Trade Off

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.

Sortino Ratio

A variation of the Sharpe ratio, the Sortino ratio is a return-versus-risk trade-off metric that uses downside deviation as its measure of risk.

How Is it Useful?

The Sortino ratio addresses a shortcoming of using standard deviation as a measure of risk in a return-versus-risk trade-off ratio. Standard deviation punishes a manager equally for “good” risk and “bad” risk. Downside deviation adjusts for this by only counting the “bad” risk and ignoring “good” observations in a return series.

Math Corner: 

The below calculation for the Sortino ratio is not complicated, as it is simply a variation of the Sharpe ratio. It is up to the user to define what the breakpoint is for minimum acceptable return (MAR) in the calculation of downside risk. Frequently used values for MAR are the risk-free rate or a hard-target value like 0%.

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.

Sharpe Ratio

The most famous return-versus-risk measurement, the Sharpe ratio represents the added value over the risk-free rate per unit of volatility risk.

How Is it Useful?

The Sharpe ratio simplifies the options facing the investor by separating investments into one of two choices: the risk-free rate or anything else. Thus, the Sharpe ratio allows investors to compare very different investments by the same criteria. Anything that isn’t the risk-free investment can be compared against any other investment. The Sharpe ratio allows for apples-to-oranges comparisons.

Math Corner: 

First proposed by William Sharpe in his landmark 1966 paper “Mutual Fund Performance,” the original version of the Sharpe ratio was known as the reward-to-variability ratio. Sharpe revised the formula in 1994 to acknowledge that the risk-free rate used as the reference point is variable, not a constant.

Zephyr K-Ratio

A return-versus-risk statistic, the Zephyr K-ratio measures the rate at which wealth is created and the consistency of the path of wealth creation.
PDF version: 

How Is it Useful?

The Zephyr K-ratio answers two questions many investors care about most: “At what rate did I grow my wealth?” and “Was that growth of wealth consistent?”

What Is a Good Number?

With the Zephyr K-ratio, a high numerator indicates a high rate of wealth creation. A low denominator indicates consistency in that rate of appreciation. Roll those two goals together and you would hope to see a high Zephyr K-ratio.

Math Corner: 

The original variant of what would eventually become the Zephyr K-ratio was proposed by Lars Kestner in 1996 using well-established statistical theories. The summary formula comparing the rate of wealth appreciation against the consistency of wealth appreciation is:

To calculate the Zephyr K-ratio, one should first replace the dates on the horizontal axis of the portfolio’s cumulative return graph with consecutive integers starting at 0. With these integers as independent x-values and the corresponding cumulative return values as dependent y-values, one can now calculate the slope of the regression line, the numerator of the Zephyr K-ratio, by the well-known formula

The standard error of the slope, the denominator of the Zephyr K-ratio, can
be calculated from the x- and y-values by the formula

Omega

Omega compares upside gains against downside risks. Omega represents the count and scale of returns above a breakpoint versus the count and scale of observations below a breakpoint.
PDF version: 

How Is it Useful?

Omega represents one useful way of understanding tail risk, the impact that extreme observations have on an overall set of numbers. If the returns of a manager are close to the minimum acceptable return (MAR) breakpoint, they don’t strongly affect omega. However, if many returns lie above or below the MAR, or if the returns are extreme, those returns will impact the value of omega significantly.

Math Corner: 

Omega was first proposed by Con Keating and William Shadwick in their 2002 paper “A Universal Performance Measure”. Omega is the ratio of two integrals: the area above the minimum acceptable return (MAR) in the numerator and the area below the MAR as the denominator. Omega captures all four moments of the distribution (return, standard deviation, skewness, and kurtosis) in a single measure.

Pain Ratio

A proprietary return-versus-risk trade-off metric, the pain ratio compares the added value over the risk-free rate against the depth, duration, and frequency of losses.
PDF version: 

How Is it Useful?

While one certainly wants to minimize losses, it is also important to make money. The pain ratio quantifies this trade-off into a single number. The pain ratio compares the gains over the risk-free investment against the losses that were suffered to obtain that return.

Math Corner: 

The return element of the pain ratio is the annualized return of the investment in excess of the risk-free investment. Typically a short-term cash investment is used as the risk-free investment. The denominator of the pain ratio is the pain index, an integral measuring the depth, duration, and frequency of losses.

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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