Differential return is a risk-adjusted performance measure that many CIPM candidates have not see before, and as a result, I get a lot of questions asking what exactly is the differential return… what does it mean and how should it be interpreted. Thus, I have decided to focus on this measure in a series of posts over the next few days.

First, it is important to recognize that differential return is a rate of return. As you might guess from its name, it is an excess return (i.e., a difference in returns). More specifically, it is a risk-adjusted excess return.

In this first post on this measure, I compare differential return to subtraction alpha. Subtraction alpha is simply the portfolio return minus the benchmark return:

Subtraction alpha is simply the portfolio return minus the benchmark return (hence the name); thus, it does not consider risk. As a result, the excess return calculated with a subtraction alpha gives the portfolio manager credit (or discredit) for the portion of returns that result from risk.

Differential return, by contrast, results in an excess return for the portfolio manager that considers risk in the form of standard deviation (the variability of past returns). Here is the formula for differential return using standard deviation:

The benchmark’s Sharpe ratio (which is the benchmark return in excess of the risk free rate, divided by the benchmark’s risk in the form of standard deviation) is multiplied by the portfolio risk to give a measure that is a rate of return (rather than the return per unit of risk, which is the basis for Sharpe ratio). The fully risk adjusted benchmark return is obtained by adding the risk free rate back in. Thus, the manager’s excess return adjusted for risk is the portfolio return minus this risk-adjusted benchmark return.

The CIPM curriculum refers to this differential return as a one-parameter performance measure (the one parameter being the Sharpe ratio, which combines return in its numerator with risk in its denominator to form a single statistic).

Differential return using standard deviation may be appropriate if the relevant risk measure is the variability of past returns. If the relevant risk measure is the volatility of portfolio returns compared to the market, then differential return using beta is more appropriate:

The same concepts apply… it’s just that now the Treynor ratio is used in calculation of the differential return rather than the Sharpe ratio.

Thus, differential return is a more appropriate excess return than a simple subtraction alpha.

Happy studying!

Can you please tell us what level and which reading this equation is described?

Than you

Hi Dan,

Differential return is covered at the CIPM Principles Level, in Reading 5. Note that one of the Learning Outcome Statements is as follows: “compare one parameter performance measures based on beta and standard deviation.” Differential return is one of these measures. On page 549, the discussion of Differential return using standard deviation begins. That section is followed by the discussion of differential return using beta.

John

John,

Is there an error in the pictures? You multiple the benchmark Sharpe ratio or Treynor ratio with portfolio risk and not benchmark risk as the box now reads? The text says portfolio risk, and the annotation suggests that portfolio risk is intended.

This must only be in the most recent CIPM curriculum, and it wasn’t in the previous one? I passed the CIPM examinations in 2012, and I don’t recall these formulas. Certainly an interesting measure which makes very much sense.

Regards,

Niklas Salminen

I think the box labeled “Benchmark Risk” should say “Portfolio risk”???

Bill Welch and Niklas Salminen: Thank you; the formula itself was correct, but the labling of the risk figures was incorrect. Indeed, as described in the text, the benchmark risk adjusted performance measure (Sharpe ratio or Treynor ratio) must be multiplied by the *portfolio* risk (standard deviation or beta, respectively.

The post has been corrected for this – good catch.

Niklas,

Differential return was added to the CIPM curriculum in 2013, if memory serves.

Your differential return calculation would be made more intuitive if you present it as the market risk premium adjusted by the ratio of the portfolio’s standard deviation relative to the benchmark standard deviation. This makes it easy to see that this is simply an application of the CAPM ( Capital Market Pricing Model.) In this context, alpha is the excess return over the required return (given the volatility risk of the portfolio.) Too often candidates are simply memorizing formulas rather than learning concepts.

Steve,

Thank you for your comment. Yes, I agree wholeheartedly, candidates should learn concepts rather than memorizing formulae. My post is not intended to encourage memorization of formulae for the sake of simply memorizing… hopefully the post does not seem that way.

The presentation of the formula for differential return is based on the reading – but during our in-person classes, I do teach it along the lines of what you mention above. This post, however, is meant simply to help candidates relate the formula to other concepts.

As an aside, CAPM itself, while mentioned several times in the CIPM curriculum, is not directly covered. Covering CAPM in greater detail would enhance the curriculum, in my opinion, and it would help candidates better connect these related concepts.