R Conference on Finance, Chicago Collinearity

slide1. R Conference on Finance, ChicagoCollinearity Visualized May 17-18, 2019 Renato Staub, Ph.D.Senior Risk Capital Analyst FOR INSTITUTIONAL USE ONLY<br>
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May produce highly volatile investment returns. 240418<br>
slide3. Inducement – Example 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 matrix1 beween the
yields of various fixed income assets
subsequent 100-day returns of the MSCI U.S. TR index (Subret)

The estimate reveals that all yields are
(very) highly mutually correlated
moderately correlated with Subret
Apparently, historical correlations do support our thesis R Commands

> load(«data.Rdata»)
> objects()
[1] "data7"
> dim(data7)
[1] 4909 7
> R1=cor(data)
> dim(R1)
[1] 7 7
> round(R1,3) 1 Source: Bloomberg, daily data between 2000.1 and 2019.1<br>
slide4. Inducement – Multiple Regression #1 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 bs of Subret vs. the various yields, based on
single regressions
a multiple regression

Obviously, the bs from the single regression
look more consistent
are easy to interpret
By contrast, the multiple bs (MB) tend to confuse
Without a doubt, intuition is lost along this path R Commands

> 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
><br>
slide5. Inducement – Multiple Regression #2 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

That is, the resulting MB have changed massively
Apparenttly, there are stability issues in the given context R Commands

> 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
><br>
slide6. Inducement – Observations Below we compare the MB due to both matrices

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 R Commands

> 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
><br>
slide7. 232814 Visualization of Risk – Key Elements A B g 2 c2=a2+b2-2abcosg
3 s2a+b=s2a+s2b+2sasbra,b
4 For the proof, see [2] Staub.<br>
slide8. 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

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 (a=90°, b≈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 A B g C=A-B a b -B C’=A+B<br>
slide9. 232814 Single Regression – Graphical Interpretation (#1) R Commands

> sigma=cos(15*pi/180)
> round(sigma,2)
[1] 0.97
> Y bX 90° Y X s Regression a b<br>
slide10. 232814 Multiple Regression – Graphical Interpretation (#2) Assume three diversely correlated standardized variables, X1, X2, and Y

The regression algorithm replicates Y by scaled versions of X1 and X2
There is
a·sina – b·sinb = 0 (green bracket)
a·cosa – b·cosb = 1= Y (orange bracket)
And
a = sinb/(cosa·sinb - cosb·sina)
b = -sina/(cosa·sinb - cosb·sina) Y X2 Regression X1 Y aX1 bX2 a b a b g<br>
slide11. Multiple Regression – Graphical Interpret. (#2, Cont.) Inevitably, highly correlated X1 and X2 result in large scalings
For instance, a=45°, b=60°, and g=15° result in
a = +3.35
b = – 2.73
In case of a=50°, b=55°, and g=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/X2 plane
We will explain this in a few instances R Commands

> 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
><br>
slide12. Multiple Regression – Contribution to Fit 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

X1 and X2 contribute 237% and –137%; their aggregate equals 100%
Positive AND negative CTRs suggests a badly conditioned regression Y CTR(X1) CTR(X2) R Commands

> CTRX1=a*ca
> round(CTRX1,2)
[1] 2.37
> CTRX2=b*cb
> round(CTRX2,2)
[1] -1.37
><br>
slide13. 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

While X1 and X2 are 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<br>
slide14. Going 3D – Solution5 X1 and X2 can only replicate vectors in the bright plane
any linear combination of X1 and X2 remains in the bright plane
But this time, Y «sticks» out of the plane
hence, it cannot be replicated perfectly by X1 and X2
only Y’s projection onto the X1/X2 plane can be replicated

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 5 For the detailed charts and formulas, see [3] Staub.<br>
slide15. Going 3D – Solution (Cont.) In case of a = 60°, b = 75°, and g =45°, the solution will be
bX1 = 1.65, bX2 = – 1.25, bRes = 0.71
CTRX1 = 82%, CTRX2 = –32%, CTRRes = 50% 6
The CTRs are the projections of the colored lines onto the solid red line

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 R Commands

> 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
> 6 For the base version of the contribution to risk formulas, see [1] Grinold and Kahn.<br>
slide16. 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<br>
slide17. 232814 Thank you veRy much for your attention! Euclid, ~300 BC<br>
slide18. 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.<br>