## Probabilistic Momentum

David Varadi has recently discussed an interesting strategy in the
Are Simple Momentum Strategies Too Dumb? Introducing Probabilistic Momentum post. David also provided the Probabilistic Momentum Spreadsheet if you are interested in doing computations in Excel. Today I want to show how you can test such strategy using the Systematic Investor Toolbox:

```###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
source(con)
close(con)
#*****************************************************************
#******************************************************************

tickers = spl('SPY,TLT')

data <- new.env()
getSymbols(tickers, src = 'yahoo', from = '1980-01-01', env = data, auto.assign = T)
bt.prep(data, align='remove.na', dates='2005::')

#*****************************************************************
# Setup
#******************************************************************
lookback.len = 60

prices = data\$prices

models = list()

#*****************************************************************
# Simple Momentum
#******************************************************************
momentum = prices / mlag(prices, lookback.len)
data\$weight[] = NA
data\$weight\$SPY[] = momentum\$SPY > momentum\$TLT
data\$weight\$TLT[] = momentum\$SPY <= momentum\$TLT
models\$Simple  = bt.run.share(data, clean.signal=T)
```

The Simple Momentum strategy invests into SPY if SPY’s momentum if greater than TLT’s momentum, and invests into TLT otherwise.

```	#*****************************************************************
# Probabilistic Momentum
#******************************************************************
confidence.level = 60/100
ret = prices / mlag(prices) - 1

ir = sqrt(lookback.len) * runMean(ret\$SPY - ret\$TLT, lookback.len) / runSD(ret\$SPY - ret\$TLT, lookback.len)
momentum.p = pt(ir, lookback.len - 1)

data\$weight[] = NA
data\$weight\$SPY[] = iif(cross.up(momentum.p, confidence.level), 1, iif(cross.dn(momentum.p, (1 - confidence.level)), 0,NA))
data\$weight\$TLT[] = iif(cross.dn(momentum.p, (1 - confidence.level)), 1, iif(cross.up(momentum.p, confidence.level), 0,NA))
models\$Probabilistic  = bt.run.share(data, clean.signal=T)
```

The Probabilistic Momentum strategy is using Probabilistic Momentum measure and Confidence Level to decide on allocation. Strategy invests into SPY if SPY vs TLT Probabilistic Momentum is above Confidence Level and invests into TLT is SPY vs TLT Probabilistic Momentum is below 1 – Confidence Level.

To make Strategy a bit more attractive, I added a version that can leverage SPY allocation by 50%

```	#*****************************************************************
# Probabilistic Momentum + SPY Leverage
#******************************************************************
data\$weight[] = NA
data\$weight\$SPY[] = iif(cross.up(momentum.p, confidence.level), 1, iif(cross.up(momentum.p, (1 - confidence.level)), 0,NA))
data\$weight\$TLT[] = iif(cross.dn(momentum.p, (1 - confidence.level)), 1, iif(cross.up(momentum.p, confidence.level), 0,NA))
models\$Probabilistic.Leverage = bt.run.share(data, clean.signal=T)

#*****************************************************************
# Create Report
#******************************************************************
strategy.performance.snapshoot(models, T)
```

The back-test results look very similar to the ones reported in the Are Simple Momentum Strategies Too Dumb? Introducing Probabilistic Momentum post.

However, I was not able to exactly reproduce the transition plots. Looks like my interpretation is producing more whipsaw when desired.

```	#*****************************************************************
# Visualize Signal
#******************************************************************
cols = spl('steelblue1,steelblue')
prices = scale.one(data\$prices)

layout(1:3)

plota(prices\$SPY, type='l', ylim=range(prices), plotX=F, col=cols[1], lwd=2)
plota.lines(prices\$TLT, type='l', plotX=F, col=cols[2], lwd=2)
plota.legend('SPY,TLT',cols,as.list(prices))

highlight = models\$Probabilistic\$weight\$SPY > 0
plota.control\$col.x.highlight = iif(highlight, cols[1], cols[2])
plota(models\$Probabilistic\$equity, type='l', plotX=F, x.highlight = highlight | T)
plota.legend('Probabilistic,SPY,TLT',c('black',cols))

highlight = models\$Simple\$weight\$SPY > 0
plota.control\$col.x.highlight = iif(highlight, cols[1], cols[2])
plota(models\$Simple\$equity, type='l', plotX=T, x.highlight = highlight | T)
plota.legend('Simple,SPY,TLT',c('black',cols))
```

David thank you very much for sharing your great ideas. I would encourage readers to play with this strategy and report back.

To view the complete source code for this example, please have a look at the bt.probabilistic.momentum.test() function in bt.test.r at github.

Mebane Faber posted another interesting blog post: Building a Simple Sector Rotation on Momentum and Trend that caught my interest. Today I want to show how you can test such strategy using the Systematic Investor Toolbox:

```###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
source(con)
close(con)

#*****************************************************************
#******************************************************************

data = new.env()
ret = cbind(temp[[1]]\$Mkt.RF + temp[[1]]\$RF, temp[[1]]\$RF)
price = bt.apply.matrix(ret / 100, function(x) cumprod(1 + x))
data\$SPY = make.stock.xts( price\$Mkt.RF )
data\$SHY = make.stock.xts( price\$RF )

ret = temp[[1]]
price = bt.apply.matrix(ret[,1:9] / 100, function(x) cumprod(1 + x))
for(n in names(price)) data[[n]] = make.stock.xts( price[,n] )

# align dates
data\$symbolnames = c(names(price), 'SHY', 'SPY')
bt.prep(data, align='remove.na', dates='2000::')

# back-test dates
bt.dates = '2001:04::'

#*****************************************************************
# Setup
#******************************************************************
prices = data\$prices
n = ncol(data\$prices)

models = list()

#*****************************************************************
# Benchmark Strategies
#******************************************************************
data\$weight[] = NA
data\$weight\$SPY[1] = 1
models\$SPY = bt.run.share(data, clean.signal=F, dates=bt.dates)

weight = prices
weight\$SPY = NA
weight\$SHY = NA

data\$weight[] = NA
data\$weight[] = ntop(weight[], n)
models\$EW = bt.run.share(data, clean.signal=F, dates=bt.dates)

#*****************************************************************
# Code Strategies
# http://www.mebanefaber.com/2013/12/04/square-root-of-f-squared/
#******************************************************************
sma = bt.apply.matrix(prices, SMA, 10)

# create position score
position.score = sma
position.score[ prices < sma ] = NA
position.score\$SHY = NA
position.score\$SPY = NA

# equal weight allocation
weight = ntop(position.score[], n)

# number of invested funds
n.selected = rowSums(weight != 0)

# cash logic
weight\$SHY[n.selected == 0,] = 1

weight[n.selected == 1,] = 0.25 * weight[n.selected == 1,]
weight\$SHY[n.selected == 1,] = 0.75

weight[n.selected == 2,] = 0.5 * weight[n.selected == 2,]
weight\$SHY[n.selected == 2,] = 0.5

weight[n.selected == 3,] = 0.75 * weight[n.selected == 3,]
weight\$SHY[n.selected == 3,] = 0.25

# cbind(round(100*weight,0), n.selected)

data\$weight[] = NA
data\$weight[] = weight
models\$strategy1 = bt.run.share(data, clean.signal=F, dates=bt.dates)

#*****************************************************************
# Create Report
#******************************************************************
strategy.performance.snapshoot(models, one.page = T)
```

Mebane thank you very much for sharing your great ideas. I would encourage readers to play with this strategy and report back.

Please note that I back-tested the strategy using the monthly observations. The strategy’s draw-down is around 17% using monthly data. If we switch to the daily data, the strategy’s draw-down goes to around 22%. There was one really bad month in 2002.

To view the complete source code for this example, please have a look at the bt.mebanefaber.f.squared.test() function in bt.test.r at github.

## Averaged Input Assumptions and Momentum

Today I want to share another interesting idea contributed by Pierre Chretien. Pierre suggested using Averaged Input Assumptions and Momentum to create reasonably quiet strategy. The averaging techniques are used to avoid over-fitting any particular frequency.

To create Averaged Input Assumptions we combine returns over different look-back periods, giving more weight to the recent returns, to form overall Input Assumptions.

```create.ia.averaged <- function(lookbacks, n.lag) {
lookbacks = lookbacks
n.lag = n.lag

function(hist.returns, index=1:ncol(hist.returns), hist.all)
{
nperiods = nrow(hist.returns)

temp = c()
for (n.lookback in lookbacks)
temp = rbind(temp, hist.returns[(nperiods - n.lookback - n.lag + 1):(nperiods - n.lag), ])
create.ia(temp, index, hist.all)
}
}
```

To create Averaged Momentum we take a look-back weighted avaerage of momentums computed over different look-back periods.

```momentum.averaged <- function(prices,
lookbacks = c(20,60,120,250) ,	# length of momentum look back
n.lag = 3
) {
momentum = 0 * prices
for (n.lookback in lookbacks) {
part.mom = mlag(prices, n.lag) / mlag(prices, n.lookback + n.lag) - 1
momentum = momentum + 252 / n.lookback * part.mom
}
momentum / len(lookbacks)
}
```

Next let’s compare using historical Input Assumptions vs Averaged Input Assumptions and Momentum vs Averaged Momentum. I will consider Absolute Momentum (not cross sectional), for more information about relative and absolute momentum, please see

```###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
source(con)
close(con)

#*****************************************************************
#******************************************************************

# 10 funds
tickers = spl('Us.Eq = VTI + VTSMX,
Eurpoe.Eq = IEV + FIEUX,
Japan.Eq = EWJ + FJPNX,
Emer.Eq = EEM + VEIEX,
Re = RWX + VNQ + VGSIX,
Com = DBC + QRAAX,
Gold = GLD + SCGDX,
Long.Tr = TLT + VUSTX,
Mid.Tr = IEF + VFITX,
Short.Tr = SHY + VFISX')

start.date = 1998

dates = paste(start.date,'::',sep='')

data <- new.env()
getSymbols.extra(tickers, src = 'yahoo', from = '1980-01-01', env = data, set.symbolnames = T, auto.assign = T)
bt.prep(data, align='keep.all', dates=paste(start.date-2,':12::',sep=''), fill.gaps = T)

#*****************************************************************
# Setup
#******************************************************************
prices = data\$prices
n = ncol(prices)
nperiods = nrow(prices)

periodicity = 'months'
period.ends = endpoints(prices, periodicity)
period.ends = period.ends[period.ends > 0]

max.product.exposure = 0.6

#*****************************************************************
# Input Assumptions
#******************************************************************
lookback.len = 40
create.ia.fn = create.ia

# input assumptions are averaged on 20, 40, 60 days using 1 day lag
ia.array = c(20,40,60)
avg.create.ia.fn = create.ia.averaged(ia.array, 1)

#*****************************************************************
# Momentum
#******************************************************************
universe = prices > 0

mom.lookback.len = 120
momentum = prices / mlag(prices, mom.lookback.len) - 1
mom.universe = ifna(momentum > 0, F)

# momentum is averaged on 20,60,120,250 days using 3 day lag
mom.array = c(20,60,120,250)
avg.momentum = momentum.averaged(prices, mom.array, 3)
avgmom.universe = ifna(avg.momentum > 0, F)

#*****************************************************************
# Algos
#******************************************************************
min.risk.fns = list(
EW = equal.weight.portfolio,
MV = min.var.portfolio,
MCE = min.corr.excel.portfolio,

MV.RSO = rso.portfolio(min.var.portfolio, 3, 100, const.ub = max.product.exposure),
MCE.RSO = rso.portfolio(min.corr.excel.portfolio, 3, 100, const.ub = max.product.exposure)
)

#*****************************************************************
# Code Strategies
#******************************************************************
make.strategy.custom <- function(name, create.ia.fn, lookback.len, universe, env) {
obj = portfolio.allocation.helper(data\$prices,
periodicity = periodicity,
universe = universe,
lookback.len = lookback.len,
create.ia.fn = create.ia.fn,
const.ub = max.product.exposure,
min.risk.fns = min.risk.fns,
)
env[[name]] = create.strategies(obj, data, prefix=paste(name,'.',sep=''))\$models
}

models <- new.env()
make.strategy.custom('ia.none'        , create.ia.fn    , lookback.len, universe       , models)
make.strategy.custom('ia.mom'         , create.ia.fn    , lookback.len, mom.universe   , models)
make.strategy.custom('ia.avg_mom'     , create.ia.fn    , lookback.len, avgmom.universe, models)
make.strategy.custom('avg_ia.none'    , avg.create.ia.fn, 252         , universe       , models)
make.strategy.custom('avg_ia.mom'     , avg.create.ia.fn, 252         , mom.universe   , models)
make.strategy.custom('avg_ia.avg_mom' , avg.create.ia.fn, 252         , avgmom.universe, models)

#*****************************************************************
# Create Report
#*****************************************************************
strategy.snapshot.custom <- function(models, n = 0, title = NULL) {
if (n > 0)
models = models[ as.vector(matrix(1:len(models),ncol=n, byrow=T)) ]

layout(1:3)
plotbt(models, plotX = T, log = 'y', LeftMargin = 3, main = title)
mtext('Cumulative Performance', side = 2, line = 1)
plotbt.strategy.sidebyside(models)
barplot.with.labels(sapply(models, compute.turnover, data), 'Average Annual Portfolio Turnover', T)
}

# basic vs basic + momentum => momentum filter has better results
models.final = c(models\$ia.none, models\$ia.mom)
strategy.snapshot.custom(models.final, len(min.risk.fns), 'Momentum Filter')

# basic vs basic + avg ia => averaged ia reduce turnover
models.final = c(models\$ia.none, models\$avg_ia.none)
strategy.snapshot.custom(models.final, len(min.risk.fns), 'Averaged Input Assumptions')

# basic + momentum vs basic + avg.momentum => mixed results for averaged momentum
models.final = c(models\$ia.mom, models\$ia.avg_mom)
strategy.snapshot.custom(models.final, len(min.risk.fns), 'Averaged Momentum')

# basic + momentum vs avg ia + avg.momentum
models.final = c(models\$ia.mom, models\$avg_ia.avg_mom)
strategy.snapshot.custom(models.final, len(min.risk.fns), 'Averaged vs Base')
```

Above, I compared results for the following 4 cases:
1. Adding Momentum filter: all algos perfrom better

2. Input Assumptions vs Averaged Input Assumptions: returns are very similar, but using Averaged Input Assumptions helps reduce portfolio turnover.

3. Momentum vs Averaged Momentum: returns are very similar, but using Averaged Momentum increases portfolio turnover.

4. historical Input Assumptions + Momentum vs Averaged Input Assumptions + Averaged Momentum: results are mixed, no consistent advantage of using Averaged methods

Overall, the Averaged methods is a very interesting idea and I hope you will experiemtn with it and share your findings, like Pierre. Pierre, again thank you very much for sharing.

The full source code and example for the bt.averaged.test() function is available in bt.test.r at github.

## Fast Threshold Clustering Algorithm (FTCA) test

Today I want to share the test and implementation for the Fast Threshold Clustering Algorithm (FTCA) created by David Varadi. This implementation was developed and contributed by Pierre Chretien, I only made minor updates.

Let’s first replicate the results from the Fast Threshold Clustering Algorithm (FTCA) post:

```###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
source(con)
close(con)

#*****************************************************************
# Load historical data for ETFs
#******************************************************************

tickers = spl('XLY,XLP,XLE,XLF,XLV,XLI,XLB,XLK,XLU')

data <- new.env()
getSymbols(tickers, src = 'yahoo', from = '1900-01-01', env = data, auto.assign = T)
bt.prep(data, align='keep.all')

#*****************************************************************
# Helper function to compute portfolio allocation additional stats
#******************************************************************
portfolio.allocation.custom.stats.clusters <- function(x,ia) {
return(list(
clusters.FTCA = cluster.group.FTCA(0.5)(ia)
))
}

#*****************************************************************
# Find clusters
#******************************************************************
periodicity = 'months'
lookback.len = 252

obj = portfolio.allocation.helper(data\$prices,
periodicity = periodicity, lookback.len = lookback.len,
min.risk.fns = list(EW=equal.weight.portfolio),
custom.stats.fn = portfolio.allocation.custom.stats.clusters
)

clusters = obj\$clusters.FTCA\$EW
clusters['2012:05::']
```

The clusters are stable and match David’s results

```           XLB XLE XLF XLI XLK XLP XLU XLV XLY
2012-05-31   1   1   1   1   1   1   1   1   1
2012-06-29   1   1   1   1   1   1   1   1   1
2012-07-31   1   1   1   1   1   1   1   1   1
2012-08-31   1   1   1   1   1   1   1   1   1
2012-09-28   1   1   1   1   1   1   1   1   1
2012-10-31   1   1   1   1   1   1   1   1   1
2012-11-30   2   2   2   2   2   2   1   2   2
2012-12-31   2   2   2   2   2   2   1   2   2
2013-01-31   2   2   2   2   2   2   1   2   2
2013-02-28   1   1   1   1   1   1   1   1   1
2013-03-28   1   1   1   1   1   1   1   1   1
2013-04-30   1   1   1   1   1   1   1   1   1
2013-05-31   1   1   1   1   1   1   1   1   1
2013-06-28   1   1   1   1   1   1   1   1   1
2013-07-31   1   1   1   1   1   1   1   1   1
2013-08-30   1   1   1   1   1   1   1   1   1
2013-09-30   1   1   1   1   1   1   1   1   1
2013-10-31   1   1   1   1   1   1   1   1   1
2013-11-26   1   1   1   1   1   1   1   1   1
```

Next let’s compare the Cluster Portfolio Allocation Algorithm using K-means and FTCA:

```	#*****************************************************************
# Code Strategies
#******************************************************************
obj = portfolio.allocation.helper(data\$prices,
periodicity = periodicity, lookback.len = lookback.len,
min.risk.fns = list(
C.EW.kmeans = distribute.weights(equal.weight.portfolio, cluster.group.kmeans.90),
C.EW.FTCA = distribute.weights(equal.weight.portfolio, cluster.group.FTCA(0.5))
)
)

models = create.strategies(obj, data)\$models

#*****************************************************************
# Create Report
#******************************************************************
barplot.with.labels(sapply(models, compute.turnover, data), 'Average Annual Portfolio Turnover')
```

Both clustering algorithms produced very similar results. One noticeable difference is turnover. Since the Fast Threshold Clustering Algorithm (FTCA) produced more stable groups, it had smaller turnover.

The full source code and example for the cluster.group.FTCA() function is available in strategy.r at github.

Categories: Asset Allocation, Cluster, R

## getSymbols Extra

The getSymbols function from the quantmod package is an easy and convenient way to bring historical stock prices into your R environment. You need to specify the list of tickers, the source of historical prices and dates. For example following commands will download historical stock prices from yahoo finance for ‘RWX’, ‘VNQ’, ‘VGSIX’ symbols:

```	data <- new.env()
getSymbols.extra(c('RWX','VNQ','VGSIX'), src = 'yahoo', from = '1980-01-01', env = data, auto.assign = T)
```

Now, the data environment contains the historical stock prices. For example, create plot for RWX using the code below:

```	plot(data\$RWX)
```

Sometimes, I find that getSymbols functionality can be extended. For example, it would be nice to rename the series. I.e. RWX is a real estate ETF, so we could ask getSymbols function to get RWX, but call output Real.Estate. Another useful feature would be ability to specify how to extend the data. I.e. RWX is only started trading in 2007, it would be convenient to extend RWX time series with VNQ and VGSIX.

I created the getSymbols.extra function to address these features. The getSymbols.extra function allows you to specify tickers in the following format:

• RWX : the original functionality. i.e get historical stock prices for RWX and it can be accessed with data\$RWX
• Real.Estate = RWX : get historical stock prices for RWX, rename it to Real.Estate, and it can be accessed with data\$Real.Estate
• RWX + VNQ + VGSIX : get historical stock prices for RWX, VNQ, VGSIX and extend RWX with VNQ, next extend it with VGSIX, and it can be accessed with data\$RWX
• Real.Estate = RWX + VNQ + VGSIX : mix and match above functionality

Let’s look at the example:

```###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
source(con)
close(con)

tickers = spl('REIT=RWX, RWX+VNQ, REIT.LONG=RWX+VNQ+VGSIX')
data <- new.env()
getSymbols.extra(tickers, src = 'yahoo', from = '1980-01-01', env = data, auto.assign = T)
bt.start.dates(data)

data\$symbolnames = spl('REIT.LONG,RWX,REIT')
bt.prep(data, align='keep.all')

plota.matplot(data\$prices)
```

The REIT.LONG is RWX extended using VNQ and next using VGSIX.

There is also a possibility to provide custom data into the getSymbols.extra function. I.e. the data which is not available through the getSymbols function.
For example, to extend GLD time series we might used historical Gold prices from Bundes Bank:

```	# Use extrenal data
raw.data <- new.env()
raw.data\$GOLD = bundes.bank.data.gold()

tickers = spl('GLD, GLD.LONG=GLD+GOLD')
data <- new.env()
getSymbols.extra(tickers, src = 'yahoo', from = '1980-01-01', env = data, raw.data = raw.data, auto.assign = T)
bt.start.dates(data)
data\$symbolnames = spl('GLD.LONG,GLD')