Bond Portfolio Immunization With
Interest Rate Futures
In the discussion on the bond cash market we analyzed
the risk associated with duration mismatches. In the
bank immunization case, a financial institution
immunized itself from interest rate risk by adjusting the
durations of its asset and liability portfolios. In the
planning period case, immunization was achieved by
setting the duration of a bond portfolio equal to the
length of the planning period. Often, such immunization
may be difficult and costly to achieve by operations in
the cash bond market alone. For example, banks cannot
turn away depositors because they wish to lengthen the
duration of their liabilities.
1
With the development of interest rate futures
markets over the past 30 years, financial managers
have a valuable tool to use in bond portfolio
immunization strategies.
Two examples of immunization with interest rate
futures, one for the planning period case and one for
the bank immunization case, are presented here.
Table A presents data on the bonds used in the
examples, along with data for T-bill and T-bond
futures contracts. The table reflects the assumption
of a flat yield curve and instruments of the same risk
level.
2
Table A: Instruments for the Immunization Analysis
Bond FV
Coupon
Maturity
Yield
Price
Duration
A $1,000
8%
4 yrs.
12%
$885.59
3.475
B $1,000
10%
10 yrs.
12%
$903.47
6.265
C $1,000
4%
15 yrs.
12%
$463.05
9.285
T-Bond Futures
8%
20 yrs.
12%
$71,875
8.674
¼ yr.
12%
$972,070
.25
$100,000
T-Bill Futures
$1,000,000
3
The 6-year Planning Period Case
Consider a $100 million invested in Bond C. This
portfolio’s duration is 9.285 years.The portfolio manager
wants to shorten the portfolio duration to 6 years in
order to match a 6-year planning period. In this case,
the decision is to sell some of Bond C and buy some of
Bond A. Mathematically, the conditions are:
WA D A  WC DC  6years
WA  WC  1
where W is the percentage of portfolio funds invested in
the corresponding asset . The solution is: put 56.54% of
the $100 million in Bond A, and 43.46% in bond C.
Call this Portfolio1 – using the cash market only.4
The 6-year Planning Period Case
Alternatively, the portfolio’s duration may be
adjusted to match the six-year planning period by
trading interest rate futures.
In Portfolio 2, the manager keeps the original
portfolio of $100,000,000 in Bond C and trades Tbill futures to adjust the duration of the combined
portfolio of Bond C and futures.
5
The 6-year Planning Period Case
Portfolio 2: Bond C and T-bill futures. Let
VP = value of the portfolio
Pi = price of Bond i;
i=A,B,C
Ni= number of Bond i;
i=A,B,C
FT-bill= T-bill futures price
NT-bill= number of T-bills contracts
FT-bond= T-bond futures price
NT-bond= number of T-bond contracts
Notice that VP = PCNC = $100,000,000. This is so,
because the futures require no initial investment.
The planning period is 6 years, thus:
6
The following equation must hold:
D P  w C DC  w T-billDT  bill  6.
The entire original $100,000,000 are
invested in Bond C. Hence, w C  1
and the above condition becomes :
6  9.285  w T-bill(.25).
The solution is :
w T-bill  13.14.
7
How many T-bill contracts?
Solving for NT-bill we obtain:
FT-billN T  bill
w T-bill 
. In our case :
VP
$972,070(N T-bill )
- 13.14 
,
$100,000,000
N T  bill  1,352.
Hold $100M of bond C and short 1,352
T - bill futures.Th e combined portfolio' s
duration is exactly 6 years.
8
The same technique used to create Portfolio 2 can be applied
using a T-bond futures contract, giving rise to Portfolio 3.
Solving:
D P  w C DC  w T-bondDT  bond  6.
The entire original $100,000,000 are
invested in Bond C. Hence, w C  1
and the above condition becomes :
6  9.285  w T-bond (8.674).
The solution is :
w T-bill  .378718.
9
The 6-year Planning Period Case
How many T-bond contracts?
Solving for NT-bond we obtain:
FT-bondN T  bond
w T-bond 
. In our case :
VP
$71,875(N T-bond )
- .378718 
,
$100,000,000
N T-bond  527.
Hold $100M in bond C and short 527 T - bond
futures. The combined portfolio' s duration is
exactly 6 years.
10
Table B Portfolio Characteristics for the 6-year
Planning Period Case
Portfolio
WA
Portfolio 1
Bonds Only
56.54%
Portfolio 2
Bonds and Futures
0
Portfolio 3
Bonds and Futures
0
Weights
WC
43.43%
1.0
1.0
WT-bill
0
-13.14
0
WT-bond
0
0
-.378718
215,959
Number of
NC
93,856
Instruments
NA
63,844
215,959
0
NT-bill
0
- 1,352contracts short
0
Value
NT-bond
0
0
- 527contracts shorts
of Each
NCPC
$43,460,021
$100,000,000
Instrument
NAPA
$56,539,608
$100,000,000
0
NT-billFT-bill
0
$1,314,238,640 short
0
NT-bondFT-bond
0
0
$37,878,125 short
$100,000,000
$100,000,000
$100,000,000
Portfolio Value
0
0
11
To see how these three immunized portfolios perform,
assume:
1. Interest rate falls from 12% to 11% for all
maturities.
2. All coupon receipts during the six-year planning
period can be reinvested at 11% until the end of
the planning period.
With the shift in interest rates the new prices are:
PA= $913.57;
PC= $504.33;
FT-bill = $974,250; FT-bond= $77,813.
12
The 6-year Planning Period Case
Table C below, shows the effect of the interest rate
shift on portfolio values, the terminal wealth at the 6year planning period end and on the total wealth
position of the portfolio holder. As the Table reveals,
each portfolio responds similarly to the shift in yields.
The table demonstrates that the annualized holding
period rate of return on every one of the three
Portfolios remains 12%. The slight differences are
due to either rounding errors or the fact that the
duration price change formula holds exactly only for
13
infinitesimal changes in yields.
Table C: The Effect of a 1% Drop in Yields on
realized Portfolio Returns
Portfolio 1
Portfolio 2
Portfolio 3
Initial Portfolio Value
$100,000,000
$100,000,000
$100,000,000
New Portfolio Value
$105,660,731
$108,914,787
$108,914,787
Gain/ Loss on Futures
0
<$2,946,808>
<$3,128,792>
Total Wealth Change
$5,660,731
$5,967,979
$5,785,995
Terminal Value of all
Funds at n = 6
$197,629,369
$198,204,050
$197,868,664
Annualized Holding
Period Return over 6
Years
1.12
1.12
1.12
Source: From R. Kolb and G. Gay, “ Immunizing Bond Portfolios with Interest Rate Futures,” Financial
Management, Summer 1982, pp. 81-89. Reprinted by permission of Financial Management Association,
University of South Florida, College of Business, Tampa, FL 33620 (813) 974-2084
14
The 6-year Planning Period Case
One important concern in the implementation of
immunization strategies is the cost involved. In
immunizing, commission charges, marketability, and
liquidity of the instruments involved become
increasingly important.
These considerations highlight the practical
usefulness of interest rate futures in bond portfolio
management.
We now analyze the cost of the 6-year planning
period case.
15
The 6-year Planning period Case
Let us analyze the cost of implementing the 6year planning period case.
The transaction costs associated with the different
portfolios for the 6-year planning period case, starting
from the initial position of $100,000,000 in Bond C,
and shortening the duration to six years. Table F
shows the trades necessary and the estimated costs
involved.
Assume:
Commission fee for bond trading: $5/bond
Cost of trading futures contracts: $20/contract
16
Table F: Transaction Costs for the 6-year Planning
Period Case
Portfolio 1
Portfolio 2
Portfolio 3
NA
Long 63,844
-
-
NC
Short 122,103
-
-
NT-bill
-
Short 1,352
-
NT-bond
-
-
Short 527
One Way
Transaction Cost
Bond A @ $5/bond
$319,220
-
-
Bond C @ $5/bond
$610,515
-
-
-
$27,040
-
-
-
$10,540
$929,735
$27,040
$10,540
T-Bill Futures
$20/contract
T-Bond Futures
$20/contract
Total Cost of
Implementation
17
The 6-year Planning Period Case
To implement Portfolio 1, one must sell 122,103
bonds of type C and buy 63,844 bonds of type A.
Assuming a commission charge of $5 per bond, the
total commission is $929,735. By contrast one could
short 1,352 T-bill futures contracts to implement
Portfolio 2, at total cost of $27,040. Alternatively,
Portfolio 3 implies a short of 527 T-bond futures at
a total cost of $10,540.
In addition, margin deposits of approximately
$2,000,000 for Portfolio 2 or, $800,000 for
Portfolio 3 are required. Of course, margin
deposits may be in the form of interest earning
assets.
18
The 6-year Planning Period Case
Clearly, there is a tremendous difference in transaction
costs between trading the bonds in the cash market
and futures contracts. The cost of shorting the 1,352
T-bill futures is a small percentage of the daily volume
or recent open interest. Likewise, the 527 T-bond
futures constitute only a trivial fraction of the volume
and open interest in that market. The evident ability of
the futures market to absorb the kind of activity
involved in this example demonstrates the practical
usefulness rate futures in managing bond portfolios.
Notice, however, that the futures will have to be rolled
over when their delivery month arrives. This roll-over
presents some risk associated with these strategies.
19
The Bank Immunization Case
We now, turn to the example of the bank immunization case.
Assume that a bank holds a $100,000,000 liability portfolio in
Bond B. The bank wishes to protect it’s wealth position from
any change which might ensue a change in yields.
Five different portfolio combinations illustrate different means to
achieve the desired result:
ASSETS
LIABILITY
Portfolio 1: Bond A and Bond C.
Bond B.
Portfolio 2: Bond C; sell T-bill futures.
Bond B.
Portfolio 3: Bond A; buy T-bond future
Bond B.
Portfolio 4: Bond A; buy T-bill futures.
Bond B.
Portfolio 5: Bond C; sell T-bond futures.
Bond B.
20
The Bank Immunization Case
For each portfolio in Table D, the full $100,000,000
is put in a bond portfolio and is balanced out by
cash.
Portfolio 1 exemplifies the traditional approach of
immunizing by holding only bonds. Portfolio 2 and
Portfolio 5 are composed of Bond C and a short
futures position. By contrast, the low volatility Bond
A is held in Portfolio 3 and Portfolio 4. In
conjunction with Bond A, the overall interest rate
sensitivity is increased by buying interest rate
futures.
21
Table 8.28: Liability Portfolio and Five Alternative Immunizing Portfolios
Liability
Portfolio 1
Portfolio 2
Portfolio 3
Portfolio 4
Portfolio 5
Portfolio
(Bonds
only)
(Short T-Bill
(Long T-Bond
(Long T-Bill
(Short TBond)
Futures)
Futures)
Futures)
Futures)
Portfolio
WA
0
51.98%
0
100%
100%
0
Weights
WB
100%
0
0
0
0
0
WC
0
48.02%
100%
0
0
100%
WCash
~0
~0
~0
~0
~0
~0
Number of
NA
0
58,695
0
112,919
112,919
0
Instruments
NB
110,684
0
0
0
0
0
NC
0
103,704
215,959
0
0
215,959
NT-Bill
0
0
<1,242,710>
0
1,148,058
0
NT-Bond
0
0
0
44,751
0
<48,441>
NAPA
0
51,979,705
0
99,999,937
99,999,937
0
NBPB
99,999,673
0
0
0
0
0
NCPC
0
48,020,137
999,999,815
0
0
99.999,815
Cash
327
158
185
63
63
185
NT-BillFT-Bill
0
0
<1,208,001,110>
0
1,115,992,740
0
NT-BondFT-Bond
0
0
0
32,164,781
0
<34,816,969>
100,000,000
100,000,000
100,000,000
Portfolio
Value
100,000,000
100,000,000
100,000,000
The Bank Immunization Case
Is the banks wealth immunized against market yield
change?
To answer this question, assume an instantaneous
drop in rates from 12% to 11% for all maturities.
Table E shows the effect of the 1% drop on the
portfolios. As the rows reporting wealth change reveal,
all five portfolios perform similarly. The small
differences stem from rounding errors and the discrete
change in interest rates. Table E below, demonstrates
that all five portfolios may serve to immunize the
bank’s wealth. For all five portfolios, the wealth
change which ensues a yield change is virtually zero.
23
Table E: The Effect of a 1% Drop in Yields on
Total Wealth
Liability
Portfolio 1
Portfolio 2
Portfolio 3
Portfolio 4
Portfolio 5
Initial Portfolio
Value
100,000,000
100,000,000
100,000,000
100,000,000
100,000,000
100,000,000
New Portfolio
Value
105,910,526
105,923,188
108,914,788
103,159,474
103,159,474
108,914,788
Profit on Futures
0
-
<2,709,108>
2,657,314
2,502,766
<2,876,427>
Total Wealth
Change:
Bonds + Futures.
5,910,526
5,923,188
6,205,680
5,816,788
5,662,240
6,038,361
-
12,622
295,154
<93,738>
<248,286>
127,835
-
.00013
.00295
<.00094>
<.00248>
.00128
Total Wealth
change:
Asset-Liability
Portfolio.
% of Wealth
Change
Source: From R. Kolb and G. Gay, “ Immunizing Bond Portfolios with Interest Rate Futures,” Financial Management,
Summer 1982, pp. 81-89. Reprinted by permission of Financial Management Association, University of South
24
Florida, College of Business, Tampa, FL 33620 (813) 974-2084
Conclusion:
Until recently, immunization strategies for bond
portfolios have traditionally focused on all bond
portfolios. The analyses of the 6-year Planning
Period case and The Bank Immunization case
have shown that interest rate futures can be used in
conjunction with bond portfolios to provide the same
kind of immunization.
Both examples assumed parallel shifting yield
curves. If the change in interest rates brings about
non-parallel shifts in the yield curve, then the
traditional, “bonds only” portfolio as well as the
“bond-with-futures” approaches will give different
results. Which method turns out to be superior
would depend upon the pattern of interest rate
25
changes that actually occurred.