Designing Portfolios of Financial Products Via Integrated Simulation and Optimization Models

Designing Portfolios of Financial Products Via Integrated Simulation and Optimization ModelsAuthor(s): Andrea Consiglio and Stavros A. ZeniosSource: Operations Research, Vol. 47, No. 2 (Mar. - Apr., 1999), pp. 195-208Published by: INFORMSStable URL: http://www.jstor.org/stable/223039 .

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DESIGNING PORTFOLIOS OF FINANCIAL PRODUCTS VIA INTEGRATED SIMULATION AND OPTIMIZATION MODELS

ANDREA CONSIGLIO University of Calabria, Italy

STAVROS A. ZENIOS University of Cyprus, Nicosia, Cyprus

(Received September 1996; revisions received September 1997, April 1998; accepted April 1998.)

We analyze the problem of debt issuance through the sale of innovative financial products. The problem is broken down to questions of designing the financial products, specifying the debt structure with the amount issued in each product, and determining an optimal level of financial leverage. We formulate a hierarchical optimization model to integrate these three issues and provide constructive answers. Input data for the models are obtained from Monte Carlo simulation procedures that generate scenarios of holding period returns of the designed products.

The hierarchical optimization model is specialized for the problem of issuing a portfolio of callable bonds to fund mortgage assets. The upper level optimization program is multimodal, and a tabu search procedure is developed for its solution. Empirical results illustrate the efficacy of the developed models in designing the appropriate structure of the callable bonds and making optimal allocations of equity and debt among the designed products. Computational results with the implementation of tabu search- on both serial and parallel computers-are also presented.

Financial innovation-the unforecastable, unantici- pated changes in financial instruments-has been

characterized as a "revolution" in financial economics by M. H. Miller (1986). Financial innovation is even associated with the printing of paper money as a reaction to the British government's prohibition of the minting of coins by the North American colonies. However, "significant and successful" financial innovation, using Miller's notions, is usually associated with the more recent stream of exotic financial instruments like futures, options, synthetic securities with embedded options, and the like.

The rapid pace of financial innovation has highlighted the need to understand this phenomenon: Why is it hap- pening and why do the securities that are used have the form they do? In a series of papers, Miller (1986), Merton (1988), and Ross (1989) provide complementary-to each other-answers on the drivers of financial innovation. In short, Miller argues that taxes and regulations are the sand in the oyster that cultivate the innovations. Merton ana- lyzes the production function for new instruments and pays attention to transaction costs as a reason for the existence of derivative securities. Ross develops an agency theoretic model to analyze the role of financial institutions and of marketing. He argues that financial innovation arises nat- urally from the supply and demand of agency constrained participants, while marketing costs shape the features of the innovation.

Following this line of research on the drivers of financial innovation, several papers were published to explain why the specific securities that are used have the form they do. See, for example, the collection of papers in the special

issue of Financial Management (Allen 1993) or the book by Allen and Gale (1994). Despite the extensive literature on this topic, and the even more extensive list of novel financial instruments that appear continuously, very little has been done to develop a scientific methodology for the design of products. The need for a scientific approach to security design becomes more pressing as the complexity of synthetic securities increases. Ross, correctly, writes:

"[Financial economists] are called upon not only to value the new

instruments and the new [dynamic trading] strategies [that make

heavy use of the new instruments] but to design them as well. Like

engineers who use physics, financial engineers use the techniques

of modern finance to build the equivalent of bridges and airplanes" (Ross 1989).

Almost 10 years after he delivered these comments in the Presidential Address to the American Finance Associa- tion, the design of financial products still remains more an art than a science. The analog of concurrent engineering for the design and manufacturing of products has not yet been adopted in the domain of finance.

In a recent paper Holmer and Zenios (1995) developed the framework of integrated financial product management (IPM), and they argued that it is the appropriate framework for analyzing financial intermediation and thereby improving its efficiency. They distinguish two major mana- gerial problems of an intermediary: the micro-product management and the macro-product management. The former deals with the process of designing, pricing, marketing, capitalizing, and "manufacturing" (i.e., funding) a single product. The later deals with the relative size of

Subject classifications: Finance: security design and pricing, innovation. Marketing: new product design and pricing. Programming: nonlinear utility maximization. Area of review: FINANcIAL SERVICES.

Operations Research Vol. 47, No. 2, March-April 1999 195

0030-364X/99/4702-0195 $05.00 C) 1999 INFORMS



196 / CONSIGLIO AND ZENIOS

different lines of business (i.e., product mix and asset allocation) and on controlling the interactions between the risk-return characteristics of equity invested in the different lines to produce a desirable overall return on equity (ROE). (See Equation (2) for definition of ROE.) They also identified several real-world settings where IPM has been adopted, and they provided empirical evidence for the positive impact of IPM on the productivity of financial intermediaries. The significance of integrating financial decisions in a common framework has been demonstrated further in more recent works, such as those of Carino et al. (1994) and Correnti (1997).

Following the original development of IPM, Consiglio and Zenios (1997) proposed specific models for the micro- product management. In particular they developed and implemented a model for designing callable bonds for the needs of a mortgage-holding agency, such as the Federal National Mortgage Association. In this paper we take an- other step toward the development of a scientific methodology for the design of financial products. In particular we develop specific models for designingportfolios of products and apply the models to design the debt structure of an agency issuing a portfolio of callable bonds. In this respect our paper provides a model for addressing the macro- product problem of IPM. It also extends the work of our previous paper from the design of a single callable bond to the design of a portfolio of such instruments.

In order to develop a model for designing a financial product we first need to specify a measure of quality. Only then we can develop a model that optimizes the design with respect to our measure of quality. Section 1 gives the finance-theoretic framework of the specific problem we address and develops the appropriate measure of quality. Section 2 develops the formal mathematical model for solving the problem. The application of the model to the design of a portfolio of callable bonds is discussed in Sec- tion 3, and the solution technique based on tabu search is discussed in Section 4. Section 5 discusses empirical results, and Section 6 concludes the paper. An appendix gives some background material on callable bonds and describes the Monte Carlo simulation procedures used in the paper.

1. PROBLEM DEFINITION

In this section we describe the specific problem we are addressing. Certainly, the problem of financial innovation for an institution is complex and multidisciplinary, and we do not attempt to capture all its aspects in a single model. For example, marketing considerations are important but are not addressed in our study. However, significant asset decisions are integrated in our framework with the decisions on leverage and product specification.

We consider an institution operating in an uncertain world, and we capture uncertainty in the form of a discrete set of scenarios denoted by Q-{1, 2, . . ., S}. The institution holds a portfolio of assets, and we use ,4 to denote 1

plus the rate of return of this portfolio under scenario s E IQ, during some holding period of interest. It is assumed here that the portfolio of assets is given a priori, and scenarios of holding period returns for this portfolio can be computed using standard pricing models. (Monte Carlo simulation models for computing scenarios of holding period returns were introduced in our earlier papers; see, for example, Consiglio and Zenios 1997 or Mulvey and Zenios 1994.) The institution funds these assets through the issue of debt (D) and the investment of shareholders' equity (E).

Debt is issued in the form of a portfolio of financial products, and the yield on this debt is denoted by RL. We assume that the yield on the debt, during a holding period, can be calculated using appropriate pricing models but we do not assume that the debt structure is given a priori. Indeed, determining the debt structure is a key aspect of the institution's problem.

The position of the institution at the end of the holding period is given by the terminal wealth:

WTs = (D + E)rs - RL. (1)

The return to shareholders is given by the return on equity (ROE):

ROEs_ WTs (D + E)rA - RL (2) RE=E E(2

This equation reveals a well-known important relation between financial leverage (i.e., the debt to equity ratio) and return on equity. For scenarios under which the return on assets exceeds the interest rate on debt, then financial leverage will increase the return on equity. When the return on assets falls below the cost of debt, then financial leverage will decrease the return on equity. However, as the expected return on assets exceeds the required yield on debt, it follows that financial leverage has two effects on the return on equity: It increases the expected return on equity, while at the same time it increases the variability of this return. Therefore, depending on the shareholders' aversion to risk, there is an optimal level of financial leverage. An empirical study verifying this effect for the property and casualty insurance firms in the United States was conducted by Babbel and Stakin (1991), Figure 1. See also the discussion of their work by Holmer and Zenios (1995). We observe from these empirical data that there is an optimal level of financial leverage (around 4) for this particular industry segment. We also observe that the market rewards firms with lower surplus duration, i.e., those firms that engage in effective asset-liability management strategies, thus reducing their interest rate exposure. It is worth pointing out that the average leverage of companies in the sample is around 3.5, very close to optimal, while the average duration surplus is around 12-13, close to the saddle point! Financial leverage is under regulatory control, and the evidence here suggests that the regulatory process has



CONSIGLIO AND ZENIOS / 197

Leverage

Market Reward

Surplus Duration

Figure 1. Market reward for different levels of leverage and for different values of duration of the gap between assets and liabilities, for a large sample of property-casualty insurance companies. (Source: Babbel and Staking 1991)

converged to a point that is best for the market. The situ- ation is not as encouraging on the asset-liability management dimension. We observe large duration gaps, which led Babbel and Staking to proclaim that "it pays to practice ALM [asset-liability management]," and Holmer and Zenios to argue in favor of an integrated view of financial product management.

In order to determine the optimal level of leverage, however, we need to determine the cost of debt under different scenarios s E fl. (Recall that in our setting the value of the other parameter that appears in the ROE calculation, i.e., the return on the asset portfolio iA is ex- ogenous to the institution's decisions.) The cost of debt is determined by the amount of debt raised through issues in different financial products. Hence, the institution has to determine simultaneously the structure of its debt (i.e., the amount issued in each product) and the optimal level of financial leverage. A model that provides an answer to this question is developed in the next section.

However, there is an additional level of flexibility for our institution. The return on the debt is a function not only of the amount issued in different types of products, but also of the yield of each particular product. Herein lies the opportunity for financial innovation and the design of new products. Since the institution is issuing the products, their yield can be adjusted with the judicious setting of some design parameters.

In conclusion, the problem of maximizing the return on equity can be broken down into three interrelated decisions: (1) Design the financial products to be used to issue debt; (2) Specify the amount issued in each product; (3) Determine the optimal level of financial leverage. The design of a product is the micro-product management of IPM, while the allocation of equity among the designed products is the macro-product management of IPM. The

above decisions have to be made in an integrated fashion, taking into account the uncertain environment-as described by the scenario set fl-and the shareholders' risk aversion.

We point out that our model assumes asymmetric information: The institution issues its own debt, which is priced at the prevailing market rates, while the amount of debt does not affect either the market rates or the cost of equity of the institution. Were we to impose market equilibrium conditions-whereby the market rates adjust in response to the new debt issued, and the institution's cost of equity increases as increasing debt raises credit risk-we would obtain the well known indifference between debt and equity implied by the Modigliani-Mifler theorem. For application of the model in a real-world setting we would need to determine if the magnitude of the issued debt would affect the cost of equity or market rates and model such effects appropriately.

1.1. Risk Aversion and Certainty Equivalent Return on Equity

We have thus far broken down our problem into the specification of a policy-which consists of the product design parameters, the structure of the debt, and the amount of leverage-but have not yet specified a measure to characterize a "good" policy. We do so in this section, thereby clarifying the notion of quality of the product design. In particular, a specific product design is considered to be of high quality if it maximizes our measure of a "good" policy.

To compare alternative policies we need to incorporate the decision maker's risk aversion for different levels of ROES. Let 'U denote the utility function of the decision maker. Then the issuing institution will prefer a policy that maximizes the expected utility

1 - 61 1 (ROEs). (3) S SEO

With each policy we associate its Certainty Equivalent Re- turn on Equity (CEROE) defined by:

cGL(CEROE) = - RL(ROEs). (4) S SEfl

Assuming nondecreasing utility functions, we can rank policies by their CEROE, and the one with the highest value of CEROE is the best policy. A set of products are then considered to be of high quality if they allow the institution to decide a debt structure and the optimal leverage with a large value of CEROE.

We note that our discussion focuses on the ranking of designs using a utility function of ROE at the end of some planning horizon. There is no guarantee that the designed products will have large utility values of ROE at earlier, intermediate periods. The problem of temporal utility as- sessment is important when consumption of wealth at each point of the planning horizon is under consideration, and it is a difficult problem. We do not address this problem




here because consumption of wealth is not a consideration for our problem, whereby an institution is issuing debt in order to hold some asset. A utility function of ROE at intermediate time periods can be incorporated into our model-if such a function were available from economic theory-although it would make the underlying optimization problem nonseparable, and hence more difficult to solve.

We also note that an alternative objective for choosing a "good" policy is by specifying a design that limits the Value-at-Risk (VaR) of the decision maker's surplus. (VaR is the maximum value of losses that can be realized within a given planning horizon with a given probability, see, e.g., the RiskMetrics Technical Document, J. P. Mor- gan, New York, 1996. The probability is typically taken to be 95%, and the time interval is one day for actively traded positions, one month for portfolio management, and 10 days for the regulatory requirements of the Bank of Inter- national Settlements. Surplus is defined as the market value of assets minus market value of liabilities.) VaR is a popular descriptor of risk, and designs that result in low VaR of surplus can be considered designs of high quality. However, VaR has zero measure, and furthermore, it tells us nothing about the decision maker's views when the portfolio looses less than VaR, but not much less. In this respect we consider the analysis with expected utility maximization as more complete and superior to VaR, but we stress that the models could be adapted to generate policies with optimal (i.e., minimal) VaR.

2. MODEL FORMULATION

This section develops the models that give constructive answers in an integrated fashion to the issues facing our institution. We formulate a hierarchy of optimization models. For a given universe of products and their associated returns, the institution can decide the optimal mix of products to be issued and the optimal level of leverage. A nonlinear program can be formulated to resolve these two decisions. New products can then be introduced, and those redefine the input data to the nonlinear optimization program. Our goal is to find the design of products that, together with an appropriate product mix and an appropriate financial leverage, yield an overall highest CEROE. This is then the design of the best quality.

2.1. Notation

First we introduce some notation:

J denotes the set of financial products {1, 2,..., N}. Each product in this set is characterized by some design parameters.

p' denotes the vector of design parameters for product j E J. For example, if the product is a callable bond the design parameters could be the lockout period and the schedule of redemption prices before maturity. We assume that n parameters

are needed to characterize each product, so that pi is a vector in R .

p = ((pl)T (p2)T (pN)T)T is the concatenated vector of design parameters for all products in J.

<C() denotes 1 plus the rate of return of the jth instrument. The return of each product depends on the design parameters pi and on the scenario s E Q.

y = 1 is the vector denoting the amount (in face value) of debt issued in each product j E J. This vector specifies the debt structure.

X = (N-),2 1 denotes the price per unit face value of each product in J.

With this notation we can express the yield on the debt portfolio by

N

RL(y, P) = E yjrj(pi). (5) j=l

The return on equity (Equation (2)) can now be written in a way that reflects its dependence on the policy of our institution:

(D + E)r' - R'(y, P) ROEs(y, P, DIE)= EA L (6)

We point out that the calculation of the return on equity involves the evaluation of the nonlinear Equation (5) in y and P. Furthermore, the values of rj((p') as a function of the design parameters p' are not available analytically but are obtained through a Monte Carlo simulation procedure, such as the one described in the appendix and in Consiglio and Zenios (1997) or Mulvey and Zenios (1994).

2.2. The Optimization Models

We now formulate two models that address the problems facing our institution. The first model assumes that a universe of designed products is given and determines the optimal debt structure (y) and the optimal level of financial leverage (E). The second model expands the universe of available products by designing new products, thereby improving the solution obtained by the first model. The two models are used hierarchically; therefore, their joint solution solves in an integrated fashion both the problem of debt structure and of financial leverage.

2.2.1. Debt Structure and Optimal Leverage. For a given value of the vector P the quantities rj(p') are parameters, evaluated using Monte Carlo simulation procedures. Let CEROE(P) denote the certainty equivalent of optimal value of the following optimization program, which determines the optimal leverage and the debt structure:

Maximize-E X t(ROEs(y, P, D/E) (7) yEV, EEW+ S sEfl

s.t. Equations (5) and (6), (8)




N

_ ujyj = D, (9) j=l

D + E = 100, (10)

N

(D +E)r' - X yjrJ(p') : 0, foralls E Q. (11) j=1

The last inequality, taken together with the nonnegativity of E, ensures that return on equity remains nonnegative under all scenarios, and hence it guarantees the solvency of the institution for any debt structure and leverage obtained by the model. Changing the objective function (7) to read MinimizeEER+ E, we can also calculate the minimum equity required to ensure solvency under extreme case scenarios.

The model is a single-period one, maximizing the expected utility of ROE over a target holding period. No decisions to rebalance the portfolio are made during this period, and any cashflows generated by the assets or liabilities are invested in the riskless rates and are captured in the calculations of holding-period returns. It is possible to formulate a multiperiod stochastic programming model for this problem (see, e.g., Censor and Zenios 1997, chapter 13 for an advanced textbook treatment of this topic). Such a model could incorporate portfolio decisions made during the target period and capture intertemporal aspects of the problem.

Several issues deserve further elaboration at this point, because they are relevant for the implementation of the model in a real-world setting. First, it must be recognized that the universe of scenarios should encompass all events that are relevant to the institution and against which the institution wants to protect itself from bankruptcy. In our empirical work later on we consider only interest rate and credit risk. An insurance firm might wish to consider also events such as large earthquakes and hurricanes that are independent from interest rate and other market risks (see, e.g., Correnti 1997). Second, the choice of a utility function could pose a problem because shareholders are a diverse group, and a single utility function might not be appropriate. The specification of the objective function should reflect such considerations, either by aggregating the diverse groups in a single (average) risk preference category, or by allowing for tradeoffs between alternative utility functions. Finally, we point out that the debt leverage decision and the equity mix decision are connected. While in our work we assume that the equity mix is exog- enous and given by a single variable E, it could also be computed as the value of an equity portfolio with stochastic returns, Es. The optimization model can incorporate the appropriate equation for the definition of Es, although we would then need to develop Monte Carlo simulation models that correlate the return of the equity portfolio with the return of the issued debt. It is important to note that the integrated framework developed here can serve as an evaluator for these macro decisions faced by the institution.

The three issues raised here can be relevant for different institutions and must be carefully addressed before the model is implemented in a real-world setting. It is important to recognize the limitations of the model as proposed here, but also to understand that these important exten- sions can be accommodated. The adoption of a multiperiod stochastic programming model would also allow us to incorporate additional real-world issues such as transaction costs, taxes and other statutory requirements.

2.2.2. Optimal Design of Products. The design parameters of the financial products are now the decision variables of the following program:

Maximize CEROE(P) (12) PE

s.t. P E X, (13)

where CEROE(P) is the certainty equivalent of the objective value of (7)-(11), and X denotes some constraints on the design parameters (e.g., maximum allowable maturity). Note that this is a global optimization program because, in general, the function CEROE(P) is not unimodal in P. This was demonstrated, even for the simple case of designing a single product, in our earlier paper (Consiglio and Zenios 1997). This problem can be solved using techniques from global optimization, bearing in mind that the objective function is not available analytically but is obtained as the solution to the optimization program (7)-(11).

A systematic procedure for solving this hierarchy of optimization models is summarized as follows:

Initialization: Parametrize the design characteristics of the financial products and assume some initial values. The specification of the holding period, generation of the scenario set fQ, and estimation of the holding period returns for the target asset, are determined during the initialization step.

Step 1. Generate holding period returns for the target product designs, using the same scenarios and holding period used to estimate the returns of the target asset. Sce- narios of holding period returns are generated using appropriate Monte Carlo simulation procedures (see, e.g., Consiglio and Zenios 1997 or Mulvey and Zenios 1994).

Step 2. Solve the nonlinear program (7)-(11) to obtain the optimal debt structure and leverage. The optimal objective value of this program is the CEROE for the product design simulated in Step 1. If termination criteria are satisfied, stop. Otherwise, proceed with Step 3.

Step 3. Adjust the design parameters of the financial products and return to Step 1.

This iterative procedure seeks product designs that allow the institution to specify an optimal debt structure and optimal leverage with the highest possible CEROE. Of particular interest is the specification of rules for adjusting the design parameters in Step 3 in order to maximize the CEROB obtained in Step 2. The rules should take into account the fact that there exist multiple locally optimal




Redemption prices (dollars)

Time 100

L K M

Figure 2. 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 security can be called at par.

solutions to the product design problem. It may be possible to find a design that is optimal only in a small neighborhood of the design parameters. A tabu search heuristic that searches for a globally optimal solution from among the local maxima is developed in Section 4.

3. APPLICATION TO THE DESIGN OF CALLABLE BOND PORTFOLIOS

We now consider an application of the general framework developed above to design a portfolio of a particular type of products, namely of 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 cal'lable 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 types and combina- tions of callable bonds that are sold to finance their assets.

We consider the design of a portfolio of N bonds. Each callable bond j = 1, 2, . .. , N, is unambiguously specified by the following four design parameters; see Figure 2.

1. Lockout period (Lj): This is the period, following issuance, during which the bond can not be called.

2. Redemption price at first call date (Rj): This is the pricep at- which;-1 thel: bo-nd canl be1 called ti the I-kfirs-t call

daeR hspiei tapeimaoepr

3. Time to maturity (Mj): This has the usual meaning for all fixed-income securities.

4. Schedule of redemption prices: 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 prices decline linearly from a starting value of Rj at the end of the lockout period, to par at some period Kj. The term during which the security can be called at a premium (i.e., Kj) 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 Lj S K1 S Mj.

Using the notation introduced in Section 2, we associate the design parameter vector pi of the jth bond with the tuple (LjR^M)Kj). The holding period return parameters rj(p') for a given value of the design parameters (LjR- jM,Kj) are computed using the Monte Carlo simulation procedures of Consiglio and Zenios (1997, Appendixes A and B). The constraint set X on the design parameters (Equation (13)) is given by

X= {p'[p': 0, Li -Kj -Mj, j = 1, 2, . . ., N}. (14)

Thus we have completely specified the model for designing portfolios of callable bonds.

4. SOLUTION METHOD USING TABU SEARCH

The upper level optimization program (12)-(13) is compli- cated by the facts that multiple locally optimal solutions exist, and that the design parameters take discrete values. In particular, the date parameters Lj, Mj, Kj are in months, while the price parameter Rj is expressed within an accu- racy of 1 bp (100 basis points (bp) 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 and 1989b for introduc- tions). A TS heuristic defines a neighborhood N(x) of the current solution x and specifies a rule (i.e., a move) by means of which each point of the neighborhood can be reached from the current solution.

In order to avoid cycling (i.e., situations where x E

N(N(x))) TS puts the most recent solutions in a tabu list. New trial solutions that are "too close" to those in the tabu list will be labeled as tabu. However, if a tabu solution has a value of the objective function better than any previously obtained solution, then this solution 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 solution.

In order for the algorithm to search regions of the solution space never investigated, an intensification strategy is adopted. This is a long-term strategy, which can be viewed as a learning process through which the heuristic becomes




aware of the structure of the solution space and drives the iterates towards new regions. The following subsection describes the elements of the TS heuristic developed to solve the portfolio design problem.

4.1. The Tabu Search Heuristic

The TS heuristic generates bond designs P for several bonds. These bonds are then passed to the lower level optimization program (7)-(11) that evaluates the objective value of the proposed design, i.e., CEROE(P). The iterates of the TS heuristic is therefore a sequence of proposed bond designs.

4.1.1. Neighborhood and Moves. The current solution P0 is a vector of design parameters as defined in Section 2 and (p')0 = (L?R5M74? )T for j = 1, 2, . .. , N. Let PE

be a vector whose components p' for all j = 1, 2, ... , N, are given by the vector (LeREMEKE)T. We define the neighborhood of P, N(P), as the set of solutions P' such that

p GE [p PIE P + PIE] (15)

The number of all possible solutions in the neighborhood N(P) is large, and an exhaustive evaluation is prohibitive. For this reason only a random sample is selected out of N(P), at each iteration of the TS heuristic. The sampling has to be performed in such a way that the temporal parameters satisfy the precedence relationships Li

- Kj S

Mj. To do so we first sample at random a value in the neighborhood of the maturity parameter

Mt = rand(Mj + M, M1- M)* (16)

The value of parameter Kj' is then selected in a way that its value does not exceed Mj, i.e.,

K; = rand(Kj + min(Kj + Ke Mj), Kj - Ke) (17)

Finally, to satisfy Lj - Kj, we select L,' as

Lj= rand(L1 + min(L1 + Le, Kj), L1 - Le). (18)

This way of generating trial solutions guarantees the feasibility of all bonds that will compose the portfolio. It is important to underline that this random sampling is restricted only to movements inside the neighborhood N(P) (i.e., it is a local strategy). The global search strategy for generating new neighborhoods is strictly based on deterministic memories, as discussed later in the section on diversification strategies.

4.1.2. Tabu Bond Designs. Let Nc be the number of trial bond designs, denoted by Si, i = 1, 2, . .. , Nc, sampled in the neighborhood of the current solution N(P?), as described above. For each of these proposed designs we calculate the optimal debt structure y by means of the model (7)-(11), thereby constructing a portfolio, and evaluate the associated CEROE(Si). These portfolios are sorted in de- scending order of their CEROE(Si) and the best nontabu portfolio is chosen as the starting point of the next iteration of TS. A portfolio is considered tabu if the distance

between the bond space of the portfolio Si and bond designs TI, 1 = 1, 2, ..., K, in the tabu list is less than a given tabu distance. (See Glover 1994 for a discussion on tabu distances.) The parameter K is the tabu tenure, i.e., the number of steps during which the bond space of a proposed solution is considered tabu.

Whenever a specific solution S' is chosen among those sampled in the neighborhood then its bond space is in- serted in the tabu list for the duration of the tabu tenure. Our TS heuristic remembers the K most recent solutions and keeps them in the tabu list. There is no formal justifi- cation about the size of the tabu list; it depends on the application. Through some preliminary testing it is possible to find out an appropriate value.

A tabu portfolio may be relaxed from being tabu if it satisfies certain aspiration criteria. In particular, the TS procedure will choose a tabu portfolio design Si if the CEROE(Si) is higher than the CEROE of any previously encountered design.

4.1.3. Diversification Strategy. In our earlier paper (Consiglio and Zenios 1997) single bond designs were selected using only a recency memory. For the portfolio design application we noticed that, due to the complexity of the solution space, it is necessary to employ a diversification strategy to ensure that the algorithm would generate high quality bond portfolios. The solution space for the portfolio design model depends on the number of bonds in the portfolio, and this enlarges its size dramatically.

A diversification strategy is based on the concept of frequency memory. In particular, it records the number of times a given attribute of the proposed solution Si has been selected during the search. This information can be used to generate solutions whose attributes have not yet been employed, i.e., solutions with low frequency counts. This diversification strategy is based on a deterministic memory and will drive the search toward regions of the solution space that were never investigated.

In our implementation, the attributes of a proposed solution are the values of each design parameter. For instance, the possible values for the temporal parameters are in the range [0, 360]. We build a structure whose components are vectors with dimension equal to 360, and we define one such vector for every temporal parameter for each bond. These vectors contain the frequencies with which a given parameter was hit, for all four parameters of a bond and for all bonds in the proposed solution.

The diversification strategy is activated when a monitored variable does not improve its value for a number of steps. Usually, the monitored variable is the value of the objective function. In particular, whenever the value of the best CEROE obtained so far does not change for a number of iterations, the diversification strategy is activated. The diversification strategy simply restarts the algorithm from the solution whose components have the lowest frequency count.




12.00- -

10.00

A

8.00 - _

AR -iff

r-iff

6.00 0

F 4.00 Bd A

-A- -Bond b

-s-Bond C

2.00

0.00

-2.00

-4 .0 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

5.50 6.50 7.50 8.50 9.50 10.50 11.50

Interest rates

Figure 3. Holding period returns of a mortgage-backed security (MBS) and three alternative callable bonds, under different interest rate scenarios. The horizontal axis denotes the geometric mean of the term structure of interest rates during the holding period (36 months); the vertical axis denotes holding period returns. 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 which has a much higher rate of return than the asset for all interest rates above 8.5%.

5. EMPIRICAL RESULTS

We applied the model described above to design a portfolio of callable bonds to fund a characteristic mortgage asset. The target asset is a mortgage-backed security with weighted average coupon 9.5% and weighted average maturity of 360 months. The holding period is taken to be 36 months. Figure 3 illustrates the holding period return of the asset under different interest rate scenarios and also illustrates the returns of three different callable bonds under the same scenarios. It is clear that bonds B and C are preferable than bond A, but the choice between B and C is not obvious; neither it is easy to see how one could com- bine these securities to get a portfolio of bonds that tracks

the asset return more closely. Such a portfolio is designed in this section. The upper level optimization program (12)-(13) is solved using the tabu search heuristic, and the lower level program (7)-(11) is solved using standard nonlinear optimization packages.

In all experiments we use a logarithmic utility function, unless stated otherwise. Note that since a minimum amount of equity is always assumed to guarantee solvency, the return on equity is always positive and in the domain of the utility function. The logarithmic utility function is the limiting case of the one-parameter family of isoelastic utility functions, qt(ROE) = ROEY/,y, as y -- 0. The use of such functions is predicated on the decision maker's current wealth equal to 1, and in our empirical investigations we set E = 100, and drop the normalization constraint (10).

5.1. Optimal Leverage

We investigate first the effects of financial leverage, assuming that the universe of products is given. In Figures 4 and 5 we illustrate the effect of leverage on the CEROE. The three diagrams in this figure plot the CEROE against financial leverage for two different bonds, and for a portfolio of bonds. First we observe a nonlinear, concave relation between financial leverage and CEROE. This is the relation we expect based on the discussion of Section 1. These diagrams allow us to identify the optimal leverage for a given debt structure. We note also that the CEROE curve for the portfolio of bonds achieves a higher peak than the curves corresponding to debt structures consisting of a single bond, thus illustrating the effect of debt structure on CEROE.

We also point out the similarity of the CEROE curves obtained by our model with the empirical results reported in Babbel and Staking (Figure 1). The CEROE curves resemble cross-sections of the Babbel-Staking diagram in the direction of "Market Rewards vs Leverage." This ob- servation is purely qualitative. Note, for instance, that the optimal leverage in our example is in the range 12 to 18, which is much higher than most financial firms will toler- ate. This is mainly because of the low level of risk-aversion implied by the logarithmic utility function employed in this experiment. Using higher risk aversion parameters-the very plausible values y = -1 to -10-in the utility function yields optimal debt-to-equity ratios comparable to those observed in the Babbel-Staking study. Figure 6 illustrates the effect of the shareholders' risk aversion on the optimal level of financial leverage. As dictated by the models in Section 1, the higher the level of risk aversion the lower is the debt to equity ratio preferred by the institution. A verification of our model vis-a-vis the Babbel- Staking study using more realistic test cases is worth further investigation.

5.2. Optimal Debt Structure

We now investigate the effects of debt structure with the optimal design of the callable bond portfolio. We consider first a debt structure whereby all debt is raised by issuing a single bond. The design of this bond is determined using




1.5 -9- Optimal Eut

1.4 CEROE =1.570

1.3

1.2 0

0.9 -U- Minimum Equity

0.8 - Equity = 0.056 CEROE =0.779

DIE

1.65 -

1.55 Equity = 0.067

1.45

1.35

W 1.25 0

~)1.15

1.05-

0.95- -U- Minimum Equity

0.85 - Equity = 0.055 CEROE = 0.798

m @0 N- em N ON @0 e4 e4 II eq m - e4 leN a @ W II 4 0

I,C ') l e - 0 0 ON ON Go Go G0 N- N- N- ~-C ~-C ~-C ~C 1

D/E

Figure 4. Effect of financial leverage on CEROE for different debt structures. These diagrams correspond to debt structures whereby all debt is raised, in each case, through issues in a single bond.

the simpler models developed in our earlier paper (Con- siglio and Zenios 1997). The CEROE of this bond was calculated to be equal to 1.482, and leverage is determined by the minimum amount of equity required to guarantee solvency. Using the models of this paper, which allow us to determine also the optimal leverage, we obtain a CEROE of 1.559 when all debt is raised through issues in a single bond. Solving the full-blown portfolio models of this paper we are able to design a portfolio (consisting in our particular example of issues in two bonds) that achieves a CEROE of 1.647. The equity required to achieve peak

CEROE is lower when we invest in the portfolio of bonds than when we issue a single product.

The composition of the portfolio depends on the amount of equity invested, as illustrated in Figure 7. In particular, when equity is large all debt can be raised through the single bond that has the lowest expected yield. As equity is decreased there is an optimal debt structure that consists of a combination of the more risky bond with the one that has lower yield. The optimal debt structure for different levels of leverage is illustrated in the figure.




1.6 - p maE

/ ~~~~Equity =0.065

1.5 CEROE = 1.647

1.4

0 1.3

1.2-

1.1

_Minimum Equity 1 - Equity = 0.047

CEROE = 0.99

N 00 W O N M II W z le 00 0 N N 0 o o II II N N l m m

-i t ?~ 0 F ?~ " "t t - 11 oi IR cI 1.0 tIo "It :o rstpi 0 0 t l N

e 0

Nq II

4 0 ON O N 00 0 0 N- Or O

D/E

Figure 5. Effect of financial leverage on CEROE for different debt structures. This diagram corresponds to a debt structure that combines issues in both bonds.

5.3. Efficacy of the Model: Duration-Matched Bond Designs and Tracking the Return of the Assets

To illustrate further the efficacy of the proposed methodology we report some additional empirical results that evaluate the

performance of the designed portfolios vis-a-vis our target asset. It is common practice in fixed-income portfolio management to manage assets and liabilities on a duration- matching basis. Duration matched portfolios maintain a constant asset-liability gap for small changes of interest rates.

------- CEROE ( gamma =-1 ) CEROE (gamma=O) - CEROE (gamma=1)

2 Max CEROE = 1.9915 D/E = 16.58

1.8

1.6

1.4 - Max CEROE = 1.55

D/E= 12.61 -

W 1.2 ------- , - - 4 Max CEROE= 1.2421

D/E= 1.68

0.8 /

0.6

0. - I I I I I I I- I F - l I N

e en 00

00ri~ eC e, --00e n

00 0WI 666

D/E

Figure 6. Effect of risk aversion on optimal financial leverage and corresponding CEROE. In this experiment we use the one-parameter family of isoelastic utility functions given by GtL(ROE) = ROEY/,y. A parameter value y = -10

corresponds to a relatively high risk aversion, while y = 1 corresponds to risk neutrality. As y -O 0 we recover the log utility function used in the rest of the paper.



CONSIGLIO AND ZENIOS I 205

Face Values Bond_1 - Face Values Bond_2 1 - ._ _ i ._ _ . ._ _ - _ _ _-_ _ _ _ _ ______

0.9 j M mimum Egiity Optimal Equity

0.9 - .. ................ -......... . ....... ....... -------........

Minimum Equity

0 .3 - - - -- - -----------

.... . ..... ....... ... ....... ...... .... ....... . ..... * -- -

0 20.12 18.86 17.75 16.75 15.85 15.04 14.38 13.85 13.22

D/E

Figure 7. Debt structure is affected by level of financial leverage. The optimal financial leverage achieves the highest CEROE and results in a balanced debt structure that consists of issues in two bonds, as shown in the diagram.

As a benchmark against which we compare our results, we evaluate the CEROE obtained when debt is raised by issuing a single bond that has the same option-adjusted duration as our target assets (such bonds were designed in Consiglio and Zenios 1997). The CEROE of the duration matched bond is 1.3313 which is much inferior than the CEROE of 1.647 obtained when raising debt with the portfolio of bonds designed with the models of this paper.

Finally, we calculate the tracking error of the yield on the issued debt against the institution's mortgage assets, under different interest rate scenarios. This calculation illustrates the tight correlation between the return of the given assets and the return of the designed bond for a wide range of interest rate scenarios. Figure 8 illustrates the results. We observe from this figure that the debt structures corresponding to issues in a single bond have, poten- tially, higher tracking error than the debt structure consisting of issues in a portfolio of bonds. The tracking error is, overall, quite small (even in the single-bond issues) providing indication of the efficacy of the developed models in designing products according to the prespecified target, even when the target is scenario dependent, and hence uncertain.

5.4. Computational Results

We report now some computational results with the solution methods to illustrate the efficiency with which the underlying models can be solved. In general the models described in this paper are solvable with standard optimization software and high-performance workstations, and such technology is widely available in financial institutions.

However, substantial improvements in performance can be realized with the utilization of parallel computing.

5.4.1. Solving the Debt Structure and Leverage Optimiza- tion Model. The optimization model (7)-(11) was formulated in our empirical experiments with 19 scenarios and 5 bonds. It is a single-period model, with a holding period of three years. Scenarios are selected using random sampling from a binomial lattice of the term-structure with the use of variance reduction techniques, and extreme case scenarios are also included. The resulting optimization models have 23 constraints and 6 variables. They are solved using the GAMS package, which is embedded in the tabu search

0.25

,f 015J

0.05 . .. ....... - ........................ . .

Intrest rates

Figure 8. Tracking error between mortgage assets and issued debt of callable bond under different interest rate scenarios. Horizontal axis denotes the geometric mean of the term structure of interest rates during the holding period (36 months).




1.65

1.63- '

1.618234

1.61 - /1.600293 /9 ~~~~~~1.593278

1 4 10 12 18 25 30 35

Tabu Tenure

Figure 9. Value of the optimal CERGE for different tabu tenure values k. Small and large values of tabu tenure force the algorithm to local optima.

1.65 |0 Recency Memory 0l Recency Memory plus Diversification|

(. 1

1 2 3 4 5 6 7 8 9 10

Experiments

Figure 10. Effect of the diversification strategy on the value of optimal CEROB. Ten experiments were carried out using different starting solutions.




1 4

14 t- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- ------ - - - - - - -- - - - - - --- -------;-- - - --- -- -- -- -- - 10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

6Pt l

4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

4 +-- -- -- -- -- -- -- --- - - -- - - -- -- -- -- -- -- -- -- ---- -- -- --- - -- - - - - - - - - - - - - - - - - - - - - - - - -

O - I I I I I I I I I I I I I I I

1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17

Number of processors

Figure 11. Relative speedup of the parallel tabu search implementation on a Parsytec CC-16 utilizing up to 16 processors.

bond design procedure by means of a call to the ANSI-C function system ( ). Each optimization model is solved in approximately 2 seconds of CPU time on a PowerPC 604.

5.4.2. Solving the Model for Optimal Design of Products. The TS heuristic was applied to the problem of designing the portfolio of callable bonds to fund a mortgage asset as discussed in Section 3. The heuristic starts from an arbi- trary (randomly generated) proposed bond design. Starting the TS from "good" initial points (for example, solutions obtained by matching the duration of the assets with the duration of the portfolio of bonds) does not produce portfolios of higher quality. Indeed, the TS heuristic would occasionally converge to a good solution quickly from a starting point with a low CEROE, while it would take longer to converge when starting from a point with higher CEROE.

The local search of TS is not exhaustive but is carried out over a random sample of 30-35 trial bond designs. We found that evaluating more points in the neighborhood of the current solution increases substantially the solution time without improving the value of CEROE. For each bond in the neighborhood of the solution the model needs to generate scenarios of holding period returns. The Monte Carlo simulation procedure takes approximately 10 seconds on a PowerPC 604. Evaluation of a trial point, i.e., simulating holding period returns for 5 bonds and determining the optimal leverage and debt structure, requires 40-60 seconds. Out of this time only approximately 2 seconds are used for the solution of the optimization model for the debt structure and leverage.

The TS heuristic evaluates trial points using the tabu procedure and the diversification strategies discussed ear-

lier. We investigate first the effect of the tabu tenure parameter K. For small values of K the algorithm gets stuck in local optima. Large values overconstrain the solution space forcing the heuristic to search in restricted areas. Figure 9 provides some insights about the best value of the tabu tenure parameter K. The length of the tabu list should be between 15 and 20. For different values TS gets stuck in local minima. In all the experiments K iS set to 20. The TS procedure terminates when the solution has not improved for a given number of iterations.

Figure 10 shows the effect of the diversification strategy on the optimal value of the CEROE. We observe that for seven experiments the diversification strategy improved the final results, generating portfolios of higher quality. However, for three experiments, diversification affected the algorithm negatively. This behavior is probably caused by the random component of TS when sampling the neighborhood of the current solution.

5.4.3. Parallel Computations. It is possible to implement a naive parallel version of the TS heuristic, whereby multiple trial points are evaluated concurrently at each step of the heuristic by multiple processors. We used a master/slave architecture for such a parallel implementation on a Parsytec CC-16 with 16 PowerPC 604. The implementation is based on a proprietary implementation of PVM 3.0. The TS procedure is terminated when the best solution does not improve for 10 iterations. The total running time is approximately 7.5 hours on a single processor of the Parsytec system, and approximately half an hour on the fully configured system. Figure 11 illustrates the speedup of our parallel implementation with increasing number of processors.




6. CONCLUSIONS

We have shown how the macro-product management of a financial intermediary can be formulated as a hierarchy of optimization models. Input data for these models are obtained from Monte Carlo simulation procedures of holding period returns. Such procedures are fairly standard in the finance literature. Our models provide the framework for integrating simulation models, which are descriptive in nature, with optimization models, which are prescrip- tive. The result is a modeling framework that provides constructive answers to problems arising in macro- product management.

The developed models have been specialized and applied to the problem of designing portfolios of callable bonds for a mortgage agency. Empirical results have demonstrated the efficacy of the models in providing constructive answers to the issues facing the agency. The tabu search heuristic procedure, that was developed to solve the difficult global optimization program, has also been proven effective in identifying product designs of high quality.

APPENDIX

The Appendix can be found at the Operations Research Home Page in the Online Collection:

http://grace.wharton.upenn. edu/-harker/opsresearch. html.

ACKNOWLEDGMENTS

This research was partially supported by an INCO-DC grant on HPC-FINANCE (No. 951139) of the Directorate General III (Industry) of the European Commission. The authors also acknowledge the comments of two anony- mous referees and the associate editor, which led to substantial sharpening of the discussion of the models. Comments are also acknowledged from the participants at the EURO Working Group meetings on Financial Model- ing in Crete and Dubrovnik and seminar attendees at Uni- versity of Geneve (CH), Universities of Bergamo and Calabria (IT), The Technion (IL), and the Wharton School (USA).

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Designing Portfolios of Financial Products Via Integrated Simulation and Optimization Models