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Need For Numerical Computing Environments Why Matlab

MATLAB is an interactive computing environment, providing both basic and sophisticated functions. You may use built-in functions to solve possibly complex but standard problems, or you may devise your own programs by writing them as M-files, i.e., as text files including sequences of instructions written in a high-level matrix-oriented language. Moreover, MATLAB has a rich set of graphical capabilities, which we will use in a very limited fashion, including the ability of quickly developing graphical user interfaces. The unfamiliar reader is referred to appendix A for a quick tour of MATLAB programming. Some classical numerical problems are readily solved by MATLAB functions. They include This and much more is included in the basic MATLAB core. More complex versions of these problems may be solved by other MATLAB ready-to-use functions, but you have to get the appropriate toolbox. A toolbox is simply a set of functions written in the MATLAB language, and it is usually provided in...

Glossary of Matlab Terms

A list of Matlab terms used in this text. For a complete list, see Matlab documentation. rand produces random uniformly distributed values on 0,1 x rand(m,n) randn produces random standard normal (Gaussian) variates x randn(m,n) realmax the largest real number representable in Matlab reshape changes the size of a matrix without changing the total number of authors, given the variety of computer languages available. On the one hand, it is useful to have working examples of code in the book and to make the code available to readers for immediate use. On the other hand, using a specific language in the text could obscure the essence of the numerical routines for those unfamiliar with the chosen language. We believe, however, that the latter concern can be substantially mitigated by conforming to the syntax of a vector processing language. Vector processing languages are designed to facilitate numerical analysis and their syntax is often simple enough that the language is transparent and...

Setting Up the Matlab Workspace

We suggest the following ways of organizing the MATLAB software environment. 2. Store your model files and any related data files in an entirely different directory, drive, or location other than the MATLAB pro gram files. You will probably install, delete, and reinstall various versions of the software many times, and keeping program files separate from data files is always good practice. 6. Document all your custom-programmed files, including Simulink models and MATLAB script and function files (m-files). See Documenting Your Models on page 53.

Matlab Scripts and Functions

Most of the work done in the MATLAB environment is performed by command files that end in the suffix .m, such as print.m. These files may have been included as part of the MATLAB software, created by the user, or created by a third party. Because of the suffix, these files are often collectively called m-files.

Drawbacks of Matlab and Simulink

In the impressive list of attributes above, there are two notable absences easy to use and inexpensive to acquire. However, economics is about trade-offs. Through this book and others recommended here, we can help the interested reader quickly use MATLAB to perform advanced analyses and create impressive graphics. It is still harder to perform simple calculations in MATLAB than it is to perform the same tasks in a spreadsheet. It is sometimes harder to perform rudimentary analysis (such as a present value, straightforward financial statement analysis, or exploratory data analysis) in MATLAB than in other software. Reporting results, especially of simple calculations, also takes more work. However, the power and other advantages have convinced us that the environment is the best available for many uses in demanding analytical tasks such as those presented in economics, finance, public policy, and business.

Conclusion Using XML in MATLAB

XML and its variants are rapidly becoming a world standard for exchanging information, and the structured nature of the files provides significant advantages for business economics. There are already rudimenatary XML tools available in recent MATLAB versions, and we expect the XML functionality in MATLAB, both in native versions and in tools programmed by the user community, to grow significantly. 403 The launchpad is an optional view within the MATLAB desktop, which appears on the screen when you start the program. If it does not appear on your screen, try the desktop command.

Introduction to Matlab Programming

We give here a brief outline of the MATLAB basics, referring to the user manual for a full treatment. You may also type demo to see a demonstration of both MATLAB and the toolboxes you are interested in. Actual use of the features we describe is illustrated in the remainder of the book. A rich online documentation is available in the MATLAB environment the reader should take advantage of this whenever a piece of code in the book is not clear. A.l MATLAB ENVIRONMENT MATLAB is an interactive computing environment. You may enter expressions and obtain an immediate evaluation

Handle Graphics in MATLAB

One of the most powerful features of MATLAB is its Handle Graphics, which allows a nearly unlimited number of graphical possibilities, each of which can be programmed meticulously. This offers careful users enormous capabilities to present the results of their analysis. The disadvantage of this powerful feature is the demand it places on users to correctly specify a large number of options whenever they create a graphic. If you have been frustrated by the inability of more common software packages to correctly illustrate data, then you will welcome the power of MATLAB'S Handle Graphics. If, on the other hand, you are satisfied with the rudimentary, preprogrammed options available in most spreadsheet programs, you will probably find MATLAB graphics much too tedious.

Troubleshooting MATLAB

Many problems in MATLAB can be resolved by looking at the MATLAB Help menu. This should be the first resource for understanding the commands, functions, and environment. Our experience indicates, however, that those using MATLAB for business economics will eventually run into the following problems, for which we provide troubleshooting suggestions below 3. Incorrect MATLAB file references These can stem from improper installations, moving files, or unexplained computer problems. In addition, problems with Excel Link are covered in Appendix B. Finally these troubleshooting tips are based on MATLAB versions 6, 6.5, and 7. Always check the help information for the specific version you are using.

Checking Matlab Paths and References Windows

We can suggest the following for MATLAB installations on Microsoft Windows machines 1. Check the matlab.ini file in your Windows directory (or equivalent) to ensure that it refers to the proper directories. 2. Check the Windows path parameter to ensure that the MATLAB root directory is listed (e.g., D MATLAB r13). 3. We do not recommend running MATLAB or, for that matter, anything important on Windows operating systems that do not have the more stable kernel that exists in the versions designed for use in business environments.418

Matlab Program Notes

Optimization software is quite common. The MATLAB function fminunc.m, for unconstrained minimization, part of the Optimization Toolbox, is the one used for the quasi-Newton gradient-based methods. It has lots of options, such as the specification of tolerance criteria and the maximum number of iterations. This function, like most software, is a minimization function. For maximizing a likelihood function, we minimize the negative of the likelihood function. The scaling functions for transforming variables to ranges between 0,1 or 1,1 are in the MATLAB Neural Net Toolbox, premnmx.m.


We have found that MATLAB and its companion simulation software, Simulink, offer unparalleled power, reliability, transparency, and flexibility to analyze complex economic phenomena and graphically display the results, as described below Power The ability to use almost every mathematical technique now in use and custom-program many more gives the MATLAB user extraordinary capabilities. Reliability MATLAB code is close to a high-level computer language such as C. It therefore does not have the undocumented features that are common in operating systems and popular office software, particularly the versions that are designed for home or light office use on Microsoft Windows platforms. Transparency All MATLAB functions, formulas, and data can be accessed by the experienced user. This means there should be no black box calculations. Even intermediate and temporary data in the midst of being calculated in a function file can be viewed through the debugger. Flexibility MATLAB was not...


MATLAB already uses XML for Help pages. You can see the files in .xml format by browsing through the Help directory.401 Thus, when you use MATLAB Help, you are already using XML to help organize your work and exchange information. The fact that appears to be seamless is part of its beauty. MATLAB in recent versions contains a small set of commands that allow file input and output of XML-formatted information. These include a limited XML parser (program to identify the information in an XML file), along with commands that serialize XML files and transform them using stylesheets.402 These features of MATLAB are not well documented but are already in use.

Matlab Example

To give the preceding regression diagnostics clearer focus, the following MATLAB code randomly generates a time series y sin(x)2 + exp( x) as a nonlinear function of a random variable x, then uses a linear regression model to approximate the model, and computes the in-sample diagnostic statistics. This program makes use of functions olsl.m, wnnestl.m, and bds.m, available on the webpage of the author.

An Alternative Analytic Framework

An economist who knows some elementary numerical methods and who can write basic Matlab code would have little difficulty solving the problem. The economist would simply write the following elementary Matlab program Consider now the rational expectations commodity market model with government intervention. The source of difficulty in solving this problem is the need to evaluate the truncated expectation of a continuous distribution. An economist who knows some numerical analysis and who knows how to write basic Matlab code, however, would have little difficulty computing the rational expectation equilibrium of this model. The economist would replace the original normal yield distribution with a discrete distribution that has identical lower moments, say one that assumes values y1 y2,. ,yn with probabilities w ,w2, , wn. After constructing the discrete distribution approximant, which would require only a single call to the CompEcon library routine...

Organization of the Book

The second chapter contrasts business economics with academic economics and discusses why simulation models in general, and MATLAB and Simulink in particular, offer advantages to the practicing economist.1 Most readers will benefit from this chapter. 3. The third chapter describes the MATLAB and Simulink environment and provides guidelines for creating and using models in this environment. Style guidelines are presented that help make such models easier to understand and debug. 4. Chapter 4 focuses on data. We present methods to get your data into the MATLAB environment and to report it out. The chapter also contains a specific introduction to the use of Microsoft Excel as an environment for easily collecting, importing, and reporting data. Three appendices to this chapter discuss XML and structured data, creating and using custom data structures, and also importing files of various formats. Most readers will again benefit from these chapters, although those with some familiarity with...

What Is the Financial Toolbox

MATLAB and the Financial Toolbox provide a complete integrated computing environment for financial analysis and engineering. The toolbox has everything you need to perform mathematical and statistical analysis of financial data and display the results with presentation-quality graphics. You can quickly ask, visualize, and answer complicated questions. In traditional or spreadsheet programming you must deal with all sorts of housekeeping details declaring, data typing, sizing, etc. MATLAB does all that for you. You just write expressions the way you think of problems. There is no need to switch tools, convert files, or rewrite applications. With MATLAB and the Financial Toolbox, you can This chapter uses MATLAB to review the fundamentals of matrix algebra you need for financial analysis and engineering applications. It contains these sections Describes acceptable formats for providing data to MATLAB and the resulting output from computations on the supplied data. This material explains...

Using Matrix Functions for Finance

Many financial analysis procedures involve sets of numbers for example, a portfolio of securities at various prices and yields. Matrices, matrix functions, and matrix algebra are the most efficient ways to analyze sets of numbers and their relationships. Spreadsheets focus on individual cells and the relationships between cells. While you can think of a set of spreadsheet cells (a range of rows and columns) as a matrix, a matrix-oriented tool like MATLAB manipulates sets of numbers more quickly, easily, and naturally.

Prerequisite Knowledge

For many of the chapters, a familiarity with (but not necessarily expertise in) the MATLAB software environment. Most of the applications presented here could be accomplished in other software environments, but we provide examples of applications in MATLAB. See References on MATLAB and Applied Economics on page 9. We suggest some resources on these topics under References on MATLAB and Applied

Accessing Files and Online Resources

Many of the MATLAB program files described in this book, including script and function m-files, and Simulink models, are available online from the author or publisher. Other information about the content of the book, including corrections of any errors, may also be posted.

Special Linear Equations

The Matlab operator will automatically employ Choleskv factorization, rather than L-U factorization, to solve the linear equation if it detects that A is symmetric positive definite. Matlab has special routines for efficiently storing sparse matrices and operating with them. In particular, the Matlab command S sparse(A) creates a version S of

Books for Other Fields

Those interested in applying MATLAB in other fields will often find multiple texts on specific applications. There are over 300 published books on using MATLAB in fields such as engineering, signal processing, mathematics, numerical methods, finite analysis, radar, communications, control systems, neural networks, mechanics, filters, earth sciences, electronics, statistics, and probability. This extensive library is one of the advantages of using MATLAB as a software environment. The MathWorks web site lists many of these books by category.

Problems With Newton Methods

Here, we assume that the user has coded a Matlab function f that returns the function value and analytic derivatives at a specified evaluation point x. Execution returns error, the highest absolute difference between an analytic and finite difference cross-partial derivative of f, and its index i and j . A large deviation indicates that the either the ijth partial derivative or the ith function value may be incorrectly coded. The following Matlab script computes the root of a function using a safeguarded Newton's method. It assumes that the user has specified a maximum number maxit of Newton iterations, a maximum number maxsteps of backstep iterations, and a convergence tolerance tol, along with the name of the function f and an initial value x

Matrix Algebra Refresher

Matrix algebra and matrix operations are fundamental to using MATLAB in financial analysis and engineering. The topics discussed in this section include William Sharpe's Macro-Investment Analysis also provides an excellent explanation of matrix algebra operations using MATLAB. It is available on the Web at

The Importance of Good Design

It is easy to underestimate the importance of good design and good design guidelines for economic models. In this chapter, we first define design, and then provide guidelines for designing MATLAB and Simulink models. We also suggest simulation parameters that will work well for most models used in business economics. This chapter provides quite specific advice for those working in the MATLAB environment for those working in other software environments, it suggests good practices that will need to be adapted to the specific environment.

Design Guidelines for mfiles

However, there are some style guidelines that we follow in creating MATLAB commands, including both scripts and functions 1. Describe your m-file. MATLAB reserves a specific part of an m-file for this purpose. The H1 or Help line which follows the line that begins the file will be displayed in MATLAB has several attributes that make it easier to use a hierarchical, modular approach. In Simulink these include the ability to organize portions of a model into a subsystem, and in MATLAB the ability to use functions and subfunctions which operate in a separate workspace.23

Update iteration fx0 fx g0 g end

The CompEcon Toolbox includes a routine qnewton that computes the maximum of a multivariate function using the quasi-Newton method. Suppose that one wished to maximize the banana function f (x) -100(x2 - xf)2 - (1 - x1)2. To apply qnewton, one first codes a stand-alone matlab function that returns the value of the objective function at an arbitrary point function y f(x)

Time and Solver Parameters in Simulink

One of the more confusing aspects of Simulink can be its time and solver parameters. For many challenges in business economics, these can be set quite simply as discrete units of time that are consistent with the conventional description of the problem at hand. Variables that are measured only at specific intervals are known as discrete variables within the mathematically precise MATLAB environment. Most business variables such as sales, revenue, profits, employees, etc., are typically measured in discrete intervals.

Operating Elementby Element

Finally, element-by-element arithmetic operations are called array operations. To indicate an array operation in MATLAB, precede the operator with a period (.). Addition and subtraction, and matrix multiplication and division by a scalar, are already array operations so no period is necessary. When using array operations on two matrices, the dimensions of the matrices must be the same. For example, given vectors of stock dividends and closing prices,

Whos winning in the crisis

As banks attempt to cut the costs of trading, more and more products are being traded electronically without the intervention of human beings. Much of this electronic trading takes the form of 'algorithms' - complicated programs that tell computers when to buy or sell. Recruiters say algorithmic trading jobs were on offer even in the depths of the financial crisis. This is unsurprising given that at Deutsche B rse, for example, 42 of trades on the exchange's electronic platform were performed by algorithmic traders in the first quarter of 2009, up from 39 in 2007 and 20 in 2004. If you're a student who wants to be sure that you're going into an area that's not going to die away next year, algorithmic trading is attractive, says Dominic Connor, director at recruitment firm P& D Quantitative Recruitment. If you want to go into algorithmic trading you will need to be good at applied mathematics. Connor says most of its practitioners studied electronic engineering, artificial...

[xw qnwtrap[10 20[1 0[2 5 integral wexpx1x2

Where x are the nodes, w are the probability weights, n is the number nodes and weights, mu the mean of the distribution, and var is the variance of the distribution. If mu and var are omitted, the mean and variance are assumed to be 0 and 1, respectively. For example, suppose one wanted to compute the expectation of exp(X) where X is normally distributed with mean 2 and variance 4. An approximate expectation could be computed using the following Matlab code

Introduction to Simulink Models

One of the major advantages of working within the MATLAB software environment, especially with Simulink, is the ability to comprehensively model and then simulate complex, dynamic economic phenomena. By dynamic we mean changing over time. Most problems in business economics and finance involve multiple variables that change over time. Thus, simulation models are a natural and powerful tool for these problems. In addition to these distinctions, Simulink models can use the vast library of MATLAB functions, as well as specialized libraries for specific purposes.

Library Functions for Business Economics

MATLAB does not provide a complete environment for all these tasks. However, it provides much more power to perform analysis than spreadsheets, statistical software, and modeling languages alone, and can often accomplish tasks in one software environment that might otherwise take two or three. Library of Tools in MATLAB and Simulink In this chapter, we recommend a number of tools for use in the MATLAB environment, some of which we have programmed specifically for the field. We call these a library of tools partially because the metaphor of shelves of books is somewhat appropriate (a toolbox metaphor is also fitting) and partially because MATLAB provides for the creation of special libraries of m-files and Simulink blocks. We present below some of the building blocks of comparative statics in economics, first discussing theory and then presenting implementations using the MATLAB environment. Once the data have been acquired, we can proceed to specify a demand equation and estimate its...

Quants in Quantitative Finance and Risk Management

For model building and risk management, you need to know how to program fluently in at least one language (C, C++, or Fortran)6. No exceptions. Fluent means that you have had years of experience and you do not make trivial mistakes. Prototyping is important and extremely useful. Prototyping can be done with spreadsheets (also Visual Basic), or with packages like Mathematica, PV Wave, Matlab, etc. However, prototyping is not a replacement for serious compiled code. Knowledge of other aspects of computer science can also be useful (GUIs, databases and SQL, hardware, networking, operating systems, compilers, Internet, etc).

Bibliographic Notes

The subject of solving initial value problems is one the most studied in numerical analysis. See discussions, for example, in Atkinson, Press et al,, and Golub and Ortega, Matlab has a whole suite of ODE solvers, of which 0DE45 and 0DE15s are good for most problems, 0DE15s is useful for stiff problems and can also handle the slightly more general problem

Solving Systems Of Linear Equations

In pencil-and-paper mathematics, when we have to solve a system of linear equations like Ax b, we use matrix inversion to get x A_1b (provided the matrix is non singular). Although MATLAB offers a function, called inv, to invert a matrix, it may sound surprising to the newcomer that this is not used to solve systems of linear equations. More efficient approaches are used. It is not our aim to dwell too deeply on this subject we limit ourselves to the basic concepts needed to understand what MATLAB offers to solve linear equations. Methods for solving linear equations can be broadly classified as direct or iterative. Direct methods have a clearly defined computational complexity, as they yield the result directly within a given number of steps iterative methods build a sequence of solutions whose limit is (under some conditions) the desired solution. For iterative methods, the number of steps is not known a priori, as it depends on convergence speed. They are useful for some large...

[XxAff SolvetaAbBCgg0X0x0

The solution should be computed at the time values specified by t. If there are m time values the outputs should be m x n and m x 1. The program may use rk4 or one of Matlab's ODE solvers. You will need to write an auxiliary function to pass to the solver. Also note that diag(z)z can be written in Matlab as z.*z.

Computational requirements

The computer package SCOM is compared with another computer package, RIOTS 95, which is also based on MATLAB. The algorithmic steps for these two packages may be summarized as follows The differential equation solver must handle discontinuities at gridpoints. Many standard solvers do not. For example, MATLAB 5.2 includes six ordinary differential equation (ODE) solvers, of which only one - designated for stiff differential equations - is useful for solving x(t) u(t) when u(.) is a step-function. A better approach is to modify slightly the well-known fourth order Runge-Kutta method. If t is in the subinterval j N, (j + 1) N , then u(t) must take the appropriate value uj, and not (for example) uj+i when t (j + 1) N. This is easily achieved by recording j as well as t. With this approach, there sequential quadratic programming (MISER, Jennings et al. 1991), MATLABs optimiser constr (MATLAB, 1997), and SCOM) they are based on a widely used mathematical package MATLAB, and

Using the SCOM package

The details of SCOM are given in the next section 3.4. It was written to use MATLAB version 5.2, and implemented on a Macintosh computer. Since MATLAB is designed for matrix computations, vector and matrix operations require very little programming. MATLAB's Optimization Toolbox includes a constrained optimizer, constr , based on a sequential quadratic programming method. Moreover, constr will use gradients if supplied, otherwise will estimate gradients by finite differences. The major part of the calculation is to solve differential equations, in order to obtain function values, and gradients if used.

An Integration Toolbox

The Matlab toolbox accompanying the textbook includes four functions for computing numerical integrals for general functions. Each takes three inputs, n, a, and b and generates appropriate nodes and weights. The functions qnwtrap and qnwsimp imple- In addition to the general integration routines, the Matlab toolbox accompanying the textbook also includes several functions for computing nodes and weights associated with common distribution functions, qnwnorm generates the quadrature nodes and weights for computing the expectations of functions of normal random variates. For univariate normal distributions, the calling sequence takes the form where x are the nodes, w are the probability weights, n is the number nodes and weights, mu the mean of the distribution, and var is the variance of the distribution. If mu and var are omitted, the mean and variance are assumed to be 0 and 1, respectively. For example, suppose one wanted to compute the expectation of exp(X) where X is normally...

Option I Borrow Depreciation Algorithms

Most human behavior, however, does not follow straight lines. Thus, a more accurate adjustment path would often be closer to double declining balance or other methods in which earlier years have faster adjustment. Using MATLAB functions, numerous adjustment paths can be programmed, but in many cases these depreciation functions will do the trick. 137 For a discussion of depreciation, including example MATLAB functions, see Chapter 12, under Auto Depreciation, on page 301.

Computational Algorithm The Steps

Set a vector of parameters par the number of the state components, number of control components, nb the number of big subintervals, ns the number of small subintervals and get the total number of the subintervals by calculating nn nb * ns. Set the MATLAB con-str function's system parameters par(13) 1 (here, 1 represents only one equation constraint in the minimization problem), par(14) the maximum number of function evaluations arbitrary starting lengths of the switching time intervals Set the vectors of upper Step 2. Call the MATLAB constr function. In turn, constr calls the Minimization Program to calculate the minimization of the calling program with respect to the optimal vector um.

Option II Specify Partial Adjustment Model

For example, investment can be considered movement toward a desired level of capital stock. Such models are sometimes called stock adjustment models. Spedfications for partial adjustment models include those based on polynomials, ratios, and geometric series.141 Such models can be readily implemented in MATLAB.

High LowClose Chart Example

First load the data and set up matrix dimensions. load and size are standard MATLAB functions. Add labels and title, and set axes with standard MATLAB functions. Use the Financial Toolbox dateaxis function to provide dates for the x-axis ticks. MATLAB produces a figure similar to this. The plotted data and axes you see may differ. Viewed online, the high-low-close bars are blue.

FFT Extreme Value Theory and Simulation to Model Non Life Insurance Claims Dependences

This paper shows an example of integrated use of three different approaches (Extreme Value Theory (EVT), two-dimensional Fast Fourier Transform (FFT) and Monte Carlo simulation) to model non-life insurance company aggregate losses, taking into account the need for Internal Risk Model development in the light of Solvency II European project. In particular EVT permits the definition of the truncation point between small and large claims. Two-dimensional FFT is used to model not only aggregate losses, but dependence between its basic components too. Finally, Monte Carlo simulation describes large claims behaviour. Collective Risk Model has been developed using Matlab software.

TABLE 41 Sample Excel Sheet for Inputting Data

Define the range of cells into which you want to place the data calculated in MATLAB. Put the ap propriate headings and related formatting items in place as well. The data cells will be blank. 2. Put a getmatrix command in a nearby cell of the Excel spreadsheet which includes the name of the variable in MATLAB and the range of cells allotted for the variables. The diary command in MATLAB The MATLAB Report Generator MATLAB Report Engine Another way to report data and document a model is by using the MATLAB or Simulink report engine. These are separate toolboxes that can be purchased and provide powerful capabilities to produce reports in various formats. The report engines can be difficult to use at first. The reader should first consult the MATLAB help information and try a few sample reports. After gaining some familiarity with the tool, the following suggestions will be helpful You will note that in many of the m-files reprinted in this book there are well-placed commands that...

Modeling Income and Payroll Taxes

Fortunately, we have programmed functions in MATLAB that estimate tax liabilities for all these taxes, given a level of wage earnings. Of course, these programs estimate liabilities, given certain assumptions, and are not suitable for preparing individual returns. However, they are especially well-suited to examining tax policy. We have collected the following functions at the end of this chapter Social Security Taxes These ubiquitous taxes can be modeled in a straight-forward fashion. A function to do so is in Code Fragment 8-1, Social Security Tax, on page 183.

Fspace fundefnbastypenaborder

Endpoints of interpolation intervals in each dimension and order is an optional input that specifies the order of the interpolating spline. On output, fspace is a structured MATLAB variable containing numerous fields of information necessary for forming approximations in the chosen function space.

Portfolio Selection Optimizing an Error

The structure of this case is as follows. In Sect. 9.1, classic MV portfolio optimization is presented together with major issues - such as lack of diversification across assets and instability of the optimal portfolio both in time and along the efficient frontier - which may arise from exposure to estimation risk. Section 9.2 details the resampling technique. Section 9.3 describes the MATLAB functions we used to perform resampling. Section 9.4 presents the results of the in-sample out-of-sample analysis.

Vjac vjac deltawkfunbasfspaceg end

At the termination of the loops, f+delta*Ev is an n by p matrix whose entries give for each state node (row) and action (column) the current reward f plus the expected value next period Ev discounted at the rate delta. The maximum value in each row of f+delta*Ev and the associated column index are the optimal value and action associated with the corresponding state node. In this implementation of vmax, the Matlab vector maximization routine max is used to perform the columnwise maximization in one call, yielding n x 1 vectors v and x that contain the optimal values and actions associated with the n state nodes. Finally, the Jacobian vjac of the collocation function is computed by executing a loop over all K possible realizations of the discrete shock. For each realization of the shock, the state next period g is computed and passed to the COMPECON library routine funbas, which returns the basis function values at that state node.

Classification Of Optimization Problems

'Derivative-freeoptimization methods are the core of a recently released MATLAB toolbox, called Genetic Algorithm and Direct Search. We outline genetic algorithms in section 12.4. Indeed, all MATLAB functions in the Optimization toolbox assume a minimization problem. Solving an optimization problem like (6.1) means finding a point x* S such that Example 6.1 A typical objective function that gives rise to local optima is a polynomial function recall that the oscillatory behavior of high-order polynomials is the reason why they are not well-suited to function interpolation (see example 3.16 on page 179). We may check this with a simple MATLAB snapshot. Consider a polynomial like and use MATLAB to plot it. The plot produced is illustrated in figure 6.1, from which it is clear that there are two local minimizers. One MATLAB function to solve a minimization We see that depending on the starting point, we get the global or the local minimizer. The MATLAB output has been cut a little, but we...

Continuous State Dynamic Programming Toolkit

In order to simplify the process of solving and analyzing continuous state Markov decision models, the CompEcon Toolbox contains a series of Matlab routines that efficiently perform many of the necessary operations. The main routine is dpsolve, which solves purely continuous state and mixed discrete-continuous state Markov decision models of arbitrary dimensional state and action spaces using the dynamic programming algorithms discussed in the preceding section. The routine, for an infinite horizon model, has a calling sequence of the form

Invariance and Noninvariance of Clustering Procedures

Let us consider a very simple example of a set of seven points symmetrically located on a circle (Fig. 1a). Initial information for clustering is given as a matrix of distances between the objects (see Appendix 1). For numeration of objects considered in Fig. 1b the average linkage clustering algorithm realized in Matlab 6.5 builds dendrogram shown in Fig. 1d. We can extract from this dendrogram a partition, for example, on two clusters. On the highest level of dendrogram we will obtain clusters 1,2,7 and 3,4,5,6 which corresponds to partition of the set of objects on subsets a,b,g and c,d,e,f . The explorer, if he does not know the geometrical structure of data, could be very happy to obtain such clear partition of objects on two clusters. But if we change numeration of objects as shown in Fig. 1c then due to a symmetry of data the input matrix will not be changed and average linkage will give the same dendrogram as in Fig. 1d but now partition 1,2,7 and 3,4,5,6 will correspond to...

Computational Algorithms for the Nonlinear Optimal Control Problem

First set the a vector of parameters p, k, r, d, c, t which includes all the parameters in this financial model, p is market capitalization rate, k maximum investment rate, r rate of return to equity, d discount on share price resulting from flotation costs, c a positive constant denoting the responsiveness of the price to changes in earnings and dividends, T planning horizon of the capital budgeting program. Then set parameter par the number of the state components, number of control components, nb the number of big subintervals, ns the number of small subintervals, parameters and get the total number of the subintervals by calculating nn nb* ns. Set the MATLAB Step2. Call the MATLAB constr function. In turn, constr calls the Model2 to calculate the minimization of the calling program with respect to the optimal vector um

Notes and Comments

KP92 discusses many methods for the approximation of paths of SDEs, and proves their convergence. An introduction is given in Pl99 . Possible orders of strongly converging schemes are integer multiples of 1 whereas the orders of weakly converging methods are whole numbers. Simple adaptions of deterministic schemes do not converge for SDEs. For the integration of random ODEs we refer to GK01 . Maple routines for SDEs can be found in CK001 , and MATLAB routines in Hig01 .

The Kendrick Taylor Model Implementation

The Kendrick-Taylor model, with the parameter values listed above, has been computed using the RIOTS 95 package on a Pentium, and also with some of the parameters varied, as listed below. For comparison, the model has also been computed using the SCOM package on an iMac computer. The latter package also uses MATLAB (version 5.2) and the constrained optimisation solver constr from MATLAB's Optimisation Toolbox, but none of the differential equation software used by RIOTS 95. Thus both computations share MATLAB's basic arithmetic and display software, but the implementations are otherwise independent.

Implementation of the Collocation Method

Let us now consider the practical steps that must be taken to implement the collocation method in a computer programming environment. Below, we outline the key operations using the Matlab vector processing language, presuming access to the function approximation and numerical quadrature routines contained in the COM-PECON library. The necessary steps can be implemented in virtually any other vector processing or high-level algebraic programming language, with a level of difficulty that will depend mainly on the availability of general code that performs the required approximation and quadrature operations.

Dynamic Programming Methods

258 See, for example, N.Chriss, Black-Scholes and Beyond (New York McGraw-Hill, 1997) which presents the venerable Black-Scholes model, as well as Cox-Ross-Rubenstein binomial trees and Black-Scholes and Beyond Interactive Toolkit (New York McGraw-Hill, 1998), developed using MATLAB. See also P.Brandimarte, Numerical Methods in Finance A MATLAB-BasedIntroduction (New York John Wiley & Sons, 2002). introduction of usable dynamic programming utilities in MATLAB will enable determined economists to apply this technique to a variety of problems without further wait. In particular, the toolbox and related text authored by Miranda and Fackler provide an excellent introduction to the use of dynamic programming methods in economics.264 264 M.Miranda and P.Fackler, Applied Computational Economics and Finance (Cambridge, MA MIT Press, 2002) the related CompEcon toolbox is available from the MIT Press Web site. In addition, the Institut National de la Recherche Agronomique has developed a...

Approximation with Polynomials and Neural Networks

We can see how efficient neural networks are relative to linear and polynomial approximations with a very simple example. We first generate a standard normal random variable x of sample size 1000, and then generate a variable y sin(x) 2 + e-x. We can then do a series of regressions with polynomial approximators and a simple neural network with two neurons, and compare the multiple correlation coefficients. We do this with the following set of MATLAB commands, which access the following functions for the orthogonal polynomials chedjudd.m, hermiejudd.m, legendrejudd.m, and laguerrejudd.m, as well as the feedforward neural network program, ffnet9.m. Matlab Program For Assessing Approximation

Applications for Finance

Half Year Convention Macrs

Checks to ensure that all variables are present and that the interest rate is entered correctly. This is particularly important with interest rates. Interest rates are often quoted in percentage terms (a mortgage rate of 9.5 typically means a 9.5 rate on the loan), while most computer programs (MATLAB included) look for interest rates in decimal form. Thus, it is a good idea to check any interest rate entered to ensure that it is in the decimal form. The if .error .end block in the function accomplishes this. 3. Calculate the periodic payment on the mortage loan. We have used the MATLAB payper function here. 4. Calculates the amortization, using a different MATLAB function other than that used for calculating the payments. We have used the MATLAB amort function here. 7. Creates a structure of values and information to provide to the user and then assign the structure into the base workspace. MATLAB structures are a powerful way to organize disparate types of data and include...

Prof Sardar M N Islam is Professor of Welfare

Chapter 3 is based (with some additional material) on Computing Optimal Control on MATLAB - The SCOM Package and Economic Growth Models, Chapter 5 in Optimisation and Related Topics, Eds, A. Rubinov et al. Volume 47 in the Series Applied Optimization, Kluwer Academic Publishers, 2001. Kluwer has given permission to reproduce this article.

Fixing Volatile Volatility

The chapter is organized in four sections. In Sect. 14.1, we briefly introduce SV models and their properties. Section 14.2.2 is dedicated to the description and calibration of the Heston model (HES) and its version with jumps (HESJ). Section 14.3 discusses the use of the FFT to compute option prices and briefly describes the Matlab code. Finally, Sect. 14.4 exhibits the empirical performance of HES and HESJ models compared to the standard BS model.

Quasi Monte Carlo Integration

There are numerous schemes for generating equidistributed sequences. The best known are the Neiderreiter, Wevl, and Haber, The following Matlab script generates equidistributed sequences of length n for the unit hvpereube The Matlab toolbox accompanying the textbook includes a function qnwequi that generates the equidistributed nodes for integration over an arbitrary bounded interval in a space of arbitrary dimension. The calling sequence takes the form

Nonlinear Optimization and Gravity Model Estimation

The specification and estimation of a gravity model requires sophisticated software. As the model is nonlinear, a simple OLS (ordinary least squares) regression will not estimate the parameters. In this case, a software environment like MATLAB is needed. In the examples presented above, we used the following methods 2. To estimate the parameters, we use a nonlinear optimization algorithm available within MATLAB. In particular, we make use of the gravity.m function and the nonlinear least squares estimation command nlinfit. Linear models can be directly estimated using a well-known formula that minimizes the sum of the squared errors. Nonlinear models typically require an initial guess at the correct parameters and then, iteratively, a search to find the parameters that minimize the sum of the squared residuals. The MATLAB nlinfit command uses the Gauss-Newton method to find these parameter estimates.348 Other optimization and numerical methods are also available within MATLAB.349 349...

Comparison Gravity and Polynomial Models

Figure 13-10, Gravity vs. polynomial specifications, shows the result of this exercise. Using the MATLAB basic curve-fitting tool, we were able to quickly overlay a cubic (third degree) polynomial equation onto the data. Eyeballing the data indicates that the cubic polynomial does a better job of explaining the variations for almost all of the ranges. The little hump upward and the tail downward in the cubic curve bring it closer to the distribution of the test data.

History and Structure of the Predictor

The predictor was one of the first fuzzy logic routines ever presented on the Internet. From just before April 15, 2000, through much of 2001, it was run on a MATLAB Web server accessible to millions of taxpayers.380 Taxpayers across the country were able to anonymously test it with real or imagined data.381

Code Fragment 151 Audit Predictor Rules

The audit predictor FIS evaluates such characteristics as resulting in a high probability of an audit. This is consistent with common sense a person who makes mistakes on his return, and whose return contains a number of audit flags, should be audited more frequently than the typical taxpayer. What cannot be shown in this figure is that the ruleviewer in the MATLAB toolbox is interactive, allowing you to see how the FIS evaluates multiple scenarios quite quickly.

Constructing a Bond Portfolio to Hedge Against Duration and Convexity

Recall that portfolio duration or convexity is a weighted average of the durations or convexities of the individual bonds in a portfolio. As in the previous example, this example uses modified duration in years and convexity in years. The hedging problem therefore becomes one of solving a system of linear equations, which is very easy to do in MATLAB. The M-file for this example is ftspex2.m.

Sensitivity of Bond Prices to Parallel Shifts in the Yield Curve

Often bond portfolio managers want to consider more than just the sensitivity of a portfolio's price to a small shift in the yield curve, particularly if the investment horizon is long. This example shows how MATLAB can visualize the price behavior of a portfolio of bonds over a wide range of yield curve scenarios, and as time progresses toward maturity. Add axis labels and set the three-dimensional viewpoint. MATLAB produces the figure. MATLAB three-dimensional graphics allow you to visualize the interest rate risk experienced by a bond portfolio over time. This example assumed parallel

Constructing Greek Neutral Portfolios of European Stock Options

Using the Black-Scholes model for European options, this example creates an equity option portfolio that is simultaneously delta, gamma, and vega neutral. The value of a particular greek of an option portfolio is a weighted average of the corresponding greek of each individual option. The weights are the quantity of each option in the portfolio. Hedging an option portfolio thus involves solving a system of linear equations, an easy process in MATLAB. This example M-file is ftspex4.m.

Calculations Databases and Dynamic Pages

All these can provide dynamic content and will continue to be developed in the future. These approaches will often be superior to one using MATLAB Webserver for most tasks. However, for some intense analytical tasks, a mathematical model run on a Webserver may be the best approach, and we consider it next.

Setting Up the Web Server Model

The MATLAB Web server toolbox comes with a User Guide and other references, and we do not repeat the information in those references here. However, we recommend a user take the following steps before beginning a Web server application 1. Develop a MATLAB application on a work station.

Postoptimality Analysis

A MATLAB routine itosimul is provided in the CompEcon Toolbox. function s itosimul(model,s0,T,N,cv,fspace) where e is a d x 1 vector of independent standard normal variates. For d 1 the MATLAB code for this is simply sqdt*sigma.*e but the vectorized version when d > 1 uses the COMPEcON toolbox function arraymult, which is coded as a mex file in C.

Example 1 Generating Help Files

For example, the command xslt instructs MATLAB to transform information in one XML-format file into a separate format as specified in a transformation stylesheet. The example in Code Fragment 16-1, Using XML to Generate Help Files, illustrates how this can be used with MATLAB'S native info.xml files to generate, in a Web browser, help files. xslt, an XML file transformer. See the MATLAB Help information under file input-output.

Example 2 Creating Custom Information Files

One of the ways MATLAB organizes information on separate toolboxes is through the use of info.xml files. One of these is referenced in the code fragment just described. Another use is to create your own information files. In current MATLAB versions, the info.xml file for each toolbox can be accessed by the MATLAB Launchpad with icons denoting various components, including Help information, specific tools, and Web sites with related information.403

Web Services Other XML Tools Available

Web Services is a generic term for applications that use the World Wide Web to access data and perform some manipulation or reformatting of that data. The SOAP protocol is one XML-based standard that MATLAB can use in the Web services applications. MATLAB sends requests to another server, and then handles the responses. Web services can be relatively simple programs that accomplish much.

Yyiykjfci2n yiy2vkxkixn J

The idea of genetic algorithms certainly has a good potential for solving quite complex problems the evident downside is that considerable experimentation may be needed to come up with the best strategy and the best setting of numerical parameters regulating the evolution mechanisms. The potential of this class of methods is also proved by the recent introduction of the Genetic Algorithm and Direct Search toolbox, which extends the functionalities of the MATLAB Optimization toolbox.

Sct I a2 S S rSPScr Scr1

This is a first order system of ordinary differential equations in r, with the known solution c(r) exp(r _1i )co, where c0 satisfies the boundary condition R(S) evaluated at the n values of the Si (note the exponential is a matrix exponential which can be computed using the Matlab function expm).

Implementation and Algorithm

Functions and market data are stored in the folder BetaEstimation. Before running the code, you must include this folder and its subfolders in the MATLAB directory list. To do so select Set Path from the File Menu and follow the instructions. The Beta Estimation folder contains the following MATLAB scripts to estimate the beta according to the different procedures To run the code from the Excel spreadsheet you need to have the Excel link. Estimating the Kalman filter requires the MATLAB Optimization Toolbox.

Business Applications for SVG Images

MATLAB in past versions has not included a native SVG print utility. However, there are a number of options available for incorporating the SVG images in your work, 1. Purchase an SVG print utility that can be used in MATLAB and other programs, such as SVGmaker, or a toolkit such as Apache Batik.

Implementation to the XploRe

Mathematical computational environments, e.g. Maple, Matlab, Mathematica and Mathcad. The reason of this implementation is a large diversity of problems to their the LW gives solution. If the LW is not implemented in some statistical software (which is the case of the XploRe) we can proceed as follows. Let us consider the selectors min(S) and max(S), where S is the set of solutions of the equation x ln(x) c, c > 1. Then we have the following relations in the terms of the Lambert W function (Stehlik, 2001) for proof

Refresher on Probability Theory and Statistics

In this appendix we recall very briefly some basic facts about probability theory and parameter estimation. This is not meant as a substitute for a thorough treatment, for which we refer the reader to the references. We will not use measure theoretic concepts and will mostly rely on intuition. We also give information on some functions provided by the MATLAB Statistics toolbox.

The Testplot Function

To capture a large number of methods used to properly set up and then customize graphics in MATLAB, we created a master graphics function called testplot. This function, which is part of the Business Economics Toolbox, creates sample data (some generated with random information, some from actual economic data), plots the data using a variety of graphic types, and then adjusts and annotates the graphics.

Format of Problem Optimization

The control function approximates the control function w(.) by a step-function, dividing 0,1 into nn equal subintervals. (Because the dynamic equation has a smoothing effect, set function controls are usually a sufficient approximation.) Function values (and often also gradients with respect to control) for the objective function are obtained by solving differential equations. They are then supplied to the optimization program constr in MATLAB's Optimization Toolbox. The computation is considerably faster if gradients are supplied, but this is not suitable for some problems, especially if the functions are not well behaved outside the feasible region of the problem. If gradients are used, then the adjoint differential equation is required. If a constraint x(l) k is required, then it is added to s a positive penalty parameter. The user must supply a calling program (defining all parameters), and user subroutines for the functions of the given control problem. This use of MATLAB...

The Lambert W function

Since the Lambert W function has many applications in pure and applied mathematics, the branches of the LW function are implemented to many mathematical computational softwares, e.g. the Maple, Matlab, Mathematica and Mathcad. For more information about the implementation and some computational aspects (Corless,, 1993).

Obsolete Bond Price and Yield Functions

The functions listed in this table are obsolete, and their descriptions have been removed from the documentation. They have been replaced with the SIA-compliant functions bndprice and bndyield. For compatibility purposes, the obsolete functions remain in the product. Type help function_name at the MATLAB command line for a description.

Note on Operating Systems

A discussion of the merits and drawbacks of operating systems is outside the scope of this book. However, we use and recommend MATLAB partially because it is reliable, robust, and can be used on multiple platforms. We suggest that users consider these factors when selecting an operating system as well. 418 At the current time, the author has successfully, and fairly reliably, used MATLAB on Windows NT, XP Pro, and 2000 versions. We do not recommend use of the earlier (3.x, 95, 98, and Me) versions. Note that The MathWorks no longer supports installations on some Windows versions.

B2 Conditional Statements And Looping

As with any programming language Matlab can evaluate boolean expression such as A> B, A> B, A< B, A< B and A B (the last one is not equal is Matlab's negation operator). Also (A> B), (A< B), etc., can be used. These need to be used with a bit of ease when A and B are not sealars, however, A> B creates a matrix of zeros and ones equal in size to A and B. If you want to know is any of the elements of A are bigger than any of the elements of B is the same as checking whether any of the elements of the matrix A> B are non-zero, Matlab provides the functions any and all to evaluate matrices resulting form boolean expressions. As with many Matlab functions, any and all operate on rows and return a rows vector with the same number of columns as the original matrix. This is true for sum and prod functions as well. The following are equivalent expressions

Appendix C Introduction to AMPL

In this brief appendix, we want to introduce the basic syntax of AMPL. We use AMPL only in the last chapters on optimization models, and the syntax is almost self explanatory. Hence, we will just describe a few basic examples, so that the reader can get a grasp of the basic language elements. The reader is referred to the original reference 1 , written by the developers of AMPL. Unlike MATLAB, AMPL is not a procedural language. There is a part of the language which is aimed at writing scripts, which behave like any program based on a sequence of control statements and instructions. But the core of AMPL is a declarative syntax to describe a mathematical programming model and the data to instantiate it. The optimization solver is separate You can write a model in AMPL, and solve it with different solvers, possibly implementing different algorithms. Actually, AMPL interfaces have been built for many different solvers in fact, AMPL is more of a language standard which has been implemented...

B3 Scripts and Functions

When you work in Matlab you are working in an interactive environment that stores the variables you have defined and allows you to manipulate them throughout a session. You do have the ability to save groups of commands in files that can be executed many times, Matlab has two kinds of command files, called m-files. The first is a script m-file. If you save a bunch of commands in a script file called MYFILE.m and then type the word MYFILE at the Matlab command line, the commands in that file will be executed just as if you had run them each from the Matlab command prompt (assuming Matlab can find where you saved the file), A good way to work with Matlab is to use it interactively, and then edit you session and save the edited commands to a script file. You can save the session either by cutting and pasting or by turning on the diary feature (use the on-line help to see how this works by typing help diary). The second type of M-files is the function file. One of the most important...

[Txrk4fTx0[additional parameters

The inputs are the name of a problem file that returns the function f, the vector of time values T and the initial conditions, xo. The fourth input is an empty matrix to make the calling syntax for rk4 compatible with the CompEcon ODE solvers. The two outputs are the vector of time values (this makes rk4 compatible with Matlab's ODE solvers) and the solution values. Unlike the suite of ODE solvers provided by Matlab, rk4 is designed to be able to compute solutions for multiple initial values. If x0 is d x k and there are n time values in T, rk4 will return x as an n x d x k array. Avoiding a loop over multiple starting points results in much faster execution when a large set of trajectories are computed. To take advantage of this feature, however, the function passed to rk4 that defines the differential equation must be able to return a d x k matrix when its second input argument is a d x k matrix (see the example below for an illustration of how this is done). 1The Matlab functions...

Common Excel Link Problems

Excel does not start MATLAB automatically. 1. Check, and if necessary reinstall, the correct Excel Link files, using the Excel Tools Add-Ins function. If you have switched MATLAB versions, try the steps listed under Run the Program with A Newer Version of MATLAB on page 456. Check MATLAB References (Windows) 1. MATLAB should start automatically with Excel if Excel Link is working properly. If it does not, a brute force method is to use the Excel command Tools Macro, and then enter matlabinit. Run the Program with a Newer Version of MATLAB 1. If you run both a current version and the R11 versions of MATLAB, note that the location of the excllink.xla file changes with R12. The R12, R13 and R14 installers put the Excel Link files within the toolbox directory. Update any references such as those discussed below. 3. Another approach is moving the old version of the excllink.xla to within the new MATLAB root directory and then pointing Excel to the relocated file. 5. Try starting the...

Handling and Converting Dates

Note The date functions datenum, datestr, datevec, eomday, now, and weekday now ship with basic MATLAB. They originally shipped only with the Financial Toolbox. Their descriptions remain in this document for your convenience. MATLAB serial date number to Excel serial date number. Thirty-second quotation to decimal. Excel serial date number to MATLAB serial date number.

Portfolios with Missing Data

The algorithm assumes that missing values are missing at random and non-ignorable. (See Little and Rubin 1 for precise definitions of these terms.) Asset data that does not exist prior to a certain date, e.g., stock price data prior to an IPO, is an example where ecmnmle is appropriate. MATLAB represents these unavailable values as NaN. For a counterexample, consider censored data, in which all values greater than some cutoff are replaced with NaNs. This type of data does not satisfy the conditions under which you can use ecmnmle. The collection of observations (or samples) is stored in a MATLAB matrix Data such that

Need For Theory

Now that we established that we are going to use MATLAB in the book, another question may arise Why should we bother learning numerical methods, when they are already available in professionally crafted, ready-to-use code Can we get rid of theory Although, in most cases, there is no need for a deep knowledge of numerical analysis in order to use MATLAB, there are at least three reasons to gain a basic understanding of the theoretical background of numerical methods. Everyone would agree that this is just a complicated way to write 10 x 8.1 81. Let us try it on a computer, using MATLAB

Intended Audience

The book may also be used by those interested in specialized topics rarely covered in economics texts, such as those listed above, and by those who have extensive background in mathematical or simulation models (such as MATLAB and Simulink) and wish to extend their knowledge.

Le20 1

Typically, computer languages such as Fortran and C allow several ways of representing a number, Matlab makes things simple by only have one representation for a number, Matlab uses what is often called a double precision floating point number. The exact details of the representation depends on the hardware but it will suffice for our purposes to suppose that floating point numbers are stored in the form rn2e, where rn and e are integers with 2h < m < 2h and 2d < e < 2d. For example, Consider now what happens when arithmetic operations are performed. If rrii2ei is multiplied by m2262, the exact result is mim22ei+e2, If mim2 is outside the range 2h, 2h), it will need to be divided by powers of 2 until it is within this range and the exponent will need to be adjusted accordingly. In the process of dividing mim2, any remainders will be lost. This means it is possible to perform the operation (x*y) y and have the result not equal x instead it may be off by 1 in its least...

Main Purpose of Code

Experienced MATLAB users will undoubtedly notice ways in which the execution of these programs could be speeded up or different ways of accomplishing the same thing. The primary purpose of including the programs is to demonstrate how to perform certain tasks and not to execute those tasks with maximum efficiency. Therefore, you will note that many of the programs include extensive comment lines (which begin with a percent-sign mark) that are ignored by the computer. If you are reviewing these code fragments in order to perform a similar task, pay attention to these comment lines. Those who can recommend a more efficient or more effective code to accomplish the same result are encouraged to send a note to the author. See the contact information at Contacting the Author on page 9.

Version Notes

The versions of MATLAB used during the development of this book included R11, R12, R13, and a prereleased version of R14. We believe that all of the routines will run, in some cases with minor modifications, on R12 and R13. Most will run on R11 as well. Based on our testing of the comparison business economics toolbox with the prereleased version of R14, we anticipate that all of the routines included here will also work under 2 A startup file is a set of instructions run by MATLAB when the program is first run. We recommend that users customize their startup files this is discussed in Chapter 3, this book, under Setting Up the MATLAB Workspace on page 30.

Key Definitions

A rectangular array of numeric or algebraic quantities subject to mathematical operations the regular formation of elements into rows and columns. Described as an m-by-n matrix, with m the number of rows and n the number of columns. The description is always row-by-column. For example, here is a 2-by-3 matrix of two bonds (the rows) with different par values, coupon rates, and coupon payment frequencies per year (the columns) entered using MATLAB notation. Vector. A matrix with only one row or column. Described as a 1-by-n or m-by-1 matrix. The description is always row-by-column. Here is a 1-by-4 vector of cash flows in MATLAB notation.

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