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					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

                f1 = 105               f4= 110
      f0 =100               f2 = 103
Long Futures Paid -110+7-2+5 = -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. .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
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     Buyers              Buyers Brokers’
          Broker              Commission Broker
 Margin


                                  Exchange
                                    (Trade)

           Margin

                                 Clearinghouse
                                     (Record)
   Margin: (p. 286-287)
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.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)]
  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
   - (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
    $111,531.25
   Expiration: March, June, Sept, Dec.
• Last trading Day: the Business Day Prior to the
  Last seven days of the expiration month.
• The First Delivery Day = The First Business Day of
  the Month
• T-Notes Futures: Same As T-Bond Except the
  maturity from 0-2 years, 4-6 and 6.5-10 years T-
  Bond or Notes
Other Futures
• Agricultural Commodity Futures
• Stock Indices Futures
• Natural Resources Futures
• Miscellaneous Commodities Futures
• Foreign Currency Futures
• T-Bills & Euro$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
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        ft
F = forward price today
f = futures price today                 0          t     T
                                        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 = vT = fT - 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 +fT) + (ft - fT) = 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 -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(S T).
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 commodity @ S and sell a futures
 contract @ 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
           0                    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 = $99.47
In General
 f = S(1+r)T -  Dt(1+r)(T-t) = Future Spot Price - FV(D)
  = [S - PV(D)](1+r)T = S + 
 For Continuous Dividends: f = Se(rc-)T = [S-PV(D)]ercT
 = S +  (where  is the dividend yield), rc = continuously
 compound risk-free rate.
 Ex. S = 85,  = 8%, rc = 10%, T = 90 day =
 0.246575yr, f = 85e(0.1-.08)0.246575
          Interest Rate Parity
• F=S(1+r)T/(1+ρ)T
• S=Spot Exchange Rate/$
• 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
 Pricing of Spreads (Different
       Expiration Dates)
f1 = S + 1
f2 = S + 2
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 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)x100]. If
  Investor Decides to close out the long futures for a gain of
  (340-338)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
                                  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 - S0 (from spot market) - (fT -
       f0) (from futures market)
       
      (long hedge) = -ST + S0 (from spot market) + (fT -
       f0) (from futures market)
     • If hedge is closed prior to expiration,
          (short hedge) = St - S0 - (ft - f0)
     • If hedge is held to expiration, St = ST = fT = ft.
Basis:                              Spread

 b0  S - f (initial basis)                  Spot
 bt  St - ft (basis @ t)
 bT ST - fT (basis @ expiration)            futures
Profit from Hedge Strategy  :                   t         T
  T Profit of long spot and short future(i.e.,Short Hedge)
    = (ST - S) + (f - fT) = f - S = - b0 (Buy @ S and Sell @ 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.
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
   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.
       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.
  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.
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 +  fNf
  – Variance of profit from short hedge:
     2 =S2 + f2Nf2 + 2SfNf
  – The optimal (variance minimizing) hedge ratio is (see
    Appendix 10A)
     • Nf = - Sf/f2
     • 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 10A for
     derivation.):


   – Note that DURS -(S/S)(1 + yS)/yS 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 = -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 = -b in the
  regression of S on f
• Effectiveness of Hedge
     e* = (2S - 2)/2S = N2f 2f /2S

• Consider: S =  + b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) + 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   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
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)]/[D ff/(1+yf)]
C.      Stock Index Futures Hedge (p. 361)

     From the Minimum Variance Hedge [S = rsS, f = rff ]
     Nf = - b s(S/f), where b s is obtained by regression of
      rs = + b 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. $.8310 for ¥ is ¥12,500,000x$.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          Spot Mkt                        Futures Mkt
   7/1    $1.319/£, F=$1.306/ £      fD=$1.278/£,
          Forward Cost                #of Contract=
          =20(35000)x1.306           20(35,000)/62,500=11.2
          =$914,200 Forward H        Buy 11 Currency Futures

   11/1    S=$1.442/£, Total Cost in $   fD=$1.4375/£,
           $700,000(1.442)=$1.009,400     Sell 11 Contracts
Cost $1,009,400-$914,200=$95,200 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
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 S=$1.362/£,F=$1.357/£   Sell £10MM Forward
                               Currency @$1.375/£

  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, ABC to Borrow $ at        Strip:
                                  On 1/2 :Sell 15 March , 45
   3/1         $15MM
                                  June, 20 Sep and 10 Dec
   6/1          45                contracts.
   9/1            20              On 3/1 Buy 15 Futures
   12/1           10              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 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
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=95.6875
B=101,Ds =7.83,      Sold $1MM Gov't Bond get
ys=.1174.yf=.1492    $956,875,(Loss $53,125 w/o Hedge)
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 =16xfx1000 =$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 b = 10.5. So, Nf = 11. (Table
  10) [Regression function: S =  + bf + ] (different Nf)

Current Date                Purchasing              Futures
                            Date                    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 = 745.45 on 11/22/0x,
  f = 745.45*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 = -b sS/f.
   Mkt Value weighted betas to get b 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 = bS/f,
 b=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 0                             f-
   S(1+r)=0
   r is the repo
      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$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 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-(100-
     94.46)(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 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 = 100-
  5.35*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    12/18                        3/19
Current date                                      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            Close out the
at $97.4142 &             Position, get
Sell a Dec.               $98.7, Yield
Futures at $98.7          = 5.94%
Buy a T-B (March) & Sell a Dec. Futures to Create a
Synthetic Dec T-Bill
Euro$ 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 $10MM
  of 180 day Euro$ 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 =
  $10MM[1+.0875(180)/360], or 9.07%
• If 90-day CD is issued @ 8.25 and sell 10 Euro$ futures @
  91.37, then need to pay 10MM [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 $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% < 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 10MM(1+8.25/4) Owe
day CD Rate 8.25 =$10,206,250          10,223,000x
Issue 90 day CD  New 90-day CD Rate    (1+7.96/4)=
for $10MM        7.96. IMM= 92.04=>    10,426,438
IMM 91.37/Dec    f= 98.01. Issue new   get $10MM
Sell 10 Futures   90-day CD for        the cost of
at $978,425 each 10,206,250 + (978425- debt 8.84%
                  980100)x10
Annual Return from 90-day CD & Furures = 8.84%
                     180 Days
 180-day CD Rate 8.75. 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)[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).
• CF0 = (.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+30+31+15 =181 days
   2/15           4/15         6/11
     13+31+15=59      15+31+11=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 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              11/15
         30     27days
        days
                                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 & 15Yrs 8%
 Coupon     Current          Coupon           Maturity     Coupon
 Payment    Time             Payment          Of Futures   Payment
      60              122              148              36
     Days             Days                             Days
                                       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 + 6x[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)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


                                                    $
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

• Repo
  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 0                f(CF)+AIT -(S+AI)(1+r)T
0 Investment 0 risk r = [(f(CF)+AIT)/(S+AI)]1/T -1
Ex. 12.5% Coupon 8/15      9/26 66 Days      12/1      2/15
                    $                                      $
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                          T
              Buy
                        Sell
  @ Time t, Get T-Bond & Pay ft(CFt)+AIt ,Finance By Repo
  Rate r. @ Time T, Deliver T-Bond & Get fT(CFT)+AIT. 0
  Net Cash Flow @ 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=108.09375, fJ=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 = .00092 (ex. P. 401) 12/1
            8/15       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+yS )
• Ex. DS=7.83, DT=4, S=1.01MM, yf=14.92% Df=7.2,
  f = 70,500, yS= 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 $1MM mkt
  value& At least 15
  Stocks transaction
Speculating on Unsystematic Risk (Individual Stock)
rS = brM + S, Or,SrS = SbrM + SS
S = Sb(M/M) + SS , M is the mkt index
Given,  = S+Nf f , and Nf = -b(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%. b =1.2 1.2x8% =9.6% on the stock. To Hedge:
  Selling S&P 500 index futures
  12/2                                            2/28


Own 100,000 shares of Bay at         Stock price is 26.25
26, S = $2,600,000                   f= 700, Buy 9 Futures
f=765.3 March,                       to Close out
                                 from Stock = $25,000
Nf=1.2(2,600,000)/765.3x500
                                 from Futures = 65.3x
=8.154, Sell 9 Futures
                                500x9= $293,850, Total
                                 = 318,850, Rate of return
                                =12.26%
Stock Mkt Timing w/Futures: (Adjust b by Futures)
• Buying or selling futures to  or  portfolio b
Given Nf = -b 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]b T, b T is the target b
  E(rS) = r + [E(rM)-r]b S and E(f /f) = E(rM)-r ,
  Nf = (b Tb S)(S/f)
   from 0-beta risk hedge ratio Nf = -b S(S/f) to target b T
  risk hedge ratio Nf = (b Tb S)(S/f)
Ex: On 12/2 current b=.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
  Current Date                       Expiration Date

                                    ST E     STE
                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
  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-rcT[fN(d1) - EN(d2)]
  where, d1 = [ln(f/E) + .52T]/sT
  d2 = d1 -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:   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
                                9.95%
A Swap is Created:      A                      B    LIBOR+1%
                    10%      LIBOR+0.05%
A pays 10%/year to Outside        B Borrows @ LIBOR+1%
Lender, Receive 9.95%/year        A Borrows @ Fixed 10% &
                                  Then Rnter a Swap to Ensure
from B, Pays LIBOR to B           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
             9.9%                    10.0%
                      Financial
       A                                      B
                      Institutio                 LIBOR
 10%         LIBOR                   LIBOR
                      n                           + 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 $1Euro$ 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

     S1          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 - (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-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
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?).

Now, Buy Buy a T-Bill Futures               Get the
a Futures          Pay f       Expired     1MMPar
 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       2/15                             June

    Given, forward           5/17            Futures T-Bill
    discount 8.94       Close Out Date       Expired Expired
    *Implied          Given IMM=92.54              Get
    forward rate 9.6%   new f=98.135,              $1MM
    IMM = 91.32         Net from futures
                                                        
    f = 97.83         = -97.83+98.135
    Buy a Futures at    = 0.305
    97.83               Buy a T-Bill @
                        discount 7.69 or
w/Hedge, the Rate of    S = 98.056 Net
Return = 9.55%=         Cost of a T-Bill
(100/97.751)365/91 -1   98.056-.305=97.751
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 $10MM
commercial paper Issue (Use: Euro$ 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
Given IMM of Sept Issue $10MM (180Days) C P Futures
88.23 =>f =97.0575 @ Spot Rate 11.34                     Expired
yf = (100/f)365/90 -1 100-11.34(180/360)
=.1288,                =94.33 per $100        (Lock in the forward
                                   365/180- 1 rate @ 11.4%)
Given 180-day C P (100/94.33)
Implied forward        =.1257 if No Hedge
Rate 10.37%, Price
100-10.37(180/360) IMM = 87.47, f = 96.8675, f = 0.19/100
ys = (100/P) 365/180-1  [100/(94.33+.38)]365/180 -1= 11.65 Cost
=.114, Nf = -19.8       of Fund if Hedge
Sell 20 Futures(Sept) Note: 1000/(943.3+3.8)=100/(94.33+.38
Contracts (Hedge)             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=(1+10.68%/12)12-1,
• Nf = - DsS(1+yf)/ Dff(1+ys)
     = -(1/12)10MM(1.0995)/ [(1/4)(9.76875)(1.1122)]
     = -3.37 and
   Nf = -(1/12)10089000(1.0995)/ [(1/4)(9.76875)(1.1122)]
   = -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
  2/3              3/2          4/6          5/4 Expired
LIBO(90days)=     IMM=90.47       IMM=89.99,      Pay Total Debt
9.68%,            f=97.6175,      f = 97.4975     $10,174,420(1+
Get 10MM          f=.07, or      f =.19, or     .1179/12)=
Loan, & Like to   700/Futures     1900/Futures    $10,274,384
Lock in the       x3 = $2,100     x3=$5,700
(1+.1068/12)12    Total Liabil.   Total Liabil.   Cost of Debt
-1 = .1122= ys    =10MM(1+        10086900(1+     (10,274,384/10
IMM=90.75, f      .1068/12) =     .11.09/12) =    MM)4= .1144
=97.6875, yf=     $10,089,000     $10,180,120     with Hedge
.0995,Nf =-3.37    -     2,100      -    5,700    w/o Hedge =[1
Sell 6 Euro$      $10,086,900     $10,174,420     +.1068/12)(1+
Futures           New LIBOR       New LIBOR       .1109/12)(1+.11
                  =10.09          =10.79          79/12)]4=.1178
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