I MATHEMATICS 1
1 LINEAR ALGEBRA 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Systems of Linear Equations and Matrices . . . . . . . . . . . . . 3
1.3 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Vectors and Vector Spaces . . . . . . . . . . . . . . . . . . . . . 11
1.6 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 14
1.10 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.11 Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.12 Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 VECTOR CALCULUS 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Vector-valued Functions and Functions of Several Variables . . . 18
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2.4 Partial and Total Derivatives . . . . . . . . . . . . . . . . . . . . 20
2.5 The Chain Rule and Product Rule . . . . . . . . . . . . . . . . . 21
2.6 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 23
2.7 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Taylor’s Theorem: Deterministic Version . . . . . . . . . . . . . 25
2.9 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . 26
3 CONVEXITY AND OPTIMISATION 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Convexity and Concavity . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Properties of concave functions . . . . . . . . . . . . . . 29
3.2.3 Convexity and differentiability . . . . . . . . . . . . . . . 30
3.2.4 Variations on the convexity theme . . . . . . . . . . . . . 34
3.3 Unconstrained Optimisation . . . . . . . . . . . . . . . . . . . . 39
3.4 Equality Constrained Optimisation:
The Lagrange Multiplier Theorems . . . . . . . . . . . . . . . . . 43
3.5 Inequality Constrained Optimisation:
The Kuhn-Tucker Theorems . . . . . . . . . . . . . . . . . . . . 50
3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
II APPLICATIONS 61
4 CHOICE UNDER CERTAINTY 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Optimal Response Functions:
Marshallian and Hicksian Demand . . . . . . . . . . . . . . . . . 69
4.4.1 The consumer’s problem . . . . . . . . . . . . . . . . . . 69
4.4.2 The No Arbitrage Principle . . . . . . . . . . . . . . . . . 70
4.4.3 Other Properties of Marshallian demand . . . . . . . . . . 71
4.4.4 The dual problem . . . . . . . . . . . . . . . . . . . . . . 72
4.4.5 Properties of Hicksian demands . . . . . . . . . . . . . . 73
4.5 Envelope Functions:
Indirect Utility and Expenditure . . . . . . . . . . . . . . . . . . 73
4.6 Further Results in Demand Theory . . . . . . . . . . . . . . . . . 75
4.7 General Equilibrium Theory . . . . . . . . . . . . . . . . . . . . 78
4.7.1 Walras’ law . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7.2 Brouwer’s fixed point theorem . . . . . . . . . . . . . . . 78
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4.7.3 Existence of equilibrium . . . . . . . . . . . . . . . . . . 78
4.8 The Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . . 78
4.8.1 The Edgeworth box . . . . . . . . . . . . . . . . . . . . . 78
4.8.2 Pareto efficiency . . . . . . . . . . . . . . . . . . . . . . 78
4.8.3 The First Welfare Theorem . . . . . . . . . . . . . . . . . 79
4.8.4 The Separating Hyperplane Theorem . . . . . . . . . . . 80
4.8.5 The Second Welfare Theorem . . . . . . . . . . . . . . . 80
4.8.6 Complete markets . . . . . . . . . . . . . . . . . . . . . 82
4.8.7 Other characterizations of Pareto efficient allocations . . . 82
4.9 Multi-period General Equilibrium . . . . . . . . . . . . . . . . . 84
5 CHOICE UNDER UNCERTAINTY 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Review of Basic Probability . . . . . . . . . . . . . . . . . . . . 85
5.3 Taylor’s Theorem: Stochastic Version . . . . . . . . . . . . . . . 88
5.4 Pricing State-Contingent Claims . . . . . . . . . . . . . . . . . . 88
5.4.1 Completion of markets using options . . . . . . . . . . . 90
5.4.2 Restrictions on security values implied by allocational ef-
ficiency and covariance with aggregate consumption . . . 91
5.4.3 Completing markets with options on aggregate consumption 92
5.4.4 Replicating elementary claims with a butterfly spread . . . 93
5.5 The Expected Utility Paradigm . . . . . . . . . . . . . . . . . . . 93
5.5.1 Further axioms . . . . . . . . . . . . . . . . . . . . . . . 93
5.5.2 Existence of expected utility functions . . . . . . . . . . . 95
5.6 Jensen’s Inequality and Siegel’s Paradox . . . . . . . . . . . . . . 97
5.7 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8 The Mean-Variance Paradigm . . . . . . . . . . . . . . . . . . . 102
5.9 The Kelly Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.10 Alternative Non-Expected Utility Approaches . . . . . . . . . . . 104
6 PORTFOLIO THEORY 105
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . 105
6.2.1 Measuring rates of return . . . . . . . . . . . . . . . . . . 105
6.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 The Single-period Portfolio Choice Problem . . . . . . . . . . . . 110
6.3.1 The canonical portfolio problem . . . . . . . . . . . . . . 110
6.3.2 Risk aversion and portfolio composition . . . . . . . . . . 112
6.3.3 Mutual fund separation . . . . . . . . . . . . . . . . . . . 114
6.4 Mathematics of the Portfolio Frontier . . . . . . . . . . . . . . . 116
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6.4.1 The portfolio frontier in <N:
risky assets only . . . . . . . . . . . . . . . . . . . . . . 116
6.4.2 The portfolio frontier in mean-variance space:
risky assets only . . . . . . . . . . . . . . . . . . . . . . 124
6.4.3 The portfolio frontier in <N:
riskfree and risky assets . . . . . . . . . . . . . . . . . . 129
6.4.4 The portfolio frontier in mean-variance space:
riskfree and risky assets . . . . . . . . . . . . . . . . . . 129
6.5 Market Equilibrium and the CAPM . . . . . . . . . . . . . . . . 130
6.5.1 Pricing assets and predicting security returns . . . . . . . 130
6.5.2 Properties of the market portfolio . . . . . . . . . . . . . 131
6.5.3 The zero-beta CAPM . . . . . . . . . . . . . . . . . . . . 131
6.5.4 The traditional CAPM . . . . . . . . . . . . . . . . . . . 132
7 INVESTMENT ANALYSIS 137
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Arbitrage and Pricing Derivative Securities . . . . . . . . . . . . 137
7.2.1 The binomial option pricing model . . . . . . . . . . . . 137
7.2.2 The Black-Scholes option pricing model . . . . . . . . . . 137
7.3 Multi-period Investment Problems . . . . . . . . . . . . . . . . . 140
7.4 Continuous Time Investment Problems . . . . . . . . . . . . . . . 140