# Taking Risk Measurement to the Extreme Nearly a decade after the global financial crisis, most methods for quantifying risk still don’t adequately account for extreme market events, but new measures are available to create a more modern risk management framework.

It’s been ten years since U.S. subprime mortgage lender New Century Financial Corp. filed for bankruptcy, after the New York Stock Exchange had delisted its shares amid revelations that federal regulators were investigating the company’s accounting practices. Three months later investment bank Bear Stearns Cos. liquidated two hedge funds that had invested in structured credit products stuffed with subprime mortgages issued by lenders like New Century — a harbinger of the growing credit troubles that would rip through Wall Street, ultimately forcing Bear Stearns to accept a government-brokered sale to JPMorgan Chase & Co., and Lehman Brothers Holdings to declare bankruptcy.

Directly or indirectly, the global financial crisis affected most financial institutions and a majority of the world’s population. Triggered by a real estate bubble and aggravated by the leverage effect of various derivative products based on securitization and collateralization, the crisis was met with an unprecedented coordinated response by regulators. Following the immediate rescue actions — most notably, central bank quantitative easing — regulators, institutions and finance professionals from both industry and academia conducted a rigorous analysis to determine what caused this disastrous chain of events. As a consequence, new regulations were adopted, implementing strict audit, control, oversight and enforcement practices. Certain activities became restricted, and new penalties for unethical business activities were introduced.

But let’s face it: Although this crisis was special because of its scope and scale, it wasn’t the first one in history. Each time a market crash occurs, there are similar responses. And later some fresh development appears — a new exchange or instrument, massive IT sector growth, securitization and derivatives, or something unknown and not yet taken into account by regulations — that leads investors to try to capitalize on the new opportunities, without realizing the risks involved. That is how the next market crash happens.

Crisis prevention is in the interest of society as a whole. The measures to ensure it are dictated by regulatory and government institutions. For their part, banks, brokerages, insurance companies, investment managers, pension funds and other financial markets participants are supposed to conduct proper risk management, consisting of two major elements: risk measurement and risk taking. Although after each market disaster the financial industry reduces its risk taking, limiting what it considers to be dangerous transactions, risk measurement practices remain largely unchanged.

**Current Practices **

The present approaches to quantifying risk may be the most outdated, if not archaic, aspect of the financial system. The two most common measures — standard deviation and beta — have been around for more than a half century. Value at risk (VaR), which measures the maximum loss of an investment portfolio under normal market conditions, has been around since the 1990s. Risk measurement, however, is not a closed subject. In recent years researchers have worked to find alternative metrics that would more adequately serve investors’ risk management needs. Despite investors’ general reluctance to embrace new methods before they have received widespread acceptance, some of these alternative measures are ready to come to light as potential successful candidates for laying the foundation for a fundamentally redesigned framework for risk management.

The standard views on quantifying risk haven’t evolved that much since Harry Markowitz and William Sharpe conducted their pioneering research in the 1950s and 1960s, respectively, for which they later shared (with Merton Miller) the Nobel Prize in economics. Known as the father of Modern Portfolio Theory, Markowitz reduced portfolio selection to two dimensions: expected investment return and its variance, or standard deviation. Sharpe was one of several academics who (independently from one another) came up with the Capital Asset Pricing Model (CAPM), which explains the relationship between a portfolio’s expected return and its riskiness. Investment managers still rely on CAPM’s concept of beta, a measurement of the systematic portion of variance of stock or portfolio returns through correlations with the market portfolio, or on multifactor extensions of CAPM, like Barra, which also focus on segregating the systematic portion of variance.

Concepts like beta, standard deviation and VaR are convenient tools for constructing portfolios in a stable economic environment, enabling investors to optimize their holdings from a risk-return perspective with a desired exposure to market trends. However, during periods of extreme asset movements, like the 2008–’09 financial crisis, estimated correlations no longer reflect the codependency of asset prices, and these risk measurement approaches fail to provide the necessary security when it’s needed the most.

#### Alternative Risk Measures

According to the assumptions of classical Markowitz portfolio theory, all investors are mean-variance optimizers — they structure their investments to minimize the variance or standard deviation of their returns. Many researchers have disputed the choice of standard deviation as a quantifier of risk, largely because it is a symmetric measure: Most investors are concerned primarily about the downside portion of variance; they’re perfectly content with deviation above the mean. Some researchers, like Javier Estrada, a finance professor at IESE Business School in Barcelona, have extolled the merits of semivariance, which measures a portfolio’s downside risk by looking only at the returns that fall below the mean. Investors can calculate semivariance and semideviation by averaging the squared deviations of returns below the mean.

Standard deviation tends not to perform well during extreme market events. Two securities with seemingly independent return streams, for example, might instead turn out to be almost perfectly correlated when the economy is in shock, as a result of a chain of bankruptcies. Switching to semideviation would hardly address this issue because it doesn’t specifically deal with tail risk when investment returns fall outside a normal distribution during extreme events like the global financial crisis.

One measure that is particularly well suited for quantifying tail-risk distributions is conditional value at risk (CVaR). Also known as tail VaR, CVaR measures the expected loss exceeding the value at risk. Over the past 15 years, CVaR has attracted growing attention from both academics and practitioners. It fits nicely into a broader theory, based on a set of axioms defining properties of functionals, that is convenient in various applications involving uncertainty. (Functionals map functions from a vector space into a field.) Mathematicians R. Tyrrell Rockafellar of the University of Washington and Stan Uryasev of the University of Florida have organized these axioms into what they call the fundamental quadrangle of risk. This concept provides logical links among four classes of functionals — measures of risk, deviation, regret and error — and a fifth, “statistic” functional that determines the theme of a particular instance of the risk quadrangle.

The risk quadrangle concept is truly brilliant; it can be seen as a map covering various optimization and statistics problems. For example, a researcher concerned about finding factors explaining some of the variance of a dependent variable could run a linear regression, which can be expressed as an *L ^{2}*-error minimization problem. The fitted model will then express an estimate of mathematical expectation of the dependent variable. Mathematical expectation, in this instance of the risk quadrangle, is a statistic associated with

*L*error and standard deviation. Another instance links CVaR deviation, Koenker-Bassett error and quantile function; this corresponds to what is known as quantile regression.

^{2}For the purposes of this overview, we will focus on just one element of the risk quadrangle: generalized deviation measures. The name of this class should not be confusing, as standard deviation — currently, the most popular functional associated with risk in portfolio management — is a member of this class. The exact list of axioms, presenting a formal definition of deviation measures and their subclasses, changed slightly across the chain of publications that led to the introduction of the risk quadrangle. For example, in Rockafellar et al. (2006a) they were defined as follows:

(D1) *D(X + C) = D(X)* for all random variables *X* and constants *C*

(D2) * D(0) = 0* and *D(λX) = λD(X)* for r.v. *X* and constants *λ > 0*

(D3) *D(X + Y) ≤ D(X) + D(Y)* for all r.v. *X* and *Y*

(D4) *D(X)* ≥ 0 for all r.v. *X* with *D* > 0 for nonconstant *X*

It can be shown that standard deviation satisfies (D1)–(D4). This class of measures has very important subclasses, defined through additional axioms.

(D5) {*X|D(X)* ≤ *C*} is closed for every constant *C*

Deviation measures satisfying (D5) are called lower-semicontinuous. Another important axiom, called lower-range dominance, is defined as follows:

(D6) *D(X)* ≤ *EX* – inf *X* for all r.v. *X*

This condition is very important, as it ensures a useful property of corresponding measures of risk, referred to as coherency. It can be shown that standard deviation doesn’t satisfy this condition.

Substituting an alternative deviation measure in place of standard deviation in Markowitz portfolio theory creates a new portfolio optimization framework. As shown in Rockafellar et al. (2006b, 2006c), a version of the one-fund theorem (stating the existence of a portfolio of risky assets with a minimum value of deviation) holds for the class of lower-range-dominated, lower-semicontinuous deviation measures. This portfolio is called the master fund: We can approximate it with a broad index fund, following the same intuition that lies behind classical CAPM. Optimality conditions for the master fund arise in the form of CAPM-like relations, expressing expected return on a security through expected return on the master fund and generalized beta. Like classical beta, the latter is a scaling factor, reflecting a portfolio’s exposure to systematic risk. Its definition, however, is different from classical beta’s and tied to the choice of the deviation measure functional.

#### Generalized Beta

Relying on classical CAPM to manage the systematic risk in a portfolio involves using traditional beta. According to CAPM, a beta-neutral portfolio is expected to perform independently of market trends. But, as we discussed, the definition of beta is manifested by the assumption that investors measure risk through standard deviation. For lower-range-dominated, lower-semicontinuous measures of deviation, a generalized version of CAPM holds, and generalized beta can be introduced.

For a lower-range-dominated, lower-semicontinuous measure of deviation *D*, dual characterization becomes possible, which means the existence of a nonempty, closed and convex class ε of random variables *Q*, called the risk envelope, with the following properties:

Random variable *Q* ∈ ε, for which *D(X)* = *EX* – *E*[*QX*], is called the risk identifier. Although this dual representation may at first sound complex, it will become more intuitive as we illustrate it with a specific class of deviation measures, quantifying the tails of distributions. This is where the risk measure CVaR comes in. CVaR can be defined for continuous distributions as *E*(–*X|X < F _{x}^{-1}(α)), *where

*F*is α-quantile of distribution of

_{x}^{-1}(α)*X*. It can be shown that the corresponding functional defined as CVaR

_{α}(

*X*–

*EX*) is a lower-range-dominated, lower-semicontinuous deviation measure.

For CVaR deviation the risk identifier of r.v.* X* takes the following form:

We are ready to introduce generalized CAPM beta. Let’s denote *R* as a security return; *R _{M}* is the return of the master fund, representing the optimal portfolio from risk-return perspective, where risk is quantified by a deviation measure

*D*. Generalized beta of a security is then defined as

where *Q _{M }*is the risk identifier

*R*for deviation measure

_{M}*D*.

If we find a deviation measure satisfying the above properties (D1)–(D6) and adequately reflecting the perception of risk, with an emphasis on the tails of distributions, the corresponding beta would become an alternative instrument for portfolio immunization, protecting against worst-case scenarios. CVaR deviation appears to be a good candidate.

#### Mixed CVaR Deviation and Tail Beta

CVaR, also known as mean excess loss and mean shortfall, has become an attractive alternative to VaR because of the concept of coherency. As Rockafellar and Uryasev have shown, investors can apply convex optimization techniques in problems involving CVaR. According to the risk quadrangle, there is a one-to-one correspondence between measures of risk and measures of deviation:

This relation immediately yields the definition of CVaR deviation as

Measures of deviation satisfy an important property: A convex combination of deviation measures (with nonnegative weights summing up to 1) is also a deviation measure, preserving each axiom. Its risk envelope would be equal to the closure of the same combination of individual risk envelopes.

Utilizing this property, we can define mixed CVaR deviation as a combination of several (or even continuous set) CVaR deviations with different parameters α. For example, an investor can assume levels *α*_{1} and *α*_{2} with weights *λ* and 1 – *λ*. If *α _{1}* <

*α*, the risk identifier for a portfolio return

_{2}*X*would then take the following form:

Because CVaR deviation measures the severity of losses associated with the tail of distribution, it is a very natural measure to consider minimizing at the time of portfolio construction. An investor immunizing a portfolio against systematic risk using CVaR deviation would also want to consider beta based on CVaR deviation. Generalized beta, based on CVaR deviation or mixed CVaR deviation, is also called tail beta. The formal definition of tail beta based on mixed CVaR deviation can be found in Kalinchenko et al. (2012). In the case with two components, it takes the following form:

where *Q _{M,α}*

_{1}and

*Q*

_{M,α2}are risk identifiers from the following class of random variables:

Let’s see why tail beta works. A risk identifier, by construction, is a step function, identifying the lower tail of the distribution of the master fund — the worst outcomes for the market. Covariance in the numerator quantifies the dependency of stock returns on whether one of these outcomes takes place. The denominator is present for a reason analogous to the presence of square variance in classical CAPM. As a result, the returns of a portfolio with zero tail beta are expected to be independent of whether or not the master fund’s return is within the lower tail.

#### A Challenging Task

Risk management practices in their current state developed largely from the utilization of standard deviation as a quantifier of risk, as first introduced by Harry Markowitz in 1952. Since then standard deviation has been the kernel of even the most recent risk management tools, and there is little evidence of alternative measures being widely adopted in the financial industry.

Regulators and consulting firms understand the need for market participants to address extreme market scenarios in their risk management practices. The Basel III accord, a set of voluntary reform measures established by the Basel Committee on Banking Supervision in the wake of the financial crisis, explicitly advises banks to develop and implement internal ratings-based approaches, while the U.S. Federal Reserve coordinates the implementation of stress-testing procedures. But neither Basel III nor the Fed offers a clear definition of any new quantifiers of risk that reflect tails of distributions. Nor are there any popularly available commercial solutions for better estimation of risk. As a consequence, many banks, asset managers, insurance companies and other financial institutions rely on in-house efforts — for example, conducting stress tests for extreme market events. But even then they typically rely on traditional risk measures like VaR.

Popularizing alternative measures like CVaR could initiate the development of new risk models and risk management solutions, create new standards in risk management and potentially lead to a more robust financial system. This is a challenging task, however, as the financial services industry is prone to inertia. More effective risk measurement approaches — cemented in audit policies, commercial risk management solutions, educational materials and standardized recommendations — would require more than just academic research. All market participants would have to realize the need to rebuild the system, assess alternative approaches and develop a road map for introducing new standards. Let’s hope it won’t take another financial crisis to convince them of the need for change.

#### References

Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath. “Coherent Measures of Risk.” *Mathematical Finance* 9, no. 3 (July 1999): 203-227.

Javier Estrada. “The Cost of Equity of Internet stocks: A Downside Risk Approach.” *European Journal of Finance* 10, no. 4 (August 2004): 239–254.

Konstantin Kalinchenko, Stan Uryasev and R. Tyrrell Rockafellar. “Calibrating Risk Preferences with the Generalized Capital Asset Pricing Model Based on Mixed Conditional Value at Risk Deviation.” *Journal of Risk* 15, no. 1 (fall 2012): 45-70.

Roger Koenker and Gilbert Bassett Jr. “Regression Quantiles.” *Econometrica* 46, no. 1 (January 1978): 33-50.

Harry Markowitz. “Portfolio Selection.” *Journal of Finance* 7, no. 1 (March 1952): 77-91.

R. Tyrrell Rockafellar, Stan Uryasev and Michael Zabarankin. “Generalized Deviations in Risk Analysis.” *Finance and Stochastics* 10 (2006): 51-74.

R. Tyrrell Rockafellar, Stan Uryasev and Michael Zabarankin. “Master Funds in Portfolio Analysis with General Deviation Measures.” *Journal of Banking and Finance* 30, no. 2 (2006): 743-778.

R. Tyrrell Rockafellar, Stan Uryasev and Michael Zabarankin. “Optimality Conditions in Portfolio Analysis with General Deviation Measures.” *Mathematical Programming* *Series B* 108, issue 2-3 (September 2006): 515-540.

R. Tyrrell Rockafellar and Stan Uryasev. “The Fundamental Risk Quadrangle in Risk Management, Optimization and Statistical Estimation.” *Surveys in Operations Research and Management Science* 18, issue 1-2 (October 2013): 33-53.

R. Tyrrell Rockafellar and Stan Uryasev. “Optimization of Conditional Value-At-Risk.” *Journal of Risk* 2, no. 3 (2000): 21-41.

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