# Momentum in R: Part 2

Many of the sites I linked to in the previous post have articles or papers on momentum investing that investigate the typical ranking factors; 3, 6, 9, and 12 month returns. Most (not all) of the articles seek to find which is the “best” look-back period to rank the assets. Say that the outcome of the article is that the 6 month look-back has the highest returns. A trading a strategy that just uses a 6 month look-back period to rank the assets leaves me vulnerable to over-fitting based on the backtest results. The backtest tells us nothing more than which strategy performed the best in the past, it tells us nothing about the future… duh!

Whenever I review the results from backtests, I always ask myself a lot of “what if” questions. Here are 3 “what if” questions that I would ask for this backtest are:

1. What if the strategy based on a 6 month look-back under performs and the 9 month or 3 month starts to over perform?
2. What if the strategies based on 3, 6, and 9 month look-back periods have about the same return and risk profile, which strategy should I trade?
3. What if the assets with high volatility are dominating the rankings and hence driving the returns?

The backtests shown are simple backtests meant to demonstrate the variability in returns based on look-back periods and number of assets traded.

The graphs below show the performance of a momentum strategy using 3, 6, 9, and 12 month returns and trading the Top 1, 4, and 8 ranked assets. You will notice that there is significant volatility and variability in returns only trading 1 asset. The variability between look-back periods is reduced, but there is still no one clear “best” look-back period. There are periods of under performance and over performance for all look back periods in the test.

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Here is the R code used for the backtests and the plots. Leave a comment if you have any questions about the code below.

```library(FinancialInstrument)
library(TTR)
library(PerformanceAnalytics)

RankRB <- function(x){
# Computes the rank of an xts object of ranking factors
# ranking factors are the factors that are ranked (i.e. asset returns)
#
# args:
#   x = xts object of ranking factors
#
# Returns:
#   Returns an xts object with ranks
#   (e.g. for ranking asset returns, the asset with the greatest return
#    receives a  rank of 1)

r <- as.xts(t(apply(-x, 1, rank, na.last = "keep")))
return(r)
}

# Converts daily data to monthly and returns only the monthly close
# Note: only used with Yahoo Finance data so far
# Thanks to Joshua Ulrich for the Monthly Ad function
#
# args:
#   x = daily price data from Yahoo Finance
#
# Returns:
#   xts object with the monthly adjusted close prices

sym <- sub("\\..*\$", "", names(x)[1])
Ad(to.monthly(x, indexAt = 'lastof', drop.time = TRUE, name = sym))
}

CAGR <- function(x, m){
# Function to compute the CAGR given simple returns
#
# args:
#  x = xts of simple returns
#  m = periods per year (i.e. monthly = 12, daily = 252)
#
# Returns the Compound Annual Growth Rate
x <- na.omit(x)
cagr <- apply(x, 2, function(x, m) prod(1 + x)^(1 / (length(x) / m)) - 1, m = m)
return(cagr)
}

SimpleMomentumTest <- function(xts.ret, xts.rank, n = 1, ret.fill.na = 3){
# returns a list containing a matrix of individual asset returns
# and the comnbined returns
# args:
#  xts.ret = xts of one period returns
#  xts.rank = xts of ranks
#  n = number of top ranked assets to trade
#  ret.fill.na = number of return periods to fill with NA
#
# Returns:
#  returns an xts object of simple returns

# trade the top n asset(s)
# if the rank of last period is less than or equal to n,
# then I would experience the return for this month.

# lag the rank object by one period to avoid look ahead bias
lag.rank <- lag(xts.rank, k = 1, na.pad = TRUE)
n2 <- nrow(lag.rank[is.na(lag.rank[,1]) == TRUE])
z <- max(n2, ret.fill.na)

# for trading the top ranked asset, replace all ranks above n
# with NA to set up for element wise multiplication to get
# the realized returns
lag.rank <- as.matrix(lag.rank)
lag.rank[lag.rank > n] <- NA
# set the element to 1 for assets ranked <= to rank
lag.rank[lag.rank <= n] <- 1

# element wise multiplication of the
# 1 period return matrix and lagged rank matrix
mat.ret <- as.matrix(xts.ret) * lag.rank

# average the rows of the mat.ret to get the
# return for that period
vec.ret <- rowMeans(mat.ret, na.rm = TRUE)
vec.ret[1:z] <- NA

# convert to an xts object
vec.ret <- xts(x = vec.ret, order.by = index(xts.ret))
f <- list(mat = mat.ret, ret = vec.ret, rank = lag.rank)
return(f)
}

currency("USD")
symbols <- c("XLY", "XLP", "XLE", "XLF", "XLV", "XLI", "XLK", "XLB", "XLU", "EFA")#, "TLT", "IEF", "SHY")
stock(symbols, currency = "USD", multiplier = 1)

# create new environment to store symbols
symEnv <- new.env()

# getSymbols and assign the symbols to the symEnv environment
getSymbols(symbols, from = '2002-09-01', to = '2012-10-20', env = symEnv)

# xts object of the monthly adjusted close prices

# monthly returns
monthly.returns <- ROC(x = symbols.close, n = 1, type = "discrete", na.pad = TRUE)

#############################################################################
# rate of change and rank based on a single period for 3, 6, 9, and 12 months
#############################################################################

roc.three <- ROC(x = symbols.close , n = 3, type = "discrete")
rank.three <- RankRB(roc.three)

roc.six <- ROC(x = symbols.close , n = 6, type = "discrete")
rank.six <- RankRB(roc.six)

roc.nine <- ROC(x = symbols.close , n = 9, type = "discrete")
rank.nine <- RankRB(roc.nine)

roc.twelve <- ROC(x = symbols.close , n = 12, type = "discrete")
rank.twelve <- RankRB(roc.twelve)

num.assets <- 4

# simple momentum test based on 3 month ROC to rank
case1 <- SimpleMomentumTest(xts.ret = monthly.returns, xts.rank = rank.three,
n = num.assets, ret.fill.na = 15)

# simple momentum test based on 6 month ROC to rank
case2 <- SimpleMomentumTest(xts.ret = monthly.returns, xts.rank = rank.six,
n = num.assets, ret.fill.na = 15)

# simple momentum test based on 9 month ROC to rank
case3 <- SimpleMomentumTest(xts.ret = monthly.returns, xts.rank = rank.nine,
n = num.assets, ret.fill.na = 15)

# simple momentum test based on 12 month ROC to rank
case4 <- SimpleMomentumTest(xts.ret = monthly.returns, xts.rank = rank.twelve,
n = num.assets, ret.fill.na = 15)

returns <- cbind(case1\$ret, case2\$ret, case3\$ret, case4\$ret)
colnames(returns) <- c("3-Month", "6-Month", "9-Month", "12-Month")

charts.PerformanceSummary(R = returns, Rf = 0, geometric = TRUE,
main = "Momentum Cumulative Return: Top 4 Assets")

table.Stats(returns)

CAGR(returns, m = 12)

print("End")```

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