## Averaged Input Assumptions and Momentum

December 5, 2013 1 comment

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)
# http://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)
# http://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

November 26, 2013 1 comment

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)
# http://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')
bt.prep(data, align='keep.all')

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

The main goal of the getSymbols.extra function is to simplify your data acquisition/preparation step. Please let me know if you run into any problems or have suggestions.

To view the complete source code for this example, please have a look at the getSymbols.extra.test() function in utils.r at github.

Categories: R

## Running Back-tests in parallel

Once you start experimenting with many different asset allocation algorithms, the computation time of running the back-tests can be substantial. One simple way to solve the computation time problem is to run the back-tests in parallel. I.e. if the asset allocation algorithm does not use the prior period holdings to make decision about current allocation, we can run many periods in parallel.

In the Update for Backtesting Asset Allocation Portfolios post, I show cased the portfolio.allocation.helper() function in strategy.r at github. The portfolio.allocation.helper() function is a user-friendly interface to evaluate multiple asset allocation algorithms over given asset universe in a sequential fashion.

Following is a sample code from the Update for Backtesting Asset Allocation Portfolios post:

```    #*****************************************************************
# Code Strategies
#******************************************************************
obj = portfolio.allocation.helper(data\$prices,
periodicity = 'months', lookback.len = 60,
min.risk.fns = list(
EW=equal.weight.portfolio,
RP=risk.parity.portfolio,
MD=max.div.portfolio,

MV=min.var.portfolio,
MVE=min.var.excel.portfolio,
MV2=min.var2.portfolio,

MC=min.corr.portfolio,
MCE=min.corr.excel.portfolio,
MC2=min.corr2.portfolio,

MS=max.sharpe.portfolio(),
ERC = equal.risk.contribution.portfolio,

# target retunr / risk
TRET.12 = target.return.portfolio(12/100),
TRISK.10 = target.risk.portfolio(10/100),

# rso
RSO.RP.5 = rso.portfolio(risk.parity.portfolio, 5, 500),

# others
MMaxLoss = min.maxloss.portfolio,
MCVaR = min.cvar.portfolio,
MCDaR = min.cdar.portfolio,
MRiskDown = min.risk.downside.portfolio,
)
)
```

To run the same strategies in parallel, I created the portfolio.allocation.helper.parallel() function in strategy.r at github. There is one extra input that you need to specify: cores – number of CPU processors used for computations.

For example, the code below will use 2 CPU processors to run back-test computations. It will run faster than the portfolio.allocation.helper() function.

```    #*****************************************************************
# Code Strategies
#******************************************************************
obj = portfolio.allocation.helper.parallel(cores = 2, data\$prices,
periodicity = 'months', lookback.len = 60,
min.risk.fns = list(
EW=equal.weight.portfolio,
RP=risk.parity.portfolio,
MD=max.div.portfolio,

MV=min.var.portfolio,
MVE=min.var.excel.portfolio,
MV2=min.var2.portfolio,

MC=min.corr.portfolio,
MCE=min.corr.excel.portfolio,
MC2=min.corr2.portfolio,

MS=max.sharpe.portfolio(),
ERC = equal.risk.contribution.portfolio,

# target retunr / risk
TRET.12 = target.return.portfolio(12/100),
TRISK.10 = target.risk.portfolio(10/100),

# rso
RSO.RP.5 = rso.portfolio(risk.parity.portfolio, 5, 500),

# others
MMaxLoss = min.maxloss.portfolio,
MCVaR = min.cvar.portfolio,
MCDaR = min.cdar.portfolio,
MRiskDown = min.risk.downside.portfolio,
)
)
```

Hopefully, I did not ruin your prolong lunch plans

## Commissions

November 5, 2013 1 comment

Today, I want to explain the commission’s functionality build in to Systematic Investor Toolbox(SIT) “share” back-test.

At each re-balance time the capital is allocated given the weight such that

```	share = weight * capital / price
cash  = capital - share * price
```

For example, if weight is 100% (i.e. fully invested) and capital = \$100 and price = \$10 then

```	share = 10 shares
cash = \$0
```

The period return is equal to

```	return = [share * price + cash] / [share * price.yesterday + cash] - 1
```

The total return is equal to

```	total.return = [1 + return.0] * [1 + return.1] * ... * [1 + return.n] - 1
```

The period returns constructed this way let me construct portfolio returns without using loops
I.e. if share, price, cash are matrices, then

```	portfolio.return = rowSums((share * price + cash) / (share * mlag(price) + cash) - 1)
```

and

```	equity = cumprod(1 + portfolio.return)
```

To introduce commissions into above framework, I had to make following assumptions.
Let’ assume that P0 and P1 are stock prices and com is commission that is very small relative to stock price then

```	[P0 - com] / P0 is close (equal) to P0 / [P0 + com] and
[P0 - com] * P1 is close (equal) to P0 * [P1 - com]
```

Given commissions, the period return formula used in SIT is equal to

```	return = [share * price + cash - commission] / [share * price.yesterday + cash] - 1
```

Now let’s look at the example of trade with commissions:

```	Let's say we are fully invested (i.e. cash = 0 and capital = share * P0)

opening trade cost = share * P0 + com
closing  trade cost = share * P1 - com

= [share * P1 - com] / [share * P0 + com] - 1
```

In SIT, these computations are equivalent to

```	([capital - com] / [capital]) * ([share * P1 + cash - com] / [share * P0 + cash]) - 1

< given that cash = 0 and capital = share * P0 >
= ([share * P0 - com] / [share * P0]) * ([share * P1 - com] / [share * P0]) - 1

< given [P0 - com] / P0 ~ P0 / [P0 + com] >
= ([share * P0] / [share * P0 + com] / ) * ([share * P1 - com] / [share * P0]) - 1

= [share * P1 - com] / [share * P0 + com] - 1
```

Hence, as long as commissions are small relative to whole trade the returns produced with SIT will be very close to the true returns.

SIT currently supports following 3 types of commissions

• cents / share commission
```	commission = abs(share - mlag(share)) * commission\$cps
```
• fixed commission
```	commission = sign(abs(share - mlag(share))) * commission\$fixed
```
• percentage commission
```	commission = price * abs(share - mlag(share)) * commission\$percentage
```

You can mix and match these commission methods in any way,

```	the total.commission = cps.commission + fixed.commission + percentage.commission
```

and period return is equal to

```	return = (share * price + cash - total.commission) / (share * mlag(price) + cash) - 1
```

Next let’s see the impact of different type of commissions. There two ways to specify the commissions.

• `commission = 0.1`

will be translated in

`commission = list(cps = 0.1, fixed = 0.0, percentage = 0.0)`
• `commission = list(cps = 0.0, fixed = 0.0, percentage = 0.0)`
```###############################################################################
# Load Systematic Investor Toolbox (SIT)
# http://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
source(con)
close(con)

#*****************************************************************
#******************************************************************
tickers = spl('EEM')

data <- new.env()
getSymbols(tickers, src = 'yahoo', from = '1970-01-01', env = data, auto.assign = T)
bt.prep(data, align='keep.all', dates='2013:08::2013:09')

#*****************************************************************
# Code Strategies
#******************************************************************
sell.date = '2013:08:15'
day.after.sell.date = '2013:08:16'

capital = 100000
prices = data\$prices

# helper function to compute trade return

#*****************************************************************
# Zero commission
#******************************************************************
data\$weight[] = NA
data\$weight[sell.date] = 0
commission = 0.0
model = bt.run.share(data, commission = commission, capital = capital, silent = T)

comp.ret( share * prices[sell.date], share * prices[buy.date] )

#*****************************************************************
# 10c cps commission
# cents / share commission
#   trade cost = abs(share - mlag(share)) * commission\$cps
#******************************************************************
data\$weight[] = NA
data\$weight[sell.date] = 0
commission = 0.1
model = bt.run.share(data, commission = commission, capital = capital, silent = T)

comp.ret( share * (prices[sell.date] - commission), share * (prices[buy.date] + commission) )

#*****************************************************************
# \$5 fixed commission
# fixed commission per trade to more effectively to penalize for turnover
#   trade cost = sign(abs(share - mlag(share))) * commission\$fixed
#******************************************************************
data\$weight[] = NA
data\$weight[sell.date] = 0
commission = list(cps = 0.0, fixed = 5.0, percentage = 0.0)
model = bt.run.share(data, commission = commission, capital = capital, silent = T)

comp.ret( share * prices[sell.date] - commission\$fixed, share * prices[buy.date] + commission\$fixed )

#*****************************************************************
# % commission
# percentage commission
#   trade cost = price * abs(share - mlag(share)) * commission\$percentage
#******************************************************************
data\$weight[] = NA
data\$weight[sell.date] = 0
commission = list(cps = 0.0, fixed = 0.0, percentage = 1/100)
model = bt.run.share(data, commission = commission, capital = capital, silent = T)

comp.ret( share * prices[sell.date] * (1 - commission\$percentage), share * prices[buy.date] * (1 + commission\$percentage) )
```

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

I recently came across a very interesting idea at the The Problem with Market Neutral (and an Answer) post by Mebane Faber. Today I want to show how you can test such strategy using the Systematic Investor Toolbox:

```###############################################################################
# Load Systematic Investor Toolbox (SIT)
# http://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 = temp[[1]]\$Mkt.RF + temp[[1]]\$RF
price = bt.apply.matrix(ret / 100, function(x) cumprod(1 + x))
data\$SPY = make.stock.xts( price )

ret = temp[[1]]
price = bt.apply.matrix(ret / 100, function(x) cumprod(1 + x))
data\$HI.MO = make.stock.xts( price\$High )
data\$LO.MO = make.stock.xts( price\$Low )

# align dates
bt.prep(data, align='remove.na')

#*****************************************************************
# Code Strategies
#*****************************************************************
models = list()

data\$weight[] = NA
data\$weight\$SPY[] = 1
models\$SPY = bt.run.share(data, clean.signal=T)

data\$weight[] = NA
data\$weight\$HI.MO[] = 1
models\$HI.MO = bt.run.share(data, clean.signal=T)

data\$weight[] = NA
data\$weight\$LO.MO[] = 1
models\$LO.MO = bt.run.share(data, clean.signal=T)

data\$weight[] = NA
data\$weight\$HI.MO[] = 1
data\$weight\$LO.MO[] = -1
models\$MKT.NEUTRAL = bt.run.share(data, clean.signal=F)

#*****************************************************************
# Modified MN
# The modified strategy below starts 100% market neutral, and depending on the drawdown bucket
# will reduce the shorts all the way to zero once the market has declined by 50%
# (in 20% steps for every 10% decline in stocks)
#*****************************************************************
market.drawdown = -100 * compute.drawdown(data\$prices\$SPY)
market.drawdown.10.step = 10 * floor(market.drawdown / 10)
short.allocation = 100 - market.drawdown.10.step * 2
short.allocation[ short.allocation < 0 ] = 0

data\$weight[] = NA
data\$weight\$HI.MO[] = 1
data\$weight\$LO.MO[] = -1 * short.allocation / 100
models\$Modified.MN = bt.run.share(data, clean.signal=F)

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

Mebane thank you very much for sharing this great observation and great strategy that works! I would encourage readers to experiment with idea and share their findings.

If you want to concentrate on the long side, one idea that comes to mind is to start not fully invested say at 90% allocation, and once the market hits say 20% draw-down to invest 100% in expectation of quick recovery.

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