Modelling returns using PCA : Evidence from Indian equity market

As my finance term paper, I investigated an interesting question where I tried to identify macroeconomic variables that explain the returns on equities. Much of the debate has already taken place on this topic which has given rise to two competing theories of asset pricing viz. CAPM (capital asset pricing theory or single factor model) and APT (arbitrage pricing theory or multi-factor model). Here is a brief discussion on the two in my previous post. In this post I would like to discuss my approach to answering this question in the context of Indian stock market.

Methodology:

• Companies that have been actively traded on NSE stock exchange for the past 10 years (218 companies) were selected and their daily stock returns data for these 10 years was taken from PROWESS. 
• Using PCA, first 10 components from the returns data of the 218 companies was extracted. More on PCA in my previous post, here. 
• These components were then separately regressed first on NIFTY returns (first regression) 
• Then these components were regressed on NIFTY returns, MIBOR rate changes, and INR/USD exchange rate changes (second regression).
• The explanatory power of the 2 regressions were compared using a F-statistic. (refer to pg. 10 in the paper attached in the end of the post)

Findings and R codes:
We start with calculating the PCA of the returns on the 218 companies daily return data, then employing the 2 regressions, then comparing the 2 regressions using a F-statistic. F-stat tells us if there is any additional explanation offered when we include macroeconomic variables (viz. MIBOR, INR/USD) in our equation.



The results that I obtained pose an interesting observation. We find that the F-stat is significant at 5% for 7 out of the 10 regressions, meaning that out of the 10 regressions (each regression with a separate component) we find statistically significant addition in the explanatory power of the model after adding the macroeconomic variables. Therefore, on statistical ground I can argue that a multi factor model (APT) is preferable over a single factor model (CAPM) for modelling stock returns in the case of Indian equity market. This assertion, if holds true, can have reaching implications for asset pricing for Indian securities. Let me explain why. The principal components (that are the dependent variables in the model) are essentially the common factor across all the companies stock returns with the idiosyncratic effects discounted, so any variables that explains this common component would be the systematic risk (think why!). Now we can relate it to the debate between the CAPM and APT guys. If the CAPM guys were correct, I would obtain no additional explanation in my model after adding the macroeconomic variables i.e their assertion that the market risk (market beta) capture the entire systematic risk holds true.

The results, however, suggest that in 7 out of 10 regressions there is statistically additional explanation offered by the macroeconomic variables. Well, so we can out-rightly reject the applicability (of the much prevalent) CAPM in the case of Indian equities. Or is there something amiss? Now if I closely look at the absolute increase in the explanatory power by looking at the Adjusted-R-squared values before and after the addition of the macroeconomic variables, the absolute increase in all the cases is < 1% (refer to pg. 11 in the paper at the end of this post). Therefore, although we obtain statistical efficiency after the addition of the variables, the economic efficiency (intuition) is called to question. Is it worth while to complicate our model with additional macroeconomic variables, when we can simply have the market rate used as a reasonable proxy for all the variables? And all this just to prove a point that we have macro-variables that can provide 0.5% additional explanation in our model? This takes us back to the eternal debate of statistical vs economic efficiency, what is more important? Is the above result robust enough (on economic intuition) to question the much used, simple and powerful CAPM? Is there a threshold even in statistical efficiency to ensure economic efficiency? These are some questions that still linger on in my mind.

If we view the above result with this caveat of economic efficiency then there is reason for us to believe that a single factor model would be a preferable way to model stock returns. There are, however, evidences in the literature to suggest that multi factor (APT) is a superior way of modelling returns, but the identification of these "multi factors" remains a contentious issue among the researchers. In some desperate attempts to refute CAPM, researcher extracted principal components from a number of macroeconomic variables as the input to the PCA. This resulted in factors that had no economic intuition at all, that were then used as independent variables in explaining the returns. The APT (Arbitrage pricing theory) is a 'theory', whereas CAPM is a 'model' that approximates reality. So even if in reality there are multiple factors that give rise to the returns signals as we see them, the identification of these factors is not a trivial exercise as we have seen above. Statistically we managed to overturn the CAPM in the context of Indian equity markets but in term of economic intuition the results do not seem to be that promising. Therefore, the above exercise tells us exactly why people still stick to the evergreen CAPM as an asset pricing model.

In case you wish to replicate the exercise the data can be obtained from here: Returns_CNX_500, Nifty_returns, MIBOR, Exchange_rates.

Here is the full text of my paper. Feedback are welcome. 

Modern Portfolio Optimization Theory: The idea

We were recently given a lecture (by Dr. Susan Thomas) on Harry Markowitz portfolio optimization theory, and I was really fascinating with the noble laureate's story of how he found it difficult to convince his guide about the importance of his thesis work. Little did anyone know that his thesis would get him the most respected award in academia 35 years down the lane. Let me try and illustrate what was the basic idea behind the modern portfolio theory in simple English language. The principal task of any financial advisory firm/individual is to try and solve the capital allocation problem of an agent, meaning how much money should be invested in what asset/security to get the desired rate of return with minimum risk. Lets try and visualize this problem of capital allocation from the individuals point of view. Suppose I want a 15% p.a rate of return on my investment and there is a security in the market that gives you an expected 15% annual rate of return with some risk(sigma or standard deviation) associated with that security/asset. One way for me to achieve the desired expected return is to invest in this security take the risk(sigma) associated with this security and hope that I am awarded for the risk I took and get a 15% return in the end. However, the beauty of financial markets is that since there are so many assets/securities being traded in the market, it is possible for me to diversify my investment, in simple words, park fraction of my money with different assets in a way that gives me approximately ~ 15% return but with substantially less risk. This is referred to as "diversification" in Finance parlance. The mathematics behind the results can be easily shown, one can find the maths in the lecture slides here. Let me try and illustrate this result through empirical simulation using R.

Suppose you are in a universe of only risky assets, i.e all assets that have a positive sigma. Then what Harry Markowitz did, with mathematics, was to compute the set of Expected returns that you could achieve with minimum risk (sigma). To illustrate this result I took stock returns for these 7 random companies for about 251 days, computed their respective expected returns, and the variance-co-variance matrix of the returns. With this I have all the necessary fodder to compute the E(r) and sigma for all the possible combination of weights that I can assign between these securities, meaning all possible different combinations of parking my money among these securities.

# Map your working directory using setwd()
# Read the relevant file
r1 <- read.csv("Markowitz.csv")


mu = colMeans(r1[,-7])  ## Calculate the column means and storing in "mu", I have removed the 7th column from the data frame as I have a risk free asset's return there, I will come back to this later in the post.


bigsig = cov(r1[,-7]) ## Variance-co-variance matrix


m = nrow(bigsig)-1
w = diff(c(0,sort(runif(m)), 1)); ## Assigning random weights between 0-1 to "w" which will have the dimension = no. of securities between which you have to divide your investment.


rb = sum(w*mu); ## Creating matrix "rb" which stores the E(r)


sb = sum(w*bigsig*w); ## Creating matrix "sb" which stores sigma's


N = 2000  ## Number of different combinations of "w" you want to look at


## Simulating the different combinations of weights "w's" and storing the E(r) and sigma^2

for (j in 2:N) {

w = diff(c(0,sort(runif(m)), 1));

r = sum(w*mu); rb = rbind(rb,r); 

s = sum(w*bigsig*w); sb = rbind(sb,s); ## Note this is sigma^2 (variance)

}

d = data.frame(rb, sb); ## Merge all the E(r) and sigmas in one data.frame.

d$sb = sqrt(d$sb); ## Square root the variance to get the sigmas

plot(d$sb, d$rb, ylab="E(r)", xlab="Sigma", col="blue",  xlim = c(0.5,10), ylim = c(-0.1,0.5), main = "E(r)- sigma (With risk free asset), N = 200000")

With only Risky assets

I did this simulation for 200000 different values of weights too, for those who are curious the plot can be obtained from here.

If we close the envelope of the above mentioned points with a smooth curve (from the left hand side) we get the efficient portfolio frontier (EFF). So what the graph suggests is that one would want to choose the weights (allocations) that lie, more specifically, on the above half of the curve. It makes intuitive sense too, why would one take on more risk to get a lesser return (which is implied by the lower half of the curve). Also another observation that we can make from the above plot is that for every additional unit of returns we will have to take on an increasingly higher risk (illustrated by the upper half of the plot). This risk-return trade off graph changes its characteristics when a risk free assets comes into the picture. Lets see what happens is we incorporate the risk free asset in our data frame. Note that a risk free assets could be thought of as a government security, whose risk (sigma) is 0, hence even its co-variances with the other risky assets would be 0. The only addition to the above codes would be to replace r1[,-7] with just r1 that is include the risk free asset too.

With risk-free asset

Simulation with N = 200000 here.


Look what happened to our risk-return trade off relationship, its linear!! So its a straight line instead of a smooth curve. Meaning for every additional unit of return you will have to take a fixed additional amount of risk, which will be given by the slope of the above half of the graph. This result, which sounds to be a trivial finding, can be extended to the entire universe of assets and one can compute the optimal portfolio that will give you a desired rate of return with minimum sigma. However the implementation of this methodology in the actual world is the real challenge. Computation of E(r), sigma, getting the up to date data for all the securities or assets in the universe are some of the challenges that one would face. But nevertheless it is inspiring to see how Harry Markowitz stuck to his idea and triumphed in the end with this path breaking research that shaped the financial advisory industry as we see it now.

P.S In case you want to replicate the exercise the data can be downloaded from here.

Principal component analysis : Use extended to Financial economics : Part 2

My previous post talked about how we can employ PCA on the data for multiple stock returns to reduce the number of variables in explaining the variance of the underlying data. But the idea was greeted with skepticism by many. A caveat to the application of PCA was that the meaning or the intuition behind the variable is lost in the computation of components, but since I was concerned with only explaining the variance in the underlying data it worked as a powerful method. This method was also adopted by Dr. Oyama in his research paper.

I came across this interesting research paper published in one of the most coveted journals, (International Journal on Finance and Economics) which has used the PCA on the macroeconomic variables. So relating it to my previous post and the question of investigating the factor affecting stock returns, what they do is that they take individual stocks and regresses it on the PCA of macroeconomic variables. At the first glance this would appear an inappropriate use of the tool, however the question that it intends to ask is also very different. There has been or infact still is this great battle between the Capital Asset Pricing (CAPM) school of thought and Arbitrage Pricing Theory (APT) school of thought. What the essential difference between these two schools is that the CAPM says that the returns to a stock are sensitive to only one factor, i.e the market rate of returns (single factor model) which captures the effect of all the various factors, however the APT guys scream that it depends on a number of factors and market rate of return would be just one of them. So in this battle to prove their point the APT folks take stock returns as a dependent variable and as independent variables they use the PCA of many macroeconomic variables. They do this essentially so that they can prove that there are more than one factors playing a significant part in the explaining the returns on a stock, they do not care whether the economic intuition is lost in the process on one-upmanship.

Well anything is fair in war, but I think this path taken by APT guys of proving their point is a fair point. The variables that are thrown into the PCA are chosen with an economic intuition in mind, so its not correct to say this entire methodology is flawed on the pretext that the PCA of these macroeconomic variables have no economics intuition. If one really digs into the intuition behind PCA as explained in this article one can visualize what the PCA of macroeconomics variables would be representing, the principal underlying components that give rise to such a macroeconomic series (signals). This is an unconventional way of doing an econometric study, as in this is more of a qualitative than a quantitative study. I cannot make sense of a statement that 1 unit increase in my first PCA results in "x" unit increase in returns, this is an absurd statement, but nevertheless the methodology of answering the underlying question is not absurd.

The methodology for calculating the PCA for macroeconomics is same as what was used in the previous post. We can have a number of macroeconomic variables, however, I have done this demonstration with only 2 variables change in Mumbai inter-bank offer rates (MIBOR) rates and change in INR/USD exchange rates.

#### Calculating the PCA of macroeconomic variables ####


# Reading the relevant files
mibor <- read.csv("MIBOR.csv", na.strings="#N/A")
exchange <- read.csv("Exchange_rates.csv", na.strings="#N/A")


# Making sure that there are no missing values in the data, the missing values are replaced by linear interpolation
mibor[, 2] <- approx(as.Date(mibor$Dates, '%d-%b-%y'), mibor[ ,2], as.Date(mibor$Dates, '%d-%b-%y'))$y   # approx() returns the interpolated values in column 'y'


exchange[, 2] <- approx(as.Date(exchange$Year,'%d-%b-%y'), exchange[ ,2], as.Date(exchange$Year, '%d-%b-%y'))$y  # Similarly for exchange rates


# Now we will have to compute the change in MIBOR and exchange rates:
for(k in 2:nrow(mibor))
{
  mibor$Change1[k] <- ((mibor$MIBOR[k] - mibor$MIBOR[k-1])/mibor$MIBOR[k-1])*100
}


for(j in 2:nrow(exchange))
{
exchange$Change[j] <- ((exchange$Exchange.rates[j] - exchange$Exchange.rates[j-1])/exchange$Exchange.rates[j-1])*100
}


# Creating matrix of the data
macro <- as.data.frame(rep(0, 2498))   


macro$ex <- exchange$Change
macro$rate <- mibor$Change1

macro.mat <- as.matrix(macro, nrow = 2498 , ncol = 2)

# Calculating the principal components 

prin.macro <- princomp(macro.mat)

barplot(prin.macro$sdev[1:2]/prin.macro$sdev[1])  # bar plot of the ratio is the standard deviation of the PC's with the standard deviation of the 1st PC, this can help us decide which PC to be used by simple eyeballing the plot.

# Extract the 1st PC in a variable

load1 <- prin.macro$loadings[,1]   # get the loadings of the 1st PC into 'load1'

pr.ma <- macro.mat %*% load1 # matrix multiply the loading with the original data to get the PC

pr1 <- as.numeric(pr.ma) # And 'pr1' has your first principal component. 

Now if you have read this article that I shared earlier, the first component gives me the principal underlying factor that gives rise to the series (signals) that I observe.

Caveats and cautions:

Before I am bombarded with question about the relevance of this methodology in this particular problem let me confess that in the above particular exercise its not the best idea to use PCA. One should use PCA only if one has many macroeconomic variables like M1, M3, CPI, IIP, WPI, etc and there is high correlation between them. I have done the above only for illustrative purposes (since I am not employing PCA on macro-variables in my project). 

Assuming I had other macro-variables in the calculation of my PCA above, I could use them as an independent variable and regress them on the stock specific returns to understand  if there are more than one factors that come out to be significant in this regression. What this would essentially mean is that are there more than one factors explaining the stock returns. For the finance students, this is the classic argument between the Capital Asset Pricing (CAPM) guys and the Arbitrage Pricing Theory (APT) guys. 

If in the regression of the PC's of macroeconomic variables on the returns we find that more than one factors are significant then we would have found evidence for or against one of the above theories.

Limitations in the above method:

1. In the above exercise I have just taken 2 variables merely for illustration, I would have to include more variables if I really intent to find evidences against the CAPM.
2. Again the caveat of interpretation applies. Its difficult to interpret the PCA on macrovariables

My take:

Despite of the above caveats I think PCA does a good job of giving us a good empirical way to answer the question of whether CAPM or APT is a superior asset pricing theory. The main idea behind these asset pricing theories is to see what factors need to be incorporated to compute the asset specific risk premiums, so on that front, its not difficult to see what people still stick to CAPM, as its a lot simple and easier to work with. APT on the other hand makes a bold assertion of multiple factors governing the returns but it has always found it difficult to list out these factors. Well in most cases you would be able to conclude that the returns can be modeled best using a multi-factor model, but whats interesting to ask is how much additional explanation is offered by factors other than the 1st one. If my multi factor model offers just a marginally higher adjusted-R-squared, is it reasonable for you to complicate your model is the question?

If it were only to prove this empirically why would you not choose the multi-factor model? The idea of modelling returns is to identify the factors that you need to take into account while computing the risk premiums, and if your factors do not have any economic intuition how can you assign risk premia looking at your principal component, that's the big question. But I would hate to impose my take here, further discussion is welcome.

P.S : In case you wish to replicate the above exercise you can find the data here, exchange rates, MIBOR. 

Principal component analysis : Use extended to Financial economics : Part 1

While working for my Financial economics project I came across this elegant tool called Principal component analysis (PCA)which is an extremely powerful tool when it comes to reducing the dimentionality of a data set comprising of highly correlated variables. This tool finds majority application in genetic research, which deals with data sets having many variables that are highly correlated.


I will try and be as explicit and refrain from using statistical/mathematical jargons to explain what/how about this tool . To state a few stylized facts PCA is used mainly for:

•  compressing the data
•  filter some of the noise in the data

Problem at hand:

I was trying to investigate the factors that affect the returns of stocks in the Indian equity market, however I wanted to take into account all the S&P CNX 500 companies. What would be really nice if I could somehow find a way of squeezing the 500 companies into say not more than 2-3 variables that can be representative of the entire set of 500 companies. This is precisely where PCA comes into play and does a fantastic job. What it gives me is just 1 variable that I can use instead of all the 500 companies!!!

Hats off and a bow of respect for the contributors/donors of packages to the CRAN servers that the above simplification can be achieved using just one line of script in R. Sounds easy, but what one really needs to do is to understand what PCA does and how the output from this script can be interpreted. Again, at the risk of over simplication (however trying hard to maintain my commandment of simplicity), I would illustrate in a crude manner the working of PCA.

What PCA does:

Let me explain this relating to the above example, if I do a PCA on the returns data for the 500 companies, I would obtain 500 principal components. These components are nothing but the linear combination of the existing 500 variables(companies) arranged in the decreasing order of their variance. So 1st principal component (PC)has the maximum variance and 500th principal component (PC)has the least variance. The variance in the PCA represent nothing but the variance in the data. So 1st PC explains the maximum amount of variance in my data. One magical feature of PCA is that all these 500 components will be orthogonal to each other, meaning these components will be uncorrelated with each other. So essentially if we look at PCA as a black box it takes inputs as data set of highly correlated variables and gives as output PC's that explain the variance in the input data and they are uncorrelated with each other.(I don't leverage this feature in this particular problem, I would illustrate this use in other part of this blog)

How PCA does it:

Since I have taken a vow of simplicity, I dont have much to say here.:-) However for the mathematically inclined and certainty freaks like Madhav, this paper does a brilliant job of illustrating the matrix algebra that goes behind PCA computations. There are essentially 2 methods of calculating PCA, one is the eigenvalue decomposition (done using princomp() command in R)and the other is singular value decomposition (done using prcomp() command using R). 

How this can be done in R:

####### Calculating Principal component of returns of S&P CNX 500 companies ########

## Access the relevant file ##

returns <- read.csv("Returns_CNX_500.csv")

One caveat that you need to keep in mind in that there should be no "NA" values in your data set. A presence of an NA would impede the computation of the var-covar matrix and hence their eigen vectors(i.e the factor loadings)

## Dealing with missing values in the returns data for companies 

for(i in 2:ncol(returns))

{

  returns1[, i] <- approx(returns$Year, returns1[ ,i], returns$Year)$y  ## approx function basically fits the value of linear approximate between the missing data points and the column $y stores the approximated values.

}    

## Convert the data into matrix ##

ret <- as.matrix(returns1, nrow = dim(returns1)[1], ncol = dim(returns1)[2])

##Computing the principal component using eigenvalue decomposition ##

princ.return <- princomp(ret) ## This is it.!!

## Identifying what components to be used ##

barplot(height=princ.return$sdev[1:10]/princ.return$sdev[1])   ## I am plotting the standard deviation of the PC's divided by standard deviation of PC 1, this can help us decide on a benchmark that we can use to select the relevant components.

Standard deviation of the first 10 components compared to 1st PC

We can clearly see from the above figure that as expected the first PC does the majority of the variance explanation in the returns data for the 500 companies. So if we want to identify factors that influence the returns of S&P CNX 500 companies I can use the 1st PC as a variable in my regression. So far we have calculated the principal components, now we will extract out 1st PC as a numeric variable from the matrix.(princ.return)

## To get the first principal component in a variable ##

load <- loadings(princ.return)[,1]   ## loadings() gives the linear combination by which our input variables will be linearly weighted to compute the components, and this command gives us the loading for 1st PC.

pr.cp <- ret %*% load  ## Matrix multiplication of the input data with the loading for the 1st PC gives us the 1st PC in matrix form. 

pr <- as.numeric(pr.cp) ## Gives us the 1st PC in numeric form in pr.

One question that might be raised is why not just use the S&P CNX 500 index returns as an input to the regression? The simple answer to that question would be that PC 1 gives you a relatively clear signal of the returns as opposed to the index which would have a lot of noise. This question would have made sense in the 1900's when the technology was not so efficient in terms of computation. Since now computational time and effort finds minimum weight in any researchers mind there is no reason to settle for anything but the best.

There is an important caveat that must be kept in mind while doing analysis using PCA, though PCA has a clear mathematical intuition it lacks an economic intuition. That is, one unit change in PC 1 of returns has a mathematical meaning but no economic meaning, you cannot make sense of this statement that PC 1 of returns for the 500 companies has gone up by "x" amount. Therefore the use of this analysis should be limited to factor analysis and not to be extended to predictive analysis.

In case you wish to replicate the above exercise the data can be obtained from here.