Introductory Mathematics and Statistics for Islamic Finance

By Abbas Mirakhor and Noureddine Krichene

A unique primer on quantitative methods as applied to Islamic finance

Introductory Mathematics and Statistics for Islamic Finance + Website is a comprehensive guide to quantitative methods, specifically as applied within the realm of Islamic finance. With applications based on research, the book provides readers with the working knowledge of math and statistics required to understand Islamic finance theory and practice. The numerous worked examples give students with various backgrounds a uniform set of common tools for studying Islamic finance.

The in-depth study of finance requires a strong foundation in quantitative methods. Without a good grasp of math, probability, and statistics, published theoretical and applied works in Islamic finance remain out of reach. Unlike a typical math text, this book guides students through only the methods that directly apply to Islamic finance, without wasting time on irrelevant techniques. Each chapter contains a detailed explanation of the topic at hand, followed by an example based on real situations encountered in Islamic finance. Topics include:

• Algebra and matrices
• Calculus and differential equations
• Probability theory
• Statistics

Written by leading experts on the subject, the book serves as a useful primer on the analysis methods and techniques students will encounter in published research, as well as day-to-day operations in finance. Anyone aspiring to be successful in Islamic finance needs these skills, and Introductory Mathematics and Statistics for Islamic Finance + Website is a clear, concise, and highly relevant guide.

• Language: English
• Publisher: Wiley
• Release date: Jun 10, 2014
• ISBN: 9781118779729

Abbas Mirakhor

Abbas Mirakhor is the former holder of the First Chair of Islamic Finance at INCEIF in Malaysia, where he mentored and supervised 17 Ph.D. candidates, amongst other duties. He was formerly Dean of the Executive Board and retired as an Executive Director of the International Monetary Fund (IMF). He has held past professorships in Alabama A&M and Florida Institute of Technology. Mirakhor has published widely across microeconomic theory, mathematical economics and Islamic economics. He is the author, contributor and editor of numerous books, journals and articles, with much of those diverse research converging towards risk-sharing.

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CONTENTS

Cover

Title Page

Copyright

Dedication

Preface

Acknowledgments

About the Authors

Part One: Mathematics

Chapter 1: Elementary Mathematics

Basic Mathematical Objects

Variables, Monomials, Binomials, and Polynomials

Equations

Equations of Higher Order

Sequences

Series

Applications of Series to Present Value of Assets

Summary

Chapter 2: Functions and Models

Definition of a Function

Functions and Models in Economics

Functions and Models in Finance

Multivariate Functions in Economics and Finance

Summary

Chapter 3: Differentiation and Integration of Functions

Differentiation

Differentiation Rules

Maximum and Minimum of a Function

Mean Value Theorem

Polynomial Approximations of a Function: Taylor's Expansion

Integration

Applications in Finance: Duration and Convexity of a Sukuk

Summary

Chapter 4: Partial Derivatives

Definition and Computation of Partial Derivatives

Total Differential of a Function with Many Variables

Directional Derivatives

Gradients

Tangent Planes and Normal Lines

Extrema of Functions of Several Variables

Extremal Problems with Constraints

Summary

Chapter 5: Logarithm, Exponential, and Trigonometric Functions

Logarithm Functions

The Exponential Function

Power Series of Logarithmic and Exponential Functions

General Exponential and Logarithmic Functions

Some Applications of Logarithm and Exponential Functions in Finance

Integration by Parts

Trigonometric Functions

Summary

Chapter 6: Linear Algebra

Vectors

Matrices

Square Matrices

The Rank of a Matrix

Determinant of a Square Matrix

Homogenous Systems of Equations

Inverse and Generalized Inverse Matrices

Eigenvalues and Eigenvectors

Stability of a Linear System

Applications in Econometrics

Summary

Chapter 7: Differential Equations

Examples of Differential Equations

Solution Methods for the Differential Equation

First-Order Linear Differential Equations

Second-Order Linear Differential Equations

Linear Differential Equation Systems

Phase Diagrams and Stability Analysis

Summary

Chapter 8: Difference Equations

Definition of a Difference Equation

First-Order Linear Difference Equations

Second-Order Linear Difference Equations

System of Linear Difference Equations

Equilibrium and Stability

Summary

Chapter 9: Optimization Theory

The Mathematical Programming Problem

Unconstrained Optimization

Constrained Optimization

The General Classical Program

Summary

Chapter 10: Linear Programming

Formulation of the LP

The Analytical Approach to Solving an LP: The Simplex Method

The Dual Problem of the LP

The Lagrangian Approach: Existence, Duality, and Complementary Slackness Theorems

Economic Theory and Duality

Summary

Part Two: Statistics

Chapter 11: Introduction to Probability Theory: Axioms and Distributions

The Empirical Background: The Sample Space and Events

Definition of Probability

Random Variable

Techniques of Counting: Combinatorial Analysis

Conditional Probability and Independence

Probability Distribution of a Finite Random Variable

Moments of a Probability Distribution

Joint Distribution of Random Variables

Chebyshev's Inequality and the Law of Large Numbers

Summary

Chapter 12: Probability Distributions and Moment Generating Functions

Examples of Probability Distributions

Empirical Distributions

Moment Generating Function (MGF)

Summary

Chapter 13: Sampling and Hypothesis Testing Theory

Sampling Distributions

Estimation of Parameters

Confidence-Interval Estimates of Population Parameters

Hypothesis Testing

Tests Involving Sample Differences

Small Sampling Theory

Summary

Chapter 14: Regression Analysis

Curve Fitting

Linear Regression Analysis

The Probability Distribution of the Estimated Regression Coefficients â and b

Hypothesis Testing of â and b

Diagnostic Test of the Regression Results

Prediction

Multiple Correlation

Summary

Chapter 15: Time Series Analysis

Component Movements of a Time Series

Stationary Time Series

Characterizing Time Series: The Autocorrelation Function

Linear Time Series Models

Moving Average (MA) Linear Models

Autoregressive (AR) Linear Models

Mixed Autoregressive-Moving Average (Arma) Linear Models

The Partial Autocorrelation Function

Forecasting Based on Time Series

Summary

Chapter 16: Nonstationary Time Series and Unit-Root Testing

The Random Walk

Decomposition of a Nonstationary Time Series

Forecasting a Random Walk

Meaning and Implications of Nonstationary Processes

Dickey-Fuller Unit-Root Tests

The Augmented Dickey-Fuller Test (ADF)

Summary

Chapter 17: Vector Autoregressive Analysis (VAR)

Formulation of the VAR

Forecasting with VAR

The Impulse Response Function

Variance Decomposition

Summary

Chapter 18: Co-Integration: Theory and Applications

Spurious Regression

Stationarity and Long-Run Equilibrium

Co-Integration

Test for Co-Integration

Co-Integration and Common Trends

Co-Integrated VARs

Representation of a Co-Integrated VAR

Summary

Chapter 19: Modeling Volatility: ARCH-GARCH Models

Motivation for Arch Models

Formalization of the Arch Model

Properties of the Arch Model

The Generalized Arch (Garch) Model

Arch-Garch in Mean

Testing for the Arch Effects

Summary

Chapter 20: Asset Pricing under Uncertainty

Modeling Risk and Return

Uncertainty and Efficient Capital Markets: Random Walk and Martingale

Market Efficiency and Arbitrage-Free Pricing

Basic Principles of Derivatives Pricing

State Prices

Martingale Distribution and Risk-Neutral Probabilities

Martingale and Complete Markets

Summary

Chapter 21: The Consumption-Based Pricing Model

Intertemporal Optimization and Implication to Asset Pricing

Asset-Specific Pricing And Correction For Risk

Relationship Between Expected Return and Beta

The Mean Variance (mv) Frontier

Risk-Neutral Pricing Implied by the General Pricing Formula pt = Et(mt+1xt+1)

Consumption-Based Contingent Discount Factors

Summary

Chapter 22: Brownian Motion, Risk-Neutral Processes, and the Black-Scholes Model

Brownian Motion

Dynamics of the Stock Price: The Diffusion Process

Approximation of a Geometric Brownian Motion by A Binomial Tree

Ito's Lemma

Discrete Approximations

Arbitrage Pricing: Black-Scholes Model

The Market Price of Risk

Risk-Neutral Pricing

Summary

References

Index

End User License Agreement

List of Tables

Table 11.1

Table 11.2

Table 11.3

Table 11.4

Table 11.5

Table 13.1

Table 14.1

Table 14.2

Table 14.3

Table 14.4

Table 17.1

Table 17.2

Table 20.1

Table 20.2

Table 20.3

Table 20.4

Table 20.5

Table 20.6

Table 20.7

Table 22.1

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 4.1

Figure 4.2

Figure 4.3

Figure 5.1

Figure 5.2

Figure 5.3

Figure 6.1

Figure 6.2

Figure 6.3

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 9.1

Figure 9.2

Figure 9.3

Figure 9.4

Figure 9.5

Figure 9.6

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 11.1

Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5

Figure 11.6

Figure 11.7

Figure 12.1

Figure 12.2

Figure 12.3

Figure 12.4

Figure 12.5

Figure 12.6

Figure 12.7

Figure 12.8

Figure 13.1

Figure 13.2

Figure 13.3

Figure 13.4

Figure 13.5

Figure 13.6

Figure 14.1

Figure 14.2

Figure 15.1

Figure 15.2

Figure 15.3

Figure 15.4

Figure 16.1

Figure 16.2

Figure 16.3

Figure 16.4

Figure 17.1

Figure 17.2

Figure 17.3

Figure 17.4

Figure 17.5

Figure 17.6

Figure 18.1

Figure 18.2

Figure 18.3

Figure 19.1

Figure 19.2

Figure 19.3

Figure 19.4

Figure 19.5

Figure 20.1

Figure 20.2

Figure 20.3

Figure 20.4

Figure 20.5

Figure 21.1

Figure 22.1

Figure 22.2

Figure 22.3

Introductory Mathematics and Statistics for Islamic Finance

Abbas Mirakhor

Noureddine Krichene

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Cover image: iStockphoto.com/amir_np

Cover design: Wiley

Copyright © 2014 by Abbas Mirakhor and Noureddine Krichene/John Wiley & Sons Singapore Pte. Ltd.

Published by John Wiley & Sons Singapore Pte. Ltd.

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All rights reserved.

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Dedication

To the memory of our respective parents

Preface

The objective of this book is to provide an introductory and unified training in mathematics and statistics for students in Islamic finance. Students enrolled in Islamic finance programs may have had different training in mathematical and statistical methods. Some students may have advanced training in some mathematical or statistical topics; however, they may not have been sufficiently exposed to some topics that are highly relevant in Islamic finance or to the applications of quantitative methods in this field. Other students may have had less advanced quantitative training. It will be therefore necessary to provide a homogenous quantitative training in mathematics and statistics for students, with a view to enhancing their command of the theory and practice of Islamic finance.

In view of the nature of Islamic finance, students or professionals should acquire adequate skills in computational mathematics and statistics in order to accomplish their duties in any financial or nonfinancial institutions where they might be employed. Without computational skills, students or professionals may not be able to manipulate economic and financial data; they may not meet the challenges of their financial career. In fact, the finance industry has reached an extremely advanced stage in terms of the quantitative methods, computerization, product innovations, and arbitrage and trading programs that are used. Many institutions such as hedge funds, pension funds, investment corporations, insurance companies, and asset management companies require advanced knowledge in actuaries, and models of investment and risk management. Professionals have to satisfy the standards required by these institutions and be able to use software, such as Microsoft Excel, EViews, Mathematica, MATLAB, and Maple, to process data and carry out computational tasks. The Internet is rich in the use of computational tools. A student can plug in data and get instantaneous answers; however, it is important for a student to understand the theory underlying the computational procedures.

While existing books on finance cover the topics of mathematics or statistics only, this book covers fundamental topics in both mathematics and statistics that are essential for Islamic finance. The book is also a diversified and up-to-date statistical text and prepares students for more advanced concepts in mathematics, statistics, and finance.

Although most of the mathematical and statistical books concentrate on traditional mathematics or statistics, this book uses examples and sample problems drawn from finance theory to illustrate applications in Islamic finance. For instance, a student will be exposed to financial products, asset pricing, portfolio selection theory, duration and convexity of assets, stock valuation, exchange rate pricing, and efficient market hypothesis. Examples are provided for illustrating these important topics.

A special feature of the book is that it starts from elementary notions in mathematics and statistics before advancing to more complex concepts. As an introductory text, no prerequisite in mathematics or statistics is required. In mathematics, this book starts from elementary notions such as numbers, vectors, and matrices, before it advances to topics in calculus and linear algebra. The same approach is applied in statistics; the book covers basic concepts in probability theory, such as events, probabilities, and distributions, and advances progressively to econometrics, time series analysis, and continuous time finance. Each chapter is aimed at an introductory level and does not go into detailed proofs or advanced concepts.

The questions at the end of each chapter repeat examples discussed in the chapter and students should be able to carry out computations using widely available software, such as Excel, Matlab, and Mathematica, online formulas, and other calculators. Internet presentations that illustrate many procedures in the book are also available. The successful resolution of these questions means that a student has a good understanding of the contents of the chapter. For self-checking, the answers have been made readily available online at www.wiley.com.

Acknowledgments

The authors acknowledge the valuable contribution of Jeremy Chia, editor at John Wiley & Sons, who added considerable value to the manuscript. They express deep gratitude to Kimberly Monroe-Hill for her hard work in the copyediting and production of the book. The authors also extend a special appreciation to Nick Wallwork and are thankful for the continuing support of John Wiley & Sons Singapore in promoting the development of Islamic finance.

Professor Abbas Mirakhor would like to thank Datuk Professor Syed Othman Al Habshi, the dean of faculty, and Professor Obiyathulla Ismath Bacha, the director of graduate studies, at INCEIF for their support. He also thanks Dr. Mohamed Eskandar, Ginanjar Dewandaru, Sayyid Aun Rizvi, and Fatemeh Kymia for their assistance.

About the Authors

Abbas Mirakhor is currently the First Holder of the Chair of Islamic Finance at the International Center for Education in Islamic Finance. He has served as the dean of the executive board of the International Monetary Fund from 1997 to 2008, and as the executive director representing Afghanistan, Algeria, Ghana, Iran, Morocco, Pakistan, and Tunisia from 1990 to 2008. He has authored numerous publications and research papers on Islamic finance; among them are the Introduction to Islamic Finance (John Wiley & Sons, 2011), Risk Sharing in Islamic Finance (John Wiley & Sons, 2011), and The Stability of Islamic Finance (John Wiley & Sons, 2010).

Noureddine Krichene received his PhD in economics from the University of California, Los Angeles in 1980. He taught Islamic finance at the Global University, International Center for Education in Islamic Finance, Malaysia. He was an economist with the International Monetary Fund from 1986 to 2009. From 2005 to 2007, he was an advisor at the Islamic Development Bank in Saudi Arabia. His areas of expertise are in the international payments system, macroeconomic policies, finance, and energy and water economics.

He is based in Laurel, Maryland.

Part One

Mathematics

Chapter 1

Elementary Mathematics

This chapter covers the measurement and presentation of economic and financial data. Data consists of numbers and graphics, which are essential for recording and understanding financial data. All financial transactions are represented by numbers. For instance, the price of a commodity in terms of commodity is a number; it is the number of units of commodity that is paid to get one unit of commodity . Usually, unknown amounts are expressed as variables, designated by symbols such as , , or any other symbol, and the equations that contain these variables may be expressed in the form of monomials, binomials, or polynomials. The applications of equations, sequences, and series are important concepts to understand in finance.

Basic Mathematical Objects

Numbers play a fundamental role in economics and finance. There are real numbers and complex or imaginary numbers. Real numbers are a subset of complex numbers. This section covers real numbers, complex numbers, the absolute value of a number, vectors and arrays, angles and directions, graphics, and the reporting of economic and financial data.

Real Numbers

The set of real numbers, denoted by R, is represented by a real line where the symbols and stand for minus infinity and plus infinity, respectively (Figure 1.1a). A real number is a value that represents a quantity along a continuous line. Real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as …(an irrational algebraic number) and …(a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. A noninteger real number has a decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real numbers are uncountable.

Figure 1.1 Real and Complex Numbers

The set of real numbers, , is a field, meaning that addition and multiplication are defined and have the usual properties. The field is ordered, meaning that there is a total order ≥ such that, for all real numbers , and :

If then

If and then .

Complex Numbers

A complex number is written in the form

(1.1) equation

where and are real numbers and is the imaginary unit, where

(1.2) equation

In this expression, is the real part of denoted by , and is a real number called the imaginary part of and is denoted by . The set of all complex numbers is denoted by . Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number

equation

can be identified with the point in the complex plane as shown in Figure 1.1b. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone. The real line can be thought of as a part of the complex plane, and, correspondingly, complex numbers include real numbers as a special case.

The set of complex numbers is a field. Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number , its additive inverse is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example, the law of commutativity of addition and multiplication for any two complex numbers and :

equation

These two laws and the other requirements on a field can be proven using the fact that the real numbers themselves form a field.

In Islamic finance, we make use of logarithm numbers, exponential numbers, and trigonometric numbers. Trigonometric numbers are important in studying the slope of a curve. These numbers will be introduced later in the book. Nonetheless, we may provide some examples.

Example: Consider number 1. Its natural logarithm is zero. Its exponential is 2.718282. Consider the number ; its cosine, is zero; its sine, is 1.

Absolute Value of a Number

Numbers are also described by their absolute value or moduli. If a number is represented by a point on the real line, then the absolute value is a measure of the length of the distance between the number and point zero. The numbers 5 and –5 have the same absolute value: . In other words, when we see a number , the corresponding number to this distance is either 2 or –2. The moduli of a complex number is the distance between the origin zero and the point represented by this number (Figure 1.1b). If , then .

Vectors and Arrays

Economic data may be represented by a point on the real line. For instance, real gross domestic product is $14 billion in 2012. We show it by a point in the real line. However, economic information may have many dimensions. We need to go from , the real line, to higher dimension Euclidian space such as , ,…, . For instance, an Islamic stock has an expected return and a risk. Traditionally, if a share has an expected return of 7 percent and a risk of 9 percent, this share is represented with a point whose coordinates are 7 percent and 9 percent (see Figure 1.2a). Further, we may be interested in the beta and alpha of the share. In this case, we have to go to . If beta is 1.2 and alpha is 3.5 percent, then we describe the share by a vector in :

(1.3) equation

Figure 1.2 Vector, Angle, and Direction

Economic and financial information may need to be presented in the form of an array. A portfolio may have four Islamic shares. We present information about these shares in the form of a matrix:

(1.4)

equation

A matrix is also used to describe the structure of international trade such as exports and imports, the structure of an economy, or the production processes of a farm or an industry.

Angles and Directions

The notions of an angle and direction are important in economics and finance. We are interested in the slope of a curve as well as the direction of economic motion. An angle shows the slope of a tangent line to a curve. A vector shows the direction of a motion along the curve (see Figure 1.2b).

Graphics

Graphics are essential tools in reporting economic and financial information and in teaching economics and finance. They facilitate economic and financial analysis. In fact, graphics are essential in all fields. For instance, Google maps show us directions in the form of a graph. A building or a house is designed in graphics before it is actually constructed. A contractor cannot build any house before he has the mapping of the house.

Consider an Islamic bank that has a portfolio composed of Murabaha (26 percent), Mudarabha (19 percent), Musharaka (25 percent), and Islamic funds (30 percent). This information is shown in Figure 1.3.

Figure 1.3 Portfolio of an Islamic Bank

A portfolio manager uses graphics to track the market value of his portfolio. Figure 1.4 shows the value of the portfolio over a period of 30 weeks.

Figure 1.4 Market Value of an Islamic Portfolio

Reporting Economic and Financial Data

Besides graphics, economic and financial data is reported in special ways. If we say the real gross domestic product (GDP) of Malaysia rose by 2 billion (RM) and that of Burundi by 550 million (BF), this information is not easy to interpret. All we can say is that real GDP did not fall in either country. However, if we say real GDP rose by 7 percent in Malaysia and 2 percent in Burundi, this information is easier to interpret because it is placed in context of the existing GDP.

Economic and financial information is reported in the form of indicators; these are percent changes, ratios, indices, elasticities, and other specific indicators. For instance, the balance sheet of a company is described in terms of ratios such as liquidity ratio, solvency ratio, and equity ratio. Macroeconomic indicators use ratios such as external deficit ratio, debt ratio, and fiscal deficit ratio. Indices are important. An index normalizes data to a base of 100, then compares the evolution of data in relation to this base. For instance, the price index measures the price of a basket of commodities in reference to a base of 100, called base year, and computes the period change in relation to this base. Elasticities are a way to describe economic and financial variables. We say that demand for bread is inelastic, implying that consumers are unable to change their demand for bread whether prices of bread go skyward or drop substantially. In contrast, the demand for apples is elastic, implying that when the price of apples increases, demand may decrease.

A percent change is defined as

(1.5) equation

where denotes change in a variable . A percent change can be positive, zero, or negative. A ratio involves two variables, one is the numerator and the other is the denominator:

(1.6) equation

For instance, per capita income is the ratio of real GDP in money terms divided by the size of the population, that is, the number of citizens of a country. An index is referred to by its abbreviation. For instance, means consumer price index. S&P 500 refers to Standard & Poor's stock price index. Elasticity is computed as the ratio of two percent changes:

(1.7) equation

It could be positive, zero, or negative. If it is close to zero, there is inelasticity of in relation to ; if it is , then there is high elasticity of in relation to .

Variables, Monomials, Binomials, and Polynomials

This section covers monomials, binomials, polynomials, polynomial lags, identities, and factorization of a polynomial.

A variable is designated by the symbol . We perform algebraic operations on the variable . We may multiply by any number , and we obtain . We may compute powers of such as

equation

We may perform operation such as

equation

For instance, we perform the following multiplications:

(1.8) equation

(1.9) equation

(1.10) equation

Monomials, Binomials, and Polynomials

A monomial is the product of nonnegative integer powers of variables. Consequently, a monomial has no variable in its denominator. It has one term (mono implies one):

equation

We notice that there are no negative exponents and no fractional exponents. The number 6 is a monomial since it can be written as .

A binomial is the sum of two monomials. It has two unlike terms (bi implies two):

equation

A trinomial is the sum of three monomials; it has three unlike terms (tri implies three):

equation

A polynomial is the sum of one or more terms (poly implies many):

equation

The degree of a polynomial is the highest exponent of its monomials. Polynomials are in the simplest form when they contain no like terms. For instance, the polynomial

equation

when simplified becomes

equation

Polynomial Lags

In statistics we use a lag operator, denoted by (lag). For instance, the price of tomatoes today is denoted as , the price of tomatoes last month is , the price two months past is , three months past is ,…. , months past is . We present this information as , , ,…., . Our notation of tomato prices can be written in a polynomial lag as

(1.11) equation

An example of a polynomial lag is

equation

A polynomial lag is very useful in performing operations on a time series, such as tomato prices, or any other time series such as the daily values of the Dow Jones Islamic stock index. We may perform operations on polynomial lag in the same way as on any regular polynomial. For instance: may be written as .

Identities

Often in Islamic finance, we need to use identities; we provide some useful identities:

(1.12) equation

(1.13) equation

(1.14) equation

(1.15) equation

The binomial identity is an important one. It is stated as

(1.16)

equation

Here is called factorial of ; it is written as .

Factorization of a Polynomial

Let us consider the following product:

(1.17)

equation

We may reverse the path and start from the polynomial and try to factorize it into . The values , , , and are called the roots of the polynomials . If we replace into the polynomial we find

equation

If we replace into the polynomial we find

equation

We observe that is different from 0 for any value of different from . For instance, for we have , and for we .

Equations

Equations are basic notions of finance. A large part of Islamic finance consists of solving equations such as computing internal rates of return, replicating portfolios, structuring products, pricing assets, and computing costs or break-even points. A simple equation is of the form

(1.18) equation

Obviously the solution is . We may have to deal with equations of the second degree, called quadratic equations, such as

(1.19) equation

This equation has two roots. Completing the square can be used to derive a general formula for solving quadratic equations. Dividing the quadratic equation by (which is allowed because a is nonzero), gives

(1.20) equation

(1.21) equation

The quadratic equation is now in a form to which the method of completing the square can be applied. To complete the square is to add a constant to both sides of the equation such that the left hand side becomes a complete square

(1.22) equation

that produces

(1.23) equation

The right side can be written as a single fraction with common denominator . This gives

(1.24) equation

Taking the square root of both sides yields

(1.25) equation

Isolating x gives

(1.26) equation

The plus-minus symbol ± indicates that

(1.27)

equation

In the above formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an uppercase Greek delta, Δ:

(1.28) equation

A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

If the discriminant is positive, then there are two distinct roots

(1.29) equation

both of which are real numbers.

If the discriminant is zero, then there is exactly one real root, sometimes called a double root:

(1.30) equation

If the discriminant is negative, then there are no real roots. Rather, there are two distinct (nonreal) complex roots, which are complex conjugates of each other. In these expressions is the imaginary unit:

(1.31) equation

Thus the roots are distinct if and only if the discriminant is nonzero, and the roots are real if and only if the discriminant is nonnegative.

Example: Complete the square for .

We apply the formula ; we obtain .

Example: Solve the following quadratic function:

equation

We compute , ,

equation

Example: Solve the following quadratic equation:1

equation

We compute ; ,

We have two complex roots: and .

Example: You want to invest in two Islamic mutual funds and , in proportions and , respectively, with , and . The risk of mutual funds and are and , respectively; the correlation coefficient between expected returns is . Find the composition that will achieve a portfolio risk of 11.2 percent.

The portfolio variance is

equation

We equate the portfolio variance to . We obtain

equation

We note that ; hence, . We replace into the equation, we obtain

equation

Or

equation

This is an equation of the form ; its roots are

equation

The latter root means 156 percent of savings are invested in mutual fund ; since we cannot invest more than 100 percent in any fund, we discard this root. The root is acceptable. It implies . Hence 36 percent of the savings are invested in fund and 64 percent are invested in fund .

Equations of Higher Order

Entrepreneurs and planners undertake investment projects with a life span of many years. They are interested in determining the profitability of the project in which they will invest considerable money. By deciding to invest in a specific project such as a textile plant, they have to renounce many other alternative or competing projects. They tend to choose the project that has the highest returns. The investment selection problem necessitates solving an equation of the order , where is an integer equal to the life span of the project or the investment horizon. Assuming a discount rate , we may summarize the cash flow (CF) of the project in table form, as shown here.

The entrepreneur is interested in solving an equation of the form

(1.32)

equation

The rate of return that solves this equation is called the internal rate of return (IRR). The equation can be rewritten in polynomial form as

(1.33)

equation

where .

The equation admits roots (real and complex). Unfortunately, there are no formulas that can be readily employed for finding the roots of the equation. We have to proceed by iteration. We may draw a chart for the equation and see the points where it intersects the horizontal line. We may also start from an initial value for such as and iterate until the polynomial becomes very close to zero. Since there are roots, we may have to discard negative roots and consider only positive roots. Nonetheless, there is software such as Microsoft Excel that can provide the solutions for an equation of order .

Example: Malay Palm Oil Corporation contemplates an investment project, which has the cash flow shown in the following table. We want to find the internal rate of return (IRR). We apply Excel formula , we find .

Sequences

A sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike in a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable, totally ordered set, such as the natural numbers.

An example of a sequence is (1, 2, 3, 5, 8). Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,…). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list the elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation can be used for infinite sequences as well. For instance, the infinite sequence of positive odd integers can be written (1, 3, 5, 7,…). Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements.

There are many important integer sequences. The prime numbers are numbers that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, 19,…). The study of prime numbers has important applications for mathematics and specifically number theory.

The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34,…).

The terms of a sequence are commonly denoted by a single variable, say , where the index indicates the nth element of the sequence. The sequence is written as

(1.34) equation

Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whose elements are related to the index (the element's position) in a simple way. For instance, the sequence of the first 10 square numbers could be written as

(1.35) equation

This represents the sequence (1, 4, 9,…, 100). The length of a sequence is defined as the number of terms in the sequence. A sequence of a finite length is also called an -tuple. Finite sequences include the empty sequence ( ) that has no elements.

Normally, the term infinite sequence refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element (a singly infinite sequence). A sequence that is infinite in both directions—it has neither a first nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence. For instance, a function from all integers into a set, such as the sequence of all even integers (…, −4, −2, 0, 2, 4, 6, 8,…), is bi-infinite. This sequence could be denoted .

A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For a sequence this can be written as for all , where is the set of all integers. If each consecutive term is strictly greater (>) than the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonically decreasing if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function.

The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively.

Example: Generate the first 10 terms of the sequence: with and . We obtain

.

A Cauchy sequence is a sequence whose terms become arbitrarily close together as gets very large. A sequence , of real numbers is called a Cauchy sequence, if for every positive real number , there is a positive integer N such that for all natural numbers

(1.36) equation

where the vertical bars denote the absolute value. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring to be infinitesimal for every pair of infinite . Figure 1.5 depicts an example of a Cauchy sequence.

Figure 1.5 Example of a Cauchy Sequence

One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Informally, a (singly-infinite) sequence has a limit if it approaches some value , called the limit, as becomes very large. That is, for an abstract sequence ( ) (with running from 1 to infinity)