Risk Analysis: A Geometric Approach

Transcription

1 Risk Analysis: A Geometric Approach Brian D. Singer, CFA Managing Director Brinson Partners, ncorporated Quantitative methods for risk management should allow investors and portfolio managers to look at, and try to manage, risk in new ways. A geometric approach can help in displaying the risk characteristics of a portfolio and its benchmark and in assessing the impact of portfolio constraints. R isk management and quantitative methods are typically considered to be almost interchangeable, even to the extent that risk management requires or depends on quantitative sophistication. Although a quantitative perspective can certainly be useful in risk management, too often quantitative methods and analytical elegance provide the illusion of controloverrisk. Such approachesmake investors feel as if they have grasped uncertainty and dealt with it simply by the act of quantifying it. But risks, by their very nature, are unexpected. So, quantitative methods should not turn confidence into arrogance; rather, what quantitative methods should do is allow investors (and portfolio managers) to look at risk in new ways-to try to manage risk in ways that they previouslycould notdo, or werenotnecessarily comfortable doing. This presentation discusses a process that uses Euclidean geometry to visualize risk. Such an approach is somewhat avant-garde for risk management and is decidedly quantitative, but the intent is to illustrate a tool that enables quantitative risk managers to communicate with nonquantitative portfolio managers and nonquantitative clients. Although the approach applies to any investment horizon, this discussion takes a relatively long-term perspective on risk-one that is typical of an investment policy perspective-and allows direct analysis of portfolio risk and portfolio constraints. Risk Estimation: Data Risk estimation relies on volatility and correlation data to construct covariance matrixes; one of the key questions, of course, is which data? Historical Data. Risk estimation typically begins with the use of historical data-the computa- tion of historical volatilities and correlations. Historical data are consistent, easy to obtain, and often easy to compute. A manager, for instance, can compute a covariance matrix very easily with Microsoft Excel. The problem with historical data is that the data are almost certain to be inappropriate representations of the future. An investor looking at a broad index of the U.S. bond market, such as the Lehman Brothers Aggregate Bond ndex, would find that the volatility of that index was in excess of 10 percent in the late 1970sand early 1980s. Currently, that same index has a volatility of 4-5 percent. That historical period (late 1970s and early 1980s) was characterized by high and volatile inflation, which is not the case now. Thus, that investor would not want, in any forward-looking sense, to rely on that period as a foundation for his or her risk estimates unless the investor believed, for example, that the U.S. Federal Reserve Board was planning to monetize, or in effect provide an inflation tax for, fiscal policy. This is a very real investment problem: Suppose at Brinson Partners we are trying to set the investment policy-the normal policy mix-for a pension plan, an endowment, or a foundation. n that instance, the client's time horizon is long term, so we do not want to know a daily or weekly value at risk estimate. History might not necessarily represent what we think could happen in the future, but an analysis of monthly or quarterly data going back several decades aids in our understanding of risk in various economic and market environments. Granted, a numberof advances in statisticalmethodology applied to historical data have occurred: the use of volatility clustering, the use of generalized autoregressive conditional heteroscedasticity, and the Association for nvestment Management and Research 73

2 Risk Management: Principles and Practices use of mean reversion for forecasting volatilities. All of those historical approaches have been beneficial for estimating risk, especially over short horizons, but investors are still faced with regime changes-some notable, some not notable; some identifiable, some not identifiable-and every regime change decreases the relevance of historical data. An interesting example comes from New Zealand, which for years suffered from high and volatile interest rates and, therefore, volatile bond returns. n an attempt to change that environment, New Zealand altered the charter for its central bank. The New Zealand central bank now provides an inflation target, and if the head of the central bank does not meet that target, he or she is fired. New Zealand's inflation volatility is now much lower than it was in the past. Forward-Looking Data. Because such regime changes are possible, forward-looking volatilities and correlations can be, although are not always, better representations of the future. That regime change in New Zealand was quick and identifiable, buthistorical data would not have predicted it. Thus, having some type of forward-looking perspective, in terms of the covariance matrix, is a good idea. But forward-looking matrixes also have problems, the biggest one being limitations on human imagination. n a forward-looking sense, people can only incorporate what they imagine,but risks, by their nature, are unexpected. Therefore, it is difficult to incorporate the appropriate forward-looking events or regime changes that might affect the covariance matrix. Geometric Representation. Other important difficulties with using a forward-looking perspective are achieving consistency and intuition, which is where Euclidean geometry comes in. A geometric interpretation of volatilities and correlations has the potential to make risk estimation consistent, practical, and more intuitive to understand and communicate, especiallybetweenquantitative-oriented people and non-quantitative-oriented people. Why take a geometric approach? The mathematician Keith Devlin has commented that mathematicians may be able to express their thoughts using the language of algebra, but generally, they do not think that way. Even a highly trained mathematician may find it hard to follow a long, algebraic argument. But every single one of us is able to manipulate mental pictures and shapes with ease. By translating a complicated problem into geometry, the mathematician is able to take advantage of this fundamental human capability. This ability to manipulate shapes starts in childhood. Children at a very young age learn to put round pegs in round holes, square pegs in square holes, and triangular pegs in triangular holes. They understand and learn to manipulate shapes long before they are able to grasp algebra and other mathematical concepts. Currency Risk Assume that from a U'S. dollar perspective the U.K. pound has a volatility of 12 percent and the German mark also has a volatility of 12 percent. The correlation between the pound and the mark is Although with this information an investor can construct a very simple covariance matrix, another way of looking at the covariance matrix is geometrically, as shown in Figure 1. Volatilities are shown as distances, and correlations are shown as angles. To construct Figure 1 from a U.S. dollar perspective, first made a point for the dollar. Because the volatility of the mark compared with the dollar is 12 percent, drew a line of length 12. That line could be 12 inches, 12 centimeters, or 12 kilometers; it does not matter, just 12 units of length. The volatility of the pound against the dollar is also 12 percent, so needed to draw another line of length 12. The question is, what is the relationship (or the angle) between those two lines? The answer is that the relationship is determined by the correlation, which is 0.71 in this example; specifically, the cosine of the angle is equal to The cosine of 45 is 0.71, so drew the second line at a 45 angle to the first line. Thus, have portrayed the same covariance matrix, but instead of Figure 1. Visual Representation of Volatilities and Correlations from a U.S. Dollar Perspective Dollar 12.0% cos(a) = P( /DM) = % Note: p is the correlation coefficient. Pound Mark 74 Association for nvestment Management and Research

3 Risk Analysis using strictly numbers, have portrayed it as part of a triangle.' One of the tools we use at Brinson Partners is what refer to as the correlation protractor. John Zerolis, one of the more quantitative-oriented people at Brinson Partners, generated the protractorby computing the correlation associated with each angle. We use the protractor for discussions in which immediate visual representations are useful. One of the interesting things people notice when looking at that correlation protractor is that not all correlation changes are created equally in risk space. Suppose a correlation goes from 0.9 to 1.0. This might not seem like a big change, but moving from 0.9 to 1.0. is a 26 angle on the protractor. Similarly, a 26 angle from zero moves the correlation from zero to about n risk space, a movement in correlation from zero to 0.45 is similar to a movement in correlation from 0.9 to 1.0. This relationship is readily apparent from a geometric representation but not at all obvious from a set of formulas. Now suppose want to know the volatility of the pound from a mark perspective. All need to do is draw a line connecting the pound and the mark and measure the length of that line. The dotted line in Figure 1 indicates that the length is 9.1 (hence the volatility is 9.1 percent). n addition, if look at the angle between the dotted line and the solid dollar/ mark line, can tell that from a mark perspective, the dollar and the pound have a correlation of This technique facilitates the ability to understand a single covariance matrix from the perspective of everyinvestorin the world, regardless of the investor's base currency. For U'S. investors, we focus on the dollar vertex of the triangle. For German investors, we focus on the mark vertex, and so on. We can use any number of different base currencies. With just three currencies, we can geometrically represent the correlation matrix on a piece of paper; with four currencies, we would need a three-dimensional tetrahedron. With five currencies, visualization must occur in triangular or tetrahedral subsegments, but the intuition is still the same. One benefit of this approach, in terms of consistency, is being able to see the implications of a covariance matrix from any base currency perspective. Suppose we think that the United Kingdom is going to join theeuropeanmonetaryunion (EMU) andthat thepound'scorrelationwith the euro (represented by the mark) will probably increase to 0.95 as the United For further discussion, see Brian D. Singer, Kevin Terhaar, and John Zerolis, "MaintainingConsistentGlobal Asset Views (with a Little Help from Euclid)," Financial Analysts Journal (January February 1998): Kingdom approaches joining the EMU. Figure 2 shows what happens if the correlation between the pound and the mark is 0.95, which corresponds to an angle of about 18 : A correlation of 0.95 means that the pound must have a volatility of 3.8 from a mark perspective. So, compared with Figure 1 (where the correlation was 0.71), the volatility dropped from 9.1 to 3.8. f we are not comfortable with that change in volatility, then we cannot be comfortable with our correlation estimate of Notice thatifthe correlation between the pound and the mark were 1.0, the line wouldessentiallybecome flat, whichimplies that the pound would have no volatility from a mark perspective. Figure 2. Dollar Effect of Correlation Change Note: p is the correlation coefficient. (T Porrjo/io cos(().) = PPortfolio, ~(T rr <TBmchmark Portfolio Pound.3.8% Mark Portfolio Risk Analysis Risk analysis of a portfolio relative to its benchmark is a simple application of this geometric approach, as shown in Figure 3. The volatility of the benchmark (benchmark risk) is represented by the base of the triangle, the volatility of the portfolio (portfolio risk) is represented by the side of the triangle drawn with the solid line, and the portfolio's tracking error is represented by the side of the triangle drawn with a dotted line. The correlation between the benchmark and the portfolio is represented by the angle o; and Figure 3. Portfolioand Risk Analysis in Geometric Terms (l-l3)u Association for nvestment Management and Research 75

4 Risk Management: Principles and Practices the vertical dashed line indicates the portfolio's residual risk. Portfolio,, and Residual Risks. n the context of a single-index model, the return of the portfolio is equal to a benchmark bet and the residual return, which is uncorrelated with the benchmark. Similarly, portfolio volatility comes from twosources-one that is perfectly correlated with the benchmark bet, or systematic risk, and one that is uncorrelated with the benclunark risk, or residual risk. So, in Figure 3, the line for residual risk is at a right angle to the line for the benchmark risk because a right angle is associated with a correlation of zero. Thus, Figure 3 shows two right triangles, and consequently, the Pythagorean theorem (the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides) can be used to help withthe risk analysis. This approachallows us to look at the risks visually. We do not have to wonder what will happen if the residual risk of our portfolio goes up by 10 percent: The residual risk line will become 10percent longer, the correlation of ourportfoliowith the benchmark will go down because the angle will increase, and the volatility of our portfolio will increase (the portfolio line will lengthen). We do not have to calculate anything to achieve that intuitive understanding. Residual Risk, Relative Bets, and Tracking Error. The right-hand, shaded triangle in Figure 3 shows residual risk, benchmark relative bets, and tracking error. Tracking error can be thought of in this context as value-added risk-the riskof the portfolio from the perspective of the benchmark, or the risk of the difference between the portfolio returns and the benchmark returns. n essence, what this figure indicates is that tracking error is a combination of two things: (1)the benchmarkrelative bet (the base of the shaded triangle), which is the portion of active management that involves an increase or decrease (as in this example) in benchmarkexposure, and (2)the residual risk, whichis that portion of the risk of active management that is not in any way correlated with the benchmark. Ex Post Analysis. n a portfolio performance sense (an ex postsense), this geometric approach is a tool that can help investors understand the overall performance of their portfolios a little more intuitively and clearly. Suppose am a plan sponsor and one of my managers comes to me and says, "The benchmark has a volatility of x, the portfolio has a volatility of y, and the beta is Z." From that information, not only can compute the correlation, determine the cosine, and create the entire triangle, including the tracking error and residual risk (even though the manager did not give me that information), but can also quickly visualize and gauge the tracking error and residual risk without doing any computations. Although could use algebraic or trigonometric formulas to calculate the tracking error, using geometry is often easier because it allows me to visualize how the various risks move relative to each other. As in Figure 3, can see that the residual risk of the portfolio is found by dropping a perpendicular line down from the point for the portfolio; can see that the base of the left-most triangle, found by multiplying the beta times the benchmark risk, is the systematic risk. Ex Ante Analysis. From an ex antestrategyperspective, the geometric view also helps in making decisions about changing the portfolio. For example, it can help investors understand how a certain strategy or change in strategy might affect a portfolio in absolute and/or relative risk terms. Say we have a portfolio that holds some cash, and we think that taking out that cash might reduce the tracking error. fwe take out the cash, we do not change the portfolio's correlation with the benchmark. All we do is increase the risk, so we have to make the line for the portfolio longer. Does increasing the portfolio line decrease the tracking error? Not necessarily. The tracking error decreases to a point but then begins to increase again. Having a risk hedge in the portfolio might reduce risk or it might increase risk relative to the benchmark, whichis easy to see from a geometric, visual perspective. Portfolio Constraints Portfolio managers more often than not operate under a variety of constraints, such as beta, tracking error, or residual risk. But implementing those constraints can cause difficulties. Once again, geometric interpretation can be used to portray feasible sets of alternative portfolios that are consistent with client constraints. Beta Constraint. Suppose a client wants an essentially defensive portfolio. Figure 4 shows four portfolios that have a beta of 0.9. can create many portfolios that have a beta of 0.9, and some of those portfolios might be considered defensive, but some would not. Portfolio A, for example, would likely be considered defensive. t has a low volatility, and it has relatively low tracking error and relatively low residual risk. When the correlation with the benchmark is decreased while maintaining the beta of 0.9, the volatility of the portfolio has to increase. Portfolio D still has a beta of 0.9, but its volatility greatly exceeds that of the benchmark, and it has a relatively 76 Association for nvestment Management and Research

5 Risk Analysis Figure 4. Portfolios with Constant Beta of (J' Ben,hmark J"Bl'lchmark 0.1 (J' Residual Risk DT C 1 B + A+ ~ = il substantial tracking error. All of the portfolios in Figure 4 have a beta of 0.9, but they are very different portfolios; the beta constraint still allows dramatically different levels of portfolio risk, residual risk, and tracking error. Tracking-Error Constraint. Suppose a manager or a plan sponsor wants a portfolio with a tracking error of 5 percent or less. Again, can create a number of portfolios that have a tracking error of 5 percent, as shown in Figure 5. simply draw a circle with a radius length of 5, representing a tracking error of 5 percent, around the benchmark position. Portfolio A, which can be formed by combining the benchmark with cash, has a tracking error of 5 percent; it also has no residual risk and a relatively low volatility. Portfolio D has a beta of 1; its residual risk Figure 5. Portfolioswith ConstantTracking Error Less than 5 Percent Portfolios D.. E B..". :.: 0. '.. ~ " ~ 2\ (1 ~.::.... F 5% would have a volatility of 10 percent and Portfolio F would have a volatility of 20 percent. Figure 5 clearly illustrates that a tracking-error constraint still permits wide variations in volatility, beta, and residual risk. Residual Risk Constraint. can also construct a number of portfolios that have the same residual risk, as shown in Figure 6, but those portfolios are decidedly different from each other. Portfolio A has relatively low volatility but relatively high tracking error. By increasing the risk of the portfolio and its correlation with the benchmark, get to Portfolio C. All three portfolios in Figure 6 have the same residual risk, but Portfolio C has the highest volatility and the lowest tracking error. Thus, a constraint on residual risk places few limits on volatility, beta, or tracking error. Figure 6. Portfolios with Constant Residual Risk (fbellchmqrk Portfolios ~ A. B. ~T 1'1-.,;-.. '. '. ". :' :'1 1..._=-L- -'-::_.~.J 1 (J'ResidualRSk Multiple Constraints. Using this type of analysis allows one to look at the interactions of the investment guidelines imposed on a manager. Figure 7 shows a simple example of this type of interaction, in which the portfolio has two constraints. The first constraint is that the total risk cannot be any greater than that of the benchmark, indicated by the circle with the center at Point A and the radius equal to the Figure 7. Feasible Portfolios with Given Total Risk and Tracking Error 0' lracking Error is equal to its tracking error of 5 percent. ts volatility is slightly greater than that of the benchmark, which is very different from Portfolio A. Portfolio F still has a tracking error of 5 percent and no residual risk, which sounds a lot like Portfolio A, but Portfolio F is much more volatile than Portfolio A. n fact, if the benchmark has a volatility of 15 percent, Portfolio A C B (T Tracking Error Association for nvestment Management and Research 77

6 Risk Management: Principles and Practices benchmark (Point B).The portfolio can be anywhere within that circle. The second constraint is that the tracking error cannot be greater than some specified amount, indicated by a semicircle around the benchmark point (B) with a radius equal to the trackingerror constraint. When combine those two constraints, the only feasible portfolios are within the cross-hatched area. Consequently, the beta cannot be greaterthan 1, and the correlationcannotbeanything less than about 0.9. Portfolio P represents maximum total risk and maximum residual risk, and the minimum risk portfolio is Portfolio C. A simple example provides a good illustration of the interaction of multiple constraints. Consider a portfolio with the same risk as the S&P 500 ndexbut with a beta, with respect to the S&P 500, of 0.8. On the surface, those constraints sound fine, but looking at the implications geometrically may indicate otherwise. First, we draw a line whose length represents the volatility of the S&P 500. Because the portfolio and the benchmark have the same volatility, the beta and the correlation both are 0.8. Second, we draw a line whose angle corresponds to a correlation of 0.8 and that has the same length as the S&P 500. That line represents the portfolio. A straight line to connectthe S&P 500 and the portfolio reveals that the tracking error is 8-9 percent. Thus, the client's constraints seem reasonable-same volatility as the benchmark and a beta of O.8--but those constraints effectively create a disguised risk: tracking error of 8-9 percent. Does the client really want a portfolio with a tracking error of 8-9 percent? Probably not. Conclusion Geometrically displaying the risk characteristics of a portfolio and its associatedbenchmarkis a simplebut powerful tool. The geometric representation of portfolio performance and portfolio strategies helps in simultaneously analyzing multiple risks-absolute risk, relative risk, systematic risk, residual risk, and tracking error. The geometric decomposition also produces an intuitive understanding of the interactions, sometimes subtle and often unintended, that result from imposing portfolio constraints. 78 Association for nvestment Management and Research

7 Question and Answer Session Brian D. Singer, CFA Risk Analysis Question: Can you represent information ratios geometrically? Singer: Yes, information ratios could be represented by drawing what are known as iso-returnlines in Figure 3. These lines might start at zero return (cash return if the analysis is in risk-premium terms) at the dollar vertex of the triangle and go out in parallel fashion at return levels of 5 percent, 10 percent, 15 percent, 20 percent, and so on. Having now superimposed these iso-return lines over the risk triangle, we can begin to do return and risk analysis simultaneously. Question: How do you incorporate fundamental factors with this approach? Singer: We use geometric risk analysis, looking for examples of the risk relationships between a portfolio basket of securities and an industry or other factor basket. n the risk triangle, the benchmark line could be thought of as the industry or factor basket, with the length of the line indicating the volatility of thatindustry or factor. The angle at the dollar vertex represents the portfolio's correlation with respect to the industry or factor and the loading on the industryor factor measuredjustas the benchmark bet (systematic risk) would be measured. This would be a univariate loading on one industry or factor, but we can also do multivariate loadings with multiple industries and factors. n fact, that is how we build our forward-looking covariance matrix-by considering country, currency, equity market, and bond market factors. We do not and could not build thousands of pairwise correlations in any consistent way. Rather, what we do is build aggregate factors, which might be regions, industries, and so on, and think about what the loading of each marketis on thosevariousfactors. Association for nvestment Management and Research 79