Performance analysis with R

PERFORMANCE
ANALYSIS WITH (I)
Jérémy Morvan
Assistant professor
January 2018
Université de Bretagne Occidentale

Introduction
• Financial performance assessment is an important issue
• An economic issue
• Financialization of the economy has led financial markets to drive
economic coordination
• A methodological issue
• How assessing portfolio performance?
• A theoretical issue
• How do financial markets work?
3

Introduction
• This course is an introduction to financial performance
analysis by using the programming language R
• What is performance in finance?
• What are the expected results according to modern portfolio
theory?
• What are the measures of performance?
4

Introduction
• Contents (I)
1. Performance in finance
2. Descriptive statistics
3. Histogram and Normality tests
4. Mean-variance analysis
5

I. WHAT IS PERFORMANCE IN
FINANCE?
6

Performance in finance
• Historical
• Performance analysis is one of the basic issues that built the
modern portfolio theory
• Bachelier (1900) highlights that the rise and the fall of prices are
equiprobable at any time
• Prices fluctuate randomly
• Cowles (1933, 1944) highlights that investment advices from financial
analysts do not allow to outperform the market
• It is impossible to predict future prices
7

• Efficient market hypothesis
• Definition
8
[…] the ideal is a market in which prices provide accurate signal for ressource
allocation: that is, a market in which firms can make production-Investment decisions,
and investors can choose among the securities that represent ownership of firms’
activities under the assumption that security prices at any time "fully reflect" all
available information. A market in which prices always "fully reflect" available
information is called "efficient". (Fama E., 1970)
[the market efficiency hypothesis is] the simple statement that security prices fully
reflect allavailable information. A precondition for this strong version of the hypothesis
is that information and tranding costs, the costs of getting prices to reflect information,
are always 0 (Grossman and Stiglitz (1980). A weaker and enconomically more
sensible version of the efficiency hypothesis says that prices reflect information to the
point where the marginal benefits of acting on information (the profits to be made) do
not exceed the marginal costs (Jensen (1978). (Fama E., 1991)

• Definition
• The definition of efficient markets varies according to the informational
set considered (Fama, 1970)
9
Present information
(semi-strong form
efficiency)
Past information
(weak-form
efficiency)
All information sets,
including private
information (strong-
form efficiency)

• Consequences
• Price is an unbiased estimate of the intrinsic value of a security
• There is no better financial analyst than the market
• Future variations are independent of past variations
• Random walk theory
• Investors cannot forecast prices successully
• The theory assumes that managers do not add value (a) to the portfolio
10

• Efficient frontier (Markowitz, 1952)
• The efficient frontier is the set of optimal portfolios that offers the
highest expected return for a given level of risk or the lowest risk for
a defined level of expected return
• Two stochastic dominance decision rules
• An efficient portfolio optimizes the risk/reward ratio
• A diversified portfolio should have at least 50 investments (see Euro
Stoxx 50)
11
At a given level of return,
investors choose the less
risky asset
At a given level of risk,
investors choose the asset
with highest returns

• Efficient frontier (Markowitz, 1952)
• Well-diversified portfolios tend to have similar return (given account
the risk) and rare extreme returns
12
Asset 1
Asset 2
Rf
0,00%
2,00%
4,00%
6,00%
8,00%
10,00%
12,00%
14,00%
0,00% 5,00% 10,00% 15,00% 20,00% 25,00%
Return (Ri)
Volatility (si)
Portfolio of two risky assets Portfolio with one risky asset and the risk free asset

• Expected results
• Financial performance integrates return and risk
• There are several measures of return (max, mean…) and risk (min, s,
VaR…)
• Returns are random
• The most common representation of randomness is the Gaussian
distribution
• A skilled manager shows a similar performance to the market given
the risk
• Outperformance is impossible
• Underperformance is possible
13
A great number of causes Causes are independant
Each cause has
a small effect

• Critics
• Behavioral Finance
• Many behavioral biases impact investment decisions
• Market anomalies
• Calendar anomalies
• "January effect" is an abnormal increase in stock prices in January
• "Weekend effect": stock returns on Monday are often significantly lower
• Other anomalies
• Small firms tend to outperfom
• There is a negative relation between PER and returns and between market to
book ratio and returns
• The Gaussian distribution imperfectly represents return ditribution
• … by underestimating risk
14

Data analysis
• Definition
• Statistical procedure to determine the main characteristics of a
dataset
• We use the programming language
16
#install packages (collections of functions which allow more
statistical techniques, and graphical devices)
install.packages("quantmode")
install.packages("fBasics")
install.packages("moments")
install.packages("PerformanceAnalytics")
install.packages("normtest")
install.packages("tseries")
install.packages("roll")
install.packages("xts")
Data are available in the data.csv file with
- Pi are daily closing prices of the stock i
- Pm are daily closing prices of the market index m
- RFR are daily closing rates of a 10-year constant maturity fictitious sovereign bond

• Calculation of daily returns
• Logarithmic returns are a better measure
17
#read data
data<-read.csv2(file= file.choose(),header = TRUE,sep = ";",dec = ",")
#Compute daily returns
library(quantmode)
data$Ri<-Delt(x1 = data$Pi,x2 = NULL,k = 1,type = "log")
data$Rm<-Delt(x1 = data$Pm,x2 = NULL,k = 1,type = "log")
data$Rf<-log(1+data$RFR/250)
#Clean up data object
#suppress 1st row
data<-data[-1,]
#suppress colums 2,3,4 (we only keep colums 1, 5 to 7)
data<-data[,c(1,5:7)]









1,
,
, ln
ti
ti
ti
R
R
R
Data analysis
If the package “quantmode” doesn’t
work (it happens !), we can compute
daily returns as follows data$Ri[2:774]<-log(data$Pi[2:774]/data$Pi[1:773])
data$Ri[2:774]<-diff(log(data$Pi))

• Descriptive statistics
• Main measures
18
Ri Rm Rf
nobs 773.000000 773.000000 774.000000
NAs 0.000000 0.000000 0.000000
Minimum -0.035737 -0.032272 0.000093
Maximum 0.035524 0.025047 0.000156
I. Quartile -0.004088 -0.004349 0.000114
3. Quartile 0.005653 0.005855 0.000142
Mean 0.000491 0.000529 0.000127
Median 0.000639 0.000844 0.000130
Sum 0.379467 0.409023 0.097973
SE Mean 0.000298 0.000299 0.000001
LCL Mean -0.000094 -0.000057 0.000125
UCL Mean 0.001075 0.001115 0.000128
Variance 0.000069 0.000069 0.000000
Stdev 0.008278 0.008300 0.000017
Skewness -0.369131 -0.391430 -0.352644
Kurtosis I.794045 I.091768 -1.073065
 


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n
t
itii RR
n 1
2
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2
1
1
s
  

n
t
tiii R
n
RRE
1
,
1
library(fBasics)
basicStats(data[,2:4])

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3
i
ii
i
RR
ESk
s
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
4
i
ii
i
RR
EK
s
Data analysis

• Main measures
19
Measures Definition
Mean
In finance, mean is a performance measure
The average is calculated on daily data. It must be annualized (n = 252 days)
Variance
In finance, the variance is a risk measure
The variance is calculated on daily data. It must be annualized
The variance is expressed in the square of the unit of measure ("%²"). The
standard deviation is the square root of the variance.
yMean.Ri<-mean(data$Ri)*252
yMean.Rm<-mean(data$Rm)*252
ySD.Ri<-sd(data$Ri)*sqrt(252)
ySD.Rm<-sd(data$Rm)*sqrt(252)
Data analysis

• Main measures
20
Measures Definition
Skewness
(Sk)
Skewness is a measure of asymmetry of the probability distribution
• If > 0, the most part of the distribution is concentrated on the left of the figure (the right tail
is longer)
• If < 0, the most part of the distribution is concentrated on the right of the figure (the left tail
is longer)
Kurtosis
(K)
Kurtosis is a measure of of the probability distribution (1 < K < ∞)
• If > 3, leptokurtic distribution (the data are heavy-tailed: extreme returns are more frequent
than predicted by the gaussian distribution)
• If < 3, platykurtic distribution (the data are light-tailed: extreme returns are less frequent
than predicted by the gaussian distribution)
Data analysis

Data analysis
• Main measures
• Some measures of kurtosis
• Kurtosis is sometimes calculated in excess of 3
21
[1] 1.794045
attr(,"method")
[1] "excess“
[1] 4.794045
[1] 1.806473
[1] 4.806473
library(moments)
kurtosis(data$Ri)
mean(((data$Ri-mean(data$Ri))/sd(data$Ri))^4)
library(PerformanceAnalytics)
kurtosis(data$Ri, na.rm = FALSE, method = "excess")
kurtosis(data$Ri, na.rm = FALSE, method = "moment")
Most explicit
command!

Data analysis
• Main measures
• Some measures of skewness
22
library(moments)
skewness(data$Ri)
mean(((data$Ri-mean(data$Ri))/sd(data$Ri))^3)
library(PerformanceAnalytics)
skewness(data$Ri, na.rm = FALSE)
Delt.1.log
-0.3698488
[1] -0.3691314
[1] -0.3698488
There are sometimes
(small) differences in
the results

Normality tests
• Histogram of returns
• For financial theory, markets evolve randomly
• A common representation of randomness is the Gaussian distribution
• The Gaussian distribution has a causal structure that defines a relatively
stable world
• ... consistent with the EMH (and perfect competition)
• Extreme returns are rare
• Returns around mean are the most frequent
24
Brownian motion Stochastic process Random walk

Normality tests
• Definition
• Graphic that represents the distribution of numerical data
• It is a graphical way to compare empirical and theoretical distributions
25
hist(data$Ri,main = "Histograms of Ri and Rm", breaks = 100, freq =
FALSE, xlab = "Ri", xlim = c(min(data$Ri,data$Rm),
max(data$Ri,data$Rm)), col = "green",axes = F)
axis(1, pos = 0, cex.axis = 0.8)
axis(2,pos = 0,cex.axis = 0.8,las = 2)
hist(data$Rm,breaks = 50, freq = FALSE, xlim =
c(min(data$Ri,data$Rm), max(data$Ri,data$Rm)), col = "red",,axes =
F,add = TRUE)
curve(dnorm(x, mean(data$Ri), sd(data$Ri)), xlim = c(min(data$Ri),
max(data$Ri)), lwd = 2,col = "grey", add = TRUE)
curve(dnorm(x, mean(data$Rm), sd(data$Rm)), xlim = c(min(data$Rm),
max(data$Rm)), lwd = 2,col = "red", add = TRUE)
legend("topleft",c("Histogram of Ri","Histogram of Rm"),lty =
c(1,1),col = c("green","red"),bty = "n")

Normality tests
26
Sharper
distribution
Less returns
than expected
on both sides
"fat" tails

Normality tests
• The Gaussian distribution poorly approximates the return
distribution
• It underestimates the probability of extreme returns
27
(min(data$Ri)-mean(data$Ri))/sd(data$Ri)
pnorm(q = (min(data$Ri)-mean(data$Ri))/sd(data$Ri),mean = 0,sd =
1, lower.tail = TRUE, log.p = FALSE)
format(x = pnorm(q = -4.376432,mean = 0,sd = 1, lower.tail =
TRUE,log.p = FALSE),scientific = FALSE, digits = 10)
format(x = 1/(pnorm(q = -4.376432, mean = 0, sd = 1,TRUE,
FALSE)*nrow(data$Ri)), scientific = FALSE, digits = 10)
[1] -4.376432
[1] 6.03189e-06
[1] "0.000006031890184"
[1] "214.4702608"
the worst return occurred 1 time
out of 773 or 214 times too
often…

• Normality test of distributions
• Jarque-Bera non-parametric test (1980) (for n >> 0)
• H0: Sk = 0 and K = 3 (the data follows a Gaussian distribution)
• H1: Sk ≠ 0 or K ≠ 3 (the data do not follow a Gaussian distribution)
28
Jarque-Bera test for normality
data: data$Ri
JB = 122.73, p-value < 2.2e-16
Title:
Jarque - Bera Normalality Test
Test Results:
STATISTIC:
X-squared: 122.7298
P VALUE:
Asymptotic p Value: < 2.2e-16
Jarque Bera Test
data: data$Ri
X-squared = 122.73, df = 2, p-value < 2.2e-16
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22
3
4
1
6
ii
i
i KSk
n
JB
Normality tests
library(normtest)
jb.norm.test(data$Ri)
library(fBasics)
jarqueberaTest(data$Ri)
library(tseries)
jarque.bera.test(data$Ri)

Normality tests
• Normality test of distributions
• Non-parametric test of Kolgomorov-Smirnov
• H0: D = D0 (the data follow the Gaussian distribution)
• H1: D ≠ D0 (the data do not follow the Gaussian distribution)
29
Title: One-sample Kolmogorov-Smirnov test
Test Results:
STATISTIC:
D: 0.4878
P VALUE:
Alternative Two-Sided: < 2.2e-16
Alternative Less: < 2.2e-16
Alternative Greater: < 2.2e-16
library(fBasics)
ksnormTest(data$Ri)
At a = 10%, criticial value = 1,223/√n
At a = 5%, criticial value = 1,358/ √n
At a = 1%, criticial value = 1,629/ √n
If oberved value D > criticial value, H0 is rejected

• Definition
• Combination of statistical hypothesis tests
• Parametric tests assume that sample data follow a probability
distribution based on a given set of parameters
• Two risks of error
• The type I error rate is the probability of rejecting the null hypothesis
given that it is true
• The type I error is the p-value (or significance level) a = 1%, 5% or 10%
• The type II error occurs when the null hypothesis is false, but is not
rejected
• The rate of the type II error (b is linked to the power of the test (1− b)
31
Mean-variance analysis

F-test of equality of
variances
Student t-test
Parameter
Null hypothesis
(H0)
Alternative
hypothesis (H1)
• Definition
• Combination of two statistical hypothesis tests on variance and
mean
32
21 mm 
21 mm 
22
1 2
ss 
22
1 2
ss 
1
1
2
2
22
1
2
11
1;1 21
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
n
Sn
n
Sn
F nn
   
 
2
11
21
21
2
22
2
11
2121
221
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
 nn
nn
SnSn
mmXX
T nn
When n1 = n2, 2
2
2
1
1;1 21
S
S
F nn 

• Parametric tests
• Decision rule
• F-test of equality of variances
• Student t-test
33
0
H0 is valid H0 is rejected
1 Critical value
at 10%
Critical value
at 5%
Critical value
at 1%
The value is more and more frequently calculated
• If p value < 1%, H0 is rejected at 1% (***)
• If p value < 5%, H0 is rejected at 5% (**)
• If p value < 10%, H0 is rejected at 10% (*)
H0 is valid H0 is rejected
Critical value
at 10%
Critical value
at 5%
Critical value
at 1%

• F-test of equality of variances
• Is the variance of Ri different from the variance of Rm?
34
F test to compare two variances
data: data$Ri and data$Rm
F = 0.99478, num df = 772, denom df = 772, p-value = 0.9421
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.8637999 1.1456325
sample estimates:
ratio of variances
0.994785
var.test(x = data$Ri,y = data$Rm)
The test
doesn’t follow
the convention
2
2
2
1 ss 
ifelse(var(data$Ri) > var(data$Rm), var(data$Ri) / var(data$Rm),
var(data$Rm) / var(data$Ri))

• Student t-test
• Is the mean of Ri different from the mean of Rm?
35
Paired t-test
data: data$Ri and data$Rm
t = -0.13619, df = 772, p-value = 0.8917
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-0.0005893457 0.0005128742
sample estimates:
mean of the differences
-3.823577e-05
t.test(x = data$Ri,y = data$Rm,paired = TRUE)

• Critics
36
Strenghts Weaknesses
The criticisms are
especially of
financial nature

• Critics
• Volatility is not stable over time
• Calculation of 20 day rolling volatility
37
#Computing 20 day rolling Std dev
library(roll)
data$SD_Ri<-roll_sd(data$Ri,width = 20)*sqrt(252)
data$SD_Rm<-roll_sd(data$Rm,width = 20)*sqrt(252)
#Converting date data to date format
date<-as.Date(data$Date,format = "%d/%m/%Y")
data<-cbind(date,data[,-1])
library(xts)
data<-xts(data[,2:6],order.by = data[,1])
#Drawing plot
windows()
plot(data$SD_Ri[23:773], xlab = "Date", ylab = "Annualized Std
Dev",type = "l", col = "green")
lines(data$SD_Rm,col = "red")
legend("topleft",c("Std dev of Ri","Std dev of Rm"),lty =
c(1,1),col = c("green","red"),bty = "n")
title(main = "20 day rolling std dev of Ri and Rm")

38
• Critics

• Critics
data<-read.csv2(file.choose(),header=T,sep=";",dec=",")
data$Ri[2:774]<-diff(log(data$Pi))
data<-data[-1,]
library(roll)
data$Ri<-as.matrix(data$Ri)
data$SD_Ri<-roll_sd(data$Ri,width = 20)*sqrt(252)
windows()
par(mfrow=c(2,1))
plot(data[,2],type="h",col="grey",xlab="Date",ylab="Price
(€)",ylim=c(round(min(data[,2])-5),round(max(data[,2])+5)),axes=F)
axis(1,pos=40)
axis(2,pos=0)
title(main="Price of i")
plot(data[,6],type="h",col="grey",xlab="Date",ylab="20 day rolling
volatility (%)",ylim=c(0,0.35),axes=F)
axis(1,pos=0)
axis(2,pos=0)
title(main="20 day rolling volatility of i")

• Critics
• Volatility is
not stable
over time

Conclusion
• Statistics is “the science of collecting, analyzing,
presenting, and interpreting data”
• Descriptive statistics summarize the population data
• The histogram and the normality test make it possible to evaluate
the adequacy between a variable and the statistical law of
reference
• The expectation variance analysis allows a comparison of the
variables
42

Conclusion
• Financial modelling seeks to improve the mathematical
representation of the behavior of securities
• Monofactorial models are the precursors
• Market model (Sharpe, 1963)
• Capital Asset Pricing Model (Lintner, 1965)
• Multifactor models try to integrate more variables
• Market timing (Treynor-Mazuy, 1966)
• Arbitrage pricing theory (Ross, 1976)
• Fama-French three factor model (1993)
• Carhart four factor model (1997)
43

References
Finance
• Bachelier L. (1900), Théorie de la spéculation, Annales scientifiques de l’ENS, (17)3, 21-86
• Carthart M. (1997), “On persistence of Mutual Fund Performance”, Journal of Finance, (52), 57-82
• Cowles A. (1933), “Can Stock Market Forecasters Forecast?”, Econometrica, (1)3, 309-324
• Cowles A. (1944), “Stock Market Forecasting”, Econometrica, (12)3-4, 206-214
• Fama E. (1970), “Efficient Capital Markets: A review of Theory and Empirical Work”, Journal of Finance,
(25)2, 383-417
• Fama E. (1991), “Efficient Capital Markets”, Journal of Finance, (46)5, 1575-1617
• Fama E., K. French (1993), “Common Risk Factors in the Returns on Stocks and Bonds”, Journal of
Financial Economics, (33), 3-56.
• Lintner J. (1965), “The valuation of risk assets and the selection of risky investments in stock portfolios and
capital budgets”, Review of Economics and Statistics, 47(1), 13–37
• Markowitz H. (1952), “Portfolio Selection”, Journal of Finance, (7)1, 77-91
• Ross S. (1976), "The Arbitrage Theory of Capital Asset Pricing". Journal of Economic Theory, (13)3, 341-
360
• Sharpe W. (1963), “A Simplified Model for Portfolio Analysis”, Management Science, (9)2, 277-293
• Treynor, J., Mazuy, K. (1966), “Can Mutual Funds Outguess the Market?” Harvard Business Review, (44),
131-136
Statistics
• Jarque C., Bera A. (1980). “Efficient tests for normality, homoscedasticity and serial independence of
regression residuals”, Economics Letters. (6) 3, 255–259
44

References
• Programming with R
45

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