Chapter 12
Futures Contracts
and Portfolio
Management
1
© 2002 South-Western Publishing
The Concept of Immunization
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2
Introduction
Bond risks
Duration matching
Bullet Immunization
Bank immunization
Duration shifting
The Concept of Immunization
(cont’d)
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3
Hedging with interest rate futures
Increasing/decreasing duration with futures
Considerations of immunization
Introduction

An immunized bond portfolio is largely
protected from fluctuations in market
interest rates
–
–
–
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:
–
–
–
5
Credit risk
Interest rate risk
Reinvestment rate risk
Bond Risks (cont’d)
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6
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
–
7
Duration is the most widely used measure of a
bond’s interest rate risk - a measure of the %
change in the price of the security for a given
change in the yield
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
–
8
The reinvestment rate will be the
prevailing interest rate at the time of
reinvestment, not some rate determined
in the past
Duration
Duration may be considered as the weighted
average maturity for a bond or other fixed
income security
 takes into consideration that some of the
cash flows i.e the coupons are received
before maturity
–
9
remember that the value of the a bond is the PV
of its future cash flows - both the coupon
payments and the face value of the bond at
maturity at prevailing market interest rates.
Duration
Discount or zero coupon bonds
 maturity equals duration
Coupon or interest bearing bonds
 duration differs from maturity because
these interest payments are received prior
to maturity
10
Duration - Calculation
N
D1 =
E
t=1
(t)(CFt)/(1+Y/2)t
P
Where:
11
t = time period until receipt of cash flow
P = current price of the bond/security
CFt = cash flow received at the end of period t
Y = yield to maturity or discount rate
N = number of discounting periods
Duration - Calculation
12

the cash flows are weighted by the time
remaining until they are received
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the weighted cash flows are discounted at
the bond’s current yield and the sum is
divided by the current price of the bond
Duration - Example
Two year 8% coupon bond, paying interest
semi-annually and selling at par
....what is the duration?
13
Duration Matching (hedging)
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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 (cash market):
–
–
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Cash market
Utilize interest rate futures to manage these risks
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).
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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 (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
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Year 6
$14,172
$12,884
$11,713
$10,648
$9,680
$8,800
$67897
$97,920
$166,725
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)

To immunize itself, a bank must reorganize
its balance sheet such that:
$ A  DA  $ 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
25
Bank Immunization (cont’d)
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A bank could have more interest-sensitive
assets than liabilities:
–
Reduce RSA or increase RSL to immunize
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A bank could have more interest-sensitive
liabilities than assets:
–
Reduce RSL or increase RSA to immunize
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reduce $value or the duration
reduce the $value or the duration
Duration Shifting - Cash Market
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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 continue
to bear some interest rate risk but at
reduced levels and will want to then shift
the portfolio duration
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 can lead to duration
matching
Duration Shifting (cont’d)
Coupon
Lower
Higher
Lower
Ambiguous
Duration
Lower
Higher
Duration
Higher
Ambiguous
Maturity
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Hedging (Managing Interest Rate
Risk) with Interest Rate Futures
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Complex area
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numerous approaches in determining the hedge
ratio
we make a number of simplifying assumptions that
enable understanding at the introductory level
Hedging With Interest Rate
Futures
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A financial institution can use futures contracts to
hedge interest rate risk
Essentially achieving the same result as duration
matching - now done with the application of futures
and not in the ‘cash market’ as previously
discussed.
The hedge ratio using the Duration Model:
Pb Db (1  YTM ctd )
HR  CFctd
Pf D f (1  YTM b )
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Hedge Ratio Using the Duration
Model
HR  CFctd
Pb Db (1  YTM ctd )
Pf D f (1  YTM b )
Pb = price of the bond portfolio as a % of par
Db = duration of the bond portfolio
Pf = price of the futures contract as a % of 100
Df = duration of the cheapest to deliver bond eligible to deliver
CFctd = correction factor for the cheapest to deliver bond
YTM ctd = yield to maturity of the cheapest to deliver bond
YTMb = yield to maturity of the bond portfolio
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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/Decreasing 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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Decreasing duration where the risk is with
interest rates increasing
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Adding long futures positions to a bond
portfolio will increase duration
selling or creating a short futures position will
decrease duration
A Model for Effectively
Changing Duration With Futures
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One method for achieving target duration
has its origin in 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
We can determine the number of futures
contracts required to adjust the duration of the
bond or portfolio using the following model
PURE BPV MODEL
BPV = $ change for a $100,000 face value
security per .01 (basis point) change in yield
Hedge Ratio = DVCc/DVCf * B
DVC - dollar value change
B = relative yield change volatility
DVCf = DVCcd/CFcd
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Changing Effective Duration
With Futures (cont’d)
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To change the effective duration of a
portfolio with the BPV method requires
calculating three BPVs:
# contracts 
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BPVtarget  BPVcurrent
BPVctd
Effectively Changing Duration
With Futures (cont’d)

The current and target BPVs are calculated
as follows:
BPVcurrent,target
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duration  portfolio size  0.0001

(1  R / 2)
Effectively Changing Duration
With Futures (cont’d)

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
Effectively Changing 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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Effectively Changing 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)
Effectively Changing Duration
With Futures (cont’d)
BPV Method Example (cont’d)
BPVfutures
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9.70  $100,000 * 0.0001

 $83.42
(1  0.0722 / 2) 1.1223
Effectively Changing 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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Duration - Simplfying Assumptions
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We have made two simplifying assumptions to
illustrate how duration can be applied to
immunize a bond portfolio
 assumed parallel shifts of a flat yield curve
 assumed stable reinvestment rates
......neither are realistic
......perfect immunization is not likely but it can
still be a very effective tool in managing
interest rate risk!
Duration - Summary
We have seen that it is possible to reduce
interest rate risk by matching the duration of a
bond or portfolio with the investment horizon including the ability to shift the duration by
replacing a bond with a different coupon and or
maturity (cash market) OR by using futures
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Immunizing - Considerations
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Opportunity cost of being wrong - always a
consideration of hedging
Lower yield - comes with shorter duration
(traditional upward sloping curve)
Transaction costs
Immunization: instantaneous only - need
for ongoing process
Opportunity Cost of Being
Wrong

An incorrect forecast can lead to an
opportunity cost/missed opportunity for
immunized portfolios - always a
consideration in hedging
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need clarity on objectives of hedging
Lower Yield

Immunization usually results in a lower
level of income generated by the funds
under management
–
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if the immunization results in a reduction of the
the portfolio duration, the portfolio return will
effectively shift to the left on the yield curve,
resulting in a lower level of income
Transaction Costs
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Costs include:
Trading fees
– Brokerage commissions
– Bid-ask spread
– Tax liabilities
.......a futures driven immunization will result in
much lower transaction costs than one done in
the cash market
–
52
Immunization: Instantaneous
Only
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Durations and yields to maturity change
every day
–
A portfolio may be immunized only temporarily
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duration changes over time
ytm’s change over time
market interest rates change