Chapter 12
Futures Contracts
and Portfolio
Management
1
© 2004 South-Western Publishing
Outline
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The concept of immunization
Altering asset allocation with futures
The Concept of Immunization
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Introduction
Bond risks
Duration matching
Duration shifting
Hedging with interest rate futures
Increasing duration with futures
Disadvantages of immunizing
Introduction

An immunized bond portfolio is largely
protected from fluctuations in market
interest rates
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4
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
Bond Risks

A fixed income investor faces three primary
sources of risk:
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Credit risk
Interest rate risk
Reinvestment rate risk
Bond Risks (cont’d)
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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
Bond Risks (cont’d)

Interest rate risk is a consequence of the
inverse relationship between bond prices
and interest rates
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7
Duration is the most widely used measure of a
bond’s interest rate risk
Bond Risks (cont’d)
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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
Duration Matching
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9
Introduction
Bullet immunization
Bank immunization
Introduction
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Duration matching selects a level of
duration that minimizes the combined
effects of reinvestment rate and interest
rate risk
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Two versions of duration matching:
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Bullet immunization
Bank immunization
Bullet Immunization
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Seeks to ensure that a predetermined
sum of money is available at a specific
time in the future regardless of
interest rate movements
Bullet Immunization (cont’d)
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Objective is to get the effects of interest
rate and reinvestment rate risk to offset
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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
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).
13
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%.
14
Bullet Immunization (cont’d)
Bullet Immunization Example (cont’d)
Panel A: Interest Rates Remain Constant
Year 1
$8,800
Year 2
$9,680
$8,800
Year 3
$10,648
$9,680
$8,800
Year 4
$11,713
$10,648
$9,680
$8,800
Year 5
$12,884
$11,713
$10,648
$9,680
$8,800
Interest
Bond
Total
15
Year 6
$14,172
$12,884
$11,713
$10,648
$9,680
$8,800
$68,805
$97,920
$165,817
Bullet Immunization (cont’d)
Bullet Immunization Example (cont’d)
Panel B: Interest Rates Fall 1 Point in Year 3
Year 1
$8,800
Year 2
$9,680
$8,800
Year 3
$10,648
$9,680
$8,800
Year 4
$11,606
$10,551
$9,592
$8,800
Year 5
$12,651
$11,501
$10,455
$9,592
$8,800
Interest
Bond
Total
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Year 6
$13,789
$12,536
$11,396
$10,455
$9,592
$8,800
$66,568
$99,650
$166,218
Bullet Immunization (cont’d)
Bullet Immunization Example (cont’d)
Panel C: Interest Rates Rise 1 Point in Year 3
Year 1
$8,800
Year 2
$9,680
$8,800
Year 3
$10,648
$9,680
$8,800
Year 4
$11,819
$10,745
$9,768
$8,800
Year 5
$13,119
$11,927
$10,842
$9,768
$8,800
Interest
Bond
Total
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Year 6
$14,563
$13,239
$12,035
$10,842
$9,768
$8,800
$69,247
$96,230
$165,477
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
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Addresses the problem that occurs if
interest-sensitive liabilities are included in
the portfolio
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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)
Bank Immunization (cont’d)
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To immunize itself, a bank must reorganize
its balance sheet such that:
$ A  DA  $ L  DL
$ A, L
where
 dollar val ue of interest sensitive assets or liabilitie s
DA, L  dollar - weighted average duration of assets or liabilitie s
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Bank Immunization (cont’d)
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A bank could have more interest-sensitive
assets than liabilities:
–
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A bank could have more interest-sensitive
liabilities than assets:
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Reduce RSA or increase RSL to immunize
Reduce RSL or increase RSA to immunize
Duration Shifting
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The higher the duration, the higher the level
of interest rate risk
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If interest rates are expected to rise, a bond
portfolio manager may choose to bear
some interest rate risk (duration shifting)
Duration Shifting (cont’d)
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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
Duration Shifting (cont’d)
Coupon
Lower
Higher
Lower
Ambiguous
Duration
Lower
Higher
Duration
Higher
Ambiguous
Maturity
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Hedging With Interest Rate
Futures
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A financial institution can use futures
contracts to hedge interest rate risk
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The hedge ratio is:
Pb Db (1  YTM ctd )
HR  CFctd 
Pf D f (1  YTM b )
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Hedging With Interest Rate
Futures (cont’d)

The number of contracts necessary is given
by:
portfolio par value
# contracts 
 hedge ratio
$100,000
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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:
0.97  9.0 1.071
HR  1.1529 
 0.9898
0.906875 11.14 1.078
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Hedging With Interest Rate
Futures (cont’d)
Futures Hedging Example (cont’d)
The number of contracts needed to hedge is:
$10,000,000
# contracts 
 0.9898  98.98
$100,000
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Increasing Duration With
Futures
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
Extending duration may be appropriate if
active managers believe interest rates are
going to fall
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Adding long futures positions to a bond
portfolio will increase duration
Increasing Duration With
Futures (cont’d)
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One method for achieving target duration is
the basis point value (BPV) method
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Gives the change in the price of a bond for a one
basis point change in the yield to maturity of the
bond
Increasing Duration With
Futures (cont’d)
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To change the duration of a portfolio with
the BPV method requires calculating three
BPVs:
# contracts 
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BPVtarget  BPVcurrent
BPVfutures
Increasing Duration With
Futures (cont’d)
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The current and target BPVs are calculated
as follows:
BPVcurrent,target
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duration  portfolio size  0.0001

(1  R / 2)
Increasing Duration With
Futures (cont’d)
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The BPV of the cheapest to deliver bond is
calculated as follows:
BPVfutures
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duration  portfolio size  0.0001

(1  R / 2)  conversion factor
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)
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BPVcurrent
4.85  $10,000,000  0.0001

 $4,652.28
(1  0.085 / 2)
BPVtarget
9.70  $10,000,000  0.0001

 $9,304.56
(1  0.085 / 2)
Increasing Duration With
Futures (cont’d)
BPV Method Example (cont’d)
9.70  $100,000  0.0001
BPVctd 
 $83.42
(1  0.0722 / 2) 1.1223
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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:
$9,304.56 - $4,652.28
# contracts 
 55.77
$83.42
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Disadvantages of Immunizing
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Opportunity cost of being wrong
Lower yield
Transaction costs
Immunization: instantaneous only
Opportunity Cost of Being
Wrong
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If the market is efficient, it is very difficult to
forecast changes in interest rates
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An incorrect forecast can lead to an
opportunity cost of immunized portfolios
Lower Yield
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Immunization usually results in a lower
level of income generated by the funds
under management
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By reducing the portfolio duration, the
portfolio return will shift to the left on the
yield curve, resulting in a lower level of
income
Transaction Costs
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Costs include:
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Trading fees
Brokerage commissions
Bid-ask spread
Tax liabilities
Immunization: Instantaneous
Only
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Durations and yields to maturity change
every day
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A portfolio may be immunized only temporarily
Altering Asset Allocation With
Futures
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Tactical changes
Initial situation
Bond adjustment
Stock adjustment
Neutralizing cash
Tactical Changes
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Investment policy statements may give the
portfolio manager some latitude in how to
split the portfolio between equities and
fixed income securities
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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
Initial Situation
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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
Initial Situation (cont’d)
Existing Asset Allocation
Desired Asset Allocation
Bonds
18%
Bonds
40%
Stock
60%
Stock
82%
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Initial Situation (cont’d)
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Stock Index Futures
September settlement = 1020.00
Treasury Bond Futures
September settlement = 91.05
Cheapest to deliver bond:
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Price = 95%
Maturity = 18 years
Coupon = 9 %
Duration = 8.60
Conversion factor = 1.3275
Initial Situation (cont’d)
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Determine:
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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
Bond Adjustment
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Using the BPV technique:
BPVcurrent
8.70  ($175,000,000 18%)  0.0001

 26,351
(1  0.080 / 2)
BPVtarget
8.70  ($175,000,000  40%)  0.0001

 58,558
(1  0.080 / 2)
BPVfutures
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8.60  $100,000  0.0001

 61.82
(1  0.0959 / 2)
Bond Adjustment (cont’d)
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The number of contracts to completely
hedge the bond portion of the portfolio is:
58,558 - 26,351
 520.98
61.82
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Thus, the manager should buy 410 T-bond
futures
Stock Adjustment
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For this portfolio, the hedge ratio is:
$175,000,000  82%
HR 
1.10  619.02
$250 1,020.00
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Selling 619 stock index futures would turn
the stock into a synthetic T-bill
Stock Adjustment (cont’d)
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
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
Stock Adjustment (cont’d)

We can use 26.83% of the stock index
futures hedge ratio:
26.83% 619.02  166.08
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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
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Buying 521 T-bond futures and
Selling 166 stock index futures
Neutralizing Cash
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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