**Michael J. Osborne**- Gulf International Bank, Bahrain
^{(note 1)}

The shortest path between two truths in the real domain passes through the complex domain.

-Jacques Hadamard (1865-1963)

Any student of economics and finance is likely to meet complex numbers. For example, they will encounter them when studying the stability of difference equations used in business cycle analysis (see Turner (1993)).

This article shows how the complex plane can be put to a new purpose - to give a novel and visual interpretation of two important financial measures, duration and the internal rate of return. Only the most cursory of descriptions of these two concepts is given here because good, orthodox descriptions can be found in any of the many texts on finance or financial economics (for example, see Fabozzi (1996) and Cuthbertson (1996)).

Consider a series of numbers, *a _{i}*. Place these numbers in the particular setting of equation [1], ie, embed them in a polynomial. Equation [1] is in the form most often found in books on finance, it is the time value of money equation, while [2] is the same thing recast in the more general form found in maths books.

[1]

where *x* = (1 + *r*) [2]

According to equation [1], we pay 1 out, subsequently receive a stream of returns, the *a _{i}*, and get 1 back alongside the last return. Any stream of numbers can be accommodated in this structure but for the moment we focus on the fixed income bond market. Equations [3] and [3a] show the algebra of the bond pricing formula in the familiar notation of a financial calculator. Equation [3] is in the same format as [1].

[3]

Equation [3] is a rearrangement of [3a] the more familiar version of the bond pricing equation.

[3a]

Setting *PV* = *FV* = 1 returns the equations [3] and [3a] to those of a par bond.

We normally think of a bond as having a stream of equal returns in which all the *a _{i}* are equal but [3] shows that we can reframe the equation in the format of [1] in which the stream of numbers is constant for (

The heart of this article is the more general analysis of a variable stream in which the *a _{i}* can take any values. In particular we examine the way in which the

The internal rate of return is that value of *r* that satisfies equations [3] or [3a] given values for all the other variables. It is a measure of the return per period on the stream of numbers, or the yield on a bond.

Duration is the first derivative of the PV (or price) with respect to *r* in [3a] divided by PV. It shows the interest elasticity of the price of a bond and is therefore a measure of risk. Duration has a number of valuable properties, for example, the duration of a weighted portfolio of assets is equal to the weighted average of their durations. It is useful to know when hedging risk.

Equation [1] is an *n ^{th}* degree polynomial therefore it has

Reference to any maths text on the subject shows that any *n ^{th}* degree polynomial with real coefficients will have

From here on, a four period example is used to illustrate most results. This simple case is chosen because it is particularly easy to visualise the results when there are only four roots. The results are easily generalised to *n* periods with *n* roots. Where it is easy to do so, the *n* period case is given.

The simplest case is where the *a _{i}*, are zero, ie, FV and PV are equal to 1 (there is no capital gain/loss) and all the returns are zero. Then equation [1] reduces to [4].

[4]

The four roots of equation [4] are evenly dispersed around the unit circle in the complex plane (Diagram A shows the relevant Argand diagram). ^{(note 2)}

This result easily generalises to the *n* period case where there are *n* roots evenly dispersed around the unit circle with equal angles (2p/*n* radians or 360/*n* degrees) between them. The general version of [4] with *n* roots is known as the cyclotomic equation [4a]. If there were 100 periods there would be 100 roots distributed around the unit circle with angles of 2p/100 radians, or 3.6 degrees, between them.

*x ^{n}* = 1 where

For the next case assume that the *a _{i}* are put into the equation and further assume, for the moment, that they are equal. The equation is that of the par bond. PV and FV are both equal to 1, and the

This result also generalises very easily. As the value of the *a _{i}*, or the coupon, rises and falls so the real root moves to the right or the left along the real axis by equal amounts. The remaining roots, no matter how many exist, are once again distributed evenly around the unit circle.

Now allow the *a _{i}* to vary in any way. For example, we can imagine the non-par bond case with all coefficients remaining constant except for the last. The last coefficient changes to reflect PV departing from the value of 1 (see equation [3]). When the

At the same time the *n ^{th}* root, the first positive real root, remains at it's customary position on the real axis showing the rate of return on the overall financial flow (the 'middle' of the

Initially it is difficult to discern patterns in the positions of the roots. Professional mathematicians researching into the subject of the roots of polynomials with varying coefficients speak of its capacity to surprise. This is partly because of the difficulty of visualising what is happening in the complex plane and partly because of the complexity of the calculations. For example Farahmand (1998) says that 'random polynomials, so simple and innocent at first sight, are among the most fascinating and mysterious in mathematics. Although some of their unexpected and amusing behaviour has been known for as long as a century, yet they still reveal their secrets so that many of those both closely and indirectly involved with the subject are aroused'.

Fortunately, there exists a theorem in complex analysis that helps. It allows duration and the internal rate of return to be given simple, intuitive, geometric interpretations involving distances in the complex plane.

The interpretation relies on a theorem by Roger Cotes who was a mathematician at Cambridge about 300 years ago (see Nahin (1998) or Needham (1998)). Consider diagram D. Place a point, *z*, on the real axis inside the unit circle. The point is at a distance *g* from the origin. The distances of *z* from the cyclotomic roots are labelled *d ^{i}* where

where the *radius* = *R* = 1 and *g* < *R* [5]

or

where *R* < *g* [5a]

The proof can be found in Nahin (1998) or Needham (1998) and the methodology of the proof is used in the results set out below.

Consider what happens if the point moves to the right along the real axis until it is outside the unit circle - equation [5a]. Furthermore, reinterpret the meaning of the point as the first positive real root, with the result that the distance *g* becomes (1 + *r*). The picture changes to that in diagram E and we can write equation [5a] as [5b].

*d*_{1}.*d*_{2}.*d*_{3}.*r* = (1 + *r*)^{4} - 1 where *r* = *d*_{4} [5b]

Equation [5b] can be further modified to [5c]. It shows that the product of the distances between the first positive real root and the other (*n* - 1) roots in the complex plane, when divided by (1 + *r*)^{n} , gives a result which is immediately recognisable. It is the sum of the discount factors. For the first time we can see a familiar financial measure in the context of the complex plane.

[5c]

But there is much more to it than this. The situation depicted in diagram E is equivalent to a par bond because the (*n* - 1) roots are the cyclotomic roots and the single real root represents the coupon (the *a _{i}* in equation [1] are equal). In this special context it can be shown that the sum of the discount factors is equal to the negative of modified duration [6].

where *D* is equal to modified duration. [6]

This immediately gives rise to a conjecture. Is the formula in the left hand side of [6] also equal to modified duration when the *a _{i}* are allowed to vary? For example, is it still true in the case of non-par bonds when the last coefficient is different from the rest, or, indeed, when we consider any stream of numbers in which each of the

Write down equation [1] *n* times in *n* rows to form a matrix, reverse the sign of each element, divide throughout by (1 + *r*) and call it equation [7]. In our particular example there are four rows.

[7]

The matrix is the null matrix because each row sums to zero. Partition the matrix diagonally to include the whole of the first column, the top three elements of the second column, the top two elements of the third column and the top element of the fourth column. The partition to the top left we call P_{1}. The partition to the lower right we call P_{2}. The elements of the matrix sum to zero, therefore P_{1}= - P_{2}.

Consider the two partitions in turn. First P_{2} because it is easier and it has the more familiar interpretation. Write down the elements of P_{2} in the form of an equation, slightly rearrange the final term, and it is recognisable as modified duration [8]. This could be for a par or a non-par bond, because the *a _{i}* have not been specified, they can take any values.

[8]

We know that P_{1}= - P_{2}, therefore P_{1} must also be equal to (the negative of) modified duration. The extra meaning of the partition P_{1} is not so obvious. We have to show it is equal to the left hand side of equation [6] - the product of the (*n* - 1) distances between the (*n* - 1) roots and the real *n ^{th}* root, divided by (1 +

We follow the line of reasoning set out in Nahin (1998). Take the universal bond pricing equation [1], reverse all the signs, multiply throughout by (1 + *r*)^{4} and set (1 + *r*) = *z*. This gives equation [9]. The notation '*z*' is used because of its universal interpretation as any complex number, and here we are recognising that the various values of the root (1 + *r*) = *z* can be any number, complex as well as real.

*z*^{4} - *a*_{1}*z*^{3} - *a*_{2}*x*^{2} - *a*_{3}*z* - (*a*_{4} + 1) = 0 [9]

We know that the equation [9] can be factorised to give equation [10].

*z*^{4} - *a*_{1}*z*^{3} - *a*_{2}*z*^{2} - *a*_{3}*z* - (*a*_{4} + 1) = 0 = (*z* - *z*_{1})(*z* - *z*_{2})(*z* - *z*_{3})(*z* - *z*_{4}) [10]

The value *z* is the point *z* in diagram D. It lies wherever we choose to set it. The *z _{i}* are the roots. The absolute value of the product on the right hand side of [10] is equal to the product of the absolute values, ie,

[11]

The absolute value of differences between complex numbers like (*z* - *z _{i}*) is the distance between the two points

Evaluating the last product is straightforward. The element (*z* - *z*_{4}) has to be eliminated from equation [10]. To do so we divide [10] throughout by (*z* - *z*_{4}) to give [12].

[12]

Only after this division do we set *z* equal to *z*_{4} by replacing *z*_{4} by (1 + *r*) in [12], and also replace *z* by (1 + *r*). Divide the result throughout by (1 + *r*)^{4} and find the absolute value of both sides. This gives the expression [13].

[13]

This is the required proof, because the left hand side of [13] is the partition P_{1} of the matrix, which is already known to be the negative of modified duration.

Equation [6] is now known to be true no matter what values the *a _{i}* take. This can be generalised to the

[6a]

where the *d _{i}* are the (

We have stated, but not proved, that the value *r* - the IRR - a component of the first positive real root, (1 + *r*), is a simple kind of average of the stream of numbers, *a _{i}*.

Simple is not the usual adjective applied to the IRR because it is usually seen through the complicated algebra of the time value of money equation. Solving for *r* in equation [1] when *n* = 1 or 2 is easy. Solving for *r* when *n* = 3 or 4 is tedious. When *n* > 4 it is impossible to solve analytically. It has to be done using numerical methods - sophisticated guessing techniques; hence the need for a first guess in most spreadsheets and calculators, and the message 'running - running - running' when finding the IRR in the professional analyst's principal tool, the HP12C financial calculator.

However, in the complex plane it is easy to see what it is, even if its calculation remains difficult. To see its simplicity, look at diagram F. The three rods linking the four roots in diagram E have been replaced by four rods linking all four roots to another point in the plane, the point (1,0) on the real axis. This is the same thing as setting *z* equal to +1 in equation [10] (instead of setting it equal to *z*_{4} as we did in the previous section). The result is equation [14] and diagram F.

[14]

On the left hand side of [14] we have equation [9] with *z* = 1. On the right hand side we have its factorisation with *z* = 1. Taking absolute values on both sides means that the sum of the coefficients, the *a _{i}*, is equal to the product of the lengths of the rods (the distances,

[15]

where the *w _{i}* are the (

The results can be summarised for the four period model with variable flow, *a _{i}* . The four roots all lie on or around the unit circle. The real root, (1 +

The observations concerning the IRR and duration have been proved true no matter what the coefficients of the polynomial. Thus, duration for a bond, par or non-par, is merely a special case of a more general model. It is the special case when *n* coefficients in the polynomial are the same (par bond), or (*n* - 1) of them are the same and the *n ^{th}* is different (non-par bond). In the general case, the new equation for duration holds true no matter what values the

Viewing the time value of money equation through the lens of its roots in the complex plane, yields novel and elegant views of two important financial concepts.

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2) Any program like Mathcad, Mathematica, Maple and Matlab can be used to manipulate the algebra and graph the results. The calculations used to generate the ideas in this article were made using Mathcad. The charts in this article have been produced with the aid of a drawing program to exaggerate and emphasise the interesting bits.

3) For readers without the appropriate software to visualise the translation of the coefficients of a polynomial into roots. but who do have access to the internet, visit `http://www.cecm.sfu.ca/organics/papers/odlyzko/support/polyform.html` . This site contains an online calculator and graphing utility that aloows the viewer to see the roots of a polynomial in the complex plane. The limitation is that the coefficients have to take the values 0 or 1, however a 'feel' for the process can be obtained from the site.

4) It is the negative of modified duration because duration is itself a negative number. The right hand side of equation [6] must be positive to agree with the left hand side. The LHS is always positive because the top line ocnsistes of absolute values in the complex plane and the bottom line is obviously positive in most conceivable circumstances. Modified duration is so called because it is a modified version of the original concept developed by Macauley (1938).