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:

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