Linear Algebra for Investment Management

This "Linear Algebra" course is designed to demonstrate the usefulness and practical relevance of working with vectors and matrices in various applications in investment management, ranging from simple portfolio calculations with assets and factors to non-parametric time series models for extracting trends and scenarios.

• Investment Managers, Both Traditional and Alternative
• Risk Managers
• Financial Economists
• Quantitative Investment Analysts
• Data Scientists

• Overcome the terminology and notation barriers of Linear Algebra
• Learn how to execute calculations involving vectors and matrices to solve practical problems in investment analytics
• Develop common sense intuition for seemingly abstract mathematical concepts
• Increase awareness for numerical issues in Linear Algebra applications

• Basic understanding of economics and financial markets, instruments and quantitative concepts like CAPM, mean-variance portfolio selection, time series analysis

Introduction

• What is Linear Algebra
• Why is Linear Algebra interesting for investment analysis?

Scalars, Vectors and Matrices

• One and Zero Objects
• Symmetrical Matrices
• Diagonals (minor/major/nth), Triangulars (upper/lower)
• Geometrical, Statistical & Analytical Interpretations of Vectors and Matrices

Transformations

• Transpose
• Vectorisations
• Hankel, Toeplitz
• Sort, Centre, Standardize
• Circular Shift
• Matrices from Lagged Data

Applications (discussed and calculated in Excel):

• Covariance Matrix Decomposition and Construction from Correlations and Volatilities
• Building Inputs for Vector Autoregressive Models
• Transition Probability Matrices in Markov Regime Switching & Credit Risk Analysis

Basic Operations with Vectors and Matrices

• Addition & Subtraction
• Multiplication: Matrix Products
• Division: Inverse of a Matrix, Pseudoinverse
• Elementwise Operations

Applications (discussed and calculated in Excel):

• Portfolio Risk and Return Calculations
• Interpreting the Inverse of a Correlation Matrix In Regression Analysis

Matrix Properties

• Unitary, Zero
• Symmetry
• Square
• Rank
• Determinant
• Definiteness: positive/negative, semi
• Singular
• Norms

Applications (discussed and calculated in Excel):

• Factor Model Analytics: Portfolio Decomposition, Asset Risk and Return from Factors
• Validity of a Correlation Matrix

Advanced Operations

• Introduction to Matrix Factorizations
  • Cholesky Decomposition
  • PCA, SVD
  • QR, LU
• Matrix Derivatives

Applications (discussed and calculated in Excel):

• Regression Analysis: Least Square Problems in Matrix Form
• Simulating Correlated Time Series Data
• Linear Model Identification: Principal Component Analysis (PCA) for S&P 500 Constituents
• Dimensionality Reduction: PCA versus SVD
• Singular Spectral Analysis: Non-Parametric Decomposition of Time Series Data

Linear Algebra in Microsoft Excel

• Built-in Functionality
• VBA Functions
• Add-Ins