Chapter 12
Futures Contracts
and Portfolio
Management
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© 2004 South-Western Publishing
Outline
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Pricing of interest rate futures
Duration
The concept of immunization
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Bank
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Hedging with interest rate futures
Pricing Interest Rate Futures
Contracts
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Interest rate futures prices come from the
implications of cost of carry:
Ft  S (1  C0,t )
where
Ft  futures price for delivery at time t
S  spot commodity price
C0,t  cost of carry from time zero to time t
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Computation
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Cost of carry is the net cost of carrying the
commodity forward in time (the carry return
minus the carry charges)
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If you can borrow money at the same rate that a
Treasury bond pays(Tr), your cost of carry is
zero
Solving for C in the futures pricing equation
yields the implied repo rate Rp (implied
financing rate)
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
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An immunized bond portfolio is largely
protected from fluctuations in market
interest rates
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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
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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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Interest rate risk (price and reinvestment)
is a consequence of the inverse
relationship between bond prices and
interest rates and the risk of reinvestment
of coupons
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Duration is the most widely used measure of a
bond’s interest rate risk
Duration Matching
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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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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
(skip details on the example)
Choose a bond with YTM=desired return and duration
matching the time you will need the money from the
investment
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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  D A  $ L  DL
where
$ A, L  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)
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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
Adding long futures positions to a bond
portfolio will increase duration
One method for achieving target duration is
the basis point value (BPV) method (the
convexity of Duration) skip BPV
Review:
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Futures – 3 theories of pricing; differences
between options&futures; futures&forwards.
Stock Index Futures –Pricing, Hedge ratio; # of
contracts to increase or decrease market risk
exposure. Beta is a linear function.
FX futures – Pricing PPP, IRP.
Interest rate futures – Pricing, discount vs. bond
equiv. yield. Hedge ratio, # of contracts, duration,
convexity of duration