Chapter 8 The Structure of Forwards Futures Markets

Chapter 8: The Structure of Forwards & Futures Markets • KEY CONCEPTS – Explanations of the Basics of Forward and Futures Contracts – More EVIL is More Beautiful – Terms and Conditions of Futures Contracts – Margins, Daily Settlements, Price Limits and Delivery – Futures Traders and Trading Styles – Reading Price Quotes

Futures Contracts • Chicago Board of Trade (CBOT) – Grains, Treasury bond futures • Chicago Mercantile Exchange (CME) – Foreign currencies, Stock Index futures, livestock futures, Eurodollar futures • New York Mercantile Exchange (NYMEX) – Crude oil, gasoline, heating oil futures • Development of new contracts – Futures exchanges look to develop new contracts that will generate significant trading volume

Futures f 0 =100, f 1 = 105, f 2= 103, f 4= 110 In Margin Account f 0 =100 +5 f 1 = 105 -2 f 2 = 103 +7 f 4= 110 Long Futures Paid -110+7 -2+5 = -100 = -f 0 to Get One Underlying Asset

Contract's Terms: (see p. 276 -277) 1. Size (see p. 276) 2. Grade, Quotation Unit 3. Delivery Months, 3, 6, 9, 12 3 rd Friday is the Last Trading day 4. Minimum Price Change (e. g. , 1/32 of 1 %, ex. . 0003125 x $100, 000 = $31. 25 for T-Bond Futures) 5. Delivery Terms: Delivery date(s), Delivery Procedure, Expiration Months, Final Trading Day, First Delivery day (see p. 277 & 288) 6. Daily Price Limits & Trading Halts 7. Margin

Futures Traders: Commission Brokers & Locals • Hedger, Speculator, Spreader (Long One & Short One), Arbitrageur. [ by Trading Strategy] • Trading Styles (Techniques): • Scalper: Holds a Few Minutes • Day Trader; Hold No More Than The Trading Day • Position Trader Cost of Seats Fig 1(p. 283), Seat can be leased monthly @1%-1. 5%of Seat price. CBT has 1402 Full members Forward Market Traders: Banks & Firms (Co. , Investment Bankers, etc. , )

Order (same as options) • • Stop Loss Order Limit Orders Good-Till-Canceled Day Orders.

Trading Procedure: (see Fig. 2, p. 285) Buyer Margin Buyers Brokers’ Commission Broker Exchange (Trade) Margin Clearinghouse (Record)

Margin: (p. 286 -287) A: Initial Margin = m + 3 d (m = the average of the daily absolute changes in the dollar value of a futures contract, d = the standard deviation, measured over some time period in the recent past). – Initial margin is used to cover all likely changes in the value of a futures contract. B: Maintenance Margin: – Equity position must be > Maintenance margin or get a margin call must deposit new $ (i. e. , variation margin)before the market opens on the next trading day. Ex. p. 287

• Open Interest: • Delivery & Cash Settlement(p. 288) • Futures Price Quotation (see p. 292 -293) T-Bond: $100, 000 (face Value in CBT), $50, 000 (Face Value in MCE), Future Price =(1/32) %x. Face Value, Ex. 102 3/32 is $102, 093. 75 in CBT • T-Bill: f utures price per $100 = 100 - (100 -IMM Index)x (90/360), Face value = $1 MM, Ex. Dec. 94. 95 by IMM, the Actual futures price = [100 -(100 -94. 95)(90/360)] x$1 MM/100= $987, 375 (will be used Chapter 11) • Note: IMM quotes based on a 90 -day T-bill w/360 -day year. • $1 MM Face Value, Interest Rate Is Discount Rate

. 1. Last Trading Date: The Business Day Prior to the Date of Issue of T-bills in the Third week of the Month 2. Delivery Day: a) Any Business Day After the Last Trading Date (During the Expiration Month). b) First Business Day of Month, c) Cash settlement 4. If Seller elects to Deliver a 91 or 92 days T-Bill, then Replace 90 by 91 or 92 in the Formula in p. 373, f = 100 - (100 -IMM Index)(90/360)

T-Bond Futures: Based on 8% Coupon & 15 Yrs' Maturity T-Bond (Face Value $100, 000) • Quoted in Dollar & 1/32 of par value of $100. Ex. 111 -17 is 111 17/32 = 111. 53125, or $111, 531. 25 Expiration: March, June, Sept, Dec. • Last trading Day: the Business Day Prior to the Last seven days of the expiration month. • The First Delivery Day = The First Business Day of the Month • T-Notes Futures: Same As T-Bond Except the maturity from 0 -2 years, 4 -6 and 6. 5 -10 years TBond or Notes

Other Futures • • • Agricultural Commodity Futures Stock Indices Futures Natural Resources Futures Miscellaneous Commodities Futures Foreign Currency Futures T-Bills & Euro$s Futures T-Notes & T-Bonds Futures Index Futures (i. e. , Equities Futures) Managed Futures: Futures Funds (Commodity Funds), Private Pools, Specialized Contract • Hedge Funds • Option on Futures Transaction Cost: Commission, Bid-Ask Spread, Delivery Cost

Chapter 9: Pinciples of Forward & Futures Pricing • KEY CONCEPTS – Difference Between Price and Value of Forward and Futures Contracts – Rationale for a Difference Between Forward and Futures Prices – Cost of Carry Futures Pricing Model – Convenience Yield, Backwardation and Contango – Risk Premium/Controversy – Role of Coupon Interest/Dividends in Futures Pricing – Put-Call Forward/Futures Parity – Pricing Options on Futures

Comparison of Forward and Futures Contracts • Forward Private contract between two parties Not standardized Usually one specified delivery date Settled at end of contract Delivery or final cash settlement usually takes place Futures Traded on an exchange Standardized contract Range of delivery dates Settled daily Contract usually closed out prior to maturity

Forward Price & Futures Price vs. Value Is Price = Value True for Futures or Forwards? Ans. No, why? Price = Value (from efficient market) ft f F = forward price today 0 t T f = futures price today F Ft Ft = forward price written at time t ft = futures price written at time t Vt = value at time t of a forward contract written today = (Ft - F)(1+r)-(T-t) = PV(Ft-F) @ time t Ex. p. 360

• Note: Value of Futures @ T = v. T = f. T - ST 0 Value of Futures @ t = vt = ft - ft-1 (before marked-to-mkt) & vt 0 once marked-to-mkt

Forward and Futures Prices (p. 308 -309) (The effect of daily settlement on forward and futures prices) Example: (A Two-Period Model) A. One day prior to expiration Buy a forward @ Ft and sell a future @ ft The profit = (-Ft +f. T) + (ft - f. T) = ft - Ft 0 -investment & 0 risk @ t => ft = Ft

B. Two days prior to expiration (interest rate r is constant for two periods) Buy a forward @ F and sell (1+r)-(T-t) futures @ f At time t, the profit = (f-ft)(1+r)-(T-t) invest in risk-free bonds. This close the futures position. Now, sell a new futures @ ft @ T, T = (ft -f. T) + [(f-ft)(1+r)-(T-t)(1+r)(T-t)] + (f. T-F) = f - F = 0 ( $0 investment & risk-free) f > (<) F if futures prices & interest rates are positively (negatively) correlated (p. 370)

A Forward and Futures Pricing Model Spot Prices, Risk Premiums, & Cost of Cary 1. Risk Neutral: A. Buy Now ($) (Paid) (1) Spot Price, S 0 (2) Storage Cost, s (3) Interest Foregone, i. S 0 B. Buy Later: (Paid) (1) Expected Future Spot Price E(ST). In Equilibrium, A = B, or S 0 + s + i. S 0 = E(ST), I. e. , S 0=E(ST)-s-i. S 0 (see p. 311)

2. Risk Aversion: (in terms of $) • Add Risk Premium E( ) to A. S 0 + s + i. S 0 + E( ) = E(ST) S 0=E(ST) -s - i. S 0 - E( ) • Cost of Carry s + i. S 0

Under no margin, mark-to-the-market etc. In Spot Market : S 0 = E(ST) - - E( ) , where, = Cost of Carry = s(Storage cost) + i. S 0 (Opp. Cost of Money), E( ) = Risk Premium(Insurance) The Cost of Carry Futures Pricing Model (Theoretical Fair Price) (p. 312) Consider buy a spot commodity @ S and sell a futures contract @ f. At time T, Closing both position and the profit is (ST-S 0 -s-i. S 0) + (f - ST) = = f-S 0 - (risk-free) = 0 ? Futures Price = Spot Price + Cost of Carry Quasi Arbitrage: Asset owner sell his Asset and Buy a Futures if f < S+ to take the Arbitrage Opp. Exists if f S+

Definition: Basis Cash price S - Futures Price f 1. If Futures Prices f < Cash Spot Prices S => Backwardation (or Inverted) Market 2. If futures Prices f > Cash Prices S=> Contango Market 3. Convenience Yield c: f = S + - c Risk Premium Controversy (mixed in empirical studies) 1. f = E(ST) [No Risk Premium] 2. f < E(f. T) = E(ST) = S + + E( ) = f + E( ) Example. p. 387 Normal Contango: E(ST) < f Normal Backwardation: f < E(ST)

The Effect of Intermediate Cash Flows on Futures Price Long a Stock S and Short a Futures at f S ST + D T 0 f-f. T = f - ST S DT+f S = (DT + f)(1+r)-T Or f = S(1+r)T - DT Ex. S = $100, DT = $2, r = 6%, T =. 25, then f = 100(1. 06). 25 -2 = $99. 47

In General f = S(1+r)T - Dt(1+r)(T-t) = Future Spot Price - FV(D) = [S - PV(D)](1+r)T = S + For Continuous Dividends: f = Se(rc- )T = [S-PV(D)]erc. T = S + (where is the dividend yield), rc = continuously compound risk-free rate. Ex. S = 85, = 8%, rc = 10%, T = 90 day = 0. 246575 yr, f = 85 e(0. 1 -. 08)0. 246575

Interest Rate Parity • • F=S(1+r)T/(1+ρ)T S=Spot Exchange Rate/$ ρ =Risk-Free Rate in US r=Foreign Risk-Free Rate F=Forward Exchange Rate/$ $(1+ ρ)F=$S(1+r) Deposit US$ in US’s Bank Us Forward Rate to Lock in and then Convert to Foreign Currency = Convert in to Foreign Currency and Deposit in Foreign Bank. • EX. See P. 327 • Arbitrage Opp. Exists If Parity is Violated (P. 328)

Pricing of Spreads (Different Expiration Dates) f 1 = S + 1 f 2 = S + 2 f 1 - f 2 = 1 - 2 = Spread Basis (Ex. p. 329

Put-Call Forward/Futures Parity • • • P=C-S+PV(E) P=C+PV(E)-PV(f) Or P(S, E, T)=C(S, E, T)+PV(E-f) Spot Price @ T vs. Exercise Price E for Options

Options On Futures: Underlying Asset is Futures • • • Call Option On Futures C(f, T, E)=IV+TV IVC=Max(0, f-E) for Call, IVP=Max(0, E-f) for Put Lower Bound for American & European Options (see P. 331 &332) • Ex. See p. 333 • Buy July call futures on Gold(100 ounces) w/E $300. Exercise Decision: If July gold futures is $340 and the most recent price=$338. The Investor receive a long Gold Futures Contract + a Cash of $3, 800 [i. e. , (338 -300)x 100]. If Investor Decides to close out the long futures for a gain of (340 -338)x 100=$200. Total Payoff from the Decision of Exercise is $4, 000

Put-Call Parity of Options on Futures • P(f, T, E)=C(f, T, E)+PV(E-f) • Ex. See p. 335 • Early Exercise of Call & Put Options on Futures? (Textbook: Possible for Both Call & Put)

B/S Option On Futures Pricing Model (p. 336) • C(f, T, E)=PV[f. N(d 1)-EN(d 2)] • Where • D 1= ln(f/E)+σ2 T/2 σ √T • D 2= D 1 - σ √T

Chapter 10: Forward and Futures Hedging Strategies • KEY CONCEPTS Why Hedge • Hedging concepts You will Get • Factors involved when constructing a hedge Rich Quick Difference Between a Short Hedge and a Long Hedge and When to Use Each Appropriate Hedging Contract to Use in a Given Situation Optimal Hedge Ratios Analysis of Specific Hedge

Why Hedge? • The value of the firm may not be independent of financial decisions because – Shareholders might be unaware of the firm’s risks. – Shareholders might not be able to identify the correct number of futures contracts necessary to hedge. – Shareholders might have higher transaction costs of hedging than the firm. – There may be tax advantages to a firm hedging. – Hedging reduces bankruptcy costs. • Managers may be reducing their own risk. • Hedging may send a positive signal to creditors. • Dealers hedge so as to make a market in derivatives.

Why Hedge? (continued) • Reasons not to hedge – Hedging can give a misleading impression of the amount of risk reduced – Hedging eliminates the opportunity to take advantage of favorable market conditions – There is no such thing as a hedge. Any hedge is an act of taking a position that an adverse market movement will occur. This, itself, is a form of speculation.

Hedging Concepts • Short Hedge and Long Hedge – Short (long) hedge implies a short (long) position in futures – Short hedges can occur because • The hedger owns an asset and plans to sell it later. • The hedger plans to issue a liability later – Long hedges can occur because • The hedger plans to purchase an asset later. • The hedger may be short an asset. – An anticipatory hedge is a hedge of a transaction that is expected to occur in the future. – See Table 10. 1, p. 348 for hedging situations.

Hedging Concepts (continued) • The Basis – Basis = spot price - futures price. – Hedging and the Basis • (short hedge) = ST - S 0 (from spot market) - (f. T f 0) (from futures market) • (long hedge) = -ST + S 0 (from spot market) + (f. T f 0) (from futures market) • If hedge is closed prior to expiration, (short hedge) = St - S 0 - (ft - f 0) • If hedge is held to expiration, St = ST = ft.

Basis: b 0 S - f (initial basis) bt St - ft (basis @ t) b. T ST - f. T (basis @ expiration) Spread Spot futures Profit from Hedge Strategy : t T T Profit of long spot and short future(i. e. , Short Hedge) = (ST - S) + (f - f. T) = f - S = - b 0 (Buy @ S and Sell @ f) T (Long Hedge) = b 0 Example: Hedging and the Basis • Buy asset for $100, sell futures for $103. Hold until expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.

Example. S = 95, f = 97, ST = x, T (Short Hedge) = $2 (why? ) t = (St - S) + (f - ft) = (St-ft) - (S-f) = S- f = bt- b 0. bt - b Is Stochastic S > f Strengthening Basis for Short Hedger S < f Weakening basis for Short Hedger Ex: @t, St = 92, ft = 90, Given S = 95, f = 97, then t(Short Hedge) = (92 -90)-(95 -97) = 2 -(2)=4

Hedging Concepts (continued) • The Basis (continued) – This is the change in the basis and illustrates the principle of basis risk. – Hedging attempts to lock in the future price of an asset today, which will be f 0 + (St - ft). – A perfect hedge is practically non-existent. – Short hedges benefit from a strengthening basis. – Everything we have said here reverses for a long hedge. – See Table 10. 2, p. 350 for hedging profitability and the basis.

Hedging Concepts (continued p. 351) • The Basis (continued) – Example: March 30. Spot gold $387. 15. June futures $388. 60. Buy spot, sell futures. Note: b 0 = 387. 15 - 388. 60 = -1. 45. If held to expiration, profit should be change in basis or 1. 45. • At expiration, let ST = $408. 50. Sell gold in spot for $408. 50, a profit of 21. 35. Buy back futures at $408. 50, a profit of -19. 90. Net gain =1. 45 or $145 on 100 oz. of gold.

Hedging Concepts (continued) • The Basis (continued) – Example: (continued) • Instead, close out prior to expiration when St = $377. 52 and ft = $378. 63. Profit on spot = -9. 63. Profit on futures = 9. 97. Net gain =. 34 or $34 on 100 oz. Note that change in basis was bt - b 0 or -1. 11 - (1. 45) =. 34. – Behavior of the Basis. See Figure 10. 1, p. 352.

Two risks exist in Hedge: • 1. Cross Hedge (commodity is not the same as the underlying commodity of futures) • 2. Quantity Risk: Size Rules for Hedging Strategies: Rule 1. High Correlated Rule 2. Expiration Date of Contract is Over and Close to the Hedge Termination Date Rule 3. If Positive Correlated => One Long and One Short , If Negative Correlated => Both are Long or Short, (Detail See 355, Table 4) Rule 4. Hedge Ratio; Nf such that some goal can achieve Portfolio consists of a long S and Nf of Futures = S + Nf f = 0 => Nf = - S/ f

Hedging Concepts (continued) • Contract Choice – Which futures commodity? • One that is most highly correlated with spot • A contract that is favorably priced – Which expiration? • The futures whose maturity is closest to but after the hedge termination date subject to the suggestion not to be in the contract in its expiration month • See Table 10. 3, p. 354 for example of recommended contracts for T-bond hedge • Concept of rolling the hedge forward

Hedging Concepts (continued) • Contract Choice (continued) – Long or short? • A critical decision! No room for mistakes. • Three methods to answer the question. See Table 10. 4, p. 355 – worst case scenario method – current spot position method – anticipated future spot transaction method

Hedging Concepts (continued) • Margin Requirements and Marking to Market – low margin requirements on futures, but – cash will be required for margin calls

Hedging Concepts (continued) • Determination of the Hedge Ratio – Hedge ratio: The number of futures contracts to hedge a particular exposure – Naïve hedge ratio – Appropriate hedge ratio should be • Nf = - S/ f • Note that this ratio must be estimated.

Hedging Concepts (continued) • Minimum Variance Hedge Ratio – Profit from short hedge: • = S + f. Nf – Variance of profit from short hedge: • 2 S 2 + f 2 Nf 2 + 2 S f. Nf – The optimal (variance minimizing) hedge ratio is (see Appendix 10 A) • Nf = - S f/ f 2 • This is the beta from a regression of spot price change on futures price change.

Hedging Concepts (continued) • Minimum Variance Hedge Ratio (continued) • Hedging effectiveness is – e* = (risk of unhedged position - risk of hedged position)/risk of unhedged position – This is coefficient of determination from regression.

Hedging Concepts (continued) • Price Sensitivity Hedge Ratio – This applies to hedges of interest sensitive securities. – First we introduce the concept of duration. We start with a bond priced at B: • where CPt is the cash payment at time t and y is the yield, or discount rate.

Hedging Concepts (continued) • Price Sensitivity Hedge Ratio – An approximation to the change in price for a yield change is – with DURB being the bond’s duration, which is a weighted-average of the times to each cash payment date on the bond, and represents the change in the bond price or yield. – Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.

Hedging Concepts (continued) • Price Sensitivity Hedge Ratio – The hedge ratio is as follows (See Appendix 10 A for derivation. ): – Note that DURS » -( S/S)(1 + y. S)/ y. S and DURf » -( f/f)(1 + yf)/ yf – Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.

Hedging Concepts (continued) • Price Sensitivity Hedge Ratio (continued) – Alternatively, • Nf = -(Yield beta)PVBPS/PVBPf – where Yield beta is the beta from a regression of spot yields on futures yields and – PVBPS, PVBPf is the present value of a basis point change in the spot and futures prices.

Hedging Concepts (continued) • Stock Index Futures Hedging – Appropriate hedge ratio is • Nf = - (S/f) • This is the beta from the CAPM, provided the futures contract is on the market index proxy. – Tailing the Hedge • With marking to market, the hedge is not precise unless tailing is done. This shortens the hedge ratio.

Hedge Ratio Determinations: A. B. C. D. Minimum Variance Hedge Ratio Price Sensitivity Hedge Ratio Stock Index Futures Hedge Tailing a Hedge

A. Minimum Variance Hedge Ratio (p. 357) 2 = 2 S + N 2 f 2 f + 2 Nf S f = Variance of Profit • Minimizing 2 => Nf = - S f/ 2 f = - in the regression of S on f • Effectiveness of Hedge e* = ( 2 S - 2 )/ 2 S = N 2 f 2 f / 2 S • Consider: S = + f + , Then The Effectiveness of the Minimum Variance Hedge e* = ( 2 S - 2 )/ 2 S = R 2 = The Coefficient of Determination in The Regression Analysis.

B. Price Sensitivity Hedge Ratio Duration-Based Hedge Strategy(p. 359) Bond Pricing: B = PV(Ci) + PV(Par) @ Yield y Note: Yield Curve is Derived from ys (IRR) 1% = 100 base points Duraion = D = Weighted Average Maturity of Bond D = -( B/B)/[ y/(1+y)] B/B -D[ y/(1+y/n)], n = # of Interest Payment/yr

Example: Given B = PV(ci) + PV(P) D = i[PV(ci)]/B, 3 years 10% Coupon Bond w/face Value $100, y= 12%, paid semiannual: Time 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 Total Payment 5 5 5 105 130 PV(ci) 4. 717 4. 450 4. 198 3. 960 3. 736 74. 021 95. 082 Weight 0. 0496 0. 0468 0. 0442 0. 0416 0. 0393 0. 7785 1. 0000 Time x Weight 0. 0248 0. 0468 0. 0663 0. 0832 0. 0983 2. 3355 2. 6549= D

Price Sensitivity Hedge Ratio(p. 359) H r = S r f f r, Portfolio H = S + ff = ( S ys r f f yf r = 0 => Nf = - ( S ys /( f yf) if ys r yf r or Nf= - ( S/ ys)/( f/ yf) In Terms of Duration Ds = -[( S/S)(1+ys)]/ ys Nf = - [Ds. S/(1+ys)]/[Dff/(1+yf)]

C. Stock Index Futures Hedge (p. 361) From the Minimum Variance Hedge [ S = rs. S, f = rff ] Nf = - s(S/f), where s is obtained by regression of rs = + srf + (Mkt Model) • D. Tailing a Hedge (p. 362) The Effect of Mark-to-the-Market is to reduce the hedge ratio below the optimum. N = Nf(1+r)-(Days to Expiration - 1)/365

Hedging Strategies: Applications • 1. Currency Hedges • 2. Intermediate & Long-term Interest Rate Futures Hedges • 3. Stock Market Hedges

3 Most Actively Traded Currency Futures • 1. Euro with size of € 125, 000 • 2 British Pound with size of £ 62, 500 • 3 Japanese Yen with size of ¥ 12, 500, 000 • In US, Futures Prices Are Stated in $. • EX. $. 8310 for ¥ is ¥ 12, 500, 000 x$. 008310/ ¥ • =$103, 875/Futures

Long Currency Hedge: A/P in £ • On 7/1, Car Dealer in US buys 20 British Car of £ 35, 000/car, A/P on 11/1. Date 7/1 11/1 Spot Mkt $1. 319/£, F=$1. 306/ £ Forward Cost =20(35000)x 1. 306 =$914, 200 Forward H Futures Mkt f. D=$1. 278/£, #of Contract= 20(35, 000)/62, 500=11. 2 Buy 11 Currency Futures S=$1. 442/£, Total Cost in $ $700, 000(1. 442)=$1. 009, 400 f. D=$1. 4375/£, Sell 11 Contracts Cost $1, 009, 400 -$914, 200=$95, 200 for No hedge than Forward $1, 009, 200 -11[(1. 4375 -1. 2780 x 62, 500]=$1, 009, 200 -109, 656. 25 = $899, 743. 75 by Futures Hedge

Short Hedge: Convert £ to $ in the Future • On 6/29, CFO in UK will Transfer £ 10 MM to NY on 9/28 (Forward Hedge) Date Spot Mkt 6/29 S=$1. 362/£, F=$1. 357/£ 11/1 S=$1. 2375/£ Forward Mkt Sell £ 10 MM Forward Currency @$1. 375/£ Exercise Forward Paid £ 10 MM & Get $13. 75 MM Paid £ 10 MM & Get $12. 375 MM for No Hedge Paid £ 10 MM & Get $13. 75 MM by Forwards Hedge

Strip Hedge & Rolling Strip Hedge On 1/2, ABC to Borrow $ at 3/1 $15 MM 6/1 45 9/1 12/1 20 10 Strip: On 1/2 : Sell 15 March , 45 June, 20 Sep and 10 Dec contracts. On 3/1 Buy 15 Futures On 6/1 Buy 45 Futures On 9/1 Buy 20 Futures On 12/1 Buy 10 Futures Rolling Hedge Strip: On 1/2 Sell 90 March Futures On 3/1 Buy 90 March Futures and Sell 75 June Futures On 6/1 Buy 75 June Futures and Sell 30 Sep Futures On 9/1 Buy 30 Sep Futures and Sell 10 Dec Futures On 12/1 Buy 10 Dec Futures

2. Intermediate & Long-term Interest Rate Futures Hedge • Intermediate and Long-Term Interest Rate Futures Hedges – First let us look at the T-note and bond contracts • T-bonds: must be a T-bond with at least 15 years to maturity or first call date • T-note: three contracts (2 -, 5 -, and 10 -year) • A bond of any coupon can be delivered but the standard is a 6% coupon. Adjustments, explained in Chapter 11, are made to reflect other coupons. • Price is quoted in units and 32 nds, relative to $100 par, e. g. , 93 14/32 is 93. 4375. • Contract size is $100, 000 face value so price is $93, 437. 50

Ex. Hedging a Long Position in a Gov't Bond (Table 7, p. 368) Hold $1 MM of Gov't Bond Today. If bond prices (interest rate ), then futures on T-Bond will . So, you should sell T-bond future today to Hedge the Risk. 3/28 2/25 T-Bond f=$66, 718. 75, B=95. 6875 Sold $1 MM Gov't Bond get $956, 875, (Loss $53, 125 w/o Hedge) B=101, Ds =7. 83, ys=. 1174. yf=. 1492 Df =7. 2, f=70. 5 w/Hedge: Closed out Futures Position at =>Nf =-16. 02, Sell $66, 718. 75, f=70. 5 -66. 71875=3. 78125 16 T-Bond Futures per $100, f =16 x fx 1000 =$60, 500 Today @ $70, 500 [T-bond futures $100, 000/Contract] Net = $956, 875 +60, 500=$1, 017, 375

Hedging a Future Purchase of a T-Notes (p. 369) • Same as the Hedging a future purchase of a T-Bill. • Buy T-note futures to hedge (why? ). Nf = - S/ f, by regression on daily data find = 10. 5. So, Nf = 11. (Table 10) [Regression function: S = + f + ] (different Nf) Current Date Purchasing Date Futures Expiration Date

Ex. Hedging a Corporate Bond Issue (21 years maturity) • • Same as the Hedging a Future Commercial Paper Issue Sell T-bond futures (why? ). Nf = -Ds. S(1+yf)/Dff(1+ys). (Table 9, p. 370)

3. Stock Index Futures Hedge (f= CME index*$250) • Note: S&P 500 Index CME = 745. 45 on 11/22/0 x, f = 745. 45*250 = $186, 362. 5/Dec. index futures Contract • Expiration: March, June, Sept, Dec. • Last Trading Day: The Thursday before the 3 rd Friday of Expiration Month • Ex. Stock Portfolio Hedge (Table 10, p 373) Hold a portfolio. Sell the S&P 500 futures to hedge his portfolio. Nf = - s. S/f. Mkt Value weighted betas to get s , Portfolio mkt value = S, Index futures times 250 = f.

Ex. Hedging a Takeover ( Table 11, p. 374, hedging a future purchase of stocks). Buy Nf S&P 500 futures Contracts, Nf = S/f, beta in CAPM

Chapter 11: Advanced Futures Strategies • KEY CONCEPTS – Cash and Carry Arbitrage – Implied Repo Rate – Delivery Option Imbedded in the T-Bond Futures Contracts – Rationale for Spread Strategies – Stock Index Futures Arbitrage and Program Trading

Short-term Interest Rate Futures Strategies • T-Bill Cash & Carry/Implied Repo • Implied Repo Rate f/S - 1 = /S [f - S = ] R =(f/S)1/t -1 = the return implied by the cost of carry relationship between spot & futures prices Sell a Futures Contracts Buy a Spot Borrow S (use Spot as Collateral) Net Cash 0 r is the repo f-ST ST -S(1+r) f-S(1+r)=0

T-Bill and Euro$ Futures Price Determination • T-Bill: f utures price per $100 = 100 - (100 -IMM Index)x (90/360), Face value = $1 MM, Ex. Dec. 94. 95 by IMM, the Actual futures price = [100(100 -94. 95)(90/360)] x$1 MM/100= $987, 375 • Note: IMM quotes based on a 90 -day T-bill w/360 -day year. • $1 MM Face Value, Interest Rate Is Discount Rate

Euro$ Futures: $1 MM Face Value, Based on LIBOR – Interest Rate of Euro$ is Called LIBOR – Note: T-bill is a discount instrument, and Euro$ is an add -on instrument. Ex. 10% quote rate on T-bill & Euro$ (Spot Market) Pay 100 -10(90/360)=97. 5 & get 100 par in 90 days Yield = (100/97. 5)365/90 -1 = 10. 81% for T-bill. Pay 97. 5 get back 97. 5(. 1)(90/365)=2. 44 interest + 97. 5 principle Yield = (1+2. 4/97. 5)365/90 -1 =10. 36% for Euro$

Euro$ Futures Price Same as T-bill Futures Price Calculation • Futures price per $100 = 100 - (100 -IMM Index)x (90/360), Face value = $1 MM, Ex. Dec. 94. 46 by IMM, the Actual futures price = [100 -(10094. 46)(90/360)] x$1 MM/100= $986, 150 • Note: IMM quotes based on a 3 -month LIBOR w/360 -day year. • Expiration months: March, June, Sept, Dec. • Last Trading Date: Second London Business Day before third Wed. of the Month • First Delivery Day: Cash Settled on Last Trading Day.

Ex. of Cash & Carry Arbitrage ( no transaction cost, Table 1, p. 386) • On 9/26, T-bill maturing on 12/18 (i. e. , 83 days to maturity) has a discount rate of 5. 19, which implied a rate of return 5. 44%. The T-bill maturing on 3/19 (i. e. 174 days to maturity) has a discount rate of 5. 35. The Dec. T-bill futures is priced by IMM index of 94. 8. (Table 1, p. 458) • Consider buy the March spot @5. 35 pay price = 1005. 35*174/360 = 97. 4142 and sell the Dec. T-bill futures @ price = 100 -5. 2*90/360 = 98. 7: Synthetic Short-term T-B • On Dec. 18, delivery the March T-bill for the futures & received 98. 7. Paid S=97. 4142 and get f=98. 7. The rate of return R = 5. 94% > 5. 44% the return on the Dec. T-bill. There is an arbitrage (why? )[(98. 7/97. 4142)365/83 -1=5. 94%] • On 9/26, Sell T-Bill Mature on 12/18 and [Buy the March

Ex 9/26 83 days Current date 12/18 3/19 174 days T-Bill Spot $98. 8034 =100 -5. 19*83/360 Yield=5. 44% March T-Bill Spot Price $97. 4142 = 100 -5. 35*174/360 Buy a T-B spot at $97. 4142 & Sell a Dec. Futures at $98. 7 Close out the Position, get $98. 7, Yield = 5. 94% Buy a T-B (March) & Sell a Dec. Futures to Create a Synthetic Dec T-Bill

Euro$ Arbitrage: (Cost of Carry relation is Violated Between Euro$ Futures & Spot) (Table 2, p. 388) • EX: On 9/16, a London bank needs either to issue $10 MM of 180 day Euro$ CD @ 8. 75 or to issue a 90 -day CD @ 8. 25 and selling a Euro$ futures contract expiring in 3 months of IMM index of 91. 37. (Table 2, p. 388) • If 180 -day Euro$CD is issued, then paid $10, 437, 500 = $10 MM[1+. 0875(180)/360], or 9. 07% • If 90 -day CD is issued @ 8. 25 and sell 10 Euro$ futures @ 91. 37, then need to pay 10 MM [1+. 0825(90/360)] on 12/16 and get 10*978, 425 from futures pay 10*980, 100 to close the futures (loss $16, 750). The firm needs to issue $10 MM x(1+. 0825/4) + $16, 750 = $10, 223, 000 on 12/6 and pays $10, 233, 000 (1+. 0796/4) = $10, 426, 438 or 8. 84% < 9. 07%

Return on furures 2. 1575% Synthetic 180 -Day CD 3 months return on CD 2. 0625% =[(100 -91. 37)/100]/4 Current Date: 90 - Owe 10 MM(1+8. 25/4) Owe 10, 223, 000 x day CD Rate 8. 25 =$10, 206, 250 (1+7. 96/4)= New 90 -day CD Rate Issue 90 day CD 10, 426, 438 7. 96. IMM= 92. 04=> for $10 MM get $10 MM f= 98. 01. Issue new IMM 91. 37/Dec the cost of 90 -day CD for Sell 10 Futures 10, 206, 250 + (978425 - debt 8. 84% at $978, 425 each 980100)x 10 Annual Return from 90 -day CD & Furures = 8. 84% 180 Days 180 -day CD Rate 8. 75. Owe $10 MM(1+8. 75 x 180/360) or the cost of debt 9. 07% > 8. 84%

Conversion Factor: Deliver a Different Coupon Rates • Ex. Find CF for delivery of the 6 5/8 of August 15, 2022, on the June 2001 T-bond future contract • On the june 1, 2001 the bond's remaining life is 21 yrs, 2 months. Rounding down to 0 (0, 3, 6, 9). • CF 0 = (. 06625/2)[1 -1. 03 -2*21]/. 03 + 1. 03 -2*21 = 1. 074067 • The Invoice price = Settlement Price on position day * CF + Accrued interest • If the settlement price on June is $104 -02 =$104. 0625 and the Accrued interest = $3404. 7, then Invoice price = $104, 062. 5*1. 074067 + $3404 = $115, 174. 07 • (Formula for CF see p. 421)

Intermediate & Long-Term Interest Rate Future Strategies • Conversion Factor: Deliver a Different Coupon Rates & T • Ex. Find CF for delivery of the 6 5/8 of August 15, 2022, on the June 2001 T-bond future contract • On the june 1, 2001 the bond's remaining life is 21 yrs, 2 months. Rounding down to 0 (0, 3, 6, 9). • CF 0 = (. 06625/2)[1 -1. 03 -2*21]/. 03 + 1. 03 -2*21 = 1. 074067 • The Invoice price = Settlement Price on position day * CF + Accrued interest • If the settlement price on June is $104 -02 =$104. 0625 and the Accrued interest = $3404. 7, then Invoice price = $104, 062. 5*1. 074067 + $3404 = $115, 174. 07 • (Formula for CF see p. 421)

• The cheapest-to-deliver bond, among all deliverable bonds, is the bond that is most profitable to deliver, where profit is measured by: [The FV of net cash flow by Selling a futures & Buying a Spot @ time t ] f(CF) + AIT - [(B+AIt)(1+r)T-t - FV of Coupon at T], where, AIT is the accrued interest on the bond at T, the delivery date, AIt is the accrued interest on the bond at time t (i. e. , today), r = risk-free rate, B = bond price

Example: Given Current date 4/15, Delivery Date 6/11, Repo Rate 2. 62%, Future Price 112. 65625 A: 12. 5% Coupon, Mature on 8/15/09, CF = 1. 4022 2/15 8/15 13+31+30+31+15 =181 days 2/15 6/11 4/15 13+31+15=59 15+31+11=57 AIt =6. 25 x 59/181=2. 04 on 4/15, AIT =6. 25 x(59+57)/181= 4. 01 from 2/15 to 6/11. Bond price is Quoted 160. 125(ask price). The Invoice Price =f(CF) + AIT=112. 65625(1. 4122)+4. 01=161. 98 on 6/11 (B+AIt)(1+r)T-t = (160. 125+2. 04)(1. 0262)57/365=162. 82 f(CF) + AIT - [(B+AIt)(1+r)T-t]=161. 98 -162. 82= -. 84

Example: Continue B: 8. 125% Coupon, Mature on 5/15/21, CF = 1. 0137, B = 116. 21875, r = 2. 62% 4/15 5/15 6/11 30 27 days 11/15 184 Days AIt = 4. 0625(181 -30)/181= 3. 39 on 4/15 from 11/15 to 4/15 AIT = 4. 0625(27/184) = 0. 60 on 6/11 from 5/15 to 6/11 FV(4. 0625)=4. 0625(1. 0262)27/365 =4. 07 on 6/11 from 5/15 -6/11 f(CF) + AIT - [(B+AIt)(1+r)T-t - FV of Coupon at T] = 112. 65625(1. 0137)+0. 6 - [(116. 21875+3. 39) (1. 0262)57/365 -4. 07 =-1. 22, 12. 5% Coupon is Cheapter-t-D Bond than 8. 125%

Rules (Determining the Quoted Futures Price) • 1. Find the Cash Spot Price (Cheapest-to-deliver Bond) from Quoted Price • 2. Find Futures Price based on on f = [S-PV(D)]er(T-t) • 3. Find Quoted Futures Price from the Cash Futures Price • 4. Divide the Quoted Futures Price by Conversion Factor to Allow the difference Between the C-t-D Bond & 15 Yrs 8% Coupon Payment 60 Days Current Time Coupon Payment 122 Days Maturity Of Futures 148 Days Coupon Payment 36 Days Suppose C-t-D T-Bond is 12%, Conversion Factor 1. 4 & Futures is 270 days to mature, Coupon Pay Semiannual, Interest rate is 10% & Current Quoted Bond Price is $120

Example: Continue • 1. The Cash Price = Quoted Bond Price + Accured Interest 120 + 6 x[60/180] = 121. 978, The PV ($6) in 122 days (0. 3342 yr) = $5. 803 2. The Futures Price for 270 days (0. 7397 yr) is (121. 978 - 5. 803)e 0. 7397 x 0. 1 = 125. 094 At Delivery, There are 148 Days of Accured Interest, The Quoted Futures Price Under 12% Coupon is 3. 125. 094 -6 x 148/183 = 120. 242 The Quoted Futures Price under 8% should be 4. 120. 242/1. 4 = 85. 887 $

Delivery Options: • 1. Wild Card Option: if S 5 < f 3*CF [note: issue notice of intention to deliver at 7 pm to clearinghouse] • 2. Quality (or Switching) Option: (switching to favorable B) • 3. The-end-of-the-month Option: (same as Wild Card Option, there are 8 Business Days in the expiration month) • 4. Timing Option(in one month; financing cost vs coupon) Implied Repo/Cost of Carry (T-B Futures) f(CF) + AIT = $ received for Delivery = $ paid for B + Cost of Carry = (S+AI)(1+r)T r = [(f(CF) + AIT)/(S+AI)]1/T - 1

Implied Repo/Cost of Carry • Repo Current Date Expiration Date Buy a Bond -(S+AI) Borrow S+AI Sell a T-Bond Futures ST+AIT -(S+AI)(1+r)T f(CF)+AIT - (ST +AIT) Net Cash Flow 0 f(CF)+AIT -(S+AI)(1+r)T 0 Investment 0 risk r = [(f(CF)+AIT)/(S+AI)]1/T -1 2/15 Ex. 12. 5% Coupon 8/15 9/26 66 Days 12/1 $ $ On 12/2/03, p. 398. Given S=141. 5, AI =1. 43 =6. 25(42/184), CF f=95. 65625, AIT =3. 669 =6. 25(108/184), r = 3. 89%

T-Bond Futures Spread: Long & Short a T-B Futures w/ Different Expiration Dates • Ex. to speculate r , if r will in short period then Sell a shorter maturity futures & Buy a longer maturity futures (see Table 5, p. 399) T-Bond Futures Spread/Implied Repo Rate t Buy T Sell @ Time t, Get T-Bond & Pay ft(CFt)+AIt , Finance By Repo Rate r. @ Time T, Deliver T-Bond & Get f. T(CFT)+AIT. 0 Net Cash Flow @ Time 0 & t & 0 risk at Time T (ft(CFt)+AIt)(1+r)T-t = f. T(CFT)+AIT, or r=[(f. T(CFT)+AIT)/(ft(CFt)+AIt)]1/(T-t)-1. If r forward rate, then Arbitrage Opportunity [i. e. Over(under)priced futures]

Ex. (T-Bond Futures Spread/ Implied Repo Rate) On 12/2/02, 16 1/4 s T-Bond Maturing on 8/15/23 is the C-TD Bond, March-June Spread & Given AI =. 35, CFM=1. 029, CFJ=1. 0289, f. M=108. 09375, f. J=108. 09375, AIM =. 35, AIJ =1. 9 =>(implied repo rate from 13/7 -6/5) r = [(108. 09375(1. 0289)+1. 9)/(108. 09375(1. 4662)+. 35 )]365/90 -1 =8/15. 00092 (ex. P. 401) 12/1 9/26 2/15 3/1 Bond Mkt Timing w/Futures: DS if r , & DS if r To change the Duration from DS to DT is decided by Nf = -[(DS-DT)S(1+yf)]/Dff(1+y. S) • Ex. DS=7. 83, DT=4, S=1. 01 MM, yf=14. 92% Df=7. 2, f = 70, 500, y. S= 11. 74%, => Nf = -7. 84 Sell 8 Futures See Table 6, p. 403

Stock Index Futures Strategies • Stock Index Arbitrage: when f = Se(rc- )T is Violated Then Buy Low Sell High, See Ex: p. 404, & Table 7 • Program Trading At least $1 MM mkt value& At least 15 Stocks transaction

Speculating on Unsystematic Risk (Individual Stock) r. S = r. M + S, Or, Sr. S = S r. M + S S S = S ( M/M) + S S , M is the mkt index Given, = S+Nf f , and Nf = - (S/f), so no systematic risk in Portfolio S + Nf f (This is a Hedge) = S S = Stock Price * Unsystemmtic Return if M/M = f/f Ex. next page

Ex. Speculating on Unsystematic Risk Table 8, p. 410 • On 12/1, Bay has a price at 26 and a beta of 1. 2, You expect Bay to by 10% by the end of Feb and the S&P 500 to 8%. =1. 2 x 8% =9. 6% on the stock. To Hedge: Selling S&P 500 index futures 12/2 Own 100, 000 shares of Bay at 26, S = $2, 600, 000 f=765. 3 March, Nf=1. 2(2, 600, 000)/765. 3 x 500 =8. 154, Sell 9 Futures 2/28 Stock price is 26. 25 f= 700, Buy 9 Futures to Close out from Stock = $25, 000 from Futures = 65. 3 x 500 x 9= $293, 850, Total = 318, 850, Rate of return =12. 26%

Stock Mkt Timing w/Futures: (Adjust by Futures) • Buying or selling futures to or portfolio Given Nf = - S(S/f), Portfilo P = S + Nf f , & p = S+Nf f , the return on the portfolio rp= ( S+Nf f )/S E(rp)= E(r. S)+Nf. E( f /S) = r + [E(r. M)-r] T, T is the target E(r. S) = r + [E(r. M)-r] S and E( f /f) = E(r. M)-r , Nf = ( T S)(S/f) from 0 -beta risk hedge ratio Nf = - S(S/f) to target T risk hedge ratio Nf = ( T S)(S/f) Ex: On 12/2 current =. 9, S = $5 MM. Portfolio Manager likes to 1. 5 for 3 months, f=765 Nf = (1. 5 -. 9)5 MM/765 x 500=7. 843, Buy 8 S&P 500 March index futures contracts Now

Put-Call-Futures Parity: Pe = Ce + (E-f)(1+r)-T vs. Pe = Ce -S + E(1+r)-T Expiration Date Current Date PV Buy a Put P Buy a Futures 0 Buy a Call Buy a Bond w/ PV(E-f) C PV(E-f) ST E E- ST ST-f E-f 0 0 ST -f ST -E E-f E-f ST -f

Chapter 12: Option on the Futures • Key Concepts – Basic Characteristics of Options on Futures – Intrinsic Values, Lower Bounds & Put-Call Parity of Options on Futures – Why Both Calls & Puts Might Be Exercised Early – Black & Binomial Option on Futures Pricing Models – Trading Strategies for Options on Futures – Diffrence Between Options on the Spot & Options on Futures

Options on Futures To give the buyer the right to buy (or sell) a futures contract @ a fixed price (E) up to a specified expiration date (T). (Commodity Options or Futures Option) Call & Put Intrinsic Value of an American Option on Futures = Max(0, f-E) for Call. = Max(0, E-f) for Put. Ex.

Black Option on Futures Pricing Model • C(f, T, 2, E, r) = e-rc. T[f. N(d 1) - EN(d 2)] where, d 1 = [ln(f/E) +. 5 2 T]/s T d 2 = d 1 -s • Ex • -.

Put-Call Parity • Ce(f, T, E) = Pe(f, T, E) + (f-E)(1+r)-T • Ex. Pe(f, T, E) = 7. 45, f = 320, E=315, r = 5. 46%, T =. 25, then Ce(f, T, E) = 7. 45 + 5(1. 0546)-. 25 = 12. 52

Chapter 14: Swaps & Other Interest Rate Agreements • Key Concepts – Interest Rate Swaps (pricing, Apllications, Termination) – Forward Rate Agreements & Similarity to Swaps – Interest Rate Options Use & Pricing – Caps, Floors, Collars Use & Pricing – The Derivative Intermediary – The Nature of Credit Risk & How It Is Managed – General Awareness of Accounting, Regulatory & Tax Issues

Basic Concepts • Swaps = Privated Agreements Between 2 Parties to Exchange Cash Flows In the Future According to a Prearranged Formula = Portfolio of Forwards Contracts • Comparative Advantage : Borrowing Fixed When it Wants Floating or Vice Versa • Prime Rate (Reference Rate of Interest for Domestic Financial Mkt) • LIBOR (Reference Rate for International Financial Mkts)

Example Borrowing Rate: Company A Company B Fixed 10% 11. 2% Floating 6 -month LIBOR +0. 3% 6 -month LIBOR +1% B pays 1. 2% more than A in Fixed & Only. 7% in Floating B has Comparative Advantage in Floating Rate Mkt, A has Comparative Advantage in Fixed Rate Mkt 9. 95% A Swap is Created: A 10% A pays 10%/year to Outside Lender, Receive 9. 95%/year from B, Pays LIBOR to B B LIBOR+1% LIBOR+0. 05% B Borrows @ LIBOR+1% A Borrows @ Fixed 10% & Then Rnter a Swap to Ensure that A Ends Up Floating Rate

Example: Company B Cash Flow: 1. Pay LIBOR+1% to Outside Lender 2. Receive LIBOR from A 3. Pays 9. 95% to A Company A Net Cash Flow with Swap -10%+9. 95%-(LIBOR) = -(LIBOR+0. 05%) Without Swap, Company A Pays LIBOR+0. 3%, Save 0. 25% Company B Net Cash Flow with Swap -(LIBOR+1%)-9. 95%+[LIBOR] = -10. 95% Without Swap, Company A Pays 11. 2%, Save 0. 25% The Total Gain = [11. 2%-10%] - [(LIBOR+1%) - (LIBOR+ 0. 3% )] = 0. 5%.

Role of Financial Intermediary (Net 0. 1%) • A: Cash Flow: (Net = LIBOR+0. 1%, Save 0. 2% ) Pay 10% to outside Lenders Receive 9. 9%/annum from Financial Intermediary Pay LIBOR to Financial Intermediary 10. 0% 9. 9% Financial A B Institution LIBOR 10% LIBOR + 1% B: Cash Flow: (Net = 11%, Save 0. 2%) Pay LIBOR + 1% to Outside Lenders Receive LIBOR from Financial Intermediary Pay 10%/annum to Financial Intermediary

Swap Valuation • VF = Value of Floating Payment = P - PV(P). Bond Sell at Par P = Notional Principal • VR = PV(Fixed Cash FLow): for Fixed Payment • Value of Swap = VF - VR , • VF (Floating Payment Discount at Euro$ Deposit Rate, i. e, the PV of Receiving $1 Euro$ at Date T) • VR (Fixed Payment Discount at T-Bill Price/$, i. e. , the PV of Receiving for Sure $1 at Date T)

Spot & Forward Rate Spot Rates Forward Rate Term Structure of Interest Rate (Based on Pure Discount Bond) Bond Pricing: B = PV(Ci) + PV(Par) @ Yield y Note: Yield Curve is Derived from ys 1% = 100 base points • Estimating the Term Structure (p. 372) • (i. e. , An Application of Forward Rates to Derive the Spot Rate) Example. See p. 372 -375

Example: Estimating the Term Structure S 1 f 2 f 3 Spot Rate = S 1 S 2 = (1+S 1 )(1+ f 1 )-1 S 3 = (1+S 2 )(1+ f 2 )-1 S 4 = (1+S 3)(1+ f 3)-1 Note: fi is derived from the T-bill Futures Price Si+1 = (1+Si)(1+fi) Annualize & then - 1

• T-Bill: f utures price per $100 = 100 - (100 IMM Index)x (90/360), Face value = $1 MM, Ex. Dec. 94. 95 by IMM, the Actual futures price = [100 -(100 -94. 95)(90/360)] = $98. 7375, Yield = [100/98. 7375]365/90 see p. 373 • Note: IMM quotes based on a 90 -day T-bill w/360 -day year.

1. Short-term Interest Rate Hedges • a. Anticipatory Hedge of a future purchase of a T-Bills (IMM), size = $1 million/contract (90 -day) (*) f = 100 - (100 -IMM index)(90/360) Ex. IMM index 92. 06 f = 100 - (7. 94)/4 = 100 -1. 985 = 98. 015 So, the futures price is $980, 150/T-bill futures

Ex. Hedging a Future Purchase of a T-bill: If you are going to buy T-bill from spot market in the future, then you should buy the T-bill futures now (why? ). Get the Now, Buy a T-Bill Futures Pay f Expired 1 MMPar a Futures If interest rate decreases, then the price of T-bill will increase => To hedge future purchase of T-Bill, BUY one (why one ? ) T-Bill futures now to capitalize the rising of the futures price due to the interest rate decrease. Because if r futures price => Losses (Table 6, p. 426) Money

Example June 2/15 Given, forward discount 8. 94 * Implied forward rate 9. 6% IMM = 91. 32 f = 97. 83 Buy a Futures at 97. 83 5/17 Close Out Date Given IMM=92. 54 new f=98. 135, Net from futures = -97. 83+98. 135 = 0. 305 Buy a T-Bill @ discount 7. 69 or S = 98. 056 Net Cost of a T-Bill 98. 056 -. 305=97. 751 Futures T-Bill Expired Get $1 MM w/Hedge, the Rate of Return = 9. 55%= (100/97. 751)365/91 -1 w/o Hedge, the Rate of Return = 8. 19% = (Lock in the forward rate @ 9. 6%) (100/98. 056)365/91 -1

b. Anticipatory Hedge of a future $10 MM commercial paper Issue (Use: Euro$ futures (IMM), size=$1 MM) • Ex. Hedging a Future Commercial Paper Issue: • If you need to issue 180 days commercial paper in the future, then you should sell the futures (why? ) (Table 7, p. 429). [Because issuing a commercial paper sell spot, if r , spot , & Interest Rate Futures Short Euro$ futures]. • Hedging Strategy: Use (*) to calculate the futures price & yield yf & use spot mkt to calculate the commercial paper's yield ys & its value. Find the hedge ratio using the Price Sensitivity Hedge Ratio (why? ): Nf = - Ds. S(1+yf)/Dff(1+ys) (p. 429)

Ex. Hedge Future Commercial Paper Issue 4/6 7/20 Sept Given IMM of Sept Issue $10 MM (180 Days) C P Futures Expired 88. 23 =>f =97. 0575 @ Spot Rate 11. 34 yf = (100/f)365/90 -1 100 -11. 34(180/360) =94. 33 per $100 (Lock in the forward =. 1288, 365/180 - 1 rate @ 11. 4%) (100/94. 33) Given 180 -day C P =. 1257 if No Hedge Implied forward Rate 10. 37%, Price 100 -10. 37(180/360) IMM = 87. 47, f = 96. 8675, f = 0. 19/100 365/180 -1= 11. 65 Cost 365/180 [100/(94. 33+. 38)] ys = (100/P) -1 of Fund if Hedge =. 114, Nf = -19. 8 Sell 20 Futures(Sept) Note: 1000/(943. 3+3. 8)=100/(94. 33+. 38 3. 8 =. 19 x 20 (Contracts) Contracts (Hedge)

Ex. Hedging a Floating Loan: (Lock in @ 10. 68%) (3 months floating loan) • Borrow $10 MM from a bank with a floating rate = LIBOR +1% for two months. If LIBOR , then futures . So, firm should sell the futures now. Given f 6 = $976, 875=> yf = 9. 95%, ys =. 1122=(1+10. 68%/12)12 -1, • Nf = - Ds. S(1+yf)/ Dff(1+ys) = -(1/12)10 MM(1. 0995)/ [(1/4)(9. 76875)(1. 1122)] = -3. 37 and Nf = -(1/12)10089000(1. 0995)/ [(1/4)(9. 76875)(1. 1122)] = -3. 4 Sell 6 futures with three to be closed out on March and three on April. (see Table 8, p. 431)

• Example: Heading a Floating Rate Loan (3 Months)Futures 3/2 4/6 5/4 Expired 2/3 LIBO(90 days)= 9. 68%, Get 10 MM Loan, & Like to Lock in the (1+. 1068/12)12 -1 =. 1122= ys IMM=90. 75, f =97. 6875, yf=. 0995, Nf =-3. 37 Sell 6 Euro$ Futures IMM=90. 47 f=97. 6175, f=. 07, or 700/Futures x 3 = $2, 100 Total Liabil. =10 MM(1+. 1068/12) = $10, 089, 000 2, 100 $10, 086, 900 New LIBOR =10. 09 IMM=89. 99, f = 97. 4975 f =. 19, or 1900/Futures x 3=$5, 700 Total Liabil. 10086900(1+. 11. 09/12) = $10, 180, 120 5, 700 $10, 174, 420 New LIBOR =10. 79 Pay Total Debt $10, 174, 420(1+. 1179/12)= $10, 274, 384 Cost of Debt (10, 274, 384/10 MM)4=. 1144 with Hedge w/o Hedge =[1 +. 1068/12)(1+. 1109/12)(1+. 1 79/12)]4=. 1178