Abstract

The target value of an immunized portfolio at the horizon date defines the portfolio's target rate of return. If interest rates change by parallel shifts for all maturities, the portfolio's realized rate of return will not be below the target value. To the extent that nonparallel rate changes occur, however, the realized return may be less than the target value.

The relative change in the end-of-horizon value of an immunized portfolio resulting from such an arbitrary rate change will be proportional to the value of its immunization risk. Immunization risk equals the weighted variance of times to payment around the horizon date, hence depends on portfolio composition. For example, immunization risk will be low if portfolio payments cluster around the end of the horizon and high if payments are widely dispersed in time. One may minimize the extent to which a portfolio's realized return differs from its target return by minimizing the portfolio's immunization risk (while keeping the portfolio's duration equal to the remaining horizon length).

Although risk minimization is the traditional objective of immunization, the immunization risk measure may also be used to optimize the risk-return tradeoff. The standard deviation of an immunized portfolio's rate of return over the investment horizon will be proportional to the value of its immunization risk. Thus an investor may choose from immunized portfolios of equal duration a portfolio with a high level of immunization risk in order to maximize his expected return.

Introduction

The traditional theory of immunization, as formalized by Fisher and Weil in 1971, assumed that a portfolio is valued at a fixed horizon date, that there are no cash inflows or outflows within the horizon, and that interest rates can change only by a parallel shift (that is, by the same amount for rates of all maturities). Under these assumptions, a portfolio is said to be immunized if its value at the end of the horizon does not fall below the target value, the target being defined as the value at the horizon date under the scenario of no change in the forward rates. The main result of this theory is that immunization is achieved if the duration of the portfolio is equal to the length of the horizon.

Immunization theory and practice have developed in a number of directions since the work by Fisher and Weil. The most significant development has been in overcoming the limitations of a fixed horizon with no contributions or withdrawals. Marshall and Yawitz (1974), for instance, demonstrated that, even if the value of an immunized portfolio declines because of interest rate changes over the investment horizon, a lower bound on the value of the portfolio exists at any point during the investment period. Fong and Vasicek (1980) and Bierwag, Kaufman, and Toevs (1983) subsequently addressed the multiple liabilities situation. Multiple liability immunization involves an investment strategy that guarantees that a specified schedule of future liabilities will be met, regardless of any parallel interest rate shifts. The amount of initial investment necessary for multiple liability immunization is equal to the present value of the liability stream under the initial interest rate structure (cf., for instance, Fong and Vasicek (1980)).

Immunization theory has also been extended to allow relaxation of the assumption of parallel changes in interest rates. Most of the work in this area—such as Bierwag (1978), Cox, Ingersoll, and Ross (1979), and Brennan and Schwartz (1983)—postulates alternative models of interest rate behavior and derives modified definitions of duration corresponding to the assumed interest rate process. A limitation of this approach is that immunity is achieved only against the assumed type of rate changes. A different approach to nonparallel rate changes described in Fong and Vasicek (1980) is to establish a measure of immunization risk for an arbitrary interest rate change; this risk can then be minimized, subject to the duration condition and other constraints, to obtain an optimally immunized portfolio. Whereas duration measures the mean time to payment on a portfolio, the proposed risk measure represents the time-to-payment variance around the liability dates, hence the exposure of the portfolio to relative changes of rates of different maturities.

Finally, immunization techniques have recently been used in conjunction with elements of active bond investment strategies. The classical objective of immunization has been risk protection, with little consideration of possible returns. Leibowitz and Weinberger (1981) proposed a scheme called contingent immunization, which provides a degree of flexibility in pursuing active strategies while ensuring a certain minimum return in the case of parallel rate shifts. With this approach, immunization serves as a fall-back strategy if the actively managed portfolio does not grow at a certain rate.

This article explores the risk-return tradeoff for immunized portfolios using a very different approach. The strategy proposed here maintains the duration of the portfolio at all times equal to the horizon length (or, in the multiple-liability case, keeps the generalized immunization conditions satisfied). Thus, the portfolio always remains fully immunized in the classical sense. However, instead of attempting to minimize the portfolio's immunization risk—that is, its vulnerability to arbitrary rate changes—this strategy aims for an optimal tradeoff between risk and return. The immunization risk measure can be relaxed if the compensation in terms of expected return warrants it.

Specifically, the strategy maximizes a lower bound on the portfolio's return. The lower bound is defined as a confidence interval on the realized return for a given probability level. The optimal portfolio therefore has the following characteristics:

1. It is completely immunized against parallel rate shifts, so the target rate of return is guaranteed as long as rates of various maturities change by the same amount.
2. Its level of immunization risk for arbitrary nonparallel rate changes is measured and minimized in the tradeoff with expected return.
3. Maximization of the expected portfolio return is included in the objective function together with risk consideration by maximizing a lower bound on return.

Portfolio Value and Interest Rate Changes

The duration of a bond portfolio is defined as the weighted average time to all the portfolio payments, the weights being the present values of the payment amounts. Mathematically, duration can be written as:

1

where:

m	=	the number of portfolio payments (interest and principal),
	=	the amount of the payment due at time ,
	=	the current discount function—that is, the present value of a unit payment expected at time t, and
	=	the initial portfolio value, equal (by definition of the discount function) to the sum of the present values of the payments.

Assume that a portfolio has been constructed to have its duration equal to the investment horizon H. If interest rates do not change (in the sense that the forward rates stay the same), the portfolio, including reinvestment of portfolio cash flows received over the horizon, will have a certain value at the horizon date. This value, denoted by I_H, is called the target value.

Now suppose that, after the portfolio has been constructed, interest rates do change, but in such a way that rates of all maturities move, up or down, by the same amount. This is called a parallel shift of interest rates. The resulting change in the portfolio value at the horizon date, , will never be negative. In other words, the portfolio's terminal value will not fall below the target value because of a parallel shift of the yield curve. This is the principal result of the traditional immunization theory.

This immunization against changes in the level of rates is achieved by balancing the changes in reinvestment rates against capital gains or losses. Suppose that rates of all maturities increase by a given amount. Because the average time to payments (the portfolio's duration) is equal to the length of the horizon, there are some portfolio payments before the horizon date and some after the horizon date. The portion of the portfolio still outstanding at the horizon date will experience a capital loss. This loss, however, will be compensated for by the reinvestment of the portfolio payments received over the horizon at rates higher than those originally expected. Conversely, if rates of all maturities decrease by a like amount, the lower value of reinvestment is balanced by capital gains on the longer portion of the portfolio. The condition that the portfolio duration be equal to the horizon length assures that the magnitudes of the opposing changes in the reinvestment amounts and the principal values are such as to keep the end-of-horizon portfolio value from falling below the target value.

This balancing act obviously works only if rates of various maturities move in the same direction and by the same amount. If, instead of a parallel shift of the initial yield curve, there is a more complicated rate change, immunization may not take place. If the yield curve twists in such a way that short rates decline while long rates increase, the portfolio will suffer both a decrease in reinvestment rates and a capital loss on the longer portion, and its value at the horizon date will be below the target. We refer to such a possibility as immunization risk.

If immunization risk could be measured, immunized portfolios could be constructed to minimize this risk. For instance, there may be a number of portfolios having the same investment horizon and duration, each having a different degree of exposure to nonparallel rate changes. A quantitative measure of this exposure, independent of assumptions about the character of the possible changes in the interest rate structure, would be desirable.

Immunization Risk

Fong and Vasicek (1984) (Chapter 23 of this volume) have shown that the change in the end-of-horizon value of an immunized portfolio resulting from an arbitrary change in interest rates is approximated by the following equation:

2

Here is the change in the slope of the term structure of interest rates; this quantity characterizes the degree of twist of the yield curve. The term is given by the formula:

3

Eq. (2) has an interesting structure. It expresses the relative change in the end-of-horizon portfolio value as a product of two terms. The first term, , depends solely on the structure of the portfolio, whereas the second term, , is a function of the interest rate change only. The investor has no control over the quantity ; it is an uncertain variable that can take any value. The investor can, however, determine the portfolio composition, hence the quantity . As a multiplier of the unknown rate change, this term measures the extent to which the portfolio can be affected by such a change. The change in the end-of-horizon value of the investment resulting from an arbitrary rate change is thus proportional to , and is a measure of immunization risk.

To gain an insight into the meaning of the risk measure , note the similarity in form between in Eq. (3) and the definition of duration in Eq. (1). Whereas duration is a weighted average of time to portfolio payments, the weights being the present values of the payments, is a similarly weighted variance of times to payment around the horizon date. If the portfolio payments occur close to the end of the horizon (as would be the case with a portfolio of deep-discount bonds maturing close to the horizon date), is low. If the payments are widely dispersed in time (as would be the case with a portfolio consisting of very short bonds and very long bonds), is high.

It is not difficult to see why a barbell portfolio composed of very short and very long bonds should be more risky than a bullet portfolio consisting of low-coupon issues with maturities close to the horizon date. Assume that both portfolios have durations equal to the horizon length, so that both portfolios are immune to parallel rate changes. When interest rates change in an arbitrary nonparallel way, however, the effect on the two portfolios is very different.

Suppose, for instance, that short rates decline and long rates increase. The end-of-horizon values of both portfolios would fall below the target, since both portfolios would experience a capital loss in addition to lower reinvestment rates. The decline, however, would be substantially higher for the barbell portfolio than for the bullet portfolio, for two reasons. First, the barbell portfolio experiences lower reinvestment rates for a longer time interval than the bullet portfolio, so its opportunity loss is much greater. Second, the portion of the barbell portfolio still outstanding at the horizon date is much longer than that of the bullet portfolio, which means that the same rate increase will result in a much steeper capital loss for the former. The low bullet portfolio has less exposure to whatever the change in the interest rate structure may be than the high barbell portfolio.

Note that the risk measure is always nonnegative. It attains its lowest possible value of zero if and only if the portfolio consists of a single discount bond with maturity equal to the length of the horizon. This is indeed the perfectly immunized portfolio; no interest rate change can affect its end-of-horizon value. Any other portfolio is to some extent vulnerable to an adverse interest rate movement. The immunization risk in effect measures how much a given portfolio differs from this ideally immunized portfolio consisting of the single discount bond.

Confidence Intervals

The target value of an immunized portfolio at the horizon date defines the target rate of return over the horizon. If immunization works as assumed, the realized rate of return R will not be below the target value. This will be the case if interest rates change only by parallel shifts for all maturities. To the extent that nonparallel rate changes occur, the realized return may be less than the target return.

Minimizing immunization risk during the horizon (while keeping the portfolio duration equal to the remaining horizon length) will minimize the extent to which the realized return differs from the target return. Unless a portfolio can be constructed with zero , however, the immunized portfolio is subject to some risk. A common way of characterizing the effect of this risk on the investment is by the variance of return (or its square root, the standard deviation).

Eq. (2) can be used in deriving an expression for the standard deviation of the rate of return over the horizon, but not without further work. This equation only characterizes the portfolio's response to a single, instantaneous change in interest rates (an interest rate shock). Over an entire investment horizon, changes in the level and shape of the yield curve can be thought of as a series of interest rate shocks, each of which affects the portfolio's terminal value. The resulting change in value over the whole horizon is then an aggregate of these individual impacts. By Eq. (2), each such impact is proportional to what the portfolio was at the time. To measure the total impact, it is necessary to establish the statistical properties of the variable , the change in the slope of the term structure.

Assuming that the subsequent values of change in the slope of the yield curve are independent random variables with a common variance , the effects of the individual rate shocks can be integrated over the total horizon, subject to a function describing how M² changes with the remaining time to horizon. This results in a formula for the standard deviation of return:

4

where:

	=	the standard deviation of the rate of return over the horizon,
	=	the standard deviation of the change in slope of the term structure (which can be estimated from historical data),
	=	the risk measure for the initial immunized portfolio, and
	=	a constant that depends only on the horizon length.

Note that the standard deviation of return in Eq. (4) is again proportional to M². A portfolio whose M² is half the value of another portfolio's can be expected to produce half the dispersion of realized returns around the target value, when submitted to a variety of interest rate scenarios, than the other portfolio.

The standard deviation of return as given by Eq. (4) can be used in the construction of confidence intervals. A confidence interval represents an uncertainty band around the target return within which the realized return can be expected with a given probability. It can be provided in the form:

5

where k is the critical value corresponding to the given confidence level. The value of k can be obtained from tables of normal distribution. For instance, to construct an interval within which the realized return can be expected with 95 percent probability, the value of k is 1.96.¹

Risk and Return

In a narrow sense, the objective of immunization is risk minimization. Given as a measure of a portfolio's exposure to general interest rate changes, construction of an immunized portfolio then becomes an optimization problem of the following structure:

Minimize the immunization risk subject to

1. immunization condition and
2. investment policy requirements.

The investment policy requirements can include restrictions such as minimum or maximum holdings of individual securities or groups of securities (for instance, issuing sector or quality requirements). It is also possible to include transaction constraints or turnover limits in the optimization.

In some situations, strict risk minimization may be deemed too restrictive. Because not all bonds are priced exactly on the current term structure of interest rates, there are yield differentials within the available universe that may be exploited to enhance the target return. If a substantial increase in the target return can be accomplished with little effect on immunization risk, then the higher yielding portfolio may be preferred in spite of its higher risk.

Consider an optimally immunized portfolio that has a target return of 13 percent over the horizon, with a 95 percent confidence interval of ±0.20 percent. This means that the minimum risk portfolio would have a 1 in 40 chance of realizing a return less than 12.8 percent. Suppose that another portfolio, less well immunized, can produce a target return of 13.3 percent with a 95 percent confidence interval of±0.30 percent. In all but one case out of 40, this portfolio would realize a return above 13 percent, compared with 12.8 percent on the minimum-risk portfolio. For many investors, this may be a preferred tradeoff.

It is possible to set up the optimization problem in such a way that, instead of risk minimization, the risk-return tradeoff is optimized. This can be accomplished by maximizing a lower bound on the realized return corresponding to a given confidence level. Since the confidence interval width in Eq. (5) is proportional to , the objective function is a linear combination of the target return and the risk measure . It could be written in an equivalent form as:

where the value of the coefficient λ depends on the desired confidence level.

This objective function represents a tradeoff between immunization risk and target return. If the parameter λ is small (corresponding to a high confidence level for the lower bound), the emphasis in the construction of the optimal portfolio is on risk. In the extreme case of λ being equal to zero, the objective would be strict risk minimization. On the other hand, if λ is high (such as for low confidence levels), the primary concern of the optimization is maximum return. The other extreme case of λ being equal to infinity would correspond to maximization of return subject only to the requirement that the portfolio be immunized against parallel rate shifts. This would mean selecting the highest return portfolio among all portfolios with durations equal to the horizon length.

By varying the coefficient λ over its range, it is in fact possible to obtain efficient frontiers for immunized portfolios, analogous to those in the mean-variance framework.

References

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2. Bierwag, G.O., G.G. Kaufman, and A. Toevs. (1983). “Immunizing Strategies for Funding Multiple Liabilities.” Journal of Financial and Quantitative Analysis, (March) 18, 113–123.
3. Brennan, M., and E. Schwartz. (1983). “Duration, Bond Pricing and Portfolio Management.” In G.O. Bierwag, G.G. Kaufman, and A. Toevs, eds., Innovations in Bond Portfolio Management: Duration Analysis and Immunization. Greenwich, CT: JAI Press.
4. Cox, J., J.E. Ingersoll, and S.A. Ross. (1979). “Duration and Measurement of Basis Risk.” Journal of Business, January, 52, 51–61.
5. Fisher, L., and R. Weil. (1971). “Coping with the Risk of Interest Rate Fluctuations: Returns to Bondholders from Naive and Optimal Strategies.” Journal of Business, October, 44 (4), 408–431.
6. Fong, H.G., and O. Vasicek. (1980).“A Risk Minimizing Strategy for Multiple Liability Immunization.” Institute for Quantitative Research in Finance.
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