Optimal Investment Strategy via Internal Arithmeic

8/7/2019 Optimal Investment Strategy via Internal Arithmeic

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International Journal of Theoretical and Applied FinanceVol. 8, No. 2 (2005) 185206c World Scientific Publishing Company

OPTIMAL INVESTMENT STRATEGY VIA INTERVAL

ARITHMETIC

BENITO STRADI and EMMANUEL HAVEN

Department of AFM and Essex Finance Center (EFiC)University of Essex, Wivenhoe Park

Colchester C04 3SQ, [email protected]

Received 9 October 2004Accepted 22 June 2004

This paper studies the optimal replacement policy of an item that experiences stochasticgeometric growth in maintenance costs. The model integrates corporate taxes, tax cred-its, depreciation, and salvage value. We extend this traditional application to cover the

cost of replacement with the payout from two bonds. The two-bond portfolio is passivelyimmunized. The intersections between the continuation and replacement boundariesare computed using the Interval-Newton Generalized-Bisection (IN/GB) method. Weallow small fluctuations of the replacement boundary. With these fluctuations, multipleintersections of the two boundaries are determined. The IN/GB method finds all theseintersections without the need for initial guesses of the problem variables. This is amajor computational improvement over traditional single-root finding implementationsthat require multiple initial guesses and provide no guarantees of existence or unique-ness. We demonstrate that without fluctuations one would expect to find a single optimalreplacement time. However with fluctuations, there are several intersections of the con-tinuation and replacement boundaries and the bond weight fractions may change by

more than 200% between intersection points. These large changes in portfolio wealthallocation highlight the fragility of the idealized solution in the realm of fluctuations inreplacement costs.

Keywords: Interval arithmetric; optimal replacement; replacement boundary.

1. Introduction

Investing under uncertainty, in an ideal setting, comprises investment actions that

maximize the value of a company in a competitive market. The decision to invest

under uncertainty means forfeiting the use of any additional information that mayarrive in the near future. In this case, there is an opportunity of investing now as

compared to investing in the future. The best timing to implement a protective

policy is decided by comparing the value of the benefits obtained by immedi-

ately implementing the policy with that of the losses from waiting. The opti-

mal investment time is critical especially in projects with considerable sunk costs

185


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186 B. Stradi & E. Haven

(i.e., hydroelectric plants) or in those that may undergo irreversible damage (i.e.,

clear-cut of a forest).There is a large body of research on making decisions under uncertainty

with applications in natural resource management, durable-goods investments, and

equipment replacements.

There are studies that expand on the theory and applications of the stochastic

dynamics of decisions under uncertainty. Dixit and Pindyck [5] study the opti-

mal stopping times for multiple applications described by stochastic processes.

Constantinides [2] develops a two-step procedure to value an asset in the pres-

ence of market risk. In the first step, the capital asset pricing model (CAPM) is

used to describe the assets rate of return. In the second, the risk-free rate is usedto compute the present value of the asset. Mauer and Triantis [18] look at how

the optimal stopping decisions interact with the capital structure of the company.

Pindyck [24] studies the problem where there are both pure technical and input

cost uncertainties, and Ingersoll and Ross [7] look at the effect of varying interest

rates upon the decision to invest.

Parallel to the interest in modeling of investment decisions under uncertainty,

there is the challenge to develop novel computational tools. These tools deal with the

inherent non-linearity of the optimal investment problem. A recent development in

applied mathematics is the use of interval arithmetic. Alefeld and Herzberger [1] andNeumaier [22] discuss, with different levels of mathematical detail, the fundamental

properties of interval mathematics. Tucker [27], Nedialkov, Jackson and Corliss [21]

and Wolfe [30] discuss applications of interval mathematics to more advanced prob-

lems concerning the solution of ordinary differential equations and optimization.

In the past, it was not considered feasible to use interval mathematics in elabo-

rate applications because of high computational needs. However, with the increase

in computing power of workstations that limitation is now manageable. An exam-

ple of interval mathematics in finance is the study by Venkataramanan, Cabot and

Winston [28]. They successfully applied interval mathematics to the determinationof the optimal-pricing multiperiod policy for a reproducible piece of software in the

presence of multiple clients who may buy or illegally copy the software. They looked

at 28 scenarios with up to seven pricing periods with computational times between

0.09 to 12.49 minutes.

In this paper we study the optimal replacement time for a piece of equipment

with an increasing stochastic maintenance cost. We assume that the costs of main-

tenance follow a geometric Brownian motion. The resulting stochastic differential

equation model of maintenance costs includes the effect of corporate taxes, tax

credits, and depreciation. Our objective is to minimize the expected present valueof maintenance costs after taxes and depreciation. We introduce a basket of two

bonds that allows paying for the future investment in the new equipment at the

optimal stopping time. These are bonds with different maturities and coupon rates.

The basket is passively immunized. After integrating the stochastic differential

equation, the model is specified by a system of non-linear equations.


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Optimal Investment Strategy Via Interval Arithmetic 187

Traditional solution methods for non-linear systems of equations require ini-

tial guesses for convergence. Instead, we introduce a different solution strategy viainterval arithmetic. Two good basic references for interval mathematics are those by

Kearfott [13] and Moore [20]. A more advanced discussion is that by Neumaier [22].

We implement the Interval-Newton Generalized-Bisection Method [13, 26]. The

inputs to the procedure are the value ranges of the variables (the cost of main-

tenance, the bond weight fractions, and the two integration constants) and the

values of model parameters (corporate tax rate, tax credit rate, depreciation rate,

risk-free rate, and correlation coefficient for the undiversifiable risk). We note that

there is no need for initial guesses of the variable values; only the ranges of their

values are needed. The interval implementation provides a guarantee of uniquenessand existence. This means that all solutions within the specified domain are located,

and if no solution is found, there is a mathematical guarantee that no solution exists

in that domain [13]. This is an important point because most of the other mathe-

matical techniques applicable to non-linear systems of equations do not provide this

guarantee. The only exceptions are those cases where the jacobian matrix is singu-

lar at the solution, but in those cases the algorithm results in a thin interval that

can be analyzed using the method discussed by Kearfott, Dian and Neumaier [14].

The solution to the optimal stopping problem gives the present value of optimal

expected maintenance cost, the value of the two integration constants, the weightfractions of the bonds in the basket, and the average replacement period. We conju-

gate a quantifiable investment model with the application of robust mathematical

techniques in which we make use of interval mathematics.

The paper is organized in the following sections: model development, interval

arithmetic, the Interval-Newton Generalized-Bisection method, modeling results

and discussion, and conclusion.

This is a novel application in that it solves the option decision and portfolio

allocation problems concurrently and deals with numerical difficulties using robust

mathematical techniques.

2. Model Development

2.1. Stochastic behavior of the optimal replacement policy

The objective of the model is to predict the optimal replacement policy, which

consist of the time and cost that are optimal to replace an operating unit. The

model includes the effects of taxes, depreciation, and salvage value. Other effects

such as recall costs and alternative levels of performance of the operating unit are

not considered.The approach consists of both maximizing the discounted, after-tax, present

value of the maintenance cost of the device and covering the cost of replacement

by using a portfolio with two bonds. We note that the portfolio allows paying for

the future investment in the new equipment at the optimal stopping time. Notice

that if instead of two bonds we used two bond indices (e.g., Goldman Sachs Euro


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High Yield, Merril Lynch UK high yield) by holding all the bonds that generate

the index, the problem would be similar only that the parameters would be theaverages of the bonds constituting the index. The use of two bonds simply reduces

the discussion to a manageable size.

The maintenance cost before taxes of the asset, C, follows a geometric Brownian

motion:

dC = Cdt + CdW, (2.1)

in which dW is a standard Wiener process, is the appreciation rate of cost, and

is a diffusion constant.

Mauer and Ott [17] have shown that the optimal stopping problem reduces to

the following differential equation (Bellman equation):

1

22C2

d2V(C)

dC2+ C

dV(C)

dC+ C(1 ) P(1 )

C

CN

Z

= rV(C), (2.2)

in which P is the price paid for the unit, [0, 1] is the investment tax credit,

P(1 ) is the net purchase price of the asset, is the exponential depreciation

rate coefficient, r, is the risk-free rate, is the corporate tax, and Z is defined as

122. CN > 0 is the operation cost and finally = . We remark thatV(C) is the value function. It is somewhat lenghty to explain how one arrives to

Eq. (2.2). The Bellman equation, in its original form, contains a term E(dV). Using

Itos Lemma on dV and using the assumption that the maintenance cost before

taxes follows a geometric Brownian motion, we arrive at Eq. (2.2).

The unsystematic risk has been diversified away, only the systematic risk remains

in terms of a risk-premium, . This is written in terms of the continuous capital

asset pricing model, CCAPM [19] as , where is the market price of risk,

and is the correlation coefficient between cost and the systematic pricing factor.

The term P(1)

CCN

Z

in the above differential equation is obtained through

the following steps. Mauer and Ott [17] model the so called expected first pas-

sage time, E(t) from CN to Ct as an approximation for t. They obtain that

E(t) = Z1 ln

CtCN

. The tax book value P(1 )et is now approximated

as P(1 )e ln

CtCN

(1/22)1and which can be written in shorter form as:

P(1 )

CtCN

Z

. As Mauer and Ott [17] argue the depreciation tax shield of

the asset over time [t, t + dt] is then P(1 ) CtCN Z

.The complete solution is the sum of the homogeneous and particular solutions:

(remark that part of Eq. (2.2) can be regarded as an Euler equation):

V(C) = K1C1 + K2C

2 +C

+

1

r C, (2.3)


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where K1C1 + K2C2 +C

+ 1

rC is the continuation boundary. We note

that: 1 is equal to12

2

+

12

2

2

+ 2r2

12

and 2 is equal to12

2

12

2

2+ 2r

2

12

and we also note that:

C

=

P(1 )

CCN

Z

r Z

12

2Z

Z

1 .

= r

The new variables , , and

= Z

are introduced for notational economy.

At the boundary between continuation and stopping regions, the value of replac-

ing the old asset must be the same as that for commissioning the new one. The first

derivatives on both sides of the boundary are required to have the same value. The

initial slope of the valuation function is zero.

Condition 1. The continuation boundary value, V(C) given in Eq. (2.3) is

equal to the replacement value at the replacement boundary.

V(C) = V(CN) + P(1 )

S(C)

S(C) P(1 )

CCN

Z

, (2.4)

where V(C) is the continuation boundary and V(CN) + P(1 ) S(C)

S(C) P(1 )

CCN

Z

is the replacement boundary.

In this equation, C is the maintenance cost at the time of replacement, and

S(C) is the salvage price of the asset.

Condition 2. At the stopping boundary the continuation and stopping func-

tions have equal slopes. This is the smooth pasting condition [5].

dV

dC|C=C = S

(C)(1 ) + P(1 )

ZC

ZN

CZ

1. (2.5)

Condition 3. The initial slope of the value function is zero.

dV

dC|C=CN = 0. (2.6)

This is a reflecting barrier condition and according to Mauer and Ott [17] it

holds at any reflecting barrier for a diffusion process.

2.2. Portfolio structure and immunization

We would like to pay with a basket of two bonds for the future replacement expen-

diture and passively immunize the portfolio. The net present value of the two bonds


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is equal to the present value of the liability, and the linearly weighted duration of

the bonds equals the duration of the liability [6]. In mathematical terms, this is,Dawa + Dbwb = T, (2.7)

wa + wb = 1, (2.8)

where Da and Db are respectively the durations of bonds a and b, which constitute

our basket of bonds. wa, wb are the portfolio-wealth fractions invested in bonds of

types a and b, respectively.

T is the expected first passage time [3, 8, 17] that gives us the expected replace-

ment cycle (that takes into account the presence of a reflecting boundary) and is

computed as:

T =ln(C) ln(CN)

122

1

2

2 12

221 C

CN

(1 22

) . (2.9)

Equations (2.4)(2.8) comprise our system of non-linear equations where the

unknowns are K1, K2, C, wa and wb. The solution to this system of equations

provides the model parameters (K1, K2), optimal stopping cost of maintenance and

the wealth distribution between two bonds. Notice that there is no limitation as

to using two bonds. Our bonds may also be portfolio indices with characteristicaverage durations, coupon payoffs, and yields-to-term. However, for clarity it is

easier to think of two types of bonds rather than portfolio averages, which may

result from having dozens of bonds in two separate indices.

Our task is to find all the solutions to the system of Eq. (2.4)(2.8) by using

the Interval-Newton Generalized-Bisection method for given domains of the model

variables.

We explore the effect of fluctuations of the stopping boundary [right-hand side

of Eq. (2.4)] upon the cost of maintenance and bond weight fractions. We compute

all the solutions simultaneously without the need for initial guesses and observethe variations in bond allocation and maintenance costs with respect to the opti-

mal bond weight fractions obtained in the absence of fluctuations of the stopping

boundary.

3. Interval Arithmetic

The equations that describe the optimal stopping problem are non-linear and

numerous techniques are used to solve non-linear systems of equations.

Our experience with commercial software packages is that they cannot easilycompute multiple solutions to a system of non-linear equations without reinitial-

ising the program and providing judicious initial guesses (e.g., Mathematica 4.0,

Matlab 4.2). There is no information as to how many solutions may be located

within the variables domains. Thus, there is no previous information to ascertain

how many initial guesses are necessary.


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We present the simultaneous solution of the dynamic optimal stopping time com-

putation and the portfolio allocation problem by the Interval-Newton Generalized-Bisection (IN/GB) method [15]. The advantages of our approach are that there is

no need for initial guesses for the cost of maintenance, parameters K1 and K2, and

weight fractions wa and wb. All solutions in a given domain can be determined with-

out reinitialising the execution of the program, and in the absence of a solution set,

there is a mathematical guarantee that there is no solution to the problem within

the initial domain of the variables. These characteristics exceed those of most other

real arithmetic methods. All computations were performed on a Sun workstation

Blade 1000 fitted with Fortran-77 interval-arithmetic software.

3.1. Definition of an interval variable

A real interval, X, is the set of all real numbers lying between an upper and lower

bound X = [x, x] = {x R : x x x} where x, x R.

An n-dimensional real-interval column vector X = {X1, X2, . . . , X n}T has n

real intervals. An interval vector is also called n-dimensional box. In Sec. 3.2, the

lower case quantities are real numbers, and the upper case (capital letter) quantities

are intervals. Similarly, in Sec. 3.3. the variables K1, K2, wa, wb and C are interval

variables.

3.2. Basic operators in interval arithmetic

Interval arithmetic provides a set of operators to work with intervals analogous to

those used in arithmetic operations with real numbers. Considering the intervals

X = [x, x] and Y =

y, y

, there are four basic operators: addition, subtraction,

multiplication and division. A detail that immediately calls our attention is that

when 0 Y, there is no definition for the division operator. We remark that zero

is understood as the degenerate interval [0, 0]. The limitation that when 0 Ythe division operator is not defined, is overcome by introducing the formulations

of the KahanNovoaRatz arithmetic [13] in which division of two intervals, with

zero contained in the denominator, is defined based upon the characteristics of

the numerator and denominator. These are used in standard interval mathematics

computations.

Table 1. Basic interval arithmetic operators.

X + Y =h

x + y, x + yi

Addition

X Y =h

x y, x yi

Subtraction

X Y =h

minn

xy,xy, xy, xyo

, maxn

xy,xy, xy, xyoi

Multiplication

X Y = X 1Y

where 1Y

=h1y

, 1y

i, if y > 0 or y < 0 Division


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There are a couple of details that are important in numerical computations.

With real numbers the multiplication is distributive with respect to the sum. Thismeans that for real numbers x, y and z, we can write: x(y + z) = xy + xz. Although

the addition of intervals is commutative and associative, distributive laws do not

hold. Interval arithmetic is subdistributive. This means that ifX, Y, Z are intervals,

then X(Y + Z) XY + Y Z. In practical terms, it means that your interval may

widen depending on the order in which operations are carried out. Similarly, even

though there is an additive neutral element, and a multiplicative neutral element,

additive and multiplicative inverses do not exist. Since cancellation subtraction is

not achieved, the resulting interval may be wider than any of the original intervals

what leads to the problem of interval widening.

3.3. Natural interval extension of a real-valued function

A real-valued function results in a real number. Similarly, the valuation of an interval

function results in an interval. To pass from a real-valued function to its equiva-

lent interval arithmetic formulation, the natural interval extension of a real valued

function is introduced [22].

Suppose a real-valued function is given in terms of n-variables, the natural

interval extension is computed by replacing each occurrence of the real variables

by their corresponding interval equivalent. For example, if f(x,y,z) = x + y + z,

where x,y,z R then the natural interval extension is f(X , Y , Z ) = X + Y + Z,

where X , Y , Z are intervals of the form X = [x, x], Y =

y, y

, Z = [z, z] withx, x,y, y,z, z

R. The interval operators were defined in the previous section.

In the following development, the arrow overhead serves to indicate a vector of ele-

ments. Using the natural interval extension of a function, we can write the interval

arithmetic equivalent of our system of non-linear equations by simply substitut-

ing the real-valued variables with an interval variable. For clarity, let us rewrite

our system of five equations (in the form f( X) = 0) and identify our five interval

variables.

K1C1 + K2C

2 +C

+

1

r C V(CN) + P(1 )

S(C)

S(C) P(1 )

C

CN

Z = 0, (3.1)

dV

dC|C=C + S

(C)(1 ) P(1 )

ZC

ZN

CZ1 = 0, (3.2)

dV

dC|C=CN = 0, (3.3)

Dawa + Dbwb ln(C) ln(CN)

122

+1

2

2 12

2

1

C

CN

1 22

= 0, (3.4)

wa + wb 1 = 0. (3.5)


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These comprise our system of non-linear equations where the unknowns are K1,

K2

, C, wa and

wb, which are now intervals instead of real numbers. In other words,we have implemented intervals into the variables K1, K2, C, wa and wb.

3.4. The Interval GaussSeidel method for the solution

of a nonlinear system of equations

Consider the general interval linear system of the form

A X = B, (3.6)

for clarity we can write the interval matrix and interval vectors more explicitly as

[a11, a11]

aij, aij

[ann, ann]

[x1, x1]

[xn, xn]

=

b1, b1

bn, bn

, where A

IRnn (this means that A is a matrix with n rows and n columns in which each

element is an interval), B IRn (B is a column vector with n elements and each

element is an interval) and X Rn.

We can solve for the elements of X iteratively using the Interval GaussSeidelmethod. The GaussSeidel method solves the system of equations stepwise. This

means that it computes one component of the vector X at a time. Consider each

interval element of X to which we apply the following algorithm.

Xi =Bi

i1j=1 AijX

j n

j=i+1 AijXj

Aii, (3.7)

where Xi is the image of the ith interval component, Xi, of the interval vector X.

Aij denotes the matrix interval element located in the ith row and jth column ofthe interval matrix A. If Xi Xi = then replace X

i by the intersection of X

i

and Xi. On the other hand if X

i Xi = then there is no intersection and there

is no solution to the system of equations.

A first impulse to solve Eq. (3.6) would have been to compute the inverse matrix,

A1. However, this is not possible because the division of two equal intervals does

not lead to the [1, 1] interval, a consequence of the fact that there are no multiplica-

tive inverses in interval mathematics.

For example, if we divide [2, 3] by itself, the result is 23 , 1.5 and not the mul-

tiplicative neutral element [1, 1]. The best that can be done is to multiply on bothsides of our system of equations [Eq. (3.6)] by a real-number preconditioning matrix.

The objective is to approximate the identity matrix on the left-hand side after mul-

tiplying by the preconditioning matrix. This makes the GaussSeidel method con-

verge faster. There are no rules of thumb to choose the best preconditioning matrix.

Kearfott [13] discusses the implementation of preconditioning matrices.


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4. The Interval-Newton Generalized-Bisection Method (IN/GB)

The logic of the algorithm is the following. There is an initial multidimensionalregion (i.e., a hyperbox) defined by the domains of the variables. In this region, we

search for all the combinations of the variables that could generate a solution.

The first step is a range test. We evaluate our interval system of equations

using the initial interval for each of the variables. The result of the evaluation

is a set of (five) intervals (one for each equation). We need to confirm that each

of these resulting intervals contains the zero element. If the zero element is not

contained, then there is no solution within the original range of the variables, and

there is no solution. If zero is contained, we cannot guarantee the existence of a

solution.The second step is to solve the system of equations generated by the Interval-

Newton Method. The solution of the system of equations is found iteratively by

the Interval GaussSeidel Method. The result is the image of the original hyperbox

generated through the Interval-Newton Method.

The third step is to determine the intersection set between the original hyperbox

with that generated by the Interval GaussSeidel solution of the Interval-Newton

method equations.

It should be clear at this point that we find our roots with the Interval-Newton

Method but solve the interval system of equations using the GaussSeidel method.The Bisection procedure comes into play if the volume of the intersection

between the initial and image boxes is larger than sixty percent of the original

box volume. If the volume of the image box is larger than sixty percent of the origi-

nal domain volume, the intersection is bisected. Each box is evaluated by the range

test, and if necessary they would be separately analysed by means of the IN/GB

method. If the intersections volume is smaller than sixty percent of the original

hyperboxs volume, then the IN/GB continues with iterations until either the image

is completely contained in the original domain or there is no solution.

Let us consider the following proposition.

Proposition 4.1. Once the image of the intersection is totally contained in the

original hyperbox, there is a mathematical guarantee that the solution can be found

with the real-valued Newton method [22]. Furthermore, the solution is unique [13,

pp. 6063].

The algorithm switches automatically from the Interval-Newton method to the

real-valued Newton method in order to speed-up solution of the problem. The com-

bination of the steps with the Interval-Newton method and the Bisection stepsgives rise to this techniques name: Interval-Newton Generalized-Bisection (IN/GB)

method.

There is a vast literature dealing with the mathematical foundations of

interval arithmetic and its applications to solving systems of non-linear equa-

tions and ordinary differential equations. The IN/GB method is discussed


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in detail by Neumaier [22], Moore [20], Schnepper and Stadtherr [26], and

Kearfott [10, 11, 12, 13].Let us consider another important proposition.

Proposition 4.2. The IN/GB method can find all of the solutions to a system

of non-linear equations with mathematical certainty and without the need of initial

guesses. The only information needed is the domain of the variables [13].

The system of equations to be solved is of the form f( X) = 0 and is given

by Eqs. (3.1)(3.5). The left hand sides are interval extensions of the real val-

ued functions occurring in the left hand sides of the corresponding real-valued

equations.

The Interval-Newton method procedure is as follows. Consider a system of non-

linear homogeneous equations f( X) = 0 and a vector x X(0)where X(0) is the

initial n-dimensional box. The k-th iteration in the IN/GB method computes N(k)

by solving (via GaussSeidel method) the following system of equations [13]:

F

X(k)

N(k) x(k)

= f(x(k)), (4.1)

where F( X(k)) is the interval extension of the real-number jacobian matrix, f (x),

over the interval vector X(k). x(k) is an array of real numbers that is usually the

midpoint of the interval vector X(k). A range test is applied before evaluating

Eq. (4.1). This test consists of evaluating the system of equations f(x) = 0 over the

desired interval. If zero is contained inside the interval image f( X) then Eq. (4.1)

proceeds. Otherwise, the interval box is discarded. N(k) is computed by the interval

GaussSeidel method and intersected with X(k). Any roots, x , inside X(k)must be

in the intersection of X(k) and N(k) : x { X(k) N(k)}. There are three possible

cases for the intersection of N(k) and X(k) as shown in Table 2.

Case 1. A unique root exists and is estimated by the NewtonRaphson method

with real arithmetic.

Case 2. There is no root in the interval box, and it can be safely discarded.

Case 3. IfN(k) X(k) is sufficiently smaller (60% of original volume) then continue

to apply IN/GB, otherwise bisect.

Table 2. Intersection of N(k) and X(k).

Case Interpretation

N(k) X(k) (1)N(k) X(k) = (2)N(k) X(k) and N(k) X(k) = (3)


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To summarize the task ahead of us, we write more explicitly our system of

equations to solve via the IN/GB method. We have:

K1

C1N C

1

+ K2

C2N C2

+

CN C

+

1

r (CN C)

+ P(1 )

S(C)

S(C) P(1 )

C

CN

Z = f1 = 0, (4.2)

K11C11N + K22C

21N +

C1N +

1

r = f2 = 0, (4.3)

K11C1

1 K22C2

1

C

1 1

r +8

C2 (1 )

+ P(1 )

ZC

Zn (C)

Z1 = f3 = 0, (4.4)

w1

P1

n1M1

(1 + y1)n1+

n1i=1

ic1

(1 + y1)i

+

w2

P2

n2M2

(1 + y2)n2+

n2i=1

ic2

(1 + y2)i

+

ln(C) ln(CN)

122

+1

2

2

12

2

1

C

CN

(1 22

)

= f4 = 0, (4.5)

wa + wb 1 = f5 = 0. (4.6)

The first two terms in Eq. (4.5) are the durations of bonds 1 and 2 where the

symbols n1, n2, c1, c2, P1 and P2 are the maturities, coupon values, and present

values of bonds 1 and 2, respectively.

5. Model Results

5.1. Optimal replacement policy

The base-case parameter values are given in Table 3. In Table 4, we have the variable

domains used for the implementation of the Interval-Newton Generalized-Bisection

(IN/GB) method. The stopping and continuation boundaries are plotted in Fig. 1.

Some comments are pertinent regarding the parameters of Table 3. The corre-

lation between cost and the systematic pricing factor is made zero, which results

in equality between the drift rate of cost and the risk-adjusted drift rate of cost.

The initial operating cost of a new machine, the purchase price of a new machine

and the bond face values are similar in magnitude. We call it similar because

the universe of values we are covering with these quantities is only from 1 to 10.We could have chosen values like P = 1000 but that would have added nothing

novel to our results. Consequently, we anticipate no awkward solutions due to large

differences between the face values of the bonds, the purchase price, and the initial

operating cost. The coupon rate and the yield-to-maturity are the same, this means

that the bonds are purchased at par.


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Table 3. Parameters for the computation of the optimal stopping timefor a geometric growth cost model.

Parameter Symbol Value

Riskless interest rate (*) r 0.07Drift rate of cost (*) 0.15Volatility of cost (*) 0.1Market price of risk (*) 0.4Correlation cost/systematic pricing factor 0Risk-adjusted drift rate of cost = r 0.15Initial operating cost of a new machine CN 1Purchase price of a new machine P 10Salvage value (*) S(C) = kC1 8C1

Proportional investment tax credit (*) 0Tax rate (*) 0.3Depreciation rate (*) 0.5Bond face value M 1Coupon rate c 0.05MYield to maturity y 0.05Maturity of bond 1 (years) n1 5Maturity of bond 2 (years) n2 25

These parameter values are used by Mauer and Ott [17].

Table 4. Domains of variables for theIN/GB method.

Lower bound Upper bound

K1 0.1 100K2 0.1 100C 1 20w1 0 1w2 0 1

The variables domains span a considerable hyperbox where meaningful answerscan be located. We have excluded the case of short sales, but it could be added by

allowing negative weight fractions.

In Fig. 1, the continuation and the replacement boundaries cross at a cost

of maintenance of C = 2.7361, with a present value of the cost of maintenance

V = 30.6947, and an average replacement time of T = 6.7038 years, which are in

agreement with the results by Mauer and Ott [17].

We analysed the effect of varying the risk-free interest rate, cost volatility, and

corporate tax rate upon the cost of maintenance, present value of the cost of main-

tenance, and the average replacement time. The results are shown in Table 5.

5.2. Discussion of initial results

The increase of the riskless interest rate causes the cost and average time to increase

while the continuation value decreases. This means that the optimal timing comes


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0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10 11 12Cost of Maintenance

PresentValueof

theCostofMaintenance

Continuation

Boundary

ReplacementBoundary

Fig. 1. Intersection of continuation and replacement boundaries.

Table 5. Average replacement time, cost of maintenance andpresent value of maintenance costs for optimal replacement.

r r r

0.02 0.07 0.12C 2.4259 2.7361 3.0888T 5.8741 6.7038 7.5399V(C) 98.4484 30.6947 19.6581w1 0.8705 0.7895 0.7080w2 0.1295 0.2105 0.2920

0.05 0.1 0.15C 2.6228 2.7361 2.9281

T 6.4259 6.7038 7.1587V(C) 30.5447 30.6947 30.9451w1 0.8166 0.7895 0.7452w2 0.1834 0.2105 0.2548

0.25 0.3 0.35C 2.7032 2.7361 2.7734T 6.6202 6.7038 6.7972

V(C) 32.6179 30.6974 28.7695w1 0.7977 0.7895 0.7804w2 0.2023 0.2105 0.2196

later in the future with higher interest rates. An increase in volatility causes the

cost, replacement time, and continuation value to increase. This indicates that the

larger the risk, the longer we use the asset. Similarly, an increase in the tax rate

causes an increase in the replacement time. The results show that the continuation


15/22


value and replacement times are more sensitive to the riskless interest rate than

to the volatility or the tax rate. The portfolio wealth distribution shifts from theearlier maturity bond (w1, 5 years) to the later one (w2, 25 years) as the average

replacement time increases. This would logically mean that as our liability extends

further into the future, we cover that expenditure with bonds of longer maturities.

As expected by its effect on the average replacement time, the riskless interest rate

causes the largest changes in wealth allocation as compared to changes in volatility

and corporate tax rate.

5.3. Replacement boundary fluctuations

In the previous section, the reference case was set up with the information from

Table 3. Now, we explore the cases where fluctuations occur on the replacement

boundary while we keep the continuation boundary static. In practice, this means

that the values for the integration constants K1(= 33.9291) and K2(= 0.5936) are

kept constant at the reference values corresponding to the base case in Table 3.

In doing so, we forfeited the use of Eqs. (4.3)(4.4). These are the smooth pasting

condition and the reflecting boundary condition.

This is a way of introducing imperfections in our model in which we allow some

error to be present in one of the boundaries. The effect is that with the oscillations,

there is a possibility that the continuation and replacement boundaries will cross.

The crossing would mean that there may be conditions under which replacement

may take place, and these intersections are not predicted when only ideal situations

are considered.

The oscillations are sinusoidal and of the form

= 0

1 + A sin

2

T0(t)

, (5.1)

where 0, T0 are the original replacement boundary [i.e., right-hand side of

Eq. (4.2)], and average replacement time of the reference case (T0 = 6.7038) in

Table 3.

The amplitude, A, is the magnitude of the fluctuation in the stopping boundary.

This will be a fraction of the original stopping boundary value. The time is the

expected time, (t) = 1 12

2 ln

CCN

. We interpret these oscillations as possible

variations in the cost of replacement of an item. This cost can increase with inflation,

but it can also decrease with improvements in technology.

The fluctuation amplitude, A, in the range from 0.02 to 0.08, causes only minor

changes in the cost of maintenance, present value of the cost of maintenance, averagereplacement time, portfolio wealth distribution (w1, w2) with their largest variation

as a percentage of the no-fluctuations case, A = 0, being 3.42, 0.3, 3.58, 3.08

and 11.16, respectively.

The situation is entirely different for amplitude fluctuations of magnitude 0.10,

0.15, and 0.20. These correspond to variations in the replacement boundary of


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10%, 15%, and 20%, respectively. There are three intersections for each value of

the amplitude A. We find the three solutions in each case by using the Interval-Newton Generalized-Bisection method. With this strategy, we only need to provide

the domain for our search of K1, K2, C, w1 and w2, which are given in Table 4.

We can look at the plot of present value of maintenance cost versus maintenance

cost to verify the presence of three intersection points between the continuation and

replacement boundaries. This is shown in Figs. 2 and 3. The three solutions can be

found with Mathematica 4.0 [31] by three separate calls to its root finder.

The intersections are the result of allowing the replacement boundary to oscil-

late. This analysis demonstrates that small deviations from an idealized model can

change considerably the expected results. In our case, the maintenance costs andaverage replacement times vary substantially from one intersection to any of the

others.

The change in the average replacement time of the portfolios makes the positions

on the two bonds to vary widely. For the case of the amplitude oscillation, A, of 0.1

the wealth portfolio allocation fraction for the 25-year maturity bond, w2, changes

from 0.1871 to 0.6391, a change of 241.58%, with similar changes for the cases where

A is 0.15 and 0.20.

Our results imply that there may exist considerable time variations in optimal

replacement time as a result of fluctuations in one of the boundaries. This meansthat rather than a point in time for replacement; there may exist an interval for

replacement, which widens as the fluctuations in replacement costs become larger.

Table 6. Replacement boundary fluctuations effects on cost of mainte-nance, present value of the cost of maintenance, average replacementtime, and portfolio wealth allocation.

A A A A A

0 0.02 0.04 0.06 0.08C 2.7361 2.6393 2.6414 2.6421 2.6424T 6.7038 6.4554 6.4609 6.4626 6.4635V(C) 30.6947 30.6013 30.6036 30.6043 30.6046w1 0.7895 0.8138 0.8132 0.8131 0.8130w2 0.2105 0.1862 0.1868 0.1869 0.1870

0.1 0.15C 2.6426 4.9182 5.1746 2.6428 4.5559 0.5809T 6.4640 10.7480 11.0985 6.4646 10.2203 11.8956V(C) 30.6048 29.2583 28.7833 30.6051 29.8440 27.4191w1 0.8129 0.3951 0.3609 0.8129 0.4465 0.2832w2 0.1871 0.6049 0.6391 0.1871 0.5535 0.7168

0.2C 2.6430 4.4717 6.0577T 6.4650 10.0912 12.1851V(C) 30.6052 29.9651 26.8168w1 0.8128 0.4591 0.2549w2 0.1871 0.5409 0.7451


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0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12

Cost of Maintenance

PresentValueoftheCostofMaintenance

Continuation

Boundary

Replacement

Boundary

Fig. 2. Verification of the presence of 3 intersection points between the continuation and replace-ment boundaries.

-25

-20

-15

-10

-5

0

5

0 1 2 3 4 5 6 7 8 9 10 11 12

Cost of Maintenance

PresentValueoftheC

ostofMaintenance-

Replaceme

ntCost

Fig. 3. Difference between the present value of the cost of maintenance and the replacement cost.Verification of the presence of three intersection points between the continuation and replacementboundaries.


18/22


The spread of the present value of cost of maintenance, cost of maintenance, and

average replacement times tells us that even with small fluctuations in one of theboundaries, we may need to hedge risk with a barbell portfolio rather than aiming

at using a safer bullet portfolio.

5.4. Replacement boundary fluctuations and variation

of parameters

In the previous section we studied the effect of fluctuations upon the optimal replace-

ment solutions, in doing so we assumed that the continuation boundary was fixed in

place (i.e., K1 and K2 were constant). We can remove this assumption and solve the

optimal replacement problem. Consequently, we reinstate the smooth pasting andreflective boundary conditions. In this case, we solve the set of five non-linear equa-

tions using the Interval-Newton Generalized-Bisection method. The interpretation

of this approach is that we may anticipate fluctuations and would like to know how

much the optimal portfolio distribution changes with the small oscillations in the

boundary just as we did in the previous section. Using the information in Tables 3

and 4, we compute the solutions for 10% amplitude fluctuations in the replacement

boundary. These are shown in Table 7. We obtained three solutions as we did in

the case of a fixed continuation boundary (previous section).

However, there is only one solution that is optimal. To see how this is possible,the solutions are plotted in Figs. 4, 5 and 6. Solution number 1 is the only optimal

solution because the replacement boundary is never below the continuation bound-

ary prior to replacement. This means that the cost of the replacement unit is the

same as the old units present value of maintenance costs only at the replacement

time, which occurs at the intersection point of the boundaries.

At any prior instant the present value of the cost of the new unit (replacement

boundary) is higher than the old units present value of maintenance costs. There

is a considerable change in portfolio composition with respect to the reference case

in Table 3. When we ignored any possible fluctuations in the replacement cost, the

Table 7. Replacement boundary fluctuations effectson the cost of maintenance, present value of the costof maintenance, average replacement time, portfoliowealth allocation, and parameters K1 and K2.

A A A

0.1 0.1 0.1Solution number 1 2 3

C 2.0474 3.3308 5.04448T 4.7041 8.0603 10.9233V(C) 21.9270 39.8719 28.9504w1 0.9846 0.6572 0.3780w2 0.0154 0.3428 0.6220K1 28.5247 38.9982 33.8924K2 0.5065 0.6753 0.5930


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0

4

8

12

16

20

24

28

32

0 1 2 3 4 5 6 7 8 9 10

Cost of Maintenance

PresentValueoftheCostofMainte

nance

Continuation Boundary

Replacement Boundary

Fig. 4. Solution 1, Table 7, optimal solution.

0

4

8

12

16

20

24

28

32

36

40

0 1 2 3 4 5 6 7 8 9 10

Cost of Maintenance

PresentValueoftheCostof

Maintenance


Replacement Boundary

Fig. 5. Solution 2, Table 7.

04

8

12

16

20

24

28

32

36

40

44

0 1 2 3 4 5 6 7 8 9 10

Cost of Maintenance

PresentValueoftheCostof

Maintenance


Stopping Boundary

Fig. 6. Solution 3, Table 7.


20/22


wealth fraction allocated to the first bond was 0.7895 in contrast with a 0.9846 with

fluctuations. This means a twenty-five percent change with respect to the idealizedmodel with no fluctuations. This substantial change in portfolio wealth allocation

highlights the need for caution in portfolio-wealth-allocation decisions based on

idealized models.

6. Conclusions

The IN/GB method was implemented to determine the average replacement time for

an item with maintenance costs that follow a geometric stochastic growth dynamics.

The solution was found with and without fluctuations of the replacement boundary.

All the computations were carried out in interval arithmetic.The cases without fluctuations result in a single intersection between the con-

tinuation and replacement boundaries. The risk-free rate has the largest effect on

the average replacement time. There is a unique portfolio wealth distribution for

each solution.

Three cases with fluctuations of the replacement boundary with a fixed contin-

uation boundary resulted in several intersections with the continuation boundary.

In the presence of a fixed continuation boundary, the replacement boundary fluctu-

ations of 10%, 15%, and 20% resulted in three intersections with different average

replacement times and portfolio wealth distributions. Our graphs confirm the resultsfrom the computations and depict the presence of multiple solutions to the optimal

stopping problem. The resulting portfolio weights for these multiple intersections

differ widely and in some cases by more than 200%. These large changes in portfolio

weight allocations are important.

They suggest that the possibility to hedge risk is greatly diminished even in the

presence of fluctuations in our simplified model. After removing the condition of a

fixed continuation boundary, we found a change of 25% in optimal portfolio wealth

allocation for a fluctuation of a 10% in replacement boundary.

We used a constant yield to maturity. In practical applications, it would beprudent to explore the variations in the optimal replacing time when there is a time

varying term structure of interest rates.

Acknowledgment

The collaboration of Dr. Albert LeMay at the University of Notre Dame (USA), the

financial support by both the University of Essex (UK) through the Oscar Arias

Award and by the British Council through the British Chevening Program are

gratefully acknowledged.

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