23.4Immunization

Immunizationis a technique for locking in the value of a portfolio at the end of a plan-

ning horizon. In a sense, immunization turns a portfolio into a zero-coupon bond.

Ordinary Immunization

Immunization was developed by F. M. Redington, an actuary, who showed that if the

durations and market values of the assets and liabilities of a financial institution were

equal, the equity of the institution would be insensitive to movements in interest rates.¹¹

We generalize this result here, allowing the ratios of the durations of the assets and lia-

bilities to be inversely proportional to their market value ratios.

Applying Immunization Techniques to Stabilize the Future Value of a Bond

Portfolio.Immunization techniques can be applied to reduce the interest rate sensi-

tivity of the equity of financial institutions or the equity stake of a corporation in a

defined benefit pension plan.¹²Today, however, immunization techniques are more

often used to stabilize the value of a bond portfolio at the horizon date, which is the

date at the end of some planning horizon.

For simplicity, assume that the term structure of interest rates is flat. In this case,

the fundamental rule for immunizing a portfolio is to match the duration of the port-

folio with the horizon date. For example, a manager with a horizon date of January 1,

2010, should, on January 1, 2004, have a portfolio duration of 6 years; on January 1,

2005, the portfolio duration should be 5 years; on June 30, 2008, it should be 1.5 years.

To implement the immunization strategy, it is important that all coupons, principal

payments, and other cash distributions received be reinvested in the portfolio. Main-

taining a duration that is matched to the horizon date means that the rate at which these

cash distributions are reinvested exactly offsets the gain or loss in the value of the port-

folio as interest rates change.

Why Immunization Locks in a Value at the Horizon Date.To see why this strat-

egy locks in the portfolio value at the horizon date, compare theportfolio with a zero-

coupon bond of identical market value which has a maturity, and hence a duration, equal

to the duration of the immunized portfolio. Along position in the portfolio and a short

position in the zero-coupon bond has both a market value and a duration of zero.

When the duration of the combined long and short position is zero, its sensitivity

to interest rate movements and hence its volatility is zero. For a brief instant, it is risk-

less. As time elapses, keeping it riskless requires adjusting the long position in the port-

folio to have the same duration as the maturity of the zero-coupon bond by shortening

the duration of the long position over time in order to maintain a duration of zero for

the combination of the immunized portfolio (the long position) and the short position

in the zero-coupon bond.

11

See Redington (1952).

¹²See

Example 23.9.

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Chapter 23

Interest Rate Risk Management

835

Since riskless self-financing investments do not appreciate or depreciate in value,

the riskless self-financing combination of the immunized bond portfolio and the short

position in the zero-coupon bond—if the former is properly updated over time—will

have a value of zero at the maturity date of the zero-coupon bond. At the maturity date,

this means that the immunized portfolio has to have a market value equal to the face

value (and market value) of the zero-coupon bond.

Result 23.5 summarizes this procedure.

Result 23.5

If the term structure of interest rates is flat, immunization “guarantees” a fixed value for animmunized portfolio at a horizon date. The value obtained is the same as the face value ofa zero-coupon bond with (1) the same market value as the original portfolio and (2) a matu-rity date equal to the horizon date selected as the target date to which the duration of theimmunized portfolio is fixed.

Viewing the portfolio to be immunized as an asset, and the zero-coupon bond as

a liability, we have done exactly what Redington suggested for financial institutions—

matching the durations of assets and liabilities with the same market value. Of course,

the zero-coupon bond was not actually sold short in order to immunize the portfolio.

Rather, we merely pretended to sell short a zero-coupon bond to help clarify what to

do to the portfolio to ensure a value at the horizon date.

Example 23.10 shows how to calculate the lock-in value at the horizon date.

Example 23.10:Computing the Lock-In Amount at Some Horizon Date

The current yield to maturity is 8 percent compounded semiannually for bonds of all matu-

rities.A bond portfolio consists of $10 million (market value) of fixed-income securities.What

amount will the portfolio manager be able to lock in three years from now?

Answer:The current market value of the portfolio is $10 million.At 8 percent, the future

value of the portfolio in three years is

1.04⁶

$10 million $12.653 million

Example 23.11 shows how to alter a portfolio to lock in its value.

Example 23.11:Using Immunization Techniques to Lock In a Payoff

The $10 million portfolio in Example 23.10 is a pension portfolio, which currently has a dura-

tion of five years.Half the portfolio’s market value consists of identical maturity zero-coupon

bonds.The remainder consists of $5 million (market value) of the 2-year straight-coupon

bonds from Example 23.6.These bonds have a duration of 1.89 years and an 8 percent

yield, compounded semiannually.

a.What is the maturity of the zero-coupon bonds in the portfolio?

b.How should the manager rebalance the portfolio between the 2-year straight-coupon

bonds and the zero-coupon bonds to immunize it at a 3-year horizon date?

Answer:a.The zero-coupon bonds have to have a maturity of 8.11 years because this

is the only maturity that makes an equal-weighted average of the 2-year bonds’duration

(1.89) and the zero-coupon bonds’duration equal five years.

b.To immunize the portfolio for a horizon of three years, the duration of the portfolio has

to be changed to three years.Find weights xand 1xthat make the weighted average of

the durations

x(1.89) (1 x)8.11 3

The approximate solution is x .82154.Hence, $8.2154 million of the 2-year straight coupon

bonds must be owned, which requires an additional purchase of $3.2154 million (face and

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Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

836Part VIRisk Management

market value) of these bonds.To finance the purchase, sell $3.2154 million (market value)

of the zero-coupon bonds, which have an aggregate face value of about

$6.07 million $3.2154 million (1.04)^16.22

Example 23.11 described what the portfolio manager must do currently to immu-

nize his portfolio for a horizon of three years. However, this manager cannot rest on

his laurels. Every day, as time elapses and interest rates change, the immunized port-

folio becomes nonimmunized if the manager acts passively. To maintain the immu-

nization, as Example 23.12 illustrates, it is important to constantly update the weight-

ing of the zero-coupon bonds and the 2-year straight-coupon bonds.

Example 23.12:Updating Portfolio Weights in an Immunized Portfolio

What must the portfolio manager do in the future to immunize the portfolio, particularly at

the maturity date of the 2-year straight-coupon bonds?

Answer:As each day elapses, the portfolio manager must shorten the duration by one

day.Some of this shortening will happen even if the manager does nothing, since the dura-

tion of both bond types is diminishing as time elapses.However, it is unlikely that this nat-

ural shortening of duration will be exactly one day.Hence, the manager must recompute

duration periodically for the new interest rate and the time to payment of the cash flows and

adjust duration accordingly.This creates a problem after two years has elapsed.At that point,

the manager would like the portfolio to have a 1-year duration, but would have only zero-

coupon bonds maturing in 6.11 years (and a duration of 6.11 years) once the straight-coupon

bonds mature.This implies that the manager must begin to use a third security in the port-

folio as the maturity date of the straight-coupon bonds nears.

The immunization strategy employed in Examples 23.10–23.12 was equivalent to

designing a hypothetical portfolio with a duration of zero. This portfolio consists of the

2-year straight-coupon bonds, 8.11-year zero coupon bonds, and a short position in

3-year zero-coupon bonds that finances the other two positions. If this hypothetical port-

folio is managed so that it has a value of zero at the horizon date, the original immu-

nized portfolio is managed so that it has a value equal to that of 3-year zero-coupon

bonds with an aggregate face value of $12.653 million.

The same insight can be used to understand Redington’s result as it applies to finan-

cial institutions. If the assets and liabilities of the financial institution have unequal

value, construct fictitious zero-coupon bonds with a market value equal to the market

value of the institution’s equity (assets minus liabilities) and a maturity equal to the

horizon date at which the equity’s value needs to be guaranteed. To immunize the equity

in this fashion over time, manage a self-financing investment that is long the assets,

short the liabilities, and short the zero-coupon bonds, so that it maintains zero dura-

tion. This is equivalent to structuring the assets and liabilities of the institution so that

they have a duration equal to the maturity of the zero-coupon bonds,¹³as Example

23.13 illustrates for a savings bank.

¹³The

self-financing investment will continue to have a value of zero if the immunization procedure

is followed, implying that the assets less the liabilities (which equals the equity) will have a value equal

to the value of the fictitious zero-coupon bond.

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Markets and Corporate		Management	Companies, 2002
Strategy, Second Edition

Chapter 23

Interest Rate Risk Management

837

Example 23.13:Using Immunization to Manage Savings Bank Assets

The assets of a savings bank consist of mortgages with a 4-year duration and a present value

of $10 billion.The bank’s liabilities consist of customer deposits and CDs with a duration of

two years and a present value of $5 billion.Interest rates are 8 percent, compounded annu-

ally at all maturities.Assume that the bank’s management wants to ensure that the bank has

a fixed amount of equity capital at the time the bank’s next regulatory examination is sched-

uled, in two years.

a.How much equity value can it guarantee at the time of the next regulatory examination?

b.Given that the savings bank can invest in commercial paper (which can be regarded

as a short-term zero-coupon bond with a 3-month maturity) and sell some or all of its

mortgage assets, what should the bank do to lock in an equity value two years from now?

Answer:a.The current $5 billion in equity can have a “lock-in”value in two years of

$5.832 billion [ (1.08)²$5 billion]

b.To lock in this value, adjust the duration of the equity to two years, and then continu-

ally shorten the duration by one day as each day elapses.The current duration of the equity

is a weighted average of the durations of the assets and liabilities.Since the assets are twice

as large as the liabilities, the asset weight must be 2 and the liability weight must be 1 for

the weights to sum to 1.This makes the current equity duration 2 (4 years) 1 (2 years)

6 years.To shorten this to two years by changing the asset portfolio, the bank should sell

some of the mortgage assets and buy commercial paper.This requires shortening the asset

duration to a duration (DUR) that satisfies

2DUR1 (2 years) 2 years

Thus, DUR 2 years.The market value of mortgage assets xand commercial paper assets

ythat have a duration of two years satisfy

4.25y

2

xy

The market values of the mortgage position xand of the commercial paper position y

mustsum to $10 billion, the current value of the bank’s assets.Thus, xand ymust also solve

xy $10 billion

Substituting this into the previous equation yields

4x.25($10 billionx)

2

$10 billion

This is solved by x $4.67 billion, implying y $5.33 billion.That is, sell $5.33 billion of

the $10 billion in mortgage assets and use the proceeds to buy 3-month commercial paper.