PerformanceAnalytics provides an package of econometric functions for performance and risk analysis of financial instruments or portfolios. This package aims to aid practitioners and researchers in using the latest research for analysis of both normally and non-normally distributed return streams.
Not all, but many of the measures in this package
require time series data. PerformanceAnalytics uses the
xts package for managing time series data for several
reasons. Besides being fast and efficient, xts includes
functions that test the data for periodicity and draw attractive and
readable time-based axes on charts. Another benefit is that
xts provides compatability with Rmetrics'
timeSeries, zoo and other time
series classes, such that PerformanceAnalytics functions that return
a time series will return the results in the same format as the object that
was passed in. Jeff Ryan and Joshua Ulrich, the authors of
xts, have been extraordinarily helpful to the
development of PerformanceAnalytics and we are very grateful for
their contributions to the community. The xts package
extends the excellent zoo package written by Achim
Zeileis and Gabor Grothendieck. zoo provides more
general time series support, whereas xts provides
functionality that is specifically aimed at users in finance.
Users can easily load returns data as time series for analysis with
PerformanceAnalytics by using the Return.read function.
The Return.read function loads csv files of returns data
where the data is organized as dates in the first column and the returns
for the period in subsequent columns. See read.zoo and
as.xts if more flexibility is needed.
The functions described below assume that input data is organized with
asset returns in columns and dates represented in rows. All of the metrics
in PerformanceAnalytics are calculated by column and return values
for each column in the results. This is the default arrangement of time
series data in xts.
Some sample data is available in the managers dataset. It is
an xts object that contains columns of monthly returns for six hypothetical
asset managers (HAM1 through HAM6), the EDHEC Long-Short Equity hedge fund
index, the S&P 500 total returns, and total return series for the US
Treasury 10-year bond and 3-month bill. Monthly returns for all series end
in December 2006 and begin at different periods starting from January 1996.
That data set is used extensively in our examples and should serve as a
model for formatting your data.
For retrieving market data from online sources, see quantmod's
getSymbols function for downloading prices and
chartSeries for graphing price data. Also see the
tseries package for the function
get.hist.quote. Look at xts's
to.period function to rationally coerce irregular price
data into regular data of a specified periodicity. The
aggregate function has methods for tseries and
zoo timeseries data classes to rationally coerce irregular data into
regular data of the correct periodicity.
Finally, see the function Return.calculate for calculating
returns from prices.
Many methods have been proposed to measure, monitor, and control the risks of a diversified portfolio. Perhaps a few definitions are in order on how different risks are generally classified. Market Risk is the risk to the portfolio from a decline in the market price of instruments in the portfolio. Liquidity Risk is the risk that the holder of an instrument will find that a position is illiquid, and will incur extra costs in unwinding the position resulting in a less favorable price for the instrument. In extreme cases of liquidity risk, the seller may be unable to find a buyer for the instrument at all, making the value unknowable or zero. Credit Risk encompasses Default Risk, or the risk that promised payments on a loan or bond will not be made, or that a convertible instrument will not be converted in a timely manner or at all. There are also Counterparty Risks in over the counter markets, such as those for complex derivatives. Tools have evolved to measure all these different components of risk. Processes must be put into place inside a firm to monitor the changing risks in a portfolio, and to control the magnitude of risks. For an extensive treatment of these topics, see Litterman, Gumerlock, et. al.(1998). For our purposes, PerformanceAnalytics tends to focus on market and liquidity risk.
The simplest risk measure in common use is volatility, usually modeled
quantitatively with a univariate standard deviation on a portfolio. See
sd. Volatility or Standard Deviation is an appropriate
risk measure when the distribution of returns is normal or resembles a random
walk, and may be annualized using sd.annualized, or the
equivalent function sd.multiperiod for scaling to an arbitrary
number of periods. Many assets, including hedge funds, commodities, options,
and even most common stocks over a sufficiently long period, do not follow a
normal distribution. For such common but non-normally distributed assets, a
more sophisticated approach than standard deviation/volatility is required to
adequately model the risk.
Markowitz, in his Nobel acceptance speech and in several papers, proposed
that SemiVariance would be a better measure of risk than
variance. See Zin, Markowitz, Zhao (2006). This measure is also
called SemiDeviation. The more general case is
DownsideDeviation, as proposed by Sortino and
Price(1994), where the minimum acceptable return (MAR) is a parameter to the
function. It is interesting to note that variance and mean return can
produce a smoothly elliptical efficient frontier for portfolio optimization
using solve.QP or
portfolio.optim or fPortfolio. Use of
semivariance or many other risk measures will not necessarily create a smooth
ellipse, causing significant additional difficulties for the portfolio
manager trying to build an optimal portfolio. We'll leave a more complete
treatment and implementation of portfolio optimization techniques for another
time.
Another very widely used downside risk measures is analysis of drawdowns, or
loss from peak value achieved. The simplest method is to check the
maxDrawdown, as this will tell you the worst cumulative loss
ever sustained by the asset. If you want to look at all the drawdowns, you
can findDrawdowns and sortDrawdowns in order from
worst/major to smallest/minor. The UpDownRatios function will
give you some insight into the impacts of the skewness and kurtosis of the
returns, and letting you know how length and magnitude of up or down moves
compare to each other. You can also plot drawdowns with
chart.Drawdown.
One of the newer statistical methods developed for analyzing the risk of
financial instruments is Omega. Omega analytically constructs
a cumulative distribution function, in a manner similar to
chart.QQPlot, but then extracts additional information from the
location and slope of the derived function at the point indicated by the risk
quantile that the researcher is interested in. Omega seeks to combine a
large amount of data about the shape, magnitude, and slope of the
distribution into one method. The academic literature is still exploring the
best manner to use Omega in a risk measurement and control process, or in
portfolio construction.
Any risk measure should be viewed with suspicion if there are not a large
number of historical observations of returns for the asset in question
available. Depending on the measure, the number of observations required
will vary greatly from a statistical standpoint. As a heuristic rule,
ideally you will have data available on how the instrument performed through
several economic cycles and shocks. When such a long history is not
available, the investor or researcher has several options. A full discussion
of the various approaches is beyond the scope of this introduction, so we
will merely touch on several areas that an interested party may wish to
explore in additional detail. Examining the returns of assets with a similar
style, industry, or asset class to which the asset in question is highly
correlated and shares other characteristics can be quite informative. Factor
analysis may be used to uncover specific risk factors where transparency is
not available. Various resampling (see tsbootstrap)
and simulation methods are available in R to construct an artificially long
distribution for testing. If you use a method such as Monte Carlo simulation
or the bootstrap, it is often valuable to use chart.Boxplot to
visualize the different estimates of the risk measure produced by the
simulation, to see how small (or wide) a range the estimates cover, and thus
gain a level of confidence with the results. Proceed with extreme caution
when your historical data is lacking. Problems with lack of historical data
are a major reason why many institutional investors will not invest in an
alternative asset without several years of historical return data available.
Traditional mean-VaR:
In the early 90's, academic literature started referring to “value at
risk”, typically written as VaR. Take care to capitalize VaR in the commonly
accepted manner, to avoid confusion with var (variance) and VAR (vector
auto-regression). With a sufficiently large data set, you may choose to use
a non-parametric VaR estimation method using the historical distribution and
the probability quantile of the distribution calculated using
qnorm. The negative return at the correct quantile (usually
95% or 99%), is the non-parametric VaR estimate. J.P. Morgan's RiskMetrics
parametric mean-VaR was published in 1994 and this methodology for estimating
parametric mean-VaR has become what people are generally referring to as
“VaR” and what we have implemented as VaR with
method="historical". See Return to RiskMetrics: Evolution of a
Standard at
https://www.msci.com/documents/10199/dbb975aa-5dc2-4441-aa2d-ae34ab5f0945.
Parametric traditional VaR does a better job of accounting for the tails of
the distribution by more precisely estimating the tails below the risk
quantile. It is still insufficient if the assets have a distribution that
varies widely from normality. That is available in VaR with
method="gaussian".
The R package VaR, now orphaned, contains methods for simulating and estimating lognormal VaR.norm and generalized Pareto VaR.gpd distributions to overcome some of the problems with nonparametric or parametric mean-VaR calculations on a limited sample size. There is also a VaR.backtest function to apply simulation methods to create a more robust estimate of the potential distribution of losses. The VaR package also provides plots for its functions. We will attempt to incorporate this orphaned functionality in PerformanceAnalytics in an upcoming release, and would welcome patches or pull requests in this direction.
Modified Cornish-Fisher VaR: The limitations of traditional mean-VaR are all related to the use of a symmetrical distribution function. Use of simulations, resampling, or Pareto distributions all help in making a more accurate prediction, but they are still flawed for assets with significantly non-normal (skewed and/or kurtotic) distributions. Huisman (1999) and Favre and Galleano (2002) propose to overcome this extensively documented failing of traditional VaR by directly incorporating the higher moments of the return distribution into the VaR calculation.
This VaR measure incorporates skewness and kurtosis via an analytical
estimation using a Cornish-Fisher (special case of a Taylor) expansion. The
resulting measure is referred to variously as “Cornish-Fisher VaR” or
“Modified VaR”. We provide this measure in function VaR
with method="modified". Modified VaR produces the same results as
traditional mean-VaR when the return distribution is normal, so it may be
used as a direct replacement. Many papers in the finance literature have
reached the conclusion that Modified VaR is a superior measure, and may be
substituted in any case where mean-VaR would previously have been used.
Note that estimates for Cornish Fisher modified VaR and ES may be unstable
with small sample sizes. See Martin and Arora (2017) for more details.
Conditional VaR and Expected Shortfall:
We have implemented Conditional Value at Risk, also called Expected Tail Loss
or Expected Shortfall
(not to be confused with shortfall probability, which is much less useful),
in function ES. Expected Shortfall attempts to measure the
magnitude of the average loss exceeding the traditional mean-VaR. Expected
Shortfall has proven to be a reasonable risk predictor for many asset
classes. We have provided traditional historical, Gaussian and modified
Cornish-Fisher measures of Expected Shortfall by using
method="historical", method="gaussian" or
method="modified". See Uryasev(2000) and Sherer and
Martin(2005) for more information on Conditional Value at Risk and Expected
Shortfall. Please note that your mileage will vary; expect that values
obtained from the normal distribution may differ radically from the real
situation, depending on the assets under analysis.
Multivariate extensions to risk measures:
We have extended all moments calculations to work in a multivariate portfolio
context. In a portfolio context the multivariate moments are generally to be
preferred to their univariate counterparts, so that all information is
available to subsequent calculations. Both the VaR and
ES functions allow calculation of metrics in a portfolio
context when weights and a portfolio_method are passed into the
function call.
Marginal, Incremental, and Component VaR:
Marginal VaR is the difference between the VaR of the portfolio without the
asset in question and the entire portfolio. The VaR function
calculates Marginal VaR for all instruments in the portfolio if you set
method="marginal". Marginal VaR as provided here may use traditional
mean-VaR or Modified VaR for the calculation. Per Artzner,et.al.(1997)
properties of a coherent risk measure include subadditivity (risks of the
portfolio should not exceed the sum of the risks of individual components) as
a significantly desirable trait. VaR measures, including Marginal VaR, on
individual components of a portfolio are not subadditive.
Clearly, a general subadditive risk measure for downside risk is required.
In Incremental or Component VaR, the Component VaR value for each element of
the portfolio will sum to the total VaR of the portfolio. Several EDHEC
papers suggest using Modified VaR instead of mean-VaR in the Incremental and
Component VaR calculation. We have succeeded in implementing Component VaR
and ES calculations that use Modified Cornish-Fisher VaR, historical
decomposition, and kernel estimators. You may access these with
VaR or ES by setting the appropriate
portfolio_method and method arguments.
The chart.VaRSensitivity function creates a chart of
Value-at-Risk or Expected Shortfall estimates by confidence interval for
multiple methods. Useful for comparing a calculated VaR or ES method to the
historical VaR or ES, it may also be used to visually examine whether the VaR
method “breaks down” or gives nonsense results at a certain threshold.
Which VaR or ES measure to use will depend greatly on the portfolio and instruments being analyzed. If there is any generalization to be made on VaR measures, we agree with Bali and Gokcan(2004) who conclude that “the VaR estimations based on the generalized Pareto distribution and the Cornish-Fisher approximation perform best”.
Most risk professionals are moving from using VaR to using Expected Shortfall preferentially. This preference has been endorsed by the Bank of International Settlements (BIS) and the Basel accord risk committee. In most cases, an ES measure with an appropriate probability target for your asset horizin will be more appropriate than a VaR measure.
The literature around the subject of performance analysis seems to have exploded with the popularity of alternative assets such as hedge funds, managed futures, commodities, and structured products. Simpler tools that may have seemed appropriate in a relative investment world seem inappropriate for an absolute return world. Risk measurement, which is nearly inseparable from performance assessment, has become multi-dimensional and multi-moment while trying to answer a simple question: “How much could I lose?” Portfolio construction and risk budgeting are two sides of the same coin: “How do I maximize my expected gain and avoid going broke?” But before we can approach those questions we first have to ask: “Is this something I might want in my portfolio?”
With the the increasing availability of complicated alternative investment strategies to both retail and institutional investors, and the broad availability of financial data, an engaging debate about performance analysis and evaluation is as important as ever. There won't be one right answer delivered in these metrics and charts. What there will be is an accretion of evidence, organized to assist a decision maker in answering a specific question that is pertinent to the decision at hand. Using such tools to uncover information and ask better questions will, in turn, create a more informed investor.
Performance measurement starts with returns. Traders may object,
complaining that “You can't eat returns,” and will prefer to look
for numbers with currency signs. To some extent, they have a point - the
normalization inherent in calculating returns can be deceiving. Most of
the recent work in performance analysis, however, is focused on returns
rather than prices and sometimes called "returns-based analysis" or RBA.
This “price per unit of investment” standardization is important for
two reasons - first, it helps the decision maker to compare opportunities,
and second, it has some useful statistical qualities. As a result, the
PerformanceAnalytics package focuses on returns. See
Return.calculate for converting net asset values or prices
into returns, either discrete or continuous. Many papers and theories
refer to “excess returns”: we implement a simple function for
aligning time series and calculating excess returns in
Return.excess.
Return.portfolio can be used to calculate weighted returns
for a portfolio of assets. The function was recently changed to support
several use-cases: a single weighting vector, an equal weighted portfolio,
periodic rebalancing, or irregular rebalancing. That replaces
functionality that had been split between that function and
Return.rebalancing. The function will subset the return
series to only include returns for assets for which weights
are provided.
Returns and risk may be annualized as a way to simplify comparison over
longer time periods. Although it requires a bit of estimating, such
aggregation is popular because it offers a reference point for easy
comparison. Examples are in Return.annualized,
sd.annualized, and SharpeRatio.annualized.
Basic measures of performance tend to treat returns as independent
observations. In this case, the entirety of R's base is applicable to such
analysis. Some basic statistics we have collected in
table.Stats include:
mean | arithmetic mean |
mean.geometric | geometric mean |
mean.stderr | standard error of the mean (S.E. mean) |
mean.LCL | lower confidence level (LCL) of the mean |
mean.UCL | upper confidence level (UCL) of the mean |
quantile | for calculating various quantiles of the distribution |
min | minimum return |
max | maximum return |
range | range of returns |
length(R) | number of observations |
sum(is.na(R)) | number of NA's |
It is often valuable when evaluating an investment to know whether the
instrument that you are examining follows a normal distribution. One of
the first methods to determine how close the asset is to a normal or
log-normal distribution is to visually look at your data. Both
chart.QQPlot and chart.Histogram will quickly
give you a feel for whether or not you are looking at a normally
distributed return history. Differences between var and
SemiVariance will help you identify skewness in
the returns. Skewness measures the degree of asymmetry in the return
distribution. Positive skewness indicates that more of the returns are
positive, negative skewness indicates that more of the returns are
negative. An investor should in most cases prefer a positively skewed
asset to a similar (style, industry, region) asset that has a negative
skewness.
Kurtosis measures the concentration of the returns in any given part of the
distribution (as you should see visually in a histogram). The
kurtosis function will by default return what is referred to
as “excess kurtosis”, where 0 is a normal distribution, other
methods of calculating kurtosis than method="excess" will set the
normal distribution at a value of 3. In general a rational investor should
prefer an asset with a low to negative excess kurtosis, as this will
indicate more predictable returns (the major exception is generally a
combination of high positive skewness and high excess kurtosis). If you
find yourself needing to analyze the distribution of complex or non-smooth
asset distributions, the nortest package has several advanced
statistical tests for analyzing the normality of a distribution.
Modern Portfolio Theory (MPT) is the collection of tools and techniques by which a risk-averse investor may construct an “optimal” portfolio. It was pioneered by Markowitz's ground-breaking 1952 paper Portfolio Selection. It also encompasses CAPM, below, the efficient market hypothesis, and all forms of quantitative portfolio construction and optimization.
The Capital Asset Pricing Model (CAPM), initially developed by
William Sharpe in 1964, provides a justification for passive or index
investing by positing that assets that are not on the efficient frontier
will either rise or fall in price until they are. The
CAPM.RiskPremium is the measure of how much the asset's
performance differs from the risk free rate. Negative Risk Premium
generally indicates that the investment is a bad investment, and the money
should be allocated to the risk free asset or to a different asset with a
higher risk premium. CAPM.alpha is the degree to which the
assets returns are not due to the return that could be captured from the
market. Conversely, CAPM.beta describes the portions of the
returns of the asset that could be directly attributed to the returns of a
passive investment in the benchmark asset.
The Capital Market Line CAPM.CML relates the excess expected
return on an efficient market portfolio to its risk (represented in CAPM by
sd). The slope of the CML,
CAPM.CML.slope, is the Sharpe Ratio for the market portfolio.
The Security Market Line is constructed by calculating the line of
CAPM.RiskPremium over CAPM.beta. For the
benchmark asset this will be 1 over the risk premium of the benchmark
asset. The slope of the SML, primarily for plotting purposes, is given by
CAPM.SML.slope. CAPM is a market equilibrium model or a
general equilibrium theory of the relation of prices to risk, but it is
usually applied to partial equilibrium portfolios, which can create
(sometimes serious) problems in valuation.
One extension to the CAPM contemplates evaluating an active manager's
ability to time the market. Two other functions apply the same notion of
best fit to positive and negative market returns, separately. The
CAPM.beta.bull is a regression for only positive market
returns, which can be used to understand the behavior of the asset or
portfolio in positive or 'bull' markets. Alternatively,
CAPM.beta.bear provides the calculation on negative market
returns. The TimingRatio uses the ratio of those to help
assess whether the manager has shown evidence that of timing skill.
Performance/Risk Ratios:
In many cases, an analyst will be looking to find a measure or performance relative to the risk of the asset under study. PerformanceAnalytics has many functions of this type.
One of the most commonly used and cited measures of the risk/reward tradeoff
of an investment or portfolio is the SharpeRatio, which
measures return over standard deviation. If you are comparing multiple
assets using Sharpe, you should use SharpeRatio.annualized. It
is important to note that William Sharpe now recommends
InformationRatio preferentially to the original Sharpe Ratio.
The SortinoRatio uses mean return over
DownsideDeviation below the MAR as the risk measure to produce
a similar ratio that is more sensitive to downside risk. Sortino later
enhanced his ideas to use upside returns for the numerator and
DownsideDeviation as the denominator in
UpsidePotentialRatio. Favre and Galeano(2002) propose
using the ratio of expected excess return over the Cornish-Fisher
VaR to produce SharpeRatio.modified.
TreynorRatio is also similar to the Sharpe Ratio, except it
uses CAPM.beta in place of the volatility measure to produce
the ratio of the investment's excess return over the beta. Use of the downside
semivariance as the denominator creates the . Use of the upside expected tail
return over the ETL creates the RachevRatio. Utilizing the
downside semivariance as the denominator produces the Downside Sharpe Ratio.
The performance premium provided by an investment over a passive strategy
(the benchmark) is provided by ActivePremium, which is the
investment's annualized return minus the benchmark's annualized return. A
closely related measure is the TrackingError, which measures
the unexplained portion of the investment's performance relative to a
benchmark. The InformationRatio of an investment in a MPT or
CAPM framework is the Active Premium divided by the Tracking Error.
Information Ratio may be used to rank investments in a relative fashion.
We have also included a function to compute the KellyRatio.
The Kelly criterion applied to position sizing will maximize log-utility of
returns and avoid risk of ruin. For our purposes, it can also be used as a
stack-ranking method like InformationRatio to describe the
“edge” an investment would have over a random strategy or
distribution.
These metrics and others such as SharpeRatio,
SortinoRatio, UpsidePotentialRatio, Spearman
rank correlation (see rcorr), etc., are all methods of
rank-ordering relative performance. Alexander and Dimitriu (2004) in
“The Art of Investing in Hedge Funds” show that relative rankings
across multiple pricing methodologies may be positively correlated with
each other and with expected returns. This is quite an important finding
because it shows that multiple methods of predicting returns and risk which
have underlying measures and factors that are not directly correlated to
another measure or factor will still produce widely similar quantile
rankings, so that the “buckets” of target instruments will have
significant overlap. This observation specifically supports the point made
early in this document regarding “accretion of the evidence” for a
positive or negative investment decision.
While PerformanceAnalytics contains many functions for computing returns based risk and performance estimators, until now there has been no convenient way to accurately compute standard errors of the estimates for independent and identically distributed (i.i.d.) returns, and no way at all to do so for returns that are serially correlated. This is no longer the case due to the existence of a new frequency domain method of accurately computing standard errors when returns are serially dependent as well as when returns are independent and identically distributed. Details are provided in Xin and Martin (2019) at https://www.ssrn.com/abstract=3085672, and to appear in December 2020 issue of Journal of Risk. The new method makes novel use of statistical influence functions borrowed from robust statistics, combined with periodogram based regularized generalized linear polynomial model (GLM) fitting for exponential distributions. Influence function for risk and performance estimators are described in Zhang, Martin and Christidis (2020) available at SSRN https://www.ssrn.com/abstract=3415903. The new method has been implemented in the RPESE package available at CRAN. The RPESE package has been integrated into PerformanceAnalytics.
Standard errors can be easily computed using the PerformanceAnalytics risk estimator functions
StdDev | Sample standard deviation |
SemiSD | Semi-standard deviation |
lpm with argument n=1 or n=2 | Lower partial moment of order 1 or 2 |
ES | Expected shortfall with tail probability \(\alpha\) |
VaR | Value-at-risk with tail probability \(\alpha\) |
and performance estimator functions
mean.arithmetic | Sample mean |
SharpeRatio with argument FUN="StdDev" | Sharpe ratio |
DownsideSharpeRatio or SharpeRatio with argument FUN="SemiSD" | Downside Sharpe ratio |
SortinoRatio | Sortino ratio with threshold MAR |
SharpeRatio with argument FUN="ES" | Mean excess return to ES ratio with tail probability \(\alpha\) |
SharpeRatio with argument FUN="VaR" | Mean excess return to VaR ratio with tail probability \(\alpha\) |
RachevRatio | Rachev ratio with lower upper tail probabilities \(\alpha\) and \(\beta\) |
Omega | Omega ratio with threshold \(c\) |
where the first columns gives the names of the functions in the PerformanceAnalytics package to compute the estimate and its standard error.
Each of the PerformanceAnalytics functions listed above have an optional argument with default SE = FALSE. By changing this default to SE = TRUE, the user obtains not only risk or performance estimate, but also a standard error for the esimate. Further details concerning the computation of standard errors for the risk and performance estimators in PerformanceAnalytics can be found in the Vignette "Standard Errors for Risk and Performance Estimators in PerformanceAnalytics" available at CRAN, where a reference to the underlying theory due to Chen and Martin (2020) may be found.
Analysis of financial time series often involves evaluating their
mathematical moments. While var and cov for
variance has always been available, as well as skewness and
kurtosis (which we have extended to make multivariate and
multi-column aware), a larger suite of multivariate moments calculations was
not available in R. We have now implemented multivariate moments and
co-moments and their beta or systematic co-moments in
PerformanceAnalytics.
Ranaldo and Favre (2005) define coskewness and cokurtosis as the
skewness and kurtosis of a given asset analysed with the skewness and
kurtosis of the reference asset or portfolio. The co-moments are useful for
measuring the marginal contribution of each asset to the portfolio's
resulting risk. As such, co-moments of an asset return distribution should
be useful as inputs for portfolio optimization in addition to the covariance
matrix. Functions include CoVariance,
CoSkewness, CoKurtosis.
Measuring the co-moments should be useful for evaluating whether or not an asset is likely to provide diversification potential to a portfolio. But the co-moments do not allow the marginal impact of an asset on a portfolio to be directly measured. Instead, Martellini and Zieman (2007) develop a framework that assesses the potential diversification of an asset relative to a portfolio. They use higher moment betas to estimate how much portfolio risk will be impacted by adding an asset.
Higher moment betas are defined as proportional to the derivative of the
covariance, coskewness and cokurtosis of the second, third and fourth
portfolio moment with respect to the portfolio weights. A beta that is less
than 1 indicates that adding the new asset should reduce the resulting
portfolio's volatility and kurtosis, and to an increase in skewness. More
specifically, the lower the beta the higher the diversification effect, not
only in terms of normal risk (i.e. volatility) but also the risk of assymetry
(skewness) and extreme events (kurtosis). See the functions for
BetaCoVariance, BetaCoSkewness, and
BetaCoKurtosis.
The functions Return.clean and clean.boudt
implement statistically robust data cleaning methods tuned to portfolio
construction and risk analysis and prediction in financial time series while
trying to avoid some of the pitfalls of standard robust statistical methods.
The primary value of data cleaning lies in creating a more robust and stable estimation of the distribution generating the large majority of the return data. The increased robustness and stability of the estimated moments using cleaned data should be used for portfolio construction. If an investor wishes to have a more conservative risk estimate, cleaning may not be indicated for risk monitoring.
In actual practice, it is probably best to back-test the out-of-sample results of both cleaned and uncleaned series to see what works best when forecasting risk with the particular combination of assets under consideration.
Summary statistics are then the necessary aggregation and reduction of
(potentially thousands) of periodic return numbers. Usually these statistics
are most palatable when organized into a table of related statistics,
assembled for a particular purpose. A common offering of past returns
organized by month and cumulated by calendar year is usually presented as a
table, such as in table.CalendarReturns. Adding benchmarks or
peers alongside the annualized data is helpful for comparing returns in
calendar years.
When we started this project, we debated whether such tables would be broadly useful or not. No reader is likely to think that we captured the precise statistics to help their decision. We merely offer these as a starting point for creating your own. Add, subtract, do whatever seems useful to you. If you think that your work may be useful to others, please consider sharing it so that we may include it in a future version of this package.
Other tables for comparison of related groupings of statistics discussed elsewhere:
table.Stats | Basic statistics and stylized facts |
table.TrailingPeriods | Statistics and stylized facts compared over different trailing periods |
table.AnnualizedReturns | Annualized return, standard deviation, and Sharpe ratio |
table.CalendarReturns | Monthly and calendar year return table |
table.CAPM | CAPM-related measures |
table.Correlation | Comparison of correlalations and significance statistics |
table.DownsideRisk | Downside risk metrics and statistics |
table.Drawdowns | Ordered list of drawdowns and when they occurred |
table.Autocorrelation | The first six autocorrelation coefficients and significance |
table.HigherMoments | Higher co-moments and beta co-moments |
table.Arbitrary | Combines a function list into a table |
Graphs and charts can also help to organize the information visually. Our
goal in creating these charts was to simplify the process of creating
well-formatted charts that are used often in performance analysis, and to
create high-quality graphics that may be used in documents for consumption by
non-analysts or researchers. R's graphics capabilities are substantial, but
the simplicity of the output of R default graphics functions such as
plot does not always compare well against graphics
delivered with commercial asset or performance analysis from places such as
MorningStar or PerTrac.
The cumulative returns or wealth index is usually the first thing displayed,
even though neither conveys much information. See
chart.CumReturns. Individual period returns may be helpful for
identifying problematic periods, such as in chart.Bar. Risk
measures can be helpful when overlaid on the period returns, to display the
bounds at which losses may be expected. See chart.BarVaR and
the prior section on Risk Analysis. More information can be conveyed when
such charts are displayed together, as in
charts.PerformanceSummary, which combines the performance data
with detail on downside risk (see chart.Drawdown).
chart.RelativePerformance can plot the relative performance
through time of two assets. This plot displays the ratio of the cumulative
performance at each point in time and makes periods of under- or
out-performance easy to see. The value of the chart is less important than
the slope of the line. If the slope is positive, the first asset is
outperforming the second, and vice verse. Affectionately known as the Canto
chart, it was used effectively in Canto (2006).
Two-dimensional charts can also be useful while remaining easy to understand.
chart.Scatter is a utility scatter chart with some additional
attributes that are used in chart.RiskReturnScatter.
Overlaying Sharpe ratio lines or boxplots helps to add information about
relative performance along those dimensions.
For distributional analysis, a few graphics may be useful.
chart.Boxplot is an example of a graphic that is difficult to
create in Excel and is under-utilized as a result. A boxplot of returns is,
however, a very useful way to instantly observe the shape of large
collections of asset returns in a manner that makes them easy to compare to
one another. chart.Histogram and chart.QQPlot
are two charts originally found elsewhere and now substantially expanded in
PerformanceAnalytics.
Rolling performance is typically used as a way to assess stability of a
return stream. Although perhaps it doesn't get much credence in the
financial literature as it derives from work in digital signal processing,
many practitioners find it a useful way to examine and segment performance
and risk periods. See chart.RollingPerformance, which is a way
to display different metrics over rolling time periods.
chart.RollingMean is a specific example of a rolling mean and
standard error bands. A group of related metrics is offered in
charts.RollingPerformance. These charts use utility functions
such as rollapply.
chart.SnailTrail is a scatter chart that shows how rolling
calculations of annualized return and annualized standard deviation have
proceeded through time where the color of lines and dots on the chart
diminishes with respect to time. chart.RollingCorrelation shows
how correlations change over rolling periods.
chart.RollingRegression displays the coefficients of a linear
model fitted over rolling periods. A group of charts in
charts.RollingRegression displays alpha, beta, and R-squared
estimates in three aligned charts in a single device.
chart.StackedBar creates a stacked column chart with time on
the horizontal axis and values in categories. This kind of chart is commonly
used for showing portfolio 'weights' through time, although the function will
plot any values by category.
We have been greatly inspired by other peoples' work, some of which is on
display at addictedtor.free.fr. Particular inspiration came from Dirk
Eddelbuettel and John Bollinger for their work at
https://addictedtor.free.fr/graphiques/RGraphGallery.php?graph=65. Those
interested in price charting in R should also look at the quantmod
package.
R is a very powerful environment for manipulating data. It can also be
quite confusing to a user more accustomed to Excel or even MatLAB. As such,
we have written some wrapper functions that may aid you in coercing data into
the correct forms or finding data that you need to use regularly. To
simplify the management of multiple-source data stored in R in multiple data
formats, we have provided checkData. This function will
attempt to coerce data in and out of R's multitude of mostly fungible data
classes into the class required for a particular analysis. The data-coercion
function has been hidden inside the functions here, but it may also save you
time and trouble in your own code and functions as well.
R's built-in apply function in enormously powerful, but is can
be tricky to use with timeseries data, so we have provided wrapper functions
to apply.fromstart and apply.rolling to make
handling of “from inception” and “rolling window” calculations
easier.
We have attempted to standardize function parameters and variable names, but more work exists to be done here.
Any comments, suggestions, or code patches are invited.
If you've implemented anything that you think would be generally useful to include, please consider donating it for inclusion in a later version of this package.
Data series edhec used in PerformanceAnalytics and
related publications with the kind permission of the EDHEC Risk and Asset
Management Research Center.
https://climateimpact.edhec.edu/retirement-investing
Kris Boudt was instrumental in our research on component risk for portfolios with non-normal distributions, and is responsible for much of the code for multivariate moments and co-moments. This works was later extended and made faster and more robust by Dries Cornilly
Jeff Ryan and Joshua Ulrich are active participants in the R finance
community and created xts, upon which much of
PerformanceAnalytics depends.
Prototypes of the drawdowns functionality were provided by Sankalp Upadhyay, and modified with permission. Stephan Albrecht provided detailed feedback on the Getmansky/Lo Smoothing Index. The late Diethelm Wuertz provided prototypes of modified VaR and skewness and kurtosis functions (and was of course the maintainer of the RMetrics suite of pricing and optimization functions). Diethelm also contributed prototypes for many other functions from Bacon's book that were incorporated into PerformanceAnalytics by Matthieu Lestel.
Thanks to Joe Wayne Byers and Dirk Eddelbuettel for comments on early versions of these functions, and to Khanh Nguyen, Tobias Verbeke, H. Felix Wittmann, and Ryan Sheftel for careful testing and detailed problem reports.
Thanks also to our Google Summer of Code students through the years for their contributions. Significant contributions from GSOC students to this package have come from Dries Cornilly, Anthony-Alexander Cristidis, Zenith "Ziheng" Zhou, Matthieu Lestel and Andrii Babii so far. We expect to eventually incorporate contributions from Pulkit Mehrotra and Shubhankit Mohan, who worked with us during the summer of 2013.
Thanks to the R-SIG-Finance community without whom this package would not be possible. We are indebted to the R-SIG-Finance community for many helpful suggestions, bugfixes, and requests.
Any errors are, of course, our own.
Maintainer: Brian G. Peterson brian@braverock.com [copyright holder]
Authors:
Peter Carl peter@braverock.com [copyright holder]
Other contributors:
Kris Boudt [contributor, copyright holder]
Ross Bennett [contributor]
Joshua Ulrich [contributor]
Eric Zivot [contributor]
Dries Cornilly [contributor]
Eric Hung [contributor]
Matthieu Lestel [contributor]
Kyle Balkissoon [contributor]
Diethelm Wuertz [contributor]
Anthony Alexander Christidis [contributor]
R. Douglas Martin [contributor]
Zeheng Zenith Zhou [contributor]
Justin M. Shea [contributor]
Dhairya Jain [contributor]
We created this package to include functionality that has been appearing in the academic literature on performance analysis and risk over the past several years, but had no functional equivalent in . In doing so, we also found it valuable to have wrappers for some functionality with good defaults and naming consistent with common usage in the finance literature.
In general, this package requires return (rather than price) data. Almost all of the functions will work with any periodicity, from annual, monthly, daily, to even minutes and seconds, either regular or irregular.
The following sections cover Time Series Data, Performance Analysis, Risk Analysis (with a separate treatment of VaR), Summary Tables of related statistics, Charts and Graphs, a variety of Wrappers and Utility functions, and some thoughts on work yet to be done.
In this summary, we attempt to provide an overview of the capabilities provided by PerformanceAnalytics and pointers to other literature and resources in useful for performance and risk analysis. We hope that this summary and the accompanying package and documentation partially fill a hole in the tools available to a financial engineer or analyst.
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