ACTSC 445: AssetLiability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 6 – Immunization
References
(recommended readings): Chap. 3 of Financial Economics (on reserve at the library: call
number HG174 .F496 1998).
What is immunization?
•
Redington (1952):
Immunization implies the investment of assets in such a way that existing
business is immune to a general change in the rate of interest.
•
FisherWeil (1971):
A portfolio of investment is immunized for a holding period if its value at
the end of the holding period, regardless of the course of rates during the holding period, must be
at least as it would have been had the interest rate function been constant throughout the holding
period.
Implication: If the realized return on an investment in bonds is sure to be at least as large as the
appropriately computed yield to the horizon, then that investment is immunized.
•
An
immunization strategy
is a risk management technique designed to ensure that for any
small change in a specified parameter, a portfolio of debt instruments (e.g., Tbills, bonds, GICs
etc) will cover a liability (or liabilities) coming due at a future date (or over a period in the future).
It is a
passive management
technique because it takes prices as given and then tries to control
the risk appropriately. (By contrast, active management techniques try to exploit changes in (1)
the level of interest rates, (2) the shape of the yield curve (3) yield spreads, by using interest rate
forecasts and identification of mispriced bonds)
⇒
asset allocation problem (i.e., must choose assets that will produce an immunized portfolio)
Singleliability case
We’ll start with the case where there is only one liability in the portfolio, with corresponding cash flow
of
L
t
at some time
t
.
The goal is to choose an asset cash flow sequence
{
A
t
, t >
0
}
that will, along with
L
t
, produce an
immunized portfolio. Let’s start with an example.
Example I:
Suppose an insurance company faces a liability obligation of
$
1 million in 5 years. The
available market instruments are: 3year, 5year and 7year zerocoupon bonds, each yielding 6% annual
e
ff
ective rate.
•
Portfolio A:
Invest
$
747,258.17 in the 5year zero coupon bond
1
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•
Portfolio B:
Invest the same amount (i.e.
$
747,258.17) in a 3year zero coupon bond.
The
maturity value at
t
= 3
is
$
889,996.44.
•
Portfolio C:
Invest
$
747,258.17 in a 7year zero coupon bond. The maturity value at
t
= 7
is
$
1,123,600.00.
If the yields remain unchanged, then the 3 portfolios have the same value of
$
1 000 000 at time 5.
To verify if these portfolios are immunized or not, we need to look at what happens if, immediately after
the portfolio is acquired, the yield changes instantaneously to
ˆ
y
and remains constant at that level.
First, note that for portfolio A, this change has no impact: its value at time 5 is still
$
1 000 000. But
this is not true for portfolios B and C, as Tables 1 and 2 show.
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 Spring '09
 ChristianeLemieux
 Bond duration, Zerocoupon bond, Redington

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