# Portfolio Management with R

Enrico Schumann        [PDF] [Code]

## 1 Introduction

This manual describes how to use the PMwR package. The aim of PMwR is to provide a small set of reliable, efficient and convenient tools that help in processing and analysing trade and portfolio data. The package does not provide a complete application that could be used as is'; the package provides building blocks for creating such an application.

PMwR grew out of various pieces of software that I have written since 2008, first at the University of Geneva, later during my work at financial firms.

The package is currently under development and changes frequently. This is mainly because the code has been written over many years and is in need of being groomed for general use.

Consequently, this manual is only a draft, and it will change as frequently as the package.1 I am grateful for comments and suggestions.

The latest version of the package is available from http://enricoschumann.net/R/packages/PMwR/index.htm. To install the package from within R, type

install.packages("PMwR",
type  = "source",
repos = c('http://enricoschumann.net/R',
getOption('repos')))


to download and install it. The package depends on several other packages, which are automatically obtained from the same repository and from CRAN. The source code is also pushed to a public repository at https://github.com/enricoschumann/PMwR.

There is currently no automatic build process for Windows. Recent versions of the package (since 0.3-4) are pure R code and can be built without any prerequisites except an R installation; older version contained C code, so you needed to have Rtools installed. If you have problems building the package for Windows, please contact me and I will provide you with a Windows version.

### 1.2 Other packages

Several other packages originated from PMwR (in older versions, much of their code had been part of PMwR).

datetimeutils
tools for handling Date and POSIXct objects

textutils
string handling for reporting (plain text, HTML, LaTeX)

tsdb
terribly-simple database for time series

## 2 Keeping track of transactions: journals

The ultimate basis of many financial computations are lists of transactions. And so many of the tools that the PMwR package provides take lists of transactions as input.

Conceptually, you can think of such lists as dataframes, but PMwR provides an S3 class journal for handling them.

A journal is a list of atomic vectors, to which a class attribute is attached. Such a list is created through the function journal. Methods should not rely on this list being sorted in any particular way: components of a journal should always be retrieved by name, never by position. (In this respect a journal differs from a dataframe, for which we can meaningfully refer to the n-th column.)

I will call these journal components, such as amount or timestamp, fields.

The simplicity of the class is intended, because it is meant for interactive analyses. The user may – and is expected to – dissect the information in a journal at will; such dissections include removing the class attribute.

What is actually stored in a journal is up to the user. A number of fields are, however, required for certain operations and so it is recommended that they be present:

amount
The notional amount that is transacted. amount is, in a way, the most important property of a journal. When functions compute something from the journal (the number of transactions, say), they will look at amount.
timestamp
When did the transaction take place? A numeric or character vector; should be sortable.
price
Well, price.
instrument
Description of the financial instrument; typically an identifier, a.k.a. ticker or symbol. That is, a string, not really a more complex object (recall that journals are lists of atomic vectors).
id
A (possibly but not necessarily unique) transaction identifier.
account
Description of the account.
...
other fields. They must be named, as in fees = c(1,2,1)

All fields except amount can be missing. Such missing values will be added back' as NA with the exception of id and account, which will be NULL. (Note that amount could be a vector with only NA values in it, but amount cannot be left out when the journal is created. This will become clearer with the examples below.) Note that when a journal has no transactions in it (which may be the case), this does not mean it has missing values. In such a case, all fields have length zero, e.g. amount would be numeric(0) and so on. Such empty journals may be created by saying journal() or by coercing a zero-row data.frame to a journal, via a call to as.journal.

Transactions in a journal can conceptually be organised in hierachies, such as account => instrument. (You may have traded stock XY for several different accounts, or as part of different strategies.) A simple scheme is to use specific characters or a pattern such as :: to introduce hierarchies into accounts,2 such as pension::equities.

### 2.1 Creating and combining journals

The function journal creates journal objects. For details about the function and methods for journal objects, see ?journal.

At is very minimum, a journal must contain amounts of something.

jnl <- journal(amount = c(1, 2, -2, 3))
jnl

   amount
1       1
2       2
3      -2
4       3

4 transactions


Actually, that is not true. On occasion it is useful to create an empty journal, one with no entries at all. You can do this by saying journal(), without any arguments.

journal()

no transactions


To see the current balance, which is nothing more than the sum over all amounts, you can use position.

position(jnl)

     [,1]
[1,]    4


Only providing amounts is, admittedly, not overly useful. You can keep track of positions, true; but a journal implies chronological information, that is, flows. (As opposed to a ledger, which gives you positions, or stocks.)

When you make sure that the amounts are actually sorted in time, then you can at least track positions over time. (But nothing in the data structure that we created above could make sure that transactions really are sorted.)

Suppose you wanted to note how many bottles of milk and wine you have stored in your basement. Whenever you add to your storage, you have a positive amount; whenever you retrieve bottles, you have a negative amount. Then, by keeping track of transactions, you may not have to take stock (apart, perhaps, from occasional checking that you did not miss a transaction), as long as you keep track of what you put into your cellar and what you take out.

There may be some analyses you can do on flows alone (without any information): perhaps checking your drinking habits for patterns, such as slow accumulation of wine, followed by rapid consumption; or the other way around.

But typically, the more information you store about transactions (when, what, at what price, etc.), the better. Journals allow you to store such information. To show how they are used, let us switch to a financial example.

J <- journal(timestamp  = as.Date("2012-01-01") + 0:3,
amount     = c(1, 2, -2, 5),
instrument = c("EUR", "EUR", "CHF", "CHF"),
comment    = c("initial balance", "",
"transfer", ""))
J

   instrument   timestamp  amount          comment
1         EUR  2012-01-01       1  initial balance
2         EUR  2012-01-02       2
3         CHF  2012-01-03      -2         transfer
4         CHF  2012-01-04       5

4 transactions


A print method defines how a journal is displayed. See ?print.journal for details. (In general, you can always get help for methods for generic functions by saying ?function_name.journal, e.g. ?print.journal or ?as.data.frame.journal.)

print(J, max.print = 2, exclude = "instrument")

    timestamp  amount          comment
1  2012-01-01       1  initial balance
2  2012-01-02       2
[ .... ]

4 transactions


A str method shows the fields in the journal.

str(J)

'journal':	 4 transactions
$instrument: chr [1:4] "EUR" "EUR" "CHF" "CHF"$ timestamp : Date[1:4], format: "2012-01-01" "2012-01-02" ...
$amount : num [1:4] 1 2 -2 5$ price     : logi [1:4] NA NA NA NA
$comment : chr [1:4] "initial balance" "" "transfer" ""  You may notice that the output is similar to that of a data.frame or list. That is because J is a list internally, with a class attribute. Essentially, it is little more than that: list(timestamp = as.Date("2012-01-01") + 0:3, amount = c(1, 2, -2, 5), instrument = c("EUR", "EUR", "CHF", "CHF"), comment = c("initial balance", "", "transfer", ""))  (But note that journal silently added a price field, even though we did not specify one.) In the example, the timestamps are of class Date. But essentially, any vector of mode character or numeric can be used, for instance POSIXct, or other classes. Here is an example that uses the nanotime package \citep{eddelbuettel2017}. require("nanotime") journal(amount = 1:3, timestamp = nanotime(Sys.time()) + 1:3)   timestamp amount 1 2017-01-23T08:10:25.404530001+00:00 1 2 2017-01-23T08:10:25.404530002+00:00 2 3 2017-01-23T08:10:25.404530003+00:00 3 3 transactions  Journals can be combined with c. J2 <- J J2$fees <- rep(0.1,4)
c(J, J2)

   instrument   timestamp  amount          comment  fees
1         EUR  2012-01-01       1  initial balance    NA
2         EUR  2012-01-02       2                     NA
3         CHF  2012-01-03      -2         transfer    NA
4         CHF  2012-01-04       5                     NA
5         EUR  2012-01-01       1  initial balance   0.1
6         EUR  2012-01-02       2                    0.1
7         CHF  2012-01-03      -2         transfer   0.1
8         CHF  2012-01-04       5                    0.1

8 transactions


But we wanted the combined journal sorted by date.

sort(c(J, J2))

   instrument   timestamp  amount          comment  fees
1         EUR  2012-01-01       1  initial balance    NA
2         EUR  2012-01-01       1  initial balance   0.1
3         EUR  2012-01-02       2                     NA
4         EUR  2012-01-02       2                    0.1
5         CHF  2012-01-03      -2         transfer    NA
6         CHF  2012-01-03      -2         transfer   0.1
7         CHF  2012-01-04       5                     NA
8         CHF  2012-01-04       5                    0.1

8 transactions


We can also sort by some other field, such as amount.

sort(c(J, J2), by = "amount", decreasing = TRUE)

   instrument   timestamp  amount          comment  fees
1         CHF  2012-01-04       5                     NA
2         CHF  2012-01-04       5                    0.1
3         EUR  2012-01-02       2                     NA
4         EUR  2012-01-02       2                    0.1
5         EUR  2012-01-01       1  initial balance    NA
6         EUR  2012-01-01       1  initial balance   0.1
7         CHF  2012-01-03      -2         transfer    NA
8         CHF  2012-01-03      -2         transfer   0.1

8 transactions


### 2.2 Selecting transactions

In an interactive session, you can use subset to select particular transactions.

subset(J, amount > 1)

   instrument   timestamp  amount  comment
1         EUR  2012-01-02       2
2         CHF  2012-01-04       5

2 transactions


With subset, you need not quote the expression that selects trades and you can directly access a journal's fields. Because of the way subset evaluates its arguments, it should not be used within functions. (See the Examples section in ?journal for what can happen then.)

More generally, to extract or change a field, use its name, either through the $ operator or double brackets [[...]].3 J$amount

[1]  1  2 -2  5


You can also replace specific fields.

J[["amount"]] <- c(1 ,2, -2, 8)
J

   instrument   timestamp  amount          comment
1         EUR  2012-01-01       1  initial balance
2         EUR  2012-01-02       2
3         CHF  2012-01-03      -2         transfer
4         CHF  2012-01-04       8

4 transactions


The [ method works with integers or logicals, returning the respective transactions.

J[2:3]

   instrument   timestamp  amount   comment
1         EUR  2012-01-02       2
2         CHF  2012-01-03      -2  transfer

2 transactions

J[J$amount < 0]   instrument timestamp amount comment 1 CHF 2012-01-03 -2 transfer 1 transaction  You can also pass a string, which is then interpreted as a regular expression that is matched against all character fields in the journal. J["eur"]   instrument timestamp amount comment 1 EUR 2012-01-01 1 initial balance 2 EUR 2012-01-02 2 2 transactions  By default, case is ignored, but you can set ignore.case to FALSE. J["Transfer"]   instrument timestamp amount comment 1 CHF 2012-01-03 -2 transfer 1 transaction  J["Transfer", ignore.case = FALSE]  no transactions  You can also specify the fields to match the string against. J["Transfer", match.against = "instrument"]  no transactions  ### 2.3 Computing balances The function position gives the current balance of all instruments. position(J)   2012-01-04 CHF 6 EUR 3  To get the position at a specific date, use the when argument. position(J, when = as.Date("2012-01-03"))   2012-01-03 CHF -2 EUR 3  If you do not like such a tabular view, consider splitting the journal. lapply(split(J, J$instrument),
position, when = as.Date("2012-01-03"))

 $CHF 2012-01-03 CHF -2$EUR
2012-01-03
EUR          3


To get a time series of positions, you can use specific keywords for when: all will print the position at all timestamps in the journal.

position(J, when = "all")

           CHF EUR
2012-01-01   0   1
2012-01-02   0   3
2012-01-03  -2   3
2012-01-04   3   3


Keywords first and last give you the first and last position. (The latter is the default; so if when is not specified at all, the last position is computed.) endofmonth prints the positions at the ends of all calendar months between the first and the last timestamp.

We are not limited to the timestamps that exist in the journal.

position(J, when = seq(from = as.Date("2011-12-30"),
to = as.Date("2012-01-06"),
by = "1 day"))

           CHF EUR
2011-12-30   0   0
2011-12-31   0   0
2012-01-01   0   1
2012-01-02   0   3
2012-01-03  -2   3
2012-01-04   6   3
2012-01-05   6   3
2012-01-06   6   3


By default, position will show you all positions, even if they are zero.

J <- c(J, journal(instrument = "EUR",
amount = -3,
timestamp = as.Date("2012-01-05")))
position(J)

    2012-01-05
CHF          6
EUR          0


You can suppress such positions with drop.zero.

position(J, drop.zero = TRUE)

    2012-01-05
CHF          6


drop.zero can also be a tolerance, which is useful in cases such as this one:

J <- c(J, journal(instrument = "USD",
timestamp  = as.Date("2012-01-05"),
amount     = c(0.1, 0.1, 0.1, -0.3)))
position(J, drop.zero = TRUE)

    2012-01-05
CHF   6.00e+00
USD   2.78e-17

position(J, drop.zero = 1e-15)

    2012-01-05
CHF          6


### 2.4 Aggregating journal information

A straightforward strategy is to call as.data.frame on a journal and then to use aggregate on the resulting data.frame.

The downside to this approach is that in aggregate.data.frame, you cannot address specific columns of the data.frame. So PMwR provides an aggregate.journal method that allows you to do that.

The by argument can be a list (as in the default method) or an atomic vector. The latter will be placed into a list automatically: this is useful for cases such as when by is format(j$timestamp, "%Y-%m"). FUN can either be a function or list. If it is function, it should expect to receive a journal and also evaluate to a journal. An example: given a journal jnl, compute monthly turnover (two-way). aggregate(jnl, by = format(jnl$timestamp, "%Y-%m"),
FUN = function(j) {
journal(timestamp = unique(j$timestamp), amount = sum(abs(j$amount)))
}
)


FUN can also be list. In this case, the elements of the list should be named functions.

jnl <- journal(instrument = c("A", "B", "A", "B"),
price  = c(100, 102, 105, 109),
amount = c(  1,   1,   1,   1))
aggregate(jnl,
by = jnl$instrument, FUN = list(price = min, amount = sum))  #### 2.4.1 Aggregate daily orders Suppose we have the following trades, collect in a journal j.  instrument timestamp amount price 1 B 2013-09-02 -3 104 2 B 2013-09-03 -4 102 3 A 2013-09-02 -3 102 4 A 2013-09-03 -1 110 5 B 2013-09-02 3 106 6 B 2013-09-02 -2 104 7 A 2013-09-03 1 104 8 B 2013-09-03 3 106 9 A 2013-09-03 5 108 10 A 2013-09-03 3 107 10 transactions  by <- list(j$instrument, sign(j$amount), as.Date(j$timestamp))
fun <- function(x) {
journal(timestamp = as.Date(x$timestamp[1]), amount = sum(x$amount),
price = sum(x$amount*x$price)/sum(x$amount), instrument = x$instrument[1L])
}
aggregate(j, by = by, FUN = fun)

   instrument   timestamp  amount     price
1           A  2013-09-02      -3  102.0000
2           B  2013-09-02      -5  104.0000
3           B  2013-09-02       3  106.0000
4           A  2013-09-03      -1  110.0000
5           B  2013-09-03      -4  102.0000
6           A  2013-09-03       9  107.2222
7           B  2013-09-03       3  106.0000

7 transactions


## 3 Computing profit and loss

In this chapter we will deal with computing profit and loss in currency units. If you are interested in computing returns, see Computing returns.

### 3.1 Simple cases

We buy one unit of an asset at a price of 100 euro and we sell it for 102 euro. We have made a profit of 2 euros. This simple case is frequent enough that we should make the required computation simple as well. The PMwR package provides a function pl, which for this case may be called like as follows.

pl(price  = c(100, 102),
amount = c(1, -1))

P/L total       2
average sell  102
volume          2

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


In principle, P/L is straightforward to compute: let $$x$$ be a vector of the absolute amounts traded, and $$p$$ be a vector of the prices at which we traded, then \textsc{P/L} is just the difference between what we gained from selling and what we paid when buying.

$$\label{eq:pl1} \sum x^{\text{sell}}_i p^{\text{sell}}_i - \sum x^{\text{buy}}_i p^{\text{buy}}_i$$

This can be simplified when we impose the convention that sold amounts are negative.

\begin{eqnarray} \label{eq:pl2} \text{P/L} &=& -\sum_{x<0} x_i p_i - \sum_{x>0} x_i p_i \\ &=& -\sum x_i p_i \end{eqnarray}

The function pl also follows this convention: note that in the code example above we had $$x = [1, -1]'$$.

There are many ways to compute PL.

amount <- rep(c(-100,100), 10)
price <- rep(100, length(amount))

amount %*% price
sum(amount*price)
crossprod(amount, price)
t(amount*price) %*% rep(1, length(amount))

require("rbenchmark")
benchmark(
amount %*% price,
sum(amount*price),
crossprod(amount, price),
t(amount*price) %*% rep(1, length(amount)),
columns = c("test",    "elapsed", "relative"),
order = "relative",
replications = 10000)


You will see shortly that pl also accepts an argument instrument, in which case P/L will be split by instrument. But if you only care about total P/L, you can do it in one computation. As an example, suppose we traded shares of Adidas (ticker ADS) and Commerzbank (ticker CBK), two German companies.

pl(amount     = c(   50,   -50,   500,  -500),
price      = c(  100,   102,     7,     8),

ADS
P/L total     100
average sell  102
volume        100

CBK
P/L total     500
average sell    8
volume       1000

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


Since the shares are denominated in the same currency (EUR) and have the same multiplier (1), total profit is the same when you leave out the tickers, though average buying and selling price become less meaningful.

pl(amount     = c(   50,   -50,   500,  -500),
price      = c(  100,   102,     7,     8))

P/L total      600
average sell  16.5
volume        1100

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


If multipliers are not the same, you need to specify them. Suppose we have traded FGBL (German Bund futures) and FESX (EURO STOXX 50 futures).

jnl <- read.table(textConnection(
"instrument, price, amount
FGBL MAR 16, 165.20,  1
FGBL MAR 16, 165.37, -1
FGBL JUN 16, 164.12,  1
FGBL JUN 16, 164.13, -1
FESX JUN 16,   2910,  5
FESX JUN 16,   2905, -5"
), header = TRUE, stringsAsFactors = FALSE, sep = ",")

jnl <- as.journal(jnl)
jnl

     instrument  amount  price
1   FGBL MAR 16       1    165
2   FGBL MAR 16      -1    165
3   FGBL JUN 16       1    164
4   FGBL JUN 16      -1    164
5   FESX JUN 16       5   2910
6   FESX JUN 16      -5   2905

6 transactions


One point of the FGBL means 1000 euros; for the FESX it is 10 euros.

pl(jnl,
multiplier = c("FGBL" = 1000, "FESX" = 10),
multiplier.regexp = TRUE)

 FESX JUN 16
P/L total     -250
average sell  2905
volume          10

FGBL JUN 16
P/L total      10
average sell  164
volume          2

FGBL MAR 16
P/L total     170
average sell  165
volume          2

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


In the examples above, we computed total P/L. As another example, suppose that a stock trader bought one share at 50, one share at 90 and sold two shares at 100. These trades resulted in a profit of 60, or an average return of more than +40% (bought at 70, sold at 100).

jnl <- journal(price  = c( 90, 50, 100),
amount = c(  1,  1,  -2))
pl(jnl)

P/L total      60
average sell  100
volume          4

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


That may appear like some pretty good trading. Yet suppose that the order of the trades was

buy at 90  =>  buy at 50  =>  sell at 100.


You may have noticed that the journal that we created above already has the trades ordered this way. We may not know what was traded and when, but there is clearly some information in the order of the trades and the drawdown that it implies: namely a drawdown (mark-to-market loss) of at least 40 before it recovered. For situations like this, the argument along.timestamp can be used.

pl(jnl, along.timestamp = TRUE)

P/L total       0 -40  60
__ realised     0   0  60
__ unrealised   0 -40   0
average sell  100
volume          1   2   4

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


Note that we do not provide an actual timestamp, in which case the function will implicitly use integers 1, 2, …, length(amount). With no further arguments, as here, the function will compute the running position and evaluate it at every trade with the trade's price. This may not be totally accurate because of bid–ask spreads or other transaction costs. But it provides more information than only computing the aggregate P/L for the trades.

At this point, it may be helpful to describe how we can access these data. The function pl always returns a list of lists – one list for each instrument. Actually, we have not yet called pl for more than one instrument, so let us do that first, to have a concrete example.

jnl <- journal(price = c(100, 102, 5, 4),
amount = c(1, -1,  20, -20),
instrument = c("Equity A", "Equity A",
"Equity B", "Equity B"))
pl(jnl)

Equity A
P/L total       2
average sell  102
volume          2

Equity B
P/L total     -20
average sell    4
volume         40

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


It is often more convenient to have the data presented as a table.

as.data.frame(pl(jnl))

          pl buy sell volume
Equity A   2 100  102      2
Equity B -20   5    4     40


Or if you like ASCII tables, with toOrg.

require("orgutils")
toOrg(as.data.frame(pl(jnl)), row.names = "instrument")

| instrument |  pl | buy | sell | volume |
|------------+-----+-----+------+--------|
| Equity A   |   2 | 100 |  102 |      2 |
| Equity B   | -20 |   5 |    4 |     40 |


In any case, as I said, pl returns a list of lists.

str(pl(jnl))

List of 2
$Equity A:List of 6 ..$ pl        : num 2
..$realised : logi NA ..$ unrealised: logi NA
..$buy : num 100 ..$ sell      : num 102
..$volume : num 2$ Equity B:List of 6
..$pl : num -20 ..$ realised  : logi NA
..$unrealised: logi NA ..$ buy       : num 5
..$sell : num 4 ..$ volume    : num 40
- attr(*, "class")= chr "pl"
- attr(*, "along.timestamp")= logi FALSE
- attr(*, "instrument")= chr [1:2] "Equity A" "Equity B"


Each such list contains numeric vectors: 'pl', 'realised', 'unrealised', 'buy', 'sell', 'volume'. Data can be extracted by standard methods. The vectors 'realised' and 'unrealised' will be NA unless along.timestamp is TRUE.

unlist(pl(jnl)[["Equity A"]])

pl   realised unrealised        buy       sell     volume
2         NA         NA        100        102          2

unlist(lapply(pl(jnl), [[, "volume"))

Equity A Equity B
2       40


(You may prefer sapply(...) instead of unlist(lapply(...)).)

Back to use cases for pl. We can also use the function for a journal in which some positions not closed yet. The simplest example is a journal of just one trade.

pl(journal(amount = 1, price = 100))

P/L total       .
average sell    .
volume          1

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.

Warning message:
In pl.default(amount, price, timestamp,
instrument, multiplier = multiplier,  :
'sum(amount)' is not zero: specify 'vprice' to compute p/l


The warning message that is thrown already tells us what to do: to close the position – only for the purpose of computing P/L; no trade really takes place –, specify vprice.

pl(journal(amount = 1, price = 100), vprice = 101)

P/L total       1
average sell  101
volume          1

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.


Note that average sell reflects the specified valuation price. But volume has remained 1 since only 1 unit was actually traded.

A quite common task is to compute P/L over a specified period of time such as one trading day. Suppose yesterday, at market close, we had the following positions.

open_position <- c(fesx201603 = -20, fgbl201603 = 10)
prices <- c(fesx201603 = 2930, fgbl201603 = 165.23)


jnl <- journal(amount = c(2,-3,-10),
price = c(2910,2938,165.80),
instrument = c("fesx201603", "fesx201603",
"fgbl201603"))
jnl

   instrument  amount  price
1  fesx201603       2   2910
2  fesx201603      -3   2938
3  fgbl201603     -10    166

3 transactions

pl(jnl, initial.position = open_position, initial.price = prices,
vprice = c(fesx201603 = 2902, fgbl201603 = 165.60))

fesx201603
P/L total      652
average sell  2931
volume           5

fgbl201603
P/L total     5.7
average sell  166
volume         10

'P/L total' is in units of instrument;
'volume' is sum of /absolute/ amounts.

Warning message:
In pl.default(amount, price, timestamp, instrument,
multiplier = multiplier,  :
all trades are closed for fgbl201603, but 'vprice' is specified


We could have simulated this computation by creating one journal of the initial position and another journal (with reversed amount signs) for the final position, merging all three journals computing P/L.

Note the warning: the final position of fgbl201603 is zero, yet we specified an vprice.

### 3.2 More-complicated cases

Unfortunately, in real life computing P/L is often more complicated:

• One asset-price unit may not translate into one currency unit: we have multipliers or contract factors. That is easy to solve by computing effective position sizes, but it may take some thinking to come up with a reusable scheme (e.g., looking up multipliers in a table).
• Asset positions may map into cashflows in non-obvious ways. The simple case is the delay in actual payment and delivery of an asset, which is often two or three days. The more problematic cases are derivatives with daily adjustments of margins. In such cases, one may need to model (i.e. keep track of) the actual account balances.
• Assets may be denominated in various currencies.
• Currencies themselves may be assets in the portfolio. Depending on how they are traded (cash, forwards, &c.), computing P/L may not be straightforward.

How – or rather, to what degree – these complications are handled is, as always, up to the user. For a single instrument, computing P/L in units of the instrument is always meaningful, though perhaps not always intuitive. But adding up the profits and losses of several assets often will often not work because of multipliers or different currencies. A simple and transparent way is then to manipulate the journal before P/L is computed (e.g., multiply notionals by their multipliers).

## We look at two examples: (i) computing the pl of ## several assets in currency units; and (ii) computing ## time-weighted returns of a portfolio of assets.

#### 3.2.1 Example 1: P/L for lists of incomes and expenses

Suppose you keep a journal just to keep track of your personal income and expenses. An aside: it is tedious to enter journals by hand, in particular if we want to update them over time. So, for practical use, we may write transactions into files, like this:

|  timestamp | amount | comment          |
|------------+--------+------------------|
| 2012-01-01 |    100 | a comment        |
| 2012-01-02 |    200 | another comment  |
| 2012-01-03 |   -200 | bought something |
| 2012-02-04 |    500 | got a present    |


In case you use Org-mode, such tables should look familiar. The org package http://enricoschumann.net/R/packages/org/index.htm provides a function readOrg to read such tables.

## extract org tables from file
use warnings;
use strict;

while (<>) {
print "$_" if /^\s*\|/; }  Some of these transactions may mean a gain or loss to us, such as a dividend payment. Others are neutral, for example a transfer between bank accounts. There are several ways to deal with this.4 One way is to switch to double-entry accounting. A second way is to add descriptions like expense and then subset the journal by these descriptions. And yet another approach is to use prices. Whenever we evaluate the balances of a cash account, we say that the price per unit is one. That is actually reasonable: if my account has a balance of 120.2, it actually means 120 euros and 20 cents, or 120.20 euros'. The price of one euro – no surprise here – is one euro. There are just two rules to remember: 1) An expense should be a negative amount; income should go with positive amounts. 2) When an entry means gain or loss to us, we make its price 0. Then, when we compute that value of a position, we make its price 1. An example: J <- journal(timestamp = c("day 1", "day 2", "day 3"), amount = c(100, 100, -200), price = c( 1, 0, 0), comment = c("neutral", "income", "expense"), account = "my account") J   timestamp amount price account comment 1 day 1 100 1 my account neutral 2 day 2 100 0 my account income 3 day 3 -200 0 my account expense 3 transactions  The current balance is 0. position(J)   day 3 [1,] 0  In terms of income and expenses we have made a loss of 100, which is exactly what the function pl reports. pl(J, current.price = 1)  P/L total -100 average buy 0.5 average sell 0 volume 400 'P/L total' is in units of instrument; 'volume' is sum of /absolute/ amounts.  #### 3.2.2 Example 2: Futures trading and a margin account #### 3.2.3 Example 3: Cash equities with splits and dividends ## 4 Computing returns ### 4.1 Simple returns The function returns computes returns from prices. The function computes what are often called simple returns:5 let $$P_t$$ be the prices in period $$t$$, then $$\label{eq:return} r_t = R_t -1 = \frac{P_t}{P_{t-1}}-1 = \frac{P_t-P_{t-1}}{P_{t-1}}\,.$$ For computing profit/loss in currency units, see Section Computing profit and (or) loss. Typically, we transform a whole series $$P_{t_1}, P_{t_2}, P_{t_3}, \ldots$$ into returns $$R_{t_1}, R_{t_2}, R_{t_3}, \ldots$$, which is a one-liner in R: simple_returns <- function(x) x[-1L]/x[-length(x)] - 1  (You may argue that these are two lines: yet even a one-liner, if used repeatedly, should be written as a function.) Given a vector of prices – here, the closing values of the DAX, the German stock-market index, for the first five business days of 2014, – the function computes returns. P <- c(9400.04, 9435.15, 9428, 9506.2, 9497.84) simple_returns(P)  [1] 0.003735 -0.000758 0.008294 -0.000879  In fact, using returns as provided by PMwR would have given the same result. returns(P)  [1] 0.003735 -0.000758 0.008294 -0.000879  PMwR's returns offers several conveniences. For instance, it will recognise when the input argument has several columns, such as a matrix or a dataframe. In such a case, it computes returns for each column. returns(cbind(P, P))   P P [1,] 0.003735 0.003735 [2,] -0.000758 -0.000758 [3,] 0.008294 0.008294 [4,] -0.000879 -0.000879  The argument pad determines how the initial observation is handled. The default, NULL, means that the first observation is dropped. It is often useful to use NA instead, since in this way the returns series keeps the same length as the original price series. data.frame(price = P, returns = returns(P, pad = NA))   price returns 1 9400 NA 2 9435 0.003735 3 9428 -0.000758 4 9506 0.008294 5 9498 -0.000879  Setting pad to 0 can also be useful, because then it is easy to 'rebuild' the original series with cumprod. (But see Section Scaling series for a description of the function scale1, which is even more convenient.) all.equal(P, P[[1L]] * cumprod(1 + returns(P, pad = 0)))  [1] TRUE  returns is a generic function, which goes along with some overhead. If you need to compute returns on simple data structures as in the examples above and need fast computation, then you may also use .returns. This function is the actual workhorse that performs the raw returns calculation. Besides methods for numeric vectors and dataframes, the function also understands zoo objects. dax <- zoo(P, as.Date(c("2014-01-02", "2014-01-03", "2014-01-06", "2014-01-07", "2014-01-08"))) returns(dax, pad = NA)  2014-01-02 2014-01-03 2014-01-06 2014-01-07 2014-01-08 NA 0.003735 -0.000758 0.008294 -0.000879  Matrices work as well: We create a second series, the prices of the REXP, a German bond-market index. Then, we combine them into a two-column matrix rexp <- zoo(c(440.5252, 440.7944, 441.5456, 441.8197, 441.7619), as.Date(c("2014-01-02", "2014-01-03", "2014-01-06", "2014-01-07", "2014-01-08"))) returns(cbind(DAX = dax, REXP = rexp))   DAX REXP 2014-01-03 0.003735 0.000611 2014-01-06 -0.000758 0.001704 2014-01-07 0.008294 0.000621 2014-01-08 -0.000879 -0.000131  In fact, zoo objects bring another piece of information – timestamps – that returns can use. ### 4.2 Holding-period returns We use two longer series for this section, included in the files PMwR_data_DAX and PMwR_data_REXP. require("tsdb") DAX <- read_ts_tables("PMwR_data_DAX", start = "1970-1-1", return.class = "zoo") colnames(DAX) <- "DAX" REXP <- read_ts_tables("PMwR_data_REXP", start = "1970-1-1", return.class = "zoo") colnames(REXP) <- "REXP"  str(DAX)  'zoo' series from 2014-01-02 to 2015-12-30 Data: num [1:505, 1] 9400 9435 9428 9506 9498 ... - attr(*, "dimnames")=List of 2 ..$ : NULL
..$: chr "DAX" Index: Date[1:505], format: "2014-01-02" "2014-01-03" "2014-01-06" ...  str(REXP)  'zoo' series from 2014-01-02 to 2015-12-30 Data: num [1:502, 1] 441 441 442 442 442 ... - attr(*, "dimnames")=List of 2 ..$ : NULL
..: chr "REXP" Index: Date[1:502], format: "2014-01-02" "2014-01-03" "2014-01-06" ...  When a timestamp is available, returns can compute returns for specific calendar periods. As an example, we look at the daily DAX levels in 2014 and 2015. returns(coredata(DAX), t = index(DAX), period = "month")   Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec YTD 2014 -1.0 4.1 -1.4 0.5 3.5 -1.1 -4.3 0.7 0.0 -1.6 7.0 -1.8 4.3 2015 9.1 6.6 5.0 -4.3 -0.4 -4.1 3.3 -9.3 -5.8 12.3 4.9 -5.6 9.6  If you work with a zoo series, you can omit the timestamp t. returns(DAX, period = "month")   Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec YTD 2014 -1.0 4.1 -1.4 0.5 3.5 -1.1 -4.3 0.7 0.0 -1.6 7.0 -1.8 4.3 2015 9.1 6.6 5.0 -4.3 -0.4 -4.1 3.3 -9.3 -5.8 12.3 4.9 -5.6 9.6  The result of the function call is a numeric vector (the return numbers), with additional information added through attributes. There is also a class attribute, which has value p_returns. The advantage of this data structure is that it is natural' to compute with the returns, e.g. computing means, extremes or similar quantities. range(returns(DAX, period = "month"))  [1] -0.0928 0.1232  Most useful, however, is probably the print method, whose results you have seen above. You may also compute monthly returns for matrices, i.e. for more than one asset. But now the print method will behave differently. Suppose we combine the prices of the DAX and of the REXP. The function's assumption is that now it would be more convenient to print the returns aligned by date in a table. returns(cbind(DAX, REXP), period = "month")   DAX REXP 2014-01-31 -1.0 1.8 2014-02-28 4.1 0.4 2014-03-31 -1.4 0.1 2014-04-30 0.5 0.3 2014-05-30 3.5 0.9 2014-06-30 -1.1 0.4 2014-07-31 -4.3 0.4 2014-08-29 0.7 1.0 2014-09-30 0.0 -0.1 2014-10-31 -1.6 0.1 2014-11-28 7.0 0.4 2014-12-30 -1.8 1.0 2015-01-30 9.1 0.3 2015-02-27 6.6 0.1 2015-03-31 5.0 0.3 2015-04-30 -4.3 -0.5 2015-05-29 -0.4 -0.2 2015-06-30 -4.1 -0.8 2015-07-31 3.3 0.7 2015-08-31 -9.3 0.0 2015-09-30 -5.8 0.4 2015-10-30 12.3 0.4 2015-11-30 4.9 0.3 2015-12-30 -5.6 -0.6  If you rather wanted the other, one-row-per-year display, just call the function separately for each series. lapply(list(DAX = DAX, REXP = REXP), returns, period = "month")  DAX
Jan Feb  Mar  Apr  May  Jun  Jul  Aug  Sep  Oct Nov  Dec YTD
2014 -1.0 4.1 -1.4  0.5  3.5 -1.1 -4.3  0.7  0.0 -1.6 7.0 -1.8 4.3
2015  9.1 6.6  5.0 -4.3 -0.4 -4.1  3.3 -9.3 -5.8 12.3 4.9 -5.6 9.6

$REXP Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec YTD 2014 1.8 0.4 0.1 0.3 0.9 0.4 0.4 1.0 -0.1 0.1 0.4 1.0 7.1 2015 0.3 0.1 0.3 -0.5 -0.2 -0.8 0.7 0.0 0.4 0.4 0.3 -0.6 0.5  See ?print.preturns for more display options. For instance: print(returns(DAX, period = "month"), digits = 2, year.rows = FALSE, plus = TRUE, month.names = 1:12)   2014 2015 1 -1.00 +9.06 2 +4.14 +6.61 3 -1.40 +4.95 4 +0.50 -4.28 5 +3.54 -0.35 6 -1.11 -4.11 7 -4.33 +3.33 8 +0.67 -9.28 9 +0.04 -5.84 10 -1.56 +12.32 11 +7.01 +4.90 12 -1.76 -5.62 YTD +4.31 +9.56  There are methods toLatex and toHTML for monthly returns. In Sweave documents, you need to use results = tex and echo = false in the chunk options: \noindent \begin{tabular}{rrrrrrrrrrrrrr} <<results=tex,echo=false>>= toLatex(returns(DAX, period = "month")) \end{tabular}  returns accepts other values for period. For yearly returns, use period "year". returns(DAX, period = "year")  2014 2015 4.3 9.6  To get annualised returns, use period ann (or actually any string that the regular expression ^ann matches; case is ignored). returns(DAX, period = "ann")  6.9% [02 Jan 2014 -- 30 Dec 2015]  Now let us try a shorter period. returns(window(DAX, end = as.Date("2014-1-31")), period = "ann")  -1.0% [02 Jan 2014 -- 31 Jan 2014; less than one year, not annualised]  The function did not annualise: it does not annualise if the time period is shorter than one year. (You can see the monthly return for January 2014 in the tables above.) To force annualising, add a !. The exclamation mark serves as a mnenomic that it is now imperative to annualise. returns(window(DAX, end = as.Date("2014-1-31")), period = "ann!")  -11.8% [02 Jan 2014 -- 31 Jan 2014; less than one year, but annualised]  There are also several other periods, such as month-to-date (mtd), year-to-date (ytd) or inception-to-date (itd). The help page of returns lists all options. ### 4.3 Returns when weights are fixed We can also compute returns from fixed weights and assumptions when the portfolio is rebalanced. For instance, we may want to see how a constant allocation $$[0.1, 0.5, 0.4]'$$ to three funds would have done, assuming that a portfolio is rebalanced once a month. If more detail is necessary, then btest can be used; see Section Backtesting. But the simple case can be done with returns already. Here is an example. prices <- c(100, 102, 104, 104, 104.5, 2, 2.2, 2.4, 2.3, 2.5, 3.5, 3, 3.1, 3.2, 3.1) dim(prices) <- c(5, 3) prices   [,1] [,2] [,3] [1,] 100 2.0 3.5 [2,] 102 2.2 3.0 [3,] 104 2.4 3.1 [4,] 104 2.3 3.2 [5,] 104 2.5 3.1  Now suppose we want a constant weight vector, $$[0.1, 0.5, 0.4]'$$, but only rebalance at times 1 and 4. (That is, we rebalance the portfolio only with the prices at timestamps 1 and 4.) returns(prices, weights = c(10, 50, 40)/100, rebalance.when = c(1, 4))  [1] -0.00514 0.06376 -0.01282 0.03146 attr(,"holdings") [,1] [,2] [,3] [1,] 0.001 0.250 0.114 [2,] 0.001 0.250 0.114 [3,] 0.001 0.250 0.114 [4,] 0.001 0.227 0.131 [5,] 0.001 0.227 0.131 attr(,"contributions") [,1] [,2] [,3] [1,] 0.000000 0.0000 0.0000 [2,] 0.002000 0.0500 -0.0571 [3,] 0.002010 0.0503 0.0115 [4,] 0.000000 -0.0236 0.0108 [5,] 0.000481 0.0435 -0.0125  The result is the return series plus two additional pieces of information, stored in attributes. holdings A matrix with the same dimensions as the price matrix we used as input. It provides the hypothetical holdings that were used to compute the returns. Note that these holdings only change at timestamps 1 and 4 in the example. contributions Another matrix; it provides the return contributions of the single assets (in columns) in each period (in rows). ### 4.4 Return contribution Let $$w(t,i)$$ be the weight of portfolio segment $$i$$ at the beginning of period $$t$$, and let $$r(t,i)$$ be the return of segment $$i$$ over period $$t$$. Then the portfolio return over period $$t$$, $$r_{\text{P}}(t)$$ is a weighted sum of the $$N$$ segment returns. $$r_{\text{P}}(t) = \sum_{i=1}^{N} r(t,i)w(t,i)\,.$$ When the weights sum to unity, we may also write $$1+ r_{\text{P}}(t) = \sum_{i=1}^{N} 1 + r(t,i)w(t,i)$$ or, defining $$1+r \equiv R$$, $$R_{\text{P}}(t) = \sum_{i=1}^{N} R(t,i)w(t,i)\,.$$ The total return contribution of segment $$i$$ over time equals $$\sum_{t=1}^{T-1} \bigg(R(t,i)w(t,i) \prod_{s=t+1}^{T}R_{\text{P}}(s) - 1\bigg) + \underbrace{r(T,i)\,w(T,i)}_{\text{final period}}\,.$$ In this way, a segment's return contribution in on period is reinvested in the overall portfolio in succeeding periods. The calculation is provided in the function rc (return contribution'). weights <- rbind(c( 0.25, 0.75), ## the assets' weights c( 0.40, 0.60), ## during three periods c( 0.25, 0.75)) R <- rbind(c( 1 , 0), ## the assets' returns c( 2.5, -1.0), ## during these periods c(-2 , 0.5))/100 rc(R, weights, segment = c("equities", "bonds"))  $period_contributions
timestamp equities    bonds    total
1         1   0.0025  0.00000  0.00250
2         2   0.0100 -0.00600  0.00400
3         3  -0.0050  0.00375 -0.00125

$total_contributions equities bonds total 0.007494363 -0.002242500 0.005251862  ### 4.5 Returns when there are external cashflows The function unit_prices helps to compute time-weighted returns of a portfolio with many in- and outflows. NAV <- data.frame(timestamp = seq(as.Date("2017-1-1"), as.Date("2017-1-10"), by = "1 day"), NAV = c(0,101:104,205:209)) cf <- data.frame(timestamp = c(as.Date("2017-1-1"), as.Date("2017-1-5")), cashflow = c(100, 100)) unit_prices(NAV, cf)   timestamp NAV price shares cashflow new_shares total_shares NAV_after_cf 1 2017-01-01 0 100.000 0.00000 100 1.000000 1.00000 100 2 2017-01-02 101 101.000 1.00000 0 0.000000 1.00000 101 3 2017-01-03 102 102.000 1.00000 0 0.000000 1.00000 102 4 2017-01-04 103 103.000 1.00000 0 0.000000 1.00000 103 5 2017-01-05 104 104.000 1.00000 100 0.961538 1.96154 204 6 2017-01-06 205 104.510 1.96154 0 0.000000 1.96154 205 7 2017-01-07 206 105.020 1.96154 0 0.000000 1.96154 206 8 2017-01-08 207 105.529 1.96154 0 0.000000 1.96154 207 9 2017-01-09 208 106.039 1.96154 0 0.000000 1.96154 208 10 2017-01-10 209 106.549 1.96154 0 0.000000 1.96154 209  ## 5 Backtesting This chapter explains how to test trading strategies with the btest function. ### 5.1 Decisions At a given instant in time (in actual life, now'), a trader needs to answer the following questions: 1. Do I want to compute a new target portfolio, yes or no? If yes, go ahead and compute the new target portfolio. 2. Given the target portfolio and the actual portfolio, do I want to rebalance (i.e. close the gap between the actual portfolio and the target portfolio)? If yes, rebalance. If such a decision is not just hypothetical, then the answer to the second question may lead to a number of orders sent to a broker. Note that many traders do not think in terms of stock (i.e. balances) as we did here; rather, they think in terms of flow (i.e. orders). Both approaches are equivalent, but the described one makes it easier to handle missed trades and synchronise accounts. During a backtest, we will simulate the decisions of the trader. How precisely we simulate depends on the trading strategy. The btest function is meant as a helper function to simulate these decisions. The logic for the decisions described above must be coded in the functions do.signal, signal and do.rebalance. Implementing btest required a number of decision, too: (i) what to model (i.e. how to simulate the trader), and (ii) how to code it. As an example for point (i): how precisely do we want to model the order process (e.g. use limit orders?, allow partial fills?) Example for (ii): the backbone of btest is a loop that runs through the data. Loops are slow in R when compared with compiled languages, so should we vectorise instead? Vectorisation is indeed often possible, namely if trading is not path-dependent. If we have already a list of trades, we can efficiently transform them into a profit-and-loss in R without relying on an explicit loop (see Section Computing profit and (or) loss). Yet, one advantage of looping is that the trade logic is more similar to actual trading; we may even be able to reuse some code in live trading. Altogether, the aim for btest is to stick to the functional paradigm as much as possible. Functions receive arguments and evaluate to results; but they do not change their arguments, nor do they assign or change other variables outside' their environment, nor do the results depend on some variable outside the function. This creates a problem, namely how to keep track of state. If we know what variables need to be persistent, we could pass them to the function and always have them returned. But we would like to be more flexible, so we can pass an environment; examples are below. To make that clear: functional programming should not be seen as a yes-or-no decision; it is a matter of degree. And more of the functional approach can help already. ### 5.2 Data structure All computations of btest will be based on one or several price series of length T. Internally, these prices are stored in numeric matrices. Prices are passed as argument prices. For a single asset, this must be a matrix of prices with four columns: open, high, low and close. For n assets, you need to pass a list of length four: prices[[1]] must be a matrix with n columns containing the open prices for the assets; prices[[2]] is a matrix with the high prices, and so on. For instance, with two assets, you need four matrices with two columns each:  open high low close +-+-+ +-+-+ +-+-+ +-+-+ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +-+-+ +-+-+ +-+-+ +-+-+  If only close prices are used, then for a single asset, use either a matrix of one column or a numeric vector. For multiple assets a list of length one must be passed, containing a matrix of close prices. For example, with 100 close prices of 5 assets, the prices should be arranged in a matrix p of size 100 times 5; and prices = list(p). The btest function runs from b+1 to T. The variable b is the burn-in and it needs to be a positive integer. When we take decisions that are based on past data, we will lose at least one data point. In rare cases b may be zero. Here is an important default: at time =t=, we can use information up to time t-1. Suppose that t were 4. We may use all information up to time 3, and trade at the open in period 4: t time open high low close 1 HH:MM:SS <--\ 2 HH:MM:SS <-- - use information 3 HH:MM:SS _________________________ <--/ 4 HH:MM:SS X <- trade here 5 HH:MM:SS  We could also trade at the close: t time open high low close 1 HH:MM:SS <-- \ 2 HH:MM:SS <-- - use information 3 HH:MM:SS _________________________ <-- / 4 HH:MM:SS X <-- trade here 5 HH:MM:SS  No, we cannot trade at the high or low. (Some people like the idea, as a robustness check, to always buy at the high, sell at the low. Robustness checks – forcing a bit of bad luck into the simulation – are a good idea, notably bad executions. High/low ranges can inform such checks, but using these ranges does not go far enough, and is more of a good story than a meaningful test.) ### 5.3 Function arguments #### 5.3.1 Available information within functions btest expects as arguments a number of functions, such as signal; see the following section for a complete list. The default is to specify no arguments to these functions, because they can all access the following objects'. These objects actually are, with the exception of Globals, themselves functions that can access certain data. These functions can only read; there are no replacement functions. The exception is Globals, which is an environment, and which can explicitly be used for writing (i.e. storing data). Open open prices High high prices Low low prices Close close prices Wealth the total wealth (cash plus positions) at a given point in time Cash cash (in accounting currency) Time current time (an integer) Timestamp the timestamp when that is specified (i.e. when the argument timestamp is supplied); if not, it defaults to Time Portfolio the current portfolio SuggestedPortfolio the currently-suggested portfolio Globals an environment (not a function) All functions take as their first argument a lag, which defaults to 1. So to get the most recent close price, say Close()  which is the same as Close(lag = 1). The lag can be a vector, too: the expression Close(Time():1)  for instance will return all available close prices. So, for instance, in period 11, you want close prices for lags 10, 9, …, 1. Hence, to receive prices in their correct order, the lag sequence must always be in reverse order. If you find it awkword to specify the lag in this reverse order, you may use the argument n instead, which specifies to retrieve the last n data points. So the above Close(Time():1) is equivalent to Close(n = Time())  and Close(n = 10)  returns the last ten closing prices. #### 5.3.2 Function arguments signal The function signal uses information until and including t-1 and returns the suggested portfolio (a vector) to be held at t. This position should be in units of the instruments; if you prefer to work with weights, then you should set convert.weights to TRUE. Then, the value returned by signal will be interpreted as weights and will be automatically converted to position sizes. do.signal do.signal uses information until and including t-1 and must return TRUE or FALSE to indicate whether a signal (i.e. new suggested position) should be computed. This is useful when the signal computation is costly and only be done at specific points in time. If the function is not specified, it defaults to function() TRUE. Instead of a function, this may also be • a vector of integers, which then indicate the points in time when to compute a position, or • a vector of logical values, which then indicate the points in time when to compute a position, or • a vector that inherits from the class of timestamp (e.g. Date), or • one of the keywords firstofmonth or lastofmonth (in this case, timestamp must inherit from Date or be coercible to Date). do.rebalance just like do.signal, but refers to the actual trading. If the function is not specified, it defaults to function() TRUE. Note that rebalancing can typically not take place at a higher frequency than implied by signal. That is because calling signal leads to a position, and when this position does not change (i.e. signal was not called), there is actually no need to rebalance. So do.rebalance is normally used when rebalancing should be done less often that signal computation, e.g. when the decision whether to trade or not is conditional on something. print.info The function is called at the end of an iteration. Whatever it returns will be ignored since it is called for its side effect: print information to the screen, into a file or into some other connection. cashflow The function is called at the end of each iteration; its value is added to the cash. The function provides a clean way to, for instance, add accrued interest to or subtract fees from a strategy. ### 5.4 Examples: A single asset It is best to describe the btest function through a number of simple examples. #### 5.4.1 A useless first example I really like simple examples. Suppose we have a single instrument, and we use only close prices. The trading rule is to buy, and then to hold forever. All we need is the time series of the prices and the signal function. As an instrument we use the EURO STOXX 50 future with expiry September 2015. timestamp <- structure(c(16679L, 16680L, 16681L, 16682L, 16685L, 16686L, 16687L, 16688L, 16689L, 16692L, 16693L), class = "Date") prices <- c(3182, 3205, 3272, 3185, 3201, 3236, 3272, 3224, 3194, 3188, 3213) data.frame(timestamp, prices)   timestamp prices 1 2015-09-01 3182 2 2015-09-02 3205 3 2015-09-03 3272 4 2015-09-04 3185 5 2015-09-07 3201 6 2015-09-08 3236 7 2015-09-09 3272 8 2015-09-10 3224 9 2015-09-11 3194 10 2015-09-14 3188 11 2015-09-15 3213  The signal function is very simple indeed. signal <- function() 1  signal must be written so that it returns the suggested position in units of the asset. In this first example, the suggested position always is 1 unit. It is only a suggested portfolio because we can specify rules whether or not to trade. Examples follow below. To test this strategy, we call btest. The initial cash is zero per default, so initial wealth is also zero in this case. We can change it through the argument initial.cash. (solution <- btest(prices = prices, signal = signal))  initial wealth 0 => final wealth 8  The function returns a list with a number of components, but they are not printed. Instead, a simple print method displays some information about the results. In this case, it tells us that the total equity of the strategy increased from 0 to 8. We arrange more details into a data.frame. suggest is the suggested position; position is the actual position. trade_details <- function(solution, prices) data.frame(price = prices, suggest = solution$suggested.position,
position = unname(solution$position), wealth = solution$wealth,
cash     = solution$cash) trade_details(unclass(solution), prices)   price suggest position wealth cash 1 3182 0 0 0 0 2 3205 1 1 0 -3205 3 3272 1 1 67 -3205 4 3185 1 1 -20 -3205 5 3201 1 1 -4 -3205 6 3236 1 1 31 -3205 7 3272 1 1 67 -3205 8 3224 1 1 19 -3205 9 3194 1 1 -11 -3205 10 3188 1 1 -17 -3205 11 3213 1 1 8 -3205  We bought in the second period because the default setting for the burnin b is 1. Thus, we lose one observation. In this particular case here, we do not rely in any way on the past; hence, we set b to zero. With this setting, we buy at the first price and hold until the end of the data. solution <- btest(prices = prices, signal = signal, b = 0) trade_details(solution, prices)   price suggest position wealth cash 1 3182 1 1 0 -3182 2 3205 1 1 23 -3182 3 3272 1 1 90 -3182 4 3185 1 1 3 -3182 5 3201 1 1 19 -3182 6 3236 1 1 54 -3182 7 3272 1 1 90 -3182 8 3224 1 1 42 -3182 9 3194 1 1 12 -3182 10 3188 1 1 6 -3182 11 3213 1 1 31 -3182  If you prefer the trades only, i.e. not the position series, the solution also contains a journal. (See Keeping track of transactions: journals for more on journals.) journal(solution)   instrument timestamp amount price 1 asset 1 1 1 3182 1 transaction  To make the journal more informative, we can pass timestamp and instrument information when we call btest. journal(btest(prices = prices, signal = signal, b = 0, timestamp = timestamp, ## defined above, ## together with prices instrument = "FESX SEP 2015"))   instrument timestamp amount price 1 FESX SEP 2015 2015-09-01 1 3182 1 transaction  Before we go to the next examples, a final remark, on data frequency. I have used daily data here, but any other frequency, also intraday data, is fine. btest will not care of what frequency your data are or whether your data are regularly spaced; it will only loop over the observations that it is given. It is your own responsibility to write signal (and other functions) in such a way that they encode a meaningful trade logic. #### 5.4.2 More-useful examples Now we make our strategy slightly more selective. The trading rule is to have a position of 1 unit of the asset whenever the last observed price is below 3200 and to have no position when it the price is above 1. The signal function could look like this. signal <- function() { if (Close() < 3200) 1 else 0 }  If you like to write clever code, you may as well have written this: signal <- function() Close() < 3200  The logical value of the comparison Close() < 3200 would be converted to either 0 or 1. But the more verbose version above is clearer.6 We call btest and check the results. solution <- btest(prices = prices, signal = signal)  trade_details(solution, prices)   price suggest position wealth cash 1 3182 0 0 0 0 2 3205 1 1 0 -3205 3 3272 0 0 67 67 4 3185 0 0 67 67 5 3201 1 1 67 -3134 6 3236 0 0 102 102 7 3272 0 0 102 102 8 3224 0 0 102 102 9 3194 0 0 102 102 10 3188 1 1 102 -3086 11 3213 1 1 127 -3086  (Yes, this strategy works better than the simple buy-and-hold, but I hope you agree that this is only because of luck.) The argument initial.position specifies the initial position; default is no position. Suppose we had already held one unit of the asset. solution <- btest(prices = prices, signal = signal, initial.position = 1)  Then the results would have looked as follows. trade_details(solution, prices)   price suggest position wealth cash 1 3182 1 1 3182 0 2 3205 1 1 3205 0 3 3272 0 0 3272 3272 4 3185 0 0 3272 3272 5 3201 1 1 3272 71 6 3236 0 0 3307 3307 7 3272 0 0 3307 3307 8 3224 0 0 3307 3307 9 3194 0 0 3307 3307 10 3188 1 1 3307 119 11 3213 1 1 3332 119  In the example above, we use the close price, but we do not access the data directly. A function Close is defined by btest and passed as an argument to signal. Note that we do not add it as a formal argument to signal since this is done automatically. In fact, doing it manually would trigger an error message: signal <- function(Close = NULL) ## argument name 'Close' 1 ## not allowed  Error in btest(prices = prices, signal = signal) : 'Close' cannot be used as an argument name for 'signal'  Similarly, we have functions Open, High and Low; see Section 5.3 above for all functions. Suppose we wanted to add a variable: a threshold that tells us when to buy. This would need to be an argument to signal; it would also need to be passed with the ... argument of btest. signal <- function(threshold) { if (Close() < threshold) 1 else 0 } solution <- btest(prices = prices, signal = signal, threshold = 3190) trade_details(solution, prices)   price suggest position wealth cash 1 3182 0 0 0 0 2 3205 1 1 0 -3205 3 3272 0 0 67 67 4 3185 0 0 67 67 5 3201 1 1 67 -3134 6 3236 0 0 102 102 7 3272 0 0 102 102 8 3224 0 0 102 102 9 3194 0 0 102 102 10 3188 0 0 102 102 11 3213 1 1 102 -3111  So far we have treated Close as a function without arguments, but actually it has an argument lag that defaults to 1. Suppose the rule were to buy if the last close is below the second-to-last close. signal could look like this. signal <- function() { if (Close(1L) < Close(2L)) 1 else 0 }  We could also have written (Close() < Close(2L)). In any case, the rule uses the close prices of yesterday and of the day before yesterday, so we need to increase b. trade_details(btest(prices = prices, signal = signal, b = 2), prices)   price suggest position wealth cash 1 3182 0 NA NA 0 2 3205 0 0 0 0 3 3272 0 0 0 0 4 3185 0 0 0 0 5 3201 1 1 0 -3201 6 3236 0 0 35 35 7 3272 0 0 35 35 8 3224 0 0 35 35 9 3194 1 1 35 -3159 10 3188 1 1 29 -3159 11 3213 1 1 54 -3159  If we want to trade a different size, we have signal return the desired value. signal <- function() if (Close() < 3200) 2 else 0 trade_details(btest(prices = prices, signal = signal), prices)   price suggest position wealth cash 1 3182 0 0 0 0 2 3205 2 2 0 -6410 3 3272 0 0 134 134 4 3185 0 0 134 134 5 3201 2 2 134 -6268 6 3236 0 0 204 204 7 3272 0 0 204 204 8 3224 0 0 204 204 9 3194 0 0 204 204 10 3188 2 2 204 -6172 11 3213 2 2 254 -6172  A often-used way to specify a trading strategy is to map past prices into +1, 0 or -1 for long, flat or short. A signal is often only given at a specified point (like in buy one unit now'). Example: suppose the third day is a Thursday, and our rule says buy after Thursday'. signal <- function() if (Time() == 3L) 1 else 0 trade_details(btest(prices = prices, signal = signal), prices)   price suggest position wealth cash 1 3182 0 0 0 0 2 3205 0 0 0 0 3 3272 0 0 0 0 4 3185 1 1 0 -3185 5 3201 0 0 16 16 6 3236 0 0 16 16 7 3272 0 0 16 16 8 3224 0 0 16 16 9 3194 0 0 16 16 10 3188 0 0 16 16 11 3213 0 0 16 16  But this is not what we wanted. If the rule is to buy and then keep the long position, we should have written it like this. signal <- function() if (Time() == 3L) 1 else Portfolio()  The function Portfolio evaluates to last period's portfolio. Like Close, its first argument sets the time lag, which defaults to 1. trade_details(btest(prices = prices, signal = signal), prices)   prices sp asset.1 wealth cash 1 3182 0 0 0 0 2 3205 0 0 0 0 3 3272 0 0 0 0 4 3185 1 1 0 -3185 5 3201 1 1 16 -3185 6 3236 1 1 51 -3185 7 3272 1 1 87 -3185 8 3224 1 1 39 -3185 9 3194 1 1 9 -3185 10 3188 1 1 3 -3185 11 3213 1 1 28 -3185  We may also prefer to specify signal so that it evaluates to a weight; for instance, after a portfolio optimisation. In such a case, you need to set convert.weights to TRUE. (Make sure to have a meaningful initial wealth: 5 percent of nothing is nothing.) signal <- function() 0.05 solution <- btest(prices = prices, signal = signal, initial.cash = 100, convert.weights = TRUE) trade_details(solution, prices)   prices sp asset.1 wealth cash 1 3182 0.00000 0.00000 100 100.0 2 3205 0.00157 0.00157 100 95.0 3 3272 0.00156 0.00156 100 95.0 4 3185 0.00153 0.00153 100 95.1 5 3201 0.00157 0.00157 100 95.0 6 3236 0.00156 0.00157 100 95.0 7 3272 0.00155 0.00155 100 95.0 8 3224 0.00153 0.00153 100 95.1 9 3194 0.00155 0.00155 100 95.0 10 3188 0.00157 0.00157 100 95.0 11 3213 0.00157 0.00157 100 95.0  Note that until now we – potentially – rebalanced in every period. If you do not want that, we need to specify do.rebalance. do.rebalance <- function() { if (sum(abs( SuggestedPortfolio(0) - SuggestedPortfolio())) > 2e-5) TRUE else FALSE } solution <- btest(prices = prices, signal = signal, initial.cash = 100, do.rebalance = do.rebalance, convert.weights = TRUE) trade_details(solution, prices)   price suggest position wealth cash 1 3182 0.0000 0.0000 100 100.00 2 3205 0.0000 0.0000 100 100.00 3 3272 0.0000 0.0000 100 100.00 4 3185 0.0306 0.0306 100 2.66 5 3201 0.0000 0.0000 100 100.49 6 3236 0.0000 0.0000 100 100.49 7 3272 0.0000 0.0000 100 100.49 8 3224 0.0000 0.0000 100 100.49 9 3194 0.0000 0.0000 100 100.49 10 3188 0.0000 0.0000 100 100.49 11 3213 0.0000 0.0000 100 100.49  do.rebalance is called after signal. Hence, the suggested position is known and the lag should be zero ('SuggestedPortfolio(0)'). The tol argument works similarly: it instructs btest to only rebalance when the maximum absolute suggested change in any single position is greater than tol. Default is 0.00001, which practically means always rebalance. solution <- btest(prices = prices, signal = signal, initial.cash = 100, tol = 2e-5, convert.weights = TRUE) trade_details(solution, prices)   prices sp asset.1 wealth cash 1 3182 0.00000 0.00000 100 100.0 2 3205 0.00157 0.00157 100 95.0 3 3272 0.00156 0.00157 100 95.0 4 3185 0.00153 0.00153 100 95.1 5 3201 0.00157 0.00157 100 95.0 6 3236 0.00156 0.00157 100 95.0 7 3272 0.00155 0.00155 100 95.0 8 3224 0.00153 0.00155 100 95.0 9 3194 0.00155 0.00155 100 95.0 10 3188 0.00157 0.00155 100 95.0 11 3213 0.00157 0.00157 100 95.0  ##### 5.4.2.1 Passing environments To keep information persistent, we can use environments. As an example, we store (and update) the most recent entry price. notepad <- new.env() notepad$entry <- numeric(length(prices))

notepad$entry[Time(0L)] <- notepad$entry[Time(1L)]
if (Close() < threshold) {
if (Portfolio() < 1)
notepad$entry[Time(0L)] <- Close(0L) 1 } else { 0 } } solution <- btest(prices = prices, signal = signal, threshold = 3200, notepad = notepad) cbind(trade_details(solution, prices), entry = notepad$entry)

   price suggest position wealth  cash entry
1   3182       0        0      0     0     0
2   3205       1        1      0 -3205  3205
3   3272       0        0     67    67  3205
4   3185       0        0     67    67  3205
5   3201       1        1     67 -3134  3201
6   3236       0        0    102   102  3201
7   3272       0        0    102   102  3201
8   3224       0        0    102   102  3201
9   3194       0        0    102   102  3201
10  3188       1        1    102 -3086  3188
11  3213       1        1    127 -3086  3188


Let us check.

subset(journal(solution), amount > 0)


btest provides an environment Globals for exactly such purposes.

signal <- function(threshold) {
Globals$entry[Time(0L)] <- Globals$entry[Time(1L)]
if (Close() < threshold) {
if (Portfolio() < 1)
Globals$entry[Time(0L)] <- Close(0L) 1 } else { 0 } } solution <- btest(prices = prices, signal = signal, threshold = 3200, include.data = TRUE) cbind(trade_details(solution, prices), entry = solution$Globals$entry)   price suggest position wealth cash entry 1 3182 0 0 0 0 NA 2 3205 1 1 0 -3205 3205 3 3272 0 0 67 67 3205 4 3185 0 0 67 67 3205 5 3201 1 1 67 -3134 3201 6 3236 0 0 102 102 3201 7 3272 0 0 102 102 3201 8 3224 0 0 102 102 3201 9 3194 0 0 102 102 3201 10 3188 1 1 102 -3086 3188 11 3213 1 1 127 -3086 3188  ### 5.5 Examples: Several assets It does not really make a difference whether btest is called with a single or with several instruments. The pattern in signal is still to call Close() and friends to obtain data, but now these functions will return matrices with more than one column. For instance, when you have 5 assets, then Close(n = 250) would return a matrix of size 250 times 5. #### 5.5.1 A simple example prices1 <- c(100,98, 98, 97, 96, 98,97,98,99,101) prices2 <- c(100,99,100,102,101,100,96,97,95,82) prices <- cbind(A = prices1, B = prices2) signal <- function() if (Close()[1L] > Close()[2L]) c(2, 0) else c(0, 1) (solution <- btest(prices = list(prices), signal = signal, b=2))   trade_details <- function(solution, prices) data.frame(price = prices, suggest = solution$suggested.position,
position = solution$position, ## do not unname wealth = solution$wealth,
cash     = solution$cash) trade_details(solution, prices)   price.A price.B suggest.A suggest.B position.A position.B wealth cash 1 100 100 0 0 NA NA NA 0 2 98 99 0 0 0 0 0 0 3 98 100 0 1 0 1 0 -100 4 97 102 0 1 0 1 2 -100 5 96 101 0 1 0 1 1 -100 6 98 100 0 1 0 1 0 -100 7 97 96 0 1 0 1 -4 -100 8 98 97 2 0 2 0 -3 -199 9 99 95 2 0 2 0 -1 -199 10 101 82 2 0 2 0 3 -199  journal(solution)   instrument timestamp amount price 1 B 3 1 100 2 A 8 2 98 3 B 8 -1 97 3 transactions  ### 5.6 Common tasks There is more than one way to accomplish a certain task. #### 5.6.1 Remembering an entry price In signal, assign the current price (with lag 0) to Globals. (That is easiest because do.rebalance may not be defined.) #### 5.6.2 Delaying signals Add a random variable to to do.rebalance: if (runif(1) > prob_of_delay) TRUE else FALSE  If TRUE, rebalancing will take place. #### 5.6.3 Specifying when to compute a signal and trade btest takes two functions, do.signal and do.rebalance, that tell the algorithm when to compute a new portfolio and when to rebalance. There are different ways to specify these dates: as a function that returns TRUE or FALSE (most general), but also as integers, logicals or actual timestamps (e.g. dates). Supplying particular timestamps is useful when you know you want to trade on a specific calendar day, say. That is OK because you know in advance when this calendar is going to be. But be careful when you use other information to specify when to trade. The following examples are not equivalent: btest(prices = prices, signal = signal, do.signal = prices > 3600)  btest(prices = prices, signal = signal, do.signal = function() Close() > 3600)  Loosely speaking, both variations compute a signal and trade only when prices is above 3600. But in the first version, there will be no time lag: if the prices exceeds 3600 at time ti, we will trade at ti. In the second example, Close() comes with a default lag of 1: if the price exceeds 3600 at ti, we will trade at ti+1, which is the more realistic case. When timestamp is of a type that can be coerced to Date, you can also use the keywords firstofmonth and lastofmonth: btest(prices = prices, signal = signal, do.signal = "firstofmonth")  #### 5.6.4 Writing a log Specify the function print.info. The function is called at the very end of an iteration, so it is best to use no time lag. An example 1: print.info <- function() { 2: cat("Time", 3: sprintf("%2d", Time(0L)), "...", 4: sprintf("%3d", Wealth(0L)), "\n") 5: flush.console() 6: }  And since cat has a file argument, you can have it write such information into a logfile. #### 5.6.5 Selecting parameters: calling btest recursively Suppose you have a strategy that depends on a parameter vector $$\theta$$. For a given $$\theta$$, the signal for the strategy would look like this. signal <- function(theta) { ## compute position as a function of theta }  Now suppose we do not know theta. We might want to test several values, and then keep the best one. For this, we need to call btest recursively: at a point in time t, the strategy simulates the results for various values for theta and chooses the best theta, according to some criterion f. A useful idiom is this: signal <- function(theta) { if (not defined theta) { - run btest with theta_1, ... \theta_n, select best theta - theta = argmin_theta f(btest(theta_i)) } compute position as a function of theta }  btest will first be invoked without θ (or NULL). When the function calls signal, θ is not defined and signal will call btest with a specified θ. Let us look at an actual example. require("tseries") require("zoo") require("runStats") ## tmp <- get.hist.quote("^GSPC", ## start = "2010-01-01", ## end = "2013-12-31", quote = "Close") signal <- function(Data) { if (is.na(Data$N)) {
message(Timestamp(0))

price <- Close(n = 500)
Ns <- c(10,20)

Data1 <- list(N = 10, hist = 200)
res1 <- btest(price, signal, Data = Data1, b = 200)

Data2 <- list(N = 20, hist = 200)
res2 <- btest(price, signal, Data = Data2, b = 200)

message("N 10 : ", round(tail(res1$wealth, 1), 2)) message("N 20 : ", round(tail(res2$wealth, 1), 2))

N <- if (tail(res1$wealth, 1) > tail(res2$wealth, 1))
10
else
20
message("N is ---> ", N, "\n")
} else  {
N <- Data$N } ## price <- Close(n = Data$hist)
MA <- runStats("mean", price, N = N)
pos <- 0
if (Close() > tail(MA, 1))
pos <- 1
pos
}

Data <- list(N = NA, hist = 200)
res <- btest(tmp$Close, signal, Data = Data, b = 500, initial.cash = 100, convert.weights = TRUE, timestamp = index(tmp)) par(mfrow = c(2,1)) plot(index(tmp), res$wealth, type = "s")
plot(tmp)


## 6 Rebalancing a portfolio

The function rebalance computes the transactions necessary for moving from one portfolio to another.

### 6.1 Usage with unnamed vectors

The current portfolio is given in currency units; the target portfolio is given in weights. To compute the required order sizes, we also need the current prices of the assets. When current, target and price are unnamed, the assets' positions in the vectors need to match.

1: current <- c(0,0,100,100)
2: prices  <- c(1,1,1,1)
3: target  <- c(0.25, 0.25, 0.25, 0.25)
4: rebalance(current, target, prices, match.names = FALSE)

  price current value    %       target value    %       order
1     1       0     0  0.0           50    50 25.0          50
2     1       0     0  0.0           50    50 25.0          50
3     1     100   100 50.0           50    50 25.0         -50
4     1     100   100 50.0           50    50 25.0         -50

Notional: 200.  Amount invested: 200.  Total (2-way) turnover: 200.


The current portfolio may also be empty, in which case current can be set to 0. Then, of course, we need to specify a notional for the target portfolio.

current <- 0
rebalance(current, target, prices,
match.names = FALSE, notional = 100)

  price current value   %       target value    %       order
1     1       0     0 0.0           25    25 25.0          25
2     1       0     0 0.0           25    25 25.0          25
3     1       0     0 0.0           25    25 25.0          25
4     1       0     0 0.0           25    25 25.0          25

Notional: 100.  Amount invested: 100.  Total (2-way) turnover: 100.


We may also specify the target portfolio as a single number.

current <- c(5, 5, 100, 100)

target <- 0    ## liquidate the portfolio
rebalance(current, target, prices, match.names = FALSE)

  price current value    %       target value   %       order
1     1       5     5  2.4            0     0 0.0          -5
2     1       5     5  2.4            0     0 0.0          -5
3     1     100   100 47.6            0     0 0.0        -100
4     1     100   100 47.6            0     0 0.0        -100

Notional: 210.  Amount invested: 0.  Total (2-way) turnover: 210.

## every assets gets a weight of 20%
target <- 0.2
rebalance(current, target, prices, match.names = FALSE, notional = 100)

  price current value     %       target value    %       order
1     1       5     5   5.0           20    20 20.0          15
2     1       5     5   5.0           20    20 20.0          15
3     1     100   100 100.0           20    20 20.0         -80
4     1     100   100 100.0           20    20 20.0         -80

Notional: 100.  Amount invested: 80.  Total (2-way) turnover: 190.


### 6.2 Usage with named vectors

More usefully, rebalance can also use the names of the vectors current, target and price. The argument match.names must be set to TRUE for this (which is the default, actually).

prices  <- c(1,1,1,1)
names(prices) <- letters[1:4]
current <- c(a = 0, b = 10)
target  <- c(a = 0, d = 0.5)
rebalance(current, target, prices)

  price current value     %       target value    %       order
b     1      10    10 100.0            0     0  0.0         -10
d     1       0     0   0.0            5     5 50.0           5

Notional: 10.  Amount invested: 5.  Total (2-way) turnover: 15.


To also show all instruments, set the argument drop.zero to FALSE.

print(rebalance(current, target, prices), drop.zero = FALSE)

  price current value     %       target value    %       order
a     1       0     0   0.0            0     0  0.0           0
b     1      10    10 100.0            0     0  0.0         -10
d     1       0     0   0.0            5     5 50.0           5

Notional: 10.  Amount invested: 5.  Total (2-way) turnover: 15.


### 6.3 Optimisation

Whenever you need to round positions, you may prefer to do an actual optimisation. The ideal place for this optimisation is the original objective function, not in rebalance. And the differences, if there are any at all, are typically small. But here is an example.

n <- 10
target <- runif(n)
target <- target/sum(target)
price <- sample(10:200, n, replace = TRUE)
s <- sample(c(1,5,10,100), n, replace = TRUE,
prob = c(0.4,0.4,0.1,0.1))
data.frame(price = price, lot.size = s)

   price lot.size
1    178        5
2     37        5
3     62        5
4     93        1
5     81        5
6    111        5
7    146        5
8    154        5
9    187        1
10   138        1


Now suppose we have only a limited budget available.

budget <- 10000
x <- rebalance(0, target, notional = budget,
price = price, match.names = FALSE)
x

   price current value   %       target value    %       order
1    178       0     0 0.0            4   712  7.1           4
2     37       0     0 0.0           40  1480 14.8          40
3     62       0     0 0.0           20  1240 12.4          20
4     93       0     0 0.0           16  1488 14.9          16
5     81       0     0 0.0           13  1053 10.5          13
6    111       0     0 0.0            6   666  6.7           6
7    146       0     0 0.0            4   584  5.8           4
8    154       0     0 0.0            6   924  9.2           6
9    187       0     0 0.0            5   935  9.3           5
10   138       0     0 0.0            7   966  9.7           7

Notional: 10000.  Amount invested: 10048.  Total (2-way) turnover: 10048.


Now we use TAopt, from the NMOF package, to find the optimal integer portfolio.

require("NMOF")
ediff <- function(x) {
tmp <- x*price/budget - target
sum(tmp*tmp)
}

neighbour <- function(x) {
i <- sample.int(length(x), size = 1L)
x[i] <- x[i] + if (runif(1) > 0.5) - s[i] else s[i]
x
}

sol <- TAopt(ediff,
algo = list(x0 = numeric(length(price)),
neighbour = neighbour,
q = 0.1,
nS = 1000,
printBar = FALSE))

Threshold Accepting.

Computing thresholds ... OK.
Estimated remaining running time: 0.23 secs.

Running Threshold Accepting...
Initial solution:  0.109341
Finished.
Best solution overall: 0.001108741

df <- data.frame(TA = sol$xbest, rounded = s*round(x$target/s))
df[apply(df, 1, function(i) any(i != 0)), ]

   TA rounded
1   5       5
2  40      40
3  20      20
4  16      16
5  15      15
6   5       5
7   5       5
8   5       5
9   5       5
10  7       7


The difference.

ediff(sol$xbest) - ediff(s*round(x$target/s))

[1] 0


### 6.4 Substituting a basket by its components

If you run tests with baskets of instruments or whole strategies, you often need to substitute the components of the basket for overall basket. PMwR provides a function replace_weight that helps with this task. Suppose we have this weight vector:

w <- c(basket_1 = 0.3,


We also know what the first two baskets represent.

b1 <- c(a = 0.5, b = 0.2, c = 0.3)
b2 <- c(d = 0.1, e = 0.2, a = 0.7)


Now we can call replace_weight.

replace_weight(w,

basket_1::a basket_1::b basket_1::c
0.15        0.06        0.09

0.05        0.10        0.35

0.20


If the names of the baskets or of the things in the baskets have spaces or other characters that cause trouble, quote them.

replace_weight(c("basket 1" = 0.3,


## 7 Looking at portfolio time-series

This chapter explains the function NAVseries.

24 Jun 2013 ==> 09 Jun 2016   (695 data points, 0 NAs)
50         68.5415

summary(tmp)

---------------------------------------------------------
24 Jun 2013 ==> 09 Jun 2016   (695 data points, 0 NAs)
50         68.5415
---------------------------------------------------------
High                   80.04  (13 Apr 2015)
Low                    50.00  (24 Jun 2013)
---------------------------------------------------------
Return (%)              11.2  (annualised)
---------------------------------------------------------
Max. drawdown (%)       23.6
_ peak                 80.04  (13 Apr 2015)
_ trough               61.18  (11 Feb 2016)
_ underwater now (%)    14.4
---------------------------------------------------------
Volatility (%)          13.1  (annualised)
_ upside                10.9
_ downside               7.4
---------------------------------------------------------

Monthly returns  ▁▃█▆█▃▁

Jan  Feb  Mar  Apr May  Jun  Jul  Aug  Sep  Oct Nov  Dec  YTD
2013                          3.9  5.3 -1.6  5.4  3.1 1.6  1.2 20.3
2014 -1.5  5.3 -1.0  0.3 2.5 -1.0 -3.1  2.4  0.8 -0.2 4.5 -0.8  8.2
2015  9.3  5.8  2.4 -1.1 1.8 -4.5  4.0 -7.7 -3.5  7.7 2.4 -3.4 12.3
2016 -5.6 -2.8  1.3 -1.1 3.7 -1.7                              -6.3


## 8 Plotting irregularly-spaced series during trading hours

### 8.1 An example

We have the following sample of prices of the Bund future contract, traded at the Eurex in Germany.

times prices
2012-10-18 20:00:09 139.82
2012-10-18 20:01:11 139.82
2012-10-18 20:01:59 139.8
2012-10-18 20:01:29 139.81
2012-10-18 20:16:49 139.77
2012-10-18 20:50:49 139.85
2012-10-18 21:23:19 139.76
2012-10-18 21:41:39 139.76
2012-10-18 21:59:59 139.77
2012-10-19 09:16:10 139.8
2012-10-19 09:49:31 139.86
2012-10-19 21:12:49 140.46
2012-10-19 21:42:31 140.39
2012-10-22 08:45:15 140.14
2012-10-22 09:05:33 140.15

Note that I have left the time zone to the operating system. Since my computer is typically located in the time zone that the tz database (http://www.iana.org/time-zones)

calls 'Europe/Berlin', the first time should be 2012-10-18 20:00:09. If, for instance, your computer is in 'America/Chicago' instead and you run the above code, the first time would be 2012-10-18 13:00:09. Which is right: it is the correct time, only translated into Chicago local time.

A plot of price against time looks like this.

plot(times, prices, type = "s")


Such a plot is fine for many purposes. But the contract for which we have prices is only traded from Monday to Friday, not on weekends, and it istraded only from 08:00 to 22:00 Europe/Berlin time. So the plot should omit those times at which no trading takes place. This is what the function plotTradingHours does.

tmp <- plotTradingHours(x = prices, t = times,
interval = "1 sec", labels = "day",
fromHHMMSS = "080000", toHHMMSS = "220000",
type = "s")


What we need for such a plot is a function that maps actual time to a point on the x-scale, while the y-scale stays unchanged. If we were talking only about days, not times, we needed something like this:

day x-position mapped x-position
Thursday 1 1
Friday 2 2
Saturday 3 <removed>
Sunday 4 <removed>
Monday 5 3

This mapping is what plotTradingHours provides. And not much more: the design goal of the function is to make it as much as possible an ordinary plot; or more specifically, to make it as similar as possible to the plot function. Indeed, plotTradingHours calls plot with a small number of default settings:

list(type = "l", xaxt = "n", xlab = "", ylab = "")


These settings can all be overridden through the ... argument, which is passed to plot. Note that we already set s as the plot's type in the last code chunk. The only required setting is suppressing the x-axis with setting xaxt to 'n', because plotTradingHours will create its own x-axis via a call to axis(1, ...). In case you wish to use your own axis specification, either set do.plotAxis to FALSE or pass settings to axis through the list axis1.par.

### 8.2 More examples

#### 8.2.1 Value of plotTradingHours

Like plot, plotTradingHours is typically called for its side effect: creating a plot. But it also returns useful information (invisibly, unless called with do.plot = FALSE).

str(tmp)

List of 6
$t : int [1:15] 1 63 81 111 1001 3041 4991 6091 7191 11763 ...$ x          : num [1:15] 140 140 140 140 140 ...
$axis.pos : num [1:2] 7193 57594$ axis.labels: chr [1:2] "19.10." "22.10."
$timegrid : POSIXct[1:61527], format: "2012-10-18 20:00:09" ...$ map        : function (t)


This information can be used to add elements to plots. An example follows.

We can add grid lines with abline. The y-axis poses no special problem. The positions of the x-axis ticks are returned from plotTradingHours.

tmp <- plotTradingHours(x = prices, t = times,
interval = "1 sec",
labels = "day",
fromHHMMSS="080000",
toHHMMSS = "220000",
type = "s")
abline(h = axTicks(2), v = tmp$axis.pos, col = "lightgrey", lty = "dotted")  If we wan to add to a specific time, say 19 October, 13:10:23, we can use the function map that the call to plotTradingHours returns. We first create the specific time with, for example, ISOdatetime or strptime. ## Again, I do not specify a time zone since time zones ## depend on the operating system. To reproduce the ## example, you may use this representation: ## ## mytime <- structure(1350645023, ## class = c("POSIXct", "POSIXt"), ## tzone = "") mytime <- ISOdatetime(2012, 10, 19, 13, 10, 23) mytime  [1] "2012-10-19 13:10:23 CEST"  Now we use map to translate this time into the appropriate x-position. tmp <- plotTradingHours(x = prices, t = times, interval = "1 sec", labels = "day", fromHHMMSS="080000", toHHMMSS = "220000", type = "s") abline(h = axTicks(2), v = tmp$axis.pos,
col = "lightgrey", lty = "dotted")
abline(v = tmp$map(mytime)$t, col = "red")


The function map returns a list with two components, t and ix.

tmp$map(mytime)  $t
[1] 25816

$ix [1] 1  The first component is the appropriate position on the x-axis; since it is a time it is called t. The second component gives the subscripts to values that should actually be plotted. As an example, suppose that we wish to plot points at several prices at 21:00:00 for several days. ## moretimes <- structure(c(1350586800, 1350673200, 1350759600), ## class = c("POSIXct", "POSIXt"), tzone = "") ## moretimes <- ISOdatetime(2012, 10, 18:20, 21, 00, 00) values <- seq(140, 140.20, length.out = length(moretimes)) data.frame(times = moretimes, weekday = format(moretimes, "%A"), values)   times weekday values 1 2012-10-18 21:00:00 Thursday 140.0 2 2012-10-19 21:00:00 Friday 140.1 3 2012-10-20 21:00:00 Saturday 140.2  But 20 October 2012 falls on a Saturday, and so it does not appear in the plot. tmp$map(moretimes)

$t [1] 3592 53993$ix
[1] 1 2


The values that should be plotted can conveniently be found by using ix.

values[tmp$map(moretimes)$ix]

[1] 140.0 140.1


## 9 Valuation

Computing the value of a position is, in principle, straightforward: multiply the prices of assets by the numbers of contracts you hold and sum the resulting values.

That immediately leads to three questions:

1. What is price?
2. What is a contract?
3. Can we really sum?

### 9.1 Prices

Valuing an instrument can mean using either a market price or a theoretical price. In the discussion that follows, I will assume that we already have prices (or net-present values).

### 10.1 Exposure

We have the following trades and times.

amount <- c(1,3,-3,1,-3,1)
time <- c(0,1,3,4,7,12)


The holding period (duration) of these trades can be computed so:

data.frame(position = cumsum(amount)[-length(amount)],
from = time[-length(time)],
to   = time[-1L],
duration = diff(time))

     position from to duration
1        1    0  1        1
2        4    1  3        2
3        1    3  4        1
4        2    4  7        3
5       -1    7 12        5


We can plot the exposure.

par(bty = "n", mar = c(4, 4, 0, 0), tck = 0.005,
las = 1, ps = 12)
plot(c(time[1], time), cumsum(c(0, amount)),
type = "s", xlab = "time", ylab = "position")


Thus, we have had a position from time zero to 12 (hours into the trading day, say), but its size varied. The function twExposure (time-weighted exposure) computes the average absolute exposure.

twExposure(amount, time)

1.75


To give a simple example: suppose we bought at the open of a trading day and sold at noon. The average exposure for the day is thus half a contract.

amount <- c(1, -1 , 0)
time   <- c(0,0.5,1)
twExposure(amount, time)

0.5


### 10.2 Splitting and rescaling

Suppose we have the following trades and impose a limit that the maximum absolute exposure for the trader should only be 2.

t <- 1:6
n <- c(-1,-1,-1,1,1,1)
p <- c(100,99,98,98,99,100)
limit(n, p, t, lim = 2)

$amount [1] -1 -1 1 1$price
[1] 100  99  99 100

$timestamp [1] 1 2 5 6  Scaling the trades. scaleToUnity(n)  [1] -0.333 -0.333 -0.333 0.333 0.333 0.333  Closing the trade at once. closeOnFirst(n)  [1] -1 -1 -1 3 0 0  ## 11 Scaling series Visual comparisons of time-series are ubiquitous in finance.7 The function scale1 helps with scaling the levels of time-series so that is becomes easier to compare them. It is a generic function, and PMwR provides methods for numeric vectors/matrices, and for zoo and NAVseries objects. ### 11.1 Examples To explain what the function does, we use two very short time-series: the values of the DAX, the German stock-market index, and the REXP, a German government-bond index, from 2 January and 8 January 2014 (just 5 days). dax <- c(9400.04, 9435.15, 9428, 9506.2, 9497.84) rex <-c(440.5252, 440.7944, 441.5456, 441.8197, 441.7619) X <- cbind(dax, rex) scale1(dax)  [1] 1.00 1.00 1.00 1.01 1.01  It may be more common to scale to a level of 100. We either multiply the whole series by 100, or use the level argument. scale1(dax, level = 100)  [1] 100 100 100 101 101  If we give a matrix to scale1, the function scales each column separately. scale1(X, level = 100)   dax rex [1,] 100 100 [2,] 100 100 [3,] 100 100 [4,] 101 100 [5,] 101 100  scale1 also works with zoo objects. (Z <-scale1(zoo(X, as.Date(c("2014-01-02", "2014-01-03", "2014-01-06", "2014-01-07", "2014-01-08"))), level = 100))   dax rex 2014-01-02 100 100 2014-01-03 100 100 2014-01-06 100 100 2014-01-07 101 100 2014-01-08 101 100  plot(Z, plot.type = "single")  The argument when defines the origin. scale1(X, when = 3, level = 100)   dax rex [1,] 99.7 99.8 [2,] 100.1 99.8 [3,] 100.0 100.0 [4,] 100.8 100.1 [5,] 100.7 100.0  With zoo objects, when should be compatible with the class of the objects index. scale1(Z, when = as.Date("2014-01-07"), level = 100)   dax rex 2014-01-02 98.9 99.7 2014-01-03 99.3 99.8 2014-01-06 99.2 99.9 2014-01-07 100.0 100.0 2014-01-08 99.9 100.0  when also understands the keyword first.complete, which is actually the default. That is useful when series have different lengths. X[1:2, 1] <- NA X   dax rex [1,] NA 441 [2,] NA 441 [3,] 9428 442 [4,] 9506 442 [5,] 9498 442  scale1(X, level = 100) ## 'first.complete' is the default   dax rex [1,] NA 99.8 [2,] NA 99.8 [3,] 100 100.0 [4,] 101 100.1 [5,] 101 100.0  When the argument centre is TRUE, the daily mean return is subtracted. scale1(Z, centre = TRUE)   dax rex 2014-01-02 1.000 1 2014-01-03 1.001 1 2014-01-06 0.998 1 2014-01-07 1.003 1 2014-01-08 1.000 1  The argument scale takes a standard deviation and scales the returns to that standard deviation. apply(returns(scale1(Z, scale = 0.02)), 2, sd)   dax rex 0.02 0.02  This may create fairer comparisons, for instance, between fund prices that exhibit very different volatilities. scale1(Z, scale = 0.02)   dax rex 2014-01-02 1.00 1.00 2014-01-03 1.02 1.02 2014-01-06 1.01 1.06 2014-01-07 1.05 1.08 2014-01-08 1.05 1.08  It should be stressed that centre and scale treat returns, but scale1 expects and returns levels (not returns). The zoo method has a further argument that affects returns: inflate. d <- seq(from = as.Date("2015-1-1"), to = as.Date("2016-1-1"), by = "1 day") z <- zoo(100, d) head(z) tail(z)  2015-01-01 2015-01-02 2015-01-03 2015-01-04 2015-01-05 2015-01-06 100 100 100 100 100 100 2015-12-27 2015-12-28 2015-12-29 2015-12-30 2015-12-31 2016-01-01 100 100 100 100 100 100  The argument should be a numeric value: the annual growth rate that is added to (or subtracted from, when negative) the time series. head(scale1(z, inflate = 0.02)) tail(scale1(z, inflate = 0.02))  2015-01-01 2015-01-02 2015-01-03 2015-01-04 2015-01-05 2015-01-06 1 1 1 1 1 1 2015-12-27 2015-12-28 2015-12-29 2015-12-30 2015-12-31 2016-01-01 1.02 1.02 1.02 1.02 1.02 1.02  ### 11.2 Scaling a series The previous section provided examples of scaling series. In this section, we are going to see how scale1 does its computations. Changing volatility (the scale argument) uses the fact that multiplying a random variable $$X$$ by a factor $$b$$ changes its variance to $$b^2$$ times the original variance. Hence, we first divide by the actual standard deviation and then multiply by the desired one. Changing returns is slightly more complicated. Suppose we want to scale the total return of the series $$X$$ such that it equals $$r^*_T$$. $$\label{eq:zret} (1+r_1)(1+r_2)(1+r_3)\cdots = 1+r_T$$ There is, clearly, an infinity of possible adjustments that would do the trick. We might, for instance, change $$X_1$$ or $$X_T$$ so that the desired return is achieved; but then we could go on an change two or more elements such there total contribution cancels. But that is probably not what we want. In fact, a reasonable requirement is that the scaling touches as few of other statistical properties as possible. Adding a constant $$z$$ to the return in every period does that. It has the additional advantage that it neither changes linear $$\label{eq:zret2} (1+r_1+z)(1+r_2+z)(1+r_3+z)\cdots = 1+r^*_T$$ ## 12 Other Tools ### 12.1 Stocks splits The function split_adjust handles stock splits. It is implemented to work on numeric vectors. ### 12.2 Treasuries quotes US treasury bonds are often quoted in 1/32nds of points. For instance, the price 110'030 would mean 110+3/32. The function quote32 provides a way to pretty-print' such prices. quote32(c("110-235", "110-237"))  110-23+ 110-23¾  Internally, quote32 will store the prices as numeric values: the fractions are only used for printing. as.numeric(quote32(c("110-235", "110-237")))  [1] 110.7812 110.8125  ## 13 FAQ/FRC (Frequently-required computations) I need to compute P/L between two points in time, for instance between yesterday's evening close and now (intraday). You need the position at $$t_1$$ and the journal of trades between $$t_1$$ and $$t_2$$. Also, you will need the valuation prices for all instruments at both points in time. Then, you can use pl: see arguments initial.position and vprice. I want to compute returns when a portfolio (specified as weights) is rebalanced at specific times. However, I only have returns for the assets in the portfolio, not prices. Compute artificial prices, and then use returns: see arguments weights and rebalance.when. I have a list of trades (bought or sold what, when and at what price) and I need to compute the profit or loss. See pl. I have a list of trades in an instrument and want to plot these trades against the price of the traded instrument. I have a signal series (+1, 0, 0, +1, …) and need to transform it into a profit-and-loss series. I have a list of trades and need to determine the profit-and-loss between two timestamps. Call the two timestamps t0 and t1. Unless the position was zero at t0 and t1, we can compute the profit/loss only if we have prices for the positions at these points in time. In case the position was indeed zero, you can use pl; the transactions are stored in a journal j. subset(j, timestamp >= t0 & timestamp <=t1)  But even the more general case is not so complicated, after all. 1. Compute the position at t0 and make it a journal j0. 2. Take all transactions at $$t > t_\mathrm{0}$$ and $$t \leq t_\mathrm{1}$$ and put them into a journal $$J$$. 3. Compute the position at $$t_\mathrm{1}$$, and make it a journal $$J_1$$, but \emph{multiply all amounts by$-1\$}.
4. Combine $$J_0$$, $$J$$, and $$J_1$$ and compute the PL.

I need to determine the month-to-date profit-and-loss.

1. compute position on last day of last month
2. make journal from position (add prices)
3. combine with journal since month start
4. use average (avg) on all instruments

btest: I want to print my current P/L in every period.

Use print.info.

btest: I invest in assets that pay accrued interest.

Directly work with the dirty prices. If the signals depend on clean prices, pass them as extra information and access them with clean_price[Time()]. Alternatively, work with the clean prices, and use cashflow to add the accrued interest to the cash account.

## 14 Appendix: R/package versions used

R version 3.3.2 (2016-10-31)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 16.04.1 LTS

locale:
[1] LC_CTYPE=en_GB.UTF-8       LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_GB.UTF-8
[5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_GB.UTF-8
[7] LC_PAPER=en_US.UTF-8       LC_NAME=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base


## Footnotes:

1

The manual itself is written in Org mode. The complete tangled code is available from the website.

2

This notation is inspired by the syntax of ledger files. See http://www.ledger-cli.org/ .

3

The behaviour of [[ may change in the future: it may then be used to iterate over the transactions in a journal, not the fields. This would be motivated by https://developer.r-project.org/blosxom.cgi/R-devel/NEWS/2016/03/09 even though the commit was reversed two days later https://developer.r-project.org/blosxom.cgi/R-devel/NEWS/2016/03/11

4

We could even do it in Perl. :-)`

5

The function never computes logarithmic returns.

6

Remember what Brian Kernighan said: Everyone knows that debugging is twice as hard as writing a program in the first place. So if you're as clever as you can be when you write it, how will you ever debug it?

7

Transformating or scaling data are a key element of exploratory data analysis in general. See Tukey's EDA (1977). TODO: find H. Simon reference on scaling (taking reciprocal value).