1. Hedging and speculative strategies using index futures Short hedge: Sell Index futures - offset market losses on portfolio by generating gains on futures Market Spot Position (Portfolio) Short Hedge (Sell Index Futures) Finance 70520 Dr. Steven C. Mann The Neeley School of Business

2. Hedging with Stock Index futures Example: Portfolio manager has well-diversified $40 million stock portfolio, with a beta of 1.20 1 % movement in S&P 500 Index expected to induce 1.20% change in value of portfolio. Scenario: Manager anticipates "bear" market (fall in value), and wishes to hedge against possibility. One Solution: Liquidate some or all of portfolio. (Sell securities and place in short-term debt instruments until "prospects brighten") n Broker Fees n Market Transaction costs (Bid/Ask Spread) n Lose Tax Timing Options (Forced to realize gains and/or losses) n Market Impact Costs n If portfolio large, liquidation will impact prices

3. Naive Short hedge: dollar for dollar Assume you hedge using the June 00 S&P 500 contract, currently priced at 1400. Dollar for dollar hedge number of contracts: V $40,000,000 = 114 contracts V (1400) ($250) = P F where: V = value of portfolio V = value of one futures contract P F

4. But: portfolio has higher volatility than S&P 500 Index ( portfolio beta=1.20) hedge initiated April 15, 2000, with Index at 1350. If S&P drops 5% by 5/2/00 to 1283, predicted portfolio change is 1.20 (-5%) = -6.0%, a loss of $2.4 million Net loss = $ 490,500 Assumes: 1) portfolio moves exactly as predicted by beta 2) futures moves exactly with spot Date IndexSpot position (equity) futures position 4/15/00 1350$40,000,000 short 114 contracts 5/2/00 1283 $37,600,000 futures drop 67 points -67 -$2,400,000 gain = (114)(67)(250) = $1,909,500

5. Minimum Risk Hedge Ratio Define hedge ratio (HR) as Futures position Spot Market Position Then the variance of returns on a position of one unit of the spot asset and HR units of futures is:   p =   S  + HR 2    F  + 2 HR  SF  S  F

6. Minimizing Hedged portfolio variance  [   S  + HR 2    F  + 2 HR  SF  S  F   Take partial derivative of variance w.r.t. HR, set = 0:    HR HR =  SF  S  F  cov SF = FF  2 HR   F + 2  SF  S  F FF This is the regression coefficient  SF

7. Estimating Minimum Variance Hedge ratio Estimate Following regression: spot return t =  SF  Futures ‘return’ t +  t Portfolio Return Futures ‘return’ (% change ) x x x x xxx x x x x x xx x x x x x xx x x x xx x line of best fit (slope =  SF ) intercept =  {

8. Calculating number of contracts for minimum variance hedging Hedge using June 00 S&P 500 contract, currently priced at 1400 Minimum variance hedge number of contracts: V P $40,000,000 = 137 contracts V F (1400)($250)  PF  where: V P = value of portfolio V = value of one futures contract F  PF = beta of portfolio against futures (1.20)

9. Minimum variance hedging results hedge initiated April 15, 2000, with Index at 1350. If S&P drops 5% by 5/2/00 to 1283, predicted portfolio change is 1.20 (-5%) = -6.0%, a loss of $2.4 million Net loss = $ 105,250 Assumes: 1) portfolio moves exactly as predicted by beta 2) futures moves exactly with spot ( note: futures actually moves more than spot : F = S (1+c) ) Date IndexSpot position (equity) futures position 4/15/00 1350$40,000,000 short 137 contracts 5/2/00 1283 $37,600,000 futures drop 67 points -67 -$2,400,000 gain = (137)(67)(250) = $2,294,750

10. Altering the beta of a portfolio where: V F = value of one futures contract V P = value of portfolio Define  PF as portfolio beta (against futures) Change market exposure of portfiolio to  new by buying (selling):  new -  PF  contracts VPVP VFVF

11. Impact of synthetic beta adjustment Return on Portfolio Return on Market Original Expected exposure Expected exposure from increasing beta Expected exposure from reducing beta

12. Changing Market Exposure: Example Begin with $40,000,000 portfolio:  PF  = 1.20 Prefer a portfolio with 50% of market risk (  New =.50) Hedge using June 00 S&P 500 contract, currently priced at 1400. V $40,000,000 (.50 - 1.20) V (1400) ($250)   New  PF  = P F = - 80 contracts

13. Reducing market exposure: results hedge initiated April 15, 2000, with Index at 1350. If S&P drops 5% by 5/2/00 to 1283, predicted portfolio change is 1.20 (-5%) = -6.0%, a loss of $2.4 million. predicted loss for  P =0.5 is (-2.5% of $40 m) = $1 million Net loss = $ 1,060,000 Assumes: 1) portfolio moves exactly as predicted by beta 2) futures moves exactly with spot ( note: futures actually moves more than spot : F = S (1+c) Date IndexSpot position (equity) futures position 4/15/00 1350$40,000,000 short 80 contracts 5/2/00 1283 $37,600,000 futures drop 67 points -67 -$2,400,000 gain = (80)(67)(250) = $1,340,000

14. Sector "alpha capture" strategies Portfolio Return Futures ‘return’ (% change) x x x x xxx x x x x x xx x x x x x xx x x x xx x line of best fit (slope =  ) intercept =  { Portfolio manager expects sector to outperform rest of market (  > 0)

15. "Alpha Capture" Return on Sector Portfolio Return on Market Expected exposure with "hot" market sector portfolio }  > 0 Slope =  SF = beta of sector portfolio against futures 0

16. "Alpha Capture" Return on Asset Return on Market Expected exposure with "hot" market sector Expected return on short hedge } expected gain on alpha 0

17. Alpha Capture example Assume $10 m sector fund portfolio with  = 1.30 You expect Auto sector to outperform market by 2% Eliminate market risk with minimum variance short hedge. ( using the June 00 S&P 500 contract, currently priced at 1400) V $10,000,000 (0 - 1.30) V (1400) ($250)   NEW  SF  = P F = - 28.6 x 1.3 = 37 contracts The desired beta is zero, and the number of contracts to buy (sell) is:

18. Alpha capture: market down 5% hedge initiated April 15, 2000, with Index at 1350. If S&P drops 5% by 5/2/00 to 1283, predicted portfolio change is 1.30(-5%) + 2% = -4.5%, a loss of $450,000. Net GAIN = $ 169,750 Assumes: 1) portfolio moves exactly as predicted by beta 2) futures moves exactly with spot ( note: futures actually moves more than spot : F = S (1+c)) Date IndexSpot position (equity) futures position 4/15/00 1350$10,000,000 short 37 contracts 5/2/00 1283 $9,550,000 futures drop 67 points -67 - $450,000 gain = (37)(67)(250) = $619,750

19. Alpha capture: market up 5% hedge initiated April 15, 2000, with Index at 1350. If S&P rises 5% by 5/2/0 to 1417, predicted portfolio change is 1.30( 5%) + 2% = 8.5%, a gain of $850,000. Net GAIN = $ 230,250 Assumes: 1) portfolio moves exactly as predicted by beta 2) futures moves exactly with spot ( note: futures actually moves more than spot : F = S (1+c)) Date IndexSpot position (equity) futures position 4/15/00 1350$10,000,000 short 37 contracts 2/2/98 1417 $10,850,000 futures rise 67 points +67 $850,000 loss = (37)(67)(250) = $619,750

20. Implied Repo rates Define: i I as implied repo rate (simple interest). i I is defined by: f (0,T) = S(0)(1+ i I T) ; f (0,T) and S(0) are current market prices. Example: asset with no dividend, storage cost, or convenience yield (example: T-bill) Given simple interest rate i s, the theoretical forward price (model price) is: f (0,T) = S(0)(1+i s T). If i I > i s market price > model price. i I < i s market price < model price. If i I > i s : arbitrage strategy : cost nowvalue at date T buy asset at cost S(0) S(0) S(T) borrow asset cost -S(0)-S(0)(1+i s T) sell forward at f(0,T) 0-[S(T) -f(0,T)] = Total 0f(0,T) - S(0)(1+i s T) = S(0)(1+ i I T) - S(0) )(1+i s T) = S(0)( i I - i s )T > 0

21. Pricing S&P 500 Index Futures Data from 11/10/97 Investor’s Business Daily

22. Index Futures to speculate on Interest rates

23. Rate speculation: rates rise and market down 30

24. Rate speculation: rates unchanged; market up 30