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 f0 =100, f1 = 105, f2= 103, f4= 110 In Margin Account +5 -2 +7
Long Futures Paid = -100 = -f0 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 3rd Friday is the Last Trading day 4. Minimum Price Change (e.g., 1/32 of 1 %, ex x $100,000 = $ 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 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 Buyers Broker Buyers Brokers’ Commission Broker Margin Exchange (Trade) Margin Clearinghouse (Record)

Margin: (p ) A:Initial Margin = m + 3d (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 ) T-Bond: $100,000 (face Value in CBT), $50,000 (Face Value in MCE), Future Price =(1/32) %xFace Value, Ex /32 is $102, in CBT T-Bill: f utures price per $100 = (100-IMM Index)x (90/360), Face value = $1 MM, Ex. Dec by IMM, the Actual futures price = [100-( )(90/360)] x$1MM/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-IMM Index)(90/360)

Quoted in Dollar & 1/32 of par value of $100.
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 is /32 = , 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 years T-Bond 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 Regulation of Futures Markets

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

Forward Price & Futures Price
Price vs. Value Is Price = Value True for Futures or Forwards? Ans. No, why? Price = Value (from efficient market) F = forward price today f = futures price today 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) = time t Ex. p.360 ft f t T F Ft

Note: Value of T = vT = fT - ST  0 Value of 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 Ft and sell a ft The profit  = (-Ft +fT) + (ft - fT) = ft - Ft 0-investment & 0 t => ft = Ft

B. Two days prior to expiration (interest rate r is constant for two periods)
Buy a F and sell (1+r)-(T-t) 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 ft @ T, T = (ft -fT) + [(f-ft)(1+r)-(T-t)(1+r)(T-t)] + (fT-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, S0 (2) Storage Cost, s (3) Interest Foregone, iS0 B. Buy Later:(Paid) (1) Expected Future Spot Price E(ST). In Equilibrium, A = B, or S0 + s + iS0 = E(ST), I.e., S0=E(ST)-s-iS0 (see p.311)

2. Risk Aversion:(in terms of $)
Add Risk Premium E() to A. S0 + s + iS0 + E() = E(ST) S0=E(ST) -s - iS0 - E() Cost of Carry s + iS0

Under no margin, mark-to-the-market etc.
In Spot Market : S0 = E(ST) -  - E() , where,  = Cost of Carry = s(Storage cost) + iS0 (Opp. Cost of Money), E() = Risk Premium(Insurance) The Cost of Carry Futures Pricing Model (Theoretical Fair Price) (p.312) Consider buy a spot S and sell a futures f. At time T, Closing both position and the profit  is (ST-S0-s-iS0) + (f - ST) =  = f-S0- (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. 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(fT) = 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 + DT f-fT = 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)]ercT = S +  (where  is the dividend yield), rc = continuously compound risk-free rate. Ex. S = 85,  = 8%, rc = 10%, T = 90 day = yr, f = 85e( )

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)
f1 - f2 = 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 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., ( )x100]. If Investor Decides to close out the long futures for a gain of ( )x100=$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[fN(d1)-EN(d2)] Where D1= ln(f/E)+σ2T/2 σ √T D2= D1- σ √T

Chapter 10: Forward and Futures Hedging Strategies
KEY CONCEPTS Why Hedge Hedging concepts Factors involved when constructing a hedge 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 You will Get Rich Quick

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 P (short hedge) = ST - S0 (from spot market) - (fT - f0) (from futures market) P (long hedge) = -ST + S0 (from spot market) + (fT - f0) (from futures market) If hedge is closed prior to expiration, P (short hedge) = St - S0 - (ft - f0) If hedge is held to expiration, St = ST = fT = ft.

Profit from Hedge Strategy :
Basis: Spread Spot b0  S - f (initial basis) bt  St - ft t) bT ST - fT expiration) Profit from Hedge Strategy : T Profit of long spot and short future(i.e.,Short Hedge) = (ST - S) + (f - fT) = f - S = - b0 S and f) T (Long Hedge) = b0 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. futures t T

S > f Strengthening Basis for Short Hedger
Example. S = 95, f = 97, ST = x, T (Short Hedge) = $2 (why?) t = (St - S) + (f - ft) = (St-ft) - (S-f) = S-f = bt- b0. 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

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 f0 + (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 $ June futures $ Buy spot, sell futures. Note: b0 = = If held to expiration, profit should be change in basis or 1.45. At expiration, let ST = $ Sell gold in spot for $408.50, a profit of Buy back futures at $408.50, a profit of Net gain =1.45 or $145 on 100 oz. of gold.

The Basis (continued) Example: (continued) Instead, close out prior to expiration when St = $ and ft = $ Profit on spot = Profit on futures = Net gain = .34 or $34 on 100 oz. Note that change in basis was bt - b0 or (-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

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

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

Margin Requirements and Marking to Market low margin requirements on futures, but cash will be required for margin calls

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 = - DS/D f Note that this ratio must be estimated.

Minimum Variance Hedge Ratio Profit from short hedge: P = DS + D fNf Variance of profit from short hedge: sP2 = sDS2 + sDf2Nf2 + 2sDSDfNf The optimal (variance minimizing) hedge ratio is (see Appendix 10A) Nf = - sDSDf/sDf2 This is the beta from a regression of spot price change on futures price change.

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.

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.

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.

Price Sensitivity Hedge Ratio The hedge ratio is as follows (See Appendix 10A for derivation.): Note that DURS » -(DS/S)(1 + yS)/DyS and DURf » -(Df/f)(1 + yf)/Dyf 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.

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.

Stock Index Futures Hedging Appropriate hedge ratio is Nf = -b(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. Minimum Variance Hedge Ratio B. Price Sensitivity Hedge Ratio C. Stock Index Futures Hedge D. Tailing a Hedge

A. Minimum Variance Hedge Ratio (p.357)
2 = 2S + N2f 2f + 2Nf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 = N2f 2f /2S Consider: S =  + f + , Then The Effectiveness of the Minimum Variance Hedge e* = (2S - 2)/2S = R2 = The Coefficient of Determination in The Regression Analysis.

B. Price Sensitivity Hedge Ratio Duration-Based Hedge Strategy(p.359)
Bond Pricing: B = PV(Ci) + 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 Payment PV(ci) Weight Time x Weight Total = D

Price Sensitivity Hedge Ratio(p.359)
Hr= Srffr, Portfolio H = S + ff = (Sysysrffyf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 = - [DsS/(1+ys)]/[Dff/(1+yf)]

Stock Index Futures Hedge (p. 361)
From the Minimum Variance Hedge [S = rsS, 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. $.8310 for ¥ is ¥12,500,000x$ / ¥ =$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 Spot Mkt Futures Mkt 7/1 fD=$1.278/£, #of Contract= 20(35,000)/62,500=11.2 Buy 11 Currency Futures $1.319/£, F=$1.306/ £ Forward Cost =20(35000)x1.306 =$914,200 Forward H S=$1.442/£, Total Cost in $ $700,000(1.442)=$1.009,400 fD=$1.4375/£, Sell 11 Contracts 11/1 Cost $1,009,400-$914,200=$95,200 for No hedge than Forward $1,009,200-11[( x62,500]=$1,009, ,656.25 = $899, by Futures Hedge

Short Hedge: Convert £ to $ in the Future
On 6/29, CFO in UK will Transfer £10MM to NY on 9/28 (Forward Hedge) Date Spot Mkt Forward Mkt 6/29 Sell £10MM Forward S=$1.362/£,F=$1.357/£ 11/1 S=$1.2375/£ Exercise Forward Paid £10MM & Get $13.75MM Paid £10MM & Get $12.375MM for No Hedge Paid £10MM & Get $13.75MM by Forwards Hedge

Strip Hedge & Rolling Strip Hedge
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 On 1/2, ABC to Borrow $ at 3/1 $15MM 6/ 9/ 12/ 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 32nds, relative to $100 par, e.g., 93 14/32 is 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 $1MM 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= Sold $1MM 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 =>Nf =-16.02, Sell 16 T-Bond Futures $70,500 w/Hedge:Closed out Futures Position at $66,718.75, f= = per $100, f =16xfx1000 =$60,500 [T-bond futures $100,000/Contract] Net = $956, ,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  = 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 = -DsS(1+yf)/Dff(1+ys). (Table 9, p. 370)

3. Stock Index Futures Hedge (f= CME index*$250)
Note: S&P 500 Index CME = on 11/22/0x, f = *250 = $186,362.5/Dec. index futures Contract Expiration: March, June, Sept, Dec. Last Trading Day: The Thursday before the 3rd 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 = -sS/f. Mkt Value weighted betas to get s , Portfolio mkt value = S, Index futures times 250 = f.

Ex. Hedging a Takeover ( Table 11, p
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 f-ST Buy a Spot ST Borrow S (use Spot as -S(1+r) Collateral) Net Cash f-S(1+r)=0 r is the repo

T-Bill and Euro$ Futures Price Determination
T-Bill: f utures price per $100 = (100-IMM Index)x (90/360), Face value = $1 MM, Ex. Dec by IMM, the Actual futures price = [100-( )(90/360)] x$1MM/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: $1MM Face Value, Based on LIBOR
Interest Rate of Euro$ 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 (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 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-IMM Index)x (90/360), Face value = $1 MM, Ex. Dec by IMM, the Actual futures price = [100-( )(90/360)] x$1MM/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 the 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 The Dec. T-bill futures is priced by IMM index of (Table 1, p. 458) Consider buy the March pay price = *174/360 = and sell the Dec. T-bill price = *90/360 = 98.7:Synthetic Short-term T-B On Dec. 18, delivery the March T-bill for the futures & received Paid S= 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/ )365/83-1=5.94%] On 9/26, Sell T-Bill Mature on 12/18 and [Buy the March Mature T-Bill & Sell Dec. T-Bill Futures]=> Arbitrage

Ex 9/26 83 days 12/18 3/19 Current date 174 days
T-Bill Spot $ = *83/360 Yield=5.44% March T-Bill Spot Price $ = *174/360 Buy a T-B spot at $ & 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 $10MM of 180 day Euro$ 8.75 or to issue a 90-day 8.25 and selling a Euro$ futures contract expiring in 3 months of IMM index of (Table 2, p. 388) If 180-day Euro$CD is issued, then paid $10,437,500 = $10MM[ (180)/360], or 9.07% If 90-day CD is 8.25 and sell 10 Euro$ 91.37, then need to pay 10MM [ (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 $10MM x( /4) + $16,750 = $10,223,000 on 12/6 and pays $10,233,000 ( /4) = $10,426,438 or 8.84% < 9.07%

Synthetic 180-Day CD Return on furures 2.1575% =[(100-91.37)/100]/4
3 months return on CD % Owe 10,223,000x (1+7.96/4)= 10,426,438 get $10MM the cost of debt 8.84% Owe 10MM(1+8.25/4) =$10,206,250 New 90-day CD Rate 7.96. IMM= 92.04=> f= Issue new 90-day CD for 10,206,250 + ( 980100)x10 Current Date: 90- day CD Rate 8.25 Issue 90 day CD for $10MM IMM 91.37/Dec Sell 10 Futures at $978,425 each Annual Return from 90-day CD & Furures = 8.84% 180 Days 180-day CD Rate Owe $10MM(1+8.75x180/360) 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). CF0 = (.06625/2)[ *21]/ *21 = The Invoice price = Settlement Price on position day * CF + Accrued interest If the settlement price on June is $ =$ and the Accrued interest = $3404.7, then Invoice price = $104,062.5* $3404 = $115,174.07 (Formula for CF see p.421)

Intermediate & Long-Term Interest Rate Futures 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). CF0 = (.06625/2)[ *21]/ *21 = The Invoice price = Settlement Price on position day * CF + Accrued interest If the settlement price on June is $ =$ and the Accrued interest = $3404.7, then Invoice price = $104,062.5* $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 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
Example:Given Current date 4/15, Delivery Date 6/11, Repo Rate 2.62%, Future Price A: % Coupon, Mature on 8/15/09, CF = 8/15 2/15 =181 days 2/15 6/11 4/15 =59 =57 AIt =6.25x59/181=2.04 on 4/15, AIT =6.25x(59+57)/181= 4.01 from 2/15 to 6/11. Bond price is Quoted (ask price). The Invoice Price =f(CF) + AIT= (1.4122)+4.01= on 6/11 (B+AIt)(1+r)T-t = ( )(1.0262)57/365=162.82 f(CF) + AIT - [(B+AIt)(1+r)T-t]= = -.84

Example: Continue B: % Coupon, Mature on 5/15/21, CF = , B = , r = 2.62% 4/15 5/15 6/11 11/15 30 days 27days 184 Days AIt = (181-30)/181= 3.39 on 4/15 from 11/15 to 4/15 AIT = (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] = (1.0137) [( ) (1.0262)57/ =-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 & 15Yrs 8% Coupon Payment Current Time Coupon Payment Maturity Of Futures Coupon Payment 60 Days 122 Days 36 Days 148 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 x[60/180] = , The PV ($6) in 122 days ( yr) = $5.803 2. The Futures Price for 270 days ( yr) is ( )e0.7397x0.1 = At Delivery, There are 148 Days of Accured Interest, The Quoted Futures Price Under 12% Coupon is x148/183 = The Quoted Futures Price under 8% should be /1.4 = $

Delivery Options: 1. Wild Card Option: if S5 < f3*CF [note: issue notice of intention to deliver at 7pm 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
Current Date Expiration Date Buy a Bond -(S+AI) ST+AIT Borrow S+AI -(S+AI)(1+r)T Sell a T-Bond Futures f(CF)+AIT - (ST +AIT) Net Cash Flow f(CF)+AIT -(S+AI)(1+r)T 0 Investment 0 risk r = [(f(CF)+AIT)/(S+AI)]1/T -1 8/15 66 Days Ex. 12.5% Coupon 9/26 12/1 2/15 $ $ On 12/2/03, p.398. Given S=141.5, AI =1.43 =6.25(42/184), CF=1.4662, f= , AIT =3.669 =6.25(108/184), r = 3.89%

T-Bond Futures Spread/Implied Repo Rate
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 T Buy Sell @ Time t, Get T-Bond & Pay ft(CFt)+AIt ,Finance By Repo Rate Time T, Deliver T-Bond & Get fT(CFT)+AIT. 0 Net Cash Time 0 & t & 0 risk at Time T  (ft(CFt)+AIt)(1+r)T-t = fT(CFT)+AIT, or r=[(fT(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 1/4s T-Bond Maturing on 8/15/23 is the C-T-D Bond, March-June Spread & Given AI =.35, CFM=1.029, CFJ=1.0289, fM= , fJ= , AIM =.35, AIJ =1.9 =>(implied repo rate from 13/7-6/5) r = [( (1.0289)+1.9)/( (1.4662)+.35 )]365/ = (ex. P. 401) 8/15 9/ /1 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+yS) Ex. DS=7.83, DT=4, S=1.01MM, yf=14.92% Df=7.2, f = 70,500, yS= 11.74%, => Nf = 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 $1MM mkt value& At least 15 Stocks transaction

Speculating on Unsystematic Risk (Individual Stock)
rS = rM + S, Or,SrS = SrM + 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 1.2x8% =9.6% on the stock. To Hedge: Selling S&P 500 index futures 2/28 12/2 Stock price is 26.25 f= 700, Buy 9 Futures to Close out Own 100,000 shares of Bay at 26, S = $2,600,000 f=765.3 March, Nf=1.2(2,600,000)/765.3x500 =8.154, Sell 9 Futures  from Stock = $25,000  from Futures = 65.3x 500x9= $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(rS)+NfE(f /S) = r + [E(rM)-r]T, T is the target  E(rS) = r + [E(rM)-r]S and E(f /f) = E(rM)-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 = $5MM. Portfolio Manager likes to  to 1.5 for 3 months, f=765 Nf = (1.5-.9)5MM/765x500=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 E- ST Buy a Futures ST-f ST -f E-f ST -f Buy a Call C ST -E Buy a Bond w/ PV(E-f) PV(E-f) E-f E-f ST E STE

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 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-rcT[fN(d1) - EN(d2)] where, d1 = [ln(f/E) + .52T]/sT d2 = d1 -s Ex - .

Ce(f,T,E) = Pe(f,T,E) + (f-E)(1+r)-T
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) = (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: Fixed Floating
Company A 10% 6-month LIBOR +0.3% Company B % 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 A Swap is Created: 9.95% A B LIBOR+1% 10% LIBOR+0.05% A pays 10%/year to Outside Lender, Receive 9.95%/year from B, Pays LIBOR to B B LIBOR+1% A 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] = % 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 Institution A B 10% LIBOR + 1% LIBOR LIBOR 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 $1Euro$ at Date T) VR (Fixed Payment Discount at T-Bill Price/$, i.e., the PV of Receiving for Sure $1 at Date T)

Term Structure of Interest Rate (Based on Pure Discount Bond)
Spot & Forward Rate Term Structure of Interest Rate (Based on Pure Discount Bond) Bond Pricing: B = PV(Ci) + 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 Spot Rates Forward Rate

Example: Estimating the Term Structure
f1 f2 f3 Spot Rate = S1 S2 = (1+S1 )(1+ f1 )-1 S3 = (1+S2 )(1+ f2 )-1 S4 = (1+S3)(1+ f3)-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-IMM Index)x (90/360), Face value = $1 MM, Ex. Dec by IMM, the Actual futures price = [100-( )(90/360)] = $ , Yield = [100/ ]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-Bill T-Bills (IMM), size = $1 million/contract (90-day) (*) f = (100-IMM index)(90/360) Ex. IMM index 92.06 f = (7.94)/4 = = 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?). 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) Buy a T-Bill Pay f Futures Expired Get the 1MMPar Now, Buy a Futures Money

Example Given, forward discount 8.94 *Implied forward rate 9.6%
2/15 June Given, forward discount 8.94 *Implied forward rate 9.6% IMM = 91.32 f = 97.83 Buy a Futures at 97.83 5/17 Futures Expired T-Bill Expired Get $1MM Close Out Date Given IMM=92.54 new f=98.135, Net from futures = = 0.305 Buy a discount 7.69 or S = Net Cost of a T-Bill =97.751  w/Hedge, the Rate of Return = 9.55%= (100/97.751)365/91 -1 w/o Hedge, the Rate of Return = 8.19% = (100/98.056)365/91 -1 (Lock in the forward 9.6%)

b. Anticipatory Hedge of a future $10MM commercial paper Issue (Use: Euro$ 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 = - DsS(1+yf)/Dff(1+ys) (p.429)

Ex. Hedge Future Commercial Paper Issue
4/6 7/20 Sept Issue $10MM (180Days) C P @ Spot Rate 11.34 (180/360) =94.33 per $100 (100/94.33)365/180- 1 =.1257 if No Hedge Futures Expired Given IMM of Sept 88.23 =>f = yf = (100/f)365/90 -1 =.1288, Given 180-day C P Implied forward Rate 10.37%, Price (180/360) ys = (100/P)365/180-1 =.114, Nf = -19.8 Sell 20 Futures(Sept) Contracts (Hedge) (Lock in the forward 11.4%) IMM = 87.47, f = , f = 0.19/100 [100/( )]365/180 -1= Cost of Fund if Hedge Note: 1000/( )=100/( ) 3.8 = .19x20 (Contracts)

Ex. Hedging a Floating Loan:(Lock in @ 10.68%) (3months floating loan)
Borrow $10MM from a bank with a floating rate = LIBOR +1% for two months. If LIBOR , then futures . So, firm should sell the futures now. Given f6 = $976,875=> yf = 9.95%, ys = .1122=( %/12)12-1, Nf = - DsS(1+yf)/ Dff(1+ys) = -(1/12)10MM(1.0995)/ [(1/4)( )(1.1122)] = and Nf = -(1/12) (1.0995)/ [(1/4)( )(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 Expired 3/2
4/6 5/4 2/3 IMM=90.47 f= , f=.07, or 700/Futures x3 = $2,100 Total Liabil. =10MM(1+ .1068/12) = $10,089,000 ,100 $10,086,900 New LIBOR =10.09 IMM=89.99, f = f =.19, or 1900/Futures x3=$5,700 Total Liabil. (1+ .11.09/12) = $10,180,120 ,700 $10,174,420 New LIBOR =10.79 Pay Total Debt $10,174,420(1+ .1179/12)= $10,274,384 LIBO(90days)=9.68%, Get 10MM Loan, & Like to Lock in the ( /12)12 -1 = .1122= ys IMM=90.75, f = , yf= .0995,Nf =-3.37 Sell 6 Euro$ Futures Cost of Debt (10,274,384/10 MM)4= .1144 with Hedge w/o Hedge =[1 +.1068/12)(1+ .1109/12)(1+.11 79/12)]4=.1178

 
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Chapter 8: The Structure of Forwards & Futures Markets

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Chapter 8: The Structure of Forwards & Futures Markets

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