Bond Portfolio Management
Notes by Day Yi
Chapter 16:
Managing Funds Against Liabilities
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I.
IMMUNIZATION STRATEGY FOR A SINGLE LIABILITY
A. Introduction
1. Classical immunization can be defined as the process by which a bond portfolio is created
having an assured return for a specific time horizon irrespective of interest rate changes
2. The fundamental principle underlying immunization is to structure a portfolio that balances
a. The change in the value of the portfolio at the end of the investment horizon
b. The return from the reinvestment of portfolio cash flows (coupon payments and maturing
bonds)
3. Immunization requires offsetting interest rate risk and reinvestment risk
4. To accomplish this balancing requires the controlling of duration
a. By setting the duration of the portfolio equal to the desired portfolio time horizon, the
offsetting of positive and negative incremental return sources can be assured
b. This is a necessary condition for effectively immunized portfolios
B. Illustration
1. The requirements for immunization can be restated in more general terms as
a. The effective duration of the portfolio must equal the effective duration of the liability
b. The initial present value of the cash flows from the bond portfolio must equal the present
value of the future liability
C. Rebalancing an Immunized Portfolio
1. In the face of changing market yields, a portfolio can be immunized if it is rebalanced
periodically so that its duration is readjusted to the shorter duration of the liability
D. Application Considerations
1. In the actual process leading to the construction of an immunized portfolio, the selection of
the universe is extremely important
a. The lower the credit quality of the securities considered, the higher the potential risk and
return
b. Immunization theory assumes there will be no defaults and that securities will be
responsive only to overall changes in interest rates
c.
Securities with embedded options such as call features or mortgage-backed prepayments
complicate and may even prevent the accurate measure of cash flows and hence
duration
d. Liquidity is a consideration for an immunized portfolio because, the portfolio must be
rebalanced over time
2. Optimization procedures can be used for the construction of an immunized portfolio
a. Typically, immunization takes the form of minimizing the initial portfolio cost subject to
the constraint of having sufficient cash to satisfy the liability at the horizon date
b. Further considerations such as average credit quality, minimum and maximum
concentration constraints, and, perhaps, issuer constraints may be included
3. Transaction costs are important in meeting the target rate for an immunized portfolio
E. Extensions of Classical Immunization Theory
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1. The sufficient condition for classical immunization is that the duration of the portfolio match
the duration of the liability
2. Classical theory is based on the following assumptions
a. Any changes in the yield curve are parallel changes
b. The portfolio is valued at a fixed horizon date and there are no cash inflows or outflows
during the horizon
c.
The target value of the investment is defined as the portfolio value at the horizon date if
the interest rate structure does not change (i.e. no change in forward rates)
3. Perhaps the most critical assumption of classical immunization techniques concerns the first
assumption — the type of interest rate change anticipated
4. Immunization risk is reinvestment risk
a. The portfolio that has the least reinvestment risk will have the least immunization risk
b. When there is a high dispersion of cash flows around the horizon date, as in the barbell
portfolio, the portfolio is exposed to higher reinvestment risk
c.
In contrast, when the cash flows are concentrated around the horizon date, as in the
bullet portfolio, the portfolio is subject to minimum reinvestment risk
5. Immunization risk measure =
PVCF1 (1  H ) 2  PVCF2 (2  H ) 2    PVCFn (n  H ) 2
Initial Investment Value
where PVCFt
= PV of CF in period t discounted at the prevailing yield
H
= length of the investment horizon
n
= time to receipt of the last portfolio CF
F. Contingent Immunization
1. Contingent immunization consists of the identification of both
a. The available immunization target rate
b. A lower safety net level return with which the client would be minimally satisfied
2. If the safety net return is violated, the manager would be obligated to completely immunize
the portfolio and lock in the safety net level return
3. Once the immunization mode is activated, the manager can no longer return to the active
mode unless the contingent immunization plan is abandoned
4. The key considerations in implementing a contingent immunization strategy are
a. Establishing accurate immunized initial and ongoing available target returns
b. Identifying a suitable and immunizable safety net
c.
Implementing an effective monitoring procedure to ensure that the safety net return is
not violated.
5. The yield level at which the immunization mode becomes operational is called the trigger
point
6. In spite of good control and monitoring procedures, attainment of the minimum target return
may not be realized due to factors beyond the control of the manager
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a. There is the possibility of a rapid adverse movement in market yields that is of sufficient
magnitude that the manager may not have enough time to shift from the active to
immunization mode at a rate needed to achieve the minimum target
b. If the immunization mode becomes operational, there is no guarantee that the
immunized rate will be achieved even if the portfolio is reconstructed at the required rate
II.
IMMUNIZATION FOR MULTIPLE LIABILITIES
A. There are two strategies that can be employed in seeking to satisfy multiple liabilities
1. An extension of the single-period immunization strategy
2. A cash flow matching strategy
B. Conditions for Immunization (for parallel shifts only)
1. The present value of the liabilities must equal the present value of the assets
2. The (composite) duration of the portfolio must equal the (composite) duration of the
liabilities
3. The distribution of durations of individual portfolio assets must have a wider range than the
distribution of the liabilities
C. Immunization risk measure =
PVCF1 (1  D) 2  PVCF2 (2  D) 2    PVCFn (n  D) 2
Initial Investment Value

PVL1 (1  D) 2  PVL2 (2  D) 2    PVLm (m  D) 2
Initial Investment Value
where PVLt
= PV of the liability at time t
m
= time of the last liability payment
D
= duration of the portfolio
D. Selecting the Appropriate Discount Rate for Liabilities
1. Treasury yield curve plus spread approach: The discount rates in this approach are the rates
from the Treasury yield curve plus a suitable spread (no theoretical basis)
2. Treasury spot rate curve plus spread approach: The discount rates in this approach are the
rates from the zero-coupon or spot rate Treasury curve plus a suitable spread (correct)
3. Yield curve derived from the portfolio of assets: The discount rates in this approach are those
obtained from a yield curve constructed from the securities in the asset portfolio (most often
used in practice)
III.
CASH FLOW MATCHING FOR MULITPLE LIABILITIES
A. Cash Flow Matching Versus Multiple Liability Immunization
1. Given typical liability schedules and bonds available for cash flow matching, perfect matching
is unlikely
2. Under such conditions, a minimum immunization risk approach would, at worst, be equal to
cash flow matching and would probably be better, because an immunization strategy would
require less money to fund liabilities
a. A relatively conservative rate of return assumption for short-term cash, which may
occasionally be substantial, must be made throughout the life of the plan in cash flow
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matching, whereas an immunized portfolio is essentially fully invested at the remaining
horizon duration
b. Funds from a cash flow matched portfolio must be available when each liability is due
and, because of the difficulty in perfect matching, usually before; an immunized portfolio
need only have sufficient value on the date of each liability because funding is achieved
by a rebalancing of the portfolio
3. Thus, even with the sophisticated linear programming techniques used in cash flow
matching, in most cases it will be technically inferior to immunization
B. Extensions of Basic Cash Flow Matching
1. A popular variation of multiple liability immunization and cash flow matching to fund liabilities
is one that combines the two strategies
a. This strategy, referred to as combination matching or horizon matching, creates a
portfolio that is duration-matched with the added constraint that it be cash matched in
the first few years, usually five years
b. The advantage of combination matching over multiple liability immunization is that
liquidity needs are provided for in the initial cash flow matched period
c.
Also, most of the positive slope or inversion of a yield curve tends to take place in the
first few years
d. Cash flow matching the initial portion of the liability stream reduces the risk associated
with nonparallel shifts of the yield curve
e. The disadvantage of combination matching over multiple liability immunization is that the
cost is greater
IV.
LIABILITY INDEXES FOR PENSION FUNDS
A. Creating a customized index for a pension fund based on the fund’s projected liabilities
1. Begins with the projected liabilities of the pension fund as determined by the actuary
2. The present value of the projected liabilities must be computed
3. The Financial Accounting Standard Board has stated that liabilities should be valued as if they
were high quality zero-coupon bonds with the same maturities as the liabilities
4. Thus, the projected liabilities should be valued as a package of zero-coupon bonds with
different maturities
5. The interest rate that should be used to determine the value of the projected liabilities would
then be the theoretical spot rates on Treasury securities or the rate on Treasury strips
6. From the value of these projected liabilities, a liability index can be created for a pension
fund
B. At the end of a quarter, the value of the liabilities is recomputed, reflecting the term structure of
interest at the time of revaluation
1. A liability return can be calculated based on the beginning of the quarter value for the index
and the value for the index at the end of the quarter
2. The objective of an active manager is then to outperform the liability return
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