Seeing portfolio risk on I-Maps

Concepts such as a portfolio's beta and the decomposition of a portfolio's risk and tracking error into a benchmark component and an off-benchmark component can be difficult to grasp.  In I-Maps they have a graphical interpretation that helps to make these concepts intuitive.

Warning:  This page is intended for readers who are familiar with portfolio beta and risk decomposition.  It is more technical and mathematical than the others.  The extent of the mathematics involved is the cosine of an angle and the Pythagorean theorem. Feel free to skip to another page.

Beta on I-Maps:  First interpretation

Beta Derivation Diagram 

Consider an I-Map which has Cash at the origin and a benchmark portfolio on the vertical axis. 

The distance of the portfolio (and the benchmark) from the origin is the volatility of the portfolio (and of the benchmark).  The volatility of a portfolio is also known as its absolute risk and also (for reasons that will become clear later) as its total risk.

The distance between the portfolio and the benchmark gives the tracking error between the two.

The correlation between the portfolio and the benchmark is dependent on the angle they make at the origin, indicated by the theta symbol on the diagram. The actual relationship is:

    Correlation(Portfolio, Benchmark) = Cosine(Portfolio, Benchmark)

In case your high school algebra is a little rusty:  Cosine of zero is 1, the maximum value it achieves.  As the angle increases, so cosine decreases, until at 90 degrees, cosine reaches 0.  As the angle increases beyond 90 degrees, cosine becomes negative, reaching its minimum of -1, at 180 degrees.

The beta between a portfolio and its benchmark is conventionally defined as:

    Beta = Correlation(Portfolio, Benchmark) x  [TotalRisk(Portfolio) / TotalRisk(Benchmark)]

In I-Maps terms this becomes

      Beta  = Cosine(Portfolio, Benchmark) x [TotalRisk(Portfolio) / TotalRisk(Benchmark)]

What this means is that a portfolio's beta is the product of two factors:

  • The correlation of the portfolio and the benchmark.
  • The ratio of the portfolio's total risk to that of the benchmark.

The interpretation in I-Maps is:

  • The smaller the angle between the portfolio and benchmark, the bigger the portfolio's beta.
  • The further the portfolio is from the origin, the bigger the portfolio's beta.

If the portfolio and the benchmark are perfectly correlated (i.e. the portfolio lies on the vertical in line with the benchmark), then beta is just the ratio of the total risks of the portfolio and the benchmark. 

One way of getting a portfolio that lies on the same line as the benchmark is to have a portfolio that consists solely of cash and the benchmark itself.  In this scenario, the beta is zero when the portfolio is all cash, while the beta is one when the portfolio is "all benchmark".  (Note that by using gearing or short selling it would be possible to get a beta above 1 or less than 0.)

As soon as a portfolio's stocks or weights differ from the benchmark, the portfolio will move away from the benchmark, and its correlation with the benchmark will decrease from 1.  The less it is correlated with the benchmark, the lower its beta.  For example, if a portfolio is at 45 degrees to the benchmark, its correlation is .7, and its beta will be 70% of the ratio of its total risk to that of the benchmark.  If the portfolio is at 90 degrees to the benchmark, meaning that the portfolio is uncorrelated with the benchmark, its correlation is 0, and so its beta is 0 too.

Another  interpretation of I-Maps is given below once the concept of benchmark and "off benchmark" risk has been introduced.

 

Decomposing a portfolio's risk

Risk Decomposition Diagram 

Consider the projection of the portfolio's total risk onto the benchmark.  This is indicated as the portfolio's Benchmark Related Risk on the diagrams.  It can be thought of as "the part of the portfolio's total risk that is in the same direction as (i.e. is correlated with) the benchmark".

Perpendicular to that, the horizontal distance of the portfolio away from the benchmark is the portfolio's Specific Risk.  It can be thought of as "the part of the portfolio's total risk that is at right angles to (i.e. uncorrelated with) the benchmark".

Because of the right angle triangles it follows from the Pythagorean theorem that

    (Portfolio Total Risk)2 = (Portfolio's Benchmark Related Risk)2 + (Portfolio's Specific Risk)2

which means that the portfolio's total risk can be decomposed into two parts:  a benchmark part and an "off-benchmark" part.

A similar decomposition can be given for tracking error, namely

    (Tracking Error)2 = (Timing Risk)2 + (Portfolio's Specific Risk)2

where

    Timing Risk = (Portfolio's Benchmark Related Risk) - (Benchmark's Total Risk)

 

Beta on I-Maps:  Second interpretation

Once a portfolio's Benchmark Related Risk is defined, some high school geometry will show that:

    Beta = (Portfolio's Benchmark Related Risk) / (Benchmark's Total Risk)

In other words, beta is the ratio  of "the benchmark part of its risk" relative to the total risk of the benchmark:  Take that part of the portfolio's risk that is "like the benchmark" (ignoring that which isn't) and compare it to the benchmark's total risk.

 

The same applies to securities

All of the above interpretations and formulae for a portfolio relative to a benchmark hold if the portfolio is replaced by any security.  That is, the beta of a security and the decomposition of its total risk into a benchmark and a specific part can be interpreted and calculated in exactly the same way as for a portfolio.  An example showing the betas calculated for securities is shown in the Positioner section of the main introduction page.