This lesson about prospect theory is based on an article by Barbaris, Mukherjee and Wang. Prospect theory is a behavioural economic theory that described the way people choose between probabilistic alternatives that involve risk, where the probabilities of outcomes are known. The main empirical prediction of the article is that […]

## Investment Sentiment in the Stock Market and Market Anomalies

The previous lesson was about noise trader risk. Noise and investor sentiment are terms that are often used interchangeably. Investor sentiment can be defined as the overall attitude of investors towards a particular asset or larger financial market. Market sentiment is the feeling or tone of a market, or its […]

## Limits to Arbitrage

The efficient market hypothesis states that prices are equal to their fundamental value. All deviations will be undone by rational arbitrageurs. In theory, arbitrage is done by an infinite number of small risk-neutral investors. This leads to a direct price adjustment. In reality, arbitrage is done by a small number […]

## Momentum and Mean Reversion

To what extend are past returns related to future returns? This question alone is already a direct attack on the weak-form theory of market efficiency. Theory about the three levels of market efficiency theoryWeak-form – The weak form suggests today’s stock prices reflect all the data of past prices and […]

## Improving the ‘Two Factor Model’ & Fama and French (1992, 1993, 2015)

Are there any other characteristics than beta that predict the expected return of an asset or portfolio? This lesson will elaborate on three important articles that introduced new parameters to the original CAPM model. Fama and French, 1992 and 1993 In this new publication, the data quality was better compared […]

## Capital Asset Pricing Model (CAPM) & Fama and MacBeth (1973)

The capital asset pricing model (CAPM) was one of the first laws in finance and it’s still widely used or, in other cases, part of new (improved) models. The CAPM describes the relationship between systematic risk and expected return for assets, generally stocks. But how do we quantify this specific […]

## Basic (Quadratic) Utility – Effect on Market Portfolio Demand

From portfolio theory, we know that investors hold a combination of the risk free asset and the market portfolio. The weights regarding the diversification depends on the so called utility function. Utility is a term in economics that refers to the total satisfaction received from consuming a good or service. […]

## Introduction to Asset Pricing and the Modern Portfolio Theory

Asset pricing is about the determinants of stock returns and how to optimise the performance of stock portfolios, not about the valuation of stocks. Asset prices are the prices for which financial instruments, such as stocks and bonds, are bought and sold. Fundamentals, risks, and sentiment may be derived from […]

## Eigenvectors and Eigenvalues

This lesson will recap the theory on eigenvectors and eigenvalues, which you most probably studies during a linear algebra or calculus course. If not, do not worry, we will elaborate on this content. Eigenvector versus eigenvalue By theorem, an eigenvector of an matrix A is a nonzero vector x, such […]

## Least-Squares in Linear Algebra

The least-squares method is very important in the area of data fitting. The lesson will give an overview of how to use the least-squares method, in order to find the best fitting model on data. Consider a stock, for which you can assume that the stock price increases linearly during […]

## Orthogonal Projections and the Gram-Schmidt Process

This lesson will elaborate on orthogonal projections and the Gram-Schmidt process, which will lay the foundation to understand least-squares problems (important in any statistical analysis). The orthogonal projection of a vector y on plane P is interesting, because it gives us the shortest distance of the vector y on plane […]

## The Inner Product and Orthogonality

Up to now, we have added and multiplied vectors and matrices with each other. There is much more to be said about vectors. Take for example a vector a and b. We might be interested in the length of a vector, the distance between vectors or the angle between vectors. […]

## Cramer’s Rule

This lesson will elaborate on how determinants can be used to solve a system of n linear equations and n variables, if the solution is unique. This technique is called the Cramer’s rule. Consider a system of n equations and n unknowns, so we have Ax = b for an […]

## Determinants for Squared Matrices

In a previous lesson about the inverse matrices, you were introduced to the determinant of a 2×2 matrix. You know that if the determinant is non-zero, the matrix is invertible. In this lesson you will learn how to calculate the determinant for any squared matrix. You can use determinants if […]

## Linear Subspaces

This lesson will start with an introduction to the superposition property. The superposition property restricts what the set of solutions to a homogeneous equation may look like. Take a look at the following homogeneous equation. (1) We know that this equation always has the trivial solution, . We have […]

## Inverse Matrices

We have seen different ways to solve the linear system Ax = b. If we look at normal numbers (so, no matrices or vectors), we can solve ax = b by simply dividing both sides by a. Our result would be . Can we do that with vectors and matrices? […]

## Matrix Transformations and Multiplication

If you have a mathematical background, you know for sure what a function is. A function takes an input, transforms it and returns an output. The set of possible inputs of the function is called the domain of the function, whereas the set of possible outputs is called the codomain. […]

## Solution Sets of Homogeneous and Inhomogeneous Equations

In the previous lesson, you were introduced to the Gaussian elimination algorithm in order to get to the reduced echelon form of a system of linear equations. This lesson, we will take a look at the solution sets of two types of linear equations, namely the homogeneous- and inhomogeneous equations. […]

## Vector Equations and Linear Combinations

As described before, linear algebra proves to be a very powerful tool for structuring and processing a large amount of data. In this lesson, you will be introduced to vectors, linear combinations and the reduced row echelon form, which are all important in the field of linear algebra. Let’s start […]

## Introduction to Linear Algebra and System of Linear Equations

Linear algebra is the study of lines and planes, vector spaces and mappings that are required for linear transformations. These are important to many areas of mathematics. This branche of mathematics also proves to be a very powerful tool for structuring and processing a large amount of data. Therefore, it is also important […]