# Stats

## PERFORMANCE METRICS

Return

The most important thing when dealing with investment returns is to bear in mind that what is measured are past returns and that it is very unlikely that the past will be ever repeated in the future.

From a calculation point of view, it is important to keep in mind that investment returns are nothing more than growth rates of the capital invested over a certain time period expressed as a percent figure.

The rate of return can be calculated over a single period, or expressed as an average over multiple periods of time.

Average Return (Arithmetic Mean)

A simple average return is calculated by summing the returns for each period and dividing the total by the number of periods. The simple average return does not consider the compounding effect of returns. Because it does not take into account the compounding effect, average returns are the best tool for calculating an expected return over short periods of time.

Where,

N = number of periods and Ri is the return for period i.

The arithmetic return for a given period is the

Logarithmic or continuously compounded return

The logarithmic return or continuously compounded return for a given period is defined as:

Geometric average rate of return

The periodic average return that assumes the same return every period that results in the equivalent compound growth rate. The geometric return is the best tool for calculating an expected return over a long-term investment horizon

Geometric average rate of return, also known as the True Time-Weighted Rate of Return, over ‘n’ periods is defined as:

Compound Annual Growth Rate – CAGR

The CAGR is defined as the year-over-year growth rate of an investment over a specified period of time.

The compound annual growth rate is calculated by taking the nth root of the total percentage growth rate, where ‘N’ is the number of years in the period being considered. This can be written as follows:

Investors can compare the CAGR in order to evaluate how well one investment performed against other investment in a peer group or against a market index. The CAGR can also be used to compare the historical returns of stocks to bonds or a portfolio. But the CAGR does not reflect investment risk, it smooth out the return thus hiding the volatility.

Standard Deviation

Standard deviation is a widely used measurement of variability or diversity used in statistics and probability theory. It shows how much variation or “dispersion” there is from the average (mean, or expected value). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

Where σ stands for standard deviation,  is the observed values of the sample items and  is the mean value of these observations.

The significance of standard deviation is that two-third of the times the annual return of the asset lies between +1 SD and -1 SD from mean, 95% of time lies between +2 SD and -2SD and 99% of time lies between +3 SD and -3 SD in a normal distribution.

Standard Error

The standard error is a method of measurement or estimation of the standard deviation of the sampling data associated with the estimation method. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate.

The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population. Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time.

Where, σ is the standard deviation of the population.

The smaller the standard error, the more representative the sample will be of the overall population. The standard error is also inversely proportional to the sample size; the larger the sample size, the smaller the standard error because the statistic will approach the actual value.

Coefficient of Variation

The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from each other. It is also known as unitized risk or the variation coefficient. The absolute value of the CV is sometimes known as relative standard deviation (RSD), which is expressed as a percentage.

The coefficient of variation (CV) is defined as the ratio of the standard deviation σ to the mean µ.

It shows the extent of variability in relation to mean of the population.

In the investing world, the coefficient of variation allows you to determine how much volatility (risk) you are assuming in comparison to the amount of return you can expect from your investment. That is, lower the ratio of standard deviation to mean, better the risk-return trade-off.

Skewness

Skewness is a term in statistics used to describe asymmetry from the normal distribution in a set of statistical data. When data is skewed to the right, the mean and the median of the set are both greater than the mode. Further, the mean is greater than the median in most cases. Conversely, when data is skewed to the left, the mean and the median are both less than the mode. In addition, as a rule, the mean is less than the median.

Positive skewness = Mode < Median < Arithmethic Mean, Negative Skewness = Mode > Median > Arithmethic Mean, Symmetrical = Arithmetic Mean = Median

The ratio of the third central moment to the cube of the standard deviation is called Pearson’s moment coefficient of skewness (or the coefficient of skewness) and is denoted by  γ or g.

Where,

m3 = ∑(x−x̄)3 / n   and    m2 = ∑(x−x̄)2 / n

is the mean and n is the sample size, as usual. m3 is called the third moment of the distribution. m2 is the second moment of the distribution, that is, variance – the square of the standard deviation.

If skewness is positive, the returns are positively skewed or skewed right, meaning that the right tail of the distribution is longer than the left. If skewness is negative, the returns are negatively skewed or skewed left, meaning that the left tail is longer.

Kurtosis

The other common measure of shape is called the kurtosis. As skewness involves the third moment of the distribution of statistical data, kurtosis involves the fourth moment. The classical interpretation of Kurtosis is that it measures both peakedness and tail heaviness of a distribution.

The reference standard is a normal distribution, which has a kurtosis of 3. In token of this, often the excess kurtosis is presented: excess kurtosis is simply kurtosis − 3.

• A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). Any distribution with kurtosis ≈3 (excess ≈0) is called mesokurtic.
• A distribution with kurtosis less than 3 (excess kurtosis <0) is called platykurtic. Compared to a normal distribution, its tails are shorter and thinner, and often its central peak is lower and broader.
• A distribution with kurtosis more than 3 (excess kurtosis >0) is called leptokurtic. Compared to a normal distribution, its tails are longer and fatter, and often its central peak is higher and sharper.

The coefficient of kurtosis of a distribution is the ratio of the fourth central moment to the fourth power of the standard deviation.

Where,

m4= ∑(x−x̄)4 / n   and   m2 = ∑(x−x̄)2 / n

m4 is called the fourth moment of the data set.  m2 is the variance, the square of the standard deviation. xi is the ith value, and   is the sample mean.

The smallest possible kurtosis is 1 (excess kurtosis −2), and the largest is ∞.  For a return distribution a negative excess kurtosis (platykurtic) is prefers over positive excess return (leptokurtic).

Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.

A positive covariance means that asset returns move together. A negative covariance means returns move inversely. Possessing financial assets that provide returns and have a high covariance with each other will not provide very much diversification.

Correlation

Correlation is the degree to which two or more quantities are linearly associated. For two random variables X and Y, the correlation is defined by

Where, σ denotes standard deviation and Cov(X,Y) is the covariance of these two variables.

Correlation is computed into what is known as the correlation coefficient, which ranges between -1 and +1. Perfect positive correlation (a correlation co-efficient of +1) implies that as one security moves, either up or down, the other security will move in lockstep, in the same direction. Alternatively, perfect negative correlation means that if one security moves in either direction the security that is perfectly negatively correlated will move in the opposite direction. If the correlation is 0, the movements of the securities are said to have no correlation; they are completely random.

Coefficient of Determination (R-Squared)

The coefficient of determination is a key output of regression analysis. It is interpreted as the proportion of the variance in the dependent variable that is predictable from the independent variable.

With a value of 0 to 1, the coefficient of determination is calculated as the square of the correlation coefficient (R) between the dependent and the independent variable.

This measure represents the percentage of an investment movement that can be explained by movement in a benchmark index.

Sharpe Ratio

The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return (or risk premium) per unit of risk in a security or portfolio, named after William Forsyth Sharpe. It is defined as

where R is the security or portfolio return, Rf is the return on a risk-free asset, E[R − Rf] is the expected value of the excess of the investment return over the risk-free return, and σ is the standard deviation of the excess of the investment return.

The Sharpe ratio is used to characterize how well the return of a security or portfolio compensates the investor for the risk taken, the higher the Sharpe ratios number the better.

Treynor Ratio

Like the Sharpe Ratio, the Treynor Ratio (sometimes called Reward-to-Variability-Ratio) is a measurement of the returns earned in excess of that which could have been earned on a security or portfolio that has no diversifiable risk (e.g., Treasury Bills or a completely diversified portfolio), per each unit of market risk assumed.

The Treynor ratio relates excess return over the risk-free rate to the additional risk taken; however, systematic risk is used instead of total risk. The higher the Treynor ratio, the better the performance of the investment under analysis.

R = Portfolio return

Rf = risk free rate

β = Portfolio Beta

M2 Measures

Modigliani Risk-Adjusted Performance or M2 or M2 or Modigliani-Modigliani measure or RAP is a risk-adjusted performance measure. It contains the same information as the Sharpe Ratio, but, being a percentage return, is easier to interpret. It measures the returns of a security or portfolio, adjusted for the risk of the security or portfolio, relative to that of some benchmark (e.g., the market).

Di be the excess return of the security or portfolio (i.e., above the risk-free rate) for some period i.

It also defined a statistic called “RAPA” (Risk Adjusted Performance Alpha). Consistent with the more common terminology of M2, this would be:

Where,   = Standard Deviation of Benchmark.

Beta

The beta coefficient is a key parameter in the capital asset pricing model (CAPM). Beta is that element of return variability from a portfolio which cannot be eliminated through diversification relative to one or several risk factors. It comprises the risk factors common to all assets in the investment universe. That’s why it is also the measure of systematic Risk.

Beta expresses the ‘sensitivity’ of the portfolio relative to the ‘market’. By definition, the “market” has a beta of 1.0. A portfolio that swings more than the market over time has a beta above 1.0. If a portfolio moves less than the market, the portfolios’ beta is less than 1.0. High-beta portfolios are supposed to be riskier but provide a potential for higher returns; low-beta portfolios pose less risk but also lower returns.

Where β is Beta,  is the volatility of the portfolio, ρ is the correlation of returns of portfolio and the market and  is market volatility.

Alpha

Jensen’s alpha (or Jensen’s Performance Index, ex-post alpha) is used to determine the return of a security or portfolio of securities over and above the theoretical expected return. The security could be any asset, such as stocks, bonds, or derivatives. The theoretical return is predicted by a market model, most commonly the Capital Asset Pricing Model (CAPM) model. The market model uses statistical methods to predict the appropriate risk-adjusted return of an asset. The CAPM for instance uses beta as a multiplier.

In the context of CAPM, calculating alpha requires the following inputs:

Jensen’s alpha = Portfolio Return − [Risk Free Rate + Portfolio Beta * (Market Return − Risk Free Rate)]

Tracking Error

Tracking error is a measure of how closely a portfolio follows the index to which it is benchmarked. The most common measure is the root-mean-square of the difference between the portfolio and index returns

Where [Rp – RB] is the difference between the portfolio return and the benchmark return.

‘Var’ means variance and ‘SD’ stands for standard deviation.

Tracking Error of a portfolio constantly underperforming its benchmark is zero, for example, and therefore hardly a desirable result.

Information Ratio

The Information Ratio (also known as Appraisal Ratio) is basically a risk-adjustment of Alpha. It measures the Alpha per unit of active risk, i.e. tracking error. It  is defined as expected active return divided by tracking error, where active return is the difference between the return of the security and the return of a selected benchmark index, and tracking error is the standard deviation of the active return.

Where,

is the portfolio return,    is the benchmark return

α = Active Return

ω= the standard deviation of the active return

Up and Down Capture Ratio

Up & down capture ratio show you whether a given security or portfolio has outperformed – gained more or lost less than – a broad market benchmark during periods of market strength and weakness, and if so, by how much.

The Up Capture Ratio measures the security or portfolio compound return when the benchmark return increased, divided by the benchmark’s compound return when the benchmark return increased. The higher the value, the better.

Ri are the fund returns when the benchmark returns are greater than or equal to zero

Bi are the benchmark returns when the benchmark returns are greater than or equal to zero

The Down Capture Ratio measures the security or portfolio compound return when the benchmark was down divided by the benchmark’s compound return when the benchmark was down. The smaller the value, the better.

Ri are the fund returns when the benchmark returns are less than zero

Bi are the benchmark returns when the benchmark returns are less than zero

A value of 100% for either ratio implies that the security or portfolio fully captures, or matches, the benchmark return during the period evaluated. A value of greater than 100% indicates that the security or portfolio captured more return than the benchmark (this is a positive for up-capture, however, a negative for down-capture). A value less than 100% means the security or portfolio captured less return than its benchmark (this is a negative for up-capture, however, a positive for down-capture).

Semi Variance (below-mean)

A measure of the dispersion of all observations that fall below the mean or target value of a data set. Semi variance is an average of the squared deviations of values that are less than the mean or target value.

Where:
n = the total number of observations below the mean
x = the observed value
= the mean value of the data set

Similarly, Semi volatility is defined as the volatility of returns below the mean return.

Downside Deviation/ Downside volatility/ Downside Risk

Downside volatility takes into account only those values of observed excess rates of return that lie below minimum acceptable return (MAR). In other words, Downside volatility is a generalization of the semi volatility as is defined as volatility below MAR.

Average Downside Deviation indicates the average size of the unacceptable returns. This statistic helps an investor judge the severity of the average ‘bad’ return. An investment that lost money twice as often as a second investment may still be preferable if it tended to lose far less than the second investment.

Downside Magnitude at the 99th Percentile

This is a worst-case scenario. An investment may lose money only occasionally, may average small losses when they do occur, and yet may prove unacceptable if the potential exists for huge losses. Downside Magnitude 99th Percentile answers the question: “When an investment is below the MAR, what is the worst case scenario?”

Lower Partial Moments

This measure of downside risk is colloquially known as “downside risk.” The main idea of the lower partial moment framework is to model moments of asset returns that fall below a minimum acceptable level of return.

LPM simply examines the moment of degree ‘a’ below a certain threshold ‘t’.

Where,

a = order of the lower partial moment

t = target return (Minimum accepted return)

LPM is a family of risk measures specified by ‘t’ and ‘a’. ‘T’ is often set to the risk free rate or simply to zero. By choosing the degree of the moment an investor can specify the measure to suit his risk aversion. Intuitively, large values of ‘a’ will penalize large deviations more than low values. Semi variance is a special case of LPM for which the degree of the moment is set to two.

Downside Potential/ Downside Factor

The possibility of making a loss in an investment

Downside Variance

Downside variance eliminates positive returns when calculating risk. Instead of using the mean return or zero, measures the variability (volatility) from the Minimum Acceptable Return.

Sortino Ratio

The Sortino ratio measures the risk-adjusted return of an investment and is similar to the Sharpe ratio, except it uses downside deviation for the denominator instead of Standard Deviation (SD). Standard deviation involves both the upward as well as the downward volatility. Whereas Downside Deviation only concerned about the volatility below MAR. The numerator is the excess return with respect to a Minimum Accepted Return (MAR) fixed by the investor.

Where,

R = Return of a security or portfolio

MAR = target return (Minimum accepted return)

DD = Downside Deviation

Downside Probability/ Downside Frequency

Likelihood or frequency of failure.

It calculates the probability that the portfolio would not achieved the target return (MAR).

While higher figure is considered bad, lower figure is considered favourable.

Downside frequency tells the user how often the returns violated the MAR (minimum acceptable return). This is important because in order to assess the likelihood of a bad outcome you need to know how often one occurred.

Upside Probability

The chance that an investment return will equal or exceed the target return (MAR).

The opposite of downside probability is Upside Probability. Upside probability is how often the investment exceeded the MAR.

Upside Potential

The average return above the target return (MAR), measuring how often and how far above the target return (MAR) a portfolio’s returns are likely to occur. Upside Potential, a term coined by Nobel Prize winner Daniel Kahneman, captures investors’ perception of risks concerning gains as opposed to risk concerning losses.

Upside Potential Ratio (U-P Ratio)

The ratio of Upside Potential to Downside Deviation at a given MAR, measuring how much upside potential is provided by a manager at a given level of downside risk.

Omega

The Omega Ratio is a risk-return performance measure of an investment. It is used to measure investment performance relative to a benchmark and also against an acceptable threshold level (MAR). Simply put, the Omega measure is a ratio of probability weighted returns above the MAR level to probability weighted returns below the MAR level.

The upside potential divided by the downside potential for a pre-defined MAR to produce a quality rating for the investment based on all return data in the period.

Maximum Drawdown

Maximum drawdown is defined as the peak-to-trough decline of an investment during a specific period and is usually quoted as a percentage of the peak value. The maximum drawdown can be calculated based on absolute returns, in order to identify strategies that suffer less during market downturns, such as low-volatility strategies. However, the maximum drawdown can also be calculated based on returns relative to a benchmark index, for identifying strategies that show steady outperformance over time.
Maximum drawdown is highly dependent on the time interval chosen (annual, monthly, daily and so on) as well as the observation period.

Where,

P = Peak value before largest drop

L = Lowest value before new high established

Preservation of capital and a steady performance are important considerations in investing. Therefore, the maximum drawdown is highly relevant. A drawdown is measured from the time a retrenchment begins to when a new high is reached. This method is used because a valley can’t be measured until a new high occurs. Once the new high is reached, the percentage change from the old high to the new trough is recorded.