COMP226 Assignment 2: Strategy Development
The latest version of this document can be found here:
https://www2.csc.liv.ac.uk/~rahul/teaching/comp226/_downloads/a2.pdf
Continuous Assessment Number
2 (of 2)
Weighting
10%
Assignment Circulated
09:00 Wednesday 27 March 2019
Deadline
17:00 Friday 3rd May 2019
Submission Mode
Electronic only
Learning Outcomes Assessed
This assignment will address the following learning outcomes:
•Understand the spectrum of computer-based trading applications and techniques, from profit-seeking trading strategies to execution algorithms.
• Be able to design trading strategies and evaluate critically their historical performance and robustness.
•Understand the common pitfalls in developing trading strategies with historical data.
• Understand methods for measuring risk and diversification at the portfolio level.
Summary of Assessment
The goal of this assignment is to implement and optimize a well-defined trading strategy within the backtester_v5.2 framework. The assignment will be assessed via the testing of 6 functions that you need to implement. The input and output behaviour of each function is fully specified and a code template is provided as a starting point.
Marking Criteria
Individual marks are attributed for each of 6 functions that should be implemented. If all 6 function implementations pass all the automated tests then a mark of 100% will be achieved. Partial credit for a function may be awarded if some but not all automated tests for that function are passed. The marks available for each function are given below.
Submission necessary in order to satisfy module requirements
No
Late Submission Penalty
Standard UoL policy; note that no resubmissions after the deadline will be considered.
Expected time taken
Roughly 8 hours
Introduction: the backtester framework
You will write a strategy that should run in the backtester framework, which is available from
http://www2.csc.liv.ac.uk/~rahul/teaching/comp226/bt.html#backtester
The first thing you should do is download and unzip backtester_v5.2.zip, which will create a directory backtester_v5.2 on your hard drive. Here is a listing of the zip file contents:
backtester_v5.2
├── DATA
│ ├── A2
│ │ ├── 01.csv
│ │ └── 02.csv
│ └── EXAMPLE
│ ├── 01.csv
│ ├── 02.csv
│ ├── 03.csv
│ ├── 04.csv
│ └── 05.csv
├── example_strategies.R
├── framework
│ ├── backtester.R
│ ├── data.R
│ ├── pdratio.R
│ ├── processResults.R
│ └── utilities.R
├── in-sample_period.R
├── main.R
├── main_optimize.R
├── main_template.R
└── strategies
├── bankrupt.R
├── bbands_contrarian.R
├── bbands_holding_period.R
├── bbands_trend_following.R
├── big_spender.R
├── copycat.R
├── extreme_limit.R
├── fixed.R
├── random.R
├── rsi_contrarian.R
└── simple_limit.R
5 directories, 28 files
Next you should open R and make sure that the working directory is the backtester_v5.2 directory on your hard drive (you can use setwd if required). You can now try the example code as follows:
If this doesn’t work, first make sure you are have set the working directory correctly, and then make sure you have installed all the required packages (see the error messages you get to figure out what the problem is). When it works it will produce a plot like the following:
source(‘main.R’)
01 : PD ratio = 0.06 / 0.03 = 1.97
Active on 100 % of days; PD ratio = −153.44
1000000 999800 999600 999400
Jan Apr Jul
02 : PD ratio = −180.2
0.06 0.04 0.02 0.00
0.0 −0.1 −0.2 −0.3 −0.4 −0.5
5
0
−5
Jan Apr Jul
Jan Apr Jul
Jan Apr Jul
0 −200 −400 −600
100 50 0 −50
Jan Apr Jul
Jan Apr Jul
03 : PD ratio = −0.19
04 : PD ratio = 23.02 / 138 = 0.17
05 : PD ratio = 3.88 / 13.7 = 0.28
There is one equity curve for each series in the data (5 of them in this case), and one aggregate equity curve.
Let’s go through main.R and see what the individual parts do.
Sourcing the framework and example strategies
First we source the framework itself.
Then we load example_strategies.R, which provides an easy way to run several examples, and which we will return to shortly.
source
source
source
source
() () () ()
‘framework/data.R’
‘framework/backtester.R’
‘framework/processResults.R’
‘framework/utilities.R’
source(‘example_strategies.R’)
Loading data
Next, we load in data that we will use via the function getData, which is defined in framework/data.R. This function returns a list of xts objects. These will be passed to the function backtester, though we may first change the start and end dates of the xts objects (which you will need to do for assignment 2).
There are 5 series in the directory backtester_5.2/DATA/EXAMPLE/, and therefore the list dataList has 5 elements too.
Each element is an xts:
All the series have the same start and end dates:
The individual series contain Open, High, Low, Close, and Volume columns:
# load data
dataList <- getData(directory="EXAMPLE")
> length(dataList)
[1] 5
> (x [1]
[1]
[1]
[1] [1]
dataList) print(class(x))
for
in
“xts”
“xts”
“xts”
“xts”
“xts”
“zoo”
“zoo”
“zoo”
“zoo”
“zoo”
> (x [1]
[1]
[1]
[1] [1]
dataList)
(paste(start(x),end(x)))
for
in
print
“1970-01-02 1973-01-05”
“1970-01-02 1973-01-05”
“1970-01-02 1973-01-05”
“1970-01-02 1973-01-05”
“1970-01-02 1973-01-05”
> head(dataList[[1]])
Open High Low Close Volume
1970-01-02
1970-01-03
1970-01-04
1970-01-05
1970-01-06
1970-01-07
0.7676
0.7689
0.7725
0.7739
0.7760
0.7738
0.7698
0.7737
0.7748
0.7756
0.7770
0.7744
0.7667
0.7683
0.7718
0.7739
0.7754
0.7728
0.7691
0.7729
0.7732
0.7751
0.7757
0.7743
3171
6311
4317
3409
2904
3514
Loading a strategy
We will now load a strategy using the load_strategy function that is defined in example_strategies.R. We can pick a strategy from a list of examples strategies that is specified at the start of example_strategies.R:
example_strategies <- c( ,
,
, ,
)
"fixed"
"big_spender"
,
"bankrupt"
"copycat"
"random"
"rsi_contrarian"
,
,
,
,
"bbands_trend_following"
"bbands_contrarian"
"bbands_holding_period"
"simple_limit"
Returning now to main.R we see that we have picked one of these (and then checked that it was a valid choice with)
# choose strategy from example_strategies
strategy <-
#strategy <
is_valid_example_strategy <- (strategy) {
strategy %in% example_strategies
}
stopifnot(is_valid_example_strategy(strategy))
"fixed"
- "bbands_contrarian"
# check that the choice is valid
function
Now we actually "load the strategy".
We used the function load_strategy from example_strategies.R. This function sources the strategy file, in this case backtester_v5.2/strategies/fixed.R, and sets a variable params using example_params from example_strategies.R:
# load in strategy and params
load_strategy(strategy)
# function from example_strategies.R
load_strategy <- (strategy) {
strategyFile <- file.path(
( ,strategyFile, )
(strategyFile)
params <<- example_params strategy]]
("Parameters:")
, paste0(strategy,'.R'))
function
# load strategy
'strategies'
cat
"Sourcing"
"\n"
source
# load in getOrders
# set params via global assignment
[[
print print
}
(params)
The structure of a strategy
Here is the contents of the strategy file backtester_v5.2/strategies/fixed.R:
# This strategy only uses market orders
# params$sizes specifies a fixed number of contracts per series
# No further orders are placed by getOrders
getOrders <- (store, newRowList, currentPos, info, params) { allzero <- (0,length(newRowList))
marketOrders <- allzero
if( (store {
marketOrders <- params$sizes
store <- 1 # not null
}
return(list(store=store,marketOrders=marketOrders,
limitOrders1=allzero,
limitPrices1=allzero,
limitOrders2=allzero,
# We take the corresponding long/short position in each series
# by placing a market order on the 1st iteration
# The backtester automatically exits all positions
# as market orders at the end when the data runs out
function
rep
is.null
))
# take position during first iteration and hold
limitPrices2=allzero))
}
The backtester framework runs a strategy by calling getOrders. The arguments to getOrders are fixed, i.e., they are the same for all strategies. In the example strategy fixed.R, getOrders is the only function. The arguments to getOrders are as follows:
• store: contains all data you choose to save from one period to the next • newRowList: new day's data (a list of single rows from the series)
• currentPos: the vector of current positions in each series
• params: a list of parameters that are sent to the function
Here's how the strategy fixed.R works. In the very first period the backtester always (for every strategy) passes store to getOrders with NULL as its value. Thus in the first period, and the first period only, the vector marketOrders will be set to the parameter params$sizes, which should be a vector of positions with length equal to the number of series, which is 5 in this case. In example_strategies.R we see this parameter params$sizes, which is the only parameter for fixed.R, set as follows:
With these sizes, we buy and hold one unit in every series.
getOrders <- function(store, newRowList, currentPos, info, params) {
list(sizes=rep(1,5))
Changing the parameters
We can change the parameters and take positions in only some series and go short in some series, e.g., with:
We can set this either in example_strategies.R, or in main.R, as long as it comes after we have called load_strategy.
params <- list(sizes=c(1,2,0,0,-1))
Active on 100 % of days; PD ratio = −364.37
1000000
999500
999000
998500
Jan Apr Jul
0.06 0.04 0.02 0.00
Jan
0.050 0.025 0.000
−0.025
−0.050
Jan
5
0
−5
Apr Jul
Apr Jul
Apr Jul
0 −500 −1000 −1500
0.050 0.025 0.000
−0.025 −0.050
Jan Apr Jul
Jan Apr Jul
03 : PD ratio = 0
05 : PD ratio = −4.02
Jan
01 : PD ratio = 0.06 / 0.03 = 1.97
02 : PD ratio = −360.4
04 : PD ratio = 0
Compare the new equity curves with the ones above. Note that for series 1 they are the same, for series 2 the new one is scaled by 2, for series 3 and 4 we no longer trade, and for series 5 we now take a short position, so the new series 5 is a reflection (in the profit and loss axis) of the old series 5 equity curve.
Market orders
The fixed example strategy enters position at the first opportunity using market orders. The backester framework supports market and limit orders and some of the example strategies use limit orders. However, we will not use limit orders for assignment 2, only market orders.
Recall that market orders specify volume and direction (but not price), and limit orders specify price, volume, and direction. In the backtester framework, trading decisions are made after the close of day k, and trades are executed on day k+1. For each day, the framework supports one market order for each series, and two limit orders for each series. These orders are returned from getOrders as follows:
return(list(store=store,marketOrders=marketOrders,
limitOrders1=limitOrders1,
limitPrices1=limitPrices1,
limitOrders2=limitOrders2,
limitPrices2=limitPrices2))
Market orders will be executed at the open on day k+1. The sizes and directions of market orders are encoded in the vector marketOrders of the return list of getOrders. For example, the vector
means place a market order for 5 units short in series 2, and 1 unit long in series 4.
To repeat, we will not use limit orders for assignment 2, so you can leave limitOrders1,
limitPrices1, limitOrders2, limitPrices2 as zero vectors when you do assignment 2. Subsetting the data
The next thing in main.R is a subsetting of the time period for the backtest as follows:
So we are only using the first 200 days.
Running the backtest
Finally we actually do the backtest and plot the results as follows:
The arguments to the function backtest are the following:
• dataList - list of (daily) xts objects (with identical indexes) • getOrders - the strategy
• params - the parameters for the strategy
• sMult - slippage multiplier (proportion of overnight gap)
Results for individual series are available in results$pnlList. The portfolio results are available in pfolioPnL, which is produced by plotResults(dataList,results). This function also automatically plots individual and aggregate equity curves, and computes variant of the Calmar Ratio that we call the Profit Drawdown Ratio (PD ratio for short) - it is the final profit divided by the maximum drawdown in terms of profit and loss, or if the
c(0,-5,0,1,0)
inSampDays <-
dataList <- (dataList, (x) x[1:inSampDays])
200
# in-sample period 1:inSampDays
lapply
function
Hint
You should adapt this use of lapply on dataList in order to define the in-sample period in assignment 2.
# Do backtest
results <- backtest(dataList,getOrders,params,sMult=0.2)
pfolioPnL <- plotResults(dataList,results)
strategy makes a loss overall the PD ratio is just that loss (which is negative). You do not need to write code to compute this, since it has already been done for you. In assignment 2 we will optimize the aggregate PD ratio - this value is stored in pfolioPnL$fitAgg, e.g., for our first example we have:
This matches up with the PD ratio that appears at the top of the aggregate equity curve produced by plotResults.
Parameter optimization
Before we move on to assignment 2, we will briefly look at an example of parameter optimization that will be useful for assignment 2. To make it easier to carry out parameter optimizations, getOrders takes an argument params. This can be used to pass a parameter combination to a strategy. This in turn can be used to do a parameter optimization as main_optimize.R demonstrates. Here is the source code for main_optmize.R, which uses the example strategy bbands_contrarian, which is an implementation of the "BBands Overbought/Oversold" strategy from the slides:
> (pfolioPnL$fitAgg) [1]
print
-153.44
(() (()
numOfDays <- 200
dataList <- getData(directory= )
dataList <- (dataList, (x) x[1:numOfDays]) sMult <-
lookbackSeq <- (from= ,to=40,by= ) sdParamSeq <- (from= ,to=2,by= ) paramsList <- (lookbackSeq,sdParamSeq) numberComb <- ( (paramsList,
resultsMatrix <- matrix(nrow=numberComb,ncol=3) colnames(resultsMatrix) <- c( ,"sdParam","PD Ratio") pfolioPnLList <- vector(mode= ,length=numberComb)
count <- 1
for (lb in lookbackSeq) {
for (sdp in sdParamSeq) {
params <- list(lookback=lb,sdParam=sdp,series=1:5,posSizes=rep(1,5)) results <- backtest(dataList, getOrders, params, sMult)
pfolioPnL <- plotResults(dataList,results)
resultsMatrix[count <- c(lb,sdp,pfolioPnL$fitAgg)
pfolioPnLList count <- pfolioPnL
source
source
'framework/data.R');
source
'framework/backtester.R'
'framework/processResults.R'
);
source
'strategies/bbands_contrarian.R'
"EXAMPLE"
lapply
0.2
# slippage multiplier
function
seq seq
20
10
1.5
0.5
list
length
))
prod
sapply
"lookback"
"list"
,]
[[
]]
( ,count,
count <- count + 1
}
}
print(resultsMatrix[order(resultsMatrix
,numberComb,"\n")
cat
"Just completed"
"out of"
#print(resultsMatrix[count,])
[,"PD Ratio"
]),])
The code template and data for assignment 2
You are now ready to start working on assignment 2. To do so you should read and work through the rest of this document very carefully.
As a first step, try to run main_template.R, which is setup to use the right data for the assignment and to load in the template strategy code strategies/a2_template.R. If you try to source main_template.R you will get an error as follows:
If you read on you will see that the final strategy requires a parameter called lookbacks. Read on to see what form this parameter should take.
The code template contains templates for the 6 functions that you need to complete. These functions are:
1. getTMA
2. getPosSignFromTMA
3. getPosSize
4. getOrders
5. getInSampleResult
6. getInSampleOptResult
The rest of the document is split into two parts. The first part describes the function requirements and marking criteria for the first 4 functions, which relate to the strategy implementation. The second part describes the function requirements and marking criteria for the last 2 functions. Hints are given on how best to implement things, so read carefully. For all 6 functions, example outputs are provided so that you can test whether you have implemented the functions correctly.
Error in if (store$iter > params$lookbacks$long) { :
argument is of length zero
Note
You can develop the first three functions without running the backtester, which may be easier.
Part 1: strategy implementation
The overall goal of the assignment is the implementation and optimization of a triple moving average crossover (TMA) trading strategy. The specification of the strategy and the functions that it should comprise are given in full detail, so the correctness of your code can and will be checked automatically.
The TMA strategy you will implement is related to Example 1 in COMP226 slides 17. However, long and short positions are swapped as compared to that example (so you will here implement a mean-reversion as opposed to a trend following type strategy).
The strategy uses three moving averages with three different lookbacks (window lengths). The short lookback should be smaller than the medium window, which in turn should be smaller than the long lookback. In every trading period, the strategy will compute the value of these three moving averages. You will achieve this be completing the implementation of the function getTMA.
The following table indicates the position that the strategy will take depending on the relative values of the three moving averages (MAs). You will compute this position (sign, but not size) by completing the function getPosSignFromTMA. The system is out of the market (i.e., flat) when the relationship between the short moving average and the medium moving average does not match the relationship between the medium moving avergage and long moving average.
The function getPosSignFromTMA should use a function getTMA. The position size, i.e., the number of units to be long or short, will be determined by the function getPosSize. Finally, as usual in the backtester framework for COMP226 and COMP396, the position sizes are given to the backtester in the function getOrders. Here are the detailed specification and marks available for these first 4 functions.
MA
MA
MA
Position
short
>
medium
>
long
short
short
<
medium
<
long
long
Function name
Input parameters
Expected behaviour
Marks available for a correct implementation
getTMA
close_prices; lookbacks. The specific form that these arguments should take is specified in the template code via the 6 checks that you need to implement.
You should first implement the checks as described in the template. Hints of how to implement them are given below.
The function should return a list with three named elements (named short, medium, and long). Each element should be equal to the value of a simple moving average with the respective window size as defined by lookbacks. The windows should all end in the same period, which should be the final row of close_prices.
18% (3% per check) for the checks; 12% for a correct return
getPosSign
FromTMA
tma_list is a list with three named elements, short, medium, and long. These correspond to the simple moving averages as returned by getTMA.
Note: You do not need to check the validity of the function argument in this case, or for the remaining functions either.
This function should return either 0, 1, or -1.
If the short value of tma_list is less than the medium value, and the medium value is less than the long value, it should return 1 (indicating a long position).
If the short value of tma_list is greater than the medium value, and the medium value is greater than the long value, it should return -1 (indicating a short position). Otherwise, the return value should be 0 (indicating a flat position).
15%
getPosSize
current_close: this is the current close for one of the series. constant: this argument should have a default value of 1000.
The function should return (constant divided by current_close) rounded down to the nearest integer.
5%
getOrders
The arguments to this function are always the same for all strategies used in the backtester framework.
This function should implement the strategy outlined above and again below in "Strategy specification".
20%
Strategy specification
The strategy should apply the following logic independently for both series.
The strategy does nothing until there have been params$lookbacks$long-many periods.
In the (params$lookbacks$long+1)-th period, and in every period after, the strategy computes three simple moving averages with window lengths equal to:
• params$lookbacks$short
• params$lookbacks$medium • params$lookbacks$long
The corresponding windows always end in the current period. The strategy should in this period send market orders to assume a position (make sure you take into account positions from earlier) according to getPosSignFromTMA and getPosSize. (Limit orders are not required at all, and can be left as all zero.)
Hints
For the checks for getTMA you may find the following functions useful:
• The operator ! means not, and can be used to negate a boolean.
• sapply allows one to apply a function element-wise to a vector or list (e.g., to a vector list c("short","medium","long")).
• all is a function that checks if all elements of a vector are true (for example, it can be used on the result of sapply).
• %in% can be used to check if a element exists inside a vector.
To compute the moving average in getTMA you can use SMA from the TTR package.
Note: The list returned by getTMA should work as input to the function getPosSignFromTMA.
For getPosSize, you can use the function floor.
As in the template, use the negative of currentPos summed with the new positions you want to take to make sure that you assume the correct position.
In order to help you check whether you have implemented the functions correctly, we next give some examples of how correct implementations of the functions will behave. These examples assume that you have correctly implemented the first 4 functions in a2_template.R and sourced the resulting code to make the functions available in the R environment.
Example output for getTMA
First you should make sure that you have correctly implemented all 6 checks on the function arguments. Here are 3 examples of expected behaviour:
> close_prices <- c(1,2,3)
> lookbacks <- list(short= (5 medium= ( long=as.integer(20)) > getTMA(close_prices,lookbacks)
Error in getTMA(close_prices, lookbacks) :
E04: close_prices is not an xts according to is.xts()
> dataList <- getData(directory="A2")
Read 2 series from DATA/A2
> close_prices <- dataList[[1]]$Close[1: ]
> getTMA(close_prices,lookbacks)
Error in getTMA(close_prices, lookbacks) :
E05: close_prices does not enough rows
> lookbacks <- list(5,10,25)
> getTMA(close_prices,lookbacks) # bad lookbacks; list elements not named
Error in getTMA(close_prices, lookbacks) :
E01: At least one of “short”, “medium”, “long” is missing from names(lookbacks)
10
),
as.integer
),
as.integer
# bad close_prices
19
# bad close_prices; too short
Here is an example where we give the function valid arguments.
10
),
> lookbacks <- list(short= (5),medium=as.integer( long=as.integer(20)) > close_prices <- dataList 1 $Close[1:20]
> getTMA(close_prices,lookbacks)
$short
[1] 169
$medium
[1] 170.4
$long
[1] 171.05
as.integer
[[
]]
Example output for getPosSignFromTMA
Example output for getPosSize
Example output for getOrders
To check your implementation of getOrders, see part 2 for examples of correct output for the function getInSampleResult below.
> getPosSignFromTMA(list(short=10,medium=20,long=30))
[1] 1
> getPosSignFromTMA(list(short=10,medium=30,long=20))
[1] 0
> getPosSignFromTMA(list(short=30,medium=20,long=10))
[1] -1
> current_close <- 100.5
> getPosSize(current_close)
[1] 9
> getPosSize(current_close,constant=100.4)
[1] 0
Part 2: in-sample tests
Warning
The last two functions require a working implementation of getOrders. If your implementation of getOrders does not work, then you will receive 0 marks for these last two functions, even if they return the correct numbers. This is a simple protection again plagiarism and collusion.
There are two more functions that you need to implement: getInSampleResult and getInSampleOptResult. For both functions you will need to compute your own in-sample period, which is based on your MWS username. This ensures that for part 2 there are different answers for different students.
To get your in-sample period you should use in-sample_period.R as follows. Source it and run the function getInSamplePeriod with your MWS username as per the following example. Then use the first number in the returned vector as the start of the in-sample period and the second number as the end.
So for this example username the start of the in-sample period is day 230 and the end is 644. Note: you may need to install the package digest to use this code.
Once you have your own in-sample period (and a correct implementation of getOrders), you are ready to complete the implementation of getInSampleResult.
> source( ) > getInSamplePeriod( ) [1] 230 644
‘in-sample_period.R’
‘x4xz1’
Function name
Input par ameters
Expected behaviour
Marks available for a correct implementation
getInSampleR
esult
None
This function should return the PD ratio that is achieved when the strategy is run with short lookback 10, medium lookback 20, and long lookback 30, on your username-specific in-sample period.
The function should not contain ANY code except the return value; it should run and complete instantanously.
10% (0 marks will be given if the function contains any code other than the return statement)
To complete the final function getInSampleOptResult you need to do an in-sample parameter optimization using the following parameter combinations for the:
• short lookback
• medium lookback • long lookback
You should not optimize the constant used with getPosSize, and leave it as 1000 as defined in the template code.
The parameter combinations are defined by two things: parameter ranges and a further restriction. Make sure you correctly use both to produce the correct set of parameter combinations. The ranges are:
The further restriction is the following:
Parameter
Minimum value
Increment
Maximum Value
short lookback
100
5
110
medium lookback
105
5
120
long lookback
110
5
130
Further restriction on parameter values
You should further restrict the parameter combinations as follows:
• The medium lookback should always be strictly greater than the short lookback. • The long lookback should always be strictly greater than the medium lookback.
You need to find the best PD ratio that can be achieved one this set of parameter combinations for the in-sample period that corresponds to your username, and set it as the return value of getInSampleOptResult.
Hint
The correct resulting number of parameter combinations is 28.
You can adapt backtester_v5.2/main_optimize.R. It is probably easiest to use three nested for loops in order to ensure that you only check valid parameter combinations (where the short < medium < long for the respective window lengths).
Function name
Input pa rameters
Expected behaviour
Marks available for a correct implementation
getInSampleO
ptResult
None
This function should return the best PD ratio than can be achieved with the stated allowable parameter combinations on your username-specific in-sample period.
The function should not contain ANY code except the return value; it should run and complete instantanously.
20% (0 marks will be given if the function contains any code other than the return statement)
Next we give some example output for these two functions.
Example output for getInSampleResult
Username
Correct return value
x1xxx
-747.6
x1yyy
-231.6
x1zzx
-639.8
Example output for getInSampleOptResult
Marks summary
Username
Correct return value
x1xxx
4.23
x1yyy
3.42
x1zzx
4.43
Function
Marks
getTMA
30
getPosSignFromTMA
15
getPosSize
5
getOrders
20
getInSampleResult
10 (0 if getOrders does not work)
getInSampleOptResult
20 (0 if getOrders does not work)
Submission
You need to submit a single R file that contains your implementation of 6 functions. The file shoud be called MWS-username.R where you should replace MWS-username by your MWS username. For example if your username is "abcd" then you should submit a file named "abcd.R".
Submission is via the department electronic submission system:
http://www.csc.liv.ac.uk/cgi-bin/submit.pl
Warning
Your code will be put through the department's automatic plagiarism and collusion detection system. Student's found to have plagiarized or colluded will likely receive a mark of zero. Do not discuss or show your work to other students.