Ch. 11 Outline
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1
Interest rate futures – yield curve
Discount yield vs. Investment Rate %”
(bond equivalent yield):
Pricing interest rate futures contracts
Spreading with interest rate futures
Speculation Example
The ED futures contract has a face value of
$1 million. Suppose the discount yield at
the time of purchase was 2.74%. In the
middle of March 2005, interest rates have
risen to 7.00%.
What is the speculator’s dollar gain or
loss?
2
Speculating With Eurodollar
Futures – Initial price
Face value - $1 million; Disc yield-2.74%
(price of 97.26; or 100-97.26)
 Discount Yield  90 
Price  Face Value 1 
360

 .0274  90 
Price  $1,000,0001 
 $993,150

360 

3
Speculating With Eurodollar
Futures (cont’d)
The price with the new interest rate of 7.00% is:
 Discount Yield  90 
Price  Face Value 1 
360
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 .0700  90 
Price  $1,000,000 1 
 $982,500.00

360 

4
Speculating With Eurodollar
Futures (cont’d)
Speculation Example (cont’d)
The speculator’s dollar loss is therefore:
$982,500.00  $993,150.00  $10,650.00
5
Hedging With Eurodollar
Futures
Hedging Example
Face value 10 mil, Disc Yield – 1.24
(at 98.76; or 100- 98.76)
6
.
Hedging With Eurodollar
Futures (cont’d)
Hedging Example (cont’d)
When you receive the $10 million in three months, assume
interest rate have fallen to 1.00%. $10 million in T-bills would
then cost:
 .01 90 
Price  $10,000,0001 
 $9,975,000.00

360 
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This is $6,000 more than the price at the time you established
the hedge.
7
Hedging With Eurodollar
Futures (cont’d)
Hedging Example (cont’d)
In the futures market, you have a gain that will offset the
increased purchase price. When you close out the futures
positions, you will sell your contracts for $6,000 more than
you paid for them.
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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
C0 , t
9
S  spot commodity price
 cost of carry from time zero to time t
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, your cost of carry is zero
Solving for C in the futures pricing equation
yields the implied repo rate (implied
financing rate)
Implied Repo or Financing rate
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Arbitrage With T-Bill Futures
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If an arbitrageur can discover a disparity between
the implied financing rate and the available repo
rate, there is an opportunity for riskless profit
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If the implied financing rate is greater than the
borrowing rate, then he/she could borrow, buy Tbills, and sell futures
NOB spread (trading the yield curve)
slope increases (long term R increases more than short term or
short term even decreases) buy notes sell bonds
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The NOB Spread
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The NOB spread is “notes over bonds”
Traders who use NOB spreads are
speculating on shifts in the yield curve
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If you feel the gap between long-term rates and
short-term rates is going to narrow ( yield curve
slope decreases or flattens), you could sell Tnote futures contracts and buy T-bond futures
Trading Spreads
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TED spread (different yield curves)
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The TED spread is the difference between
the price of the U.S. T-bill futures contract
and the eurodollar futures contract, where
both futures contracts have the same
delivery month (T-bill yield<ED yield)
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If you think the spread will widen, buy the
spread (buy T-bill, sell ED)
Chapter 12
Futures Contracts
and Portfolio
Management
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© 2004 South-Western Publishing
Outline
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The concept of immunization
Altering portfolio duration with futures
Duration as a convex function as opposed
to market risk measure beta
Introduction
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An immunized bond portfolio is largely
protected from fluctuations in market
interest rates
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–
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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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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)
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Interest rate risk is a consequence of the
inverse relationship between bond prices
and interest rates
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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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Bullet immunization
Change of portfolio duration with interest
rate futures
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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–
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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
$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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Duration Shifting
34
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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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Summary of Immunization and
duration hedging
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Bullet immunization (bond with target yield and
duration = target date)
Duration as a measure of sensitivity to interest
rate changes
Duration is a convex function hedge ratio does
not change linearly (BPV)
Examples for review
Spot rate is $1.33 per 1€. The US 3m T-bill rate is 2.7%
and the Forward 3m rate is 1.327011. What is the risk
free rate of the European Central Bank if the interest
rate parity condition determined this forward rate?
(3.6%)
The spot rate is CAD 2.2733 per 1£. If the inflation rate in
Canada is 3.4% a year and the inflation rate in UK is
2.3% per year, according to the purchasing power
parity the forward exchange rate should be……….?
(2.285838)
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