The role of risk management in financial firms has evolved far beyond the simple insurance of identified risks. Today it is recognised that risks cannot be properly managed unless they are quantified. And the assessment of risk requires mathematics. Take, for instance, a large portfolio of stocks. The relationship between the portfolio returns and the market returns – and indeed other potential risk factor returns – is typically estimated using a statistical regression analysis.
And the systematic risk of the portfolio is then determined by a quadratic form, a fundamental concept in matrix algebra that is based on the covariance matrix of the risk factor returns.
Volatility is not the only risk metric that financial risk managers need to understand. During the last decade value-at-risk (VaR) has become the ubiquitous tool for risk capital estimation. To understand a VaR model, risk managers require knowledge of probability distributions, simulation methods and a host of other mathematical and statistical techniques. Market VaR is assessed by mapping portfolios to their risk factors and forecasting the volatilities and correlations of these factors. The diverse quantitative techniques that are commonly applied in the assessment of market VaR include eigenvectors and eigenvalues, Taylor expansions and partial derivatives. Credit VaR can be assessed using firm-value models that are based on the theory of options, or statistical and/or macro-econometric models. Probability distributions are even applied to operational risks, though they are very difficult to quantify because the data are sparse and unreliable. Indeed, the actuarial or loss model approach has been adopted as industry standard for operational VaR models.
Even if not directly responsible for designing and coding a risk capital model, middle office risk managers must understand these models sufficiently well to be competent to assess them. And the risk management role encompasses many other responsibilities. Ten years ago my best students aspired to become traders because of the high salaries and status – risk management was viewed (by some) as a ‘second-rate’ job that did not require very special expertise. Now, this situation has definitely changed. Today, the middle office risk manager’s responsibility has expanded to include the independent validation of traders’ models, as well as risk capital assessment. And the role of risk management in the front office itself has expanded, with the need to hedge increasingly complex options portfolios. So today, the hallmark of a good risk manager is not just having the statistical skills required for risk assessment – a comprehensive knowledge of pricing and hedging financial instruments is equally important.
No wonder, therefore, that the PRM qualification includes an entire exam on mathematical and statistical methods. However, we do recognise that many students will not have degrees in mathematics, physics or other quantitative disciplines. So this section of the Handbook is aimed at students having no quantitative background at all. It introduces and explains all the mathematics and statistics that are essential for financial risk management. Every chapter is presented in a pedagogical manner, with associated Excel spreadsheets explaining the numerous practical examples. And, for clarity and consistency, we chose two much respected authors of the highly acclaimed textbook Quantitative Methods in Finance to write the entire section. Keith Parramore and Terry Watsham have put considerable effort into making the PRM material accessible to everyone, irrespective of their quantitative background.
The first chapter, II.A (Foundations), reviews the fundamental mathematical concepts: the symbols used and the basic rules for arithmetic, equations and inequalities, functions and graphs, etc. Chapter II.B (Descriptive Statistics) introduces the descriptive statistics that are commonly used to summarise the historical characteristics of financial data: the sample moments of returns distributions, ‘downside’ risk statistics, and measures of covariation (e.g. correlation) between two random variables. Chapter II.C (Calculus) focuses on differentiation and integration, Taylor expansion and optimisation. Financial applications include calculating the convexity of a bond portfolio and the estimation of the delta and gamma of an options portfolio. Chapter II.D (Linear Mathematics and Matrix Algebra) covers matrix operations, special types of matrices and the laws of matrix algebra, the Cholesky decomposition of a matrix, and eigenvalues and eigenvectors. Examples of financial applications include: manipulating covariance matrices;
calculating the variance of the returns to a portfolio of assets; hedging a vanilla option position;
and simulating correlated sets of returns. Chapter II.E (Probability Theory) first introduces the concept of probability and the rules that govern it. Then some common probability distributions for discrete and continuous random variables are described, along with their expectation and variance and various concepts relating to joint distributions, such as covariance and correlation, and the expected value and variance of a linear combination of random variables. Chapter II.F (Regression Analysis) covers the simple and multiple regression models, with applications to the capital asset pricing model and arbitrage pricing theory. The statistical inference section deals with both prediction and hypothesis testing, for instance, of the efficient market hypothesis.
Finally, Chapter II.G (Numerical Methods) looks at solving implicit equations (e.g. the Black–
Scholes formula for implied volatility), lattice methods, finite differences and simulation.
Financial applications include option valuation and estimating the ‘Greeks’ for complex options.
Whilst the risk management profession is no doubt becoming increasingly quantitative, the quantification of risk will never be a substitute for good risk management. The primary role of a
financial risk manager will always be to understand the markets, the mechanisms and the instruments traded. Mathematics and statistics are only tools, but they are necessary tools. After working through this part of the Handbook you will have gained a thorough and complete grounding in the essential quantitative methods for your profession.
Carol Alexander, Chair of PRMIA’s Academic Advisory Council and co-editor of The PRM Handbook