### Blog Contributors

Ryan Nauman
VP, Product and Market Strategist
Stephen Berei
VP, Client Services & Implementation
Jeremy Poulin
Senior Client Consultant

# Alpha Star

Sep 23, 2011
I think this statistic is one that may not be so popular, but could prove to be very useful. The statistic is a measure of risk-adjusted excess return first introduced by Richard C. Marston in 2004.

The premise behind using risk-adjusted return is to ensure that we compare apples to apples versus apples to oranges. When comparing the manager versus benchmark, we do not take into account the risk levels, which in most cases is different. We would expect a higher risk to reward a higher return.

Therefore, the solution is to use risk-adjusted excess return. We replace the given benchmark with a blended version to equal out the risk levels of the manager and benchmark. When doing this we take a blend of the benchmark returns and a fraction of cash in a way that the final blended benchmark risk level matches that of the manager.

What is Alpha Star?
There are three main factors that need to be taken into account when looking at alpha star:
• The first is to understand what risk measurement is being used. Two standard options are available, standard deviation or beta. For alpha star, we use standard deviation.
• Next, we need to know what measure of return we would use for excess return, whether that is arithmetic or geometric. For alpha star, we use arithmetic.
• Finally, we can assume there is a constant risk-free rate used as cash (StyleADVISOR defaults to the Citigroup 3-month T-Bill).
Without getting deep into the math, I’ll explain what is done to calculate the statistic. We take the standard deviation of the manager over the standard deviation of the benchmark. This is known as our factor value. Then we take 1 minus the factor value multiplied by cash plus the factor value multiplied by the benchmark. This gives us our new blended benchmark return series. Finally, we calculate the difference between the arithmetic means of the manager return series and benchmark return series.

Alpha Star compared to similar statistics.
There are a few other similar statistics to alpha star. Each one of these statistics relates to one another in that they all blend the benchmark series with cash to create a new benchmark. The other statistics include Jensen Alpha (currently not available in StyleADVISOR) and Zephyr’s Risk-Adjusted Excess Return. Below is a matrix with two of the three main factors mentioned above to show the differences between each statistic.

Our Mathematician, Thomas Becker, Ph.D. has written a paper on Alpha Star here.