© 2004 South-Western Publishing 1

Chapter 12

Futures Contracts and Portfolio Management

2

Outline

The concept of immunization Altering asset allocation with futures

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The Concept of Immunization

Introduction Bond risks Duration matching Duration shifting Hedging with interest rate futures Increasing duration with futures Disadvantages of immunizing

4

Introduction

An immunized bond portfolio is largely protected from fluctuations in market interest rates– Seldom possible to eliminate interest rate risk

completely – A portfolio’s immunization can wear out, requiring

managerial action to reinstate the portfolio– Continually immunizing a fixed-income portfolio can

be time-consuming and technical

5

Bond Risks

A fixed income investor faces three primary sources of risk:– Credit risk– Interest rate risk– Reinvestment rate risk

6

Bond Risks (cont’d)

Credit risk is the likelihood that a borrower will be unable or unwilling to repay a loan as agreed– Rating agencies measure this risk with

bond ratings– Lower bond ratings mean higher expected

returns but with more risk of default– Investors choose the level of credit risk

that they wish to assume

7

Bond Risks (cont’d)

Interest rate risk is a consequence of the inverse relationship between bond prices and interest rates– Duration is the most widely used measure of a

bond’s interest rate risk

8

Bond Risks (cont’d)

Reinvestment rate risk is the uncertainty associated with not knowing at what rate money can be put back to work after the receipt of an interest check– The reinvestment rate will be the

prevailing interest rate at the time of reinvestment, not some rate determined in the past

9

Duration Matching

Introduction Bullet immunization Bank immunization

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Introduction

Duration matching selects a level of duration that minimizes the combined effects of reinvestment rate and interest rate risk

Two versions of duration matching:– Bullet immunization– Bank immunization

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Bullet Immunization

Seeks to ensure that a predetermined sum of money is available at a specific time in the future regardless of interest rate movements

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Bullet Immunization (cont’d)

Objective is to get the effects of interest rate and reinvestment rate risk to offset– If interest rates rise, coupon proceeds can be

reinvested at a higher rate– If interest rates fall, proceeds can be reinvested

at a lower rate

13

Bullet Immunization (cont’d)

Bullet Immunization Example

A portfolio managers receives $93,600 to invest in bonds and needs to ensure that the money will grow at a 10% compound rate over the next 6 years (it should be worth $165,818 in 6 years).

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Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)

The portfolio manager buys $100,000 par value of a bond selling for 93.6% with a coupon of 8.8%, maturing in 8 years, and a yield to maturity of 10.00%.

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Bullet Immunization Example (cont’d)Panel A: Interest Rates Remain Constant

Bullet Immunization (cont’d)

 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6$8,800 $9,680 $10,648 $11,713 $12,884 $14,172

  $8,800 $9,680 $10,648 $11,713 $12,884    $8,800 $9,680 $10,648 $11,713      $8,800 $9,680 $10,648        $8,800 $9,680         

        Interest $68,805        Bond         Total $165,817

$8,800

$97,920

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Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)Panel B: Interest Rates Fall 1 Point in Year 3

 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6$8,800 $9,680 $10,648 $11,606 $12,651 $13,789

  $8,800 $9,680 $10,551 $11,501 $12,536    $8,800 $9,592 $10,455 $11,396      $8,800 $9,592 $10,455        $8,800 $9,592         

        Interest $66,568        Bond        Total $166,218

$8,800

$99,650

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Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)Panel C: Interest Rates Rise 1 Point in Year 3

 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6$8,800 $9,680 $10,648 $11,819 $13,119 $14,563

  $8,800 $9,680 $10,745 $11,927 $13,239    $8,800 $9,768 $10,842 $12,035      $8,800 $9,768 $10,842        $8,800 $9,768         

        Interest $69,247        Bond         Total $165,477

$8,800

$96,230

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Bullet Immunization (cont’d)

Bullet Immunization Example (cont’d)

The compound rates of return in the three scenarios are 10.10%, 10.04%, and 9.96%, respectively.

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Bank Immunization

Addresses the problem that occurs if interest-sensitive liabilities are included in the portfolio– E.g., a bank’s portfolio manager is concerned

with the entire balance sheet– A bank’s funds gap is the dollar value of its

interest rate sensitive assets (RSA) minus its interest rate sensitive liabilities (RSL)

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Bank Immunization (cont’d)

To immunize itself, a bank must reorganize its balance sheet such that:

sliabilitieor assets ofduration average weighted-dollar

sliabilitieor assets sensitiveinterest of uedollar val$

where

$$

,

,

LA

LA

LLAA

D

DD

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Bank Immunization (cont’d)

A bank could have more interest-sensitive assets than liabilities:– Reduce RSA or increase RSL to immunize

A bank could have more interest-sensitive liabilities than assets:– Reduce RSL or increase RSA to immunize

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Duration Shifting

The higher the duration, the higher the level of interest rate risk

If interest rates are expected to rise, a bond portfolio manager may choose to bear some interest rate risk (duration shifting)

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Duration Shifting (cont’d)

The shorter the maturity, the lower the duration

The higher the coupon rate, the lower the duration

A portfolio’s duration can be reduced by including shorter maturity bonds or bonds with a higher coupon rate

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Duration Shifting (cont’d)

Maturity

Coupon

Lower Higher

Lower Ambiguous Duration Lower

Higher Duration Higher

Ambiguous

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Hedging With Interest Rate Futures

A financial institution can use futures contracts to hedge interest rate risk

The hedge ratio is:

)1(

)1(

bff

ctdbbctd YTMDP

YTMDPCFHR

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Hedging With Interest Rate Futures (cont’d)

The number of contracts necessary is given by:

ratio hedge000,100$

par value portfolio contracts #

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Hedging With Interest Rate Futures (cont’d)

Futures Hedging Example

A bank portfolio holds $10 million face value in government bonds with a market value of $9.7 million, and an average YTM of 7.8%. The weighted average duration of the portfolio is 9.0 years. The cheapest to deliver bond has a duration of 11.14 years, a YTM of 7.1%, and a CBOT correction factor of 1.1529.

An available futures contract has a market price of 90 22/32 of par, or 0.906875. What is the hedge ratio? How many futures contracts are needed to hedge?

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Hedging With Interest Rate Futures (cont’d)

Futures Hedging Example (cont’d)

The hedge ratio is:

9898.0078.114.11906875.0

071.10.997.01529.1

HR

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Hedging With Interest Rate Futures (cont’d)

Futures Hedging Example (cont’d)

The number of contracts needed to hedge is:

98.989898.0000,100$

0$10,000,00 contracts #

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Increasing Duration With Futures

Extending duration may be appropriate if active managers believe interest rates are going to fall

Adding long futures positions to a bond portfolio will increase duration

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Increasing Duration With Futures (cont’d)

One method for achieving target duration is the basis point value (BPV) method– Gives the change in the price of a bond for a one

basis point change in the yield to maturity of the bond

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Increasing Duration With Futures (cont’d)

To change the duration of a portfolio with the BPV method requires calculating three BPVs:

futures

currenttarget

BPV

BPVBPVcontracts #

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Increasing Duration With Futures (cont’d)

The current and target BPVs are calculated as follows:

)2/1(

0.0001size portfoliodurationBPV targetcurrent, R

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Increasing Duration With Futures (cont’d)

The BPV of the cheapest to deliver bond is calculated as follows:

factor conversion)2/1(

0.0001size portfoliodurationBPVfutures

R

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Increasing Duration With Futures (cont’d)

BPV Method Example

A portfolio has a market value of $10 million, an average yield to maturity of 8.5%, and duration of 4.85. A forecast of declining interest rates causes a bond manager to decide to double the portfolio’s duration. The cheapest to deliver Treasury bond sells for 98% of par, has a yield to maturity of 7.22%, duration of 9.7, and a conversion factor of 1.1223. Compute the relevant BPVs and determine the number of futures contracts needed to double the portfolio duration.

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Increasing Duration With Futures (cont’d)

BPV Method Example (cont’d)

28.652,4$)2/085.01(

0.0001000,000,10$4.85BPVcurrent

56.304,9$)2/085.01(

0.0001000,000,10$9.70BPVtarget

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Increasing Duration With Futures (cont’d)

BPV Method Example (cont’d)

42.83$1223.1)2/0722.01(

0.0001000,100$9.70BPVctd

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Increasing Duration With Futures (cont’d)

BPV Method Example (cont’d)

The number of contracts needed to double the portfolio duration is:

77.55$83.42

$4,652.28-$9,304.56contracts #

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Disadvantages of Immunizing

Opportunity cost of being wrong Lower yield Transaction costs Immunization: instantaneous only

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Opportunity Cost of Being Wrong

If the market is efficient, it is very difficult to forecast changes in interest rates

An incorrect forecast can lead to an opportunity cost of immunized portfolios

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Lower Yield

Immunization usually results in a lower level of income generated by the funds under management

By reducing the portfolio duration, the portfolio return will shift to the left on the yield curve, resulting in a lower level of income

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Transaction Costs

Costs include:– Trading fees– Brokerage commissions– Bid-ask spread – Tax liabilities

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Immunization: Instantaneous Only

Durations and yields to maturity change every day– A portfolio may be immunized only temporarily

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Altering Asset Allocation With Futures

Tactical changes Initial situation Bond adjustment Stock adjustment Neutralizing cash

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Tactical Changes

Investment policy statements may give the portfolio manager some latitude in how to split the portfolio between equities and fixed income securities

The portfolio manager can mix both T-bonds and S&P 500 futures into the portfolio to adjust asset allocation without disturbing existing portfolio components

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Initial Situation

Portfolio market value = $175 million Invested 82% in stock (average beta = 1.10)

and 18% in bonds (average duration = 8.7; average YTM = 8.00%)

The portfolio manager wants to reduce the equity exposure to 60% stock

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Initial Situation (cont’d)

Existing Asset Allocation

Stock82%

Bonds18%

Desired Asset Allocation

Stock60%

Bonds40%

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Initial Situation (cont’d)

Stock Index Futures  September settlement = 1020.00

 Treasury Bond Futures

September settlement = 91.05 Cheapest to deliver bond:

– Price = 95%– Maturity = 18 years– Coupon = 9 %– Duration = 8.60– Conversion factor = 1.3275

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Initial Situation (cont’d)

Determine:– How many contracts will remove 100% of each

market and interest rate risk– What percentage of this 100% hedge matches

the proportion of the risk we wish to retain

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Bond Adjustment

Using the BPV technique:

351,26)2/080.01(

0.0001%)18000,000,175($8.70BPVcurrent

82.61)2/0959.01(

0.0001000,100$8.60BPVfutures

558,58)2/080.01(

0.0001%)40000,000,175($8.70BPVtarget

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Bond Adjustment (cont’d)

The number of contracts to completely hedge the bond portion of the portfolio is:

Thus, the manager should buy 410 T-bond futures

98.52082.61

26,351-58,558

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Stock Adjustment

For this portfolio, the hedge ratio is:

Selling 619 stock index futures would turn the stock into a synthetic T-bill

02.61910.100.020,1250$

%82000,000,175$

HR

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Stock Adjustment (cont’d)

The current equity investment is $143,500,000

The desired equity investment is $105,000,000, which is 26.83% less than the current level

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Stock Adjustment (cont’d)

We can use 26.83% of the stock index futures hedge ratio:

08.16602.619%83.26

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Stock Adjustment (cont’d)

The portfolio manager can change the asset allocation from 82% stock, 18% bonds to 60% stock, 40% bonds by– Buying 521 T-bond futures and– Selling 166 stock index futures

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Neutralizing Cash

It is harder to “beat the market” with the downward bias in relative fund performance due to cash

Cash can be neutralized by offsetting it with long positions in stock index futures

Cash can be neutralized by offsetting it with long positions in interest rate futures