## Abstract

My first goal is to present the basic immunization problem (BIP) as it is understood in finance. BIP relies on a construction of such a bond portfolio (BP), meaning a selection of individual bonds, that the single liability to pay L dollars q years from now will be discharged by means of BP (a patient will return to health at time q), no matter what random shift a(t) (a particular disease) will occur in the future. What kind of a function is a shift of interest rates is critically important because both present and future values of BP depend solely on underlying interest rates. Having identified shifts (diseases) against which a BP is immunized, the natural question arises how to find among such immunized (immune) portfolios the best ones. In the context of finance, it means bond portfolios with maximal unanticipated rate of return. My second goal is to trigger interest among medical scientists by suggesting that certain finance notions, such as duration and convexity of a bond portfolio, might give extra insight to medical researchers working in the immunization area both into BIP and into similar problems in medicine. A considerable attention is also paid to certain mathematical notions (base of a linear space, a Hilbert space, triangular functions) because of their successful applications to problem-solving occurring in bond portfolio immunization.

### Keywords

- immunization
- immunity
- active immunity
- passive immunity
- best immunization strategy
- duration
- convexity
- barbell immunization strategy
- focused immunization strategy

## 1. Introduction

In this chapter, I present one of the research areas existing in finance called bond portfolio immunization (BPI). My goal is to make it known to medical researchers dealing with immunity (resistance) of human organisms to diseases. I feature not only basic notions, problems, and solutions occurring in BPI, but also selected mathematical concepts and tools which proved to be instrumental in developing BPI. I do believe that such information has a good chance to be useful in creation of immunity against particular diseases. Bond investors are called immunizers if, possessing *C* dollars today, they must achieve an investment goal of *L* dollars *q* years from now (a human organism or a particular human organ must achieve a certain level of health *q* years from now); here *L* is the future value of *C* at time *q* under the current interest rates. This investment goal must be accomplished by means of an appropriately selected bond portfolio, even despite unfavorable sudden change (shift) in interest rates (appearance of a disease), having in mind that the present and future prices of all bonds depend solely on interest rates.

Although, as it will be demonstrated in Sections 3.1, 3.2, and 3.3, immunization against all shifts is never possible, there are many results giving sufficient, or necessary and sufficient, conditions for immunization against a certain classes of shifts (certain diseases). It is worth to know that in the financial immunization, there is no such thing as *acquired* immunity (immunity that develops in a human after exposure to a suitable agent) and *active* immunity (acquired through production of antibodies within the organism in response to the presence of antigens).

Such types of immunization might theoretically take place on a bond market only if a bond holder had the right to change the coupon payments, which is completely out of the question. In other words, the immunization in financial reality has features of *passive* immunity, being in fact a short-acting immunity. On the other hand, however, a BP manager can achieve the state of a BP being all the time immune against a specific class of shifts, provided the manager regularly (every week or so) performs (if necessary) subsequent adjustments of his/her BP according to their expertise in the area of immunization theory.

Theorem 1 (Section 2.4.2) as well as Theorems 3 and 4 allows one to look at immunization from a different perspective. They enable one to identify all shifts a(t) (diseases) of the term structure s(t) of interest rates against which BP is already (fully) immunized, that is, protected against loss of its value at time *q*. Finally, having identified immunized (immune) bond portfolios, the natural question arises how to find among them the best ones. This topic is dealt with in Section 4.

Below, I shall present (i) what are bonds and bond portfolios; (ii) what is meant by standard (and general) immunization problem; (iii) historical development of immunization theory; (iv) overview of some recent results; (v) the concept of a Hilbert space, and a base in a linear space; (vi) application of orthogonal polynomials to description of the class IMMU of all shifts (diseases) against which a given bond portfolio is immunized; (vii) triangular functions as a base for the linear space IMMU; and (viii) the crucial role of the notions of duration and convexity in choosing the “best” immunized (immune) bond portfolios.

## 2. Immunization in finance

Below, we will introduce the concept of bonds, formulate the standard and general immunization problem, and outline the development of immunization theory in finance, from the beginning to the latest achievements.

### 2.1. What are bonds?

Each bond with a face value (par value) of F dollars is a financial instrument which generates to its buyer (holder) specified payments every 6 months (or every quarter, every year) in the form of coupons, plus par value paid only at the termination (maturity) of the bond. The face value represents the amount borrowed by the seller (issuer) of a bond from the bond buyer. The coupons represent a predetermined percentage, say 3%, of face value F; if F = $10,000, then all coupons paid per year sum up to $300. Each bond has its own life span (maturity) of *n* years (3, 5, 10, 20 years, etc.)

A bond portfolio (BP), by its definition, is a collection of different bonds with various maturities. Thus, each BP generates a more complicated cash flow pattern than a single bond does. A cash flow generated by a BP consists of various size payments

The present and future value of each bond, and consequently each bond portfolio, depends solely on current interest rates *t*, that is,

### 2.2. Standard and general immunization problem

The standard immunization problem relies on a construction of such a bond portfolio with the present value of *C* dollars that the single liability to pay *L* dollars *q* years from now (L is the future value of *C*) by means of the cash flow generated by BP will be secured regardless of how adverse changes in interest rates will occur in a future. This nontrivial problem is automatically solved by each zero-coupon bond maturing at time *q* with par value of *L* dollars. Thus, using medical terminology, one may say that such a zero-coupon bond possesses an innate (natural) immunity. Unfortunately, in practice, such zero-coupon bonds rarely exist.

Besides, an investor may already possess bonds and would like to buy additional ones so that the created, in this way, portfolio BP with the present value of *C* dollars would secure the payment of *L* dollars *q* years from now. Having built such a portfolio, the investor would immunize (hedge) their own investment against a loss of its value at time *q*. We assume that the new term structure will always be of the form s*(t) = s(t) + a(t), where a(t) belongs to a certain class of shifts (diseases).

On the other hand, the general immunization problem relies on a construction of such a bond portfolio BP with the present value of *C* dollars that multiply liabilities to pay

### 2.3. Beginnings of immunization

Immunization as a concept dates back as far as to articles [1, 2]. However, not until work [3] of Fisher and Weil was the impact of interest shifts on the design of immunization strategies rigorously studied. In a vast majority of publications, immunization was based on a specific stochastic process governing interest rate shifts a(t). In [2], Redington discussed immunization in the context of an actuarial company which had projected liability outflows *L*(t) at some finite number *M* of instances (dates) *N* (typically) different dates

In such a situation, the company’s task was to choose inflows *A*(t) in such a manner that the outflows *L*(t) would be discharged if the interest rates *A*(t) occurring at instances *L*(t). Redington introduced the notion of a “mean term” having in mind the weighted average of the dates when the flows are to be received (in case of assets) or have to be discharged (in case of liabilities). This “mean term” was nothing different than the concept of *duration* introduced by Macaulay in [1]. These two authors understood duration as:

where *A*(t) in terms of today’s money. It was proved in [2] that any *parallel* movement (shift) of the *flat* term structure *A*(*t*) were equal to duration *L*(*t*), and additionally, the so-called *convexity* of the assets would exceed that of the liabilities.

### 2.4. Assumptions concerning term structure of interest rates and admissible shifts: historical development

Twenty years later, Fisher and Weil [3] restricted themselves to a single liability at a specified date *q*, but significantly weakened the adopted so far assumption that the term structure was flat, that is,

They applied (popular already at that time) continuous compounding of cash flows *A*(t) and *L*(t), which in their approach represented instantaneous rate of payments per one unit of time rather than payments themselves, so that the present value of the assets

while the duration

It was stated in [3] that immunization was secured if the duration of the assets *q*: *q* equals *q* since all weights, except for the one at date *q*, are equal to 0 (zero).

#### 2.4.1. Further developments of immunization theory

In subsequent 20–30 years of development of immunization theory, the strong assumption made so far that interest rates were subject to random shifts of the form

Few years later, it was demonstrated in [5] that these differences may be significant. For example, when a multiplicative stochastic process

and the additive one

were studied in [7], where suitable implicit formulas for the respective immunizing durations were derived.

Another approach, called contingent immunization, was developed in [8]. It consists of building a bond portfolio with a duration shorter or longer than the investor’s planning horizon, taking into account “personal” expectations of a bond manger with regard interest rates. The idea standing behind such approach is to take advantage of the manager’s ability to forecast interest rate movements (diseases) as long as his/her predictions are accurate.

#### 2.4.2. Latest developments of immunization theory

The contingent immunization was implemented in many situations for various term structures of interest rates; see [9]. Other than mentioned in Eqs. (4) and (5), stochastic process governing admissible shifts was analyzed in [10].

Striving to offer a more general approach, the authors of article [11] did not confine ourselves to a specific process governing shifts, but allowed them to belong to a certain class of functions, such as, for example, polynomials of degree less than some specified number *n* (pp. 858–861). In this way, they did not expose themselves to any model misspecification risk, similarly as Zheng in [12].

The larger class of shifts (diseases) against which the immunization will work, the better. Having this in mind, the interval *n* equal nonoverlapping subintervals

with

**Theorem 1.** If *q* denotes the date when the single liability of L dollars has to be discharged by means of the cumulative value of assets *A(t)*, then the immunization is secured against all adverse piecewise constant shifts *h(0,t)* if and only if

where *A(t)*.

**Remark 1.** When *n* = 1, then Theorem 1 gives a sufficient and necessary condition

for immunization when the term structure *h(0,t)* is subject to shifts

**Remark 2.** Theorem 1 can also be looked at from a different perspective. Namely, one may be interested in identification of such a set of shifts a(t) of the term structure s(t) of interest rates, say IMMU, against which a bond portfolio BP is already immunized, that is, protected against loss of its value at time *q*. In this context, Theorem 1 offers a sufficient and necessary condition for a shift a(t) to belong to set IMMU.

### 2.5. One cannot immunize against all possible shifts of interest rates development

As of today, no one was successful in building up a bond portfolio BP immunized (immune) against all shifts of interest rates (diseases). What is more, in Section 3, we demonstrate that the set IMMU is always a proper subset of all admissible shifts, being in fact a linear subspace of all shifts.

## 3. Overview of some recent results

In a recent paper [13], the authors found a strong evidence that momentum across various asset classes is caused by macroeconomic variables. By properly modifying their portfolio, in response to changes in macroeconomic environment, their strategy performed particularly well in times of economic distress. The obtained results allowed them to establish a link between momentum and sophisticated predictive regressions.

Aiming at securing higher effectiveness of their investment in fixed income bonds, the authors of [14] successfully used simulations of the portfolio surplus, measuring the inherent risk by means of the value-at-risk methodology. In another very recent publication [15], the authors studied immunization assuming that shifts were parallel or symmetric. A quite different approach to immunization was proposed in [16]. The authors concentrated on hedging risk inherent in bond portfolio. They divided the entire problem into two parts, by formulating a two-step optimization problem. They focused first on immunization risk, and next maximized the portfolio wealth.

In this section, it is proved that the set of all continuous shocks *a(t)* against which a bond portfolio BP is immunized is an *m*-dimensional linear subspace in the *(m + 1)*-dimensional linear space of all continuous shifts a(t), with *m* standing for the number of instances when BP promises to pay cash (coupons or par values generated by bonds forming BP). The main mathematical concept used below is the notion of a Hilbert space and the concept of a base in a Hilbert space.

### 3.1. When polynomials are admissible shifts

From now on, we assume that

generate only payments *A(t)* is no longer given by Eq. (2), but by

One of two classes of admissible shifts studied in [14] was the class of polynomials.

The new term structure was assumed to be of the form:

with

**Definition 1.** (see also [11], p. 859). A set S is said to be a linear space if the sum of its arbitrary 2 elements *r* the product of *r* and any element

The most well-known linear spaces are probably the set of all real numbers R, a two-dimensional Cartesian plane *n*-dimensional linear spaces.

**Definition 2.** A set of *k* vectors

**Definition 3.** A set of linearly independent vectors from a linear space S is called a base for S if each vector

All bases have the same size and there are many of them in each linear space S. *r*. The most popular base in *n* vectors:

**Remark 3.** The below Formula (13), being a counterpart of Formula 8, gives a necessary and sufficient condition for immunization against shifts a(t) in case when a bond portfolio BP generates payments

with weights

**Remark 4**. The class of polynomials of the form (11), with fixed *n*, is a linear space. What is more, the subset of these polynomials satisfying Eq. (13) is a linear space, too.

**Proof**. Assume that a*(t)* satisfies Eq. (13). For any real number *r*, Eq. (13) implies

**Theorem 2.** (see [11], Theorem 2.1). Let *q* denote the date when the single liability of L dollars has to be discharged by means of the cumulative value of assets (9) despite additive adverse shifts (11) (*n* is fixed) of interest rates s(t) so that the new interest rates will be of the form (12). Then, the subclass of shifts (11) for which immunization is secured is a *(n−1)*-dimensional linear space, denote it by IMMU, of the space of all polynomials (11), which itself has dimension *n*.

How to determine IMMU is demonstrated in [11] in pp. 858–860.

### 3.2. Continuous functions are admissible shifts: a Hilbert space approach

The other class of admissible shifts studied in [11] was the class of all continuous functions (CF) defined as always on interval *A(t)* are given by Formula 9. It is easy to notice that the class of CF is a linear space with ordinary addition of two functions and ordinary multiplication of a function by a real number. However, it has an infinite number of independent vectors!

We shall demonstrate that the notion of a Hilbert space is very useful in the study of immunization theory. It was named after a German mathematician David Hilbert (1862–1943) who is recognized as one of the most influential and universal mathematicians of the nineteenth century and the first half of twentieth century. By definition, a Hilbert space is a linear space, say H, which is additionally equipped with so-called scalar product (a generalization of the scalar product of two vectors from

A specific Hilbert space *H** of all CF (shifts) defined on interval *H** had dimension *m*. However, the shifts of interest rates should be considered on interval *H** would be *(m + 1)*, which is really the case. So, in this chapter, we correct and simplify the definition of a scalar product of two arbitrary continuous functions (shifts) f(t) and g(t), by letting

It is good to know that in each Hilbert space H, one can measure a distance between any two elements *h*. In space *H**, the norm is therefore defined as follows:

Clearly, *h(t)* belonging to *H** is treated as an element (vector) 0 (zero) if and only if

In Theorem 3 to follow, we identify a base for *H** among polynomials. This approach is rather complicated since it involves the use of Gram-Schmidt orthogonalization procedure to determine base polynomials. In Section 3.3, a far more straightforward and easier to implement approach is presented where there is no need to identify base functions (shifts) because they are already given by Formulas (20)–(23).

**Theorem 3.** (compare Theorem 3.1 in [11]). Suppose a bond portfolio BP has been bought, and admissible shifts a(t) of a term structure s(t) are allowed to be continuous functions on interval *(m + 1)*-dimensional Hilbert space *H**, where *m* is the number of instances when portfolio BP generates cash. The subset of these shifts, say IMMU, against which a holder of BP is immune (will be able to discharge the liability of L dollars to be paid at time *m*-dimensional subspace (depending to a large extent on BP) of the form

where the *m + 1* polynomials *H**. This base may be determined by the Gram-Schmidt orthogonalization procedure, while the coefficients

It is worth to notice that after determination of polynomials

### 3.3. Identification of continuous shifts against which a bond portfolio is immunized: the triangular functions approach

A strict definition of triangular functions is given by Eqs. (20)–(23) below. Roughly speaking, a triangular function (sometimes called a tent function, or a hat function) is a function whose graph takes the shape of a triangle. Among our *(m + 1)* tent functions employed in this chapter, *(m* − *1)* are isosceles triangles with height 1 and base 2, while the other two are perpendicular triangles with height 1 and base 1. Triangular functions have been successfully employed in signal processing as representations of idealized signals from which more realistic signals can be derived, for example, in kernel density estimation.

They also have applications in pulse code modulation as a pulse shape for transmitting digital signals, and as a matched filter for receiving the signals. Triangular functions are used to define the so-called triangular window, also known as the Bartlett window. Since they occur in the formula for Lagrange polynomials used in numerical analysis for polynomial interpolation, they are also called Lagrange functions. Their other applications include the Newton-Cotes method of numerical integration, and Shamir’s secret sharing scheme in cryptography.

In the financial context, tent functions were employed in [17] for modeling shifts of the term structure of interest rates. The framework and assumptions made in this section are the same as in Section 3.2. Our purpose is to characterize the subspace IMMU of the Hilbert space *H** by means of triangular functions based on results presented in [18].

In this section, *q* when the liability to pay L dollars has to be discharged. Below, we define *m + 1* triangular functions

The following result is well known.

**Remark 5.** Each continuous function a(t) defined on *(m + 1)* triangular functions) at all points *H** with the piecewise linear function b(t) because the distance between a(t) and b(t) in *H** is zero: ||b(t) – a(t)|| = 0.

It is a nice exercise to prove the following result.

**Remark 6.** The Lagrange functions *H** of all admissible (continuous) shifts defined on

**Theorem 4.** The set IMMU of all shifts (continuous functions) against which a bond portfolio BP with payouts represented by (9) and the new term structure given by (12) is immunized constitutes an *m*-dimensional linear subspace in the *(m + 1)*-dimensional Hilbert space *H**.

Two examples illustrating how to identify IMMU are worked out in detail in [18], pp. 531–537. A special attention is given to continuity properties of subspace IMMU; see [18], pp. 534–537.

## 4. Maximizing the unanticipated rate of return among immunized bond portfolios

The natural question arises of how to select the “best” portfolios among those which are (have been) protected (immunized) against admissible shifts (movements) of interest rates? In finance, by best portfolios are meant those which yield the highest rate of return (the highest increase in the present value of a BP), resulting from a sudden shift of interest rates. Below, we present the results obtained in [19]. Rewriting a sufficient and necessary condition (13) and (14) for immunization of portfolio BP generating payouts (9), one obtains:

and

For each vector *a(t)* for which (25) holds was defined in [19]. When vector

**Theorem 5.** The immunization of a bond portfolio BP against shifts a(t) from class

With s(t) standing for the current interest rates,

### 4.1. Convexity of a bond portfolio

Set *BP*; for more details, see [19], p. 105. It is easy to notice that convexity of a zero-coupon bearing bond maturing at *a(t)* of interest rates

where lim O(a) = 0 when

**Assumption 1.** All zero-coupon bearing bonds

**Definition 4.** Following [20], p. 552, a bond portfolio *BP* is said to be a barbell strategy (barbell portfolio) if it is built up of two bonds, say *BP* is said to be a focused strategy (focused portfolio) if it consists of several bonds whose dedicated durations *q* in our context).

**Theorem 6.** (see [19], Theorem 1). If Assumption 1 holds then the bond portfolio BP*** with the highest unanticipated rate of return is a barbell strategy built up of zero-coupon bearing bonds

**Comment 1.** Suppose that instead of dedicated duration and dedicated convexity, we employ the classic notions of duration and convexity derived for additive shifts only. Then,

Finally, another interesting question arises, to what extend does the dedicated duration of the best immunized portfolio BP*** differ from its Macaulay’s counterpart? That is, what is the difference between

For a specific situation, when shifts a(t) of interest rates s(t) satisfied the “proportionality” condition

## 5. Concluding remarks

Let us summarize what we have said so far. Each bond portfolio BP (a human body? or a human body organ?) generates cash at various dates

In bond portfolio theory, the greater payouts generated by BP, the higher is the present value (PV) and future value (FV) of BP. An analogous statement is therefore expected in the medical context. Having settled what is Z, it would be probably easy to find out what is the counterpart in medicine of the duration concept defined for the first time by Macaulay (in 1938) and independently by Redington (in 1952); see Formula (1).

Let us formulate the following hypothesis: the higher values (levels) of Z, the more healthy is a human body (a human body organ).

In the financial immunization context, there is a fixed date *q* when BP must attain at least a certain value *L*, called liability. In the medical context, one might say that there is a fixed date *q* when the quality of human health must attain at least a certain level L.

In the financial theory context, when interest rates s(t) change due to a shift a(t), that is, *q* may fall below *L* dollars. In the medical context, the appearance of disease may cause a deterioration of health at date *q*.

We still do not know what should (could) be substituted for interest rates s(t), knowing that changes (movements, shifts) in interest rates mean a disease.

Using the concept of *duration* (and dedicated duration), we identified the set IMMU of all shifts (diseases) a(t) against which BP is immunized. By means of notion of duration and *convexity* (dedicated convexity), we determined the best immunizing portfolios for a large class of shifts (continuous functions). In the financial context, the best portfolios meant portfolios generating the highest (unanticipated) rate of return. In the medical context, the best would probably mean the fastest rate of health improvement.