Derivatives Why? Allow easier methods to short sell a stock without a broker lending it. Facilitates hedging easily Allows the ability to take long/short position on less available commodities (Rice, Cotton, etc.) Three types of traders Hedger Attempts to reduce exposure to operating risk within firm Speculator Profiting from a bet that markets will move in a certain direction Arbitrageurs Profiting without any risk Forward contracts Forward A binding agreement (obligation) to buy/sell an asset or a commodity in the future, at a price set today Contract specifies: Features and quantity of the asset to be delivered Delivery logistics (time/date/place) Underlying Asset exchanged at maturity Long = S T K Short = K - S T Forward Price with No storage cost: F 0 = S 0 (1 + r ) T F 0 = Spot price + Opportunity cost Assumptions: No transaction cost, same rate for borrowing/lending, no default/counterparty risk E.g. Spot price is $450, risk free rate = 4%, F 0 = $477. Is there arbitrage opportunity? Theoretical forward price = 450 e (4% * 1) = 468.34 => There is an arbitrate opportunity. How? Time 0 1 Year Sell 1 forward contract 0 477 - S T Borrow $450 450 (468.36) Buy gold spot and sell it at time T (450) S T 0 8.64 If F 0 = $460 Time 0 1 Year Buy 1 forward contract 0 S T 460 Short 1oz. gold 450 -S T Borrow/repay the bank (450) 468.36 0 8.36
E.g. Bond is 1,074 and Forward price is 1,060. There are two $50 coupon payments semi-annually. Rates are 8% (6mo.) and 9% (1yr) with continuous compounding. Theoretical forward price: 1074 * e Arbitrage follows: Time 0 6 month 1 Year Buy Forward 0 0-1060 + S T Sell bond 1,074-50 - 50 - S T Borrow/repay the bank - 48.04 50 0 Borrow/repay the bank -1,025.96 0 1,122.58 0 0 12.58 Spot $600, rate is 5% (1 year continuous compounding). Storage cost is 2% continuously compounding proportionate to spot price. What is 1 year forward price? F 0 = 600e (5% + 2%)(1) = 643.50 If F 0 = 650, how is arbitrage realized? F theoretical < F0 so we take a short position Today 1 year Sell 1 forward contract 0 650 ST Borrow $612.12 612.12-612.12e 5% = -643.50 Buy underlying including storage fees 600*(e 2% -1) -612.12 ST 0 6.50 Convenience yield Ownership of physical commodity provides benefits that are not obtained by the holders of contracts for future delivery. (c y )T F 0 = S 0 e S0 < F0: Contango occurs when c > y S0 > F0: Backwardation occurs when y >c Currency Forwards F 0 = S 0 e (r-rf)t r : T-year domestic risk-free interest rate rf : T-year foreign risk-free interest rate S 0 : Spot exchange rate e.g. S0 = 2.30 r = 0.04 r f = 0.05, T = 1 F 0 = 2.30e (0.04 0.05) = 2.277 C$/ Synthetic Long Currency Forward Today One Year 1 2
e.g. The 8 month rate in the U.S. is 5% with semi-annual compounding. The 8 month interest rate in France is 6% compounded continuously. The spot exchange rate is 1.89 US$/. The 8 month forward exchange rate is 1.95 is there an arbitrage opportunity? S 0 = $1.89/ F 0 = $1.95/ R C = mln(1+rm/m) = 2 ln(1+0.05/2) = 0.0494 (convert to domestic continuous rate) F hypothetical = S 0 e (r-rf)(t) = 1.89e (0.0494 0.06) (8/12) = 1.8767 $/ Today 8 Months Sell Forward 0 1.95 - ST Borrow 0.9608 * 1.89 = 1.8159USD @ 4.94% continuous 1.8159-1.8767 Convert borrowed USD and invest in for 8 months -1.8159 +ST 0 0.0733 Valuing a Forward Contract Value of a long forward contract Value of a short forward contract f = (F 0 K)e -rt f = (K F 0 )e -rt e.g. 1 year long forward contract on non-dividend paying stock was entered when stock is at $40 and the risk free rate is 10% p.a., continuously compounded. What is forward price and initial value of forward contract? f 0 = 0 F 0 = 40e (10%) (1) = 44.20 Six months later stock price is $45, what is forward price and value of forward contract? F 6m = 45e (10%)(0.5) = 47.31 f 6m = F 6m F 0 e rf = 47.31 44.20 e -(10%)(.5) = 2.96 Margin Requirements Margin requirements are established based on the risk level of daily transactions. Underlyings which are highly volatile and prone to large daily fluctuations in spot price will have larger margin requirements relative to a more stable underlying. e.g. A company enters into a short futures contract to sell 5,000 bushels of wheat for 250 cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin call? Under what circumstance could they withdraw $1500 from the margin account? Initial value: 250 cents / 100 * 5000 = 12,500 3000/12500 = MI% Trigger a margin call 1000/12500 = 8% 250 * 8% = 20 cent drop $1500 margin withdrawn 1500 /12,500 = 12% 250 * 12% = 30 cent gain
e.g. Suppose there are no storage costs for crude oil and the interest rate for borrowing or lending is 5% per annum. How could you have made money on January 8, 2007 by trading June 2007 and December 2007 contracts on crude oil? Prices below Open High Low Settle Change Jun 60.20 61.33 59.08 60.01 0.10 Dec 62.78 64.08 62.10 62.94 0.28 June 30 th S 0 = 60.01 F hypo for June = S June (1+5%) 1/2 = 60.01(1.05) 1/2 = 61.49 January June December Long 1 June future 0 S June 60.01 0 Short 1 Dec future 0 0 62.94 - S Dec Borrow 60.01 @ 5% buy oil in June 0 60.01-61.49 Dec delivery 0-60.01 S Dec 0 0 1.45 Why do we need derivatives? CDS Trading Initiation (event) Creditworthiness exchange rate drops, foreign reserves, credit rating Risk regional CDS index, global CDS index Ex Debt GDP Hypothesis Effect driven by opacity & riskiness Benefits of CDS Initiation Facilitates risk sharing expansion of the risk return space Allows hedging of adverse selection risk Adverse selection, systematic risk Costs Many more to infer asset value Not that options push down the stock price but true value MSCI Emerging Markets Exposure ETF vs EM forwards 1. Forward Stack & Roll a. Liquidity b. Transaction costs 2. Full capitals w/ ETF vs. Margin a. Cost of Capital needs to be put up front for ETFs 3. Tracking Error a. Trading timing b. Premium/discount on ETF trading c. Dividend forecast error 4. Short position access is easier for forward contracts
Hedging example 1 Portfolio worth 100M, B = 1.2, index futures price = 1000, contract is $250 times index Change the beta to 0.5. a) What position should the company take? # contracts = (B-B*)P N /F N = (1.2.5)(100,000,000/(250)(1000)) = 280 contracts b) Company wants to increase beta to 1.5, what position should they take? # contracts = B*-B P N /F N = (1.5-1.2)(100,000,000/(250)(1000)) = 120 contracts Hedging example 2 Portfolio of 50 million, Beta of 0.87, the manager is concerned about the performance of the market over the next two months and plans to use three month future contracts on the S&P 500 to hedge the risk. The current index is 1,250, one contract is on $250 times the index, the risk free rate is 6% per annum and the dividend yield on the index is 3% per annum. The current three month futures price is 1,259. a) What position should fund manager take to hedge exposure to market over next 2 months. # contracts = B PN/FN = 0.87(50,000,000 / (250)(1254)) = 138.2 => 138 b) Calculate effect of strategy if index in 2 months is 1100 or 13000. Assume 1mo future price is 0.25% higher than index level at this time. Index drops 1250 -> 1100 Gain on short position = (1250 1102.75) * 250 * 135 = 5,390,625 Loss of Portfolio = 3% * 2/12 = 0.5% (1100 1250)/1250 = -11.5% actual return R P 1% =.87 ( 11.5% -1%) = -9.875% Portfolio return = -9.875% => 50mil * -0.09875 = -4937500 Net result: Gain of 5390625 4937500 = 453,125 Interest rate forwards and futures Conversion Factor Example Maturity 21years 3m + days => round to 21 years 3 months Value today = (3.5 + 42 i=1 3.5 /1.03 i + 100/1.03 42 ) / 1.03 (1/2) = 113.66 Adjust for actual interest 3.5/2 = 1.75 113.66 1.75 = 111.91 Conversion Factor 111.91/100 = 1.1191 Cash price for bond = (118 + 23/32) * 1.1191 = $132.86 Example: Cheapest bond to deliver. Given: most recent settle = 93 08 = 93.25 Bond 1 = 99.50 (93.25*1.0382) = 2.69 Bond 2 = 143.50 (93.25 * 1.5188) = 1.87 Bond 3 = 119.75 (93.25*1.2615) = 2.12 Accrued interest for T-bonds : Actual/Actual ratio Corporate & Municipal : 30/360
Forward Rate Agreement (FRA) Hedging Interest rate Risk Firm expects to receive 1 million in 6 months, they plan to invest the money for 3 months firm enters into a contract with a bank. Under the contract the firm will earn 5% per annum w/ quarterly compounding for the three money period starting in six months on a principal of 1 million. N = 1,000,000 Forward Rate = 5% Assume @ 6mo LIBOR = 4.4% (annual rate w/ quarterly comp) No hedge 1,000,000 * 4.4/4 = 11,000 interest With hedge Long pay 11,000 Short pay 1,000,000 * (0.05/4) = 12,500 Diff of 1500 / (1 + (0.044/4)) = 1483.67 Settle (1mill + 1483.67) (0.044/4) = 12,500 OR Settle = [L(R k R)(T 2 -T 1 )] / [1 + R(T 2 -T 1 )] 1,000,000 (0.05 0.044) 0.25 / (1 + (0.044)(0.25) 1,483.67 Value of FRA Difference between Rk & R: 2 scalars: i) L (notional) ii)horizon (T2 T1), good rate for 1 year better than good rate for 1 month. Rk = 5% L = 1,000,000 3 months from now, invest for 3months Rf = R 2 T 2 R 1 T 1 / T 2 -T 1 = (0.045)(0.5) (0.043)(0.25) /.25 = 0.047 continuously comp. Convert to 1F2 = 0.04728 Value to the party receiving Rk is 1,000,000(0.05 0.04728)(0.25)e -0.045*0.5 = 665.55 US Treasury Bonds Quoted in dollars and thirty-seconds e.g. a bond price of 112-04 is equal to 112.125 where 4/32 = 0.125 e.g. Party Long: Receives 6% coupon US T-Bond Long <= Bond 6% Coupon $$$ => Short Conversion factor required for fairness. Consider 2 bonds Bond Coupon Yield FV Mat Value Conversion factor 1 7% 6.4% 100 20 year 106.72 111.56 = PV (coupons + FV) w/ r = 6% 2 5% 6.4% 100 15 year 84.33 88.44 Payment bond 1 = (95 21/32 * 1.1156) = 106.71 Payment bond 2 = (95 21/32 * 0.8844) = 84.59
Bond 1 2 Market Price 106.72 84.33 Invoice Price 106.71 84.59 Invoice Market -0.01 0.27 Bond 2 is cheapest to deliver for short party Conversion rules 1) 15 years to maturity 2) Discount rate for calculation is always 6% per annum with semi-annual compounding 3) Round down to nearest 3 months Duration-Based Hedge Ratio Optimal number of contracts to use for hedging is N* = PD P / F C D F FC = contract price for the interest-rate future contract DF = duration of the asset underlying the futures contract at maturity of the future contract PL = value of the portfolio being hedged DP = duration of asset being hedged at maturity of the hedge e.g. Portfolio A consists of a one-year zero coupon bond with a face value of $2000 and a 10-year zero coupon bond with a face value of $6000. Portfolio B consists of a 5.95-year zero-coupon bond with a face value of $5000. The current yield on all bonds is 10% per annum. a. Show that both portfolios have the same duration. b. Show the % changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same. c. What are the % changes in the values of the two portfolios for a 5% per annum increase in yields? Method 1 D B = 5.95 (Since you get your payment at a later date) D A = 2000e -(.1)(1) *1 + 6000e -(.1)(10) *10 / 2000e -(.1)(1) + 6000e -(.1)(10) = 5.95 Method 2 Portfolio Proportion A 1 : D A1 = 1year PV = 2000e -(.1)(1) = 1809.67 45% A 2 : D A2 = 10 years PV = 6000e -(.1)(10) = 2207.28 55% D A = (.45)1 + (.55) 10 = 5.95 e.g. Suppose that on January 20 a corporate treasurer learns that US$10 million will be received on May 5. The funds will be needed for a major capital investment in November. The treasurer therefore plans to make a six-month Eurodollar deposit as soon as the funds are received. The treasurer is concerned that Eurodollar rates may decline between January 20 and May 5 and decides to hedge using the one June Eurodollar futures. On January 20 the June Eurodollar futures is quoted at 95.20. a) What Eurodollar futures position should the company take? Explain. How many June Eurodollar futures contracts should the company use to hedge its exposure? b) On May 5, the June Eurodollar futures was quoted at 96.00, and the six-month Eurodollar deposit rate was 4.20% per annum with semiannual compounding. Determine the firm s profit or loss on the Eurodollar futures position.
Q is a Eurodollar futures quote, (100-Q)% is the annualized Eurodollar futures interest rate for a threemonth period beginning on the 3 rd Wed. of the delivery month Price of 1 contract = 10,000[100-0.25(100-Q)] Position of Treasurer Treasurer wants to be long for the protection from rate decrease Optimal Number of Contracts = N* = PD P / F C D F = 10(.5)/1(.25) = 20 Jan20: Price of 1 future = 10,000(100-25(100-95.20)) = 988,000 May5: Price of 1 future = 10,000(100-25(100-96)) = 990,000 Gain = 20(990,000-988,000) = 40,000 (10mil 40,000) (0.042/2) = 10,250,840 250840/10mil = 2.51% => 5.02% per annum Suppose the term structure of interest rates is flat in the US and Australia. The USD interest rate is 7% per annum and the AUD rate is 9% per annum. The current value of the AUD is 0.62 USD. Under the terms of a swap agreement, a financial institution pays 8% per annum in AUD and receives 4% per annum in USD. The principles in the two currencies are $12 million USD and $20 million AUD. Payments are exchanged every year, with one exchange having just taken place. The swap will last two more years. What is the value of the swap to the financial institution? Assume all interest rates are continuously compounded. Solution 1 - Difference in Value between USD & AUD Bonds Institution: Short USD Bond & Long AUD Bonds PV USD Bond = 0.48e -0.07(1) + 12.48e -0.07(2) = 11.297USD USD Coupon = 12million * 0.04 = 0.48 PV AUD Bond = 1.6e-0.09(1) + 21.6e-0.09(2) = 19.504AUD Coupon =20million * 0.08 = 1.6 Value of swap = B 0 -S 0 B F = 11.297 0.62(19.504) = -0.795 => -795,000 Solution 2 Value as Series of Forward Exchange Agreements 1 st year forward exchange F 0 = S 0 e (RD-Rf)t = 0.62e (0.07-0.09)(1) = 0.6077 2 nd year forward F 0 =S 0 e (RD-Rf)t = 0.62e (0.07-0.09)(2) = 0.5957 Value of swap = (.48 (1.6*0.6077))e -0.07(1) + (12.48-(21.6*.5957)e -0.07(2) = -795,000 Valuation of Equity Swaps To lower risk, the fund manager agrees to pay a dealer S&P 100 return for 8.75% fixed Index moves as follows I 0 =2500 I 6mo = 2600 I 12mo = 2570 R 6mo = 2600/2500-1 = 4% Fund manager pays => Swap dealer 0.04(1,000,000) = 40,000 Swap Dealer => Fund manager 0.0875(1,000,000)(182/365) = 43,630 Period 1 Net payment: Swap dealer to fund manager = 3630 R 12mo 2570/2600 1 = -1.15% => 1,000,000 (0.0115) = 11,500 Period 2 Net payment: Swap dealer to fund manager = 11500 + 43630