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This self-contained book presents the main techniques of quantitative portfolio management and associated statistical methods in a very didactic and structured way, in a minimum number of pages. The concepts of investment portfolios, self-financing portfolios and absence of arbitrage opportunities are extensively used and enable the translation of all the mathematical concepts in an easily interpretable way.

All the results, tested with Python programs, are demonstrated rigorously, often using geometric approaches for optimization problems and intrinsic approaches for statistical methods, leading to unusually short and elegant proofs. The statistical methods concern both parametric and non-parametric estimators and, to estimate the factors of a model, principal component analysis is explained. The presented Python code and web scraping techniques also make it possible to test the presented concepts on market data.

This book will be useful for teaching Masters students and for professionals in asset management, and will be of interest to academics who want to explore a field in which they are not specialists. The ideal pre-requisites consist of undergraduate probability and statistics and a familiarity with linear algebra and matrix manipulation. Those who want to run the code will have to install Python on their pc, or alternatively can use Google Colab on the cloud. Professionals will need to have a quantitative background, being either portfolio managers or risk managers, or potentially quants wanting to double check their understanding of the subject.

### Chapter 1. Returns and the Gaussian Hypothesis

Abstract
In this book, the problem of finding optimal portfolios is mathematically solved under the assumption that the returns of the risky assets follow a Gaussian distribution. In this section, we give the definition of a price return and of a total return and describe some tools to analyse these returns and to statistically test the hypothesis of normality on them. The hypothesis does not always appear to be satisfied, depending on the stock or on the period considered, nevertheless, even in these cases, the methods of portfolio optimisation may still teach some useful lessons.
Pierre Brugière

### Chapter 2. Utility Functions and the Theory of Choice

Abstract
When payouts are deterministic, investor preferences are easy to determine, and if the payout of asset A is twice the payout of asset B, then the price of A is twice the price of B. Now, when payouts are random, determining the criteria of choice between two investments is more complex, and if the expected payout of asset A is twice the expected payout of asset B, the price of A is not necessarily twice the price of B. Furthermore, when choosing between two assets, beyond the mathematical expectations of the payouts, the variances and the whole distribution of the payouts are usually considered. For a random payout X, this analysis leads us to look not only at its expectation E(X) but at something of the form E(u(X)), where u is called a utility function . In this context, E(u(X)) is the objective to maximise when choosing between different random payouts X. The function u is chosen to reflect the preferences of the investors and it will be shown in this chapter that the appetite or aversion to risk is linked to the convexity or concavity of the function u. An extensive literature exists in economics about utility functions and their applications. John von Neumann and Oskar Morgenstern proved that, from a mathematical point of view, individuals whose preferences satisfy four particular axioms have a utility function. Gerard Debreu (Nobel price in Economics 1983) and Kenneth Arrow (Nobel price in Economics 1972) produced additional landmark results, defining the equilibrium in an economy where agents act upon some utility functions. This chapter is only a very brief introduction to the topic, and is included to make the link with the mean-variance criteria which is used in the rest of the book.
Pierre Brugière

### Chapter 3. The Markowitz Framework

Abstract
We present here the mathematical framework under which the “Markowitz problem” of maximising the expected return of a portfolio under a risk constraint is solved.
Pierre Brugière

### Chapter 4. Markowitz Without a Risk-Free Asset

Abstract
In this chapter we solve the Markowitz problem of finding the investment portfolios which, for a given level of expected return, present the minimum risk. The assumption is made that the returns of the assets (and consequently of the portfolios) follow a Gaussian distribution, and the risk is defined as the standard deviation of the returns. Except in the case where all the risky assets have the same returns, the solution portfolios $$\mathcal {F}$$ of this mean-variance optimisation problem define a hyperbola when representing in a plane the set $$\mathcal {F}(\sigma ,m)$$ of their standard deviations and expected returns. This hyperbola also determines the limit of all the investment portfolios that can be built. Its upper side $$\mathcal {F}^+(\sigma ,m)$$ corresponds to the efficient portfolios and is called the efficient frontier , while its lower side $$\mathcal {F}^-(\sigma ,m)$$ is called the inefficient frontier . The two fund theorem demonstrated here proves that, when taking any pair of distinct portfolios from $$\mathcal {F}$$, any other portfolio from $$\mathcal {F}$$ can be constructed through an allocation between these two portfolios. As a consequence, when two optimal portfolios are found, the subsequent problem of finding other optimal portfolios is just a problem of allocation between these two funds.
Pierre Brugière

### Chapter 5. Markowitz with a Risk-Free Asset

Abstract
In this chapter, a risk-free asset is added to the set of investable securities and the optimal portfolios are now derived in this augmented economy. One of the results obtained is that, in the risk/return analysis, the parameters (σ, m) of any investment portfolio lay either on or inside a certain cone $$\mathcal {C}(\sigma , m)$$. The upper side of the cone represents the efficient investment portfolios and is called the Capital Market Line . We show that the portfolios on the Capital Market Line can be built as an allocation between the risk-free asset and a particular investment portfolio, made of risky assets only, called the Tangent Portfolio . The Tangent Portfolio appears to define the tangent point between the cone $$\mathcal {C}(\sigma , m)$$ and the hyperbola $$\mathcal {F}(\sigma , m)$$ delimiting the (σ, m) of all investment portfolios made of risky assets only. Based on some economic reasoning, the Tangent Portfolio is sometimes assimilated to the Market Portfolio, for which the investment in each asset is proportional to its relative market capitalisation. We also show in this chapter that the problem of optimal allocation can be segmented into two steps. First, the investor decides on the risk exposure he is ready to take, secondly, he calculates the allocation to the Tangent Portfolio which gives him this level of risk (the rest of the money being invested in the risk-free asset). This paradigm for investing is known as the Separation Theorem of James Tobin (Nobel Prize in Economics in 1981, see Tobin) and shows that whatever the risk appetite is, there is only one way to take risk exposure efficiently.
Pierre Brugière

### Chapter 6. Performance and Diversification Indicators

Abstract
This chapter describes some statistics which, when screening a large number of funds, are useful to classify them and to automatically select the ones which seem to be particularly relevant. Some of these indicators depend on the leverage used by the funds while others measure the intrinsic quality of the fund, i.e. its engine of performance independently from any potential leverage artefacts. The Diversification ratio is also explained, as it is linked to many new alternative methods of asset allocation such as risk parity investing.
Pierre Brugière

### Chapter 7. Risk Measures and Capital Allocation

Abstract
Risk measures are widely used in risk management, and to calculate capital requirements when investing or conducting banking or insurance activities. In this chapter, we study risk measures in the context of asset allocation, and explain the notions of Value at Risk , Expected Shortfall and Return on Risk-Adjusted Capital (RORAC ). We provide some explicit formulas in the Gaussian framework and an example of calculation based on historical data, without any model assumptions. Euler’s formula is presented, for standard homogeneous risk measures, as well as its applications for capital allocation between risky positions. We also prove that, when the capital is allocated according to Euler’s formula, each position produces the same RORAC.
Pierre Brugière

### Chapter 8. Factor Models

Abstract
In the Security Market Line theorem, the Tangent Portfolio happens to be a single factor, which explains alone all the excess expected returns of all the assets to the risk-free rate, and incidentally explains a portion of their risks, which is called the systematic risk. If now the aim is to explain the risk, i.e. the standard deviation of the returns of all the assets, then the Tangent Portfolio may not be the best instrument to consider, as that is not the specific purpose of this factor. In this chapter, we study techniques to find the best factors to explain the risks, and do not limit ourselves to searching for a single factor. Ideally, the set of factors identified should explain most of the risky assets’ variances and correlations, and potentially leave the residual unexplained variations as independent “noises”. Two types of factors can be considered: endogenous factors, which are statistically derived from the observed variables, i.e. from the observed returns of the assets, and exogenous factors, which are explanatory variables added to the model, such as inflation or macro-economic indicators. The normal distribution assumption is maintained here, keeping us in the Markowitz framework. When a factor model follows an additional condition, called the APT condition, it is called an APT model . For these models the fundamental APT theorem links each factor to a risk premium. In APT models the factors explain all the common sources of risks of the risky assets, which was the primary objective, but also the expected excess returns of the risky assets to the risk-free rate.
Pierre Brugière

### Chapter 9. Identification of the Factors

Abstract
In this chapter we use Principal Component Analysis to study the returns of a set of risky assets and to identify the most relevant factors explaining their variations. The residual risks will be uncorrelated with the factors by construction, and therefore the general conditions of a factor model are satisfied. The number of factors chosen will be based on the percentage of the total variance they explain. If a large portion of the total variance is explained then for most stocks the residual risks will be small.
Pierre Brugière

### Chapter 10. Exercises and Problems

Abstract
In this chapter some midterm and final exam subjects are presented with their solutions. Many of the subjects deal with alternative methods to demonstrate some important results from the course.
Pierre Brugière