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

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., 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

1

•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

$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 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.

2

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)−(3−t)+ 561 800.00(1 + ˆy)−(7−t)

for t= 1, . . . , 10 and diﬀerent ˆy’s.

3

**Unlock Document**

###### Document Summary

Department of statistics and actuarial science, university of waterloo. 3 of financial economics (on reserve at the library: call number hg174 . f496 1998). Asset allocation problem (i. e. , must choose assets that will produce an immunized portfolio) We"ll start with the case where there is only one liability in the portfolio, with corresponding cash ow of lt at some time t. The goal is to choose an asset cash ow sequence {at, t > 0} that will, along with lt, produce an immunized portfolio. Example i: suppose an insurance company faces a liability obligation of million in 5 years. The available market instruments are: 3-year, 5-year and 7-year zero-coupon bonds, each yielding 6% annual e ective rate: portfolio a: invest ,258. 17 in the 5-year zero coupon bond. 1: portfolio b: invest the same amount (i. e. ,258. 17) in a 3-year zero coupon bond.

## More from OC26778

###### ACTSC445 Lecture Notes - United States Treasury Security, Unsecured Debt, Money Market

Lecture Note

###### ACTSC445 Lecture Notes - Interest Rate Risk, General American, Credit Risk

Lecture Note

###### ACTSC445 Lecture Notes - Vasicek Model, Callable Bond, Arbitrage

Lecture Note

## Similar documents like this

###### HLSC 3022C Chapter 2: free body diagram example 14

Textbook Note

###### HLSC 3022C Chapter 2: free body diagram example 13

Textbook Note

###### HLSC 3022C Chapter 2: free body diagram example 16

Textbook Note

###### HLSC 3022C Chapter 2: free body diagram example 5

Textbook Note

###### HLSC 3022C Chapter 2: free body diagram example 3

Textbook Note

###### HLSC 3022C Chapter 2: free body diagram example 1

Textbook Note

###### HLSC 3022C Chapter 2: free body diagram examples 2 and 4

Textbook Note

###### HLSC 3022C Chapter Notes - Chapter 2: William Trelease

Textbook Note

###### HLSC 3022C Chapter 2: how to set up a free body diagram

Textbook Note

###### HLSC 3022C Chapter Notes - Chapter 1: Swiss 1Jj Tarot

Textbook Note

###### HLTH 2011 Chapter 1: intro to health ethics and regulations

Textbook Note

###### HLTH 2011 Chapter 1: basics of healthcare ethics

Textbook Note

###### HLSC 4015C Lecture 12: cancer notes

Lecture Note

###### HLSC 4015C Lecture 12: brain health notes

Lecture Note

###### HLSC 4015C Lecture 12: Stroke notes

Lecture Note