Conference on Finance, Chicago: Collinearity Visualized by Renato Staub, Ph.D.

May 17-18, 2019 R Conference on Finance, Chicago Collinearity Visualized Renato Staub, Ph.D. Senior Risk Capital Analyst FOR INSTITUTIONAL USE ONLY

240418 Compliance Disclosure This material is provided by William Blair s Dynamic Allocation Strategies team for informational purposes only and is not intended as investment advice. Any discussion of particular topics is not meant to be comprehensive and may be subject to change. Any investment or strategy mentioned herein may not be suitable for every investor. Factual information has been taken from sources we believe to be reliable, but its accuracy, completeness or interpretation cannot be guaranteed. Information and opinions expressed are those of the author(s) and may not reflect the opinions of other investment teams within William Blair & Company, L.L.C. s Investment Management division. Information is current as of the date appearing in this material only and subject to change without notice. There is no guarantee that expected return or risk expectations indicated will equal actual return or risk for any capital market or investment strategy. Return and risk information contained herein is intended to illustrate potential expectations for various capital markets and should not be considered any representation of past or expected future return or risk for any William Blair investment strategy or product. Expected returns are provided are for informational purposes only and not intended to be reflective of results a person should expect to achieve. Expected returns have been calculated based on historical index returns and are not based on actual investment portfolios. Actual results will vary and may be higher or lower than the values indicated. Differences between expected and actual results may be exaggerated in volatile market environments. Alternative investments, including options, futures and hedge funds, are speculative and typically involve a high degree of risk. These investments are intended for experienced and sophisticated investors who are willing to bear the loss of their entire investment and may not be suitable for all investors. Performance of these products may be volatile, and while they may provide the potential for positive returns in both rising and declining markets, the potential for loss is equal. Some alternative investments can be highly illiquid, may not be required to provide periodic pricing or valuation to investors, and may involve complex tax structures and delays in distribution of important tax information. Certain alternatives are not subject to the same regulatory requirements, charge higher fees and may have limited opportunity for early redemption or transference of interests. Alternative investment strategies are not intended as a complete investment program. Each investor should consult their own advisors regarding the legal, tax, and financial suitability of alternative investments. William Blair s Dynamic Allocation Strategies employ sophisticated investment strategies that may not be suitable for all investors, and an investor could lose all, or a substantial amount of their investment. These strategies: Are speculative and involve a substantial degree of risk; May use leverage to achieve potentially higher returns through proportionally higher ex-ante risk exposures through, but not limited to, the direct use of swaps, options, foreign exchange contracts, exchange traded funds, futures contracts, and/or by borrowing money to purchase investments; Are subject to other investment risks including those associated with high yield securities, emerging markets, non-U.S. securities, currency markets and fixed income securities; Expect to incur, but not target, equity-like risk, over periods of five years or longer but may experience risk and returns significantly different than expectations; and May produce highly volatile investment returns. 2

Inducement Example R Commands R Commands High interest rates suggest a future reversion, that is, a decrease in interest rates Hence if stocks are interest rate sensitive a rising stock market Let us calculate the historical correlation matrix1beween the yields of various fixed income assets subsequent 100-day returns of the MSCI U.S. TR index (Subret) > load( data.Rdata ) > objects() [1] "data7" > dim(data7) [1] 4909 7 > R1=cor(data) > dim(R1) [1] 7 7 > round(R1,3) MSCI U.S. US Bond US Tbill US TIPS US BAA (Subs.) US HY US IG MSCI U.S. (Subs.) 1.000 0.347 0.330 0.325 0.253 0.208 0.367 US Bond 0.347 1.000 0.948 0.769 0.817 0.477 0.830 US TIPS 0.330 0.948 1.000 0.871 0.757 0.669 0.903 US BAA 0.325 0.769 0.871 1.000 0.599 0.907 0.986 US Tbill 0.253 0.817 0.757 0.599 1.000 0.282 0.690 US HY 0.208 0.477 0.669 0.907 0.282 1.000 0.845 US IG 0.367 0.830 0.903 0.986 0.690 0.845 1.000 The estimate reveals that all yields are (very) highly mutually correlated moderately correlated with Subret Apparently, historical correlations do support our thesis 1Source: Bloomberg, daily data between 2000.1 and 2019.1 3

R Commands R Commands Inducement Multiple Regression #1 > Rx=R1[2:7,2:7] > dim(Rx) [1] 6 6 > Rxy=R1[2:7,1] > dim(Rxy)=c(6,1) > Beta1=solve(Rx)%*%Rxy > dim(Beta1) [1] 6 1 > round(Beta1,2) > [,1] V1 -0.35 V2 0.24 V3 -1.44 V4 -0.57 V5 -0.78 V6 2.91 > In order to predict Subret we might rely on a regression And to use all information, we might regress vs. all yields Below you see the s of Subret vs. the various yields, based on single regressions a multiple regression Multiple Single Beta Beta 0.35 -0.35 0.33 0.24 0.32 -1.44 0.25 -0.57 0.21 -0.78 0.37 2.91 Obviously, the s from the single regression look more consistent are easy to interpret By contrast, the multiple s (MB) tend to confuse Without a doubt, intuition is lost along this path 4

Inducement Multiple Regression #2 R Commands R Commands Next we manipulate the matrix Apparently, the manipulation is of negligible extent The biggest positive and negative changes equal 0.003 and -0.005 Rounded to two comma positions, the matrix is unchanged Below, we see the new matrix and the resulting Beta estimates > R2=edit(R1) > dim(R2) [1] 7 7 > Rx=R2[2:7,2:7] > dim(Rx) [1] 6 6 > Rxy=R2[2:7,1] > dim(Rxy)=c(6,1) > Beta2=solve(Rx)%*%Rxy > dim(Beta2) [1] 6 1 > round(Beta2,2) > [,1] V2 1.23 V3 -0.83 V4 -10.72 V5 -0.49 V6 2.72 V7 8.74 > MSCI U.S. Multiple US Bond US Tbill US TIPS US BAA Single (Subs.) US HY US IG Beta Beta MSCI U.S. (Subs.) 1.000 0.350 0.330 0.320 0.250 0.210 0.370 US Bond 0.350 1.000 0.950 0.770 0.820 0.480 0.830 0.35 1.23 US TIPS 0.330 0.950 1.000 0.870 0.760 0.670 0.900 0.33 -0.83 US BAA 0.320 0.770 0.870 1.000 0.600 0.909 0.988 0.32 -10.72 US Tbill 0.250 0.820 0.760 0.600 1.000 0.280 0.690 0.25 -0.49 US HY 0.210 0.480 0.670 0.909 0.280 1.000 0.840 0.21 2.72 US IG 0.370 0.830 0.900 0.988 0.690 0.840 1.000 0.37 8.74 That is, the resulting MB have changed massively Apparenttly, there are stability issues in the given context 5

R Commands R Commands Inducement Observations Below we compare the MB due to both matrices > Diff=(Beta2-Beta1)/Beta1 > round(Diff,2) [,1] V2 -4.52 V3 -4.40 V4 6.42 V5 -0.14 V6 -4.49 V7 2.00 > Manipulated Beta from Beta from Relative Original Change Matrix Matrix -0.35 1.23 -452% 0.24 -0.83 -440% -1.44 -10.72 642% -0.57 -0.49 -14% -0.78 2.72 -449% 2.91 8.74 200% Wide MB swings look familiar to many of us We may have encountered them in college already typically when regressing without a concept i.e. regression gone wild Many people know when such patterns have to be expected However, fewer of them understand why they happen Ultimately, they result from enforced data fitting This is why many multiple regressions are of little value Fortunately, geometry can help to better illuminate 6

232814 Visualization of Risk Key Elements From geometry classes we know the Cosine rule2; it defines the relationships between the three sides and the three angles of a triangle From statistics we know the Covariance formula3; it defines the risk of any linear combination of two random variables Structural identity of 1 and 2 allows risk to be shown geometrically4 The risk of a random variable is marked by a line of a length proportional to it The correlation between two random variables is represented by an angle whose cosine equals their correlation In the example below the random variable A has a risk about twice as big as has B correlation between A and B equals about 1/ 2 because angle equals approximately 45 A 2 c2=a2+b2-2abcos 3 2a+b= 2a+ 2b+2 a bra,b 4For the proof, see [2] Staub. B 7

232814 Visualization of Risk Example Due to structural indentity with geometry, risks are added vectorial Let us make A and B vectors, that is, we give them directions Then we add or subtract them like vectors A B -B Visualization tells us that C=A-B has a smaller risk than A and a bigger risk than B is uncorrelated with B and meaningfully correlated with A ( =90 , 45 ) Further, visualization tells us that C =A+B has a bigger risk than both A and B is meaningfully correlated with both A and B 8

232814 Single Regression Graphical Interpretation (#1) R Commands R Commands X and Y are two moderately correlated, standardized variables > sigma=cos(15*pi/180) > round(sigma,2) [1] 0.97 > Regression 90 Y Y Due to less than perfect correlation X cannot explain Y entirely Hence, the regression Y= X + entails an error term represents Y s portion that is unexplained by X and X are uncorrelated by definition Geometrically speaking, this means a 90 angle between and X In case of =75 and =15 , is 97% the size of Y 9

232814 Multiple Regression Graphical Interpretation (#2) Assume three diversely correlated standardized variables, X1, X2, and Y Regression Y Y The regression algorithm replicates Y by scaled versions of X1and X2 There is a sin b sin = 0 (green bracket) a cos b cos = 1= Y (orange bracket) And a = sin /(cos sin cos sin ) b = sin /(cos sin cos sin ) 10

Multiple Regression Graphical Interpret. (#2, Cont.) R Commands R Commands Inevitably, highly correlated X1and X2result in large scalings For instance, =45 , =60 , and =15 result in a = +3.35 b = 2.73 In case of =50 , =55 , and =5 we find a = 9.40 >> +3.35 b = 8.79 << 2.73 That is, the constellation in case is subject to considerable instability X1, X2, and Y being laid into a plane allows perfect explanation of Y Quite naturally, this is a fairly unusual constellation The more general case deals with Y s projection onto the X1/X2plane We will explain this in a few instances > ca=cos(45*pi/180) > sa=sin(45*pi/180) > cb=cos(60*pi/180) > sb=sin(60*pi/180) > a=sb/(sa*sb-cb*sa) > round(a,2) [1] 3.35 > b=-sa/(sa*sb-cb*sa) > round(b,2) [1] -2.73 > 11

Multiple Regression Contribution to Fit R Commands R Commands A classical regression report entails the goodness of fit But I have neever seen a report breaking down the fit In other words, what are the individual contributions (CTR) to the fit? CTR corresponds to the projections onto Y s direction > CTRX1=a*ca > round(CTRX1,2) [1] 2.37 > CTRX2=b*cb > round(CTRX2,2) [1] -1.37 > Y CTR(X1) CTR(X2) X1and X2contribute 237% and 137%; their aggregate equals 100% Positive AND negative CTRs suggests a badly conditioned regression 12

Going 3D Constellation In the previous example, all three variables were in the same plane In other words, there is a linear dependency between them It means that a variable can be replicated entirely by the other two Let us go now from 2D to 3D w 1 v 1 u While X1and X2are laid into the bright plane, Y is laid into the darker plane The two planes are orthogonal This is the easiest way of setting a constellation without loss in generality 13

Going 3D Solution5 X1and X2can only replicate vectors in the bright plane any linear combination of X1and X2remains in the bright plane But this time, Y sticks out of the plane hence, it cannot be replicated perfectly by X1and X2 only Y s projection onto the X1/X2 plane can be replicated w 1 v 1 u Going for the projection is cogent; it means that the residual portion ( ) is uncorrelated and hence truly residual the shortest possible Setting the two planes non-orthogonal does not alter the idea of projection renders the chart unfortunately more complicate 5For the detailed charts and formulas, see [3] Staub. 14

Going 3D Solution(Cont.) R Commands R Commands In case of = 60 , = 75 , and =45 , the solution will be X1= 1.65, X2= 1.25, Res= 0.71 CTRX1= 82%, CTRX2= 32%, CTRRes = 50% 6 The CTRs are the projections of the colored lines onto the solid red line > round(R3,2) V1 V2 V3 V4 1 1.00 0.50 0.26 0.71 2 0.50 1.00 0.92 0.00 3 0.26 0.92 1.00 0.00 4 0.71 0.00 0.00 1.00 > Rx=R3[2:4,2:4] > Rxy=R3[2:4,1] > Beta3=solve(Rx)%*%Rxy > round(Beta3,2) [,1] V2 1.65 V3 -1.25 V4 0.71 > CTR=Beta3*(Rx%*%Beta3) > round(CTR,2) [,1] 2 0.82 3 -0.32 4 0.50 > w v u However, note that visualization is mainly a didactical tool it helps framing intuition and triggers the click experience we often discuss portfolio constellations through visualization here, we are supposed to show how it comes to large Beta estimates however, >3 dimensions cannot be managed by visualization but this is not our intention anyway In case of more difficult constellations and higher dimensions the solution can be inferred on the basis of formulas this has been done in our intruductory example on pages 4-6 For equations and more numerical detail see [3] Staub 6For the base version of the contribution to risk formulas, see [1] Grinold and Kahn. 15

232814 Summary and Conclusions Strongly correlated independent variables make multiple regressions problematic The point is that the resulting estimates are far from stable and not intuitive In the end, the estimates are due to enforced data fitting, that is, optimization Fortunately, risk and correlation can be represented graphically This allows us to portray low-dimensional constellations through visualization Without a doubt, intuition is supported much better by visualization The reason is easy there is no need for (tedious) formulas By means of visulation we can easily explain how it comes to large and unstable Betas Nonetheless, visualization is mainly a didactical tool But practical work based on larger data structures should be performed by formulas The according formulas do exist and they are at our disposal In contrast to visualization, formulas are not constrained to 3D cases 16

232814 Thank you veRy much for your attention! Euclid, ~300 BC 17

232814 References [1] Grinold, Richard C. and Ronald N. Kahn (1999) Active Portfolio Management. MCGRAW-HILL Professional, Second Edition. [2] Staub, Renato (2006) Multilayer modeling of a covariance matrix. Journal of Portfolio Management 32(3): 33 44. [3] Staub, Renato (2018) Collinearity Visualized. Working Paper, William Blair & Company, LLC, Chicago. 18