Category Archives: financial econometrics

Statistical models for risk management and asset management.

1 Missing stock price data

When downloading historic stock price data it happens quite frequently that some observations in the middle of the sample are missing. Hence the question: how should we cope with this? There are several ways how we could process the data, each approach coming with its own advantages and disadvantages, and we want to compare some of the most common approaches in this post.

In any case, however, we want to treat missing values as NA and not as Julia’s built-in NaN (a short justification on why NA is more suitable can be found here). Hence, data best should be treated through DataFrames or – if the data comes with time information – through type Timenum from the TimeData package. In the following, we will use these packages in order to show some common approaches to deal with missing stock price data using an artificially made up data set that shall represent logarthmic prices.

The reason why we are looking at logarthmic prices and returns instead of normal prices and net returns is just that logarithmic returns are defined as simple difference between logarithmic prices of successive days:

$\displaystyle r_{t}^{\log}=\log(P_{t}) - \log(P_{t-1})$

This way, our calculations simply involve nicer numbers, and all results equally hold for normal prices and returns as well.

We will be using the following exemplary data set of logarthmic prices for the comparison of different approaches:

using TimeData
using Dates
using Econometrics

df = DataFrame()
df[:stock1] = @data([100, 120, 140, 170, 200])
df[:stock2] = @data([110, 120, NA, 130, 150])

dats = [Date(2010, 1, 1):Date(2010, 1, 5)]


 idx stock1 stock2 2010-01-01 100 110 2010-01-02 120 120 2010-01-03 140 NA 2010-01-04 170 130 2010-01-05 200 150

One possible explanation for such a pattern in the data could be that the two stocks are from different countries, and only the country of the second stock has a holiday at January the 3rd.

Quite often in such a situation, people just refrain from any deviations from the basic calculation formula of logarithmic returns and calculate the associated returns as simple differences. This way, however, each missing observation NA will lead to two NAs in the return series:

simpleDiffRets = originalPrices[2:end, :] .- originalPrices[1:(end-1), :]

 idx stock1 stock2 2010-01-02 20 10 2010-01-03 20 NA 2010-01-04 30 NA 2010-01-05 30 20

For example, this also is the approach followed by the PerformanceAnalytics package in R:

library(tseries)
library(PerformanceAnalytics)

stockPrices1 <- c(100, 120, 140, 170, 200)
stockPrices2 <- c(110, 120, NA, 130, 150)

## combine in matrix and name columns and rows
stockPrices <- cbind(stockPrices1, stockPrices2)
dates <- seq(as.Date('2010-1-1'),by='days',length=5)
colnames(stockPrices) <- c("A", "B")
rownames(stockPrices) <- as.character(dates)
(stockPrices)

returns = Return.calculate(exp(stockPrices), method="compound")

 nil nil 20 10 20 nil 30 nil 30 20

When we calculate returns as the difference between successive closing prices $P_{t}$ and $P_{t-1}$, a single return simply represents all price movements that happened at day $t$, including the opening auction that determines the very first price at this day.

Thinking about returns this way, it obviously makes sense to assign a value of NA to each day of the return series where a stock exchange was closed due to holiday, since there simply are no stock price movements on that day. But why would we set the next day’s return to NA as well?

In other words, we should distinguish between two different cases of NA values for our prices:

1. NA occurs because the stock exchange was closed this day and hence there never were any price movements
2. the stock exchange was open that day, and in reality there were some price changes. However, due to a deficiency of our data set, we do not know the price of the respective day.

For the second case, we really would like to have two consecutive values of NA in our return series. Knowing only the prices in $t$ and $t+2$, there is no way how we could reasonably deduce the value that the price did take on in $t+1$. Hence, there are infinitely many possibilities to allocate a certain two-day price increase to two returns.

For the first case, however, it seems unnecessarily rigid to force the return series to have two NA values: allocating all of the two-day price increase to the one day where the stock exchange was open, and a missing value NA to the day that the stock exchange was closed doesn’t seem to be too arbitrary.

This is how returns are calculated by default in the (not yet registered) Econometrics package.

logRet = price2ret(originalPrices, log = true)

 idx stock1 stock2 2010-01-02 20 10 2010-01-03 20 NA 2010-01-04 30 10 2010-01-05 30 20

And, the other way round, aggregating the return series again will also keep NAs for the respective days, but otherwise perform the desired aggregation. Without specified initial prices, aggregated prices will all start with value 0 for logarithmic prices, and hence express something like normalized prices that allow a nice comparison of different stock price evolutions.

normedPrices = ret2price(logRet, log = true)

 idx stock1 stock2 2010-01-01 0 0 2010-01-02 20 10 2010-01-03 40 NA 2010-01-04 70 20 2010-01-05 100 40

To regain the complete price series (together with a definitely correct starting date), one simply needs to additionally specify the original starting prices.

truePrices = ret2price(logRet, originalPrices, log = true)

 idx stock1 stock2 2010-01-01 100 110 2010-01-02 120 120 2010-01-03 140 NA 2010-01-04 170 130 2010-01-05 200 150

In some cases, however, missing values NA may not be allowed. This could be a likelihood function that requires real values only, or some plotting function. For these cases NAs easily can be removed through imputation. For log price plots, a meaningful way would be:

impute!(truePrices, "last")

 idx stock1 stock2 2010-01-01 100 110 2010-01-02 120 120 2010-01-03 140 120 2010-01-04 170 130 2010-01-05 200 150

However, for log returns, the associated transformation then would artificially incorporate values of 0:

impute!(logRet, "zero")

 idx stock1 stock2 2010-01-02 20 10 2010-01-03 20 0 2010-01-04 30 10 2010-01-05 30 20

As an alternative to replacing NA values, one could also simply remove the respective dates from the data set. Again, there are two options how this could be done.

First, one could remove any missing observations directly in the price series:

originalPrices2 = deepcopy(originalPrices)
noNAprices = convert(TimeData.Timematr, originalPrices2, true)

 idx stock1 stock2 2010-01-01 100 110 2010-01-02 120 120 2010-01-04 170 130 2010-01-05 200 150

For the return series, however, we then have a – probably large – multi-day price jump that seems to be a single-day return. In our example, we suddenly observe a return of 50 for the first stock.

logRet = price2ret(noNAprices)

 idx stock1 stock2 2010-01-02 20 10 2010-01-04 50 10 2010-01-05 30 20

A second way to eliminate NAs would be to remove them from the return series.

logRet = price2ret(originalPrices)
noNAlogRet = convert(TimeData.Timematr, logRet, true)

 idx stock1 stock2 2010-01-02 20 10 2010-01-04 30 10 2010-01-05 30 20

However, deriving the associated price series for this processed return series will then lead to deviating end prices:

noNAprices = ret2price(noNAlogRet, originalPrices)

 idx stock1 stock2 2010-01-01 100 110 2010-01-02 120 120 2010-01-04 150 130 2010-01-05 180 150

as opposed to the real end prices

originalPrices

 idx stock1 stock2 2010-01-01 100 110 2010-01-02 120 120 2010-01-03 140 NA 2010-01-04 170 130 2010-01-05 200 150

The first stock now suddenly ends with a price of only 180 instead of 200.

2 Summary

The first step when facing a missing price observation is to think whether it might make sense to treat only one return as missing, assigning the complete price movement to the other return. This is perfectly reasonable for days where the stock market really was closed. In all other cases, however, it might make more sense to calculate logarithmic returns as simple differences, leading to two NAs in the return series.

Once there are NA values present, we can chose between three options.

2.1 Keeping NAs

Keeping NA values might be cumbersome in some situations, since some functions might only be working for data without NA values.

2.2 Replacing NAs

In cases where NAs may not be present, there sometimes might exist ways of replacing them that perfectly make sense. However, manually replacing observations in some way means messing with the original data, and one should be cautious to not incorporate any artificial patterns this way.

2.3 Removing NAs

Obviously, when dates with NA values for only some variables are eliminated completely, we simultaneously lose data for those variables where observations originally were present. Furthermore, eliminating cases with NAs for returns will lead to price evolutions that are different to the original prices.

2.4 Overview

Possible prices:

 idx simple diffs single NA replace \w 0 rm NA price rm NA return 2010-01-01 100, 110 100, 110 100, 100 100, 110 100, 110 2010-01-02 120, 120 120, 120 120, 120 120, 120 120, 120 2010-01-03 140, NA 140, NA 140, 120 2010-01-04 170, 130 170, 130 170, 130 170, 130 150, 130 2010-01-05 200, 150 200, 150 200, 150 200, 150 180, 150

Possible returns:

 idx simple diffs single NA replace \w 0 rm NA price rm NA return 2010-01-02 20, 10 20, 10 20, 10 20, 10 20, 10 2010-01-03 20, NA 20, NA 20, 0 2010-01-04 30, NA 30, 10 30, 10 50, 10 30, 10 2010-01-05 30, 20 30, 20 30, 20 30, 20 30, 20

Real world quantitative risk management: estimation risk and model risk

Often, when we learn about a certain quantitative model, we solely focus on deriving quantities that are implied by the model. For example, when there is a model of the insurance policy liabilities of an insurance company, usually the only issue is the derivation of the risk that is implied by the model. Implicitly, however, this is nothing else than the answer to the question: “Given that the model is correct, what is it that we can deduce from it?” One very crucial point in quantitative modeling, however, is that in reality, our model will never be completely correct!

This should not mean that quantitative models are useless. As George E. P. Box has put it: “Essentially, all models are wrong, but some are useful.” Deducing properties from wrong models, like for example value-at-risk, may still help us to get a quite good notion about the risks involved. In our measurement of risk, however, we need to take into account one additional component of risk: the risk that our model is wrong, which would make any results deducted from the model wrong as well. This risk arises from the fact that we do not have full information about the true underlying model. Ultimately, we can distinguish between three different levels of information.

First, we could know the true underlying model, and hence the true source of randomness. This way, exact realizations from the model are still random, but we know the true probabilities, and hence can perfectly estimate the risk involved. This is how the world sometimes appears to be in textbooks, when we only focus on the derivation of properties from the model, without ever reflecting about whether the model really is true. In this setting, risk assessment simply means the computation of certain quantities from a given model.

On the next level, with slightly less information, we could still know the true structure of the underlying model, while the exact parameters of the model now remain unobserved. Hence, the parameters need to be estimated from data, and the exact probabilities implied by the true underlying model will not be known. This way, we will not be able to manage risks exactly the way that we would like to, as we are forced to act on estimated probabilities instead of true ones. In turn, this additionally introduces estimation risk into the risk assessment. The actual size of estimation risk depends on the quality of data, and in principle could be eliminated through infinite sample size.

And, finally, there is the level of information that we encounter in reality: neither the structure of the model nor the parameters are known. Here, large sample sizes might help us to detect the deficiencies of our model. Still, however, we will never be able to retrieve the real true data generating process, so that we have to face an additional source of risk: model risk.

Hence, quantitative risk management is much more than just calculating some quantities from a given model. It is also about finding realistic models, assessing the uncertainty associated with unknown parameters, and validating the model structure with data. We now want to clarify the consequences of risk management under incomplete information about the underlying probabilities. Therefore, we will look at an exemplary risk management application, where we gradually loosen the assumptions towards a more realistic setting. The general setup will be such that we want to quantify the risk of an insurance company under the following simplifying assumptions:

• fix number of policy holders
• deterministic claim size

Quantitative Risk Management: Value-at-risk for discrete random variables

Insolvency occurs, when the total value of all assets is smaller than the value of liabilities. And, due to deviations of assets and liabilities from their expected values, any financial institution does face some risk of insolvency. It is the purpose of risk management to correctly estimate this risk, and to adjust the portfolio composition to appropriately match the company’s risk appetite. Here, we will derive the definition of value-at-risk – presumably the most important risk measure at present – and apply it to an exemplary setting with discrete random variables.

Analyzing volatility patterns for SP500 stocks

Now that we already have a cleaned high-dimensional stock return dataset at our hands, we want to use it in order to examine stock market volatility in a high-dimensional context. Thereby, we are searching for patterns in the joint evolution of individual stock volatilities. We have in mind the following goal: to reduce the high-dimensional set of individual volatilities to some low-dimensional common factors.

What we will find is that all of the assets’ volatilities seem to follow some common market trend of volatility. However, the variation explained through this market trend is about 50 percent only. Additionally, we find that, during each financial crisis, the one sector where the crisis originates will show levels of volatility above the market trend. Our sample contains two examples for this: the IT sector during the dot-com crash, and the financial sector during the financial crisis in 2008. However, based on findings of principal component analysis, there still is a large part of variation in volatilities that remains unexplained by common factors. This indicates that we should think about volatility as consisting of two parts: one part arising from common factors, and one part that is specific to the individual asset.

1 Motivation

In a portfolio context, individual asset volatility levels are probably still the most important risk factor, together with the components’ dependence structure. However, in large dimensional portfolios, it becomes extremely difficult to keep track of all of the individual volatility levels. Hence, if there was some pattern inherent in the joint evolution of volatilities, this could enable us to simplify risk management approaches through reduction of dimensionality. Whenever multiple stochastic variables show some common patterns, this is caused by dependency. And, given that this dependency is large enough, there is some redundancy in keeping track of all of the variables separately. A better way then would be to relate the components of the high-dimensional problem to some lower dimensional common factors.

So, why should there be any patterns in the joint evolution of individual volatilities? Well, as a first indication for this we can take financial crises, where usually almost every individual stock will exhibit an increased level of volatility. Slightly related to this, we later examine the explanatory power of shifts in mean market volatility. The justification for this is that usually market volatility indices react quite sensitive to financial crises, with index levels high above normal times. And, in turn, volatility indices show large dependency with mean market volatility levels. This way, mean market volatility can be seen as a proxy for financial crises.

When analyzing volatility, we are not dealing with a directly observable quantity. Hence, the first step of the analysis will be to derive some estimates for the individual asset volatility time paths. Of course, any shortcomings that we bring in at this step will directly influence the subsequent analysis. Nevertheless, we will refrain from overly sophisticated approaches for now, since we only want to get a feeling about the data first. Hence, we will derive estimates for individual stock volatilities simply by assuming a GARCH(1,1) process for each individual stock. If necessary, this assumption could easily be relaxed, and volatility paths could be estimated by different volatility models like EGARCH or some multi-dimensional extensions. Or, we could refrain from hard model assumptions, and simply estimate volatility with high-frequency data as realized volatility.

2 Estimation of volatility paths with GARCH(1,1)

As already explained, we first need to derive some estimates of the unobservable volatility evolutions over time. Hence, we will now estimate a GARCH(1,1) process with $t$-distributed innovations for each of the individual assets of our S&P500 dataset. Therefore, we make use of the garchFit function from the fGarch package. As we are primarily interested in the derivation of the unobservable volatility paths, we will neither take a look at the estimated parameters, nor save them to disk. The only two things that we extract are the estimated sigma series and the standardized residuals, which are the residuals divided by their respective estimated volatility.

## clean workspace
rm(list=ls())

library(zoo)
library(fGarch)

logRet <- coredata(z)

####################
## fit GARCH(1,1) ##
####################

## get dimensions
nAss <- length(logRet[1, ])
nObs <- length(logRet[, 1])

## preallocation
stdResid <- data.frame(matrix(data=NA, nrow=nObs, ncol=nAss))
vola <- data.frame(matrix(data=NA, nrow=nObs, ncol=nAss))

## fitting procedure
for (ii in 1:nAss) {
## display current iteration step
print(ii)

## sometimes garchFit failed
result <- try(gfit <- garchFit(data=logRet[, ii], cond.dist = "std"))
if(class(result) == "try-error") {
next
}
else {
## extract relevant information
stdResid[, ii] <- gfit@residuals / gfit@sigma.t
vola[, ii] <- gfit@sigma.t
}
}

## label results
colnames(stdResid) <- colnames(logRet)
rownames(stdResid) <- index(z)
colnames(vola) <- colnames(logRet)
rownames(vola) <- index(z)

## round to five decimal places to reduce file size
stdResid <- round(stdResid, 5)
vola <- round(vola, 5)

##################
## data storage ##
##################

write.csv(stdResid, file = "data/standResid.csv", row.names=T)
write.csv(vola, file = "data/estimatedSigmas.csv", row.names=T)

3 Visualization of volatility paths

Now that we have an estimate for each asset’s volatility path, we want to get a first impression about the data through visualization. Therefore, we now will set up our workspace first, and load required data and packages.

######################
## set up workspace ##
######################

library(zoo)
library(reshape)
library(ggplot2)
library(fitdistrplus)
library(gridExtra)
library(plyr)

## clean workspace
rm(list=ls())

## source multiplot function
source("rlib/multiplot.R")

## load cleaned logarithmic percentage returns as dataframe
names(logRetDf)[names(logRetDf) == "Index"] <- "time"

## get dimensions and dates
nObs <- nrow(logRetDf)
nAss <- ncol(logRetDf)-1
dates <- as.Date(logRetDf$time) ## load volatilities volaDf <- read.csv(file = "data/estimatedSigmas.csv", header = T) names(volaDf)[names(volaDf) == "X"] <- "time" As first visualization, we now plot all volatility paths that where estimated with GARCH(1,1). Therefore, we define the function plotLinePaths, which we can reuse later on. ############################################## ## make multi-line plot of individual paths ## ############################################## ## plot multi-line path of df with column "time" or row.names plotLinePaths <- function(data, alpha=0.2){ "plot dataframe with or without time column" ## test for time column and rename if required if(any(colnames(data) == "time") | any(colnames(data) == "Time")){ if(any(colnames(data) == "Time")){ ## rename to lower case names(data)[names(data)=="Time"] <- "time" } }else{ ## take row.names as x-index data$time <- row.names(data)
}

## convert index to date format
data$time <- as.Date(data$time)

## melt data to use ggplot
dataMt <- melt(data, id.vars = "time")

## create plot
p <- (ggplot(dataMt) +
geom_line(aes(x=time, y=value, group=variable),
alpha = alpha))
}

## calculate mean volatility
colnames(meanVola) <- c("time", "value")
meanVola$time <- dates plotOnly <- plotLinePaths(volaDf,0.2) p1 <- plotOnly + xlab("time") + ylab("volatility") p1 <- p1 + geom_line(mapping = aes(x=time, y=value), data = meanVola, color = "red") As can be seen, this graphics is not very enlightening, as the volatility paths remain in a very small interval at the bottom most of the time. Hence, we will immediately proceed by looking at logarithmic volatility paths, so that differences at the bottom will get shown in higher resolution. ############################################# ## plot log-transformation of volatilities ## ############################################# logVola <- data.frame(time=dates, log(as.matrix(volaDf[, -1]))) p2 <- plotLinePaths(logVola, 0.2) + xlab("time") + ylab("logarithmic volatility") + geom_line(mapping = aes(x=time, y=log(value)), data = meanVola, color = "red") p2 What appeared as one big black hose of nearly constant height before, now exhibits some shifts in its level over time. Clearly, times of crises do stand out from the plot: both in the aftermath of the dot-com bubble and during the financial crisis 2008, volatility paths were significantly above average levels. Even more remarkable is the fact that not only most volatility paths and their mean value are affected, but even the minimum of all volatility paths shifts significantly upwards. Hence, this is most likely not only a phenomenon that holds for most assets, but it really seems to affect all assets without exception. Furthermore, we observe some noisy border at the top, where there are assets that individually enter high-volatility regimes in calm market periods as well. However, this also could be some artificial effect due to the estimation of volatilities with GARCH. Due to the structure of GARCH, a single large return of an asset will always automatically let its volatility estimate spike up. In contrast to the upper border, the lower border of all volatilities seems rather stable, without extreme outliers. This is at large due to the fact that volatility is bounded downwards by zero, and even the logarithmic transformation is not strong enough in order to fully compensate for this. Still, however, even with logarithmic volatility paths, it is not possible to get more detailed insights into the individual evolutions, as they still overlap too much. This way, it is not possible to draw any conclusion of whether, for example, most paths are rather above the mean volatility, so that volatility levels would be asymmetrically distributed. Therefore, we now will take a look at the empirical quantiles of the volatilities, also. Again, we show them on normal scale, as well as on logarithmic scale. ############################# ## get empirical quantiles ## ############################# ## specify quantiles alphas <- c(0, 0.05, 0.25, 0.5, 0.75, 0.95, 1) nQs <- length(alphas) ## preallocation of empirical quantiles quants <- data.frame(matrix(data=NA, nrow=nObs, ncol=nQs), row.names=as.Date(dates)) names(quants) <- as.character(alphas) ## for each day get empirical quantiles for (ii in 1:nObs) { quants[ii, ] <- quantile(volaDf[ii, -1], probs = alphas) } ## process results for plotting quantsDf <- data.frame(time=as.Date(dates), quants) names(quantsDf) <- c("time", as.character(alphas)) logQuantsDf <- data.frame(time=as.Date(dates), log(quants)) names(logQuantsDf) <- c("time", as.character(alphas)) p3 <- plotLinePaths(quantsDf, 0.99) + ylab("quantiles of volatilities") p4 <- plotLinePaths(logQuantsDf, 0.99) + ylab("quantiles of logarithmic volatilities") p5 <- multiplot(p1, p2, p3, p4, cols=2) Besides the variation in height, the logarithmic quantiles look quite well-behaved and almost symmetric around the median. The symmetry only seems to fade out at the 5 and 95 percent quantiles. This could still be related to the fact that volatility, on a normal scale, is bounded from below by zero, while it is unbounded from above. Again, the shift in height due to crises periods also becomes apparent for the empirical quantiles, and again it looks like all quantiles are affected similarly. Looking at this picture, one might be tempted to conclude that all individual volatility paths strongly follow some overall market volatility, which is represented through the median in this case. However, this conclusion is premature at this point. Even if we can get a quite good guess about the overall distribution of volatilities once we know the market volatility, it is still unsure how much this will tell us about any individual stock. We will further clarify what is meant by this in some later part. For now, we first want to look at an at least somewhat finer resolution than the complete overall distribution of all assets. Therefore, we will split up individual assets into groups. This way, we decrease overlap of volatility paths, as the total number of paths becomes significantly lower. Hence, patterns and changes will become better visible. The division into groups that we will apply here shall follow the classification of assets provided by sector affiliation. Hence, for each of the remaining assets in our SP500 dataset, we need to determine its sector. Usually, you can access this information from any big proprietary data provider. Here, we will rely on the classification provided by Standard & Poor’s, where you can download a list of constituents for almost all sector indices. I already brought this information together in a csv file, where each of the remaining assets of our dataset is listed together with its sector. However, by the time of writing, Standard & Poor’s did not provide lists of constituents for sectors materials, telecommunication services and utilities. Hence, in the csv file, any asset that was not mentioned in one of the other sectors simply bears the label “matTelUtil”. In order to plot volatility paths for the individual sectoral subgroups, we need to calculate the mean value of each subgroup for each day of our sample period. This task exactly matches the split-apply-combine procedure that Hadley Wickham describes in [Wickham], and that he implemented in the plyr package. Using this package, we now can easily split the data into subgroups determined by date and sector, and apply the meanValue function to each group. ## refine volatility for incorporation of sectors volaMt <- melt(volaDf, id.vars="time") names(volaMt) <- c("time", "symbol", "value") ## load sectors sectors <- read.table(file="data/sectorAffiliation.csv", header=T) ## merge volaSectorsMt <- merge(volaMt, sectors, by="symbol") volaSectors <- volaSectorsMt[names(volaSectorsMt) != "symbol"] ## define function to get mean on each subgroup meanValue <- function(data){ meanValue <- mean(data$value)
}

## for each day and sector, get mean volatility
sectorVola <- ddply(.data=volaSectors,
.variables=c("time", "sector"), .fun=meanValue)
sectorVola$time <- as.Date(sectorVola$time)

p <- ggplot(sectorVola) + geom_line(aes(x=time, y=V1, color=sector))
p

As we can see, at large, all sectors follow some joint market volatility evolution. However, there exist some slight differences over time. From 1995 until 2005, the information technology sector did exhibit the highest volatility of all sectors, while its volatility level is rather average afterwards. Then, in 2005, the energy sector temporarily becomes the sector with highest volatility, before it gets replaced again by the financial sector in 2008. Remarkably, although the financial sector exhibits large volatility in the last third of the sample, its volatility level did reside near the bottom of all sector volatilities for more than the first half. Hence, there clearly is some further variation in volatility levels besides the variation that is caused through up- and downwards shifting market volatility. The relative position of individual sectors changes over time: sectors that where beneath the market volatility in the first half, suddenly show higher levels than average in the second half. Whether this influences the explanatory power of the mean market volatility is what we want to see in the next part.

4 Explanatory power of shifting mean

So far, we have seen that there is some large variation in the level of stock volatilities during periods of market turmoil. In these periods, not only the average market volatility increases, but even the minimum of all volatilities increases. Hence, knowledge of the market volatility clearly reveals some information about the current distribution of volatilities. However, the market volatility might still reveal very little information about any given single volatility.

Let’s just assume for a second, that individual logarithmic volatilities are normally distributed around the market volatility. Then, with knowledge of the market volatility, you also know the overall distribution. However, without any further assumptions, you still do not know anything about the individual volatilities within this distribution. Any given single stock still could be anywhere around the mean, provided that it follows the probabilities determined by the normal distribution. You only know that it will be within a range with a given probability, but you still can not predict whether it is above or below the mean. Hence, without further structure, the market volatility might still be a bad estimator for any given single volatility.

However, through imposing some additional structure, one could easily extend this example into a system, where the mean of the normal distribution tells you a lot about the individual stocks as well. Let’s now additionally assume that the individual volatilities follow some ranking. Hence, the volatility of some stock A now will always be larger than the volatility of some stock B. Suddenly, we now will be able to pin down the exact location of any individual volatility very accurately. Let’s just convince ourselves of this effect with a little simulated example.

For that, we will artificially create a first increasing and then decreasing sequence of (logarithmic) market volatility levels. This can be achieved by letting the market volatility follow the first half of a sine oscillation. From this course of oscillation, we extract 100 points in time. Then, for each of the 100 points, we simulate 30 values of a normal distribution with mean value given by the market volatility sequence. This already represents the first case, where only the overall distribution is affected by the market volatility, but where still a large variation of the individual volatilities around the mean exists.

Then, for the second case, we now impose the additional requirement that individual volatilities follow a certain ranking. In order to achieve this, we simply need to sort all the simulated observations at a given point in time. Thereby, we make use of the plyr package, so that we can easily apply the sort function to each of the 100 rows of the matrix of simulated values. We then make use again of the plotLinePaths function in order to plot the resulting 30 simulated volatility paths for both cases. Clearly, the second case imposes much more structure on the evolution of joint volatilities.

For the sake of completeness, we now additionally want to examine whether this increased structure really results in higher predictive power of the market volatility. Therefore, we extract the sample correlation between each of the simulated individual volatility paths and the market volatility. Hence, for each of the two market situations we get 30 correlations, which we compare in a kernel density plot. Clearly, the situation where volatility paths are sorted results in larger correlations. The market volatility thus will have greater explanatory power in this case.

########################################
## get correlations with market trend ##
########################################

corrs <- vector(length = nAss)
for(ii in 1:nAss){
corrs[ii] <- cor(simVals[, ii], mus)
}

for(ii in 1:nAss){
}

corrsDf <- data.frame(index = 1:nAss, unSorted = corrs,
corrsMt <- melt(corrsDf, id.vars = "index")

p <- ggplot(corrsMt) + geom_density(aes(x=value, color=variable))
p

So which of the two situations is closer to the behavior of real world markets? We will try to find out in the next part.

5 Principal component analysis

We now want to add some quantitative rigor to the qualitative analysis of the preceding parts. So far, we have narrowed our focus down to financial crises and market volatility as common underlying factors of the evolution of stock market volatilities. The reason for this is twofold. First, the relation with these factors became obvious in the first pictures of joint volatility paths. And second, they have the big advantage of being nicely interpretable. However, with the help of quantitative methods, we now have the chance to extend our search of common factors to uninterpretable factors also. Interpretable factors would only be the best case that we could hope for, and even unobservable and uninterpretable factors would be welcome in risk and asset management applications. Hence, we will now use a more general approach to check volatility paths for commonalities. And, as we will see, the market volatility will play a big role here again.

Without going too much into details, you can think of principal component analysis as a technique that tries to explain the variation of a high-dimensional dataset as efficient as possible. Therefore, you construct synthetic and uncorrelated factors (called principal components) from the original data sample. Each component is a linear combination of the original variables of the dataset. The components are calculated successively, and in each step you choose the orthogonal linear combination that explains as much of the remaining variation as possible. This way, the individual principal components will show decreasing proportions of variation that they explain. Ideally, you will end up with a situation, where most of the variation of the high-dimensional data is already explained through a small number of principal components.

For the computation of the principal components we can simply rely on the implementation of the base R package “stats”. Then, in order to see how much the high-dimensional setting boils down to some factors only, we plot the cumulated variation explained by the first components.

## get volatilities without time for PCA analysis

## get first two principal components
PCAresults <- princomp(x=volaNoTime)
PC1 <- PCAresults$scores[, 1] PC2 <- PCAresults$scores[, 2]

## get cumulative proportion of variance
explainedByPCA <- summary(PCAresults)
props <- explainedByPCA$sdev^2 / sum(explainedByPCA$sdev^2)

## plot proportions of first k values
k <- 10
propsDf <- data.frame(PC=1:k, proportion=props[1:k],
cumulated=cumsum(props[1:k]))
propsMt <- melt(propsDf, id.vars = "PC")
p <- ggplot(propsMt, aes(x=factor(PC), y=value, fill=variable)) +
geom_bar(position="dodge") +
xlab("Principal Components") + ylab("proportion of variance")
p

As the picture shows, the first component already explains roughly 50 percent of the total variation of the data. However, the explanatory power of the following components decreases substantially, with the second component already explaining only 15 percent of the variation. Subsequently, the next three components explain roughly 5 percent each. From the sixth component onwards the explanatory power is already so low that they probably capture already very specific variation that concerns a few assets only. And, even with 10 components, we hardly exceed a variation of 80 percent.

Hence, it looks like if the evolution of volatility paths was driven by too many different factors to be efficiently reduced to a low-dimensional system. One way we could think about this is that asset volatilities roughly do follow one, two or maybe even five common factors. However, at any point in time, there also will be a substantial number of assets that temporarily follow some individual dynamics, deviating from the larger trend. For example, even if the common factors indicate a calm market period, there still will be assets that individually exhibit stressed behavior and high volatility. Or, following a more traditional way of thinking, the evolution of a given asset is driven by two different factors: a common market factor, and an idiosyncratic, firm specific component. Sadly, the idiosyncratic components seem too large to be neglected.

Nevertheless, even though it is clear that we will not achieve the dimensionality reduction that we where striving for, we will now take a more detailed look at the first components. Maybe we can at least give them some meaningful interpretation, so that we get a feeling about when we have to expect a high average market volatility.

The first thing to do is to look at the evolution of the first two principal components. Thereby, we want to plot the path of each principal component individually, as well as the path that they follow when considered together. As ggplot does not support three-dimensional graphics yet, we will express the third dimension, time, through colors. To simplify matters, we do only map numerical indices to the color axes, instead of real dates. However, you can still read off the date-color mapping from the respective one-dimensional paths of the individual components. For reasons that will become obvious in subsequent parts, we also plot the first component with negative sign.

## scatterplot of first two PCs: first one with negative sign
pCs <- data.frame(time=1:nObs, cbind(PC1, PC2))
pCsMt <- melt(pCs, id.vars=c("PC1","PC2"))

p1 <- ggplot(pCsMt, aes(x=-PC1, y=PC2), colour=value) +
geom_point(aes(colour=value))

## plot individual time series PC1 and PC2
pC3 <- data.frame(time=as.Date(dates), time2=1:nObs, -PC1, PC2)
names(pC3) <- c("time", "time2", "-PC1", "PC2")
pC3Mt <- melt(pC3, id.vars=c("time", "time2"))

## plot individual time series PC1 and PC2
p4 <- ggplot(pC3Mt, aes(x=time, y=value)) +
geom_path(aes(colour=time2)) + facet_grid( variable ~ .)

grid.arrange(p1, p4, ncol=2)

Looking at the first principal component, one should already recognize some familiar pattern: it’s peaks coincide with financial crises. As we will show later, this first principal component has an exceptional high correspondence with the market volatility. We can recognize crisis periods also in the path of the second principal component. Here, the component takes on extreme values during both financial crises, too. However, the values differ in sign for the two crises periods: the component shows negative values in the aftermath of the dot-com bubble, while it takes on positive values during the financial crisis of 2008. Hence, when considering both components together, crisis periods can be found in the upper right and lower right corners of a two-dimensional space of both components.

As the first component’s sensitivity to financial crises is equally strong as the market volatility’s sensitivity, they both should also exhibit some mutual relationship. Hence, we now want to examine the strength of the similarity between first component and market volatility. Thereby, we simply use mean and median of all volatility paths as a proxy for the market volatility. Additionally, we also allow the principal component to be linearly transformed (shifted and rescaled) to better match the path of the market volatility. We regress each market volatility proxy on the first component, in order to get a best linear transformation. Afterwards, we choose the market volatility proxy with the largest $R^{2}$ value, and plot it together with the rescaled version of the first principal component.

In order to get mean volatility and median for each given date, we again use the plyr package of Hadley Wickham. This way, we can easily split the data into subsets, and then apply a function to each part.

library("plyr")

## get first two principal components
PCAresults <- princomp(x=volaNoTime)
PC1 <- PCAresults$scores[, 1] PC2 <- PCAresults$scores[, 2]

## get mean of volatilities as market volatility proxy
marketVola <- aaply(volaMatrix, .margins=1, .fun=mean)
marketMedian <- aaply(volaMatrix, .margins=1, .fun=median)

## apply linear rescaling according to regression coefficients
fit <- lm(marketVola ~ PC1)
fit2 <- lm(marketMedian ~ PC1)

## get r squared for both mean and median
rSquared1 <- summary(fit)$r.squared rSquared2 <- summary(fit2)$r.squared

(sprintf("R^2 for mean=%s, R^2 for median=%s",
round(rSquared1, 3), round(rSquared2, 3)))

rescaled_PC1=fit$fitted.values, marketVola=marketVola) pC1MarketVolaMt <- melt(pC1MarketVolaDf, id.vars="time") p1 <- ggplot(pC1MarketVolaMt, aes(x=time, y=value, colour=variable)) + geom_line() [1] "R^2 for mean=0.983, R^2 for median=0.966" Comparing the $R^{2}$ values of both regressions, we choose the mean volatility as proxy for the market volatility. The $R^{2}$ value of 0.983 already shows the strong relation between principal component and market volatility. Something that is also confirmed visually: As the first principal component nearly exactly matches the market volatility, we can relate its properties to the market volatility. Hence, we now can conclude that roughly 50 percent of the variation of individual asset volatilities is explained through a general market volatility level. The rest of the variation thus arises from fluctuations around the mean, and we will try to find some explanation for this, too. First, however, we want to find out how the influence of the market volatility exactly affects the individual assets. Therefore, we need to take a more detailed look at the factor loadings, which will be distinguished according to sectoral affiliation again. As with the principal component itself, we also change the sign of its loadings, so that the graphics will be consistent, and the component becomes directly interpretable as market volatility. While the first plot shows a histogram of the loadings in absolute terms, the second plot shows relative proportions for the individual sectors. ## get loadings for first component loadings1 <- PCAresults$loadings[, 1]

## adjust bin size to 20 bins
binw <- diff(range(loadSectorDf$loadings))/10 ## colorize according to sector p1 <- ggplot(loadSectorDf, aes(x=-loadings, fill=sector)) + geom_histogram(binwidth=binw) + xlab("loadings of 1st principal component") p2 <- ggplot(loadSectorDf, aes(x=-loadings, fill=sector)) + geom_bar(position="fill", binwidth=binw) + xlab("loadings of 1st principal component") multiplot(p1, p2, cols=1) The most important point here certainly is the fact that all individual assets have a positive loading for the first component. Hence, each single asset follows the variations of the market volatility. Thereby, companies from the financial sector have loadings that are larger than average, while sectors consumer staples, health care and industrials are below average. Still, this does not directly translate into the strength of the market volatility’s influence on individual sectors. To measure this strength, we additionally regress each asset’s volatility path on the path of the first principal component, and extract the values of $R^{2}$. ## preallocation rSquared1 <- vector(mode="logical", length=nAss) ## regress each volatility on the first principal component for(ii in 1:nAss){ dataDf <- data.frame(y=volaDf[, ii+1], x=PC1) lmFit <- lm(y~x, data=dataDf) temp <- summary(lmFit) rSquared1[ii] <- temp$r.squared
}

## plot R^2
rSquaredDf1 <- data.frame(symbol=row.names(PCAresults$loadings), rSquared=rSquared1) rSquaredSectors <- merge(rSquaredDf1, sectors, sort=F) ## adjust bin size to 10 bins binw <- diff(range(rSquared1))/10 ## colorize according to sector p1 <- ggplot(rSquaredSectors, aes(x=rSquared, fill=sector)) + geom_histogram(binwidth=binw) + xlab("R squared for individual assets and PC1") p2 <- ggplot(rSquaredSectors, aes(x=rSquared, fill=sector)) + geom_bar(position="fill", binwidth=binw) + xlab("R squared for individual assets and PC1") multiplot(p1, p2, cols=1) As we can see, the values of $R^{2}$ nearly spread over the complete range from zero to one, with a majority of assets between values of 0.25 and 0.75. The effect of the market volatility differs for sectors, and is more pronounced for financial and industrial companies. On the other hand, companies from the consumer staples sector are influenced by the market volatility only slightly. This can be seen even better with boxplots: ## preallocation rSquared1 <- vector(mode="logical", length=nAss) ## regress each volatility on the first principal component for(ii in 1:nAss){ dataDf <- data.frame(y=volaDf[, ii+1], x=PC1) lmFit <- lm(y~x, data=dataDf) temp <- summary(lmFit) rSquared1[ii] <- temp$r.squared
}

## plot R^2
rSquaredDf1 <- data.frame(symbol=row.names(PCAresults$loadings), rSquared=rSquared1) rSquaredSectors <- merge(rSquaredDf1, sectors, sort=F) ## boxplot p <- ggplot(rSquaredSectors) + geom_boxplot(aes(x=sector, y=rSquared)) + xlab("sectors") + theme(axis.text.x=element_text(angle=90)) + ylab(expression(R^{2})) p As next step, we now want to find some interpretation for the second component as well. However, the path of the second component did not immediately suggest any easy interpretation, so that we now will need to take a look at the loadings first. ## get loadings for first component loadings2 <- PCAresults$loadings[, 2]

fill=sector))

## adjust bin size to 20 bins
binw <- diff(range(loadSectorDf$loadings))/10 ## colorize according to sector p1 <- ggplot(loadSectorDf, aes(x=loadings, fill=sector)) + geom_histogram(binwidth=binw) + xlab("loadings of 2nd principal component") p2 <- ggplot(loadSectorDf, aes(x=loadings, fill=sector)) + geom_bar(position="fill", binwidth=binw) + xlab("loadings of 2nd principal component") multiplot(p1, p2, cols=1) As it turns out, the loadings of the assets are almost fairly cut in halves with respect to their sign: approximately one half of the loadings has a positive sign, while the other half has a negative sign. However, this fair split does not hold for individual sectors. While financial companies nearly show a positive loading exclusively, companies from health care and information technology sectors have negative signs only. Hence, the second component picks up differences in volatility between these sectors. Let’s see, what these different signs of the loadings will actually mean for the volatilities of the sectors. Therefore, we need to look at loadings and the path of the second component simultaneously. ## evolution 2nd factor PC2df <- data.frame(time=dates, PC2=PC2) p2 <- ggplot(PC2df, aes(x=time, y=PC2)) + geom_line() As we can see from the figure, the second principal component takes on negative values constantly for roughly the first half of the sample period, before it shifts into positive numbers until the end of the sample. Due to the loadings, a negative second component coincides with decreased volatility levels for the financial sector, and increased levels for health care and information technology sectors. This effect becomes even more pronounced during financial crises, where the component takes on its most extreme values. In other words, the first component corresponds to the level of the market volatility, and it shifts the overall distribution of asset volatilities up and down. The second component, however, only captures changes within a given distribution. For the first half of the sample, it adds some extra volatility on the information technology sector, while it decreases the volatility of the financial sector. In the second half, it is the other way round. To further clarify the role of the second component, we create another simulated example as visualization of this effect. ## simulate points from normal distribution nSim <- 20 mu <- 0 xx <- rnorm(nSim, mu, 1) ## points below the mean value are listed as group1 indSign <- factor(xx<mu, levels=c("TRUE", "FALSE"), labels=c("group1", "group2")) ## append points with changed sign - groups affiliation unchanged xxDf <- data.frame(vals=c(xx,-xx), time=factor(c(rep(1, nSim), rep(2, nSim)), levels=c(1, 2), labels=c("positive factor", "negative factor")), group=rep(indSign, 2), y=rep(0,(2*nSim))) ## show points for different realizations of factor p <- ggplot(xxDf, aes(x=vals, y=y, color=group)) + geom_point() + facet_grid(time ~ .) + stat_function(fun=dnorm, color="black") + scale_x_continuous(limits = c(-5, 5)) + labs(title="variation explained by PC2") p The figure shall convey the following intuition: the realization of the second component does not change the overall distribution, but it only explains variation within the distribution. Hence, for positive values, the first group of points will be located to the left, while they are located to the right for negative realizations. As with the first component, we now want to see how the loadings translate into real influence. Therefore, we regress each asset’s volatility path on the path of the second component, and extract the $R^{2}$ values of the individual regressions. ## preallocation rSquared2 <- vector(mode="logical", length=nAss) ## regress each volatility on the first principal component for(ii in 1:nAss){ dataDf <- data.frame(y=volaDf[, ii+1], x=PC2) lmFit <- lm(y~x, data=dataDf) temp <- summary(lmFit) rSquared2[ii] <- temp$r.squared
}

## plot R^2
rSquaredDf2 <- data.frame(symbol=row.names(PCAresults$loadings), rSquared=rSquared2) rSquaredSectors <- merge(rSquaredDf2, sectors, sort=F) ## adjust bin size to 10 bins binw <- diff(range(rSquared2))/10 ## colorize according to sector p1 <- ggplot(rSquaredSectors, aes(x=rSquared, fill=sector)) + geom_histogram(binwidth=binw) + xlab("R squared for individual assets and PC2") p2 <- ggplot(rSquaredSectors, aes(x=rSquared, fill=sector)) + geom_bar(position="fill", binwidth=binw) + xlab("R squared for individual assets and PC2") multiplot(p1, p2, cols=1) ## preallocation rSquared2 <- vector(mode="logical", length=nAss) ## regress each volatility on the first principal component for(ii in 1:nAss){ dataDf <- data.frame(y=volaDf[, ii+1], x=PC2) lmFit <- lm(y~x, data=dataDf) temp <- summary(lmFit) rSquared2[ii] <- temp$r.squared
}

## plot R^2
rSquaredDf2 <- data.frame(symbol=row.names(PCAresults$loadings), rSquared=rSquared2) rSquaredSectors <- merge(rSquaredDf2, sectors, sort=F) ## boxplot p <- ggplot(rSquaredSectors) + geom_boxplot(aes(x=sector, y=rSquared)) + xlab("sectors") + theme(axis.text.x=element_text(angle=90)) + ylab(expression(R^{2})) p As we can see from the pictures, the influence of the second component measured by the $R^{2}$ values is substantially lower than it was for the first component. However, as already indicated by the loadings, it does have a great influence on the IT sector nevertheless. Now let’s try to embed these quantitative findings into a more qualitative perception that we have of the sample period, and think about their relevance for practical applications. First, the quantitative findings do match very well with the respective origins of the crises periods. During the dot-com crash, which was labeled after its origin in the IT sector, our quantitative findings also do show some enlarged volatility levels of the same sector. And, for the financial crisis of 2008, the second component assigns a higher volatility level to the financial sector, while the IT sector simply follows some more general market conditions. 6 Conclusion How much should we rely on these findings when trying to assess future risks? Well, in my opinion, we probably should focus rather on the larger patterns pointed out by the analysis, instead of sticking too much with the details. For example, although the second component captures differences between both crises very appropriately, I would not bet too much on its predictive power in future. The point is, it overfits the exact origin of the crises, such that it mainly distinguishes between IT and financial sector. However, I’d rather focus on the bigger story that this component tells: crises may originate in one sector, which then will exhibit volatility levels above the general market trend. Nevertheless, the exact sector, where the next crisis will originate, could be any. Our data simply did contain only crises that were caused by IT and financial sector. Another pattern that seems rather robust to me is the variation captured by the first principal component. Hence, the general market volatility is not constant over time, but it increases during crises periods. And these shifts in market volatility seem to affect many, if not all, individual assets. Besides, however, at each point in time, some individual assets will always deviate from the larger trend of volatility. Hence, there must be some idiosyncratic component to volatility, too. References  [Wickham(2011)] Hadley Wickham. The split-apply-combine strategy for data analysis. Journal of Statistical Software, 400 (1):0 1-29, 4 2011. ISSN 1548-7660. URL http://www.jstatsoft.org/v40/i01. Downloading SP500 stock price data with R 1 Summary In this blog post I want to show how to download data of all constituents of the S&P500 index, which is one of the leading stock market indices for US equity, from Yahoo!Finance. According to Standard and Poor’s, “the index includes 500 leading companies and captures approximately 80% coverage of available market capitalization.” The target will be to get a clean and broadly applicable dataset, which then can be used for many types of financial applications, such as risk management and asset management. Instead of running the R code snippets on your own, you can also directly download the most recent version of the resulting dataset which I keep for my own research: historic prices and clean return series. These datasets, however, could slightly deviate from the results obtained in this blog post, since they probably build on a more recent version of the input files. Update: Due to the Yahoo!Finance data disclaimer, I will not provide the application-ready and cleaned-up final dataset. I am sorry, but in order to get the data, you need to run the steps on your own. 2nd Update: Meanwhile I did replicate the whole exercise in my new statistical software language of choice: julia. Not only does this dramatically speed up the procedure, but I also did process the data with regards to missing values in a slightly improved manner. You can check out this new version here, and in addition you can find a comparison of different approaches to deal with missing values at this blog post. 2 Data considerations The best way to test the adequacy of a model is testing its performance on out-of-sample data. With this application in mind, the dataset should be large enough so that it can be divided into a part that is used to fit the model and another part that can be used for out-of-sample testing. The problem with financial data is that asset return distributions vary over time, thereby posing additional challenges to risk and asset management. This most easily can be seen in times of crisis, where markets clearly exhibit higher levels of volatility. Any econometric model now should be able to replicate reality at any given point in time, and therefore needs to be able to capture both calm and turbulent market periods. Therefore, your model ideally should also be tested on both market states, so that your out-of-sample period should entail a crisis period as well. Similarly, of course, your model ideally should be fitted to both types of market states as well. For example, fitting a model to calm market times only will most likely lead to improper results during financial crisis, since crisis would be one region of the unconditional distribution that the model has never seen before. For a very extreme example, think about fitting a model to positive returns only. You would clearly fail to capture the overall unconditional distribution, and not to speak of the conditional distribution of negative returns. According to this argumentation, we will pick January the 1st, 1995, as the beginning of the data sample. The data sample hence entails two financial crisis: the bursting of the dot-com bubble in 2001, and the 2008 financial crisis following the bursting of the US housing bubble. The first crisis now could be used to fit some econometric model, while we can test its performance in both risk or asset management applications out-of-sample during the second crisis. 3 Data download The first step – of course – is downloading the data from Yahoo!Finance. This means that we need to get all the individual assets that comprise the index. A list of all current constituents can be accessed on the S&P homepage, and can be directly downloaded as xls-file here, or – if the direct link is broken – by manually clicking on the full constituents list on the S&P 500 homepage. However, to make things easier, you can download this already processed csv-file, which then can be directly read into R. Following my convention, you should copy the file in the subfolder raw_data of the project’s directory, to indicate that this list comprises externally created data. In the past, this list was where I did encounter the first problem with the data: the list of 500 components was already missing the symbol of one stock. A problem that seems to be fixed by now. After reading in the list of constituents, the function get.hist.quote from the tseries package is used to download the associated data in a for-loop. Note, that I did not conduct any linearization or preallocation with this code, so that the code’s processing time could be further decreased in general. Anyways, since individual historic price series need to be merged to common dates, acceleration most likely would result in significantly less readable code. ## downloads historic prices for all constituents of SP500 library(zoo) library(tseries) ## read in list of constituents, with company name in first column and ## ticker symbol in second column spComp <- read.csv("raw_data/sp500_constituents.csv" ) ## specify time period dateStart <- "1995-01-01" dateEnd <- "2013-05-01" ## extract symbols and number of iterations symbols <- spComp[, 2] nAss <- length(symbols) ## download data on first stock as zoo object z <- get.hist.quote(instrument = symbols[1], start = dateStart, end = dateEnd, quote = "AdjClose", retclass = "zoo", quiet = T) ## use ticker symbol as column name dimnames(z)[[2]] <- as.character(symbols[1]) ## download remaining assets in for loop for (i in 2:nAss) { ## display progress by showing the current iteration step cat("Downloading ", i, " out of ", nAss , "\n") result <- try(x <- get.hist.quote(instrument = symbols[i], start = dateStart, end = dateEnd, quote = "AdjClose", retclass = "zoo", quiet = T)) if(class(result) == "try-error") { next } else { dimnames(x)[[2]] <- as.character(symbols[i]) ## merge with already downloaded data to get assets on same dates z <- merge(z, x) } } ## save data write.zoo(z, file = "data/all_sp500_price_data.csv", index.name = "time")  Although this code did work for me, please note that it is not implemented in a completely robust way, as the download of the first asset is not executed in a try block. Hence, if problems occur with this asset, you might encounter an error. 4 Visualizing missing values The best way to identify any data errors with such high-dimensional data always is through visualization. As first step, we will identify assets and dates with missing values. One easy way of looking at this is to order assets with respect to the number of occurring NAs, and highlighting entries with missing values. Usually, I tend to create my visualizations with ggplot2 whenever possible. However, given the size of the dataset, this will be quite time-consuming. Hence, I will also implement a comparable version of the visualization using only low-level graphical tools of R with faster execution. Missing values are shown in light blue and pink respectively. ## load libraries library(zoo) library(ggplot2) library(lattice) library(reshape) ## read in sp500 prices z <- read.zoo(file = "data/all_sp500_price_data.csv", header = T) nAss <- ncol(z) ## decode missing values with numeric ones miss <- is.na(coredata(z))*1 # multiplication converts # logical to numeric matrix ## summing up missing values per column and ordering with respect to ## number of missing values ranks <- rank(colSums(miss), ties.method = "random") miss[, ranks] <- miss ################################ ## visualization with ggplot2 ## ################################ ## missing values as dataframe missDf <- data.frame(miss) ## append time index as first column missDf$time <- factor(index(z))
missDf <- missDf[c(nAss+1, 1:nAss)]

## processing dataframe for ggplot2
missMt <- melt(missDf, id.vars = c("time"))

## plotting values
p <- (ggplot(missMt, aes(x = time, y = variable, colour = value)) +
geom_tile(aes(fill = value)))

## getting rid of date factor labels
p2 <- p + theme(axis.line = element_blank(),
axis.text.x = element_blank(),
axis.text.y = element_blank(),
axis.ticks = element_blank(),
legend.position = "none",
panel.background = element_blank(),
panel.border = element_blank(),
panel.grid.major = element_blank(),
panel.grid.minor = element_blank(),
plot.background = element_blank()) +
xlab("time") + ylab("stocks")

##################################
## visualization with levelplot ##
##################################

## getting rid of column names
colnames(miss) <- NULL
levelplot(miss, aspect = 1, col.regions = cm.colors,
xlab = "time", ylab = "Stocks", main = "Missing values",
colorkey = F)


As can be seen, there are a significant number of missing values for about one fifth of the 500 S&P components. In addition, there are also a few dates where data is missing for a substantial share of stocks. We now want to simply eliminate stocks with multiple missing values, since we assume that these data problems are only arising due to deficiencies of the data provider. In other words, it generally should be possible to get clean data on the chosen sample period through usage of a proprietary and more comprehensive database. We hence do not tackle stocks with large ratios of missing values in any ad-hoc way, since we do not want to include any spurious effects into the subsequent analysis.

As threshold value we choose to exclude all assets with more than nine missing values, and afterwards any dates with more than two missing values. Eliminated dates are shown together with the number of missing observations in a Google Charts html table. After completely removing stocks and dates with multiple NAs, all occasionally occurring remaining NAs will simply be set to 0 in the associated return series. For example, for a case of one missing observation for one asset, like

incompleteData <- cbind(c(4, NA, 6), c(6, 9, NA))

 4 6 nil 9 6 nil

this procedure will lead to a return of zero for the first pair of days, and a return corresponding to the full price change for the second pair of days.

missingValues <- which(is.na(incompleteData))

filledValues <- incompleteData
filledValues[missingValues] <- incompleteData[missingValues-1]
(filledValues)
(diff(filledValues))

     [,1] [,2]
[1,]    4    6
[2,]    4    9
[3,]    6    9
[,1] [,2]
[1,]    0    3
[2,]    2    0


At the end, we will have a matrix of approximately 4600 observations for each of the approximately 350 remaining stocks.

## load packages
library(zoo)
library(fGarch)

rm(list=ls())

######################################################
## eliminate stocks with more than 2 missing values ##
######################################################

## get indices of stocks with max 9 missing values
miss <- is.na(coredata(z))*1
indices <- colSums(miss) < 10
suitableStocks <- z[, indices]

## show number of remaining stocks
print(length(suitableStocks[1, ]))

## create table with missing values to show dates with many NAs
miss <- is.na(coredata(suitableStocks))*1
datesWithNAs <- rowSums(miss)
missingValues <- data.frame(dates = index(z)[datesWithNAs > 3],
number = datesWithNAs[datesWithNAs > 3])

## show missing values in html file
p <- gvisTable(missingValues)
write(p$html$chart, file = "pics/table_of_missingValues.html")

####################################################
## remove dates with more than two missing values ##
####################################################

datesToKeep <- rowSums(miss) < 3

#############################################################
## calculate percentage log returns and replace NAs with 0 ##
#############################################################

## show number of remaining NAs

## avoid logical indexing with zoo object
missingValues <- which(is.na(allPrices))

## fill missing values with value of previous day
filledValues <- allPrices
filledValues[missingValues] <- allPrices[missingValues - 1]
logRetNumeric <- 100*diff(log(filledValues))

## create zoo object again
logRet <- zoo(x = logRetNumeric, order.by = returnDates)

## check that no NAs exist anymore
if(sum(is.na(logRet)) != 0){
stop("still NAs in return series!!")
## could be due to NAs on consecutive days
}

## round to five decimal places
logRet <- round(logRet, 5)

## store data
write.zoo(logRet, file = "data/all_sp500_clean_logRet.csv")


Again, at this point the code is not implemented in a completely robust manner. If you have an NA at the first day of the sample, you would try to access an element at index 0. This, of course, would lead to an error. Furthermore, I do not deal explicitly with the case of consecutive NAs. If this case occurred, you would be warned, as the last if-statement would throw an exception. For both cases one should rather easily find some go-around with for loops. However, this most likely would kill your performance. And any more sophisticated solution I did skip in order to keep the code simple.

5 Zoo object pitfalls

When removing all of the remaining NAs, bear in mind that zoo objects can not be accessed by logical indexing! Hence, the following assignment does not work:

logRet[is.na(logRet)] <- 0


coredata(logRet)[is.na(logRet)] <- 0


Analytical solutions in risk management

In risk management, most traditional approaches are built around a fundamental desire for analytical tractability. The reason for this is obvious: with only minor computational power of computers the only way to guarantee quick solutions was through mathematics. And in practice, when risk assessment and portfolio optimization becomes part of your business, you wouldn’t want to have a model that needs hours or days in order to come up with some insights into your risks.

However, in reality the challenges on your analytical risk management model are rather high. It is almost sure that your model will not have to deal with one-dimensional risks only. Whether high-dimensionality arises through multiple assets or multi-period risk assessment of one asset only: both cases force you to deal with interactions of multiple random variables, thereby complicating an analytical solution enormously. And, of course: in most real world situations, you will encounter both cases. Life just ain’t easy…

In this post we will try to work out some circumstances that need to prevail in order that we could still hope to find an analytical solution. Still, whether such convenient circumstances also do occur in reality, is something that we will need to put some further thought on afterwards.

Motivation

Often when we examine financial data, we find that logarithmic returns exhibit fatter tails than the normal distribution. Hence, we need a distribution with more probability mass in the tails, and – coming from traditional financial mathematics with normally distributed log-returns – the natural first choice is the student’s $t$-distribution.

However, the standard student’s $t$-distribution has zero mean, and a variance determined through its degrees of freedom parameter. This is usually too restrictive, since one major characteristic of financial data is changing volatility, which ideally should be captured without further changes of the shape of the distribution. To gain such extended flexibility, the student’s $t$-distribution can be extended through linear transformation, so that mean, standard deviation and tails can be adapted flexibly.

Standard student’s t-distribution

For $X\sim t$ the variance depends on the degrees of freedom parameter $\nu$:

$\displaystyle \mathbb{V}(X)=\frac{\nu}{\nu-2}$

## set parameter
nu <- 8

## number of samples
n <- 10000

## simulate
t <- rt(n,nu)

## analytically calculated variance
variance <- nu/(nu-2)
sprintf("True variance: %f",variance)

## get sample variance
sprintf( "Sample variance: %f",var(t))

: [1] “True variance: 1.333333” : [1] “Sample variance: 1.291841”

Location-scale transformation

Through application of a linear transformation, we can easily shift the center of the distribution and affect the divergence from its mean.

\displaystyle \begin{aligned} Y&=aX + b,\quad X\sim t \\ \mathbb{E}[Y]&=b \\ \mathbb{V}(Y)&=a^{2}\mathbb{V}(X) \\ &=a^{2} \frac{\nu}{\nu-2} \end{aligned}

Note, that the standard deviation of the new random variable $Y$ does not equal the scaling factor $a$, but is a composition of scaling factor and original standard deviation determined by $\nu$.

The cumulative distribution function after the transformation can easily be obtained through:

\displaystyle \begin{aligned} \mathbb{P}(Y\leq x)&= \mathbb{P}(aX+b\leq x)\\ &= \mathbb{P}\left(X\leq \frac{x-b}{a}\right)\\ &= F_{X}\left(\frac{x-b}{a}\right) \end{aligned}

Taking this result, we can derive the inverse cdf:

\displaystyle \begin{aligned} F_{Y}(x)&=F_{X}\left(\frac{x-b}{a}\right)=p \\ \Leftrightarrow &\frac{x-b}{a}=(F_{X})^{-1}(p)\\ \Leftrightarrow &x=a(F_{X})^{-1}(p)+b \\ \Leftrightarrow &(F_{Y})^{-1}(p)=a(F_{X})^{-1}(p)+b \end{aligned}

Getting the density of $Y$ is slightly more demanding, since we need an additional rescaling in order to guarantee that integration of the density function still yields an area of 1. Hence, we need to apply the transformation theorem:

$\displaystyle f_{Y}(z)=f_{X}(g^{-1}(z)) |(g^{-1})'(z)|,$

with $g$ given by $g(X)=aX+b$.

Calculating $g^{-1}$ requires exchanging $X$ and $Y$ and solving for $Y$:

$\displaystyle X=aY+b \Leftrightarrow Y=\frac{X-b}{a}$

Taking the derivative:

$\displaystyle (g^{-1})'(X)=\left(\frac{X-b}{a}\right)'=\frac{1}{a}$

Putting the parts together, we get:

$\displaystyle f_{Y}(x)=f_{X}\left(\frac{x-b}{a}\right)\left|\frac{1}{a}\right|$

Implementation

The functions now can be implemented in R. We use the code provided by wikipedia:

dt_ls <- function(x, df, mu, a) 1/a * dt((x - mu)/a, df)
pt_ls <- function(x, df, mu, a) pt((x - mu)/a, df)
qt_ls <- function(prob, df, mu, a) qt(prob, df)*a + mu
rt_ls <- function(n, df, mu, a) rt(n,df)*a + mu

We can now try to fit the t-location-scale distribution to simulated data.

library(MASS)

## set parameters
n <- 10000
mu <- 0.4
a <- 1.2
nu <- 4

## simulate values
simVals <- rt_ls(n, nu,mu,a)
fit <- fitdistr(simVals,"t")
print(fit)
There were 12 warnings (use warnings() to see them)
m            s            df
0.40849216   1.20630991   4.00772609
(0.01427656) (0.01498895) (0.16515696)