ELSEVIER Journal of Economic Dynamics and Control

21 (1997) 1445-1470

A model for designing callable bonds and its solution using tabu search

Andrea Consiglio, Stavros A. Zenios*

Department of Public and Business Administration, University of Cyprus, Nicosia, Cyprus

Abstract

We formulate the problem of designing callable bonds as a non-linear, global, opti- mization problem. The data of the model are obtained from simulations of holding-period returns of a given bond design, which are used to compute a certainty equivalent return, viz., some target assets. The design specifications of the callable bond are then adjusted so that the certainty equivalent return is maximized.

The resulting problem is multi-modal, and a tabu search procedure, implemented on a distributed network of workstations, is used to optimize the bond design. The model is compared with the classical portfolio immunization model, and the tabu search so- lution technique is compared with simulated annealing for solving the global optimiza- tion program. It is shown that the global optimization model yields higher returns than portfolio immunization. It is also shown that tabu search is computationally more ef- ficient than simulated annealing in solving the model, and it produces better solutions.

Keywords: Financial innovation; Product design; Simulation; Tabu search; Callable bonds; Parallel computations JEL clussijication: C61; C63; G12; G13; G31; G32

1. Introduction

Recent years have witnessed a rapid pace in financial innovation. The tradition

of financing a firm’s activities with a limited number of securities - equity, debt, preferred stock and convertibles - gave way to the introduction of innovative

* Corresponding author.

0165-1889/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved

PIISO165-1889(97)00034-l

1446 A. Consiglio, S. A. Zeniosl Journal of Economic Dynamics and Control 2I (1997) 1445-1470

instruments, such as options, bonds with embedded options, securitized assets,

etc. Miller (1986), Ross (1989) and Merton (1990) analyze the rationale of financial innovation.

In a recent special issue of Financial Management, Allen (1993) writes:

“This [rapid pace of financial innovation] has highlighted the need to under-

stand why the securities that are used have the particular form they do. As a result, financial economics has paid increasing attention to the issue of security

design”.

In this paper we develop a model for designing the particular form of a specific

type of securities, namely, callable bonds. Such bonds are a major borrowing instrument for corporations, utilities and government agencies with interest rate sensitive assets. The rationale for this choice is simple: the bonds can be called

as interest rates drop and assets repay. Agencies like Fannie Mae and Freddie Mac fund more than 95% of their mortgage assets with the issue of callable bonds and indexed sinking fund debentures. As interest rates drop and mortgages

prepay, the bonds are called. The problem faced by the agencies is to specify the kind of a callable bond that is sold to finance a particular asset. This is the problem of integrated jinancial product management, analyzed in Holmer and Zenios (1995), and a nonlinear, global, optimization model is proposed here for this problem. Good - in a sense to be made specific later - solutions to the problem of integrated product management improve the efficiency of the financial intermediary, with potentially measurable rewards by the market. Holmer and

Zenios, and references therein, address this issue in depth. Another contribution of this paper is in developing a heuristic procedure, based

on tabu search for solving the global optimization model. Empirical results show that the proposed, global optimization, model produces better returns than the classical model of portfolio immunization. It is also shown, through computa- tional experiments, that the tabu search heuristic is more efficient and accurate than a global optimization algorithm based on simulated annealing.

Section 2 presents the problem and develops the nonlinear programming model in global optimization. Section 3 develops and implements the technique for solv- ing the model using tabu search, the metaheuristic procedure of Glover ( 1989a, b). Empirical results are presented and analyzed. The technique is suitable for imple- mentation on parallel computer architectures. Alternative parallelization strategies are discussed, and computational results are reported. Section 4 discusses the parallel implementation of the procedure. Section 5 summarizes empirical results.

2. Global optimization model for designing callable bonds

A callable bond is unambiguously specified by the following four design para- meters parameters; see Fig. 1.

A. Cons&o. S.A ZmioslJournul of Economc Dynamics and Control 71 (1997) 1445-1470 1447

1 Redemption prices (dollars)

Time

L K M

Fig. I. The four parameters of the bond design problem: (1) L, lockout period, (2) R, redemption

price at first call date, (3) M, time to maturity, and (4) K, time after which the security can be

called at par.

Lockout period (L): This is the period, following issuance, during which the bond cannot be called. Redemption price at first call date (R): This is the price at which the bond can be called at the first call date. This price is at a premium above par. Time to maturity (M): This has the usual meaning for all fixed-income se- curities. Schedule of redemption price: The redemption price at the first call date is

at a premium above par. This premium declines as the security approaches maturity. A schedule of redemption prices during the term of the bond is

specified by the issuer. We assume, for simplicity, that the redemption price declines linearly from a starting value of R at the end of the lockout period, to par at some period K. The term during which the security can be called at a premium (i.e., K) is the design parameter. Other schedules of redemption prices - instead of the linear decline - can be easily specified as well. The time parameters are constrained to satisfy the precedence relationship L 5 K<M. We now develop a model for selecting values of the design parameters. The

- model accounts for uncertainty in interest rates, and for the cashflows generated by the designed bond and the target funded asset, for different realizations of the interest rates. The stochasticity of interest rates is described using a discrete prob- ability space, denoted by a set of scenarios Q = { 1,2,3,. . , S}. (Models for gen- erating the scenarios abound in the finance literature, see, e.g., Black et al. 1990).

1448 A. Consiylio, S. A. Zrnios I Journal of Economic Dynamics and Control 21(1997) 1445-1470

A target distribution of holding period returns, denoted by F”, s = 1,2,. S, is assumed given. This is the return that the designed bond should replicate.

It could be, for example, the return of an asset whose purchase is funded in part by the issue of a callable bond. Herein lies an important assumption of

our modeling process, namely that an accurate pricing model is available for the target asset. If the pricing model is inaccurate the model we propose next

will track closely a target that may not reflect the true market movements. For instruments such as the mortgage securities we consider as targets in this paper very good pricing models are currently available. Readers are referred

to Bartlett (1994) for the fundamentals, Kang and Zenios ( 1992) for prepay-

ment models and Ben-Dov et al. (1992) for option adjusted pricing. The papers by Golub et al. (1995) and Worzel et al. (1994) provide substantial empirical

evidence that the mortgage pricing models can track, on the average, market

movements very well. The distribution of holding-period returns of the callable bond is a function of

the design parameters:

rs= f(L,R,M,K). (1)

This distribution can be adjusted by changing the design parameters. However, the function f is not specified explicitly. It is obtained from a simulation procedure that relates the cashflows generated from a callable bond with given parameters L, R, M, K to the interest rate scenarios in 8. Appendix A describes the simulation

procedure. We consider now the problem faced by a financial institution that raises $100

in funds by issuing callable bonds at par prices. In addition to these borrowed funds it invests E dollars from shareholders’ equity to purchase $( 100 + E) in the target assets. At the end of the holding period the liabilities realize a return of 1 + rS and the assets a return of 1 + 7. The position of the institution at the end of this holding period is given by the terminal wealth:

WTS=lOO(r’S-rS)+E(l +r’“).

The return to shareholders is given by the Return on Equity (ROE)

(2)

FS - rs ROES=(l +i”)+ 100E

=(l +iS)+ 100 7 - f(L,R,M,K)

E ’

The design parameters L, R,M, K should be selected in such a way that the shareholders’ return on equity (cf. Eq. (3)) is maximized. Note, however, that return on equity is scenario dependent. No single setting of the design parame- ters will have maximal ROE” for all scenarios. Fig. 2 illustrates the return of three bond designs and of a target asset - in this example, a mortgage backed

A. Consiylio, S.A. ZenioslJournnl of Economic Dynamics ond Control 21 (1997) 144S-1470 1449

g 4.00 -+

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Interest rates

Fig. 2. Holding period returns of a mortgage-backed security (MBS) and three alternative callable

bonds, under different interest rate scenarios. In this example it is obvious that bonds B and C are

preferable liabilities to hold against the mortgage asset, as opposed to bond A.

1450 A. Consiylio, S.A. ZeniosIJournal of Economic Dynamics and Control ZI(1997) 1445-1470

security - under different interest rate scenarios. It is obvious that bonds B and C

are preferable for the shareholders than bond A, but the right choice between B

and C is not clear.

2.1. Risk aversion and certainty equivalent return on equity

To compare alternative bond designs we need to incorporate the decision maker’s risk aversion for different levels of ROE”. Let % denote the utility

function of the decision maker. Then the issuing institution will prefer the bond that maximizes the expected utility:

1

s C( % ROES).

SER (4)

With each bond we associate its Certainty Equivalent Return on Equity2 (CEROE) defined by

%(CEROE) = f c %(ROES). SE0

(5)

Assuming non-decreasing utility functions we can rank bonds by their CEROE, and the one with the highest value of CEROE has the best design. Combining

now (4), (5) and (3) we can write the problem of optimal bond design as the following optimization program:

Maximize {~,R,M,K~L<KsM}

@(L,R,M,K) A

CEROE = W’ (l+iS)+l(@‘S-f(~M’K) (6)

We reiterate that the function @ is not explicitly specified, but it can be evaluated for different settings of L, R, M, K using the simulations of Appendix A to compute rS (Eq. (1)) and evaluating the utility function (5). A systematic procedure to solve (6) is summarized as follows:

Initialization: Parametrize the design characteristics of the bond and assume some initial values. The specification of the holding period, generation of the interest

’ The certainty equivalent return is the level of return (known with complete certainty) that makes the

decision maker indifferent between accepting this level of certain return, versus taking her chances

with returns ROES. For a risk neutral investor the certainty equivalent is the expected value of

ROES. A risk averse individual is willing to accept a lower certain return than the expected value,

in order to avoid playing the odds.

rate scenarios, and estimation of the holding period returns for the target asset,

are computed during the initialization step.

Step 1: Estimate the coupon rate (C) that prices the bond at par. The procedure for estimating C is given in Appendix B.

Step 2: Generate returns for the bond, using the same interest rate scenarios

and holding period used to estimate the returns of the target asset. Scenarios of holding period returns are generated using the simulation procedure described in Appendix A.

Step 3: Compute the CEROE of the specified bond design using Eq. (5). If termination criteria are satisfied, Stop. Otherwise proceed with Step 4.

Step 4: Adjust the bond design parameters to improve its CEROE and return to Step 1.

This iterative procedure seeks designs that have maximum CEROE. Of particular interest is the specification of rules for adjusting the design parameters in Step 4, in order to maximize the CEROE. The rules should take into account the fact that there exist multiple, locally optimal, solutions to the bond design problem. It may be possible to find a design that is optimal only in a small neighborhood of

the design parameter. For example, applying Powell’s method, Press et al. ( 1989) to the solution of (6) and for the target mortgage return illustrated in Fig. 2 we get two locally optimal bonds, with CEROE 1.23 16 and 1.4056 (Fig. 3).

In the next section we develop a tabu search heuristic that searches for a globally optimal solution from among the local maxima.

3. Optimal bond design using tabu search

The optimization model defined by (6) is complicated by the facts that multiple locally optimal solutions exist, and that the design parameters take discrete values. In particular, the date parameters L,A4, K are in months, while the price parameter

R is expressed within an accuracy of 1 bp (100 basis points is 1%). To solve this optimization problem we design a tabu search (TS) procedure. TS is a metaheuristic framework introduced by Glover (1987). See Glover (1989a, b) for introductions. A TS heuristic defines a neighborhood N(x) of the current solution x and specifies a set of moues P, that allow the current point to reach points in its neighborhood. The neighborhood N(x) is not, necessarily, the set of points that are close to x by some distance measure. It is the set of points that can be reached from x by some move from the set P. Neighborhood and moves are designed in such a way that the heuristic can move away from the current iterate x into regions of the search space where good solutions can be found, without

1452 A. Consigiio, S. A. Zeniosl Journal of’ Economic Dynamics and Control 21 (1997) 1445-1470

HPR

11.00

10.00

9.00

8.00

7.00

6.00

5.00

4.00

3.00

2.00

1.00

0.00

-1.00

-2.00

-3.00

-4.00

-5.00

-6.00

I I I \ ‘. ‘b * .

6.00 7.00 8.00 9.00 10.00 11.00 12.00

Fig. 3. Holding period return of two bonds designed using local optimization methods. Their CEROE are 1.2316 and 1.4056.

staying necessarily close to x. Points close to x should also be explored, if they

appear to contain good solutions. In order to avoid cycling (i.e. situations where x E N(N(x))) TS puts some

moves in a tabu list. These moves cannot be performed for a number of steps, equal to the tabu tenure. A memory function keeps track of performed moves and determines if a move should be labelled tabu. However, if a tabu move would lead to a solution better than any previously obtained solution, then this move would not lead to cycling and should be taken. Some aspiration criteria are specified, and the tabu restriction is overwritten if these criteria are satisfied by a tabu move.

Quite often the number of possible moves is large, and the evaluation of all of them in order to select the next move is expensive. TS allows the specification

A. Consiylio, S. A. ZenioslJournal ofEconomic D]vzamirs and Control 21 (1997) 1445-1470 1453

(1, 1, -1)

Fig. 4. Illustration of allowable directions in a three-dimensional parameter space. From the current

point x0 moves are allowed through the eight vertices and the mid-points of the eight edges of the

cube. The step size along each direction takes discrete values, as illustrated above for the direction

(1,-1,-l).

of a candidate list of moves. Only moves in the candidate list are evaluated, but the rules for creating a candidate list should allow the exploration of all possible moves. The following subsection describes the elements of our TS heuristic.

3.1. The tabu search heuristic

3.1. I. Neighborhood and moves The current solution x0 is a tuple of given parameter values L”,Ro,Mo,Ko.

A move from x0 specifies a direction for changing these parameters (dj, j E {L, R, M,K}) and a step length that determines the magnitude of the change (A,,

j E {L, R, IU, K}). The direction vector takes values dj E { + 1, 0, - 1) for every j. That is, the value of each design parameter can either increase, or remain at its current value, or decrease. There are altogether 81 directions possible from each solution. Fig. 4 illustrates the direction vectors for a three-dimensional parameter space.

The step length parameter determines the magnitude of the change. Each pa- rameter can change in discrete increments. The time parameters L, K,M change in monthly steps (6,) while R changes in basis points (8,). Aj is determined by se- lecting the number of increments for all parameters, A, and setting AK = AL = AM

1454 A. Consiylio. S. A. Zenios I Journal of’ Economic Dynamics and Control 21 (1997) 1445-1470

= AS, and AM = Ah,. The parameter A takes values from the set { 1,2,3,4,6, 12, 20,36,50}. This sequence is specified so that an intensive search is performed near the current solution, i.e. for A = 1,2,3,4,6, while the search can also diver-

sify to regions away from the current solution, i.e. A = 12,20,36,50. There is no formal justification for these parameters, except that they allow for both intensive

searches in the neighborhood of the current solution, and they also allow the al-

gorithm to move away from local optima. Other values may work as well. Fig. 4 illustrates the move for a given direction in a three-dimensional parameter space.

With these definitions the neighborhood of the current point x0 is given by

P/(X’) is the set of tuples (L’, R’,M’,K’) such that

L’ = Lo + ALdL, R’ = R” •k ARdR,

M’ = MO + AMdM, K’ = K” + AKdK.

(7)

(8) A final remark on the definition of a move deals with the constraint L G K GM.

A move Ajdj that violates these inequalities is called infeasible and is not an

allowable move.

3.1.2. Tabu moves

Whenever a move Ajdj is used to move from a point x0 into the neighborhood N(x”) then the direction dj is marked as tabu. It remains tabu for the tabu tenure K, which is defined in terms of recency memory. That is, our TS heuristic

remembers the rc most recent directions and keeps them in the tabu list. In our implementation we use a value of K = 3 or 4, which was found to produce the best result for our experiments. Our definition of tabu moves specifies a tabu region, Glover (1994), of the search space which is inaccessible for the next K steps.

Smaller values of K would produce worst objective values, as the algorithm would get stuck in local optima, while larger values result in larger execution times as

the algorithm is restricted from searching larger parts of the search region. There is no formal justification for these settings, but our values are consistent with Glover’s (1994) recommendations, based on experiences with tabu search with many diverse applications.

The tabu region may be relaxed from being tabu if some point in it satisfies certain aspiration criteria. In particular, the TS procedure will move from a bond design x0 to a design x’ in the tabu region if the CEROE of x’ is higher than the CEROE of any previously encountered design.

3.1.3. Candidate list We now specify how to select a move from the list of allowable moves. One

strategy would be to evaluate all non-tabu moves and select the one that pro- duces the best CEROE value. (Recall that tabu moves are also evaluated, and

A. Corzsiq/io. S. A. ZaGosl Journal of’EcorwM L?,vmtnics and Control 21 (1997) 1445-1470 1455

a tabu move is selected if it satisfies the aspiration criteria.) This strategy is com-

putationally very expensive: the evaluation of the CEROE for a potential move

requires the evaluation of the function @(L, R, M, K) via a simulation procedure. A function evaluation takes, on average, 1 minute of CPU time on a DEC- station 5100. To speedup the execution of the algorithm the selection of a move

is guided by a list of candidate directions. Our candidate list strategy is specified by a tuple (N,, N,), where N, is the size of the candidate list, and N, is the

number of local searches performed from the list of candidate directions. The candidate list is created by searching, in cyclic order, and adding to the

list the first N, directions that lead to a local move (i.e., with A = 1) with

a CEROE > 1. Once a list is created the TS heuristic will start making moves along the directions in this list. To select a move the heuristic evaluates the step size for each direction in the candidate list, and selects the best non-tabu move.

(It will select a tabu move if it satisfies the aspiration criteria.) The process is repeated either until all moves in the candidate list are tabu, or until a number of N, moves has been performed. The candidate list is then refreshed, and the

process repeats. This candidate list strategy is quite general. For example, the strategy

(card(N(x”)), 1)) will pick up the best move from all eligible moves around the current point. This strategy is computationally expensive, but will lead to a good solution within few steps. Strategy (1,1) will take the first move that

produces a CEROE 2 1. Each step of this strategy is computationally inex- pensive, but most likely a large number of steps will be required to obtain a good solution. In the following sections we experiment with alternative settings (N,,N,) and identify strategies that are both efficient and effective.

3. I. 4. Empirical results The TS heuristic was applied to the problem of designing a callable bond to

fund a mortgage asset. The target asset is a mortgage backed security (MBS) with weighted average coupon (WAC) 9.5% and weighted average maturity (WAM) of 360 months. The holding period return of the mortgage security, during a 36-month holding period, is shown in Fig. 2.

The heuristic starts from an arbitrary (randomly generated) bond design. The

solution trajectory is sensitive to the starting point, and in our numerical results we report averages of ten repetitions of each experiment. (It is tempting to attempt to find a ‘good’ starting point by using, for example, the bond designed with a simpler model, such as the classical portfolio immunization model, Section 5.1. Due to the random search nature of the heuristic there is no guarantee that a global optimum will be reached faster from such a starting point. Indeed, the TS heuristic would occasionally converge to a good solution quickly from a starting point with low CEROE, while it would take longer to converge when starting from a point with higher CEROE.) The tabu tenure parameter K is set equal to 4 for all experiments. The TS procedure terminates when the solution

1456 A. Consiylio, S. A. ZenioslJournal oiEconomic Dynamics and Control 21 (1997) 1445-1470

____................... . _” . . . __ . . . . . . _ . . . . . “.” . . . . . . __--__._ . . . . . . . . . . . .._...._............... __ . 1.483

. ..___. _ . . . . __..___.,

I.482

I.481

I .48

1.47Y

I.478

I.477

1.476

I .4?5

I .474

1.473

Fig. 5. The effect of the candidate list strategy on the average and standard deviation of CEROE obtained from ten different starting points. Cx_y denotes a candidate list with NE =x and NS = y.

has not improved for a given number of moves. These moves may be expensive to execute, depending on the candidate list strategy. In our results we report averages for two different solution times: the total running time, and the running time to the best solution.

Fig. 5 shows the effect of the candidate list strategy on the average and standard deviation of the CEROE obtained from different starting points. We observe that large candidate lists (i.e. NC >> 1) produce better designs and are less sensitive

A. Comi(q/io, S. A. Zmrosl Journal o/ Econor~~~ Dynamics and Couirol 21 (1997) 1445-1470 1457

8.24.00 *

I .476 I .477 I .478 1,479 I ,48 1.481 1.482 I.483

CEROE

Fig. 6. Trade-off between average CEROE and total running time and time to the best solution.

CEROE in the range (1.4765, 1.4795) were obtained with candidate list of length NC = 5, in the

range ( 1.4785, I .48 19) with NC = IO, and in the range (I .48 I6,1.4823) with NC = 20.

to the starting point. Note, however, that improvements tail off as we go from NC = 10 to NC = 20. The number of searches, N,, for a given candidate list size should be somewhat smaller than the size of the list.

While larger candidate lists produce better results, they are also computationally more expensive. Fig. 6 illustrates the trade-off between CEROE and execution time, averaged over ten runs. Note that exploring candidate list of size NC = 20

produced marginal improvements in CEROE (at the third decimal point) while it doubles the solution time. In particular, note that the total running time for NC = 20 is much higher than the time it took to reach the best solution. A strategy

such as (10,5) is the overall best for this model.

4. Parallel tabu search

The fact that the computer runs are computationally intensive motivates an

investigation into parallel computing as a way to speed up the execution of the model. Parallelism could be exploited at two levels. At a low level we could parallelize the simulations required in computing @(L., R, M, K). Parallelism in

1458 A. Cunsiglio, S. A. Zenios I Journal of Economic Dynamics and Conrrol 21 (1997) 1445-1470

computing prices and holding period returns has already been investigated - using different environments - by Hutchinson and Zenios ( 199 1) and Cagan et al.

(1993). In both cases the results were very encouraging and these approaches are directly applicable to the bond design simulations as well.

At a higher level we could parallelize the generation and selection of moves in the TS algorithm. We investigate this form of parallelism here since it is more

general. In particular, we could substitute the function @ that evaluates a callable bond design with functions that evaluate other types of instruments, and the parallelized TS methodology would remain applicable. We also point out that

TS is considered, in general, well suited for parallel computations, but research contributions in this directions are scant. See, for example, Crainic et al. (1993)

for some recent work. Our investigation adds to this line of research.

The parallel computing environment we use is that of heterogeneous networked workstations, using software systems like Linda from Scientific Computing Associates ( 1993) or PVM of Geist et al. (1993). Such environments are read-

ily available to financial institutions. Furthermore, the use of Linda or PVM en- sures portability of the code to alternative parallel environments. Our experiments

were performed on a network of IBM RS6000 running PVM at the University of Cyprus, Nicosia, and a similar network running Linda at Yale University, New Haven, CT.

4.1. Parallelization strategies

We use a master/slave scheme for parallel computations, see e.g. Censor and Zenios (1997). A master processor allocates tasks to slaves processors, and co-

ordinates the exchange of information. In the parallel TS heuristic the master determines if the prescribed number of candidates N, have been obtained, con- trols that the prescribed local searches, N,, are performed, and updates the current solution and the tabu list.

The distribution of work to slaves is considered by the master processor during the following phases of the algorithm:

1. The allocation of eligible directions to slaves, in order to select those directions that are added to the candidate list.

2. The allocation of directions from the candidate list to slaves for evaluation of a move.

The allocation of tasks can be done statically - i.e., before the slaves start processing their tasks - or dynamicalfy - i.e., as the slaves complete their tasks. We distinguish the following parallelization strategies:

1. Static/static: In this strategy the master allocates the 81 possible directions among the slaves. Each slave proceeds to evaluate its own directions, and add suitable ones to a local candidate list. The master is notified when a new direction is added to the candidate list. When a total of N, candidates are found

A. Corwiglio. S. A. Zerlios I Journal of’ Economic Dwmnics and Control 21 (I 997) 1445-1470 1459

the master notifies the slaves to start executing searches along the directions

in its own members of the candidate list. This parallelization strategy has very

low communication requirements. However, it does not ensure load balancing

among the slaves since they may have to search along a different number of candidates from their local lists.

2. Static/dynamic: In this strategy each slave sends its own candidates to the master. Once the candidate list is completed the master assigns, in a cyclic

order, candidates to the slaves who execute the local searches. This paralleliza- tion strategy has higher communication requirements than the static/static.

However, it improves load balancing among slaves since all of them evaluate a similar number of candidates from the global list.

3. Dynamic/dynamic: In this strategy the master assigns directions to any idle slave to determine whether they should be added to the candidate list. Once NC candidates are found the evaluation of local searches is parallelized dy- namically as in the static/dynamic strategy.

The static/static strategy was implemented on a network of IBM RS6000 work-

stations running PVM. It achieves a relative speedup of 4.5-5.0 on 8 processors (see Fig. 7). Adding more processors does not improve performance. We point out, here, that evaluation of speedups for a natural algorithm, such as TS, is a dif- ficult task since the algorithm follows different trajectories when implemented in parallel.

The static/dynamic strategy achieves better speedups, up to 6 using 12 pro- cessors, and performance can be improved even further with the addition of more processors. Fig. 8 shows the speedup of this strategy as a function of the number of processors. The dynamic/dynamic strategy has the highest po- tential for achieving high speedups. In general our implementation using the dynamic/dynamic strategy was 20% more efficient than the implementation based on the static/dynamic strategy.

We comment now on the quality of the solutions obtained by the sequential

and the parallel TS implementations. For sufficiently large number of iterations the sequential and the parallel TS achieve the same solution (see Fig. 9). The parallel TS implementations, however, reach a better solution in fewer iterations than the serial implementation. Since there are no known bounds on the solution of the model, and TS is usually terminated after some fixed number of iterations, there is more confidence in the solution obtained by a parallel implementation after only few iterations.

5. Comparisons with other methods

We now report experimental results to establish the effectiveness of the bond design model, and the efficiency of tabu search in solving this model. First, we analyze the performance of bonds designed using the classical portfolio

1460 A. Consiglio. S. A. Zeniosl Journal of Economic Dynamics and Control 21 (1997) 1445-1470

7, --__

-c-6Proc.

1 5 9 13 17 21 25 29

5,

I I-+2Prw.l

I 5 9 13 I? 21 2s 29

Fig. 7. Relative speedup vs iteration number using TS. The two figures were obtained using two

different starting points.

immunization approach, and compare it to the performance of the bonds obtained using the global optimization model. Second, we compare the solutions and ex- ecution times of the tabu search heuristic and a simulated annealing algorithms for solving the global optimization model.

4. Cansi~~lio. S. A. Zenios I Journul of Economic Dynamics und Conir.01 21 (1997) 1445-1470 146 1

12

IO

8

16

4

2

0

1 2 3 4 5 6 7 a

Number of workstations

9 10 11 12

Fig. 8. Relative speedup vs number of processors using the static/dynamic parallelization strategy

of TS.

1.60

1.00

ri 080

E'

0.60

0.40

0.20

0.00 -4

0 1 2 3 4 5 6 7 8 9 10 II I2 13 14 I5

lterrtioa number

Fig. 9. CEROE vs iteration number for serial and parallel implementations of TS. The parallel

versions approach the best solution in 4-5 iterations, while the sequential version approaches the

solution in 11 iterations.

i 462 A. Consiglio, S. A. Zenios I Journal of Economic Dynamics and Control 21 (I 997) 1445-1470

5.1. Modeling benchmark: duration-matched bond design

It is common in fixed-income portfolio management to match assets and lia-

bilities on a duration basis. A duration matched portfolio implies that the asset- liability gap will remain invariant, for small changes of interest rates. Such port-

folios are called immunized. While the shortcomings of portfolio immunization are well understood - and we do not stop to repeat them here - this approach

remains quite popular. We designed several bonds and for each one we calculated its option adjusted

duration (see, e.g., Golub et al. 1995). Fig. 10 illustrates the duration of bonds

with varying lockout and maturity parameters. From this figure we can easily identify those bonds that have the same duration as the target mortgage assets. For each such bond we can then calculate its CEROE. The best CEROE thus obtained was 1.3313. The CEROE achieved when solving the global optimiza-

tion bond design problem (see the next subsection) is 1.482. This experiment illustrates the effectiveness of the developed model in generating good bond de-

signs.

Bond thn-ation

Fig. 10. (1) Option adjusted duration of bonds with varying lockout and maturity parameters.

(2) The combination of maturity and lockout periods that result into callable bonds with identical

option adjusted duration like the target mortgage asset.

A Consiglio. S. A. Zenios I Journal qf‘ Economic D)wamics and Control 21 (I 997) 1445-1470 1163

Lcckoul

Maturity

Fig. 10. Continued

5.2. Solution technique benchmark: simulated annealing

Recall that the application of a nonlinear optimization algorithm (Powell’s method) produced two localy optimal designs with CEROE 1.23 16 and 1.4056, and this observation motivated an investigation into global optimization algo- rithms. A general purpose, and robust, method for solving global optimization

problems is simulated annealing, see, e.g., Kirkpatrick et al. (1983). The global optimization bond design problem - for an identical test problem - was solved

by Holmer et al. (1993) using simulated annealing. They report an overall best bond design with a CEROE of 1.477. The simulated annealing would reach this solution in 8-48 h, depending on the starting point. The TS algorithm achieves an overall best design with a CEROE of 1.482. This solution is reached in 2-3 h. Both codes were implemented in C, use the same simulation codes for evaluating

@(L, K, M, R) and were run on a DECstation 5 100. This experiment illustrates both the effectiveness and the efficiency of our TS

heuristic in solving the global optimization bond design problem. Fig. 11 il- lustrates the difference between the return of the target mortgage asset and the callable bonds designed using TS and simulated annealing, respectively.

1464 A. Consiglio. S.A. Zeniosl Journal oj’ Economic Dynamics and Control 21 (1997) 14451470

Interest rates

Fig. 11. Difference between the return of the target mortgage backed security and return of the callable

bonds designed using TS and simulated annealing, respectively, with changing interest rates.

6. Conclusions

We have presented a general framework for designing callable bonds in order to fund a specified collection of assets. The developed framework is based on the ideas of holding period returns. It analyzes these returns for both the asset and the liability side of the balance sheet in an integrated fashion. The decision maker’s risk preference is taken explicitly into consideration through the use of a utility function and the concept of certainty equivalent of return. This is an example of integrated jinancial product management.

Our results have shown that bond designs based on the ideas of duration matching do not necessarily reflect the decision makers’ best preference under their utility function and for the holding period scenarios. Superior designs can be obtained using the global optimization model developed in this paper. Inspite of the complexity of the developed model the TS heuristic has generated good solutions, that improve upon previous results obtained using simulated annealing. We also point out that the computational complexity of these models can be somewhat alleviated with the use of distributed computations.

Finally, we want to bring up an important issue deserving ftnther investigation. There is no reason to expect that a single bond will be used as the liability to

A. Consigiio, S. A. Zrriios I Journal of Econot17rc Dynamics and Conrrol 171 (1997) 1445-1470 I465

fund the assets. Instead, a portfolio of bonds will typically be used. Deciding

the composition of the portfolio - assuming that the bond designs are given a priori - is in itself a difficult problem. This problem can be solved using models of stochastic optimization, such as those developed by Golub et al. (1995).

However, it is possible to integrate the bond design framework developed here with stochastic optimization models to design the best combination of bonds to

be put in the portfolio. Preliminary results, applied to the exact same problem addressed in this paper, show that even better CEROE can be expected by the optimal design of a portfolio of bonds.

Appendix A. Computing scenarios of holding period returns

In earlier publications (Golub et al., 1995; Mulvey and Zenios, 1992) we described the simulation procedure for computing scenarios of holding-period returns of mortgage backed securities. A similar procedure is developed here for computing scenarios of returns for callable bonds. We start first with a model for pricing a callable bond. Once a pricing model is developed, the generation of scenarios of holding period returns is straightforward.

A. 1. Pricing a callable bond

The value of a callable bond depends on three basic components: The term structure of the Treasury (i.e., risk free) interest rates, the credit risk of the issuing institution, and the value of the embedded option.

A.I.1. The term structure model Risk-free interest rates are generated by the binomial lattice model of Black

et al. (1990). This model generates distributions of interest rates that are arbitrage

free, and that price correctly the points on the spot yield curve. The volatility of these rates is also consistent with the term structure of volatility. The bi-

nomial lattice can be described as a series of base rates {q”, t = 0, 1,2,. . . , T} and volatilities {k,, t = 0, 1,2,. . . , T}, where T denotes the end of the planning horizon (typically 360 months). The short-term rate at any state cr of the bino- mial lattice at some time instance t is given by 5” =q”(k,)Q. Fig. 12 illustrates a simple binomial lattice.

The price of a future stream of risk free cashflows - such as those generated, for example, by a US Treasury security - can be obtained as the net present

value of these cashflows, with discounting done at the rates Yp.

A.l.2. Credit risk The cashflows generated by a callable bond cannot be priced simply by dis-

counting them at the risk free rate. Market participants assume the default, liquidity

1466 A. Consiglio. S. A. Zenios / Journal of‘ Economic Dynamics and Control 21 (1997) 1445-1470

and other risks inherent in callable bonds issued by corporations, utilities or government agencies. The discount rates should reflect these risks. Therefore the

risk free rates must be adjusted by appropriate risk premia. Of primary interest to an institution issuing callable bonds is the credit risk premium. A model for the term structure of credit risk has been developed by Litterman and Iben ( 1991).

In particular, they show how to obtain a vector of risk premia {m} that are used to adjust the risk free rates so that the discounted cashflows of non-callable

corporate bonds are priced correctly. The term structure of the credit risk for an institution is obtained using the

Litterman-Iben model from existing traded securities issued by the institution.

These risk premia are applied to the pricing of new issues of callable bonds.

A.1.3. Value of the option and of the callable bond We can now value the bond. A callable security can be viewed as a long

position in a non-callable bond, and a short position in a call option on that

underlying bond. The price of the callable security is the difference between the non-callable bond value and the option premium.

The non-callable bond can be priced starting at the end of the binomial lattice - when the security is priced at par - discounting its price backwards, and com- puting the average value. Fig. 12 illustrates the procedure on a simple binomial

lattice. If Pi: denotes the price of the non-callable bond at time period t and state cr (with risk-free discount rate r,O =qO(k,)“), then Ph;c” can be computed by applying the recursive equation

p’,. _ 1 P;‘,” + P$‘,O+’ NC - 2 1 + $p* ’

(A.11

where P$’ = 100 (par) for all states Q at maturity T. The value of the call option is calculated using the same discounting method

used above for the non-callable bond. After the lockout period - when the call may be exercised - its value cannot fall below the difference between the non- callable bond and the redemption price; see Fig. 12. If PO”” denotes the price of

the call option at time period t and state 0, and R’ denotes the redemption price, then PA” can be computed by applying the recursive equation:

(A.21

Note that the redemption price is infinite before the lockout period expires, and is par at maturity.

Finally, the value of the callable bond Pda, at period t and state cr, is given as the difference between the non-callable bond price and the value of the call option. That is,

A. Corzsiyliu, S.A. ZeJliosI Journal oj Economic Dynurnics and Control 21 (1997) 1445-1470 1467

(1) Pricing a non-callable bond.

(2) Pricing a call option, callable at RI in one year and at Rz = 100 in two years.

(3) Subtracting the values of the call option from the price of the non-callable bond at each node of the tree we get the price of the callable bonds.

Fig. 12. Pricing a non-callable bond, a call option and a callable bond using a binomial lattice.

I468 A. Consiglio, S. A. Zenios I Journal of Economic Dynamics and Control 21 (I 997) 144%1470

(A.3 1

A.2. Holding period returns

The return of a callable bond during a holding period (r), under an interest

rate scenario s, is given by

(A.4)

Here Ps is the value of the bond at the end of the holding period (assuming it has not been called), FS is the accrued value of any coupon payments made during the holding period plus the accrued value of the cashflow generated if

the bond were called, and PO is the current market price. In order to evaluate the holding period return we need to specify the interest rate scenario (s) that was realized during the holding period. This scenario will determine whether the bond is called or not during the holding period.

To generate scenarios of holding period returns we sample paths from the

binomial lattice starting at the origin t = 0 and ending at t = 2. Each path corre- sponds to a scenario S. We then traverse the path on the binomial lattice, check whether the call option has been exercised or not, and compute the value of FS taking also into account any coupon payments. (All relevant information has al- ready been generated during the pricing calculations described above). The price of the bond at the end of the holding period under the given scenario, Ps, is also available from the pricing calculations.

Appendix B. Estimating a coupon rate (C) that prices the bond at par

The coupon rate C of a callable bond - while it can be considered as a design parameter - is typically set at the value that prices the security at par. Par- issuance of interest paying bonds simplifies buyers’ tax accounting by eliminating

any original issue discount. To find a bond’s par-issue coupon rate we have to apply an iterative algorithm: Starting from some initial guess of the coupon we price the security using the equations (A.l)-(A.3). If the security is priced at a premium the coupon is decreased (and if it is priced at a discount the coupon is increased) by a suitable amount. The pricing procedure is then repeated, and further coupon adjustments are made until the security is priced within 0.01 basis points of par. If g(C) denotes the function that maps coupon rates into prices (i.e., using Eqs. (A.l)-(A.3)), then we are searching for a value Co such that I = 100. Any algorithm for solving nonlinear equations can be used to solve this equation for CO. In our work we use the method of van Wijingaarden, Dekker and Brent, see Press et al. (1989).

A. Consrylio. S. A. Zenios I Journal of Economic Dynamics and Control 21 (I 997) 1445-14 70 I469

Acknowledgements

We would like to acknowledge the support we received from Scientific Com-

puting Associates and the Computer Science Department at Yale University in carrying out the experiments with Linda. Discussions with N. Carriero, L. Cagan and M. Schultz are acknowledged too. Partial funding provided by NFS grant

CCR-91-04042, HPC-Finance grant of the European Union, and the GNAFA and GNIM groups of Consiglio Nazionale delle Ricerche (CNR) and MURST 40%, Italy.

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