Benchmark

Excess Return

The simplest of the benchmark-relative statistics, excess return measures the difference between the manager return and the benchmark return.

How Is it Useful?

Excess return is simple to understand and doesn’t require any sophisticated statistical knowledge. One calculates excess return using nothing more complicated than subtraction.

What Is a Good Number?

One would hope to outperform the benchmark, resulting in an excess return greater than zero. Negative excess return indicates the investor would have been better off investing in a low-cost index product. The higher the excess return, the better.

Math Corner: 

Excess return is one of the simplest metrics to calculate. Its simplicity is both an advantage and a disadvantage. It is easy to understand but does not take into consideration any form of risk.

Batting Average

An indicator of consistency, batting average measures the percentage of time an active manager outperformed the benchmark.

How Is it Useful?

Batting average is conceptually easy to understand. It is simply the percentage of periods when the manager outperformed the benchmark. The higher the batting average, the more consistent the outperformance.

Math Corner: 

The calculation for batting average is quite simple. Its relative simplicity is both its strength and weakness. It is easy to understand but limited in what it tells you.

Treynor Ratio

A return-versus-risk tradeoff metric, the Treynor ratio measures the added value per unit of market risk, with beta defined as risk.

How Is it Useful?

The Treynor ratio is similar to the Sharpe ratio. In both cases the measure of return is the excess over the risk-free investment. The two differ in their definitions of risk. The Sharpe ratio uses standard deviation to define volatility risk, whereas the Treynor ratio uses beta as a measure of market or systematic risk.

Math Corner: 

The Treynor ratio actually pre-dates its more famous cousin, the Sharpe ratio. The first version appeared in early 1965 in the Harvard Business Review under the title “How to Rate Management of Investment Funds.” Treynor originally wanted to examine portfolio performance with the market impact neutralized. Eventually the formula below became that standard definition of Treynor ratio.

Up/Down Capture

Up capture measures the percentage of market gains captured by a manager when markets are up. Down capture measures the percentage of market losses endured by a manager when markets are down.

How Is it Useful?

It’s said that the two elements that drive the market are “greed” and “fear.” The up capture and down capture ratios are a useful way of separating the two so they can be analyzed independently. Also, up capture and down capture address the shortcomings of beta, which fails to distinguish between up and down markets.

Math Corner: 

For up capture, the first step identifies all the periods in which the market was up. For those up-market periods the returns for both the manager and the benchmark are geometrically compounded and then annualized. Finally, a ratio between the two is calculated.

The down capture process is the same, but for down market periods.

It is possible for the manager to have a negative return in a period when the market is up. By the same token, it is possible for the manager to have a positive return during a period when the market is down. Obviously, the latter is preferred to the former.

It is also worth noting that up capture and down capture values can differ significantly if the underlying periods used in the calculation are monthly or quarterly. For example, assume that over the course of a quarter the monthly returns were -1.2%, +5.2%, and -0.8%. Compounded, the quarterly return was +3.1%. Using monthly data would result in one up period and two down periods. However, using quarterly data in the calculation of the capture ratios results in one up period and no down periods.

Beta

Beta measures the sensitivity of the manager to movements in an underlying benchmark.
PDF version: 

How Is it Useful?

Beta answers the question, “When markets go up or down, does the manager typically go up or down more than the market or less than the market?” This is what is meant by market sensitivity. Beta is also used to quantify market risk, sometimes known as “systematic risk”.

Math Corner: 

The numerator of beta measures how the manager return moves relative to the benchmark movements. The denominator scales the results of the numerator so that the point of reference of beta is 1.0.

R-squared

R-squared represents the “goodness of fit” of a manager to its benchmark. R-squared is the percentage of variation in a manager’s returns explained by the benchmark’s returns.
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How Is it Useful?

R-squared is used primarily as a cross-check on the appropriateness of the benchmark. Many other statistics such as alpha, beta, information ratio, and up/down capture use a passive benchmark as a reference point. If the R-squared of a manager to the benchmark is too low, the usefulness of all these other benchmark-relative metrics diminishes.

Math Corner: 

R-squared literally takes the correlation of a manager versus a benchmark and squares it. Squaring correlation removes the directional aspect of correlation. It is impossible to have a negative R-squared. This is intentional, as the sole point of R-squared is to determine what percentage, from 0% to 100%, of the variation of a manager’s return is explained by the benchmark.

Tracking Error

Also known as the standard deviation of excess returns, tracking error measures how consistently a manager outperforms or underperforms the benchmark.

How Is it Useful?

Tracking error measures the consistency of excess returns. It is created by taking the difference between the manager return and the benchmark return every month or quarter and then calculating how volatile that difference is. Tracking error is also useful in determining just how “active” a manager’s strategy is. The lower the tracking error, the closer the manager follows the benchmark. The higher the tracking error, the more the manager deviates from the benchmark.

Math Corner: 

Calculating tracking error is a three-step process. First, an excess return series is created by calculating the periodic differences between the manager and the benchmark. Next, the mean of that excess return series is calculated. Finally, the dispersion of individual observations from the mean excess return is calculated.

Information Ratio

A benchmark-relative return-versus-risk metric, the information ratio measures the excess return against the benchmark divided by tracking error, where tracking error is a measure of consistency.

How Is it Useful?

The information ratio answers the two most important questions for an active manager. First, did the manager outperform the passive benchmark? Second, was the manager able to outperform the benchmark consistently? If the answer to either of these is “no” then a low-cost passive product like an index fund or an ETF might make sense. Therefore, the information ratio stands as a great way to justify an active manager’s existence.

Math Corner: 

The numerator of the information ratio is quite easy to calculate. It is simply the difference between the manager return and its benchmark return. The denominator is calculated by taking the standard deviation of the numerator. It is the volatility of that excess return series. The standard deviation of excess return is known as tracking error.

Alpha

Alpha measures the risk-adjusted added value an active manager adds above and beyond the passive benchmark.
PDF version: 

How Is it Useful?

Alpha is often described as a measure of a manager’s skill, or ability to add value over a passive benchmark. It is important to remember that alpha first adjusts for the degree of market risk undertaken by the manager. Alpha is what remains after the market risk, or beta, is netted out.

Math Corner: 

The simpler, standard definition of alpha is to treat a manager’s total returns as a combination of two components: a portion that is a function of market movements and a portion that is unique to the individual manager. Rearranging the terms, alpha can be expressed as:

Another common version of alpha is known as Jensen’s alpha or cash-adjusted alpha. This version first subtracts out a risk-free rate from both the manager returns and the benchmark returns before proceeding with the standard alpha calculation. Jensen’s alpha is more in-sync with the Capital Asset Pricing Model (CAPM). It is written:

 
 

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