = (100-F0)/100 = (100-94)/100 = 6%

10. Using Interest Rate Futures Contracts to Convert Floating Rate Loans to Fixed Rate Loans ��To Convert Floating --> Fixed, SELL FUTURES ��To Convert Fixed --> Floating, BUY FUTURES For Example: Suppose a firm has a project that will cost $300,000,000 to complete over a 6-month period. A 6-month loan can be obtained at a floating rate of LIBOR+2%, reset each quarter. The current 90-day LIBOR rate is 5% and a 3-month Eurodollar Futures contract is quoted at 94.70 (implying a 3-month forward rate on 90-day LIBOR of 5.3%) Thus, if the current market expectations are realized, the firm can expect to pay 7% for the first quarter, and 7.3% for the second quarter on the $300,000,000 loan. The First Quarter’s interest rate of 7% is CERTAIN because the 90-day LIBOR rate is now 5%. But, the 7.3% EXPECTED Rate for the 2nd quarter is uncertain. The firm fears that in 3 months, when the rate is re-set, the 90-day LIBOR rate may be above 5.3%. Therefore, the firm wishes to lock in the 2nd quarter expected borrowing rate of 7.3%. In deciding what to do, the firm reasons it should take a position in the Eurodollars futures contract that would generate a profit if interest rates rise above current expectations. When rates rise, futures prices fall. So, to profit from a rise in the rate, the firm should SELL SHORT. Next, it must decide how many futures to sell short. NF = -HR (Amt. To Hedge/KSize) Assuming the loan can be treated as SIMPLE Interest (discount), the Hedge ratio should be HR =[ (t/360)/.25](Yield β) = [(90/360)/.25](1.0) = 1.0 NF = -1.0(300,000,000/1,000,000) = -300 Contracts

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For Example: Suppose the same firm was offered a $300,000,000 6-month loan at a fixed rate of 7.15%with interest payments payable every quarter and all principle due at maturity. Currently LIBOR is 5% and the Eurodollar futures prices are quoted at K maturity Euro Futures Implied Forward Rate on 90-day LIBOR 3-months 94.70 5.3% 6-months 94.50 5.5% 9-months 94.40 5.6% The firm believes that rates will fall and thus would like to turn its fixed-rate loan into a floating rate loan. But, the lender refuses, so the firm wants to deal with the futures market. NF = -HR(Amt. To be Hedged/Ksize) HR = [(t/360)/.25](Yield β) = [(90/360)/.25](1.0) = 1.0 NF = -1(-300,000,000/1,000,000) = 300 ��NOTE: these examples are from the perspective of the BORROWER. ON the Exam,

they might ask from the perspective of the LENDER (in which case, things are reversed).

��The KEY is to ask; what is feared? If fear rates fall (prices go up), go long; if fear rates rising (prices falling) go short

11. Strip and Stack Hedges

��Stalla thinks there is a 30% chance this will be on the Exam For Example: Suppose a construction firm is arranging $500,000,000 of financing for a 1-year project. The bank providing the lending insists that it be a FLOATING rte loan, resetting quarterly at the 90-day LIBOR rate plus 4%. The Current 90-day LIBOR rate is 5% and the Eurodollar futures prices are: K Maturity Euro Futures Implied 90-day Forward Rate 3-months 94.70 5.3% 6-months 94.50 5.5% 9-months 94.40 5.6% Fearing rates may rise more than implied by the futures market expectations, the firm decides to hedge the rates by locking in the forward rates implied by the current yield curve. Reasoning that interest rate futures contract prices fall when rates rise, the firm knows it will profit from a rise in rates above the current expectations if it SELLS. There are 2 Approaches the firm could take: STRIP Hedge or STACK Hedge

��A STRIP Hedge, in which the firm sells an equal number of 3-month, 6-month and 9-month futures contracts. This is the number of contracts needed to hedge each quarter’s interest cost, because each quarter’s interest is hedged with a contract whose underlying duration is .25. NF = -HR(Amt to Hedge) / KSize

HR = [(tD/360)/.25](Yield β) = [(90/360)/.25](1.0) = 1.0 NF = -1 (500,000,000)/1,000,000) = -500 The firm should sell 500 contracts short for each the 3-month, 6-month, and 9-month contract. This procedure matches the hedge to dates when the quarterly floating interest rates will be reset. This will hedge every quarter against adverse interest rate shifts that might occur in that quarter. Thus, the hedge aligns the hedges with the firm’s actual risk exposure.

��MAIN ADVANTAGE of a STRIP – it works even when the Yield Curve SHIFTS and/or RESHAPES

��A STACK Hedge occurs when the firm attempts to protect itself against a SHIFT in the Yield Curve by ONLY selling 3-month futures contracts. Just before these contracts expire, the firm will cover its short position and roll the hedge over into the next three-month contract. Thus, all the futures are STACKED in one contract maturity and ROLLED into the next maturity when the original contract expires

��SINCE the Stack hedge does NOT match the maturity of the hedging vehicles with the cash flows being hedged, it is NOT considered as good a method of hedging cash flows of a strip hedge.

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��DISADVANTAGES of the STACK (and advantages of the Strip) o STACKS will only work perfectly when the Yield Curve Shifts in a

PARALLEL manner. When re-shaping occurs, Stacks won’t work perfectly o Stacks require the use of MORE futures contracts than a STRIP. For Example: in the Strip example, the hedge used 3 sets of 500 contracts, for 1,500 total. If the firm implemented a Stack, it would need the following number of contracts (a total of 3,000). NF = -HR (Amt./1,000,000) HR = [(tD/360)/.25](Yield β) = [(270/360)/.25](1.0) = 3.0 NF = -3(500,000,000/1,000,000) = -1,500 After the first 3 months, when it comes time to rollover, then need to NF = -HR (Amt./1,000,000) HR = [(tD/360)/.25](Yield β) = [(180/360)/.25](1.0) = 2.0 NF = -2(500,000,000/1,000,000) = -1,000 Then, after another 3-months, will need to rollover NF = -HR (Amt./1,000,000) HR = [(tD/360)/.25](Yield β) = [(90/360)/.25](1.0) = 1.0 NF = -1(500,000,000/1,000,000) = -500 o STACKS require CONTINUAL Rollovers, while Strips are more “Set It &

Forget it” o Stacks can be used to HEDGE BULLET interest payments, because THEN

they will match the cash flow being hedged. Else, strips should be preferred as they are easier to implement and hedge better where the slope of the yield curve might change or some other reshaping may occur.

��ADVANTAGES of a STACK (and disadvantages of the Strip) o It might require the use of some very long-term contracts in order to match

cash flow requirements in the Distant future. Generally, there is less liquidity in long-term futures contracts making it a problem to sell the required securities with long maturities. Stacks, then, using liquid-short-term contracts and continually rolling them over make sense

o Since Strips are Set It & Forget IT, the Hedger cannot take advantage of anticipated mispricings of futures contracts. Stacks, when selling contracts that are over-priced, to be repurchased later, can take advantage of these mispricings where strips can’t.

12. Tailing the Hedge

��Since futures markets are marked-to-market daily, there is a Tailing the Hedge problem. To Tail the Hedge, the Hedger adjusts the number of contracts that effectuate the hedge slightly to compensate for the interest that can be earned (or lost) on funds gained (or lost) due to daily settlement.

��Usually, the Tailing Factor can be approximated Tailing Factor = 1 / [1+(rFtM/360)]

��Usually, this factor (less than 1.0) is multiplied by the number of contracts required to effectuate a hedge calculated using formulas that ignore the tailing factor, resulting in the use of FEWER contracts than would be calculated from the usual hedging formulas.

��Everyday, however, tM in the tailing factor declines by 1 day, causing the tailing factor to rise slightly. This results in a small daily increase in the number of contracts required to effectuate a hedge. Eventually, at maturity, tM reaches 0 and the tailing factor reaches 1.0

��As Tailing factors are generally small, they are usually ignored. Usually, they are used only to fine-tune large hedges involving hundreds or thousands of futures

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contracts. Usually, hedges this large are found only in interest rate futures contracts

13. Hedging with Foreign Currency Futures Contracts

��Foreign Currency Futures can be used to Hedge BOTH TRANSACTION (economic risk) and TRANSLATION (accounting risk) Risks.

��The Basic Hedging Relationship is NF = -HR(Qcurrency/KSize)

��Because Forward contracts are generally used to hedge currency and the currency underlying the contract is usually the currency being hedged, the Hedge Ratio is CLOSE to or EQUAL to 1.0

For Example: The XYZ Corp. has sold a machine to a German buyer for DM 5,000,000 with payment expected in 3 months. The spot exchange rate is DM1 = $0.50. The futures quote for the 3-month forward DM is $0.5005. The contract size is DM 125,000. How many DM contracts are needed to perform this hedge? If the hedge is lifted at the time of settlement, what exchange rate will be locked in by the hedge? If the DM 5,000,000 is received in 80 days and the hedge is lifted at that time when the spot rate is DM1 = $0.48 and the futures rate is $0.4801, what will be the value of the hedged positing v. the unhedged position? Answer: NF = -HR (Qcurrency/KSize) NF = -1(5,000,000/125,000) = -40 contracts The company should SELL 40 contracts If the 40 contracts are held ‘til settlement, the effective exchange rate realized by the firm will be the RATE at which the contracts were sold ($0.5005). This is because, on settlement day, the firm will receive DM 5,000,000 at some unknown rate. But, due to the convergence to spot, the price of the futures will equal the spot rate on settlement day. Thus, the portfolio of the firm will consist of a long position in the DM and the profit/loss from the short position in the futures market VPS = SS + (F0-FS) = SS +(F0-SS) = F0

Therefore, the U.S. dollar value of the hedged DM on settlement day will be: DM 5,000,000($0.5005/DM) = $2,502,500 If the DM 5,000,000 is received in 80 days and the hedge is lifted at that time, the firm will receive the following US $ Value of DM 5,000,000 @ $0.48 $2,400,000 Profit on Short (40)(DM125,000)(.5005-.4801 $/DM) 102,000 Value of DM in US $ $2,502,000 The small $500 difference is the result of Basis Risk TRANSLATION RISK EXAMPLE A U.S. Corp. has a French subsidiary that is expected to report a NI of fr 645,678,000 at the end of its fiscal year, which is less than 3 months away. The Spot Exchange rate between the dollar and franc is $0.20/fr. The 3-month futures contract for francs (size fr 500,000) is at $.2020. Design a hedge to protect the firm from translation exposure. If by the end of the fiscal year, the exchange rate is $.18/fr and the futures is at $.1816, determine the net income measured in US$ with and without the hedge WITHOUT the HEDGE NI = (fr 645,778,000* $.18/fr) = $116,222,040 WITH the HEDGE NF = -1(fr645,780,000/500,000) = -1,291.36 The firm should sell 1,291 futures contracts (not a total hedge) When Hedge is Lifted NI (fr 645,780.000*$.18/fr) $116,222,040 Gain on Short (1291)(500,000)(.2020-.1816 $/fr) 13,168,200 Net Value of Income $129,390,240

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14. “Using Interest Rate Futures in Portfolio Management” by the Chicago Board of Trade ��Stalla says this WILL be on EXAM ��This is a slightly different methodology than the KOLB book ��Futures can be used as a low-cost and efficient method of altering the risk and return

characteristics of securities and portfolios, with a minimum of portfolio disruption. Bond futures are very good at this as they track the bond market quite well

��The Conventional formula to determine the number of futures contracts required to hedge a commodity is NF = -HR(Q/Ksize)

��This can be modified for the special case of hedging a bond with treasury bond futures by the following HR = (PVBPB/PVBPF)(Yield β) = (.0001*PBD*

B/.0001*F0D*MD)(Yield β)

��Thus, the number of treasury bond futures required to hedge a number of bonds (NB) is NF = -(.0001*PB*D*

BNB/.0001*F0*D*MD*100)(Yield β)

NF = - (.0001*VBD*B/.0001*VFD*

MD)(Yield β) ��Note: the numerator & denominator of the first term of the equation represent the

amounts by which the VALUES of the bonds and futures will change if interest rates change by 1 basis point This can be defined as the BASIS POINT VALUE. When extended to entire bond portfolios, one may obtain the following definitions BPVP = .0001VPD*

P BPVF = (.0001VMDD*

MD/Conversion FactorMD) = BPVMD/Conversion FactorMD ��Using this BPV formulation, the number of bond futures contracts required to totally

hedge a bond portfolio can be written as: NF hedge = - (BPVP/BPVF)(Yield β) For Example: A bond manager has a $104,350,000 bond portfolio whose DURATION (note, need to change to modified duration) is 4.5 years and whose YTM is 5.5%. The manager fears interest rates will rise (temporarily) and thus decides to totally hedge the portfolio. A 6-month treasury bond futures contract is trading at 95.55. Its basis point value is $79.62. How many of these futures contracts should the manager use to hedge the portfolio? Answer: NF hedge = - (BPVP/BPVF)(Yield β) BPVP = .0001VPD*

P D*

P = DP/[1+(rP/2)] = 4.5/[1+(.055/2)] = 4.3796562 BPVP = - (.0001*104,350,000*4.379562) = $45,701 As no Yield β is given, assume it is 1.0 NF hedge = - (45,701)/(79.62) * (1.0) = -574 contracts Thus, the manager should sell 574 treasury bond futures contracts short against the portfolio a) Using Treasury Bond Futures to Change the Duration of Bond Portfolios

o Portfolio managers often want to change the duration of their bond portfolios, usually as a means of immunization. This can be done by selling some bonds and buying others with different durations. But, this is costly. An alternative, one can INCREASE the DURATION of a bond Portfolio by BUYING Treasury Bond Futures contracts, and DECREASE the DURATION of a bond portfolio by SELLING Treasury bond futures.

o If a bond portfolio is completely hedged by selling a sufficient amount of treasury futures, the result will be a portfolio that will produce the risk-free rate (i.e., a portfolio with a duration equal to the risk-free security).

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NF duration = [(BPVdesired – BPVcurrent)/BPVfutures]{Yield β} For Example: A bond manager holds $40,000,000 in bonds with a duration of 5.5 years (note: need to change to modified duration) and a Yield of 6.0%. The manager expects interest rates will decline and want to take advantage of this by lengthening the maturity of the portfolio to 15 years. The Treasury Bond futures contract maturing in 3 months is quoted at 98. The manager calculates that the basis point value of the futures contract is $80.56. Indicate how many futures contracts will be required to change the duration of the portfolio to the desired level. BPVdesired = .0001VPD*

P = (.0001)($40,000,000)[15/(1+(.06/2))] = $58,252.43 BPVcurrent = .0001VPD*

current = (.0001)($40,000,000)(5.5/(1+.(.06/2))) = $21,359.22 NF = (BPVdesired –BPVcurrent / BPVF) (Yield β) = [(58,252.43 – 21,359.22)/80.56](1.0) = 458 contract The Manager should buy 458 treasury futures contracts to increase the duration of the portfolio from 5.5 years to 15

��NOTE: To INCREASE DURATION, BUY LONG, To REDUCE DURATION, SELL SHORT

��For a TOTAL HEDGE, Duration is 0 DP = 0, D*

P = 0, BPVP = 0 NF = [(0-BPVcurrent)/BPVF](Yield β) NF = - (BPVcurrent/BPVF)(Yield β)

b) Using Futures to Adjust the Mix of a Portfolio

��Portfolio managers often want to change the way a portfolio is allocated between stocks, bond and other assets on a temporary basis. This can be done at a low cost without disturbing the underlying portfolio holdings by using futures. Again, the BPV concept can be used to determine the number of contracts required to reallocate a Portfolio’s exposure in BONDS.

For Example: A portfolio manager holds a $100,000,000 portfolio that is 60% invested in equities and 40% in bonds. The stock portion has a β of 1.0 and the bond portion has a duration of 4.2 and a yield of 6.5%. The manager fears a stock market decline. Consequently, he wants to change the mix to 25% equities and 75% bonds. How can this be done using futures if the 6-month S&P 500 futures is trading at 1,250.60 and the 6-month treasury bond futures has a Basis Point Value estimated to be $75.33? Answer: The portfolio now contains $60,000,000 in stock and $40,000,000 in bonds. The desired mix is $25,000,000 in stock and $75,000,000 in bonds. Thus, the manager wants to decrease the equity holdings by $35,000,000 and increase his bond holding by a like amount. The number of S&P 500 futures contracts required to do this is NF (S*P) = βP[Desired Change in Exposure / ($multiplierF0 (S&P))] NF (S&P) = (1.0)[-35,000,000 / (250)(1250.60)] = -112 Contracts The number of Treasury Bond Futures contracts required to do this is NF = [(BPVdesired – BPVCurrent)/BPVF](Yield β) BPVP = .0001VPD*

P D*

P = (DP / (1+ (rP/2))) = [4.2/(1+(.065/2))] = 4.0677966 BPVdesired = (.0001)(75,000,000)(4.0677966) = $30,508.47 BPVCurrent = (.0001)(40,000,000)(4.0677966) = $16,271.19 NF (bonds) = [(30,508.47-16,271.19)/75.35] (1.0) = 189 Contracts Thus, the manager should SELL 112 S&P 500 Futures contracts and BUY 189 Treasury Futures contracts

c) Creating a Synthetic Bond Portfolio using Treasury Bond Futures ��If a short-term bond or T-bill portfolio is owned, and treasury bond futures are

purchased, a SYNTHETIC Long-Term Bond Portfolio is created ��This is merely lengthening the duration of a portfolio by buying futures

contracts ��This method works under ideal conditions, but there are several disadvantages

o The futures contracts may be mispriced o The number of contracts required must be an even number

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o There might be basis risk or tracking error if the futures are not held ‘til maturity

o The time until a futures contract expires is relatively short For Example: A manager has $10,000,000 invested in 90-day Treasury bills. The manager believes that interest rates will be falling temporarily and would like exposure to a longer-term, 10-year bond. But, it would be too costly to buy an actual 10-year coupon bond for only a few days and then re-sell it to invest the proceeds into T-bills. Thus, the manager decides to create a synthetic exposure to a 10-year bond by using treasury bond futures contracts. Given the following data, how many futures contracts should be purchased to create a synthetic 10-year coupon bond? 90-day T-Bill Yield 4.5% 10-year Coupon Bond Yield 6.0% T-Bond Futures Price 97.75 BPV of $10,000,000 Bills $232.00 BPV of $10,000,000 10 Year Bonds $7,095.00 BPV of Futures $79.67 Answer: Employing the same principles used to change the DURATION of a portfolio, the number of contracts required to increase the duration of the portfolio from .25 (the T-Bill duration) to that of a 10-year bond is NF = [(BPV10 year – BPV.25) / BPVF] = [(7095-232)/79.67] = 86 contracts Thus, the Manager should Purchase 86 Futures Contracts

��This technique is used when o Exposure to a longer-term bond is desired for only a short period of

time, so trading costs are lower by using futures o Mispricings occur between the spot and futures bond market, making

it advantageous to own (or short) the futures contract, rather than the spot bond of equal duration

o A reshaping of the yield curve is expected to result in a flattening of longer-term yields compared to short-term yields

futures - angelfireand a generic 20-year, 8% coupon bond that is used as the basis for quoting...

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