1 Chapter 20 Benching the Equity Players Portfolio Construction, Management, & Protection, 4e, Robert A. Strong Copyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.

2 Don’t be discouraged by a failure. It can be a positive experience. Failure is, in a sense, the highway to success, inasmuch as every discover of what is false leads us to seek earnestly after what is true, and every fresh experience points out some form of error which we shall afterwards carefully avoid. John Keats

3 Outline u Introduction u Using Options u Using Futures Contracts u Dynamic Hedging

4 Introduction u Portfolio protection involves adding components to a portfolio in order to establish a floor value for the portfolio using: Equity or stock index put options Futures contracts Dynamic hedging

5 Using Options u Introduction u Equity options with a Single Security u Index Options

6 Introduction u Options enable the portfolio manager to adjust the characteristics of a portfolio without disrupting it u Knowledge of options improves the portfolio manager’s professional competence

7 Equity Options with a Single Security u Importance of Delta u Protective Puts u Protective Put Profit and Loss Diagram u Writing Covered Calls

8 Importance of Delta u Delta is a measure of the sensitivity of the price of an option to changes in the price of the underlying asset:

9 Importance of Delta (cont’d) u Delta: Equals N(d 1 ) in the Black-Scholes OPM Allows us to determine how many options are needed to mimic the returns of the underlying security Is positive for calls and negative for puts Has an absolute value between 0 and 1

10 Protective Puts u A protective put is a long stock position combined with a long put position u Protective puts are useful if someone: Owns stock and does not want to sell it Expects a decline in the value of the stock

11 Protective Put Profit and Loss Diagram u Assume the following information for ZZX:

12 Protective Put Profit and Loss Diagram (cont’d) u Long position for ZZX stock: $50 0 –50 Stock Price at Option Expiration Profit or Loss Stock purchased at $50.

13 Protective Put Profit and Loss Diagram (cont’d) u Long position for SEP 45 put ($1 premium): 0 44 $45 Stock Price at Option Expiration Profit or Loss Maximum Gain = $44 Maximum Loss = $1 –1–1

14 Protective Put Profit and Loss Diagram (cont’d) u Protective put diagram: 0 $45 Stock Price at Option Expiration Profit or Loss Maximum Gain is Unlimited Maximum Loss = $6 –6–6 Stock bought at $50; 45 put bought at $1.

15 Protective Put Profit and Loss Diagram (cont’d) u Observations: The maximum possible loss is $6 The potential gain is unlimited

16 Protective Put Profit and Loss Diagram (cont’d) u Selecting the striking price for the protective put is like selecting the deductible for your car insurance The more protection you want, the higher the premium

17 Writing Covered Calls u Writing covered calls is an alternative to protective puts Appropriate when an investor owns the stock, does not want to sell it, and expects a decline in the stock price An imperfect form of portfolio protection

18 Writing Covered Calls (cont’d) u The premium received means no cash loss occurs until the stock price falls below the current price minus the premium received u The stock price could advance and the option could be exercised

19 Index Options u Investors buying index put options: Want to protect themselves against an overall decline in the market or Want to protect a long position in the stock

20 Index Options (cont’d) u If an investor has a long position in stock: The number of puts needed to hedge is determined via delta (as part of the hedge ratio) He needs to know all the inputs to the Black- Scholes OPM and solve for N(d 1 )

21 Index Options (cont’d) u The hedge ratio is a calculated value indicating the number of puts necessary:

22 Index Options (cont’d) Example OEX 315 OCT puts are available for premium of $3.25. The delta for these puts is – How many puts are needed to hedge a portfolio with a market value of $150,000 and a beta of 1.20?

23 Index Options (cont’d) Example (cont’d) Solution: You should buy 24 puts to hedge the portfolio:

24 Using Futures Contracts u Importance of Financial Futures u Stock Index Futures Contracts u S&P 500 Stock Index Futures Contract u Hedging with Stock Index Futures u Calculating a Hedge Ratio u Hedging in Retrospect

25 Importance of Financial Futures u Financial futures are the fastest-growing segment of the futures market u The number of underlying assets on which futures contracts are available grows every year

26 Stock Index Futures Contracts u Stock index futures contracts are similar to the traditional agricultural contracts except for the matter of delivery

27 S&P 500 Stock Index Futures Contract

28 Hedging with Stock Index Futures u With the S&P 500 futures contract, a portfolio manager can attenuate the impact of a decline in the value of the portfolio components u S&P 500 futures can be used to hedge: Endowment funds Mutual funds Other broad-based portfolios

29 Hedging with Stock Index Futures (cont’d) u To hedge using S&P stock index futures: Take a position opposite to the stock position –e.g., if you are long in stock, short futures Determine the number of contracts necessary to counteract likely changes in the portfolio value using: –The value of the appropriate futures contract –The dollar value of the portfolio to be hedged –The beta of your portfolio

30 Hedging with Stock Index Futures (cont’d) u Determine the value of the futures contract The CME sets the size of an S&P 500 futures contract at $250 times the value of the S&P 500 index The difference between a particular futures price and the current index is the basis

31 Calculating A Hedge Ratio u Computation u The Market Falls u The Market Rises u The Market is Unchanged

32 Computation u A futures hedge ratio indicates the number of contracts needed to mimic the behavior of a portfolio u The hedge ratio has two components: The scale factor –Deals with the dollar value of the portfolio relative to the dollar value of the futures contract The level of systematic risk –i.e., the beta of the portfolio

33 Computation (cont’d) u The futures hedge ratio is:

34 Computation (cont’d) Example You are managing a $90 million portfolio with a beta of The portfolio is well-diversified and you want to short S&P 500 futures to hedge the portfolio. S&P 500 futures are currently trading for How many S&P 500 stock index futures should you short to hedge the portfolio?

35 Computation (cont’d) Example (cont’d) Solution: Calculate the hedge ratio:

36 Computation (cont’d) Example (cont’d) Solution: The hedge ratio indicates that you need 1,530 S&P 500 stock index futures contracts to hedge the portfolio.

37 The Market Falls u If the market falls: There is a loss in the stock portfolio There is a gain in the futures market

38 The Market Falls (cont’d) Example Consider the previous example. Assume that the S&P 500 index is currently at a level of Over the next few months, the S&P 500 index falls to Show the gains and losses for the stock portfolio and the S&P 500 futures, assuming you close out your futures position when the S&P 500 index is at

39 The Market Falls (cont’d) Example (cont’d) Solution: For the $90 million stock portfolio: –6.81% × 1.50 × $90,000,000 = $9,193,500 loss For the futures: (353 – 325) × 1,530 × $250 = $10,710,000 gain

40 The Market Rises u If the market rises: There is a gain in the stock portfolio There is a loss in the futures market

41 The Market Rises (cont’d) Example Consider the previous example. Assume that the S&P 500 index is currently at a level of Over the next few months, the S&P 500 index rises to to Show the gains and losses for the stock portfolio and the S&P 500 futures, assuming you close out your futures position when the S&P 500 index is at

42 The Market Rises (cont’d) Example (cont’d) Solution: For the $90 million stock portfolio: 4.66% × 1.50 × $90,000,000 = $6,291,000 gain For the futures: (365 – 353) × 1,530 × $250 = $4,590,000 loss

43 The Market Is Unchanged u If the market remains unchanged: There is no gain or loss on the stock portfolio There is a gain in the futures market –The basis will deteriorate to 0 at expiration (basis convergence)

44 Hedging in Retrospect u Futures hedging is never perfect in practice: It is usually not possible to hedge exactly –Index futures are available in integer quantities only Stock portfolio seldom behave exactly as their betas say they should u Short hedging reduces profits in a rising market

45 Dynamic Hedging u Definition u Dynamic Hedging Example u The Dynamic Part of the Hedge u Dynamic Hedging with Futures Contracts

46 Definition u Dynamic hedging strategies: Attempt to replicate a put option by combining a short position with a long position to achieve a position delta equal to that which would be obtained via protective puts

47 Dynamic Hedging Example u Assume the following information for ZZX:

48 Dynamic Hedging Example (cont’d) u You own 1,000 shares of ZZX stock u You are interested in buying a JUL 50 put for downside protection u The JUL 50 put expires in 60 days u The JUL 50 put delta is –0.435 u T-bills yield 8 percent u ZZX pays no dividends u ZZX stock’s volatility is 30 percent

49 Dynamic Hedging Example (cont’d) u The position delta is the sum of all the deltas in a portfolio: (1,000 × 1.0) + (1,000 × –0.435) = 565 –Stock has a delta of 1.0 because it “behaves exactly like itself” –A position delta of 565 behaves like a stock-only portfolio composed of 565 shares of the underlying stock

50 Dynamic Hedging Example (cont’d) u With the puts, the portfolio is 56.5 percent as bullish as without the puts u You can sell short 435 shares to achieve the position delta of 565: (1,000 × 1.0) + (435 × –1.0) = 565

51 The Dynamic Part of the Hedge u Suppose that one week passes and: ZZX stock declines to $49 The delta of the JUL 50 put is now –0.509 The position delta has changed to: –(1,000 × 1.0) + (1,000 × –0.509) = 491

52 The Dynamic Part of the Hedge (cont’d) u To continue dynamic hedging and to replicate the put, it is necessary to sell short 74 shares ( = 509 shares)

53 The Dynamic Part of the Hedge (cont’d) u Suppose that one week passes and: ZZX stock rises to $51 The delta of the JUL 50 put is now –0.371 The position delta has changed to: –(1,000 × 1.0) + (1,000 × –0.371) = 629

54 The Dynamic Part of the Hedge (cont’d) u To continue dynamic hedging and to replicate the put, it is necessary to cover 64 of the 435 shares you initially sold short

55 Dynamic Hedging with Futures Contracts u Appropriate for large portfolios u Stock index futures have a delta of +1.0

56 Dynamic Hedging with Futures Contracts (cont’d) u Assume that: We wish to replicate a particular put option with a delta of –0.400 We manage an equity portfolio with a beta of 1.0 and $52.5 million market value A futures contract sells for 700 –The dollar value is $250 × 700 = $175,000

57 Dynamic Hedging with Futures Contracts (cont’d) u We must sell enough futures contracts to pull the position delta to u The hedge ratio is:

58 Dynamic Hedging with Futures Contracts (cont’d) u If the hedge ratio is 300 contracts, we must sell 40% × 300 = 120 contracts to achieve a position delta of 0.600