By: Alpha Male 

Vectors have been identified as an intuitive, graphical way of representing the combination of alpha and beta. (see article by William Ferrell and John Podewils discussed elsewhere in this blog).  This posting goes under the hood on vector analysis to provide a more in depth picture of how vectors provide an excellent intuitive model of market correlation and beta. 

Vector analysis is particularly useful in answering the following questions: 

1. What is the resulting volatility and correlation when an active fund is added to a passive portfolio?
2. How can an active portfolio be decomposed into a passive ETF and a market neutral hedge fund?
3. What is the volatility of this embedded market neutral hedge fund?
4. What are the correlations between embedded market neutral hedge funds?
5. How much passive management (e.g. market ETF) would need to be added to a given hedge fund into order to replicate a given actively managed mutual fund.

Vectors – An Intuitive Way to Visualize Correlation

Before addressing these questions, let’s review how vectors can be used to analyze active management. 

We use straight lines (vectors) to represent different funds.  Each line has 2 properties: its length and its direction.  The following observations can be made: 

1. The length of each vector will represent the volatility of the fund (e.g. monthly standard deviation).
2. The angle between two vectors will represent their correlation.  Of course, angles are relative, so we need a benchmark to measure against.  So,
3. The market vector is perfectly vertical and interpret market correlations and angles away from this (vertical) market vector.  And finally,
4. The addition of two vectors represents the result (standard deviation and correlation) of combining the two funds.

Special Cases – Perfect Positive and Negative Correlation & Zero Correlation

Let’s take the example of benchmark A and active fund B.  As we know, C is the sum of A and B.  First, let’s assume A and B are perfectly correlated.  That is to say, fund B is tracking the index.

As you can see from Figure 1, the length (standard deviation) of C is simply the sum of the lengths (standard deviations) contributed by the benchmark A and fund B.

If fund B had a 0.0 market correlation, we would tilt vector B by 90 degrees as in Figure 2.  Off the top, we can see two things about Figure 2:

1. C is shorter, and,
2. C is at an angle to the market (A)

If fund B had a market correlation of -1.0, then vectors A and B would be pointing in opposite directions as in Figure 3.  Fund B’s monthly changes would perfectly off-set the market’s monthly changes by about half (according to Figure 3).  The result would be a portfolio that:
 
1) has a much lower standard deviation than the market (about half the volatility according to Figure 3), and,
2) is perfectly correlated with the market.
  
There are two useful conclusions one can drawn from these diagrams:

1) Ceterus paribus, a low or negatively correlated fund is a better addition to a passive portfolio than a positively correlated one, and,
2) The greatest benefit from diversification comes along the arc created by B at the point of tangency with vector C. 

If this all seems just a little too neat, then let’s do a sanity check using the formula for adding standard deviations:

 

                                             

Recall that when Fund B was perfectly correlated with the benchmark, the standard deviation of the combined portfolio (“C”) was simply the sum of the standard deviations contributed by the benchmark (A) and another hypothetical fund (B).  In other words:

                                                     

When Fund B had a 0.0 market correlation, it was represented by a vector at 90 degrees to the benchmark.  Since vectors A and B are at right angles to each other, we can use the Pythagorean Theorem to calculate the length of vector C.  If we used the standard formula for adding standard deviations in this zero-correlation scenario, we would get the same result:   

                                                   

Finally, when fund B had a -1.0 correlation to the benchmark, we saw from our vector analysis that the length of C was simply A minus B.  When we use a -1.0 correlation in the standard formula for adding standard deviations we get:

                                                    

Non-Integer Correlations

But what about correlations between -1.0 and 0.0 and between 0.0 and 1.0? 

William Ferrell’s article graphically depicts a 0.5 correlation with a line at approximately a 45% angle from the horizontal plane (as in Figure 4).  At first blush, this makes intuitive sense.  When fund B had a 1.0 correlation to the benchmark, its vector was at 90 degrees from the vertical plane (i.e. parallel with Fund A) and when fund B had a 0.0 correlation to Fund A, its vector was at 0 degree angle to the vertical plane.  So why shouldn’t a correlation of 0.5 be 45%, precisely half-way between 90% and 0%?

Ferrell computes non-integer correlations using the traditional formula, but stops short of expanding on vector analysis to graphically represent such cases.  It turns out the angle between fund B and the horizontal plane is the arcsine of the correlation (quick, grab the grade 10 trigonometry textbook!).  In other words, if the angle of vector B from the vertical plane were theta (see Figure 5), then:

                                                     

This makes intuitive sense when you consider that sine is defined as:

                                                 
 
and market correlation is defined as the proportion a fund’s volatility derived from the volatility of the market (ie. D as a proportion of B).

Given only the correlation between fund B and benchmark A and the standard deviation of fund B, we are able to quickly work backwards to calculate D.  But we are also able to calculate E since:

                                                    

The Meaning Behind the Horizontal and Vertical Axes

So what?  What are E and D anyway?  D is the amount of fund B’s volatility derived from the market (ie. beta) and E is fund B’s remaining volatility derived from some inexplicable source that we might call manager skill (ie. alpha).

I use the terms beta and alpha generally here to identify each form of volatility with its source.  But to know the actual beta of fund B, we need to know the market’s volatility (M).  If we simply divide D by M we get a normalized number for the amount of market-derived volatility in fund B – that is, fund B’s beta.

Another way of showing that Beta equals D divided by M is to define beta in terms of correlation and volatility.  The following re-arrangement of the mathematical definition for beta is a useful way to calculate beta and understand the relationship between beta and correlation.  It also helps us see that Beta is simply Vector D divided by the market vector (M):

                                      

In other words, beta is just the market correlation grossed up (or down) by the fund’s normalized volatility (i.e. its volatility vs. the market’s volatility).

When we plug our vectors (A, B, C, D, and M for the overall market) into this equation, we get: 

                                                     

Multiple Ways to Represent the Same Active Fund

We have proven that the vector approach to adding correlations perfectly aligns with the traditional algebraic approach.  But the vector approach can also help us intuitively understand another important axiom:  that any fund with a non-zero market correlation can be represented as the combination of two funds – one with a zero correlation and one with a perfect correlation.  In other words, every mutual fund is a combination of an ETF (the perfectly correlated portion) and a zero-correlation fund (often called a market neutral or hedge fund). 

In other words, fund C is the sum of funds A and B.  So fund C can also be broken down into its constituent parts – its genes if you will.  In fact, fund C could be represented by an infinite number of funds A and B – each with different lengths and different correlations. 

Fund C could be represented as an ETF (A) and a long-biased low-market correlation fund (B) as in Figure 7. 

Or it might be represented as an ETF and a pure alpha hedge fund as in Figure 8.

Or it could be represented by a larger ETF allocation and a negatively correlated fund as in Figure 9

It could even be represented by an ETF sold short and a long investment in a highly volatile fun with a high market correlation as in Figure 10.

   

 

 

 

Deriving the Volatility of the Embedded Market Neutral Hedge Fund

With vectors, an active mutual fund can be represented as a combination of a market ETF and another active fund with its own unique market correlation and volatility.  One special case reviewed above sub-divides a fund into a passive (“beta”) portion and a new fund with a market correlation of zero (i.e. where vector B is perfectly horizontal).  This perfectly horizontal zero-correlation fund can be thought of as an “embedded market neutral hedge fund“.  We might think of the original active mutual fund as the host mutual fund. 
 
We know that the length of fund B represents its volatility.  So we can use vector analysis to quickly calculate the volatility of the embedded market neutral fund.

                                                                        

Deriving the required volatility of a hedge fund given a specific amount of ETF

But what is we want to calculate the required volatility of fund B in cases where fund B has a non-zero market correlation. 

If we know the volatility of a fund (C) and it’s market correlation (sin(delta), then we can calculate the volatility of B for any amount of A.

                               
We can also use this equation to work backwards to calculate the amount of A (market beta) required given various B’s (volatilities of active portion) if we wanted to replicate C (the host fund).

Since…                                                           

…we can define the relationship between the host fund’s market correlation and the embedded active fund’s correlation:

                                        

Deriving Correlations Between Embedded Hedge Funds

As above, we can now calculate the volatility of an embedded hedge fund.  But we can also calculate the cross-correlations of different embedded hedge funds if all we have is a correlation matrix for the mutual funds themselves?  Mathematicians call these partial correlations.  Partial correlations are the correlations between two sets of data removing the effect that a third variable has on both sets of data.  For our purposes, the partial correlation is effectively the correlation between two sets of returns as if the market had been removed from the equation.  In other words, the correlation between two funds as if 100% of the beta had been shorted out of each of the funds.  

Let’s say we were analyzing mutual funds j and k.  We have learned that these funds have a high degree of correlation, but we suspect that this is simply due to the fact that both funds have a high market correlation.  Do determine the extent to which j and k are correlated with each other without the effect of the market, we would calculate their partial correlation.  The partial correlation between j and k without regard to the market (m) would be:

                                                                          

Intuitively, we can see that the partial correlation between j and k is the same as the correlation between the embedded hedge funds within funds j and k.

If we construct a correlation matrix of the embedded hedge funds we would see that the correlations are generally much lower than those of their host mutual funds.  This suggests that a combination of these embedded hedge funds would provide extensive benefits from diversification since the cross-correlations from the embedded hedge funds can often be very different from the cross-correlations from the host mutual funds.

Partial correlations can also be explained using vectors.  Let’s assume a fund (fund B) has a zero market correlation. 

Now let’s assume a second fund (fund C) also has a zero market correlation.  What can this tell us about the relationship between the two funds?  Since both have a zero market correlation, are they both perfectly correlated?  Do they have a zero correlation?  The answer is: we cannot tell.  In the same way that a 3 by 3 correlation matrix contains 3 cross-correlations (AB, BC, AC), our vectors must also now have three dimensions:

As we can see, fund B may have a zero (90 degree) correlation with the market, but it could also have a -0.5, a 1.0 or a +0.5 correlation with fund A (or any other non-zero correlation).  Thus, by neutralizing the market volatility, we isolate the cross- correlation between the two funds.  It would be as if we were looking directly down at the chart above. 
 

Higher Moments of Embedded Hedge Funds

We can also remove the skewedness and kurtosis of the market out of the active mutual fund.  There are formulae similar to the one we use for adding standard deviations, but they are virtually impossible for mortals to comprehend.  Like the formula for adding standard deviations (ie. adding co-variances), the formula for adding skews relies on a measure of co-skewedness.  Similarly, the formula for adding kurtoses relies on a measure of co-kurtosis.  Like co-variances, co-skews and co-kurtoses need their own matrices.   

But regardless of our mathematical acumen, we cannot derive the returns from the embedded hedge fund (either their contribution to mutual fund returns or their returns as stand alone funds) directly from the data we have on their host mutual fund.  Volatility measures ignore the timing of deviation from the mean, they just add up the deviations.  Instead, we must re-construct the returns from each embedded hedge fund to identify the embedded hedge fund that would have provided.

Once we have calculated the actual returns of the embedded hedge fund, we could calculate the third and fourth moments directly from this information – thus avoiding formulae involving co-skewedness and co-kurtosis.  While we’re at it, we could also confirm that the partial correlations and volatilities calculated directly for the embedded hedge funds are, in fact, correct.
   
Conclusions

Vectors provide an intuitive way of understanding correlations.  They also show us that there are an infinite number of ways to represent any active mutual fund using combinations of a market ETF and another active fund of differing correlation.  One special case would be a combination of a market ETF and an embedded market neutral hedge fund.

If we know the market correlation and volatility of an active mutual fund, we can use vector analysis to calculate the volatility of the embedded market neutral hedge fund.  We can even select any amount of ETF and calculate the volatility and market correlation of the requisite active fund that would replicate our original active mutual fund.

Further, we can calculate the correlation between two embedded market neutral funds given only the correlation between their host active mutual funds.

Vector analysis helps us see beyond superficial returns and discover what lies beneath the surface of mutual funds.  This is a critical first step toward analyzing and comparing the true costs of active management.    

- Alpha Male

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