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ACTSC445 Lecture Notes - Reinvestment Risk, Yield Curve, 0 (Year)

Actuarial Science
Course Code
Jiahua Chen

of 20
ACTSC 445: Asset-Liability 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.
Fisher-Weil (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
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., T-bills, 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)
Single-liability case
We’ll start with the case where there is only one liability in the portfolio, with corresponding cash flow
of Ltat some time t.
The goal is to choose an asset cash flow sequence {At, t > 0}that will, along with Lt, 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: 3-year, 5-year and 7-year zero-coupon bonds, each yielding 6% annual
effective rate.
Portfolio A: Invest $747,258.17 in the 5-year zero coupon bond
Portfolio B: Invest the same amount (i.e. $747,258.17) in a 3-year zero coupon bond. The
maturity value at t= 3 is $889,996.44.
Portfolio C: Invest $747,258.17 in a 7-year zero coupon bond. The maturity value at t= 7 is
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 ˆyand 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.
Table 1: Value of Portfolio B for different yields
Value of Portfolio B Capital Gain Implied
ˆy(%) at time 0 at time 5 at time 0 Yield (%)
4.00 791203.5944 962620.1495 43945.4215 5.20
5.00 768812.3874 981221.0751 21554.2146 5.60
5.90 749377.0511 998114.0975 2118.8782 5.96
6.00 747258.1729 1000000.0000 0.0000 6.00
6.10 745147.2753 1001887.6824 2110.8975 6.04
7.00 726502.2044 1018956.9242 20755.9684 6.40
8.00 706507.8685 1038091.8476 40750.3044 6.80
So for portfolio B, if the yields go up, then we realize a gain at time 5, because we can reinvest the
proceeds obtained at time 3 at a high yield. But if the rates drop, then we realize a loss at time 5. The
problem here is the reinvestment risk.
Table 2: Value of Portfolio C for different yields
Value of Portfolio C Capital Gain Implied
ˆy(%) in year 0 at time 5 at time 0 Yield (%)
4.00 853843.6549 1038831.3609 106585.4820 6.81
5.00 798521.5425 1019138.3220 51263.3697 6.40
5.90 752211.5711 1001889.4658 4953.3982 6.04
6.00 747258.1729 1000000.0000 0.0000 6.00
6.10 742342.0181 998115.8742 4916.1548 5.96
7.00 699721.6100 981395.7551 47536.5629 5.60
8.00 655609.8081 963305.8985 91648.3647 5.21
The situation here is opposite from what we face with Portfolio B: if the rates drop, then we can sell
the 7-year zero bond at a higher price at time 5, which results in a gain. But a yield increase produces
a loss. The problem here is the interest rate or price risk.
Observations from Example I
With a single liability, the best immunization strategy is the one for which the asset cash flow
coincides with the liability cash flow
When asset cash flows occur prior to (or after) the liability cash flow, the portfolio is subject
to reinvestment risk (or market/interest rate/price risk).
A valid question is: could we construct a portfolio containing cash flows occuring before and after the
liability due date that could be immunized? Motivation:
Any initial capital loss may be offset in time by greater returns from reinvestment.
Similarly, any initial capital gain may be offset in time by lower returns from reinvestment.
Does there exist an “optimum” trade-off? I.e., a way to construct a portfolio like this that
maximizes (in some sense) the gain?
The following example studies this idea.
Example II: Consider Portfolio D, which consists in an investment of $373 629.0864 in 3-year zero-
coupon bonds, and $373 629.0864 in 7-year zero-coupon bonds. Their maturity values are, respectively,
444,998.22 and 561,800.00. Note that the Macaulay duration of this portfolio is 5.
If the yields remain unchanged, then at t= 5 we have 373,629.0864 ×2×(1.06)3= 1 000 000.
If the rates change, then we get the following results:
Value of Portfolio D Capital Gain Implied
ˆy(%) at time 0 at time 5 at time 0 Yield (%)
4.0 822523.6247 1000725.7552 75265.4518 6.01538
5.0 783666.9650 1000179.6986 36408.7921 6.00381
5.9 750794.3111 1000001.7817 3536.1382 6.00004
6.0 747258.1729 1000000.0000 0.0000 6.00000
6.1 743744.6467 1000001.7783 -3513.5262 6.00004
7.0 713111.9072 1000176.3396 -34146.2657 6.00374
8.0 681058.8383 1000698.8731 -66199.3346 6.01481
Hence with this portfolio, a gain is realized at time 5 for all alternative yields ˆyconsidered...
Note that at time 0, there is a capital loss for portfolio D. More generally, we can look at the value of
this portfolio at time tif the initial yield goes from 6% to ˆy. That is, we can consider the value
Vt= 444 998.22(1 + ˆy)(3t)+ 561 800.00(1 + ˆy)(7t)
for t= 1, . . . , 10 and different ˆy’s.