From Markowitz to modern risk management

Abstract

Nobel Laureate Harry Markowitz is often referred to as the ‘founder of Modern portfolio theory’ and deservedly so given his enormous influence on the money management industry.¹ However, it is my contention that he should also be referred to as the ‘founder of Modern Risk Management’ since his contributions to portfolio theory formed the basis for how risk is currently viewed and managed. More specifically, Markowitz argued that a portfolio of securities should be viewed through the lens of statistics where the probability distribution of its rate of return is evaluated in terms of its expected value and standard deviation. Since the ultimate selection of a portfolio involves the evaluation and management of risk as measured by standard deviation, it is clear that Markowitz's process of portfolio selection represents the birth of modern risk management whereby risk is quantified and controlled. In this paper, I will first, introduce value-at-risk as a measure of risk and how it relates to standard deviation, the risk measure at the heart of the model of Markowitz. Second, I will similarly introduce conditional value-at-risk (also known as expected shortfall) as a measure of risk and compare it with VaR. Third, I will briefly introduce stress testing as a supplemental means of controlling risk and will then present my conclusions.²

Keywords:

• risk management
• value-at-risk
• conditional value-at-risk
• stress testing

Acknowledgements

This paper is based on the Keynote Address given at the Asset Management and International Capital Markets Conference that was held in Frankfurt on 29 May 2008. The author is indebted to Alexandre M. Baptista for comments and many stimulating discussions on the topic.

Notes

Wikipedia, for example, makes such a reference to Markowitz: http://en.wikipedia.org/wiki/People_known_as_the_father_or_mother_of_something#Economics . Also see the press release announcing his being awarded the 1990 Alfred Nobel Memorial Prize in Economic Sciences: http://nobelprize.org/nobel_prizes/economics/laureates/1990/press.html.

My comments draw largely from Alexander and Baptista (2002, 2004, 2006). An excellent book on VaR, CVaR, and stress testing is Jorion (2007). Also see Crouhy, Galai, and Mark (2006) and Hull (2007).

Technically VaR is equal to E _p −zS _p . However, this typically produces a negative number. The desire to express VaR as a positive number results in Equation (1). It is also important to point out that Equation (1) and the subsequent analysis is based on the assumption that the portfolio's returns have a normal distribution (it is straightforward to show that similar results are obtained with a t-distribution).

Note that MVaRP might have an infinite expected return depending on the shape of the hyperbola and confidence level. I assume it has a finite expected return in all of my examples.

Motives for controlling the tail risk of the bank's trading portfolio are two-fold. First, it is a means of limiting the probability of default. Second, the bank's capital requirement is based in part on the VaR of its trading portfolio. For example, see Crouhy, Galai, and Mark (2006, 154–61) and the Basle Committee on Banking Supervision (2006), respectively. Also see Ball and Fang (2006) for a survey of the literature on VaR and bank regulation.

Due to the practice of expressing VaR and CVaR as positive numbers, technically −CVaR is the portfolio's expected return conditioned on the return being less than or equal to −VaR.

It is possible for a portfolio's VaR and CVaR to be equal when the lower mass of the probability density function is concentrated at the portfolio's VaR. That is, VaR=CVaR when the probability of observing a return less than the portfolio's VaR is zero. Since such distributions are highly unlikely, for expository purposes it is assumed that CVaR.

Note that MCVaRP might have an infinite expected return depending on the shape of the hyperbola and confidence level. I assume it has a finite expected return in all of my examples.

Banks are currently required to stress test their portfolios; see Basle Committee on Banking Supervision (2006) and Committee on the Global Financial System (2005).

The bounds can be of different values for the various scenarios; a constant bound is used here simply for expository purposes.