Chapter 8: The Structure of Forwards

1
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

2
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

3
Futures

• f0 100, f1 105, f2 103, f4 110

In Margin Account
5
-2
7
f1 105
f4 110
f0 100
f2 103
Long Futures Paid -1107-25 -100 -f0 to
Get One Underlying Asset
4
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.
  .0003125x 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

5
Futures TradersCommission 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 _at_1-1.5of Seat price. CBT has 1402 Full
  members
• Forward Market Traders Banks Firms (Co.,
  Investment Bankers, etc.,)

6
Order (same as options)

• Stop Loss Order
• Limit Orders
• Good-Till-Canceled
• Day Orders.

7
Trading Procedure (see Fig. 2, p. 285)

• Buyer

Buyers Broker
Buyers Brokers Commission Broker
Margin
Exchange (Trade)
Margin
Clearinghouse (Record)
8
Margin (p. 286-287)

• AInitial 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 gt 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

9

• 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) xFace
  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) x1MM/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

10
.

• 1. Last Trading DateThe 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)

11
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
  T-Bond or Notes

12
Other Futures

• Agricultural Commodity Futures
• Stock Indices Futures
• Natural Resources Futures
• Miscellaneous Commodities Futures
• Foreign Currency Futures
• T-Bills Euros 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

13
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

14
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

15
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)(1r)-(T-t) PV(Ft-F) _at_ time t
• Ex. p.360

ft
f
t
0
T
F
Ft
16

• Note
• Value of Futures _at_ T vT fT - ST ? 0
• Value of Futures _at_ t vt ft - ft-1 (before
  marked-to-mkt)
• vt ? 0 once marked-to-mkt

17
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 _at_ Ft and sell a future _at_ ft
• The profit ? (-Ft fT) (ft - fT) ft - Ft
• 0-investment 0 risk _at_ t gt ft Ft

18

• B. Two days prior to expiration (interest rate r
  is constant for two periods)
• Buy a forward _at_ F and sell (1r)-(T-t) futures _at_
  f
• At time t, the profit ? (f-ft)(1r)-(T-t)
  invest in risk-free bonds. This close the futures
  position. Now, sell a new futures _at_ ft
• _at_ T, ?T (ft -fT) (f-ft)(1r)-(T-t)(1r)(T-t)
  (fT-F)
• f - F 0 ( 0 investment risk-free)
• f gt (lt) F if futures prices interest rates are
  positively (negatively) correlated (p. 370)

19
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20
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.,
• S0E(ST)-s-iS0
• (see p.311)

21

• 2. Risk Aversion(in terms of )
• Add Risk Premium E(?) to A.
• S0 s iS0 E(?) E(ST)
• S0E(ST) -s - iS0 - E(?)
• Cost of Carry ??s iS0

22

• 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 commodity _at_ S and sell a
  futures contract _at_ 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 lt S? to take the Arbitrage
  Opp.
• Arbitrage Opp. Exists if f ??S?

23
Definition Basis ? Cash price S - Futures Price
f

• 1. If Futures Prices f lt Cash Spot Prices S gt
  Backwardation (or Inverted) Market
• 2. If futures Prices f gt Cash Prices Sgt 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 lt E(fT) E(ST) S ? E(?) f E(?)
• Example. p. 387
• Normal Contango E(ST) lt f
• Normal Backwardation f lt E(ST)

24
The Effect of Intermediate Cash Flows on Futures
Price

• Long a Stock S and Short a Futures at f
• S ST DT
• 0 f-fT f - ST
• S DTf
• S (DT f)(1r)-T
• Or f S(1r)T - DT
• Ex. S 100, DT 2, r 6, T .25,
• then f 100(1.06).25-2 99.47

25
In General

• f S(1r)T - ?Dt(1r)(T-t) Future Spot Price
  - FV(D)
• S - PV(D)(1r)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
  0.246575yr, f 85e(0.1-.08)0.246575

26
Interest Rate Parity

• FS(1r)T/(1?)T
• SSpot Exchange Rate/
• ? Risk-Free Rate in US
• rForeign Risk-Free Rate
• FForward Exchange Rate/
• (1 ?)FS(1r)
• Deposit US in USs 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)

27
Pricing of Spreads (Different Expiration Dates)

• f1 S ?1
• f2 S ?2
• f1 - f2 ?1 - ?2 Spread Basis (Ex. p.329

28
Put-Call Forward/Futures Parity

• PC-SPV(E)
• PCPV(E)-PV(f)
• Or
• P(S,E,T)C(S,E,T)PV(E-f)
• Spot Price _at_ T vs. Exercise Price E for Options

29
Options On Futures Underlying Asset is Futures

• Call Option On Futures
• C(f,T,E)IVTV
• IVCMax(0, f-E) for Call,
• IVPMax(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 price338. The Investor
  receive a long Gold Futures Contract a Cash of
  3,800 i.e., (338-300)x100. If Investor Decides
  to close out the long futures for a gain of
  (340-338)x100200. Total Payoff from the
  Decision of Exercise is 4,000

30
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)

31
B/S Option On Futures Pricing Model (p. 336)

• C(f,T,E)PVfN(d1)-EN(d2)
• Where
• D1 ln(f/E)s2T/2
• s vT
• D2 D1- s vT

32
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
33
Why Hedge?

• The value of the firm may not be independent of
  financial decisions because
• Shareholders might be unaware of the firms
  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.

34
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.

35
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.

36
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.

37
Basis
Spread
Spot

• b0 ? S - f (initial basis)
• bt ? St - ft (basis _at_ t)
• bT? ST - fT (basis _at_ expiration)
• Profit from Hedge Strategy ?
• ?T? Profit of long spot and short
  future(i.e.,Short Hedge)
• (ST - S) (f - fT) f - S - b0 (Buy _at_ S
  and Sell _at_ 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
38
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 gt ?f ??Strengthening Basis for Short Hedger
• ?S lt ?f ??Weakening basis for Short Hedger
• Ex
• _at_t, St 92, ft 90, Given S 95, f 97,
• then ?t(Short Hedge) (92-90)-(95-97) 2-(-2)4

39
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 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.

40
Hedging Concepts (continued p. 351)

• The Basis (continued)
• Example March 30. Spot gold 387.15. June
  futures 388.60. Buy spot, sell futures. Note
  b0 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.

41
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 - b0 or -1.11 - (-1.45) .34.
• Behavior of the Basis. See Figure 10.1, p. 352.

42
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 gt One Long and
  One Short , If Negative Correlated gt 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 gt Nf -?S/?f

43
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

44
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

45
Hedging Concepts (continued)

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

46
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 - DS/D f
• Note that this ratio must be estimated.

47
Hedging Concepts (continued)

• 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.

48
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.

49
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.

50
Hedging Concepts (continued)

• Price Sensitivity Hedge Ratio
• An approximation to the change in price for a
  yield change is
• with DURB being the bonds 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 bonds price
  to its yield.

51
Hedging Concepts (continued)

• 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
  options delta and, like delta, it will not
  capture the risk of large moves.

52
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.

53
Hedging Concepts (continued)

• 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.

54
Hedge Ratio Determinations

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

55
A. Minimum Variance Hedge Ratio (p.357)

• ?2? ?2?S N2f ?2?f 2Nf??S?f Variance of
  Profit ?
• Minimizing ?2? gt 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.

56
B. Price Sensitivity Hedge Ratio
Duration-Based Hedge Strategy(p.359)

• Bond Pricing B ?PV(Ci) PV(Par) _at_ 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/(1y)
• ?B/B? -D?y/(1y/n), n of Interest
  Payment/yr

57
Example Given

• B ?PV(ci) PV(P)
• D ?iPV(ci)/B, 3 years 10 Coupon Bond w/face
  Value 100, y 12, paid semiannual
• Time Payment PV(ci) Weight Time x Weight
• 0.5 5 4.717 0.0496 0.0248
• 1.0 5 4.450 0.0468 0.0468
• 1.5 5 4.198 0.0442 0.0663
• 2.0 5 3.960 0.0416 0.0832
• 2.5 5 3.736 0.0393 0.0983
• 3.0 105 74.021 0.7785 2.3355
• Total 130 95.082 1.0000 2.6549 D

58
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
• gt Nf - (?S??ys?/(?f??yf) if ?ys??r????yf??r
• ??or
• Nf - (?S/?ys)/(?f/?yf)
• In Terms of Duration
• Ds -(?S/S)(1ys)/?ys
• Nf - DsS/(1ys)/Dff/(1yf)

59
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(1r)-(Days to Expiration - 1)/365

60
Hedging Strategies Applications

• 1. Currency Hedges
• 2. Intermediate Long-term Interest Rate
  Futures Hedges
• 3. Stock Market Hedges

61
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.008310/
• 103,875/Futures

62
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
fD1.278/, of Contract 20(35,000)/62,50011.
2 Buy 11 Currency Futures
1.319/, F1.306/ Forward Cost 20(35000)x1.30
6 914,200 Forward H
S1.442/, Total Cost in 700,000(1.442)1.00
9,400
fD1.4375/, Sell 11 Contracts
11/1
Cost 1,009,400-914,20095,200 for No hedge
than Forward 1,009,200-11(1.4375-1.2780x62,500
1,009,200-109,656.25 899,743.75 by Futures
Hedge
63
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 Currency _at_1.375/
S1.362/,F1.357/
11/1
S1.2375/
Exercise Forward Paid 10MM Get 13.75MM
Paid 10MM Get 12.375MM for No Hedge Paid
10MM Get 13.75MM by Forwards Hedge
64
Strip Hedge Rolling Strip Hedge
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

• On 1/2, ABC to Borrow at
• 3/1 15MM
• 6/1 45
• 9/1 20
• 12/1 10

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
65
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 93.4375.
• Contract size is 100,000 face value so price is
  93,437.50

66

• 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 f66,718.75,B95.6875 Sold 1MM Gov't
Bond get 956,875,(Loss 53,125 w/o Hedge)
B101,Ds 7.83, ys.1174.yf.1492 Df 7.2,
f70.5 gtNf -16.02, Sell 16 T-Bond Futures Today
_at_ 70,500
w/HedgeClosed out Futures Position
at 66,718.75, ?f70.5-66.718753.78125 per
100, ?f 16x?fx1000 60,500 T-bond futures
100,000/Contract Net 956,875
60,5001,017,375
67
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
Futures Expiration Date
Purchasing Date
68
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(1yf)/Dff(1ys).
• (Table 9, p. 370)

69
3. Stock Index Futures Hedge (f CME index250)

• Note SP 500 Index CME 745.45 on 11/22/0x,
• f 745.45250 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 SP 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.

70
Ex. Hedging a Takeover ( Table 11, p. 374,
hedging a future purchase of stocks).

• Buy Nf SP 500 futures Contracts, Nf ?S/f,
  ????beta in CAPM

71
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

72
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(1r) Collateral) Net Cash 0 f-S(1r)0 r
is the repo
73
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) x1MM/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

74
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 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 (12.4/97.5)365/90 -1 10.36 for Euro

75
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-(100-94.46)(90/360) x1MM/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.

76
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 _at_5.35 pay price
  100-5.35174/360 97.4142 and sell the Dec.
  T-bill futures _at_ price 100-5.290/360
  98.7Synthetic Short-term T-B
• On Dec. 18, delivery the March T-bill for the
  futures received 98.7. Paid S97.4142 and get
  f98.7. The rate of return R 5.94 gt 5.44 the
  return on the Dec. T-bill. There is an arbitrage
  (why?)(98.7/97.4142)365/83-15.94
• On 9/26, Sell T-Bill Mature on 12/18 and Buy
  the March
• Mature T-Bill Sell Dec. T-Bill Futuresgt
  Arbitrage

77

• Ex

9/26
12/18
3/19
83 days
Current date
174 days
T-Bill Spot 98.8034 100-5.1983/360
Yield5.44
March T-Bill Spot Price 97.4142
100-5.35174/360
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
78
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 CD _at_ 8.75 or to issue a
  90-day CD _at_ 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 EuroCD is issued, then paid
  10,437,500 10MM1.0875(180)/360, or 9.07
• If 90-day CD is issued _at_ 8.25 and sell 10 Euro
  futures _at_ 91.37, then need to pay 10MM
  1.0825(90/360) on 12/16 and get 10978,425
  from futures pay 10980,100 to close the futures
  (loss 16,750). The firm needs to issue 10MM
  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 lt 9.07

79
Return on furures 2.1575 (100-91.37)/100/4

• Synthetic 180-Day CD

3 months return on CD 2.0625
Owe 10,223,000x (17.96/4) 10,426,438 get
10MM the cost of debt 8.84
Owe 10MM(18.25/4) 10,206,250 New 90-day CD
Rate 7.96. IMM 92.04gt f 98.01. Issue new
90-day CD for 10,206,250 (978425- 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 8.75. Owe 10MM(18.75x180/360) or
the cost of debt 9.07 gt 8.84
80
Conversion FactorDeliver 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)1-1.03-221/.03 1.03-221
  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 104,062.51.074067 3404
  115,174.07
• (Formula for CF see p.421)

81
Intermediate Long-Term Interest Rate Futures
Strategies

• Conversion FactorDeliver 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)1-1.03-221/.03 1.03-221
  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 104,062.51.074067 3404
  115,174.07
• (Formula for CF see p.421)

82

• 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 _at_ time t
• f(CF) AIT - (BAIt)(1r)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

83
ExampleGiven 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

8/15
2/15
13313031303115 181 days
2/15
6/11
4/15
13311559
15311157
AIt 6.25x59/1812.04 on 4/15, AIT
6.25x(5957)/181 4.01 from 2/15 to 6/11. Bond
price is Quoted 160.125(ask price). The Invoice
Price f(CF) AIT112.65625(1.4122)4.01161.98
on 6/11 (BAIt)(1r)T-t (160.1252.04)(1.0262)57
/365162.82 f(CF) AIT - (BAIt)(1r)T-t161.98
-162.82 -.84
84
Example Continue

• B 8.125 Coupon, Mature on 5/15/21, CF
  1.0137,
• B 116.21875, r 2.62

11/15
5/15
6/11
4/15
30 days
27days
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 -
(BAIt)(1r)T-t - FV of Coupon at T
112.65625(1.0137)0.6 - (116.218753.39)
(1.0262)57/365-4.07 -1.22,??12.5 Coupon is
Cheapter-t-D Bond than 8.125
85
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
86
Example Continue

• 1. The Cash Price Quoted Bond Price Accured
  Interest

120 6x60/180 121.978, 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)e0.7397x0.1 125.094 At
Delivery, There are 148 Days of Accured Interest,
The Quoted Futures Price Under 12 Coupon is
3. 125.094-6x148/183 120.242 The Quoted
Futures Price under 8 should be
4. 120.242/1.4 85.887

87
Delivery Options

• 1. Wild Card Option if S5 lt f3CF 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 (SAI)(1r)T
• r (f(CF) AIT)/(SAI)1/T - 1

88
Implied Repo/Cost of Carry

• Repo

Current Date
Expiration Date
Buy a Bond -(SAI) STAIT Borrow SAI
-(SAI)(1r)T Sell a T-Bond Futures
f(CF)AIT - (ST AIT)
Net Cash Flow 0 f(CF)AIT -(SAI)(1r)T 0
Investment 0 risk ??r (f(CF)AIT)/(SAI)1/T -1
8/15
66 Days
9/26
12/1
Ex. 12.5 Coupon
2/15


On 12/2/03, p.398. Given S141.5, AI 1.43
6.25(42/184), CF1.4662, f95.65625, AIT 3.669
6.25(108/184), r 3.89
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
T
Buy
Sell
_at_ Time t, Get T-Bond Pay ft(CFt)AIt ,Finance
By Repo Rate r. _at_ Time T, Deliver T-Bond Get
fT(CFT)AIT. 0 Net Cash Flow _at_ Time 0 t 0
risk at Time T ? (ft(CFt)AIt)(1r)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
90
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, CFM1.029, CFJ1.0289, fM108.09375,
fJ108.09375, AIM .35, AIJ 1.9 gt(implied repo
rate from 13/7-6/5)r (108.09375(1.0289)1.9)/(
108.09375(1.4662).35 )365/90 -1 .00092 (ex.
P. 401)
8/15 9/26 12/1 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(1yf)/Dff(1yS)
• Ex. DS7.83, DT4, S1.01MM, yf14.92 Df7.2, f
  70,500, yS 11.74, gt Nf -7.84 Sell 8
  Futures See Table 6, p. 403

91
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

92
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, ? ?SNf ?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

93
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 SP 500 to ? 8. ? 1.2 ?1.2x8 9.6 on the
  stock. To Hedge Selling SP 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 f765.3 March, Nf1.2(2,600,000)/765.3
x500 8.154, Sell 9 Futures
? from Stock 25,000 ? from Futures
65.3x 500x9 293,850, Total ? 318,850, Rate of
return 12.26
94
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
  ?SNf ?f ,
• the return on the portfolio rp (?SNf ?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, f765
• Nf (1.5-.9)5MM/765x5007.843, Buy 8 SP 500
  March index futures contracts Now

95
Put-Call-Futures Parity

• Pe Ce (E-f)(1r)-T vs. Pe Ce -S
  E(1r)-T

Expiration Date
Current Date
PV Buy a Put P E- ST 0 Buy a
Futures 0 ST-f ST -f E-f ST
-f Buy a Call C 0 ST -E Buy a Bond w/
PV(E-f) PV(E-f) E-f E-f E-f
ST -f
ST ??E
ST???E
96
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

97
Options on Futures

• To give the buyer the right to buy (or sell) a
  futures contract _at_ 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.

98
Black Option on Futures Pricing Model

• C(f,T,?2,E, r) e-rcTfN(d1) - EN(d2)
• where, d1 ln(f/E) .5?2T/s?T
• d2 d1 -s??
• Ex
• - .

99
Put-Call Parity

• Ce(f,T,E) Pe(f,T,E) (f-E)(1r)-T
• Ex. Pe(f,T,E) 7.45, f 320, E315, r 5.46,
  T .25, then Ce(f,T,E) 7.45 5(1.0546)-.25
  12.52

100
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

101
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)

102
Example

• Borrowing Rate Fixed Floating
• Company A 10 6-month LIBOR 0.3
• Company B 11.2 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
LIBOR1
LIBOR0.05
10
B Borrows _at_ LIBOR1 A Borrows _at_ Fixed 10 Then
Rnter a Swap to Ensure that A Ends Up Floating
Rate
A pays 10/year to Outside Lender, Receive
9.95/year from B, Pays LIBOR to B
103
Example

• Company B Cash Flow
• 1. Pay LIBOR1 to Outside Lender
• 2. Receive LIBOR from A
• 3. Pays 9.95 to A
• Company A Net Cash Flow with Swap
• -109.95-(LIBOR) -(LIBOR0.05)
• Without Swap, Company A Pays LIBOR0.3, Save
  0.25
• Company B Net Cash Flow with Swap
• -(LIBOR1)-9.95LIBOR -10.95
• Without Swap, Company A Pays 11.2, Save 0.25
• The Total Gain 11.2-10 - (LIBOR1) -
  (LIBOR 0.3 ) 0.5.

104
Role of Financial Intermediary (Net 0.1)

• A Cash Flow (Net LIBOR0.1, 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
LIBOR 1
10
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
105
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)

106

• Spot Forward Rate
• Term Structure of Interest Rate (Based on Pure
  Discount Bond)
• Bond Pricing B ?PV(Ci) PV(Par) _at_ 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

Spot Rates
Forward Rate
107
Example Estimating the Term Structure
S1
f1
f2
f3
Spot Rate S1
S2 (1S1 )(1 f1 )-1
S3 (1S2 )(1 f2 )-1
S4 (1S3)(1 f3)-1
Note fi is derived from the T-bill Futures
Price Si1 (1Si)(1fi) Annualize then - 1
108

• 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
  100/98.7375365/90 see p.373
• Note IMM quotes based on a 90-day T-bill
  w/360-day year.

109
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 - (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

110
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 gt 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???gt???Losses (Table 6, p. 426)

Buy a T-Bill Pay f
Get the 1MMPar
Futures Expired
Now, Buy a Futures
Money
111
June

• Example

2/15
Futures Expired
5/17
Given, forward discount 8.94 ??Implied forward
rate 9.6 IMM 91.32 ??f 97.83 Buy a Futures
at 97.83
T-Bill Expired Get 1MM
Close Out Date Given IMM92.54 ??new f98.135,
Net from futures -97.8398.135 0.305 Buy a
T-Bill _at_ discount 7.69 or S 98.056 ?Net Cost
of a T-Bill 98.056-.30597.751
?
w/Hedge, the Rate of Return 9.55 (100/97.751)3
65/91 -1 w/o Hedge, the Rate of Return 8.19
(100/98.056)365/91 -1
(Lock in the forward rate _at_ 9.6)
112
b. Anticipatory Hedge of a future 10MM
commercial paper Issue (Use Euro futures (IMM),
size1 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(1yf)/Dff(1ys) (p.429)

113
Ex. Hedge Future Commercial Paper Issue
4/6
7/20
Sept
Issue 10MM (180Days) C P _at_ Spot Rate
11.34 100-11.34(180/360) 94.33 per
100 (100/94.33)365/180- 1 .1257 if No Hedge
Futures Expired
Given IMM of Sept 88.23 gtf 97.0575 yf
(100/f)365/90 -1 .1288, Given 180-day C
P Implied forward Rate 10.37,
Price 100-10.37(180/360) ys (100/P)365/180-1 .1
14, Nf -19.8 Sell 20 Futures(Sept) Contracts
(Hedge)
(Lock in the forward rate _at_ 11.4)
IMM 87.47, f 96.8675, ?f 0.19/100 100/(94.3
3.38)365/180 -1 11.65 Cost of Fund if
Hedge Note 1000/(943.33.8)100/(94.33.38)
3.8 .19x20 (Contracts)
114
Ex. Hedging a Floating Loan(Lock in _at_ 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 976,875gt yf 9.95, ys
  .1122(110.68/12)12-1,
• Nf - DsS(1yf)/ Dff(1ys)
• -(1/12)10MM(1.0995)/ (1/4)(9.76875)(1.1122)
• -3.37 and
• Nf -(1/12)10089000(1.0995)/ (1/4)(9.76875)(1.1
  122)
• -3.4
• Sell 6 futures with three to be closed out on
  March and three on April.(see Table 8, p.431)

115

• Example Heading a Floating Rate Loan (3 Months)

Futures Expired
3/2
4/6
5/4
2/3
IMM90.47 f97.6175, ?f.07, or 700/Futures x3
2,100 Total Liabil. 10MM(1 .1068/12)
10,089,000 - 2,100 10,086,900 New
LIBOR 10.09
IMM89.99, f 97.4975 ?f .19,
or 1900/Futures x35,700 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) 10,274,3
84
LIBO(90days)9.68, Get 10MM Loan, Like to Lock
in the (1.1068/12)12 -1 .1122 ys IMM90.75,
f 97.6875, 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
116

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