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7. Derivatives Part1 Pdf

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7. Derivatives Part1 Pdf

  1. 1. Derivatives – Part 1 (LOs 27.x – 31.x) Forwards (LO 27.x) 1 Hedging Strategies (LO 28.x) 2 3 Interest Rate Futures (LO 29.x) 4 Commodity Futures (LO 30.x) 5 Swaps (LO 31.x)
  2. 2. Forwards/Futures rT F0 E(ST ) F0 S0 e F0 F0 FT-1 ST ST=FT ST-1 S0 Time (T)
  3. 3. Forwards/Futures (Commodity with High F0 = E(ST) “Convenience Yield” or High-dividend Financial Asset) S0 ST-1 F0 F0 FT-1 ST ST=FT Time (T)
  4. 4. Cost-of-carry model LO 27.1: State & explain cost-of-carry model for forward prices with & without interim cash flows Risk-free Storage Rate (r) Cost (U, u) Income/ Convenience Dividend (y) Forward (q) (F0) Spot (S0) Time (T)
  5. 5. Cost-of-carry: Question A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk- free rate is 5% for all maturities. What the price of a (long) forward contract (F0) to purchase the stock in one year?
  6. 6. Cost-of-carry: Question A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk- free rate is 5% for all maturities. What the price of a (long) forward contract (F0) to purchase the stock in one year? rT F0 ( S0 I )e F0 ( 0.05)(6/12) (.05)(1) ($50 [($1)e ])e $51.538
  7. 7. Derivatives LO 27.2: Compute the forward price given both the price of the underlying and the appropriate carrying costs of the underlying. Commodity ( r u q y )T F0 S0 e Financial asset (e.g., stock index) ( r q )T F0 S0 e
  8. 8. Cost-of-Carry Model Cost of carry = interest to finance asset (r) + storage cost (u) - income earned (q) Commodity ( r u q y )T ( r y )T F0 S0 e F0 ( S0 U I )e constant rates as % Present values U = Present value, storage costs u = storage costs I = Present value, income q = income (dividend) y = convenience yield
  9. 9. Cost-of-Carry Model Cost of carry = interest to finance asset (r) + storage cost (u) - income earned (q) Financial asset (e.g., stock index) ( r q )T rT F0 S0 e F0 ( S0 I )e constant rates as % Present values q = income (dividend) I = Present value, income
  10. 10. Cost-of-Carry: Question The spot price of corn today is 230 cents per bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum. What is the forward price in four (4) months?
  11. 11. Cost-of-Carry: Question The spot price of corn today is 230 cents per bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum. What is the forward price in four (4) months? ( r u)T (6%/12 1.5%)(4) F0 S0 e (230)e (.02)(4) 230e 249.16
  12. 12. Derivatives LO 27.3: Calculate the value of a forward contract. rT f ( F0 K )e
  13. 13. Value of a forward contract A long forward contract on a non dividend-paying stock has three months left to maturity. The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%. What is the value of the forward contract?
  14. 14. Value of a forward contract A long forward contract on a non dividend-paying stock has three months left to maturity. The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%. What is the value of the forward contract? rT (5%)(0.25) F0 S0 e 10e $10.126 rT (5%)(0.25) f ( F0 K )e (10.126 8)e $2.153
  15. 15. Derivatives LO 27.4: Describe the differences between forward and futures contracts. Forward vs. Futures Contracts Forward Futures Trade over-the-counter Trade on an exchange Not standardized Standardized contracts One specified delivery date Range of delivery dates Settled at contract’s end Settled daily Delivery or final cash Contract usually closed settlement usually occurs out prior to maturity
  16. 16. Derivatives LO 27.5: Distinguish between a long futures position and a short futures position. A long-futures position agrees to buy in the future  A short-futures position agrees to sell in the future.  Price mechanism maintains a balance between buyers and  sellers.(market equilibrium) Most futures contracts do not lead to delivery, because  most trades ―close out‖ their positions before delivery. Closing out a position means entering into the opposite type of trade from the original.
  17. 17. Derivatives LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled. An (underlying) asset  A Treasury bond futures contract is on underlying U.S. Treasury with maturity of at least 15 years and not callable within 15 years (15 years ≤ T bond).  A Treasury note futures contract is on the underlying U.S. Treasury with maturity of at least 6.5 years but not greater than 10 years (6.5 ≤ T note ≤ 10 years).  When the asset is a commodity (e.g., cotton, orange juice), the exchange specifies a grade (quality).
  18. 18. Derivatives LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled. Contract size varies by type of futures contract  Treasury bond futures: contract size is a face value of $100,000  S&P 500 futures contract is index $250 (multiplier of 250X)  NASDAQ futures contract is index $100 (multiplier of 100X) Recently, ―mini contracts‖ have been introduced:  S&P 500 ―mini‖ = $50 x S&P Index  NASDAQ ―mini‖ = $20 x NASDQ  (each contract is one-fifth the price, to attract smaller investors)
  19. 19. Derivatives LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled. Delivery Arrangements  The exchange specifies delivery location. Delivery Months  The exchange must specify the delivery month; this can be the entire month or a sub-period of the month.
  20. 20. Derivatives LO 27.7: Describe the marking-to-market procedure, the initial margin, and the maintenance margin. Margin account: Broker requires deposit.  Initial margin: Must be deposited when contract is  initiated. Mark-to-market: At the end of each trading day, margin  account is adjusted to reflect gains or losses.
  21. 21. Derivatives LO 27.7: Describe the marking-to-market procedure, the initial margin, and the maintenance margin. Maintenance margin: Investor can withdraw funds in the  margin account in excess of the initial margin. A maintenance margin guarantees that the balance in the margin account never gets negative (the maintenance margin is lower than the initial margin). Margin call: When the balance in the margin account falls  below the maintenance margin, broker executes a margin call. The next day, the investor needs to ―top up‖ the margin account back to the initial margin level. Variation margin: Extra funds deposited by the investor  after receiving a margin call.
  22. 22. Derivatives LO 27.8: Compute the variation margin. There is only a variation margin if and when there is a  margin call. Variation margin = initial margin – margin account balance The maintenance margin is a trigger level—once  triggered, the investor must ―top up‖ to the initial margin, which is greater than the maintenance level.
  23. 23. Derivatives LO 27.9: Explain the role of the clearinghouse. The exchange clearinghouse is a division of the exchange  (e.g., the CME Clearing House is a division of the Chicago Mercantile Exchange) or an independent company. The clearinghouse serves as a guarantor, ensuring that the obligations of all trades are met.
  24. 24. Derivatives LO 27.9: Explain the role of the clearinghouse. Market order: Execute the trade immediately at the  best price available. Limit order: This order specifies a price (e.g., buy at $30  or less)—but with no guarantee of execution. Stop order: (aka., stop-loss order) An order to execute  a buy/sell when a specified price is reached.
  25. 25. Derivatives LO 27.9: Explain the role of the clearinghouse. Stop-limit: Requires two specified prices, a stop and a  limit price. Once the stop-limit price is reached, it becomes a limit order at the limit price. Market-if-touched: Becomes a market order once  specified price is achieved. Discretionary (aka., market-not-held order): A  market order, but the broker is given the discretion to delay the order in an attempt to get a better price.
  26. 26. Derivatives LO 28.1: Differentiate between a short hedge and a long hedge, and identify situations where each is appropriate. A short forward (or futures) hedge is an agreement to sell in the future and is appropriate when the hedger already owns the asset. Classic example is farmer who wants to lock in a sales price:  protects against a price decline. A long forward (or futures) hedge is an agreement to buy in the future and is appropriate when the hedger does not currently own the asset but expects to purchase in the future. Example is an airline which depends on jet fuel and enters into a  forward or futures contract (a long hedge) in order to protect itself from exposure to high oil prices.
  27. 27. Derivatives LO 28.2: Define and calculate the basis. Basis = Spot Price Hedged Asset –  Futures Price Futures Contract = S0 – F0 Basis =  Futures Price Futures Contract – Spot Price Hedged Asset = F0 – S0 Hull says first is correct but second is common for financial assets (either is okay)
  28. 28. Basis risk No hedge Spot = -$0.50 SPOT $2.50 $2.00 Short 1.67 F Basis $0.30 $0.10 Spot = -$0.50 Forward $2.20 Future = $0.30 (1.67) $1.90 Net = 0 T0 T1 Time (T) Weakening of the basis = Futures price increases more than spot
  29. 29. Basis risk No hedge Spot = -$0.50 SPOT $2.50 $2.00 Short 1.67 F Basis $0.30 $0.30 Spot = -$0.50 Forward $2.20 Future = $0.50 (1.67) $1.90 Net = +33.5 $1.70 T0 T1 Time (T) Basis unchanged. But unexpected strengthening= Hedger improved!
  30. 30. Basis risk T1 $2.60 $2.60 SPOT $2.50 $0.0 Basis $0.30 Short 1.67 F Spot = +$0.10 Forward $2.20 $1.90 Future = -$0.40 (1.67) Net = -56.8 T0 Time (T) Basis declines Unexpected weakening= Hedger worse!
  31. 31. Derivatives LO 28.3: Define the types of basis risk and explain how they arise in futures hedging. Spot price increases by more than the futures price   basis increases. This is a ―strengthening of the basis‖ When unexpected, strengthening is favorable for a short  hedge and unfavorable for a long hedge Futures price increases by more than the spot price   basis declines. This is a ―weakening of the basis‖ When unexpected, weakening is favorable for a long hedge  and unfavorable for a short hedge
  32. 32. Derivatives LO 28.3: Define the types of basis risk and explain how they arise in futures hedging. But basis risk arises because often the characteristics  of the futures contract differ from the underlying position. Contract ≠ Commodity  Contract is standardized (e.g., WTI oil futures)  Commodities are not exactly commodities (e.g., hedger has a  position in different grade of oil) Basis risk higher with cross-hedging
  33. 33. Derivatives LO 28.3: Define the types of basis risk and explain how they arise in futures hedging. But basis risk arises because often the characteristics  of the futures contract differ from the underlying position. Contract ≠ Commodity.  Contract is standardized (e.g., WTI oil futures)  Commodities are not exactly commodities (e.g., hedger has a  position in different grade of oil) Trade-off Liquidity Basis risk (exchange)
  34. 34. Derivatives LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio. The optimal hedge ratio (a.k.a., minimum variance hedge  ratio) is the ratio of futures position relative to the spot position that minimizes the variance of the position. Where is the correlation and is the standard deviation, the optimal hedge ratio is given by: S h* F
  35. 35. Derivatives LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio. For example, if the volatility of the spot price is 20%, the  volatility of the futures price is 10%, and their correlation is 0.4, then: 20% S h* h* (0.4) 0.8 10% F
  36. 36. Derivatives LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio. For example, if the volatility of the spot price is 20%, the  volatility of the futures price is 10%, and their correlation is 0.4, then: 20% S h* h* (0.4) 0.8 10% F h * NA Number of N* contracts QF
  37. 37. Derivatives LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.  Given a portfolio beta ( ), the current value of the portfolio (P), and the value of stocks underlying one futures contract (A), the number of stock index futures contracts (i.e., which minimizes the portfolio variance) is given by: P N A
  38. 38. Derivatives LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock.  Byextension, when the goal is to shift portfolio beta from ( ) to a target beta ( *), the number of contracts required is given by: P N(* ) A
  39. 39. Futures: Question LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock. Assume: Value of S&P 500 Index is 1240  Value of portfolio is $10 million  Portfolio beta ( ) is 1.5  How do we change the portfolio beta to 1.2? Hint: Contract = ($250 Index) P N (* ) and # of futures is given by: A
  40. 40. Futures: Question LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock. Assume: • Value of S&P 500 Index is 1240 • Value of portfolio is $1 million • Portfolio beta ( ) is 1.5 P N (* ) A $10,000,000 (1.2 1.5) 9.7 (1240)(250) We short about 10 contracts. (-) indicates short, (+) long…
  41. 41. Derivatives LO 28.6: Identify situations when a rolling hedge is appropriate, and discuss the risks of such a strategy. When the delivery date of the futures contract occurs  prior to the expiration date of the hedge, the hedger can roll forward the hedge: close out a futures contract and take the same position on a new futures contract with a later delivery date. Exposed to:  Basis risk (original hedge)  Basis risk (each new hedge) = ―rollover basis risk‖ 
  42. 42. Derivatives LO 29.1: Identify and apply the three most common day count conventions. Actual/actual U.S. Treasuries 30/360 U.S. corporate and municipal bonds Actual/360 U.S. Treasury bills and other money market instruments
  43. 43. Derivatives LO 29.2: Explain the U.S. Treasury bond (T-bond) futures contract conversion factor. The Treasury bond futures contract allows the party with  the short position to deliver any bond with a maturity of more than 15 years and that is not callable within 15 years. When the chosen bond is delivered, the conversion factor defines the price received by the party with the short position: Cash Received = Quoted futures price Conversion factor + Accrued interest = (QFP CF) + AI
  44. 44. Derivatives LO 29.3: Calculate the Eurodollar futures contract convexity adjustment. The convexity adjustment assumes continuous  compounding. Given that ( ) is the standard deviation of the change in the short-term interest rate in one year, t1 is the time to maturity of the futures contract and t2 is the time to maturity of the rate underlying the futures contract: 1 2 Forward = Futures t 1 t2 2
  45. 45. Derivatives LO 29.4: Formulate a duration-based hedging strategy using interest rate futures. The number of contracts required to hedge against an  uncertain change in the yield, given by y, is given by: PDP N* FC DF FC = contract price for the interest rate futures contract. DF = duration of asset underlying futures contract at maturity. P = forward value of the portfolio being hedged at the maturity of the hedge (typically assumed to be today’s portfolio value). DP = duration of portfolio at maturity of the hedge
  46. 46. Derivatives LO 29.4: Formulate a duration-based hedging strategy using interest rate futures. Assume a portfolio value of $10 million.  The fund manager will hedge with T-bond futures (each  contract is for delivery of $100,000) with a current futures price of 98. She thinks the duration of the portfolio at hedge maturity will  be 6.0 and the duration of futures contract with be 5.0. How many futures contracts should be shorted? 
  47. 47. Derivatives LO 29.4: Formulate a duration-based hedging strategy using interest rate futures. Assume a portfolio value of $10 million.  The fund manager will hedge with T-bond futures (each  contract is for delivery of $100,000) with a current futures price of 98. She thinks the duration of the portfolio at hedge maturity will  be 6.0 and the duration of futures contract with be 5.0. How many futures contracts should be shorted?  PDP ($10 million)(6) N* 122 FC DF (98,000)(5)
  48. 48. Derivatives LO 29.5: Identify the limitations of using a duration- based hedging strategy. Portfolio immunization or duration matching is when a  bank or fund matches the average duration of assets with the average duration of liabilities. Duration matching protects or ―immunizes‖ against  small, parallel shifts in the yield (interest rate) curve. The limitation is that it does not protect against nonparallel shifts. The two most common nonparallel shifts are: A twist in the slope of the yield curve, or  A change in curvature 
  49. 49. Derivatives LO 30.1: Explain the derivation of the basic equilibrium formula for pricing commodity forwards and futures. Risk-free Rate (r) (r )T F0,T E0 (ST )e Discount Forward rate ( ) (F0) Spot (S0) Time (T)
  50. 50. Derivatives LO 30.1: Explain the derivation of the basic equilibrium formula for pricing commodity forwards and futures. (r )T F0,T E0 (ST )e E0 (ST ) Spot price of S at time T, as expected at time 0 F0,T Forward price r Risk-free rate Discount rate for commodity S
  51. 51. Derivatives LO 30.2: Define lease rates, and discuss the importance of lease rates for determining no- arbitrage values for commodity futures and forwards.  Lease rate = commodity discount rate – growth rate  Lease rate dividend yield (r )T F0,T S0e Financial asset ( r q )T F0 S0 e
  52. 52. Derivatives LO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation. Contango refers to an upward-sloping forward curve  which must be the case if the lease rate is less than the risk-free rate. Backwardation refers to a downward- sloping forward curve which must be the case if the lease rate is greater than the risk-free rate.
  53. 53. Derivatives LO 30.3: Explain how lease rates determine whether a forward market is in contango or backwardation. Forward (F0) Spot E(ST) (S0) Forward (F0) Time (T) Research says normal backwardation is “normal:” speculators want compensation (risk premium) for buying the futures contract
  54. 54. Derivatives LO 30.4: Explain how storage costs impact commodity forward prices, and calculate the forward price of a commodity with storage costs. Risk-free Storage Cost ( ) Rate (r) negative dividend Lease rate ( ) Convenience (y) Forward dividend dividend (F0) Spot (S0) Time (T)
  55. 55. Derivatives LO 30.4: Explain how storage costs impact commodity forward prices, and calculate the forward price of a commodity with storage costs. Storage Risk-free Cost ( ) Rate (r) (r c )T F0 S0 e Convenience Lease Forward (y) rate ( ) (F0) Spot (S0) Time (T)
  56. 56. Derivatives LO 30.5: Explain how a convenience yield impacts commodity forward prices, and determine the no-arbitrage bounds for the forward price of a commodity when the commodity has a convenience yield. Risk-free Storage Risk-free Storage Rate Cost Rate Cost (r c )T (r )T S0 e F0 S0 e Convenience Yield
  57. 57. Commodity Futures S&P 500 Index 1640 1620 1600 1580 1560 1540 1520 Rational forward curve rises by cost 1500 of capital (risk free + premium) less 1480 dividends 1460
  58. 58. Commodity Futures LO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures. Gold futures 900 850 800 750 700 650 Durable, (relatively) cheap to store. 600 550 Forward curve is “uninteresting” 500 Jul-07 Nov-08 Mar-10 Aug-11 Dec-12
  59. 59. Commodity Futures LO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures. Corn 450 400 350 300 250 200 Jun-09 Jun-08 Jun-10 Mar-09 Mar-08 Mar-10 Sep-07 Sep-08 Sep-09 Sep-10 Dec-07 Dec-08 Dec-09 Dec-10
  60. 60. Commodity Futures LO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures. Natural Gas 10 9 8 7 Costly to transport. Costly to store 6 (storage costs). Highly seasonal 5 May-08 May-09 May-10 Aug-07 Aug-08 Aug-09 Aug-10 Nov-07 Feb-08 Nov-08 Feb-09 Nov-09 Feb-10 Nov-10
  61. 61. Commodity Futures LO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures. Crude oil 76 75 74 73 72 71 70 Compared to natural gas, easier to store and 69 transport. Global market. Long-run forward price less (<) volatile than short-run forward. 68 Jan-10 Jan-08 Mar-08 Jan-09 Mar-09 Mar-10 Sep-07 Nov-08 Nov-07 Sep-08 Sep-09 Nov-09 Sep-10 Nov-10 May-08 May-09 May-10 Jul-08 Jul-09 Jul-10
  62. 62. Commodity Futures LO 30.7: Describe and calculate a commodity spread. If we can take a long position on one commodity that is an  input (e.g., oil) into another commodity that is an output (e.g., gas or heating oil), then we can take a short position in the output commodity and the difference is the commodity spread. Assume oil is $2 per gallon, gasoline is $2.10 per gallon and  heating oil is $2.50 per gallon. If we take a long position in 2 gallons of gasoline and one  gallon of heating oil, plus a short position in three gallons of oil, the commodity spread = (2 long gasoline $2.10) + (1 long heating oil $2.50) – (3 oil $2) = +$0.70
  63. 63. Commodity Futures LO 30.8: Define basis risk, and explain how basis risk can occur when hedging commodity price exposure. The basis is the difference between the price of the  futures contract and the spot price of the underlying asset. Basis risk is the risk (to the hedger) created by the uncertainty  in the basis. The futures contract often does not track exactly with  the underlying commodity; i.e., the correlation is imperfect. Factors that can give rise to basis risk include: Mismatch between grade of underlying and contract  Storage costs  Transportation costs 
  64. 64. Commodity Futures LO 30.9: Differentiate between a strip hedge and a stack hedge. Oil producer to deliver  10K barrels per month Strip hedge: contract for  each obligation Stack hedge: Single maturity,  ―stack and roll‖ <120 <110 <100 Jan Feb Mar 10 10 10 10 10 10 10 10 10 10 10 10 Jan Feb Mar
  65. 65. Commodity Futures LO 30.9: Differentiate between a strip hedge and a stack hedge. A strip hedge is when we hedge a stream of obligations by offsetting each  individual obligation with a futures contract that matches the maturity and quantity of the obligation. For example, if a producer must deliver X number of commodities per month, then the strip hedge entails entering into a futures contract for X commodities, to be delivered in one month; plus a futures contract for X commodities to be delivered in two months. The strip hedger matches a series of futures to the obligations. A stack hedge is front-loaded: the hedger enters into a large future with a  single maturity. In this case, our hedger would take a long position in a near-term futures contract for 12X commodities (i.e., a year’s worth). The stack hedge may have lower transaction costs but it entails speculation (implicit or deliberate) on the forward curve: if the forward curve gets steeper, the stack hedger may lose. On the other hand, if the forward curve flattens, then the stack hedger gains because he/she has locked in the commodity at a relatively lower price.
  66. 66. Swaps LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap. A swap is an agreement to exchange future cash  flows “Plain vanilla” swap: company pays fixed rate on • notional principal and receives floating rate (pay fixed receive floating) Interest rate swap: principal not exchanged • (i.e., that’s why it is called notional) Currency swap: principal is (typically) exchanged • at beginning (inception) and end (maturity)
  67. 67. Swaps LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap. Receive Fixed “Plain-vanilla” Add Your Pay LIBOR Text here Counterparty Add Your Add Your Add Your Receive Receive Receive Text here Text here Text here LIBOR LIBOR LIBOR Pay Pay Pay Fixed Fixed Fixed
  68. 68. Swaps LO 31.1: Illustrate the mechanics and compute the cash flows of a plain vanilla interest rate swap. Notional principal: $100 million (notional principal is not exchanged)  Swap agreement: Pay fixed rate of 5% and receive LIBOR  Term: 3 years with payments every six months  End of LIBOR at the Receive Period Start of Pay Fixed Floating Net Cash (6 months) Period Cash Flow Cash Flow Flow 1 5.0% -2.5 +2.5 0.0 2 (Year 1) 5.2% -2.5 +2.6 +0.1 3 5.4% -2.5 +2.7 +0.2 4 (Year 2) 5.0% -2.5 +2.5 0.0 5 4.8% -2.5 +2.4 -0.1 6 (Year 3) 4.6% -2.5 +2.3 -0.2
  69. 69. Swaps LO 31.2: Explain how an interest rate swap can be combined with an existing asset or liability to transform the interest rate risk.  Intel borrowing fixed-rate @ 5.2%  MSFT borrowing floating-rate @ LIBOR + 10 bps
  70. 70. Swaps LO 31.3: Explain the advantages and disadvantages of the comparative advantage argument often used for the existence of the swap market. Fixed Floating BetterCreditCorp 4% LIBOR + 1% WorseCreditCorp 6% LIBOR + 2%
  71. 71. Swaps LO 31.4: Explain how the discount rates in a swap are computed. LIBOR/swap zero given: six-month = 3%, 1 year = 3.5%, 1.5 year = 4%.  The 2 year swap rate is 5% which implies that a $100 face value bond with  a 5% coupon will sell exactly at par (why? Because the 5% coupons are discounted at 5%) We can solve for the two year zero rate (R) because it is the unknown  Present LIBOR/swap Value of Period Cash flow zero rates Cash Flow 0.5 $2.5 3.0% $2.46 1.0 $2.5 3.5% $2.41 1.5 $2.5 4.0% $2.35 102.5e2R 2.0 $102.50 X? Total PV $100.00
  72. 72. Swaps LO 31.4: Explain how the discount rates in a swap are computed. Present LIBOR/swap Value of zero rates Period Cash flow Cash Flow 0.5 $2.5 3.0% $2.46 1.0 $2.5 3.5% $2.41 1.5 $2.5 4.0% $2.35 102.5e2R 2.0 $102.50 X? Total PV $100.00 2.5e( .5)(3%) 2.5e( 1)(3.5%) 2.5e( 1.5)(4%) 2R 102.5e 100 2.46 2.41 2.35 102.5e 2 R 100 e( 2 R) 0.90506 R 4.99%
  73. 73. Swaps LO 31.5: Explain how a swap can be interpreted as two simultaneous bond positions or as a sequence of forward rate agreements (FRAs).  Iftwo companies enter into an interest rate swap arrangement, then one of the companies has a swap position that is equivalent to a long position in floating- rate bond and a short position in a fixed-rate bond. VSWAP = BFL - BFIX  The counterparty to the same swap has the equivalent of a long position in a fixed-rate bond and a short position in a floating-rate bond: VSWAP Counterparty = BFIX -BFL
  74. 74. Swaps LO 31.6: Calculate the value of an interest rate swap. Assumptions Notional 100 Receive Fixed 7.0% Time Time Time LIBOR Rates 0.25 0.75 1.25 3 Months (0.25) 5.0% 6 Months (0.5) 5.5% Receive Receive Receive 9 Months (0.75) 6.0% 12 Months (1.0) 6.5% ½ of 7% ½ of 7% ½ of 7% AddPay AddPay Pay Your Add Your Your Text here Text here Text here ½ LIBOR ½ LIBOR ½ LIBOR
  75. 75. Assumptions Swaps Notional 100 Receive Fixed 7.0% LO 31.6: Calculate the value of an interest rate swap. LIBOR Rates 3 Months (0.25) 5.0% 6 Months (0.5) 5.5% 9 Months (0.75) 6.0% 12 Months (1.0) 6.5% Fixed Floating LIBOR Disc. Cash Flows Cash Flows Time Rates Factor FV PV FV PV 0.25 5.0% 0.988 $3.5 $3.46 $102.75 $101.47 0.75 6.0% 0.956 $3.5 $3.35 1.25 6.5% 0.922 $103.5 $95.42 Total $102.23 $101.47 Value (swap) = $102.23 - $101.47 = $0.75
  76. 76. Swaps LO 31.7: Explain the mechanics and calculate the value of a currency swap. Assumptions Principal, Dollars ($MM) 10 Principal, Yen (MM) Y 1,000 FX rate 120 US rate 5.0% Japanese rate 2.0% SWAP: PAY dollars @ 5% RECEIVE yen @ 9%
  77. 77. Assumptions Principal, Dollars ($MM) 10 Swaps Principal, Yen (MM) Y 1,000 FX rate 120 LO 31.7: Explain the mechanics and US rate 5.0% calculate the value of a currency swap. Japanese rate 2.0% SWAP: PAY dollars @ 5% RECEIVE yen @ 9% Dollars (MM) Yen (MM) Time FV PV FV PV 1 0.5 $0.48 90 Y 88 2 0.5 $0.45 90 Y 86 3 0.5 $0.43 90 Y 85 3 10 $8.61 1000 Y 942 $9.97 Y 1,201 Yen bond Y 1,201 Yen bond in US dollars $10.01 Dollar bond $9.97 Swap, yen bond - dollar bond $0.04
  78. 78. Swaps LO 31.8: Explain the role of credit risk inherent in an existing swap position.  Because a swap involves offsetting choir position, there is no credit risk when the swap has negative value. Credit risk only exists when the swap has positive value.  Further, because principal is not exchanged at the end of the life of an interest rate swap, the potential default losses are much less than those on an equivalent loan. On the other hand, in a currency swap, the risk is greater because currencies are exchanged at the end of the swap.
  79. 79. Forwards/Futures Derivatives – Part 1 (LOs 27.x – 31.x) S0 erT F0 E(ST ) F0 Forwards (LO 27.x) 1 F0 F0 FT-1 ST Hedging Strategies (LO 28.x) 2 ST=FT 3 Interest Rate Futures (LO 29.x) ST-1 S0 4 Commodity Futures (LO 30.x) 5 Swaps (LO 31.x) Time (T) Forwards/Futures Cost-of-carry model (Commodity with High LO 27.1: State & explain cost-of-carry model for F0 = E(ST) forward prices with & without interim cash flows “Convenience Yield” or High-dividend Financial Asset) Risk-free Storage Rate (r) Cost (U, u) S0 ST-1 F0 F0 FT-1 ST Income/ Convenience Dividend (y) ST=FT Forward (q) (F0) Spot (S0) Time (T) Time (T) 1
  80. 80. Cost-of-carry: Question Cost-of-carry: Question A stock’s price today is $50. The stock will pay A stock’s price today is $50. The stock will pay a $1 (2%) dividend in six months. The risk- a $1 (2%) dividend in six months. The risk- free rate is 5% for all maturities. free rate is 5% for all maturities. What the price of a (long) forward contract What the price of a (long) forward contract (F0) to purchase the stock in one year? (F0) to purchase the stock in one year? I )erT F0 ( S0 F0 ($50 [($1)e( 0.05)(6/12) ])e(.05)(1) $51.538 Derivatives Cost-of-Carry Model Cost of carry = interest to finance asset (r) LO 27.2: Compute the forward price given both the + storage cost (u) - income earned (q) price of the underlying and the appropriate carrying costs of the underlying. Commodity ( r u q y )T I )e(r y )T Commodity F0 S0 e F0 ( S0 U ( r u q y )T F0 S0 e constant rates as % Present values U = Present value, storage costs u = storage costs I = Present value, income q = income (dividend) Financial asset (e.g., stock index) y = convenience yield S 0 e( r q )T F0 2
  81. 81. Cost-of-Carry Model Cost-of-Carry: Question Cost of carry = interest to finance asset (r) The spot price of corn today is 230 cents per + storage cost (u) - income earned (q) bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum. Financial asset (e.g., stock index) ( r q )T I )e rT What is the forward price in four (4) months? F0 S0 e F0 ( S0 constant rates as % Present values q = income (dividend) I = Present value, income Cost-of-Carry: Question Derivatives The spot price of corn today is 230 cents per LO 27.3: Calculate the value of a forward contract. bushel. The storage cost is 1.5% per month. The risk-free interest rate is 6% per annum. rT f ( F0 K )e What is the forward price in four (4) months? S 0 e( r u)T (230)e(6%/12 1.5%)(4) F0 230e(.02)(4) 249.16 3
  82. 82. Value of a forward contract Value of a forward contract A long forward contract on a non dividend-paying stock has A long forward contract on a non dividend-paying stock has three months left to maturity. three months left to maturity. The stock price today is $10 and the delivery price is $8. The stock price today is $10 and the delivery price is $8. Also, the risk-free rate is 5%. Also, the risk-free rate is 5%. What is the value of the forward contract? What is the value of the forward contract? S0 e rT 10e(5%)(0.25) F0 $10.126 rT (10.126 8)e(5%)(0.25) f ( F0 K )e $2.153 Derivatives Derivatives LO 27.4: Describe the differences between forward LO 27.5: Distinguish between a long futures position and futures contracts. and a short futures position. A long-futures position agrees to buy in the future  Forward vs. Futures Contracts A short-futures position agrees to sell in the future.  Forward Futures Price mechanism maintains a balance between buyers and  Trade over-the-counter Trade on an exchange sellers.(market equilibrium) Not standardized Standardized contracts Most futures contracts do not lead to delivery, because  One specified delivery date Range of delivery dates most trades ―close out‖ their positions before delivery. Settled at contract’s end Settled daily Closing out a position means entering into the opposite Delivery or final cash Contract usually closed type of trade from the original. settlement usually occurs out prior to maturity 4
  83. 83. Derivatives Derivatives LO 27.6: Describe the characteristics of a futures LO 27.6: Describe the characteristics of a futures contract and explain how futures positions are settled. contract and explain how futures positions are settled. Contract size varies by type of futures contract An (underlying) asset  Treasury bond futures: contract size is a face value of  A Treasury bond futures contract is on underlying $100,000 U.S. Treasury with maturity of at least 15 years and not  S&P 500 futures contract is index $250 (multiplier of 250X) callable within 15 years (15 years ≤ T bond).  NASDAQ futures contract is index $100 (multiplier of  A Treasury note futures contract is on the underlying 100X) U.S. Treasury with maturity of at least 6.5 years but not greater than 10 years (6.5 ≤ T note ≤ 10 years). Recently, ―mini contracts‖ have been introduced:   When the asset is a commodity (e.g., cotton, orange S&P 500 ―mini‖ = $50 x S&P Index  NASDAQ ―mini‖ = $20 x NASDQ juice), the exchange specifies a grade (quality).  (each contract is one-fifth the price, to attract smaller investors) Derivatives Derivatives LO 27.6: Describe the characteristics of a futures LO 27.7: Describe the marking-to-market procedure, the contract and explain how futures positions are settled. initial margin, and the maintenance margin. Delivery Arrangements Margin account: Broker requires deposit.   The exchange specifies delivery location. Initial margin: Must be deposited when contract is  initiated. Delivery Months Mark-to-market: At the end of each trading day, margin  The exchange must specify the delivery month; this can  account is adjusted to reflect gains or losses. be the entire month or a sub-period of the month. 5
  84. 84. Derivatives Derivatives LO 27.8: Compute the variation margin. LO 27.7: Describe the marking-to-market procedure, the initial margin, and the maintenance margin. Maintenance margin: Investor can withdraw funds in the There is only a variation margin if and when there is a   margin account in excess of the initial margin. A maintenance margin call. margin guarantees that the balance in the margin account Variation margin = initial margin – margin never gets negative (the maintenance margin is lower than the account balance initial margin). Margin call: When the balance in the margin account falls The maintenance margin is a trigger level—once   below the maintenance margin, broker executes a margin call. triggered, the investor must ―top up‖ to the initial The next day, the investor needs to ―top up‖ the margin margin, which is greater than the maintenance level. account back to the initial margin level. Variation margin: Extra funds deposited by the investor  after receiving a margin call. Derivatives Derivatives LO 27.9: Explain the role of the clearinghouse. LO 27.9: Explain the role of the clearinghouse. The exchange clearinghouse is a division of the exchange Market order: Execute the trade immediately at the   (e.g., the CME Clearing House is a division of the best price available. Chicago Mercantile Exchange) or an independent Limit order: This order specifies a price (e.g., buy at $30  company. The clearinghouse serves as a or less)—but with no guarantee of execution. guarantor, ensuring that the obligations of all trades are Stop order: (aka., stop-loss order) An order to execute  met. a buy/sell when a specified price is reached. 6
  85. 85. Derivatives Derivatives LO 28.1: Differentiate between a short hedge and a LO 27.9: Explain the role of the clearinghouse. long hedge, and identify situations where each is appropriate. Stop-limit: Requires two specified prices, a stop and a  A short forward (or futures) hedge is an agreement to limit price. Once the stop-limit price is reached, it sell in the future and is appropriate when the hedger already becomes a limit order at the limit price. owns the asset. Classic example is farmer who wants to lock in a sales price: Market-if-touched: Becomes a market order once   protects against a price decline. specified price is achieved. A long forward (or futures) hedge is an agreement to buy Discretionary (aka., market-not-held order): A  in the future and is appropriate when the hedger does not market order, but the broker is given the discretion to currently own the asset but expects to purchase in the future. delay the order in an attempt to get a better price. Example is an airline which depends on jet fuel and enters into a  forward or futures contract (a long hedge) in order to protect itself from exposure to high oil prices. Basis risk Derivatives LO 28.2: Define and calculate the basis. No hedge Spot = -$0.50 Basis = Spot Price Hedged Asset –  SPOT $2.50 Futures Price Futures Contract = S0 – F0 $2.00 Short 1.67 F Basis $0.30 $0.10 Basis = Spot = -$0.50  Forward $2.20 Futures Price Futures Contract – Future = $0.30 (1.67) $1.90 Spot Price Hedged Asset = F0 – S0 Net = 0 T0 T1 Time (T) Hull says first is correct but second is common for financial assets (either is okay) Weakening of the basis = Futures price increases more than spot 7
  86. 86. Basis risk Basis risk No hedge T1 Spot = -$0.50 $2.60 $2.60 SPOT SPOT $2.50 $2.50 $2.00 $0.0 Short 1.67 F Basis Basis $0.30 $0.30 Short 1.67 F $0.30 Spot = -$0.50 Spot = +$0.10 Forward $2.20 Forward $2.20 Future = $0.50 (1.67) $1.90 $1.90 Future = -$0.40 (1.67) Net = +33.5 $1.70 T0 T0 Net = -56.8 T1 Time (T) Time (T) Basis unchanged. Basis declines But unexpected strengthening= Hedger improved! Unexpected weakening= Hedger worse! Derivatives Derivatives LO 28.3: Define the types of basis risk and explain how LO 28.3: Define the types of basis risk and explain how they arise in futures hedging. they arise in futures hedging. But basis risk arises because often the characteristics Spot price increases by more than the futures price    basis increases. This is a ―strengthening of the basis‖ of the futures contract differ from the underlying position. When unexpected, strengthening is favorable for a short  hedge and unfavorable for a long hedge Contract ≠ Commodity  Futures price increases by more than the spot price  Contract is standardized (e.g., WTI oil futures)   basis declines. This is a ―weakening of the basis‖ Commodities are not exactly commodities (e.g., hedger has a  position in different grade of oil) When unexpected, weakening is favorable for a long hedge  and unfavorable for a short hedge Basis risk higher with cross-hedging 8
  87. 87. Derivatives Derivatives LO 28.3: Define the types of basis risk and explain how LO 28.4: Define, calculate, and interpret the minimum they arise in futures hedging. variance hedge ratio. But basis risk arises because often the characteristics The optimal hedge ratio (a.k.a., minimum variance hedge   of the futures contract differ from the underlying ratio) is the ratio of futures position relative to the spot position. position that minimizes the variance of the position. Where is the correlation and is the standard Contract ≠ Commodity.  deviation, the optimal hedge ratio is given by: Contract is standardized (e.g., WTI oil futures)  Commodities are not exactly commodities (e.g., hedger has a  position in different grade of oil) S h* Trade-off Liquidity Basis risk F (exchange) Derivatives Derivatives LO 28.4: Define, calculate, and interpret the minimum LO 28.4: Define, calculate, and interpret the minimum variance hedge ratio. variance hedge ratio. For example, if the volatility of the spot price is 20%, the For example, if the volatility of the spot price is 20%, the   volatility of the futures price is 10%, and their correlation volatility of the futures price is 10%, and their correlation is 0.4, then: is 0.4, then: 20% 20% S S h* h* h* (0.4) 0.8 h* (0.4) 0.8 10% 10% F F h * NA Number of N* contracts QF 9
  88. 88. Derivatives Derivatives LO 28.5: Calculate the number of stock index futures LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or contracts to buy or sell to hedge an equity portfolio or individual stock. individual stock.  By extension, when the goal is to shift portfolio beta from  Given a portfolio beta ( ), the current value of the portfolio (P), and the value of stocks underlying one ( ) to a target beta ( *), the number of contracts futures contract (A), the number of stock index futures required is given by: contracts (i.e., which minimizes the portfolio variance) is given by: P P N(* ) N A A Futures: Question Futures: Question LO 28.5: Calculate the number of stock index futures contracts to LO 28.5: Calculate the number of stock index futures contracts to buy or sell to hedge an equity portfolio or individual stock. buy or sell to hedge an equity portfolio or individual stock. Assume: Assume: • Value of S&P 500 Index is 1240 Value of S&P 500 Index is 1240  • Value of portfolio is $1 million Value of portfolio is $10 million • Portfolio beta ( ) is 1.5  Portfolio beta ( ) is 1.5 P  N (* ) How do we change the portfolio beta to 1.2? A $10,000,000 (1.2 1.5) 9.7 Hint: Contract = ($250 Index) P (1240)(250) and # of futures is given by: N (* ) A We short about 10 contracts. (-) indicates short, (+) long… 10
  89. 89. Derivatives Derivatives LO 28.6: Identify situations when a rolling hedge is LO 29.1: Identify and apply the three most common appropriate, and discuss the risks of such a strategy. day count conventions. When the delivery date of the futures contract occurs  prior to the expiration date of the hedge, the hedger can Actual/actual U.S.Treasuries roll forward the hedge: close out a futures contract and 30/360 U.S. corporate and take the same position on a new futures contract with a municipal bonds later delivery date. Actual/360 U.S.Treasury bills and Exposed to:  other money market Basis risk (original hedge)  instruments Basis risk (each new hedge) = ―rollover basis risk‖  Derivatives Derivatives LO 29.2: Explain the U.S. Treasury bond (T-bond) LO 29.3: Calculate the Eurodollar futures contract futures contract conversion factor. convexity adjustment. The Treasury bond futures contract allows the party with The convexity adjustment assumes continuous   the short position to deliver any bond with a maturity of compounding. Given that ( ) is the standard deviation of more than 15 years and that is not callable within 15 the change in the short-term interest rate in one year, t1 years. When the chosen bond is delivered, the conversion is the time to maturity of the futures contract and t2 is factor defines the price received by the party with the the time to maturity of the rate underlying the futures short position: contract: 1 Cash Received = Quoted futures price Conversion 2 Forward = Futures t 1 t2 factor + Accrued interest 2 = (QFP CF) + AI 11
  90. 90. Derivatives Derivatives LO 29.4: Formulate a duration-based hedging strategy LO 29.4: Formulate a duration-based hedging strategy using interest rate futures. using interest rate futures. The number of contracts required to hedge against an Assume a portfolio value of $10 million.   uncertain change in the yield, given by y, is given by: The fund manager will hedge with T-bond futures (each  contract is for delivery of $100,000) with a current futures PDP price of 98. N* She thinks the duration of the portfolio at hedge maturity will FC DF  be 6.0 and the duration of futures contract with be 5.0. FC = contract price for the interest rate futures contract. How many futures contracts should be shorted?  DF = duration of asset underlying futures contract at maturity. P = forward value of the portfolio being hedged at the maturity of the hedge (typically assumed to be today’s portfolio value). DP = duration of portfolio at maturity of the hedge Derivatives Derivatives LO 29.4: Formulate a duration-based hedging strategy LO 29.5: Identify the limitations of using a duration- using interest rate futures. based hedging strategy. Assume a portfolio value of $10 million.  Portfolio immunization or duration matching is when a  The fund manager will hedge with T-bond futures (each  bank or fund matches the average duration of assets with contract is for delivery of $100,000) with a current futures the average duration of liabilities. price of 98. Duration matching protects or ―immunizes‖ against  She thinks the duration of the portfolio at hedge maturity will  small, parallel shifts in the yield (interest rate) curve. The be 6.0 and the duration of futures contract with be 5.0. limitation is that it does not protect against nonparallel How many futures contracts should be shorted?  shifts. The two most common nonparallel shifts are: A twist in the slope of the yield curve, or PDP ($10 million)(6)  N* 122 A change in curvature  FC DF (98,000)(5) 12
  91. 91. Derivatives Derivatives LO 30.1: Explain the derivation of the basic equilibrium LO 30.1: Explain the derivation of the basic equilibrium formula for pricing commodity forwards and futures. formula for pricing commodity forwards and futures. Risk-free Rate (r) E0 (ST )e(r )T F0,T E0 (ST )e(r )T F0,T E0 (ST ) Spot price of S at time T, as expected at time 0 F0,T Forward price r Risk-free rate Discount Forward Discount rate for commodity S rate ( ) (F0) Spot (S0) Time (T) Derivatives Derivatives LO 30.2: Define lease rates, and discuss the LO 30.3: Explain how lease rates determine whether a importance of lease rates for determining no- forward market is in contango or backwardation. arbitrage values for commodity futures and forwards. Contango refers to an upward-sloping forward curve   Lease rate = commodity discount rate – growth rate which must be the case if the lease rate is less than  Lease rate dividend yield the risk-free rate. Backwardation refers to a downward- sloping forward curve which must be the case if the lease rate is greater than the risk-free rate. S0e(r )T F0,T Financial asset S 0 e( r q )T F0 13
  92. 92. Derivatives Derivatives LO 30.3: Explain how lease rates determine whether a LO 30.4: Explain how storage costs impact commodity forward market is in contango or backwardation. forward prices, and calculate the forward price of a commodity with storage costs. Forward (F0) Risk-free Storage Cost ( ) Rate (r) negative dividend Spot E(ST) (S0) Forward (F0) Lease rate ( ) Convenience (y) Forward dividend dividend (F0) Spot Time (T) (S0) Research says normal backwardation is “normal:” speculators Time (T) want compensation (risk premium) for buying the futures contract Derivatives Derivatives LO 30.5: Explain how a convenience yield impacts commodity LO 30.4: Explain how storage costs impact commodity forward prices, and determine the no-arbitrage bounds for the forward prices, and calculate the forward price of a forward price of a commodity when the commodity has a commodity with storage costs. convenience yield. Storage Risk-free Cost ( ) Rate (r) Risk-free Storage Risk-free Storage Rate Cost Rate Cost (r c )T F0 S0 e S 0 e( r c )T S 0 e( r )T F0 Convenience Lease Forward (y) rate ( ) (F0) Convenience Spot Yield (S0) Time (T) 14
  93. 93. Commodity Futures Commodity Futures LO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures. S&P 500 Index 1640 Gold futures 1620 1600 900 1580 850 1560 800 1540 750 1520 Rational forward curve rises by cost 700 1500 of capital (risk free + premium) less 650 1480 Durable, (relatively) cheap to store. dividends 600 1460 550 Forward curve is “uninteresting” 500 Jul-07 Nov-08 Mar-10 Aug-11 Dec-12 Commodity Futures Commodity Futures LO 30.6: Discuss the factors that impact the pricing of LO 30.6: Discuss the factors that impact the pricing of gold, corn, natural gas, and oil futures. gold, corn, natural gas, and oil futures. Corn Natural Gas 450 10 9 400 8 350 7 300 Costly to transport. Costly to store 6 250 (storage costs). Highly seasonal 5 200 Nov-08 Nov-09 Aug-07 Nov-07 Aug-08 Aug-09 Aug-10 Nov-10 May-08 May-09 May-10 Feb-08 Feb-09 Feb-10 Mar-08 Mar-09 Mar-10 Dec-07 Dec-08 Dec-09 Dec-10 Sep-07 Sep-08 Sep-09 Sep-10 Jun-10 Jun-08 Jun-09 15

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