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Chapter 22Benching the Equity Players

Portfolio Construction, Management, & Protection, 5e, Robert A. StrongCopyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.

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Outline Introduction Using options Using futures contracts Dynamic hedging

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Introduction 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

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Hedging Hedging removes risk. Hedging involves

establishing a second position whose price behavior will likely offset the price behavior of the original portfolio.

The objective of portfolio protection is the temporary removal of some or all the market risk associated with a portfolio. Portfolio protection techniques are generally more economic in terms of commissions and managerial time than the sale and eventual replacement of portfolio components.

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Using Options Introduction Equity options with a single security Index options

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Introduction Options enable the portfolio manager to

adjust the characteristics of a portfolio without disrupting it

Knowledge of options improves the portfolio manager’s professional competence

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Equity Options with A Single Security

Importance of delta Protective puts Protective put profit and loss diagram Writing covered calls

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Black-Scholes Formula(European Options)

0 1 2

2 0 1

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1

2 1

( ) ( )

( ) ( )

where

ln( / ) ( / 2)

qT rT

rT qT

C S e N d Ke N d

P Ke N d S e N d

S K r q Td

T

d d T

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Importance of Delta Delta is a measure of the sensitivity of the

price of an option to changes in the price of the underlying asset:

Delta

where option premium

stock price

P

S

P

S

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Importance of Delta (cont’d)

Delta enables the portfolio manager to figure out the number of option contracts necessary to mimic the returns of the underlying security. This statistic is important in the calculation of many hedge ratios.

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Importance of Delta (cont’d) Delta:

• Equals N(d1) in the Black-Scholes Call price.

• Equals -N(-d1) in the Black-Scholes Put price.

• 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

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Protective Puts A protective put is a long stock position

combined with a long put position

Protective puts are useful if someone:• Owns stock and does not want to sell it• Expects a decline in the value of the stock

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Protective Put Profit and Loss Diagram

Assume the following information for ZZX:

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Protective Put Profit & Loss Diagram (cont’d) Long position for ZZX stock:

0

-50$50

Stock Price at Option Expiration

Profit or Loss

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Protective Put Profit & Loss Diagram (cont’d) 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

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Protective Put Profit & Loss Diagram (cont’d) Protective put diagram:

0$45

Stock Price at Option Expiration

Profit or Loss Maximum Gain is unlimited

Maximum Loss = $6

-6

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Protective Put Profit & Loss Diagram (cont’d)

Observations:• The maximum possible loss is $6

• The potential gain is unlimited

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Protective Put Profit & Loss Diagram (cont’d) Selecting the striking price for the

protective put is like selecting the deductible for your stock insurance• The more protection you want, the higher the

premium

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Writing Covered Calls 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

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Writing Covered Calls (cont’d) The premium received means no cash loss

occurs until the stock price falls below the current price minus the premium received

The stock price could advance and the option could be called

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Differences

Protective puts provide protection against large price declines, whereas covered calls provide only limited downside protection. Covered calls bring in the option premium, while the protective put requires a cash outlay.

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Index Options 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

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Index Options (cont’d)

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(d1)

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Index Options (cont’d) The hedge ratio is a calculated value

indicating the number of puts necessary:

Portfolio value 1Portfolio beta

Contract "value" Delta

where contract value = index level 100

HR

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Portfolio Insurance Example #1: S&P 100 index Options (OEX)

The OEX contract is the tool of choice for many professional portfolio risk managers.

While the S&P 100 futures contract is similar to traditional agricultural futures, delivery does not occur, nor does it need to occur for this to be an effective hedging tool.

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Index Options (cont’d)Example

OEX 315 OCT puts are available for premium of $3.25. The delta for these puts is –0.235, and the current index level is 327.19.

How many puts are needed to hedge a portfolio with a market value of $150,000 and a beta of 1.20?

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Index Options (cont’d)Example (cont’d)

Solution: You should buy 23 puts to hedge the portfolio:

Portfolio value 1Portfolio beta

Contract "value" Delta

$150,000 11.20

$32,719 0.235

23.41

HR

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Financial Futures Financial futures are the fastest-growing segment

of the futures market

The number of underlying assets on which futures contracts are available grows every year• Futures markets and indexes exist in many nations

Stock index futures contracts are similar to the traditional agricultural contracts except for the matter of delivery

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Characteristics of the S&P 500 Stock Index Futures

Contract size = $250 x index value Minimum price change is 0.10 ($25) Initial good faith deposit for a speculator is $20,625

(subject to change) Contracts are marked to the market daily Contracts are settled in cash Contracts do not earn dividends Trading hours: 9:30 a.m. – 4:15 p.m. EST Settlement months:

• March (H), JUNE (M), SEPT (U), December (Z) Expiration: Third Friday of contract month

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Hedging with Stock Index Futures

With the S&P 500 futures contract, a portfolio manager can attenuate the impact of a decline in the value of the portfolio components

S&P 500 futures can be used to hedge:• Endowment funds• Mutual funds• Other broad-based portfolios

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Hedging with Stock Index Futures (cont’d)

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

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Hedging with Stock Index Futures (cont’d)

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

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Hedge Ratio Computation A futures hedge ratio indicates the number

of contracts needed to mimic the behavior of a portfolio

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

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Hedge Ratio Computation (cont’d) The futures hedge ratio is:

Dollar value of portfolio Beta

Dollar value of S&P contractHR

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Hedge Ratio Computation (cont’d)Example

You are managing a $90 million portfolio with a beta of 1.50. 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 353.00.

How many S&P 500 stock index futures should you short to hedge the portfolio?

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Hedge Ratio Computation (cont’d)Example (cont’d)

Solution: Calculate the hedge ratio:

75.529,1

50.1353250$

0$90,000,00

Betacontract P&S of ueDollar val

portfolio of ueDollar val

HR

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Hedge Ratio 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.

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The Market Falls If the market falls:

• There is a loss in the stock portfolio

• There is a gain in the futures market

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The Market Falls (cont’d)Example

Consider the previous example. Assume that the S&P 500 index is currently at a level of 348.76. Over the next few months, the S&P 500 index falls to 325.00.

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 325.00.

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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

Note: Hedge is not perfect!

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The Market Rises If the market rises:

• There is a gain in the stock portfolio

• There is a loss in the futures market

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The Market Rises (cont’d)Example

Consider the previous example. Assume that the S&P 500 index is currently at a level of 348.76. Over the next few months, the S&P 500 index rises to to 365.00.

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 365.00.

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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:

(353 – 365) × 1,530 × $250 = $4,590,000 loss

[This position also ends with a gain because the index striking price exceeds the index value.]

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The Market Is Unchanged 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)

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Hedging in Retrospect Futures hedging is never perfect in practice:

• It is usually not possible to hedge exactly– Index futures are available in integer quantities only

• Stock portfolios seldom behave exactly as their betas say they should

Short hedging reduces profits in a rising market

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Single Stock Futures Promise to buy or to deliver 100 shares of a

single stock Physical delivery required 550 different stocks in 2008 2,400 different stocks in 2013 Allows seller to hedge risk arising from the

decline in a specific stock

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Speculating with Single Stock Futures

If you expect a stock to increase you could: a. Buy stock: In a cash account requires 100% of

value b. Buy call option: Pay price to seller, may end up

being worthless c. Buy futures contract:

• Requires only 20% margin– This good faith deposit is still investor’s money

• Option premium does not exist and is not paid to seller• But, Liable for all losses as stock price sinks below

striking price

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Dynamic Hedging 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

• Avoids the cost of a protective put

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Dynamic Hedging Example (cont’d)

You own 1,000 shares of ZZX stock You are interested in buying ten JUL 50 puts for

downside protection The JUL 50 put expires in 60 days The JUL 50 put delta is –0.435 T-bills yield 8 percent ZZX pays no dividends ZZX stock’s volatility is 30 percent

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Dynamic Hedging Example (cont’d)

The position delta is the sum of all the deltas in a portfolio:• (1,000 × 1.0) + (100 x 10 × –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

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Dynamic Hedging Example (cont’d)

With the puts, the portfolio is 56.5 percent as bullish as without the puts

You can sell short 435 shares to achieve the position delta of 565:• (1,000 × 1.0) + (435 × –1.0) = 565

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The Dynamic Part of the Hedge 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 491

– (1,000 × 1.0) + (10 x 100 × –0.509) = 491

To continue dynamic hedging and to replicate the put, it is necessary to sell short another 74 shares (435 + 74 = 509 shares)

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The Dynamic Part of the Hedge (cont’d)

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 629

– (1,000 × 1.0) + (1,000 × –0.371) = 629

To continue dynamic hedging and to replicate the put, it is necessary to cover 64 of the 435 shares you initially sold short

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Dynamic Hedging with Futures Contracts

Alternative to stock options (and their related costs) and short sales (and related margins)

Appropriate for large portfolios

Stock index futures have a delta of +1.0

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Dynamic Hedging with Futures Contracts (cont’d)

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

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Dynamic Hedging with Futures Contracts (cont’d)

We must sell enough futures contracts to pull the position delta to 0.600

The hedge ratio is:

Dollar value of portfolio Beta

Dollar value of S&P contract$52,500,000

1.0$250 700

300 contracts

HR

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Dynamic Hedging with Futures Contracts (cont’d)

If the hedge ratio is 300 contracts, we must sell 40% × 300 = 120 contracts to achieve a position delta of 0.600