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Chapter 22

Integrating Derivative Assets and Portfolio Management

Portfolio Construction, Management, & Protection, 4e, Robert A. StrongCopyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.

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Life wasn’t designed to be risk-free. They key is not to eliminate risk, but to estimate it accurately and manage

it wisely.

William A. Schreyer, former chairman and CEO, Merrill Lynch & Company

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Outline Introduction Setting the Stage Meeting an Income Constraint Risk Management Managing Cash Drag

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Introduction Futures and options:

• Can be used in risk management and income generation

• Can be integrated into the portfolio management process

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Setting the Stage Portfolio Objectives Portfolio Construction

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Portfolio Objectives Portfolio objectives must be set with or

without derivatives

Futures and options can be used to adjust the fixed-income portfolio, the equity portfolio, or both to accomplish the objectives

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Portfolio Objectives (cont’d) Assume:

• You are newly responsible for managing a corporate in-house scholarship fund

• The fund consists of corporate and government bonds and bank CDs

• The fund has growth of income as the primary objective and capital appreciation as the secondary objective

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Portfolio Objectives (cont’d) Assume (cont’d):

• A one-time need requires income generation of $75,000 during the next year

• An account is opened with the deposit of cash and the existing fixed-income securities for a value of about $1.5 million

• Trading fees are paid out of a small, separate trust fund

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Portfolio Construction Fixed-Income Securities Stocks

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Fixed-Income Securities The fund holds ten fixed-income securities:

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Stocks You decide to include stocks in the

portfolio for $1,000,000 so that:• The portfolio beta is between 1.05 and 1.15• The investment in each stock is between 4 and

7 percent of the total• You avoid odd lots

Linear programming can be used to determine the solution (see next slide)

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Stocks (cont’d) The final portfolio consists of:

• $495,002 in bonds

• $996,986 in stocks

• $3,014 in cash

• A total value of $1,495,002

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Meeting an Income Constraint Determining Unmet Income Needs Writing Index Calls

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Determining Unmet Income Needs

The existing portfolio should yield:• $33,350 from bonds• $25,026 from dividends

You are $16,624 short relative to the $75,000 goal

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Writing Index Calls You want to write index call options to

generate the additional needed income:• Write short-term out-of-the-money calls to

avoid exercise• Determine implied volatilities of the options• Use the implied volatilities to determine the

option deltas• Determine the number of options you can write

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Writing Index Calls (cont’d) Eligible options are identified (all with

August expiration):

Striking Price Premium Delta

305 4.13 0.435

310 3 0.324

315 1.75 0.228

320 1 0.151

Current level of the Index = 298.96

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Writing Index Calls (cont’d) You determine the maximum contracts you

can write using stock as collateral:

Striking Price Premium Delta

Maximum Contracts Income

305 4.875 0.435 171 $83,362

310 3.00 0.324 203 60,900

315 1.75 0.228 244 42,700

320 1.00 0.151 301 30,100

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Writing Index Calls (cont’d) You decide to write 56 AUG 310 index

calls:• Generates $3 × 56 × 100 = $16,800 in income

immediately

• The delta of 0.324 indicates that these options will likely expire worthless

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Risk Management Stock Portfolio Hedging Company Risk Fixed-Income Portfolio

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Stock Portfolio Determining the Portfolio Delta and Beta Caveats about Prices from the Popular Press Caveats about Black-Scholes Prices for

Away-from-the-Money Options

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Determining the Portfolio Delta and Beta

The equity portion of the portfolio has a beta of 1.08

Writing index call options always reduces the portfolio beta• Short calls carry negative deltas

It is important to know the risk level of the portfolio

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Determining the Portfolio Delta and Beta (cont’d)

First, determine the hedge ratio for the stock portfolio:

Portfolio value Beta

Contract value

$996,9751.08 36.02

$298.96 100

HR

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Determining the Portfolio Delta and Beta (cont’d)

The stock portfolio is theoretically equivalent to 36.02 at-the-money contracts of the index

Next, calculate the delta of a hypothetical index option with a striking price of 298.96• Assume the delta is 0.578

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Determining the Portfolio Delta and Beta (cont’d)

Determine the delta contributions of the stock and the short options:

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Determining the Portfolio Delta and Beta (cont’d)

Lastly, estimate the resulting portfolio beta:

Initial portfolio delta Final portfolio delta

Initial portfolio beta Final portfolio beta

2,081.96 267.56

1.08 BetaBeta 0.14

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Determining the Portfolio Delta and Beta (cont’d)

The stock portfolio combined with the index options:• Has a slightly positive position delta• Has a slightly positive beta

The total portfolio is slightly bullish and will benefit from rising market prices

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Caveats about Prices from the Popular Press

Nonsynchronous trading is the phenomenon whereby comparative prices come from different points in time• Prices for less actively traded issues may have

been determined hours before the close of the market

• When you consider strategies involving away from the money options, you should verify the actual bid/ask prices for a security

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Caveats about Black-Scholes Prices for Away-from-the-Money

Options

The Black-Scholes Option Pricing Model:• Works well for near-the-money options • Works less accurately for options that are

substantially in the money or out of the money

To calculate delta, it may be preferable to calculate implied volatility for the option you are investigating

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Hedging Company Risk Introduction Buying Puts Buying Puts and Writing Calls

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Introduction Equity options can be used to hedge

company specific risk• Company specific risk is in additional to overall

market risk– e.g., a lawsuit

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Buying Puts To hedge against a small price change, it is

necessary to determine how many option contracts are necessary to bring the position delta to zero:

100deltaput

0.1sharescontracts of #

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Buying Puts (cont’d)Example

You own 1,000 shares of a stock currently selling for $56 per share. Put options are available with a premium of $0.45 and a $55 striking price. The put delta is –0.18.

How many options should you purchase to hedge your position in the stock from a downfall due to company specific risk?

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Buying Puts (cont’d)Example (cont’d)

Solution: Calculate the number of contracts needed:

56.5510018.0

0.1000,1

100deltaput

0.1sharescontracts of #

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Buying Puts and Writing Calls Hedging a long stock position involves adding

negative deltas to the positive deltas from the shares• Long puts and short calls both have negative deltas

Major market movements up or down can result in significantly different ending portfolio values for the puts-only scenario compared with the puts-and-calls alternative

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Fixed-Income Portfolio Hedging the Bond Portfolio Value with T-

Bond Futures Hedging the Bond Portfolio with Futures

Options

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Hedging the Bond Portfolio Value with T-Bond Futures

T-bond futures can be used to reduce interest rate risk by reducing portfolio duration• If interest rates rise, the value of a fixed-income

portfolio declines

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Hedging the Bond Portfolio Value with T-Bond Futures (cont’d)

Determine the hedge ratio:

where price of bond portfolio as a percentage of par

duration of bond portfolio

price of futures contract as a percentage

duration of cheapest-to-deliver bond eligible

b bctd

f f

b

b

f

f

P DHR CF

P D

P

D

P

D

for delivery

conversion factor for the cheapest-to-deliver bondctdCF

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Hedging the Bond Portfolio Value with T-Bond Futures (cont’d)

Determine the number of contracts you need to sell to hedge:

Portfolio valueNumber of contracts

$100,000HR

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Hedging the Bond Portfolio Value with T-Bond Futures (cont’d)

Example

A fixed-income portfolio has a value of $495,002. Using the cheapest-to-deliver bond, you determine a hedge ratio of 0.8215.

How many T-bond futures do you need to sell to completely hedge this portfolio?

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Hedging the Bond Portfolio Value with T-Bond Futures (cont’d)

Example (cont’d)

Solution: You need to sell 5 contracts to hedge completely:

contracts 91.4

9914.0000,100$

002,495$

000,100$

valuePortfoliocontracts ofNumber

HR

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Hedging the Bond Portfolio with Futures Options

A futures option is an option giving its owner the right to buy or “sell” a futures contract• A futures call gives its owner the right to go

long a futures contract

• A futures put gives its owner the right to go short a futures contract

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Hedging the Bond Portfolio with Futures Options (cont’d)

The buyer of a futures option has a known and limited maximum loss• Buying only the futures contract can result in

large losses

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Hedging the Bond Portfolio with Futures Options (cont’d)

Futures options do not require the good faith deposit associated with futures

You could buy T-bond futures puts instead of going short T-bond futures to hedge the bond portfolio

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Hedging the Bond Portfolio with Futures Options (cont’d)

The appropriate hedge ratio for futures options is:

Portfolio value 1

$100,000 DeltaHR CF

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Hedging the Bond Portfolio with Futures Options (cont’d)

Example

A fixed-income portfolio has a value of $495,002. MAR 98 T-bond futures calls are available with a premium of 2-44 and a delta of 0.583. The underlying futures currently sell for 91.

How many calls do you need to write to hedge? What is the income this strategy generates?

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Hedging the Bond Portfolio with Futures Options (cont’d)

Example (cont’d)

Solution: The hedge ratio indicates you need to write 9 contracts to hedge:

Portfolio value 1

$100,000 Delta

$495,002 10.91

$100,000 0.583

8.933

HR CF

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Hedging the Bond Portfolio with Futures Options (cont’d)

Example (cont’d)

Solution (cont’d): Writing 9 calls will generate $24,187.50:

2 44/64% × $100,000 × 9 = $24,187.50

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Managing Cash Drag A portfolio suffers a cash drag when it is

not fully invested• Cash earns a below-market return and dilutes

the portfolio return

A solution is to go long stock index futures to offset cash holdings

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Managing Cash Drag (cont’d) The hedge ratio is:

Portfolio size Beta

Futures sizeHR

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Managing Cash Drag (cont’d)Example

You are managing a $600 million portfolio. 93% of the portfolio is invested in equity, and 7% is invested in cash. Your equity beta is 1.0. During the last year, the S&P 500 index (your benchmark) earned 8 percent, with cash earning 2.0 percent.

What is the return on your portfolio?

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Managing Cash Drag (cont’d)Example (cont’d)

Solution: The return on your total portfolio is 7.58% (42 basis points below the market return):

(0.93 x 0.08) + (0.07 x 0.02) = 7.58%

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Managing Cash Drag (cont’d)Example (cont’d)

Assume a distant SPX futures contract settles for 1150.00.

How many futures contracts should you buy to make your portfolio behave like a 100 percent equity index fund?

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Managing Cash Drag (cont’d)Example (cont’d)

Solution: The hedge ratio indicates you should buy 146 SPX futures:

Portfolio size Beta

Futures size0.07 $600,000,000

1.01150.00 $250

146.09

HR