Building financial optimization models

The financial work presents interesting challenges to those who build optimization models and wish to implement them by developing the appropriate software. These challenges fall into two interrelated categories: first, the environment in which the model operates is, in most cases, a very risky one and the model's output is expected to be used almost instantaneously. Second, the software based on the model is most often required to be run by the end-user (e.g., a financial manager or a trader), without the intervention and expert advice of optimization specialists. The implication of the risky environment, as far as model development is concerned, is that failing to provide a correct recommendation for optimal action can have very serious financial consequences for the model's user. It is, therefore, vital that the model perform in a foolproof manner in all circumstances. The development of such a model is by no means trivial. For example, without some precautionary measures a very small change in the data input of an otherwise perfectly operating linear program can result in failing to find an optimal (or any feasible) solution and, as far as the end-user is concerned, this means failure. Such failures can be avoided by careful model formulation and by appropriate preprocessing of the problem data that detects inadmissible inputs.

Introduction

Bond portfolio immunization and dedication are two widely used methods for controlling interest-rate risk, see, e.g., Platt (1986), Bierwag (1987), Fabozzi and Pollack (1987) and chapter 1 in this volume. The two approaches are seemingly very different. Under the immunization scheme, interest-rate risk is controlled by matching durations on assets and liabilities. When the interest rate moves, both sides of the balance sheet are affected in the same manner, leaving net present value virtually unaltered. Under the dedication scheme, the aim is to match cashflows to a degree which is economical under particular reinvestment and borrowing assumptions. As long as reinvestment rates do not fall below the scenario rates and borrowing rates do not rise above the ones assumed, the strategy ensures that enough cashflow is generated to gradually withdraw liabilities as they occur.

Which strategy should a particular investor choose? The traditional answer is that an immunized portfolio is riskier than a dedicated portfolio, hence also promises higher returns. If the investor is willing to accept some risk, he should establish an immunized portfolio, otherwise he should dedicate. However, this answer leads to additional questions. For example, to decide on the strategy, the investor may wish to know which risks are assumed in immunization, how large they are, and whether they agree with the investor's market views.

Introduction

Traditionally portfolio immunization has referred to the problem of finding a set of bonds whose present value matches that of a predefined set of liabilities. This problem arises, for example, in the context of pension fund management where one seeks a way of investing a portion of a fund in a manner that will protect its value relative to the fund's projected liabilities. In this way, regardless of external factors such as interest-rate changes, the fund's assets and liabilities will have similar values. Any surplus funds may then be used to ensure capital growth.

In the early 1980s an extremely volatile interest-rate environment and high levels of interest rates gave prominence to models for portfolio immunization. In particular, pension fund managers found that they were valuing assets and liabilities inconsistently. The effect was dramatic when short-term rates approached 20% whereas internal actuarial discount rates were conservatively set to approximately 5%.

Since that time the use of “proper evaluation techniques” has been legislated in the United States and, more recently, in certain European countries as well. Optimization models for portfolio immunization are now used routinely for managing fixed-income portfolios.

It is curious to note that immunization models are almost identical across investment banking environments.

Introduction

The use of optimization techniques to design or modify bond portfolios is over twenty years old. The development of these techniques parallels the growth in computer power over the same time frame. Large-scale problems solved on the most powerful computers in the late 1960s may now be solved on run-of-the-mill mainframes and in some cases on personal computers. Bradley and Crane's paper (1972) provides a view of one of the first such models and reviews earlier published research. Hodges and Schaefer (1977) formulate a linear program (LP) that minimizes the cost of a portfolio while requiring that a series of cash outlays be met on time. Income taxes are assessed by multiplying coupons by the marginal tax rate. Alexander and Resnick (1985) formulate a linear goal program that maximizes the portfolio yield-to-maturity while immunizing the portfolio with a given duration and placing bounds on bond quality and on the mean absolute deviation of individual bond durations. More recently Ronn (1987) developed an LP approach which seeks to buy “underpriced” bonds and sell “overpriced” bonds while simultaneously considering taxes and bond portfolio cashflows. Leibowitz (1986a and b) presents a general overview of the issues involved in bond portfolio dedication and immunization procedures.

Introduction

Mortgage-backed securities — MBS for short — have emerged in the 1980s as an important class of securities. The total size of outstanding mortgage debt, the potential for MBS market growth, and the complexity of mortgage securities present unique opportunities and challenges for financial analysts. At the end of 1985 outstanding mortgage debt in the United States was approximately $2.2 trillion, of which nearly 70% was in residential mortgages. This sum dwarfs the more established corporate and government debt markets. To date only some 20% to 30% of the outstanding residential debt has been securitized via the issuance of mortgage-backed securities, but MBS represent the fastest growing segment of the debt markets. Growth in outstanding MBS was dramatic into the mid 1980s — from $3 billion in MBS outstanding in 1979 to $500 billion by the end of 1986. Trading in MBS after issuance has also increased significantly in the last few years — from $243 billion in 1981 to $1.2 trillion in 1985. Interest in MBS is not restricted to the US alone; for example, 90% of all residential debt in Denmark has been securitized.

MBS facilitate the flow of funds from the ultimate lenders in the capital markets to the mortgage borrower.

Introduction

An old adage states that “where you stand depends on where you sit.” In the context of asset allocation, this translates into: sound investment advice must be based on the investor's unique situation. Some investors accept great risk for hopefully greater rewards. Others attempt to immunize their portfolios for fear of loss, however slight. Most investors fit somewhere between these two extremes.

A related issue involves the costs for making changes to an existing portfolio. Several asset categories require a substantial payment for either entry or exit, for example, real-estate or venture capital. Other investment categories generate small commissions, but trade in a relatively thin market. Therefore, institutions and other large investors may pay substantial market impact costs whenever a change is made in the makeup of their portfolios. Smaller capitalized US stocks display this feature — estimates range from 80 to over 400 basic points for each side of these transactions.

Despite the importance of turnover and transactions costs, most asset allocation programs treat the rebalancing issue in a simplistic fashion. Recommendations do not depend upon the investor's current portfolio; for example, “average” transaction costs are subtracted from expected returns. The rebalancing costs are often ignored.

This chapter develops a systematic approach for rebalancing a portfolio.

Development—general

This book is concerned with indicators of the thrust, direction and impact of socioeconomic change, which can be identified with development. The term is often used as a synonym for growth or advancement, but it is useful to look at the distinction made by Daly and Cobb (1989: 71) between growth, seen as quantitative expansion in the scale of the physical dimensions of the economic system, and development, which should refer to the qualitative change of a physically non-growing economic system in dynamic disequilibrium with the environment.

Growth in that sense has natural limits, and it remains arguable whether such limits also apply to development as far as technology and social organisation are concerned. Both growth and development are forward looking; but as we approach the limits or potential of either, more attention is given to the reverse process of retrogression and ‘de-development’.

Economic, social and technological development can be broadly distinguished, but usually these aspects interact and should preferably be considered together. Social development cannot be separated from the economic limitations imposed by scarce resources; it is often expressed in terms of manpower, equipment or budgets. Social implications of the distribution of income and wealth, or of the impact on national welfare and the environment, are never far below the surface of economic analysis.

Economic activity

Social scientists sometimes look with envy on economic indicators because of their seeming certainty in expressing changing phenomena in fixed and evenly spaced quantitative or value terms presented in graphs or series that lend themselves to further statistical analysis. Statistical expression and illustration are usually more concise and persuasive than verbal description. There is no denying that the multitude of statistics and graphs published in general and specialised publications, or presented on radio and television, have a major role in communicating economic changes to business people and the general public. Any criticism of their occasional inadequacies should be seen in the context of their general usefulness.

It seems to be appropriate to begin a discussion of economic indicators with this proviso in mind, because in a general textbook there is a tendency to dwell on shortcomings and limitations that are not negligible but do not vitiate necessarily the main purpose for which indicators are and can be used. This is not meant to excuse the types of errors that Huff has slated under the telling title How to Lie with Statistics (1954), with examples such as the sample with the built-in bias or the gee-whiz graph, nor the interpretative errors that are discovered so often on close scrutiny of comments on published tables and graphs.

‘Social indicators’

The term ‘social indicators’ has long been used for statistics that are relevant for the analysis of the situation in a particular social field or for society as a whole, similarly as statistics for economic analysis are referred to as economic indicators. We need not enter here into the debate about a precise definition of ‘social’, beyond noting that in some ways it overlaps with ‘economic’ because social demands are subject to economic restraints and because economic processes are linked to their social and societal environment. At most we can say that some indicators belong mainly to the social sphere (e.g. school performance, sporting performance), while others (e.g. exchange rate, productivity) are mainly economic phenomena, or that economic indicators deal mainly with things and money while social indicators are more concerned with people.

Social indicators, in this general sense, go back thousands of years. The early enumerations of the population mentioned in the Bible, and ancient registers of land titles, had a social element of ascertaining the population structure and military preparedness connected with the economic purpose of establishing a tax base. The British poverty studies since the seventeenth century, and two centuries later the family budget studies of European statisticians (Brussels Congress, 1853), mark the beginning of social reporting and the application of indicative statistics. They set the pace for the government collections published in yearbooks that translate social phenomena into the neat numerical language so persuasively developed for economic analysis.

Indicators help us to gain knowledge and understanding. They condense and transform information so that it can be applied to analysis and policy making. They are used in all sciences for their particular purposes. In the case of social indicators, which, broadly interpreted, have been our main subject here, their function has been described as tracing pathways through the maze of society's interconnections (Rice, 1967), or as illuminating the topography of the human landscape (L'Inguiste, 1978). Even monetary or other quantitative indicators should not stand alone as statistical abstractions but should be viewed within a human and social setting that can be described by further indication.

One major theme of this book has been reference to time: what has happened, is happening or will happen. Another has been structure: the analysis of an existing pattern and comparison with other patterns located elsewhere in time or space. A further recurring feature has been the extension of indicative measurement into an ordering function that gives indicators a role in determining and describing what is being measured.

The book has tried to avoid technical obfuscation and to demystify jargon by explaining its meaning. This applies also to the sections on techniques, which select from the general texts a few major processes that are often misapplied or mismatched in their mechanical application to indicators (e.g. averaging, weighting, scaling).

The sections on application represent no more than part of the larger repertory of economic and social types.