CHAPTER 8
Interest Rate Futures Refinements
In this chapter, we extend the discussion of interest rates
futures. This chapter is organized into the following
sections:
1. The T-Bond Futures Contract
2. Seller’s Options for T-Bond Futures
3. Interest Rate Futures Market Efficiency
4. Hedging with T-Bond Futures
Chapter 8
1
T-Bond Futures Contract
In this section, the discussion of T-bond futures is
extended by analyzing the cheapest-to-deliver bond.
Recall that a number of candidate bonds can be delivered
against a T-bond future contract. Recall further that short
traders choose when to deliver and which combination of
bonds to deliver.
Some bonds are cheaper to obtain than others. In this
section, we learn techniques to identify the cheapest-todeliver bond, including:
1. Cheapest-to-deliver bond with no intervening coupons.
2. Cheapest-to-deliver bond with intervening coupons.
3. Cheapest-to-deliver and the implied repo rate.
Chapter 8
2
Cheapest-to-Deliver with No Intervening
Coupons
Assume today, September 14, 2004, a trader observes that
the SEP 04 T-bond futures settlement price is 107-16 and
thus decides to deliver immediately. That is, the trader
selects today, September 14, as her Position Day.
Therefore, she will have to deliver on September 16.
The short is considering the following bonds with $100,000
face value each for delivery. The short wishes to determine
if delivering one or the other bond will produce a larger profit
for her.
How much should the short receive?
Which bond should the short deliver?
Maturity
Coupon Price
November 15, 2028 5.25
November 15, 2021 8.00
93-15
127-13
SEP 04
CF
0.9052
1.2113
Days
May-Nov
184
184
Days
May-Set
122
122
To answer these two questions, we need to determine the
invoice amount and then which bond is cheapest-todeliver.
Chapter 8
3
Cheapest-to-Deliver with No Intervening
Coupons
Recall that the total price of a bond depends upon the
quoted price plus the accrued interest (AI).
Invoice Amount  DSP ($ 100 , 000 )( CF )  AI
Where:
DSP = decimal settlement price
the decimal equivalent of the quoted price
CF = conversion factor
the conversion factor as provided by the CBOT
AI
= accrued interest
the Interest that has accrued since the last coupon payment
on the bond
Pi
= cash market price
Chapter 8
4
Cheapest-to-Deliver with No Intervening
Coupons
The accrued interest (AI) is computed as follows:
 Days Since Last Coupon  

1


AI  
  # Coupons per year  Coupon Rate
Days
in
Half
Years



FaceValue 
The days in half-year can be obtained from Table 8.1.
Table 8.1
Days in HalfBYears
Days in HalfBYear
Interest Period
January to July
February to August
March to September
April to October
May to November
June to December
July to January
August to February
September to March
October to April
November to May
December to June
1 year
(any 2 consecutive
halfByears)
Interest Paid on
1st or 15th
Regular
Year
181
181
184
183
184
183
184
184
181
182
181
182
365
Leap
Year
182
182
184
183
184
183
184
184
182
183
182
183
366
Interest Paid on
Last Day
Regular
Year
181
184
183
184
183
184
184
181
182
181
182
181
365
Leap
Year
182
184
183
184
183
184
184
182
183
182
183
182
366
Source: Treasury Circular No. 300, 4th Rev.
Chapter 8
5
Cheapest-to-Deliver with No Intervening
Coupons
Step 1: compute the cash price and invoice price.
5.25% Bond
AI = (122/184) (0.5) (0.0525) ($100,000) = $1,740.49
Invoice Amount= 1.0750 ($100,000) (0.9052) + $1,740.49
Invoice Amount = $99,049.49
8.00% Bond
AI = (122/184) (0.5) (0.08) ($100,000) = $2,652.17
Invoice Amount= 1.0750 ($100,000) (1.2113) + $2,652.17
Invoice Amount = $132,866.92
The 8% bond has an invoice amount 34% greater than the
5.25% bond.
Chapter 8
6
Cheapest-to-Deliver with No Intervening
Coupons
Sept 2: compute the cheapest-to-deliver bond.
The bond that is most profitable to deliver is the
cheapest-to-deliver bond. The short’s profit is the
difference between the invoice amount and the cash
market price.
For a particular bond I, the profit πi is:
πi = Invoice Amount - (Pi + AIi)
Recall that the invoice amount is:
Invoice Amount  DSP ($ 100 , 000 )( CF )  AI
Substituting the formula for the invoice amount into the
profit equation gives:
πi = (DFPi) ($100,000) (CFi) + AIi - (Pi + AIi)
And simplifying:
πi = DFPi ($100,000) (CFi) - Pi
Chapter 8
7
Cheapest-to-Deliver with No Intervening
Coupons
The cheapest-to-deliver is:
5.25% Bond
π = 1.0750 ($100,000) (0.9052) - $93,468.75 = $3,840.25
8.00% Bond
π = 1.0750 ($100,000) (1.2113) - $127,093.75 = $3,121.00
Thus, in this case the cheapest-to-deliver bond is the
5.25% bond.
Chapter 8
8
Cheapest-to-Deliver with No Intervening
Coupons
General rules based on interest rates:
1. When interest rates are below 6%, there is an incentive
to deliver short maturity/high coupon bonds.
2. When interest rates exceed 6%, there is an incentive to
deliver long maturity/low coupon bonds.
General rules based on duration:
1. A trader should deliver low duration bonds when
interest rates are below 6%.
2. A trader should deliver high duration bonds when
interest rates are above 6%.
Chapter 8
9
Cheapest-to-Deliver with Intervening
Coupons
This section examines, cheapest-to-deliver bonds when a
bond pays a coupon between the beginning of the cashand-carry holding period and the futures expiration.
To find the cheapest-to-deliver bond before expiration, the
cash-and-carry strategy is used.
The bond with the greatest profit at delivery from following
the cash-and-carry strategy will be the cheapest-to-deliver
bond.
For this analysis Assume that:
1. A trader buys a bond a today and carries it until delivery.
2. Interest rates and futures prices remain constant.
3. Consider the estimated invoice amount plus the estimate
of the cash flows associated with carrying the bond to
delivery.
Chapter 8
10
Cheapest-to-Deliver with Intervening
Coupons
The estimated invoice amount depends on three factors:
1.
Today's quoted futures price.
2.
The conversion factor for the bond we plan to deliver.
3.
The accrued interest on the bond at the expiration date.
Acquiring and carrying a bond to delivery involves three
cash flows as well:
1.
The amount paid today to purchase the bond.
2.
The finance cost associated with obtaining money today
to buy a bond in the future.
3.
The receipt and reinvesting of coupon payment.
Figure 8.1 brings all these factors together.
Chapter 8
11
Cheapest to Delivery and Bond Yield
Insert Figure 8.1 here
Chapter 8
12
Cheapest-to-Deliver with Intervening
Coupons
Estimated Invoice Amount =
DFP0 $100,000 (CF) + AI2
Estimated Future Value of the Delivered Bond =
(P0 + AI0)(1 + C0,2) - COUP1(1 + C1,2)
For bond I, the expected profit from delivery is the
estimated invoice amount minus the estimated value of
what will be delivered:
π = DFP0 ($100,000) (CF) + AI2 - {(P0 + AI0)(1 + C0,2) COUP1(1 + C1,2)}
where:
P0
AI0
C0,2
COUP1
C1,2
DFP0
CF
AI2
=
=
=
=
=
=
=
quoted price of the bond today, t = 0
accrued interest as of today, t = 0
interest factor for t = 0 to expiration at t = 2
coupon to be received before delivery at t = 1
interest factor from t = 1 to t = 2
decimal futures price today, t = 0
conversion factor for a particular bond and the
specified futures expiration
= accrued interest at t = 2
Chapter 8
13
Cheapest-to-Deliver with Intervening
Coupons
To illustrate these computation consider the following
situation.
Suppose that today is Sept 14, 2004, and you want to find
the cheapest-to-deliver bond for the DEC 04 futures
expiration. The bond has a $100,000 face value and a
target delivery date of Dec 31, 2004. The futures contract
is the DEC 04. The T-bond contract had a settlement price
of 106-23 today. The coupon invested repo rates is 7%.
Summary
Today
=
Bond face value
=
Target delivery date
=
Futures contract
=
Coupon invested repo rate =
Settlement price Sept 14 =
Sept 14
$100,000
Dec 31
DEC 04 T-bond
7%
106-23
You are considering two bonds for delivery. The bonds are
as follows:
Chapter 8
14
Cheapest-to-Deliver with Intervening
Coupons
Maturity
Coupon Price
November 15, 2028 5.25
November 15, 2021 8.00
SEP 04
CF
93-15 0.9056
127-13 1.2094
Accrued
Interest days
122
122
Accrued
interest
$1,740.49
$2,652.17
Step 1: estimate the value of AI.
5.25% Bond
P0 + AI0 = $93,468.75 + $1,740.49 = $95,209.24
8% Bond
P0 + AI0 = $ 127,093.75 + $2,652.17 = $129,745.92
Step 2: estimate the accrued interest that will accumulate
from the next coupon date, Nov 15, 2004 if the
planned delivery date is Dec 31, 2004 (46 days).
5.25% Bond
AI2 = (46/181) (0.5) (0.0525) ($100,000) = $667.13
8% Bond
AI2 = (46/181) (0.5) (0.08) ($100,000) = $1,016.57
Chapter 8
15
Cheapest-to-Deliver with Intervening
Coupons
Step 3: compute the estimated invoice amounts.
5.25% Bond
1.0671875 ($100,000) (0.9056) + $667.13 = $97,311.63
8% Bond
1.0671875 ($100,000) (1.2094) + $1,016.57 = $130,082.23
Step 4: compute financing rates.
Period: Sept 15 until Dec 31 (108 days)
C0,2 = 0.07 (108/360) = 0.0210
Period: Nov 15 until Dec 31 (46 days)
C1,2 = 0.07 (46/360) = 0.008944
Table 8.2 summarizes these calculations.
Chapter 8
16
Cheapest-to-Deliver with Intervening
Coupons
Table 8.2
Data for CheapestBtoBDeliver Bonds
Bond
P0
AI0
C0,2
5.25%
8.00%
$93,468.75
$127,093.75
$1,740.49
$2,652.17
.0210
.0210
C1,2
DFP0
.008944 1.0671875
.008944 1.0671875
CF (DEC 04)
AI2
0.9056
1.2094
$667.13
$1,016.57
Step 5: Compute expected profit for each bond.
5.25% Bond
π = (1.06718750) ($100,000) (0.9056) + $667.13- [($ 93,468.75
+ $1,740.49) (1.0210) - ($2,625) (1.008944)]
π = $2,751.48
8% Bond
π= (1.06718750) ($100,000) (1.2094) + $1,016.57[($127,093.75 + $2,652.17) (1.0210) - $4,000 (1.008944)]
π =$1,647.43
The profit from the 5.25% bond is higher, so it is the
cheapest- to-deliver.
Chapter 8
17
Cheapest-to-Deliver Bond and The
Implied Repo Rate
We can analyze the same situation using the implied repo
rate. The implied repo rate for a given period equals the net
cash flow at delivery divided by the net cash flow when the
carry starts.
Repo Rate General Rules
1. A cash-and-carry arbitrage nets a zero profit if the
actual borrowing cost equals the implied repo rate.
2. If the effective borrowing rate is less than the implied
repo rate, one can earn an arbitrage profit by using
cash-and-carry arbitrage (i.e., buy a cash bond and sell
a futures).
3. If the effective borrowing rate exceeds the implied repo
rate and if one can sell bonds short, then one can earn
an arbitrage profit by using a reverse cash-and-carry
arbitrage ( i.e., sell a bond short, buy the futures, and
cover the short position at the expiration of the futures).
Chapter 8
18
Cheapest-to-Deliver Bond and The
Implied Repo Rate
Implied Repo Rate =
Net Cash Flow Over Horizon
Net Cash Flow at Inception
The numerator consists of cash inflows of the Invoice
Amount, plus the future value of the coupons at the time of
delivery, less the cost of acquiring the bond initially.
The denominator consists of the cost of buying the bond.
Thus, the Implied repo rates is:
Implied
Repo Rate 
DFP 0 ($100,000)
(CF ) + AI
2
+ funcCOUP
( P 0 + AI
0
1
(1 + C 1,2 ) - ( P 0 + AI
0
)
)
For the 5.25% bond, we have:
Implied
Repo Rate 
1 . 0671875 $ 100 , 000 0 . 9056   $ 667 . 13  $ 2 , 625 1 . 008944   $ 93 , 468 . 75
$ 93 , 468 . 75  $ 1, 740 . 49 
 $ 1, 740 . 49 
Implied Repo Rate = 0.0499
Annualized, the implied repo rate is:
0.0499(360/180) = 16.63%
Chapter 8
19
Cheapest-to-Deliver Bond and The
Implied Repo Rate
For the 8% bond, we have:
Implied
Repo Rate 
(1.0671875
) ($100,000)
(0.9056)
+ $667.13
($93,468.7
+ $2,625 (1.008944)
- ($93,468.7
5 + $1740.49)
5 + $1,740.49)
Implied Repo Rate = 0.0337
Annualized, the implied repo rate is:
0.0337(360/180) = 13.23%
The cheapest-to-deliver bon has the highest repo rate in a
cash-and-carry arbitrage, so we should deliver the 5.25%
bond.
Chapter 8
20
Cheapest-to-Deliver Bond and Implied
Repo Rate
Table 8.3 shows how financing a cash-and-carry arbitrage
at the implied repo rate yields a zero profit.
Table 8.3
Transactions Showing Implied Repo Rates
September 14, 2004
Borrow $129,745.92 for 108 days at implied repo rate of 11.23 percent.
Buy $100,000 face value of 8.00 T-bonds maturing on Nov. 15, 2021, for a total
price of $129,745.92 including accrued interest.
Sell one DEC 04 T-bond futures contract at the current price of 106-23.
November 15, 2004
Receive coupon payment of $4,000 and invest for 46 days at 7.00 percent.
December 31, 2004 (Assuming futures is still at 106-23)
Deliver the bond and receive invoice amount of $130,082.23
From the invested coupon receive $4,000 + $4,000 (0.07) (46/360) = $4,035.78
Repay debt: $129,745.92 + $129,745.92 (0.1123) (108/360) = $134,117.06
Net Profit = $130,082.23 + $4,035.78 B $134,117.06 = $.95 0 (given rounding
error)
Chapter 8
21
T-Bond Risk Arbitrage
Arbitrage in the T-bond futures market is really risk
arbitrage. Risks stem from three sources:
1. Intervening coupon payments that must face
reinvestment.
2. The use of conversion factors.
3. The seller options.
Chapter 8
22
T-Bond Risk Arbitrage
Table 8.3
Transactions Showing Implied Repo Rates
September 14, 2004
Borrow $129,745.92 for 108 days at implied repo rate of
11.23 percent.
Buy $100,000 face value of 8.00 T-bonds maturing on Nov.
15, 2021, for a total
price of $129,745.92 including accrued interest.
Sell one DEC 04 T-bond futures contract at the current price
of 106-23.
November 15, 2004
Receive coupon payment of $4,000 and invest for 46 days at
7.00 percent.
December 31, 2004 (Assuming futures is still at 106-23)
Deliver the bond and receive invoice amount of $130,082.23
From the invested coupon receive $4,000 + $4,000 (0.07)
(46/360) = $4,035.78
Repay debt: $129,745.92 + $129,745.92 (0.1123) (108/360) =
$134,117.06
Net Profit = $130,082.23 + $4,035.78 B $134,117.06 = $.95 
0
(given rounding error)
A closer examination of Table 8.3 shows some potentially
risky elements of the cash-and-carry arbitrage. Notice that:
1. The debt was financed at a constant rate throughout the
108-day carry period.
2. The trader was able to invest the coupon at the
reinvestment rate of 7%.
3. The futures price did not change over the horizon.
Chapter 8
23
T-Bond Risk Arbitrage
Cash-and-Carry Strategy
Changes in the futures price can affect the cash flow from
the cash-and-carry strategy, as Table 8.4 illustrates. In this
case, the futures price drops from 106-23 to 104-23 over
the life of the contract.
Table 8.4
Transactions Showing Implied Repo Rates
September 14, 2004
Borrow $129,745.92 for 108 days at implied repo rate of 11.23 percent.
Buy $100,000 face value of 8.00 T-bonds maturing on Nov. 15, 2021, for a total
price of $129,745.92 including accrued interest.
Sell one DEC 04 T-bond futures contract at the current price of 106-23.
November 15, 2004
Receive coupon payment of $4,000 and invest for 46 days at 7.00 percent.
December 31, 2004 (Assuming futures has fallen to 104-23)
From September 14 to December 31, the futures price has fallen from 106-23 to
104-23, generating cash inflows of $2,000.
Deliver the bond and receive invoice amount of $127,663.43
From the invested coupon receive $4,000 + $4,000 (0.07) (46/360) = $4,035.78
Repay debt: $129,745.92 + $129,745.92 (0.1123) (108/360) = $134,117.06
Net Profit = $2,000 + $127,663.43 + $4,035.78 B $134,117.06 = B$417.85
Thus, the cash-and-carry strategy now produces a
negative profit.
Chapter 8
24
T-Bond Risk Arbitrage
Reserve Cash-and Carry
Here we examine an attempt to earn a profit using a
reserve cash-and-carry strategy. Recall that the trades
used in a reverse cash-and-carry are as follows:
Buy futures
Sell bond short
Invest proceeds
until futures exp.
Realize profit
Repay short sale
obligation
Take delivery
Chapter 8
25
T-Bond Risk Arbitrage
Reserve Cash-and Carry
Utilizing the same information from Table 8.3, we have:
Table 8.5
Transactions Showing Implied Repo Rates
September 14, 2004
Sell short $100,000 face value of 8.00 T-bonds maturing on Nov.
15, 2021, for a total price of $129,745.92 including accrued
interest.
Buy one DEC 04 T-bond futures contract at the current price of
106-23.
Lend $129,745.92 for 108 days at implied repo rate of 11.23
percent.
November 15, 2004
Borrow $4,000 for 46 days at 7 percent and make coupon
payment of $4,000.
December 31, 2004 (Assuming futures is still at 106-23)
Collect investment: $129,745.92 + $129,745.92 (0.1123) (108/360) = $134,117.06
Accept delivery of the bond and pay invoice amount of
$130,082.23
Pay debt from funds borrowed to make coupon payment:
$4,000 + $4,000 (0.07) (46/360) = $4,035.78
Net Profit = $134,117.06 B $130,082.23 B $4,035.78 = -$.95 
0
(given rounding error)
As expected ,there is no arbitrage profit in this case.
Chapter 8
26
T-Bond Futures Seller’s Options
The structure of T-bond futures contract gives sellers
timing and quality options.
1. Timing option
The seller’s right to choose the time of delivery.
2. Quality option
The seller’s right to select which bond to deliver.
These two main seller's options become entangled in the
actual T-bond futures contract. The timing and quality
options are commonly present in what the futures markets
refers to as:
1. The wildcard option.
2. The end-of-the-month option.
Chapter 8
27
Wildcard Option
The settlement price is determined at 2:00 PM. However,
the short seller has until 8:00 PM to notify the exchange of
his/her intent to deliver. Thus, the seller can observe what
happens between 2:00 PM and 8:00 PM before making
his/her decision.
If interest rates jump between 2:00 PM and 8:00 PM, the
short trader notifies the exchange his/her intent to deliver
at the 2:00 PM price.
If interest rates stay the same or go down, the short seller
waits for the next day to notify the exchange of an intent to
deliver.
Chapter 8
28
The End-of-the-Month Option
Recall that the last trading day for T-bond futures is the 8th
of the month.
The settlement price established on the final trading day is
the settlement price used in all invoice calculations for all
deliveries in the month. Thus, the seller can still make two
choices:
1. The seller can choose the delivery date.
2. The seller can choose the bond to deliver.
Assuming that interest rates are stable, then the seller may
apply the following general rules:
1. If the coupon yield on the bond exceeds the financing
rate to hold the bond, the seller should deliver on the
last day.
2. If the financing rate exceeds the coupon yield, the seller
should deliver immediately.
Chapter 8
29
Value of The Seller’s Options
Recall that under perfect market conditions, the Cost-ofCarry Model concludes that the futures price is equal to:
F = S (1 + C)
If the seller's options have value, then market equilibrium
requires that the following equation holds:
F + SO = S (1 + C)
where:
SO = value of seller's options
This implies that:
F = S ( 1 + C ) - SO
This implies that the futures price observed in the market
should be below the cost of carry by an amount equal to
the seller’s options.
Chapter 8
30
Interest Rate Futures Market Efficiency
There are three commonly distinguished forms of the
market efficiency hypothesis:
– The weak form.
– The semi-strong form.
– The strong form.
While many studies neglect the full magnitude of
transaction charges, more recent studies find potential for
arbitrage even after transaction costs.
Pure Arbitrage
For a pure arbitrage, the yield discrepancy must be large
enough to cover all transaction costs faced by a market
outsider.
Quasi-Arbitrage
Occurs when a trader with an initial portfolio can
successfully engage in an arbitrage. For quasi-arbitrage,
the trader faces less than full transaction costs.
Chapter 8
31
Pure Arbitrage
Table 8.9 is from a famous study on the efficiency of the Tbill futures market conducted by Elton, Gruber and
Rentzler. They found large arbitrage profits exist, many
with single contract profits in excess of $800.
Table 8.9
Pure Arbitrage Results for TBBill Futures
Immediate Execution
Size of
Filter
$ 0
100
200
300
400
500
600
700
800
Number Expected
of Trades Profit
2,304
2,093
1,902
1,738
1,595
1,469
1,332
1,190
1,063
$ 894
980
1,064
1,142
1,212
1,279
1,352
1,437
1,519
Delayed Execution
Actual
Profit
$ 889
975
1,058
1,135
1,206
1,271
1,346
1,432
1,516
Standard Number Expected
Error of Trades Profit
$15
16
16
16
17
17
18
18
18
1,725
1,569
1,428
1,301
1,206
1,107
1,005
890
789
$ 893
977
1,059
1,137
1,199
1,267
1,339
1,429
1,517
Actual
Profit
$ 880
964
1,041
1,117
1,176
1,244
1,315
1,401
1,490
Standard
Error
$18
19
19
20
20
21
22
23
24
Source: E. Elton, M. Gruber, and J. Rentzler, AIntraBDay Tests of the Efficiency of
the Treasury Bill Futures Market,@The Review of Economics and Statistics,
February 1984, 66, pp. 129B137.
Chapter 8
32
Pure Arbitrage in T-Bond Futures
Kolb, Gay and Jordan conducted a study on T-bond
futures. They investigated the possibility of a pure
arbitrage for all T-bond contracts from December 1977
through June 1981. Figure 8.4 shows the profitability of
deliverable bond for 15 contracts maturities.
Insert Figure 8.4 here
Chapter 8
33
Alternative Risk Management Strategy
In this section, alternative risk management strategies
using short-term interest rate futures are explored,
including:
1. Changing the Maturity of an Investment
Shortening the maturity of a T-bill investment
Lengthening the maturity
2. Fixed and Floating Loan Rates
3. Strip and Stack Hedges
4. Tailing Hedge
Chapter 8
34
Changing The Maturity of an Investment
Shortening the Maturity
Many investors find themselves holding a portfolio with
undesirable maturity characteristics.
Spot market transaction costs are relatively high, and
many investors prefer to alter the maturities of investment
by trading futures.
Consider a firm that has invested in a T-bill with a
$1,000,000 face value. Today, March 20, the T-bill has a
maturity of 180 days. The firm’s manager learns that the
company will need cash in 90 days. Assume that the shortterm yield is flat with all rates at 10% and a 360-day year.
Chapter 8
35
Changing The Maturity of an Investment
Shortening the Maturity
Table 8.10 illustrate the process of shortening the
maturity.
Table 8.10
Transactions to Shorten Maturities
Date
March 20
Cash Market
Futures Market
Holds six-month T-bill with a
face value of $10,000,000, worth
$9,500,000. Wishes a threemonth maturity.
June 20
Sell 10 JUN T-bill futures contracts at 90.00, reflecting the
10% discount yield.
Deliver cash market T-bills
against futures; receive
$9,750,000.
The price of a bill is given by:
P = FV - [DY(FV)(DTM)]/360
P= $1,000,000- [(.10)($10,000,000(180)]/360
P= $1,000,000 - $50,000 = $9,500,000,000
By making the above trades, the firm has effectively
shortened the maturity from 6 months to three months.
Chapter 8
36
Changing The Maturity of an Investment
Lengthening the Maturity
On August 21, an investor holds a T-bill with a $100 million
face value. The T-bill matures in 30 days (September 20).
The investor plans to reinvest for another 3 months after
the T-bill matures. The investor fears that interest rates
might fall. The investor finds the current SEP T-Bill futures
yield of 9.8% attractive and would like to lengthen the
maturity of the T-bill investment. The transaction necessary
to do so are presented in Table 8.11.
Table 8.11
Transactions to Lengthen Maturities
Date
Cash Market
Futures Market
August 21
Holds 30Bday TBbill with a face
value of $100,000,000. Wishes
to extend the maturity for 90
days.
Buy 102 SEP TBbill futures
contracts, with a yield of
9.8%.
September 20
30Bday TBbill matures and
investor receives $100,000,000.
Invest
$499,000 in money market fund
at 9.8%
Accept delivery on 102 SEP
futures, paying
$99,501,000.
December 19
TBbills received on SEP futures
mature for $102,000,000.
Receive proceeds of $511,533
from investment.
These transactions locked in a 9.8% rate over the four
months (Aug-Dec). Thereby, lengthening the maturity of
the individuals investments.
Chapter 8
37
Fixed and Floating Loan Rates
This section examines:
1. Converting a Floating Rate to a Fixed Rate Loan
– How a borrower holding a floating rate loan can effectively
convert this loan into a fixed rate loan.
2. Converting a Fixed Rate to a Floating Rate Loan
– How a lender who feels compelled to offer fixed rate loans
can use the futures markets to make the investment
perform like a floating rate loan.
Chapter 8
38
Converting a Floating Rate to a Fixed
Rate Loan
Converting a floating rate loan to a fixed rate loan, also
known as creating a synthetic fixed rate loan, occurs when
you start with a floating rate loan and transact to fix the
interest rate.
Today is Sept 20th, assume that a construction company
has planned a project which will take 6 months to
complete. The cost of the project is $100,000,000. The
firm’s bank offers the following conditions on a loan.
Rates
First 3 months=
Last 3 months=
LIBOR + 200 basis point
DEC 20 LIBOR + 200 basis point
The bank insists that the second 3-month rate be based on
the LIBOR prevailing 3 months from today. This is a risky
preposition for the construction company.
Chapter 8
39
Converting a Floating Rate to a Fixed
Rate Loan
The construction company wishes to lock in a fixed rate
loan for the entire period. The company has accumulated
the following information:
Sept 20
DEC Eurodollar futures
LIBOR
Loan
7%
7.3%.
9.0%
9.3%
These rates give the following cash flows on the loan:
Sept 20 Receive principal
+ $100,000,000
Dec 20 Pay interest
- 2,250,000
Mar 20 Pay interest and principal
- 102,325,000
The cash flows for September and December are certain
but the cash flow for March is unknown.
Using the above information, construct a synthetic fixed
rate loan.
Chapter 8
40
Converting a Floating Rate to a Fixed
Rate Loan
To convert to a variable rate loan to a fixed rate loan,
the following transaction are completed.
Table 8.12
Synthetic Fixed Rate Borrowing
Date
Cash Market
Futures Market
September 20
Borrow $100,000,000 at 9.00%
for three months and commit
to extend the loan for three
additional months at a rate
200
basis points above the threeB
month LIBOR rate prevailing
at that time.
Sell 100 DEC Eurodollar
futures contracts at 92.70,
reflecting the 7.3% yield.
December 20
Pay interest of $2,250,000.
LIBOR is now at 7.8%, so borrow $100,000,000 for three
months at 9.8%.
Offset 100 DEC Eurodollar
futures at 92.20, reflecting the
7.8% yield. Produces profit of
$125,000 = 50 basis points 
$25 per point  100 contracts.
March 20
Pay interest of $2,450,000 and
repay principal of
$100,000,000.
Total Interest Expense:
$4,700,000
Futures Profit: $125,000
Net Interest Expense After Hedging: $4,575,000
By engaging in the above transactions, the company
knows with certainty the interest expense that it will pay
over the life of the loan. As such, it has created a fixed
rate loan.
Chapter 8
41
Converting a Fixed Rate to a Floating
Rate Loan
From the bank’s perspective, it can grant the fixed rate
loan. However, doing so exposes the bank to risks.
The bank expects to obtain money to make to the loan by
borrowing at LIBOR 7% today and 7.3% for the next
quarter, for an average of 7.15%. The bank makes a fixed
rate loan at 9.15%. The bank sources of funds are as
follows:
BANK
Sep 20
Dec 20
Mar 20
Borrow principal + $100,000,000
Make loan to
- $100,000,000
Pay interest
- $ 1,750,000
Receive principal
& interest
+ $104.575,000
Pay principal
& interest
- $101,825,000
Chapter 8
42
Converting a Fixed Rate to a Floating
Rate Loan
To reduce its risks and lock in a profit, the bank trades as
follows:
Table 8.13
Synthetic Floating Rate Lending
Date
September 20
December 20
Cash Market
Futures Market
Borrow $100,000,000 at 7.00%
for three months and lend it
for six months at 9.15%.
Pay interest of $1,750,000.
LIBOR is now at 7.8%, so borrow $100,000,000 for three
months at 7.8%.
Sell 100 DEC Eurodollar
futures contracts at 92.70,
reflecting the 7.3% yield.
Offset 100 DEC Eurodollar
futures at 92.20, reflecting the
7.8% yield. Produces profit of
$125,000 = 50 basis points 
$25 per point  100 contracts.
March 20
Pay interest of $1,950,000 and
repay principal of
$100,000,000.
Total Interest Expense:
Futures Profit: $125,000
$3,700,000
Net Interest Expense After Hedging: $3,575,000
Thus, the bank has locked in a profit fo $1,000,000
($4,575,000 - $3,575,000). The bank has also effectively
crated a fixed rate loan.
Chapter 8
43
Strip and Stack Hedges
Using the same example. Now assume that the
construction company needs a one year loan instead of 6month loan.
The bank sets the rates to be LIBOR plus 200 basis points.
The rate will be adjusted every 3 months to reflect any
LIBOR rate changes.
On September 15, the construction company observes the
following rates:
Three-month LIBOR
DEC Eurodollar
MAR Eurodollar
JUN Eurodollar
7.00%
7.30
7.60
7.90
The company estimates that it can finance $100,000,000
at the following rates, for an average rate of 9.45%.
I Quarter
II Quarte r
III Quarter
IV Quarter
9.0%
9.3
9.6
9.9
Chapter 8
44
Stack Hedges
A stack hedge occurs when futures contracts are
concentrated or stacked in a single future expiration.
The construction company enters into a stacked
hedge by transacting as shown in Table 8.14.
Table 8.14
Results of a Stack Hedge
Date
Cash Market
Futures Market
September 20
Borrow $100,000,000 at 9.00,
for three months and commit
to roll over the loan for three
quarters at 200 basis points
over the prevailing LIBOR
rate.
Sell 300 DEC Eurodollar
futures contracts at 92.70,
reflecting the 7.3% yield.
December 20
Pay interest of $2,250,000.
LIBOR is now at 7.8 percent,
so borrow $100,000,000 for
three months at 9.8 percent.
Offset 300 DEC Eurodollar
futures at 92.20, reflecting the
7.8% yield. Produces profit of
$375,000 = 50 basis points 
$25 per point  300 contracts.
March 20
Pay interest of $2,450,000 and
borrow $100,000,000 for three
months at 10.10 percent.
June 20
Pay interest of $2,525,000 and
borrow $100,000,000 for three
months at 10.40 percent.
September 20
Pay interest of $2,600,000 and
principal of $100,000,000.
Total Interest Expense:
$9,825,000
Futures Profit: $375,000
Interest Expense Net of Hedging: $9,450,000
The hedge worked perfectly by locking in the cost of
borrowing regardless of the future course of interest
rates.
Chapter 8
45
Stack Hedges
Notice that in the above example all interest rates change
by 50 basis points.
Stack hedges may perform poorly if interest rates change
in differing amounts. That is, the yield curve shifts. Figure
8.5 illustrates this situation.
Insert Figure 8.5 here
Chapter 8
46
Strip Hedge
A strip hedge uses an equal number of contracts for each
futures expiration over the hedging horizon. By doing so,
the futures market hedge is aligned with the actual risk
exposure. The transactions necessary to implement a strip
hedge are demonstrated in Table 8.15.
Table 8.15
Results of a Strip Hedge
Date
Cash Market
Futures Market
September 20
Borrow $100,000,000 at 9.00%
for three months and commit
to roll over the loan for three
quarters at 200 basis points
over the prevailing LIBOR
rate.
Sell 100 Eurodollar futures for
each of: DEC at 92.70, MAR at
92.40, and JUN at 92.10.
December 20
Pay interest of $2,250,000.
LIBOR is now at 7.4%, so borrow $100,000,000 for three
months at 9.4%.
Offset 100 DEC Eurodollar
futures at 92.60. Produces
profit of $25,000 = 10 basis
points  $25 per point  100
contracts.
March 20
Pay interest of $2,350,000 and
borrow $100,000,000 for three
months at 10.30%.
Offset 100 MAR Eurodollar
futures at 91.70. Produces
profit of $175,000 = 70 basis
points
 $25 per point  100
contracts.
June 20
Pay interest of $2,575,000 and
borrow $100,000,000 for three
months at 10.60%.
Offset 100 JUN Eurodollar
futures at 91.40. Produces
profit of $175,000 = 70 basis
points  $25 per point  100
contracts.
September 20
Pay interest of $2,650,000 and
principal of $100,000,000.
Total Interest Expense:
$9,825,000
Futures Profit: $375,000
Interest Expense Net of Hedging: $9,450,000
The performance of a strip hedge is superior to the stack
hedge because the interest rates adjust every quarter.
Chapter 8
47
Advantages of Stacked and Striped
Hedge
Advantages of Stack Hedges
1. Works better when the cash position has a single
horizon.
2. Requires trading a single contract.
Advantage of Strip Hedges
1. Can provide a more aligned hedge and better results
with a multiple-maturity cash position.
Chapter 8
48
Tailing The Hedge
In a tailing hedge the trader slightly adjusts the hedge to
compensate for the interest that can be earned from daily
resettlement profits or paid on daily resettlement losses.
Thus, the tail of the hedge is the slight reduction in the
hedge position to offset the effect of daily resettlement
interest.
Tail Factor
The tail factor is the present value of $1 at the hedging
horizon discounted to the present (plus one day) at the
investment rate for the resettlement cash flows.
Tailed Hedge = Untailed Hedge  Tailing Factor
.
Chapter 8
49
Hedging with T-Bond Futures
The effectiveness of a hedge depends on the gain or loss
on both the spot and futures sides of the transaction. The
change in the price of any bond depends on the shifts in
the levels of:
– Interest rates
– Changes in the shape of the yield curve
– The maturity of the bond
– Bond coupon rate
Table 8.16 and 8.17 illustrate to effect of maturity and
coupon rates on hedging performance.
Chapter 8
50
Hedging with T-Bond Futures
A manager learns on March 1 that he will receive $5 million
on June 1 to invest in AAA corporate bonds with a 5%
coupon rate and 10 years to maturity. The yield curve is flat
and will remain so. The current yield on AAA bonds as well
as forward rates are 7.5%. So the manager expects to
acquire the bonds at 7.5%. However, fearing a drop in
rates, he decides to hedge in the futures market to lock-in
the forward rate of 7.5%.
The manager considers hedging with T-bills or T-bonds.
The AAA bonds have a 5% coupon rate and a 10-year
maturity, which do not match the characteristics of either
the T-bill or T-bond futures contracts. The deliverable Tbills have a zero coupon and a maturity of only 90 days,
and the T-bonds have a maturity of at least 15 years and
an assortment of semi-annual coupons. Assume that the
cheapest-to-deliver T-bond will have a 20-year maturity at
the target date of June 1, and a 6% coupon.
The manager will hedge the AAA position with T-bill or Tbond futures with yields of 6 and 6.5%, respectively. The
manager plans to invest in 6,051 bonds each with a price
of $826.30.
Chapter 8
51
Hedging with T-Bond Futures
Table 8.16 illustrates the transactions and results of
hedging with T-bill futures.
Table 8.16
A CrossBHedge Between Corporate Bonds
and TBBill Futures
Date
Cash Market
Futures Market
March 1
A portfolio manager learns he
will receive $5 million to invest
in 5%, 10-year AAA bonds in 3
months, with an expected yield
of 7.5% and a price of $826.30.
The manager expects to buy
6,051 bonds.
The portfolio manager buys $5
million face value of T-bill futures (5 contracts) to mature on
June 1 with a futures yield of
6.0% and a futures price, per
contract, of
$985,000.
June 1
AAA yields have fallen to 7.08%,
causing the price of the bonds to
be $852.72. This represents a
loss, per bond, of $26.42. Since
the plan was to buy 6,051 bonds,
the total loss is (6,051  $26.42)
= B$159,867.
The T-bill futures yield has
fallen to 5.58%, so the futures
price = spot price = $986,050 per
contract, for a profit of $1,050
per contract. Since 5 contracts
were traded, the total profit is
$5,250.
Loss = B$159,867
Gain = $5,250
Net wealth change = B$154,617
Notice that this loss occurs despite the fact that rates
changed by the same amount on both investments.
Chapter 8
52
Hedging with T-Bond Futures
Table 8.17 illustrates the results of hedging with Tbond futures.
Table 8.17
A Cross Hedge Between Corporate Bonds
and TBBond Futures
Date
Cash Market
Futures Market
March 1
A portfolio manager learns he
will receive $5 million to invest
in 5%, 10-year AAA bonds in 3
months, with an expected yield
of 7.5% and a price of $826.30.
The manager expects to buy
6,051 bonds.
The portfolio manager buys $5
million face value of T-bond
futures (50 contracts) to mature
on June 1 with a futures yield of
6.5% and a futures price, per
contract, of $94,448.
June 1
AAA yields have fallen to 7.08%,
causing the price of the bonds to
be $852.72. This represents a
loss, per bond, of $26.42. Since
the plan was to buy 6,051 bonds,
the total loss is (6,051  $26.42)
= B$159,867.
The T-bond futures yield has
fallen to 6.08%, so the futures
price = spot price = $99,081 per
contract, for a profit of $4,633
per contract. Since 50 contracts
were traded, the total profit is
$231,650.
Loss = B$159,867
Gain = $231,650
Net wealth change = +$71,783
Again the hedge did not produce the desired results of
isolating the portfolio.
Chapter 8
53
Hedging with T-Bond Futures
Simple approaches to hedging interest rate risk often give
unsatisfactory results due to mismatches of coupon and
maturity characteristics, as demonstrated in the previous
examples.
This section examines some of the major alternative
strategies for hedging interest rate risk:
– Face Value Naive (FVN) Model
– Market Value Naive (MVN) Model
– Conversion Factor (CF) Model
– Basis Point (BP) Model
– Regression (RGR) Model
– Price Sensitivity (PS) Model
Chapter 8
54
Face Value Naive (FVN) Model
According to FVN Model, the hedger should hedge $1 of
face value of the cash instrument with $1 face value of the
futures contract.
Disadvantages
Neglects potential differences in market values between
the cash and futures positions.
Neglects the coupon and maturity characteristics that
affect duration for both the cash market good and the
futures contract.
Chapter 8
55
Market Value Naïve (MVN) Model
The MVN Model recommends hedging $1 of market value
in the cash good with $1 of market value in the futures
market.
Disadvantages
Neglects to make adjustments for price sensitivity.
Advantages
Consider potential differences in market values between
cash and futures positions.
Chapter 8
56
Conversion Factor (CF) Model
The CF Model applies only to futures contracts that use
conversion factors to determine the invoice amount,
such as T-bond and T-note futures.
The intuition behind this model is to adjust for differing
price sensitivities by using the conversion factor as an
index of the sensitivity.
The CF Model recommends hedging $1 of face value
of a cash market security with $1 of face value of the
futures good times the conversion factor.
 Cash Market Principal
HR = _ 
 Futures Market Principal

 (Conversio

Chapter 8
n Factor)
57
Basic Point (BP) Model
The BP Model focuses on the price effect of a one basis
point change in yields on different financial instruments.
To correct for the differences in sensitivity, the BP Model
can be used to compute the following hedge ratio:
HR = Where
BPCS
BPCF
BPC
S
BPC
F
= dollar price change for a 1 basis point change
in the spot instrument.
= dollar price change for a 1 basis point change
in the futures instrument.
Chapter 8
58
Basic Point (BP) Model
Today, April 2, a firm plans to issue $50 million of 180-day
commercial paper in 6 weeks. For a one basis point yield
change, the price of 180-day commercial paper will change
twice as much as the 90-day T-bill futures contract,
assuming equal face value amounts.
The cash basis price change (BPCS) is twice as great as the
futures basis price change (BPCF), so the hedge ratio is
-2.0. With a -2.0 hedge ratio and a $50 million face value
commitment in the cash market, the firm should sell 100 Tbill futures contracts. Table 8.18 illustrates the hedging
results.
Table 8.18
Hedging Results with the BP Model
for the Commercial Paper Issuance
April 2
Cash Market
Futures Market
Firm anticipates issuing $50 million in
180-day commercial paper in 45 days
at a yield of 11%.
Firm sells 100 T-bill June futures contracts yielding 10% with an index value
of 90.00.
May 15
Spot market and futures market rates have both risen 45 basis points. The spot
rate is now 11.45% and the futures market yield is 10.45%.
Cash Market Effect
Futures Market Effect
Each basis point move causes a price
change of $50 per million-dollar face
value. Firm will receive $112,500 less
for the commercial paper, due to the
change in rates. (45 basis points 
B$50  50 = B$112,500)
Each basis point increase gives a
futures market profit of $25 per
contract.
Futures Profit = 45 basis points  +$25
 100 contracts = +$112,500
Net wealth change = 0
Chapter 8
59
Basic Point (BP) Model
Sometimes rates do not change by the same amounts. In
our previous example, suppose that the commercial
paper rate is 25% more volatile than the T-bill futures
rate. To consider differences in volatility in determining
the hedge ratio. The hedge ratio is recomputed as:
 BPC
HR = - 
 BPC
where:
RV =

 RV
F 
S
volatility of cash market yield relative to futures
yield. Normally found by regressing the yield of
the cash market instrument on the futures
market yield.
Assume a RV equal to 1.25. Now the hedge ratio is:
 $50 
HR = - 
 1.25 = - 2.5
$25


Table 8.19 shows these transactions.
Chapter 8
60
Basic Point (BP) Model
Table 8.19
Hedging Results with the BP Model
Adjusted for Relative Yield Variances
for the Commercial Paper Issuance
April 2
Cash Market
Futures Market
Firm anticipates issuing $50 million in
180Bday commercial paper in 45 days
at a yield of 11%.
Firm sells 125 T-bill June futures contracts yielding 10% with an index value
of 90.00.
May 15
Spot market rates have risen 56 basis points to 11.56% and futures rates have
risen 45 basis points to 10.45%
Cash Market Effect
Futures Market Effect
Each basis point move causes a price
change of $50 per million-dollar face
value. Firm will receive $140,000 less
for the commercial paper, due to the
change in rates.
56 basis points  B$50  50 =
B$140,000
Each basis point increase gives a
futures market profit of $25 per
contract.
Futures Profit = 45 basis points  +$25
 125 contracts = +$140,625
Net wealth change = $625
Because more T-bill futures were sold, the futures
profit still almost exactly offsets the commercial paper
loss.
Chapter 8
61
Regression (RGR) Model
The hedge ratio found by regression minimizes the
variance of the combined futures-cash position during the
estimation period. This estimated ratio is applied to the
hedging period.
For the RGR Model the hedge ratio is:
HR = -
COV
S , F
F
2
where:
COVS,F = covariance between cash and futures
σF2
= variance of futures
Recall from Chapter 4, the hedge ratio is the negative of
the regression coefficient found by regressing the change
in the cash position on the change in the futures position.
These changes can be measured as dollar price changes
or as percentage price changes.
 S t =  +   Ft +  t
Chapter 8
62
Price Sensitivity Model
The PS Model assumes that the goal of hedging is to
eliminate unexpected wealth changes at the hedging
horizon, as defined in the following equation:
dPi + dPF (N) = 0
where:
dPi =
unexpected change in the price of the cash
market instrument
dPF =
unexpected change in the price of the futures
instrument
N
number of futures to hedge a single unit of the
cash market asset
=
Chapter 8
63
Price Sensitivity Model
The correct number of contracts (N) is calculated using
the Modified Duration MD:
MD x =
N=-
Dx
1+ rx
P i MD
i
FP F MD
where:
FPF
Pi
MDi
DF
RYC
RYC
F
= the futures contract price.
= the price of asset I expected to prevail at the
hedging horizon.
= the modified duration of asset I expected to
prevail at the hedging horizon.
= the modified duration of the asset underlying
futures contract F expected to prevail at the
hedging horizon.
= for a given change in the risk-free rate, the
change in the cash market yield relative to
the change in the futures yield, often
assumed to be 1.0 in practice.
Chapter 8
64
Price Sensitivity Model
Suppose that you have accumulated the data in Table
8.20. Assume that the cash and futures markets have
the same volatility.
Table 8.20
Data for the Price Sensitivity Hedge
Cash Instrument
Pi
MDi
TBBill Futures
TBBond Futures
$826.30
FPi
$985,000
FPF
$94,448
7.207358
MDF
0.235849
MDF
10.946953
N
-0.025636
N
-0.005760
Number of Contracts
to Trade
-155.12
Number of Contracts
to Trade
-34.85
For the T-bill hedge, the number of contracts to trade is
given by:
N=-
($826.30) (7.207358)
= - 0.025636
($985,000) (0.235849)
For the T-bond hedge the number of contracts to be traded
is:
N=-
($826.30) (7.207358)
= - 0.005760
($94,448) (10.946953 )
Chapter 8
65
Price Sensitivity Model
The performance of the T-bond and T-bill hedges are
presented in Table 8.21.
Table 8.21
Performance Analysis of Price Sensitivity Futures Hedge
Cash Market TBBill Hedge TBBond Hedge
Gain/Loss
Hedging Error
B$159,867
B
Percentage Hedging Error
+$162,876
+$161,460
$3,009
$1,593
1.8822%
0.9965%
The T-bond hedge is slightly more effective as it
produces a lower hedging error.
Chapter 8
66
Summary of Alternative Hedging
Strategies
Table 8.22
Summary of Alternative Hedging Strategies
Hedging Model
Face Value Naive
(FVN)
Market Value Naive
(MVN)
Conversion Factor
(CF)
Basis Point (BP)
Regression (RGR)
Price Sensitivity
(PS)
Basic Intuition
Hedge $1 of cash instrument face value with $1 of futures
instrument face value.
Hedge $1 of cash instrument market value with $1 of
futures instrument market value.
Find ratio of cash market principal to futures market
principal. Multiply this ratio by the conversion factor for
the cheapestBtodeliver instrument.
For a 1 basis point yield change, find the ratio of the cash
market price change to the futures market price change.
(Sometimes weighted by the relative volatility of interest
rates on the cash market instrument compared to the
futures instrument interest rate.)
For a given cash market position, use regression analysis
to find the futures position that minimizes the variance of
the combined cash/futures position.
Using duration analysis, find the futures market position
designed to give a zero wealth change at the hedging horizon. (Sometimes weighted by the relative volatility of
interest rates on the cash market instrument compared to
the futures instrument interest rate.)
Chapter 8
67
Immunization
In bond investing, maturity mismatches result in exposure
to interest rate risk.
Consider the case of a bank.
when the asset duration is higher than the liability duration, a
sudden rise in interest rates will cause the value of the
portfolio to decline.
When the asset duration is less than the liability duration a
sudden rise in interest rates will cause the value of the
portfolio to rise.
By matching the duration of asset and liabilities, it is
possible for the bank to immunize itself from changes in
interest rates.
We consider two examples of immunization:
1. Planning Period Case
2. Bank Immunization case
Chapter 8
68
Immunization with Interest Rate Futures
Planning Period Case
A portfolio manager has collected the following
information:
Table 8.23
Instruments for the Immunization Analysis
Coupon
Maturity
Yield
Price
Duration
Bond A:
8%
4 yrs.
12% 875.80 3.4605
Bond B:
10%
10 yrs.
12% 885.30 6.3092
Bond C:
4%
15 yrs.
12% 449.41 9.2853
6%
20 yrs.
12% 548.61 9.0401
TBBond Futures*
B
12% 970.00 0.2500
TBBill Futures*
3 yr.
*For comparability, face values of $1,000 are assumed for these instruments.
The portfolio manager has a $100 million bond portfolio of
bond C with a duration of 9.2853 years and is considering
two alternatives. The manager has a 6-year planning
period.
The manager wants to shorten the portfolio duration to six
years to match the planning period, and is considering two
alternatives to do so.
Chapter 8
69
Immunization with Interest Rate Futures
Planning Period Case
Alternative 1
The shortening could be accomplished by selling Bond C
and buying Bond A until the following conditions are met:
W
A
D A + W C D C = 6 years
Subject to:
WA + WC =1
Where:
WI = percent of portfolio funds committed to asset I.
Chapter 8
70
Immunization with Interest Rate Futures
Planning Period Case
Alternative 2
The manager could also adjust the portfolio's duration to
match the six-year planning period by trading interest rate
futures and keeping bond C.
If bond C and a T-bill futures comprise the portfolio, the Tbill futures position must satisfy the condition:
PP = PC NC + FPT-bill NT-bill
where:
Pp
Pc
FPT-bill
Nc
NT-bill
=
=
=
=
=
value of the portfolio
price of bond C
t-bill futures price
number of C bonds
number of T-bills
Expressing the change in the price of a bond as a function
of duration and the yield on the asset:
dP = -D{d(1 + r)/(1 + r)}P
Chapter 8
71
Immunization with Interest Rate Futures
Planning Period Case
Applying the equation to the portfolio value, bond C, and
the T-bill futures, the following immunization condition is
obtained:
 d(1 + r) 
 d(1 + r)
 d(1 + r) 
 P c N c + - D T- bill 
-Dp
 P P = - D c 

 (1 + r)
 1+ r 
 (1 + r) 


 FP T- bill N T- bill


This can be simplified to:
DP PP = DC PC NC + DT-bill FPT-bill NT-bill
Chapter 8
72
Immunization with Interest Rate Futures
Planning Period Case
Because immunization requires mimicking alternative 1,
which has a total value of $100,000,000 and a duration of
six years, it must be that:
Pp =
Dp =
DC =
PC =
NC=
DT-bill =
FPT-bill =
$100,000,000
6
9.2853
$449.41
222,514
0.25
$970.00
Solving by:
DP PP = DC PC NC + DT-bill FPT-bill NT-bill
Or alternatively for a T-bond:
DP PP = DC PC NC + DT-bond FPT-bond NT-bond
Table 8.24 shows the relevant data for each of the three
scenarios.
Chapter 8
73
Immunization with Interest Rate Futures
Planning Period Case
Table 8.24
Portfolio Characteristics for the Planning Period Case
Portfolio
Weights
Number
of
Instruments
Value
of Each
Instrument
Portfolio
Value
WA
WC
WCash
NA
NC
NT-bill
NT-bond
NAPA
NCPC
NT-billFPT-bill
NT-bondFPT-bond
Cash
NAPA +
NCPC +
Cash
Portfolio 1
(Bonds Only)
Portfolio 2
(Short T-Bill Fut.)
Portfolio 3
(Short T-Bond Fut.)
56.39%
43.61%
~0
64,387
97,038
B
B
B
B
100%
~0
0
222,514
(1,354,764)
B
B
100%
~0
B
222,514
B
(66,243)
B
56,390,135
43,609,848
B
B
100,000,017
1,314,121,080
B
17
(17)
36,341,572
(17)
100,000,000
100,000,000
100,000,000
100,000,017
B
Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@
Financial Management, Summer 1982, pp. 81-89.
Chapter 8
74
Immunization with Interest Rate Futures
Planning Period Case
To see how the immunized portfolio performs, assume that
rates drop from 12 to 11 percent for all maturities. Assume
also that all coupon receipts during the six-year planning
period can be reinvested at 11 percent, compounded semiannually, until the end of the planning period. With the shift
in interest rates the new prices are:
PA
= $904.98
PC
= $491.32
FPT-bill
= $972.50
FPT-bond = $598.85
Table 8.25 shows the effect of the interest rate shift on
portfolio values, terminal wealth at the horizon (year 6),
and on the total wealth position of the portfolio holder.
Chapter 8
75
Immunization with Interest Rate Futures
Planning Period Case
Table 8.25
Effect of a 1% Drop in Yields
on Realized Portfolio Returns
Portfolio 1
Original Portfolio Value
New Portfolio Value
Gain/Loss on Futures
Total Wealth Change
Terminal Value of all Funds
at t = 6
Annualized Holding Period
Return over 6 Years
100,000,000
105,945,674
B0B
5,945,674
$201,424,708
1.120180
Portfolio 2
Portfolio 3
100,000,000 100,000,000
109,325,562 109,325,562
(3,386,910) (3,328,048)
5,938,652 5,997,514
$201,411,358 $201,523,27
1.120168
1.120266
Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@
Financial Management, Summer 1982, pp. 81-89. Terminal values and holding period returns assume
semi-annual compounding at 11 percent.
Notice that each portfolio responds similarly.
Chapter 8
76
Transaction Costs for Planning Period
Case
While each of the portfolios are equally effective in
immunizing, the cost of obtaining the immunization
varies as demonstrated in Table 8.28.
Table 8.28
Transaction Costs for the Planning Period Case
Portfolio 1
Portfolio 2
Portfolio 3
64,387
B
B
125,476
B
B
B
B
B
1,355
B
Bond A @ $5
321,935
B
B
Bond C @ $5
627,380
B
B
B
B
B
27,100
B
13,240
$949,315
$27,100
$13,240
Number of Instruments Traded
Bond A
Bond C
TBBill Futures Contracts
TBBond Futures Contracts
662
One Way Transaction Cost
TBBill Futures $20
TBBond Futures @ $20
Total Cost of Becoming Immunized
Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@Finan-
cial Management, Summer 1982, pp. 81-89.
Notice that the cost of becoming immunized varies from
$949,315 to $13,240 depending upon the strategy
selected.
Chapter 8
77
Immunization with Interest Rate Futures
Bank Immunization Case
Assume that a bank holds a $100,000,000 liability portfolio
in Bond B, the composition of which is fixed. The bank
wishes to hold an asset portfolio of Bonds A and C that will
protect the wealth position of the bank from any change as
a result of a change in yields. Five different portfolio
combinations illustrate different means to achieve the
desired result:
Portfolio 1:
Hold Bond A and Bond C (the traditional
approach)
Portfolio 2:
Hold Bond C; Sell T-bill futures
Portfolio 3:
Hold Bond A; Buy T-bond futures
Portfolio 4:
Hold Bond A; Buy T-bill futures
Portfolio 5:
Hold Bond C; Sell T-bond futures
The portfolios are presented in Table 8.26. For each
portfolio, the full $100,0000,000 is put into a bond portfolio
and is balanced out by cash.
Chapter 8
78
Immunization with Interest Rate Futures
Bank Immunization Case
Table 8.26
Liability Portfolio and Five Alternative Immunizing Portfolios
Portfolio 2
Portfolio 3 Portfolio 4
(Short
(Long
(Long
Liability
Portfolio 1
TBBill
TBBond
TBBill
Portfolio
(Bonds
Futures)
Futures)
Futures)
Only)
Portfolio
Weights
WA
WB
WC
WCash
Number of NA
InstruNB
ments
NC
NTBbill
NTBbond
NAPA
NBPB
NCPC
Cash
NTBbillPTBbill
NTBbond
PTBbond
Portfolio
Value
Portfolio 5
(Short
TBBond
Futures)
0
100%
0
~0
51.0936%
0
48.9064%
~0
0
0
100%
~0
100%
0
0
~0
100%
0
0
~0
0
0
100%
~0
0
112,956
58,339
0
0
0
114,181
0
114,181
0
0
0
0
0
0
108,824
0
0
222,514
(1,227,258)
0
0
0
57,440
0
1,174,724
0
222,514
0
(60,008)
0
99,999,947
0
53
0
0
51,093,296
0
48,906,594
110
0
0
0
0
100,000,017
(17)
99,999,720
0
0
280
0
31,512,158
99,999,720
0
0
280
100,000,000
100,000,000
(1,190,440,260)
0
100,000,000
0
0
100,000,017
(17)
1,139,482,280
0
0
(32,920,989)
100,000,000 100,000,000 100,000,000
Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@
Financial Management, Summer 1982, pp. 81-89.
Chapter 8
79
Immunization with Interest Rate Futures
Bank Immunization Case
Now consider a drop in rates from 12% to 11% for all
maturities. The effect on the portfolio is presented in
Table 8.27.
Table 8.27
Effect of a 1% Drop in Yields on Total Wealth
Liability
Original Port.
Value
New Port.
Value
Profit on
Futures
Total Wealth
Change
(Port. and
Futures)
Total Wealth
Change
(AssetBLiability Port.)
% Wealth
Change
Portfolio 1
Portfolio 2
Portfolio 3
Portfolio 4
Portfolio 5
100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000
106,206,932 106,263,146 109,325,578 103,331,521 103,331,521 109,325,579
0
B (3,068,145)
2,885,786
2,936,810 (3,014,802)
6,206,932
6,263,146
6,257,416
6,217,587
6,268,611
6,310,760
B
56,214
50,484
10,655
61,679
103,828
B
.00056
.00050
.00011
.00062
.00104
Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,
Financial Management, Summer 1982, pp. 81-89.
Notice that all 5 methods perform similarly.
Chapter 8
80