Chapter
APPLICATIONS OF SERIES TO PRESENT VALUE OF ASSETS
Applications of Series to Present Value Computation
Chapter 2: Functions and Models
Parametric Form of a Function
FUNCTIONS AND MODELS IN ECONOMICS
The Market Model: Demand and Supply Functions
The Production Possibility Frontier (PPF)
Other Functions in Economics
FUNCTIONS AND MODELS IN FINANCE
The Present Value Function
The Capital Asset Pricing Model (CAPM)
Payoff of a Futures Contract
Payoff of an Option Contract
The Forward Exchange Rate
MULTIVARIATE FUNCTIONS IN ECONOMICS AND FINANCE
Parametric Representation
Chapter 3: Differentiation and Integration of Functions
MAXIMUM AND MINIMUM OF A FUNCTION
POLYNOMIAL APPROXIMATIONS OF A FUNCTION: TAYLOR’S EXPANSION
The First Fundamental Theorem of Calculus
Second Fundamental Theorem of Calculus
Change in Variables in Indefinite Integrals
APPLICATIONS IN FINANCE: DURATION AND CONVEXITY OF A SUKUK
Application of Taylor Expansion to the Convexity of Sukuk’s Price
Chapter 4: Partial Derivatives
DEFINITION AND COMPUTATION OF PARTIAL DERIVATIVES
Derivatives of Implicit Functions
TOTAL DIFFERENTIAL OF A FUNCTION WITH MANY VARIABLES
TANGENT PLANES AND NORMAL LINES
EXTREMA OF FUNCTIONS OF SEVERAL VARIABLES
EXTREMAL PROBLEMS WITH CONSTRAINTS
Chapter 5: Logarithm, Exponential, and Trigonometric Functions
The Natural Logarithmic Function
POWER SERIES OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS
GENERAL EXPONENTIAL AND LOGARITHMIC FUNCTIONS
SOME APPLICATIONS OF LOGARITHM AND EXPONENTIAL FUNCTIONS IN FINANCE
Simple Compounding and Continuous Compounding of Returns
The Present Value Formula
Chapter 6: Linear Algebra
Multiplication of Vectors
Linear Combinations of Vectors
Linear Dependence and Linear Independence of Vectors
DETERMINANT OF A SQUARE MATRIX
HOMOGENOUS SYSTEMS OF EQUATIONS
INVERSE AND GENERALIZED INVERSE MATRICES
Generalized Inverse of a Matrix
EIGENVALUES AND EIGENVECTORS
Similarity of Square Matrices
STABILITY OF A LINEAR SYSTEM
APPLICATIONS IN ECONOMETRICS
Chapter 7: Differential Equations
EXAMPLES OF DIFFERENTIAL EQUATIONS
SOLUTION METHODS FOR THE DIFFERENTIAL EQUATION
Method of Indefinite Integrals
Method of Separable Variables
FIRST-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
Homogeneous Linear Differential Equation
Nonhomogeneous Linear Differential Equations
LINEAR DIFFERENTIAL EQUATION SYSTEMS
Transforming the System into a Second-Order Homogeneous Differential Equation
Method of Eigenvalues and Eigenvectors
PHASE DIAGRAMS AND STABILITY ANALYSIS
Phase Line of an Ordinary Differential Equation
Phase Diagram of a Linear Differential Equation System
Chapter 8: Difference Equations
DEFINITION OF A DIFFERENCE EQUATION
FIRST-ORDER LINEAR DIFFERENCE EQUATIONS
Solutions of the First-Order Difference Equation
The Impulse Response Function
SECOND-ORDER LINEAR DIFFERENCE EQUATIONS
Homogeneous Second-Order Difference Equations
Nonhomogeneous Second-Order Difference Equations
The Multiplier-Accelerator Model
SYSTEM OF LINEAR DIFFERENCE EQUATIONS
EQUILIBRIUM AND STABILITY
Stability of the Linear Difference System
Chapter 9: Optimization Theory
THE MATHEMATICAL PROGRAMMING PROBLEM
Formulation of the Programming Problem
The Geometry of Optimization
UNCONSTRAINED OPTIMIZATION
One Variable Function y= F(x)
Function of Two Variables z= F(x,y)
The Method of Lagrange Multipliers
THE GENERAL CLASSICAL PROGRAM
The Geometry of Constrained Optimization
Interpretation of the Lagrangian Multipliers
The Case of No Inequality Constraints
The Kuhn-Tucker (K-T) Conditions
Chapter 10: Linear Programming
Standard Form and Canonical Form of the LP
THE ANALYTICAL APPROACH TO SOLVING AN LP: THE SIMPLEX METHOD
Notion of Technical Equivalence
THE DUAL PROBLEM OF THE LP
THE LAGRANGIAN APPROACH: EXISTENCE, DUALITY, AND COMPLEMENTARY SLACKNESS THEOREMS
Interpretation of the Dual Variables
ECONOMIC THEORY AND DUALITY
Chapter 11: Introduction to Probability Theory: Axioms and Distributions
THE EMPIRICAL BACKGROUND: THE SAMPLE SPACE AND EVENTS
DEFINITION OF PROBABILITY
TECHNIQUES OF COUNTING: COMBINATORIAL ANALYSIS
CONDITIONAL PROBABILITY AND INDEPENDENCE
PROBABILITY DISTRIBUTION OF A FINITE RANDOM VARIABLE
Probability Distribution and Histogram
Cumulative Distribution Function
Continuous Random Variables
MOMENTS OF A PROBABILITY DISTRIBUTION
First Moment of the Random Variable
Second Moment of the Random Variable: Variance and Standard Deviation
Third Moment of a Random Variable: Skewness
Fourth Moment of a Random Variable: Kurtosis
JOINT DISTRIBUTION OF RANDOM VARIABLES
Independent Random Variables
CHEBYSHEV’S INEQUALITY AND THE LAW OF LARGE NUMBERS
The Central Limit Theorem
Chapter 12: Probability Distributions and Moment Generating Functions
EXAMPLES OF PROBABILITY DISTRIBUTIONS
The Bernoulli Distribution
The Binomial Distribution
The Chi-Square Distribution
MOMENT GENERATING FUNCTION (MGF)
Examples of Moment Generating Functions
Chapter 13: Sampling and Hypothesis Testing Theory
Sampling Distribution of the Mean
Sampling Distribution of Proportions
Sampling Distribution of Differences
Point Estimates and Interval Estimates
CONFIDENCE-INTERVAL ESTIMATES OF POPULATION PARAMETERS
Confidence Intervals for Means
Confidence Intervals for Proportions
Confidence Intervals for Differences
Confidence Intervals for Standard Deviations
Type I and Type II Errors
Probability Value: p-Value
TESTS INVOLVING SAMPLE DIFFERENCES
Tests Based on the Student’s t-Distribution
Confidence Intervals for t Distribution
Tests Based on the Chi-Square Distribution
Confidence Intervals for χ²
Tests Based on the F Distribution
Chapter 14: Regression Analysis
Equations of Approximating Curves
The Linear Regression Line
The Principle of Estimation: Method of Least Squares
LINEAR REGRESSION ANALYSIS
Formulation of the Regression Model
Estimation of the Regression Model
The Method of the Maximum Likelihood
Estimate of the Variance of the Error Term
The Coefficient of Determination and the Coefficient of Correlation
THE PROBABILITY DISTRIBUTION OF THE ESTIMATED REGRESSION COEFFICIENTS â AND b
HYPOTHESIS TESTING OF â AND b
Test of Significance of the Regression Intercept â
Test of Significant of the Regression Slope b
Test of the Simultaneous Significance of the Regression Coefficients
DIAGNOSTIC TEST OF THE REGRESSION RESULTS
Standard Error of Regression (SER) and the R-Squared
Serial Correlation: Durbin-Watson Statistic
Chapter 15: Time Series Analysis
COMPONENT MOVEMENTS OF A TIME SERIES
Components of a Time Series
Modeling the Trend of Time Series
Properties of Stationary Processes
CHARACTERIZING TIME SERIES: THE AUTOCORRELATION FUNCTION
Stationarity and the Autocorrelation Function
LINEAR TIME SERIES MODELS
Wold Decomposition of a Stationary Process
MOVING AVERAGE (MA) LINEAR MODELS
AUTOREGRESSIVE (AR) LINEAR MODELS
MIXED AUTOREGRESSIVE-MOVING AVERAGE (ARMA) LINEAR MODELS
Invertibility of the ARMA(1,1)
THE PARTIAL AUTOCORRELATION FUNCTION
FORECASTING BASED ON TIME SERIES
Minimum Mean-Squared-Error Forecasts
Forecast Confidence Interval (CI)
Forecast of the AR(1) Process
Forecast of the MA(1) Process
Forecast of the ARMA(1,1) Process
Chapter 16: Nonstationary Time Series and Unit-Root Testing
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)
Chapter 17: Vector Autoregressive Analysis (VAR)
THE IMPULSE RESPONSE FUNCTION
Chapter 18: Co-Integration: Theory and Applications
STATIONARITY AND LONG-RUN EQUILIBRIUM
CO-INTEGRATION AND COMMON TRENDS
REPRESENTATION OF A CO-INTEGRATED VAR
Vector Error-Correction (VEC) Representation
Co-Integration and Moving Average Representation
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
TESTING FOR THE ARCH EFFECTS
Chapter 20: Asset Pricing under Uncertainty
UNCERTAINTY AND EFFICIENT CAPITAL MARKETS: RANDOM WALK AND MARTINGALE
Relationship between the Random Walk and Martingale Models
MARKET EFFICIENCY AND ARBITRAGE-FREE PRICING
Pricing of Assets by Arbitrage
BASIC PRINCIPLES OF DERIVATIVES PRICING
Principles of Derivatives Pricing Theory
Fundamental Principle for Pricing Derivatives
MARTINGALE DISTRIBUTION AND RISK-NEUTRAL PROBABILITIES
MARTINGALE AND COMPLETE MARKETS
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
Chapter 22: Brownian Motion, Risk-Neutral Processes, and the Black-Scholes Model
DYNAMICS OF THE STOCK PRICE: THE DIFFUSION PROCESS
APPROXIMATION OF A GEOMETRIC BROWNIAN MOTION BY A BINOMIAL TREE
The Futures Contract Model
ARBITRAGE PRICING: BLACK-SCHOLES MODEL
Introductory Mathematics and Statistics for Islamic Finance
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