FINC621 Winter 2018-19 Lab Worksheet 03

Variance, Covariance, Correlation & Causality (finc621-lab03)

About

In this worksheet we look at different variance, covariance, volatility, and causality calculations. We finish with a short matematical proof (no R required).

Setup

Remember to always set your working directory to the source file location. Go to ‘Session’, scroll down to ‘Set Working Directory’, and click ‘To Source File Location’. Read carefully the below and follow the instructions to complete the tasks and answer any questions. Submit your work to RPubs as detailed in previous notes.

Note

For clarity, tasks/questions to be completed/answered are highlighted in red color (color visible only in preview mode) and numbered according to their particular placement in the task section. Type your answers outside the red color tags!

Quite often you will need to add your own code chunk. Execute sequentially all code chunks, preview, publish, and submit link on Sakai following the naming convention. Make sure to add comments to your code where appropriate. Use own language!

Any sign of plagiarism, will result in dissmissal of work!

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Task 1: Variance, Covariance, and Volatility

This task follows the two examples in the book R Example 2.5/p. 58 and R Example 2.6/p. 66

# Require will load the package only if not installed 
# Dependencies = TRUE makes sure that dependencies are install
if(!require("quantmod",quietly = TRUE))
  install.packages("quantmod",dependencies = TRUE, repos = "https://cloud.r-project.org")

Attaching package: ‘zoo’

The following objects are masked from ‘package:base’:

    as.Date, as.Date.numeric

Version 0.4-0 included new data defaults. See ?getSymbols.
Learn from a quantmod author: https://www.datacamp.com/courses/importing-and-managing-financial-data-in-r

##### 1A) Calculate the correlation and covariance matrix of the adjusted daily log returns for four different stocks of your choice. Explain your observations in terms of potential relationships.

# Once you have obtained the adjusted daily log returns for your stocks, omitting the time index, you will need to combine them to create a matrix. Below is an example.  For more details see the Help command in R on cbind, cov, and cor.
# M <- cbind(A,B,C) # create a matrix where each column is an array/vector of numerical values 
# cov(M) # compute the covariance matrix
# cor(M, method="pearson") # compute the correlation matrix based on the Pearson method

getSymbols("WFC",src="yahoo",from="2000-01-01", to="2000-11-30")

[1] "WFC"

WFCAd=WFC$WFC.Adjusted["2000-01-01/2000-11-30"]
WFCRd= periodReturn(WFCAd,period="daily", type = "log")
getSymbols("BAC",src="yahoo",from="2000-01-01", to="2000-11-30")

[1] "BAC"

BACAd=BAC$BAC.Adjusted["2000-01-01/2000-11-30"]
BACRd= periodReturn(BACAd,period="daily", type = "log")
getSymbols("C",src="yahoo",from="2000-01-01", to="2000-11-30")

[1] "C"

CAd=C$C.Adjusted["2000-01-01/2000-11-30"]
CRd= periodReturn(CAd,period="daily", type = "log")
getSymbols("JPM",src="yahoo",from="2000-01-01", to="2000-11-30")

[1] "JPM"

JPMAd=JPM$JPM.Adjusted["2000-01-01/2000-11-30"]       
JPMRd= periodReturn(JPMAd,period="daily", type = "log")
M <- cbind(WFCRd,BACRd,CRd,JPMRd) # create a matrix where each column is an array/vector of numerical values
cov(M) # compute the covariance matrix

                daily.returns daily.returns.1 daily.returns.2 daily.returns.3
daily.returns    0.0007178985    0.0005150312    0.0003581854    0.0004182091
daily.returns.1  0.0005150312    0.0008050273    0.0005111028    0.0005656111
daily.returns.2  0.0003581854    0.0005111028    0.0006621937    0.0004907226
daily.returns.3  0.0004182091    0.0005656111    0.0004907226    0.0008162787

cor(M, method="pearson")

                daily.returns daily.returns.1 daily.returns.2 daily.returns.3
daily.returns       1.0000000       0.6774803       0.5194980       0.5463145
daily.returns.1     0.6774803       1.0000000       0.7000199       0.6977396
daily.returns.2     0.5194980       0.7000199       1.0000000       0.6674585
daily.returns.3     0.5463145       0.6977396       0.6674585       1.0000000

# compute the correlation matrix based on the Pearson method

##### 1B) Calculate the three types of volatility for a particular stock of your choice. Consider a time window extending one year back from most recent obtainable closing day price. Order the three estimates from low to high volatility and explain how the ordering makes sense.

getSymbols("F",src="yahoo",from="2017-10-01", to="2018-10-02")

[1] "F"

m=length(F$F.Close)
ohlc <-F[,c("F.Open","F.High","F.Low","F.Close")]
vClose <- volatility(ohlc, n= m,calc="close",N=252)
vParkinson <- volatility(ohlc, n= m,calc="parkinson",N=252)
vGK <- volatility(ohlc, n= m,calc="garman",N=252)
vClose[m];vParkinson[m]; vGK[m]

              [,1]
2018-10-01 0.22557
                [,1]
2018-10-01 0.1922442
                [,1]
2018-10-01 0.1939152

# For this task make sure you understand well what the variables n,m represent in the book's referenced example.

Task 2: Auto-Correlation and Auto-Regression

Follow the example in the book R Example 3.2/p. 74 and R Example 4.1/p. 115

##### 2A) Calculate the ACF for a stock of your choice. Consider both the log return and squared log return. Interpret your results in terms of possible existence of autocorrelation.

acf(na.omit(BACRd),main="acf of BAC",ylim=c(-0.2,0.2))

acf(na.omit((BACRd)^2),main="acf of BAC",ylim=c(-0.2,0.2))

##### 2B) Plot the exchange rate for USD versus another currency of your choice. Interpret your results in terms of behavior.

getFX("USD/EUR")

[1] "USDEUR"

plot(USDEUR)

The USD is stengthening against the EURO consistently trending above 0.80. Outlook on the economy is improving following the FED’s beliefs that the economy has more room for growth. Given that trade wars have not yet set in motion, investors can expect the dollar to strengthen in the future.

##### 2C) Test for the possible existence of an underlying AR(1) – Markov process in your exchange rate currency pair. To this end, plot the ACF and the partial ACF (PACF). Interpret your results. Clearly refer to the lags, and their impacts in determining the order.

acf(USDEUR)

pacf(USDEUR)

The magnitude of the correlation is decreasing over time.

Task 3: Granger Causality Test

To conduct this test the package lmtest will be required, as already done in the code chunk below.

# Require will load the package only if not installed 
# Dependencies = TRUE makes sure that dependencies are install
if(!require("lmtest",quietly = TRUE))
  install.packages("lmtest",dependencies = TRUE, repos = "https://cloud.r-project.org")

##### 3A) Include below the code chunk to solve for 3.5.7 R Lab/p. 106. Write your conclusions.

## Which came first: the chicken or the egg?
data(ChickEgg)
## chickens granger-cause eggs?
grangertest(egg ~ chicken, order = 3, data = ChickEgg)

Granger causality test

Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3)
Model 2: egg ~ Lags(egg, 1:3)
  Res.Df Df      F Pr(>F)
1     44                 
2     47 -3 0.5916 0.6238

## eggs granger-cause chickens?
grangertest(chicken ~ egg, order = 3, data = ChickEgg)

Granger causality test

Model 1: chicken ~ Lags(chicken, 1:3) + Lags(egg, 1:3)
Model 2: chicken ~ Lags(chicken, 1:3)
  Res.Df Df     F   Pr(>F)   
1     44                     
2     47 -3 5.405 0.002966 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The egg comes before the chicken.

# More information about the data used in testing for causality can be obtained by typing the name of the data set `ChickEgg` in the R Help menu.

##### 3B) Briefly describe the data in terms of time range and variables. Similar to the linear autoegressive model described in class, write the mathematical regression model solved in each Granger test, including the proper order. Use naming conventions, and notations more reflective of the data set considered for ChickEgg. \[M_F: Y_t =a_0 + a_1 Y_{t-1} + ...+ a_h Y_{t-h} + b_1 X_{t-1} +...+ b_h X_{t-h} + \epsilon_{t}\] \[M_r: Y_t =a_0 + a_1 Y_{t-1} + ...+ a_h Y_{t-h} + \epsilon_{t}\]

Task 4: Mathematical Proof

##### 4A) Prove the two results in Eq (2.32)/p. 53. No R-coding is needed here. Clearly show your steps. Hint: Use the definition of \(E(X^n)\) for X-log normally distributed. Observe also that \(Var(X) = E(X^2)-E^2(X)\) for any random variable X.

Starting with \(E(X^n)=exp(n\mu + \frac{1}{2}n^2\sigma^2)\) and \(r_t=ln(R_t+1) \sim N(\mu_r), \sigma_r^2\) we have,

\(E(R_t)=exp(1*\mu_r + \frac{1}{2}1^2\sigma_r^2)-1\)

therefore,

\(E(R_t)=exp(\mu_r + \frac{1}{2}\sigma_r^2)-1\).

Last, we have,

\(E(R_t)=e^{\mu_r + \frac{\sigma_r^2}{2}}-1\)

Starting with \(var(x) = E(X^2) - E^2(X)\),

we have,

\(var(R_t) = E(R_t^2) - E^2(R_t)\),

so,

\(E(R_t^2) = e^{\mu_r^2 + \frac{\sigma_r^2}{2}}-1\)

and

\(E^2(R_t) = (e^{\mu_r + \frac{\sigma_r^2}{2}}-1)^2\)

Therefore,

\(var(R_t) = e^{\mu_r^2 + \frac{\sigma_r^2}{2}}- (e^{2\mu_r + \sigma_r^2} + 2e^{\mu_r+\sigma^2/2}-2)\)

and last

\(var(R_t) = e^{2\mu_r + \sigma_r^2}(e^{ \sigma_r^2}-1)\)

*http://computationalfinance.lsi.upc.edu