# Momentum, Markowitz, and Solving Rank-Deficient Covariance Matrices — The Constrained Critical Line Algorithm

```
```

This post will feature the differences in the implementation of my constrained critical line algorithm with that of Dr. Clarence Kwan’s. The constrained critical line algorithm is a form of gradient descent that incorporates elements of momentum. My implementation includes a volatility-targeting binary search algorithm.

First off, rather than try and explain the algorithm piece by piece, I’ll defer to Dr. Clarence Kwan’s paper and excel spreadsheet, from where I obtained my original implementation. Since that paper and excel spreadsheet explains the functionality of the algorithm, I won’t repeat that process here. Essentially, the constrained critical line algorithm incorporates its lambda constraints into the structure of the covariance matrix itself. This innovation actually allows the algorithm to invert previously rank-deficient matrices.

Now, while Markowitz mean-variance optimization may be a bit of old news for some, the ability to use a short lookback for momentum with monthly data has allowed me and my two coauthors (Dr. Wouter Keller, who came up with flexible and elastic asset allocation, and Adam Butler, of GestaltU) to perform a backtest on a century’s worth of assets, with more than 30 assets in the backtest, despite using only a 12-month formation period. That paper can be found here.

Let’s look at the code for the function.

```CCLA <- function(covMat, retForecast, maxIter = 1000,
verbose = FALSE, scale = 252,
weightLimit = .7, volThresh = .1) {
if(length(retForecast) > length(unique(retForecast))) {
sequentialNoise <- seq(1:length(retForecast)) * 1e-12
retForecast <- retForecast + sequentialNoise
}

#initialize original out/in/up status
if(length(weightLimit) == 1) {
weightLimit <- rep(weightLimit, ncol(covMat))
}
rankForecast <- length(retForecast) - rank(retForecast) + 1
remainingWeight <- 1 #have 100% of weight to allocate
upStatus <- inStatus <- rep(0, ncol(covMat))
i <- 1
while(remainingWeight > 0) {
securityLimit <- weightLimit[rankForecast == i]
if(securityLimit < remainingWeight) {
upStatus[rankForecast == i] <- 1 #if we can't invest all remaining weight into the security
remainingWeight <- remainingWeight - securityLimit
} else {
inStatus[rankForecast == i] <- 1
remainingWeight <- 0
}
i <- i + 1
}

#initial matrices (W, H, K, identity, negative identity)
covMat <- as.matrix(covMat)
retForecast <- as.numeric(retForecast)
init_W <- cbind(2*covMat, rep(-1, ncol(covMat)))
init_W <- rbind(init_W, c(rep(1, ncol(covMat)), 0))
H_vec <- c(rep(0, ncol(covMat)), 1)
K_vec <- c(retForecast, 0)
negIdentity <- -1*diag(ncol(init_W))
identity <- diag(ncol(init_W))
matrixDim <- nrow(init_W)
weightLimMat <- matrix(rep(weightLimit, matrixDim), ncol=ncol(covMat), byrow=TRUE)

#out status is simply what isn't in or up
outStatus <- 1 - inStatus - upStatus

#initialize expected volatility/count/turning points data structure
expVol <- Inf
lambda <- 100
count <- 0
turningPoints <- list()
while(lambda > 0 & count < maxIter) {

#old lambda and old expected volatility for use with numerical algorithms
oldLambda <- lambda
oldVol <- expVol

count <- count + 1

#compute W, A, B
inMat <- matrix(rep(c(inStatus, 1), matrixDim), nrow = matrixDim, byrow = TRUE)
upMat <- matrix(rep(c(upStatus, 0), matrixDim), nrow = matrixDim, byrow = TRUE)
outMat <- matrix(rep(c(outStatus, 0), matrixDim), nrow = matrixDim, byrow = TRUE)

W <- inMat * init_W + upMat * identity + outMat * negIdentity

inv_W <- solve(W)
modified_H <- H_vec - rowSums(weightLimMat* upMat[,-matrixDim] * init_W[,-matrixDim])
A_vec <- inv_W %*% modified_H
B_vec <- inv_W %*% K_vec

#remove the last elements from A and B vectors
truncA <- A_vec[-length(A_vec)]
truncB <- B_vec[-length(B_vec)]

#compute in Ratio (aka Ratio(1) in Kwan.xls)
inRatio <- rep(0, ncol(covMat))
inRatio[truncB > 0] <- -truncA[truncB > 0]/truncB[truncB > 0]

#compute up Ratio (aka Ratio(2) in Kwan.xls)
upRatio <- rep(0, ncol(covMat))
upRatioIndices <- which(inStatus==TRUE & truncB < 0)
if(length(upRatioIndices) > 0) {
upRatio[upRatioIndices] <- (weightLimit[upRatioIndices] - truncA[upRatioIndices]) / truncB[upRatioIndices]
}

#find lambda -- max of up and in ratios
maxInRatio <- max(inRatio)
maxUpRatio <- max(upRatio)
lambda <- max(maxInRatio, maxUpRatio)

#compute new weights
wts <- inStatus*(truncA + truncB * lambda) + upStatus * weightLimit + outStatus * 0

#compute expected return and new expected volatility
expRet <- t(retForecast) %*% wts
expVol <- sqrt(wts %*% covMat %*% wts) * sqrt(scale)

#create turning point data row and append it to turning points
turningPoint <- cbind(count, expRet, lambda, expVol, t(wts))
colnames(turningPoint) <- c("CP", "Exp. Ret.", "Lambda", "Exp. Vol.", colnames(covMat))
turningPoints[[count]] <- turningPoint

#binary search for volatility threshold -- if the first iteration is lower than the threshold,
#then immediately return, otherwise perform the binary search until convergence of lambda
if(oldVol == Inf & expVol < volThresh) {
turningPoints <- do.call(rbind, turningPoints)
threshWts <- tail(turningPoints, 1)
return(list(turningPoints, threshWts))
} else if(oldVol > volThresh & expVol < volThresh) {
upLambda <- oldLambda
dnLambda <- lambda
meanLambda <- (upLambda + dnLambda)/2
while(upLambda - dnLambda > .00001) {

#compute mean lambda and recompute weights, expected return, and expected vol
meanLambda <- (upLambda + dnLambda)/2
wts <- inStatus*(truncA + truncB * meanLambda) + upStatus * weightLimit + outStatus * 0
expRet <- t(retForecast) %*% wts
expVol <- sqrt(wts %*% covMat %*% wts) * sqrt(scale)

#if new expected vol is less than threshold, mean becomes lower bound
#otherwise, it becomes the upper bound, and loop repeats
if(expVol < volThresh) {
dnLambda <- meanLambda
} else {
upLambda <- meanLambda
}
}

#once the binary search completes, return those weights, and the corner points
#computed until the binary search. The corner points aren't used anywhere, but they're there.
threshWts <- cbind(count, expRet, meanLambda, expVol, t(wts))
colnames(turningPoint) <- colnames(threshWts) <- c("CP", "Exp. Ret.", "Lambda", "Exp. Vol.", colnames(covMat))
turningPoints[[count]] <- turningPoint
turningPoints <- do.call(rbind, turningPoints)
return(list(turningPoints, threshWts))
}

#this is only run for the corner points during which binary search doesn't take place
#change status of security that has new lambda
if(maxInRatio > maxUpRatio) {
inStatus[inRatio == maxInRatio] <- 1 - inStatus[inRatio == maxInRatio]
upStatus[inRatio == maxInRatio] <- 0
} else {
upStatus[upRatio == maxUpRatio] <- 1 - upStatus[upRatio == maxUpRatio]
inStatus[upRatio == maxUpRatio] <- 0
}
outStatus <- 1 - inStatus - upStatus
}

#we only get here if the volatility threshold isn't reached
#can actually happen if set sufficiently low
turningPoints <- do.call(rbind, turningPoints)

threshWts <- tail(turningPoints, 1)

return(list(turningPoints, threshWts))
}
```

Essentially, the algorithm can be divided into three parts:

The first part is the initialization, which does the following:

It creates three status vectors: in, up, and out. The up vector denotes which securities are at their weight constraint cap, the in status are securities that are not at their weight cap, and the out status are securities that receive no weighting on that iteration of the algorithm.

The rest of the algorithm essentially does the following:

It takes a gradient descent approach by changing the status of the security that minimizes lambda, which by extension minimizes the volatility at the local point. As long as lambda is greater than zero, the algorithm continues to iterate. Letting the algorithm run until convergence effectively provides the volatility-minimization portfolio on the efficient frontier.

However, one change that Dr. Keller and I made to it is the functionality of volatility targeting, allowing the algorithm to stop between iterations. As the SSRN paper shows, a higher volatility threshold, over the long run (the *VERY* long run) will deliver higher returns.

In any case, the algorithm takes into account several main arguments:

A return forecast, a covariance matrix, a volatility threshold, and weight limits, which can be either one number that will result in a uniform weight limit, or a per-security weight limit. Another argument is scale, which is 252 for days, 12 for months, and so on. Lastly, there is a volatility threshold component, which allows the user to modify how aggressive or conservative the strategy can be.

In any case, to demonstrate this function, let’s run a backtest. The idea in this case will come from a recent article published by Frank Grossmann from SeekingAlpha, in which he obtained a 20% CAGR but with a 36% max drawdown.

So here’s the backtest:

```symbols &amp;lt;- c("AFK", "ASHR", "ECH", "EGPT",
"EIDO", "EIRL", "EIS", "ENZL",
"EPHE", "EPI", "EPOL", "EPU",
"EWA", "EWC", "EWD", "EWG",
"EWH", "EWI", "EWJ", "EWK",
"EWL", "EWM", "EWN", "EWO",
"EWP", "EWQ", "EWS", "EWT",
"EWU", "EWW", "EWY", "EWZ",
"EZA", "FM", "FRN", "FXI",
"GAF", "GULF", "GREK", "GXG",
"IDX", "MCHI", "MES", "NORW",
"QQQ", "RSX", "THD", "TUR",
"VNM", "TLT"
)

getSymbols(symbols, from = "2003-01-01")

prices &amp;lt;- list()
entryRets &amp;lt;- list()
for(i in 1:length(symbols)) {
prices[[i]] &amp;lt;- Ad(get(symbols[i]))
}
prices &amp;lt;- do.call(cbind, prices)
colnames(prices) &amp;lt;- gsub("\\.[A-z]*", "", colnames(prices))

returns &amp;lt;- Return.calculate(prices)
returns &amp;lt;- returns[-1,]

sumIsNa &amp;lt;- function(col) {
return(sum(is.na(col)))
}

appendZeroes &amp;lt;- function(selected, originalSetNames) {
zeroes &amp;lt;- rep(0, length(originalSetNames) - length(selected))
names(zeroes) &amp;lt;- originalSetNames[!originalSetNames %in% names(selected)]
all &amp;lt;- c(selected, zeroes)
all &amp;lt;- all[originalSetNames]
return(all)
}

computeStats &amp;lt;- function(rets) {
stats &amp;lt;- rbind(table.AnnualizedReturns(rets), maxDrawdown(rets), CalmarRatio(rets))
return(round(stats, 3))
}

CLAAbacktest &amp;lt;- function(returns, lookback = 3, volThresh = .1, assetCaps = .5, tltCap = 1,
returnWeights = FALSE, useTMF = FALSE) {
if(useTMF) {
returns\$TLT &amp;lt;- returns\$TLT * 3
}
ep &amp;lt;- endpoints(returns, on = "months")
weights &amp;lt;- list()
for(i in 2:(length(ep) - lookback)) {
retSubset &amp;lt;- returns[(ep[i]+1):ep[i+lookback],]
retNAs &amp;lt;- apply(retSubset, 2, sumIsNa)
validRets &amp;lt;- retSubset[, retNAs==0]
retForecast &amp;lt;- Return.cumulative(validRets)
covRets &amp;lt;- cov(validRets)
weightLims &amp;lt;- rep(assetCaps, ncol(covRets))
weightLims[colnames(covRets)=="TLT"] &amp;lt;- tltCap
weight &amp;lt;- CCLA(covMat = covRets, retForecast = retForecast, weightLimit = weightLims, volThresh = volThresh)
weight &amp;lt;- weight[[2]][,5:ncol(weight[[2]])]
weight &amp;lt;- appendZeroes(selected = weight, colnames(retSubset))
weight &amp;lt;- xts(t(weight), order.by=last(index(validRets)))
weights[[i]] &amp;lt;- weight
}
weights &amp;lt;- do.call(rbind, weights)
stratRets &amp;lt;- Return.portfolio(R = returns, weights = weights)
if(returnWeights) {
return(list(weights, stratRets))
}
return(stratRets)
}
```

In essence, we take the returns over a specified monthly lookback period, specify a volatility threshold, specify asset caps, specify the bond asset cap, and whether or not we wish to use TLT or TMF (a 3x leveraged variant, which just multiplies all returns of TLT by 3, for simplicity). The output of the CCLA (Constrained Critical Line Algorithm) is a list that contains the corner points, and the volatility threshold corner point which contains the corner point number, expected return, expected volatility, and the lambda value. So, we want the fifth element onward of the second element of the list.

Here are some results:

```config1 &amp;lt;- CLAAbacktest(returns = returns)
config2 &amp;lt;- CLAAbacktest(returns = returns, useTMF = TRUE)
config3 &amp;lt;- CLAAbacktest(returns = returns, lookback = 4)
config4 &amp;lt;- CLAAbacktest(returns = returns, lookback = 2, useTMF = TRUE)

comparison &amp;lt;- na.omit(cbind(config1, config2, config3, config4))
colnames(comparison) &amp;lt;- c("Default", "TMF instead of TLT", "Lookback 4", "Lookback 2 and TMF")
charts.PerformanceSummary(comparison)
computeStats(comparison)
```

With the following statistics:

```&amp;gt; computeStats(comparison)
Default TMF instead of TLT Lookback 4 Lookback 2 and TMF
Annualized Return           0.137              0.146      0.133              0.138
Annualized Std Dev          0.126              0.146      0.125              0.150
Annualized Sharpe (Rf=0%)   1.081              1.000      1.064              0.919
Worst Drawdown              0.219              0.344      0.186              0.357
Calmar Ratio                0.625              0.424      0.714              0.386
```

The variants that use TMF instead of TLT suffer far worse drawdowns. Not much of a hedge, apparently.

Here’s the equity curve:

Taking the 4 month lookback configuration (strongest Calmar), we’ll play around with the volatility setting.

Here’s the backtest:

```config5 &amp;lt;- CLAAbacktest(returns = returns, lookback = 4, volThresh = .15)
config6 &amp;lt;- CLAAbacktest(returns = returns, lookback = 4, volThresh = .2)

comparison2 &amp;lt;- na.omit(cbind(config3, config5, config6))
colnames(comparison2) &amp;lt;- c("Vol10", "Vol15", "Vol20")
charts.PerformanceSummary(comparison2)
computeStats(comparison2)
```

With the results:

```&amp;gt; computeStats(comparison2)
Vol10 Vol15 Vol20
Annualized Return         0.133 0.153 0.180
Annualized Std Dev        0.125 0.173 0.204
Annualized Sharpe (Rf=0%) 1.064 0.886 0.882
Worst Drawdown            0.186 0.212 0.273
Calmar Ratio              0.714 0.721 0.661
```

In this case, more risk, more reward, lower risk/reward ratios as you push the volatility threshold. So for once, the volatility puzzle doesn’t rear its head, and higher risk indeed does translate to higher returns (at the cost of everything else, though).

Here’s the equity curve.

Lastly, let’s try toggling the asset cap limits with the vol threshold back at 10.

```config7 &amp;lt;- CLAAbacktest(returns = returns, lookback = 4, assetCaps = .1)
config8 &amp;lt;- CLAAbacktest(returns = returns, lookback = 4, assetCaps = .25)
config9 &amp;lt;- CLAAbacktest(returns = returns, lookback = 4, assetCaps = 1/3)
config10 &amp;lt;- CLAAbacktest(returns = returns, lookback = 4, assetCaps = 1)

comparison3 &amp;lt;- na.omit(cbind(config7, config8, config9, config3, config10))
colnames(comparison3) &amp;lt;- c("Cap10", "Cap25", "Cap33", "Cap50", "Uncapped")
charts.PerformanceSummary(comparison3)
computeStats(comparison3)
```

With the resulting statistics:

```&amp;gt; computeStats(comparison3)
Cap10 Cap25 Cap33 Cap50 Uncapped
Annualized Return         0.124 0.122 0.127 0.133    0.134
Annualized Std Dev        0.118 0.122 0.123 0.125    0.126
Annualized Sharpe (Rf=0%) 1.055 1.002 1.025 1.064    1.070
Worst Drawdown            0.161 0.185 0.186 0.186    0.186
Calmar Ratio              0.771 0.662 0.680 0.714    0.721
```

Essentially, in this case, there was very little actual change from simply tweaking weight limits. Here’s an equity curve:

To conclude, while not exactly achieving the same aggregate returns or Sharpe ratio that the SeekingAlpha article did, it did highlight a probable cause of its major drawdown, and also demonstrated the levers of how to apply the constrained critical line algorithm, the mechanics of which are detailed in the papers linked to earlier.

Thanks for reading

# A Basic Logical Invest Global Market Rotation Strategy

This may be one of the simplest strategies I’ve ever presented on this blog, but nevertheless, it works, for some definition of “works”.

Here’s the strategy: take five global market ETFs (MDY, ILF, FEZ, EEM, and EPP), along with a treasury ETF (TLT), and every month, fully invest in the security that had the best momentum. While I’ve tried various other tweaks, none have given the intended high return performance that the original variant has.

Here’s the link to the original strategy.

While I’m not quite certain of how to best go about programming the variable lookback period, this is the code for the three month lookback.

```require(quantmod)
require(PerformanceAnalytics)

symbols <- c("MDY", "TLT", "EEM", "ILF", "EPP", "FEZ")
getSymbols(symbols, from="1990-01-01")
prices <- list()
for(i in 1:length(symbols)) {
prices[[i]] <- Ad(get(symbols[i]))
}
prices <- do.call(cbind, prices)
colnames(prices) <- gsub("\\.[A-z]*", "", colnames(prices))
returns <- Return.calculate(prices)
returns <- na.omit(returns)

logicInvestGMR <- function(returns, lookback = 3) {
ep <- endpoints(returns, on = "months")
weights <- list()
for(i in 2:(length(ep) - lookback)) {
retSubset <- returns[ep[i]:ep[i+lookback],]
cumRets <- Return.cumulative(retSubset)
rankCum <- rank(cumRets)
weight <- rep(0, ncol(retSubset))
weight[which.max(cumRets)] <- 1
weight <- xts(t(weight), order.by=index(last(retSubset)))
weights[[i]] <- weight
}
weights <- do.call(rbind, weights)
stratRets <- Return.portfolio(R = returns, weights = weights)
return(stratRets)
}

gmr <- logicInvestGMR(returns)
charts.PerformanceSummary(gmr)
```

And here’s the performance:

```> rbind(table.AnnualizedReturns(gmr), maxDrawdown(gmr), CalmarRatio(gmr))
portfolio.returns
Annualized Return                  0.287700
Annualized Std Dev                 0.220700
Annualized Sharpe (Rf=0%)          1.303500
Worst Drawdown                     0.222537
Calmar Ratio                       1.292991
```

With the resultant equity curve:

While I don’t get the 34% advertised, nevertheless, the risk to reward ratio over the duration of the backtest is fairly solid for something so simple, and I just wanted to put this out there.

Thanks for reading.

# Advertising a Few Systematic ETFs (Strictly Of My Own Volition)

This post will introduce several ETFs from Alpha Architect and Cambria Funds (run by Meb Faber) that I think readers should be aware of (if not so already) in order to capitalize on systematic investing without needing to lose a good portion of the return due to taxes and transaction costs.

So, as my readers know, I backtest lots of strategies on this blog that deal with monthly turnover, and many transactions. In all instances, I assume that A) slippage and transaction costs are negligible, =B) there is sufficient capital such that when a weighting scheme states to place 5.5% of a portfolio into an ETF with an expensive per-share price (EG a sector spider, SPY, etc.), that the issue of integer shares can be adhered to without issue, and C) that there are no taxes on the monthly transactions. For retail investors without millions of dollars to deploy, one or more of these assumptions may not hold. After all, if you have \$20,000 to invest, and are paying \$50 a month on turnover costs, that’s -3% to your CAGR, which would render quite a few of these strategies pretty terrible.

So, in this short blurb, I want to shine a light on several of these ETFs.

First off, a link to a post from Alpha Architect that essentially states that there are only two tried-and-true market “anomalies” when correcting for data-mining: value, and momentum. Well, that and the durable consumption goods factor. The first, I’m not quite sure how to rigorously test using only freely available data, and the last, I’m not quite sure why it works. Low volatility, perhaps?

In any case, for people who don’t have institutional-grade investing capabilities, here are some ETFs that aim to intelligently capitalize on the value and momentum factors, along with one “permanent portfolio” type of ETF.

Momentum:
GMOM: Global Momentum. Essentially, spread your bets, and go with the trend. Considering Meb Faber is a proponent of momentum (see his famous Ivy Portfolio book), this is the way to capitalize on that.

Value:
QVAL: Alpha Architect’s (domestic) Quantitative Value ETF. The team at Alpha Architect are proponents of value investing, and with a team of several PhDs dedicated to a systematic value investing research process, this may be a way for retail investors to buy-and-hold one product and outsource the meticulous value research necessary for the proper implementation of such a strategy.

IVAL: an international variant of the above.

GVAL: The Cambria Funds quantitative value fund.

Asset Allocation (permanent portfolio):

GAA: Global Asset Allocation. My interpretation? Take the good old stocks, bonds, and real assets portfolio, and spread it out across the globe.

Now, let’s just do a quick rundown and see how these strategies have performed over the small time horizon the latest one has been in existence.

```symbols <- c("GMOM", "QVAL", "IVAL", "GVAL", "GAA")

getSymbols(symbols, from = "1990-01-01")
prices <- list()
for(i in 1:length(symbols)) {
prices[[i]] <- Ad(get(symbols[i]))
}
prices <- do.call(cbind, prices)
colnames(prices) <- gsub("\\.[A-z]*", "", colnames(prices))

coolEtfReturns <- Return.calculate(prices)
coolEtfReturns <- na.omit(coolEtfReturns)
charts.PerformanceSummary(coolEtfReturns, main = "Quant investing for retail people.")

stats <- rbind(table.AnnualizedReturns(coolEtfReturns),
maxDrawdown(coolEtfReturns),
CalmarRatio(coolEtfReturns),
SortinoRatio(coolEtfReturns) * sqrt(252))
round(stats, 3)
```
```                           GMOM  QVAL  IVAL  GVAL   GAA
Annualized Return         0.038 0.237 0.315 0.323 0.106
Annualized Std Dev        0.082 0.138 0.123 0.192 0.066
Annualized Sharpe (Rf=0%) 0.466 1.709 2.556 1.680 1.595
Worst Drawdown            0.039 0.046 0.046 0.069 0.028
Calmar Ratio              0.981 5.189 6.816 4.678 3.737
Sortino Ratio (MAR = 0%)  0.665 2.598 3.742 2.274 2.407
```

In other words, aside from momentum, which is having a flat-ish series of months, the performances are overall fairly strong, in this tiny sample (not at all significant).

The one caveat I’d throw out there, however, is that these instruments are not foolproof. For fun, here’s a plot of GVAL (that is, Cambria’s global value fund) since its inception.

And the statistics for it for the whole duration of its inception.

```                          GVAL.Adjusted
Annualized Return                -0.081
Annualized Std Dev                0.164
Annualized Sharpe (Rf=0%)        -0.494
Worst Drawdown                    0.276
Calmar Ratio                     -0.294
```

Again, tiny sample, so nothing conclusive at all, but it just means that these funds may occasionally hurt (no free lunch). That stated, I nevertheless think that Dr. Wesley Gray and Mebane Faber, at Alpha Architect and Cambria Funds, respectively, are about as reputable of money managers as one would find, and the idea that one can invest with them, as opposed to god knows with what mutual fund, to me, is something I think that’s worth not just pointing out, but drawing some positive attention to.

That stated, if anyone out there has hypothetical performances for these funds that goes back to a ten year history in a time-series, I’d love to run some analysis on those. After all, if there were some simple way to improve the performances of a portfolio of these instruments even more, well, I believe Newton had something to say about standing on the shoulders of giants.

Thanks for reading.

NOTE: I will be giving a quick lightning talk at R in finance in Chicago later this month (about two weeks). The early bird registration ends this Friday.

# The JP Morgan SCTO strategy

This strategy goes over JP Morgan’s SCTO strategy, a basic XL-sector/RWR rotation strategy with the typical associated risks and returns with a momentum equity strategy. It’s nothing spectacular, but if a large bank markets it, it’s worth looking at.

Recently, one of my readers, a managing director at a quantitative investment firm, sent me a request to write a rotation strategy based around the 9 sector spiders and RWR. The way it works (or at least, the way I interpreted it) is this:

Every month, compute the return (not sure how “the return” is defined) and rank. Take the top 5 ranks, and weight them in a normalized fashion to the inverse of their 22-day volatility. Zero out any that have negative returns. Lastly, check the predicted annualized vol of the portfolio, and if it’s greater than 20%, bring it back down to 20%. The cash asset–SHY–receives any remaining allocation due to setting securities to zero.

For the reference I used, here’s the investment case document from JP Morgan itself.

Here’s my implementation:

Step 1) get the data, compute returns.

```require(quantmod)
require(PerformanceAnalytics)
symbols <- c("XLB", "XLE", "XLF", "XLI", "XLK", "XLP", "XLU", "XLV", "XLY", "RWR", "SHY")
getSymbols(symbols, from="1990-01-01")
prices <- list()
for(i in 1:length(symbols)) {
prices[[i]] <- Ad(get(symbols[i]))
}
prices <- do.call(cbind, prices)
colnames(prices) <- gsub("\\.[A-z]*", "", colnames(prices))
returns <- na.omit(Return.calculate(prices))
```

Step 2) The function itself.

```sctoStrat <- function(returns, cashAsset = "SHY", lookback = 4, annVolLimit = .2,
topN = 5, scale = 252) {
ep <- endpoints(returns, on = "months")
weights <- list()
cashCol <- grep(cashAsset, colnames(returns))

#remove cash from asset returns
cashRets <- returns[, cashCol]
assetRets <- returns[, -cashCol]
for(i in 2:(length(ep) - lookback)) {
retSubset <- assetRets[ep[i]:ep[i+lookback]]

#forecast is the cumulative return of the lookback period
forecast <- Return.cumulative(retSubset)

#annualized (realized) volatility uses a 22-day lookback period
annVol <- StdDev.annualized(tail(retSubset, 22))

#rank the forecasts (the cumulative returns of the lookback)
rankForecast <- rank(forecast) - ncol(assetRets) + topN

#weight is inversely proportional to annualized vol
weight <- 1/annVol

#zero out anything not in the top N assets
weight[rankForecast <= 0] <- 0

#normalize and zero out anything with a negative return
weight <- weight/sum(weight)
weight[forecast < 0] <- 0

#compute forecasted vol of portfolio
forecastVol <- sqrt(as.numeric(t(weight)) %*%
cov(retSubset) %*%
as.numeric(weight)) * sqrt(scale)

#if forecasted vol greater than vol limit, cut it down
if(as.numeric(forecastVol) > annVolLimit) {
weight <- weight * annVolLimit/as.numeric(forecastVol)
}
weights[[i]] <- xts(weight, order.by=index(tail(retSubset, 1)))
}

#replace cash back into returns
returns <- cbind(assetRets, cashRets)
weights <- do.call(rbind, weights)

#cash weights are anything not in securities
weights\$CASH <- 1-rowSums(weights)

#compute and return strategy returns
stratRets <- Return.portfolio(R = returns, weights = weights)
return(stratRets)
}
```

In this case, I took a little bit of liberty with some specifics that the reference was short on. I used the full covariance matrix for forecasting the portfolio variance (not sure if JPM would ignore the covariances and do a weighted sum of individual volatilities instead), and for returns, I used the four-month cumulative. I’ve seen all sorts of permutations on how to compute returns, ranging from some average of 1, 3, 6, and 12 month cumulative returns to some lookback period to some two period average, so I’m all ears if others have differing ideas, which is why I left it as a lookback parameter.

Step 3) Running the strategy.

```scto4_20 <- sctoStrat(returns)
getSymbols("SPY", from = "1990-01-01")
spyRets <- Return.calculate(Ad(SPY))
comparison <- na.omit(cbind(scto4_20, spyRets))
colnames(comparison) <- c("strategy", "SPY")
charts.PerformanceSummary(comparison)
apply.yearly(comparison, Return.cumulative)
stats <- rbind(table.AnnualizedReturns(comparison),
maxDrawdown(comparison),
CalmarRatio(comparison),
SortinoRatio(comparison)*sqrt(252))
round(stats, 3)
```

Here are the statistics:

```                          strategy   SPY
Annualized Return            0.118 0.089
Annualized Std Dev           0.125 0.193
Annualized Sharpe (Rf=0%)    0.942 0.460
Worst Drawdown               0.165 0.552
Calmar Ratio                 0.714 0.161
Sortino Ratio (MAR = 0%)     1.347 0.763

strategy         SPY
2002-12-31 -0.035499564 -0.05656974
2003-12-31  0.253224759  0.28181559
2004-12-31  0.129739794  0.10697941
2005-12-30  0.066215224  0.04828267
2006-12-29  0.167686936  0.15845242
2007-12-31  0.153890329  0.05146218
2008-12-31 -0.096736711 -0.36794994
2009-12-31  0.181759432  0.26351755
2010-12-31  0.099187188  0.15056146
2011-12-30  0.073734427  0.01894986
2012-12-31  0.067679129  0.15990336
2013-12-31  0.321039353  0.32307769
2014-12-31  0.126633020  0.13463790
2015-04-16  0.004972434  0.02806776
```

And the equity curve:

To me, it looks like a standard rotation strategy. Aims for the highest momentum securities, diversifies to try and control risk, hits a drawdown in the crisis, recovers, and slightly lags the bull run on SPY. Nothing out of the ordinary.

So, for those interested, here you go. I’m surprised that JP Morgan itself markets this sort of thing, considering that they probably employ top-notch quants that can easily come up with products and/or strategies that are far better.

Thanks for reading.

NOTE: I am a freelance consultant in quantitative analysis on topics related to this blog. If you have contract or full time roles available for proprietary research that could benefit from my skills, please contact me through my LinkedIn here.

# The Logical Invest Enhanced Bond Rotation Strategy (And the Importance of Dividends)

This post will display my implementation of the Logical Invest Enhanced Bond Rotation strategy. This is a strategy that indeed does work, but is dependent on reinvesting dividends, as bonds pay coupons, which means bond ETFs do likewise.

The strategy is fairly simple — using four separate fixed income markets (long-term US government bonds, high-yield bonds, emerging sovereign debt, and convertible bonds), the strategy aims to deliver a low-risk, high Sharpe profile. Every month, it switches to two separate securities, in either a 60-40 or 50-50 split (that is, a 60-40 one way, or the other). My implementation for this strategy is similar to the ones I’ve done for the Logical Invest Universal Investment Strategy, which is to maximize a modified Sharpe ratio in a walk-forward process.

Here’s the code:

```LogicInvestEBR <- function(returns, lowerBound, upperBound, period, modSharpeF) {
count <- 0
configs <- list()
instCombos <- combn(colnames(returns), m = 2)
for(i in 1:ncol(instCombos)) {
inst1 <- instCombos[1, i]
inst2 <- instCombos[2, i]
rets <- returns[,c(inst1, inst2)]
weightSeq <- seq(lowerBound, upperBound, by = .1)
for(j in 1:length(weightSeq)) {
returnConfig <- Return.portfolio(R = rets,
weights = c(weightSeq[j], 1-weightSeq[j]),
rebalance_on="months")
colnames(returnConfig) <- paste(inst1, weightSeq[j],
inst2, 1-weightSeq[j], sep="_")
count <- count + 1
configs[[count]] <- returnConfig
}
}

configs <- do.call(cbind, configs)
cumRets <- cumprod(1+configs)

#rolling cumulative
rollAnnRets <- (cumRets/lag(cumRets, period))^(252/period) - 1
rollingSD <- sapply(X = configs, runSD, n=period)*sqrt(252)

modSharpe <- rollAnnRets/(rollingSD ^ modSharpeF)
monthlyModSharpe <- modSharpe[endpoints(modSharpe, on="months"),]

findMax <- function(data) {
return(data==max(data))
}

#configs\$zeroes <- 0 #zeroes for initial periods during calibration
weights <- t(apply(monthlyModSharpe, 1, findMax))
weights <- weights*1
weights <- xts(weights, order.by=as.Date(rownames(weights)))
weights[is.na(weights)] <- 0
weights\$zeroes <- 1-rowSums(weights)
configCopy <- configs
configCopy\$zeroes <- 0

stratRets <- Return.portfolio(R = configCopy, weights = weights)
return(stratRets)
}
```

The one thing different about this code is the way I initialize the return streams. It’s an ugly piece of work, but it takes all of the pairwise combinations (that is, 4 choose 2, or 4c2) along with a sequence going by 10% for the different security weights between the lower and upper bound (that is, if the lower bound is 40% and upper bound is 60%, the three weights will be 40-60, 50-50, and 60-40). So, in this case, there are 18 configurations. 4c2*3. Do note that this is not at all a framework that can be scaled up. That is, with 20 instruments, there will be 190 different combinations, and then anywhere between 3 to 11 (if going from 0-100) configurations for each combination. Obviously, not a pretty sight.

Beyond that, it’s the same refrain. Bind the returns together, compute an n-day rolling cumulative return (far faster my way than using the rollApply version of Return.annualized), divide it by the n-day rolling annualized standard deviation divided by the modified Sharpe F factor (1 gives you Sharpe ratio, 0 gives you pure returns, greater than 1 puts more of a focus on risk). Take the highest Sharpe ratio, allocate to that configuration, repeat.

So, how does this perform? Here’s a test script, using the same 73-day lookback with a modified Sharpe F of 2 that I’ve used in the previous Logical Invest strategies.

```symbols <- c("TLT", "JNK", "PCY", "CWB", "VUSTX", "PRHYX", "RPIBX", "VCVSX")
suppressMessages(getSymbols(symbols, from="1995-01-01", src="yahoo"))
etfClose <- Return.calculate(cbind(Cl(TLT), Cl(JNK), Cl(PCY), Cl(CWB)))
etfAdj <- Return.calculate(cbind(Ad(TLT), Ad(JNK), Ad(PCY), Ad(CWB)))
mfClose <- Return.calculate(cbind(Cl(VUSTX), Cl(PRHYX), Cl(RPIBX), Cl(VCVSX)))
mfAdj <- Return.calculate(cbind(Ad(VUSTX), Ad(PRHYX), Ad(RPIBX), Ad(VCVSX)))
colnames(etfClose) <- colnames(etfAdj) <- c("TLT", "JNK", "PCY", "CWB")
colnames(mfClose) <- colnames(mfAdj) <- c("VUSTX", "PRHYX", "RPIBX", "VCVSX")

etfClose <- etfClose[!is.na(etfClose[,4]),]
etfAdj <- etfAdj[!is.na(etfAdj[,4]),]
mfClose <- mfClose[-1,]
mfAdj <- mfAdj[-1,]

etfAdjTest <- LogicInvestEBR(returns = etfAdj, lowerBound = .4, upperBound = .6,
period = 73, modSharpeF = 2)

etfClTest <- LogicInvestEBR(returns = etfClose, lowerBound = .4, upperBound = .6,
period = 73, modSharpeF = 2)

mfAdjTest <- LogicInvestEBR(returns = mfAdj, lowerBound = .4, upperBound = .6,
period = 73, modSharpeF = 2)

mfClTest <- LogicInvestEBR(returns = mfClose, lowerBound = .4, upperBound = .6,
period = 73, modSharpeF = 2)

fiveStats <- function(returns) {
return(rbind(table.AnnualizedReturns(returns),
maxDrawdown(returns), CalmarRatio(returns)))
}

etfs <- cbind(etfAdjTest, etfClTest)
colnames(etfs) <- c("Adjusted ETFs", "Close ETFs")
charts.PerformanceSummary((etfs))

mutualFunds <- cbind(mfAdjTest, mfClTest)
colnames(mutualFunds) <- c("Adjusted MFs", "Close MFs")
charts.PerformanceSummary(mutualFunds)
chart.TimeSeries(log(cumprod(1+mutualFunds)), legend.loc="topleft")

fiveStats(etfs)
fiveStats(mutualFunds)
```

So, first, the results of the ETFs:

Equity curve:

Five statistics:

```> fiveStats(etfs)
Adjusted ETFs Close ETFs
Annualized Return            0.12320000 0.08370000
Annualized Std Dev           0.06780000 0.06920000
Annualized Sharpe (Rf=0%)    1.81690000 1.20980000
Worst Drawdown               0.06913986 0.08038459
Calmar Ratio                 1.78158934 1.04078405
```

In other words, reinvesting dividends makes up about 50% of these returns.

Let’s look at the mutual funds. Note that these are for the sake of illustration only–you can’t trade out of mutual funds every month.

Equity curve:

Log scale:

Statistics:

```                          Adjusted MFs Close MFs
Annualized Return           0.11450000 0.0284000
Annualized Std Dev          0.05700000 0.0627000
Annualized Sharpe (Rf=0%)   2.00900000 0.4532000
Worst Drawdown              0.09855271 0.2130904
Calmar Ratio                1.16217559 0.1332706
```

In this case, day and night, though how much of it is the data source may also be an issue. Yahoo isn’t the greatest when it comes to data, and I’m not sure how much the data quality deteriorates going back that far. However, the takeaway seems to be this: with bond strategies, dividends will need to be dealt with, and when considering returns data presented to you, keep in mind that those adjusted returns assume the investor stays on top of dividend maintenance. Fail to reinvest the dividends in a timely fashion, and, well, the gap can be quite large.

To put it into perspective, as I was writing this post, I wondered whether or not most of this was indeed due to dividends. Here’s a plot of the difference in returns between adjusted and close ETF returns.

```chart.TimeSeries(etfAdj - etfClose, legend.loc="topleft", date.format="%Y-%m",
main = "Return differences adjusted vs. close ETFs")
```

With the resulting image:

While there may be some noise to the order of the negative fifth power on most days, there are clear spikes observable in the return differences. Those are dividends, and their compounding makes a sizable difference. In one case for CWB, the difference is particularly striking (Dec. 29, 2014). In fact, here’s a quick little analysis of the effect of the dividend effects.

```dividends <- etfAdj - etfClose
divReturns <- list()
for(i in 1:ncol(dividends)) {
diffStream <- dividends[,i]
divPayments <- diffStream[diffStream >= 1e-3]
divReturns[[i]] <- Return.annualized(divPayments)
}
divReturns <- do.call(cbind, divReturns)
divReturns

divReturns/Return.annualized(etfAdj)
```

And the result:

```> divReturns
TLT        JNK        PCY        CWB
Annualized Return 0.03420959 0.08451723 0.05382363 0.05025999

> divReturns/Return.annualized(etfAdj)
TLT       JNK       PCY       CWB
Annualized Return 0.453966 0.6939243 0.5405922 0.3737499
```

In short, the effect of the dividend is massive. In some instances, such as with JNK, the dividend comprises more than 50% of the annualized returns for the security!

Basically, I’d like to hammer the point home one last time–backtests using adjusted data assume instantaneous maintenance of dividends. In order to achieve the optimistic returns seen in the backtests, these dividend payments must be reinvested ASAP. In short, this is the fine print on this strategy, and is a small, but critical detail that the SeekingAlpha article doesn’t mention. (Seriously, do a ctrl + F in your browser for the word “dividend”. It won’t come up in the article itself.) I wanted to make sure to add it.

One last thing: gaudy numbers when using monthly returns!

```> fiveStats(apply.monthly(etfs, Return.cumulative))
Adjusted ETFs Close ETFs
Annualized Return            0.12150000   0.082500
Annualized Std Dev           0.06490000   0.067000
Annualized Sharpe (Rf=0%)    1.87170000   1.232100
Worst Drawdown               0.03671871   0.049627
Calmar Ratio                 3.30769620   1.662642
```

Look! A Calmar Ratio of 3.3, and a Sharpe near 2!*

*: Must manage dividends. Statistics reported are monthly.

Okay, in all fairness, this is a pretty solid strategy, once one commits to managing the dividends. I just felt that it should have been a topic made front and center considering its importance in this case, rather than simply swept under the “we use adjusted returns” rug, since in this instance, the effect of dividends is massive.

In conclusion, while I will more or less confirm the strategy’s actual risk/reward performance (unlike some other SeekingAlpha strategies I’ve backtested), which, in all honesty, I find really impressive, it comes with a caveat like the rest of them. However, the caveat of “be detail-oriented/meticulous/paranoid and reinvest those dividends!” in my opinion is a caveat that’s a lot easier to live with than 30%+ drawdowns that were found lurking in other SeekingAlpha strategies. So for those that can stay on top of those dividends (whether manually, or with machine execution), here you go. I’m basically confirming the performance of Logical Invest’s strategy, but just belaboring one important detail.

Thanks for reading.

NOTE: I am a freelance consultant in quantitative analysis on topics related to this blog. If you have contract or full time roles available for proprietary research that could benefit from my skills, please contact me through my LinkedIn here.

# The Downside of Rankings-Based Strategies

This post will demonstrate a downside to rankings-based strategies, particularly when using data of a questionable quality (which, unless one pays multiple thousands of dollars per month for data, most likely is of questionable quality). Essentially, by making one small change to the way the strategy filters, it introduces a massive performance drop in terms of drawdown. This exercise effectively demonstrates a different possible way of throwing a curve-ball at ranking strategies to test for robustness.

Recently, a discussion came up between myself, Terry Doherty, Cliff Smith, and some others on Seeking Alpha regarding what happened when I substituted the 63-day SMA for the three month SMA in Cliff Smith’s QTS strategy (quarterly tactical strategy…strategy).

Essentially, by simply substituting a 63-day SMA (that is, using daily data instead of monthly) for a 3-month SMA, the results were drastically affected.

Here’s the new QTS code, now in a function.

```qts <- function(prices, nShort = 20, nLong = 105, nMonthSMA = 3, nDaySMA = 63, wRankShort=1, wRankLong=1.01,
movAvgType = c("monthly", "daily"), cashAsset="VUSTX", returnNames = FALSE) {
cashCol <- grep(cashAsset, colnames(prices))

#start our data off on the security with the least data (VGSIX in this case)
prices <- prices[!is.na(prices[,7]),]

#cash is not a formal asset in our ranking
cashPrices <- prices[, cashCol]
prices <- prices[, -cashCol]

#compute momentums
rocShort <- prices/lag(prices, nShort) - 1
rocLong <- prices/lag(prices, nLong) - 1

#take the endpoints of quarter start/end
quarterlyEps <- endpoints(prices, on="quarters")
monthlyEps <- endpoints(prices, on = "months")

#take the prices at quarterly endpoints
quarterlyPrices <- prices[quarterlyEps,]

#short momentum at quarterly endpoints (20 day)
rocShortQtrs <- rocShort[quarterlyEps,]

#long momentum at quarterly endpoints (105 day)
rocLongQtrs <- rocLong[quarterlyEps,]

#rank short momentum, best highest rank
rocSrank <- t(apply(rocShortQtrs, 1, rank))

#rank long momentum, best highest rank
rocLrank <- t(apply(rocLongQtrs, 1, rank))

#total rank, long slightly higher than short, sum them
totalRank <- wRankLong * rocLrank + wRankShort * rocSrank

#function that takes 100% position in highest ranked security
maxRank <- function(rankRow) {
return(rankRow==max(rankRow))
}

#apply above function to our quarterly ranks every quarter
rankPos <- t(apply(totalRank, 1, maxRank))

#SMA of securities, only use monthly endpoints
#subset to quarters
#then filter
movAvgType = movAvgType[1]
if(movAvgType=="monthly") {
monthlyPrices <- prices[monthlyEps,]
monthlySMAs <- xts(apply(monthlyPrices, 2, SMA, n=nMonthSMA), order.by=index(monthlyPrices))
quarterlySMAs <- monthlySMAs[index(quarterlyPrices),]
smaFilter <- quarterlyPrices > quarterlySMAs
} else if (movAvgType=="daily") {
smas <- xts(apply(prices, 2, SMA, n=nDaySMA), order.by=index(prices))
quarterlySMAs <- smas[index(quarterlyPrices),]
smaFilter <- quarterlyPrices > quarterlySMAs
} else {
stop("invalid moving average type")
}

finalPos <- rankPos*smaFilter
finalPos <- finalPos[!is.na(rocLongQtrs[,1]),]
cash <- xts(1-rowSums(finalPos), order.by=index(finalPos))
finalPos <- merge(finalPos, cash, join='inner')

prices <- merge(prices, cashPrices, join='inner')
returns <- Return.calculate(prices)
stratRets <- Return.portfolio(returns, finalPos)

if(returnNames) {
findNames <- function(pos) {
return(names(pos[pos==1]))
}
tmp <- apply(finalPos, 1, findNames)
assetNames <- xts(tmp, order.by=as.Date(names(tmp)))
return(list(assetNames, stratRets))
}
return(stratRets)
}
```

The one change I made is this:

```  movAvgType = movAvgType[1]
if(movAvgType=="monthly") {
monthlyPrices <- prices[monthlyEps,]
monthlySMAs <- xts(apply(monthlyPrices, 2, SMA, n=nMonthSMA), order.by=index(monthlyPrices))
quarterlySMAs <- monthlySMAs[index(quarterlyPrices),]
smaFilter <- quarterlyPrices > quarterlySMAs
} else if (movAvgType=="daily") {
smas <- xts(apply(prices, 2, SMA, n=nDaySMA), order.by=index(prices))
quarterlySMAs <- smas[index(quarterlyPrices),]
smaFilter <- quarterlyPrices > quarterlySMAs
} else {
stop("invalid moving average type")
}
```

In essence, it allows the function to use either a monthly-calculated moving average, or a daily, which is then subset to the quarterly frequency of the rest of the data.

(I also allow the function to return the names of the selected securities.)

So now we can do two tests:

1) The initial parameter settings (20-day short-term momentum, 105-day long-term momentum, equal weigh their ranks (tiebreaker to the long-term), and use a 3-month SMA to filter)
2) The same exact parameter settings, except a 63-day SMA for the filter.

Here’s the code to do that.

```#get our data from yahoo, use adjusted prices
symbols <- c("NAESX", #small cap
"PREMX", #emerging bond
"VEIEX", #emerging markets
"VFICX", #intermediate investment grade
"VFIIX", #GNMA mortgage
"VFINX", #S&P 500 index
"VGSIX", #MSCI REIT
"VGTSX", #total intl stock idx
"VUSTX") #long term treasury (cash)

getSymbols(symbols, from="1990-01-01")
prices <- list()
for(i in 1:length(symbols)) {
prices[[i]] <- Ad(get(symbols[i]))
}
prices <- do.call(cbind, prices)
colnames(prices) <- gsub("\\.[A-z]*", "", colnames(prices))

monthlySMAqts <- qts(prices, returnNames=TRUE)
dailySMAqts <- qts(prices, wRankShort=.95, wRankLong=1.05, movAvgType = "daily", returnNames=TRUE)

retsComparison <- cbind(monthlySMAqts[[2]], dailySMAqts[[2]])
colnames(retsComparison) <- c("monthly SMA qts", "daily SMA qts")
retsComparison <- retsComparison["2003::"]
charts.PerformanceSummary(retsComparison["2003::"])
rbind(table.AnnualizedReturns(retsComparison["2003::"]), maxDrawdown(retsComparison["2003::"]))
```

And here are the results:

Statistics:

```                          monthly SMA qts daily SMA qts
Annualized Return               0.2745000     0.2114000
Annualized Std Dev              0.1725000     0.1914000
Annualized Sharpe (Rf=0%)       1.5915000     1.1043000
Worst Drawdown                  0.1911616     0.3328411
```

With the corresponding equity curves:

Here are the several instances in which the selections do not match thanks to the filters:

```selectedNames <- cbind(monthlySMAqts[[1]], dailySMAqts[[1]])
colnames(selectedNames) <- c("Monthly SMA Filter", "Daily SMA Filter")
differentSelections <- selectedNames[selectedNames[,1]!=selectedNames[,2],]
```

With the results:

```           Monthly SMA Filter Daily SMA Filter
1997-03-31 "VGSIX"            "cash"
2007-12-31 "cash"             "PREMX"
2008-06-30 "cash"             "VFIIX"
2008-12-31 "cash"             "NAESX"
2011-06-30 "cash"             "NAESX"
```

Now, of course, many can make the arguments that Yahoo’s data is junk, my backtest doesn’t reflect reality, etc., which would essentially miss the point: this data here, while not a perfect realization of the reality of Planet Earth, may as well have been valid (you know, like all the academics, who use various simulation techniques to synthesize more data or explore other scenarios?). All I did here was change the filter to something logically comparable (that is, computing the moving average filter on a different time-scale, which does not in any way change the investment logic). From 2003 onward, this change only affected the strategy in four places. However, those instances were enough to create some noticeable changes (for the worse) in the strategy’s performance. Essentially, the downside of rankings-based strategies are when the overall number of selected instruments (in this case, ONE!) is small, a few small changes in parameters, data, etc. can lead to drastically different results.

As I write this, Cliff Smith already has ideas as to how to counteract this phenomenon. However, unto my experience, once a strategy starts getting into “how do we smooth out that one bump on the equity curve” territory, I think it’s time to go back and re-examine the strategy altogether. In my opinion, while the idea of momentum is of course, sound, with a great deal of literature devoted to it, the idea of selecting just one instrument at a time as the be-all-end-all strategy does not sit well with me. However, to me, QTS nevertheless presents an interesting framework for analyzing small subgroups of securities, and using it as one layer of an overarching strategy framework, such that the return streams are sub-strategies, instead of raw instruments.

Thanks for reading.

NOTE: I am a freelance consultant in quantitative analysis on topics related to this blog. If you have contract or full time roles available for proprietary research that could benefit from my skills, please contact me through my LinkedIn here.

# The Logical-Invest “Universal Investment Strategy”–A Walk Forward Process on SPY and TLT

I’m sure we’ve all heard about diversified stock and bond portfolios. In its simplest, most diluted form, it can be comprised of the SPY and TLT etfs. The concept introduced by Logical Invest, in a Seeking Alpha article written by Frank Grossman (also see link here), essentially uses a walk-forward methodology of maximizing a modified Sharpe ratio, biased heavily in favor of the volatility rather than the returns. That is, it uses a 72-day moving window to maximize total returns between different weighting configurations of a SPY-TLT mix over the standard deviation raised to the power of 5/2. To put it into perspective, at a power of 1, this is the basic Sharpe ratio, and at a power of 0, just a momentum maximization algorithm.

The process for this strategy is simple: rebalance every month on some multiple of 5% between SPY and TLT that previously maximized the following quantity (returns/vol^2.5 on a 72-day window).

Here’s the code for obtaining the data and computing the necessary quantities:

```require(quantmod)
require(PerformanceAnalytics)
getSymbols(c("SPY", "TLT"), from="1990-01-01")
returns <- merge(Return.calculate(Ad(SPY)), Return.calculate(Ad(TLT)), join='inner')
returns <- returns[-1,]
configs <- list()
for(i in 1:21) {
weightSPY <- (i-1)*.05
weightTLT <- 1-weightSPY
config <- Return.portfolio(R = returns, weights=c(weightSPY, weightTLT), rebalance_on = "months")
configs[[i]] <- config
}
configs <- do.call(cbind, configs)
cumRets <- cumprod(1+configs)
period <- 72

roll72CumAnn <- (cumRets/lag(cumRets, period))^(252/period) - 1
roll72SD <- sapply(X = configs, runSD, n=period)*sqrt(252)
```

Next, the code for creating the weights:

```sd_f_factor <- 2.5
modSharpe <- roll72CumAnn/roll72SD^sd_f_factor
monthlyModSharpe <- modSharpe[endpoints(modSharpe, on="months"),]

findMax <- function(data) {
return(data==max(data))
}

weights <- t(apply(monthlyModSharpe, 1, findMax))
weights <- weights*1
weights <- xts(weights, order.by=as.Date(rownames(weights)))
weights[is.na(weights)] <- 0
weights\$zeroes <- 1-rowSums(weights)
configs\$zeroes <- 0
```

That is, simply take the setting that maximizes the monthly modified Sharpe Ratio calculation at each rebalancing date (the end of every month).

Next, here’s the performance:

```stratRets <- Return.portfolio(R = configs, weights = weights)
rbind(table.AnnualizedReturns(stratRets), maxDrawdown(stratRets))
charts.PerformanceSummary(stratRets)
```

Which gives the results:

```> rbind(table.AnnualizedReturns(stratRets), maxDrawdown(stratRets))
portfolio.returns
Annualized Return                 0.1317000
Annualized Std Dev                0.0990000
Annualized Sharpe (Rf=0%)         1.3297000
Worst Drawdown                    0.1683851
```

With the following equity curve:

Not perfect, but how does it compare to the ingredients?

Let’s take a look:

```stratAndComponents <- merge(returns, stratRets, join='inner')
charts.PerformanceSummary(stratAndComponents)
rbind(table.AnnualizedReturns(stratAndComponents), maxDrawdown(stratAndComponents))
apply.yearly(stratAndComponents, Return.cumulative)
```

Here are the usual statistics:

```> rbind(table.AnnualizedReturns(stratAndComponents), maxDrawdown(stratAndComponents))
SPY.Adjusted TLT.Adjusted portfolio.returns
Annualized Return            0.0907000    0.0783000         0.1317000
Annualized Std Dev           0.1981000    0.1381000         0.0990000
Annualized Sharpe (Rf=0%)    0.4579000    0.5669000         1.3297000
Worst Drawdown               0.5518552    0.2659029         0.1683851
```

In short, it seems the strategy performs far better than either of the ingredients. Let’s see if the equity curve comparison reflects this.

Indeed, it does. While it does indeed have the drawdown in the crisis, both instruments were in drawdown at the time, so it appears that the strategy made the best of a bad situation.

Here are the annual returns:

```> apply.yearly(stratAndComponents, Return.cumulative)
SPY.Adjusted TLT.Adjusted portfolio.returns
2002-12-31  -0.02054891  0.110907611        0.01131366
2003-12-31   0.28179336  0.015936985        0.12566042
2004-12-31   0.10695067  0.087089794        0.09724221
2005-12-30   0.04830869  0.085918063        0.10525398
2006-12-29   0.15843880  0.007178861        0.05294557
2007-12-31   0.05145526  0.102972399        0.06230742
2008-12-31  -0.36794099  0.339612265        0.19590423
2009-12-31   0.26352114 -0.218105306        0.18826736
2010-12-31   0.15056113  0.090181150        0.16436950
2011-12-30   0.01890375  0.339915713        0.24562838
2012-12-31   0.15994578  0.024083393        0.06051237
2013-12-31   0.32303535 -0.133818884        0.13760060
2014-12-31   0.13463980  0.273123290        0.19637382
2015-02-20   0.02773183  0.006922893        0.02788726
```

2002 was an incomplete year. However, what’s interesting here is that on a whole, while the strategy rarely if ever does as well as the better of the two instruments, it always outperforms the worse of the two instruments–and not only that, but it has delivered a positive performance in every year of the backtest–even when one instrument or the other was taking serious blows to performance, such as SPY in 2008, and TLT in 2009 and 2013.

For the record, here is the weight of SPY in the strategy.

```weightSPY <- apply(monthlyModSharpe, 1, which.max)
weightSPY <- do.call(rbind, weightSPY)
weightSPY <- (weightSPY-1)*.05
align <- cbind(weightSPY, stratRets)
align <- na.locf(align)
chart.TimeSeries(align[,1], date.format="%Y", ylab="Weight SPY", main="Weight of SPY in SPY-TLT pair")
```

Now while this may serve as a standalone strategy for some people, the takeaway in my opinion from this is that dynamically re-weighting two return streams that share a negative correlation can lead to some very strong results compared to the ingredients from which they were formed. Furthermore, rather than simply rely on one number to summarize a relationship between two instruments, the approach that Frank Grossman took to actually model the combined returns was one I find interesting, and undoubtedly has applications as a general walk-forward process.

Thanks for reading.

NOTE: I am a freelance consultant in quantitative analysis on topics related to this blog. If you have contract or full time roles available for proprietary research that could benefit from my skills, please contact me through my LinkedIn here.

# A Closer Update To David Varadi’s Percentile Channels Strategy

So thanks to seeing Michael Kapler’s implementation of David Varadi’s percentile channels strategy, I was able to get a better understanding of what was going on. It turns out that rather than looking at the channel value only at the ends of months, that the strategy actually keeps track of the channel’s value intra-month. So if in the middle of the month, you had a sell signal and at the end of the month, the price moved up to intra-channel values, you would still be on a sell signal rather than the previous month’s end-of-month signal. It’s not much different than my previous implementation when all is said and done (slightly higher Sharpe, slightly lower returns and drawdowns). In any case, the concept remains the same.

For this implementation, I’m going to use the runquantile function from the caTools package, which contains a function called runquantile that works like a generalized runMedian/runMin/runMax from TTR, once you’re able to give it the proper arguments (on default, its results are questionable).

Here’s the code:

```require(quantmod)
require(caTools)
require(PerformanceAnalytics)
require(TTR)
getSymbols(c("LQD", "DBC", "VTI", "ICF", "SHY"), from="1990-01-01")

prices <- cbind(Ad(LQD), Ad(DBC), Ad(VTI), Ad(ICF), Ad(SHY))
prices <- prices[!is.na(prices[,2]),]
returns <- Return.calculate(prices)
cashPrices <- prices[, 5]
assetPrices <- prices[, -5]

require(caTools)
pctChannelPosition <- function(prices,
dayLookback = 60,
lowerPct = .25, upperPct = .75) {
leadingNAs <- matrix(nrow=dayLookback-1, ncol=ncol(prices), NA)

upperChannels <- runquantile(prices, k=dayLookback, probs=upperPct, endrule="trim")
upperQ <- xts(rbind(leadingNAs, upperChannels), order.by=index(prices))

lowerChannels <- runquantile(prices, k=dayLookback, probs=lowerPct, endrule="trim")
lowerQ <- xts(rbind(leadingNAs, lowerChannels), order.by=index(prices))

positions <- xts(matrix(nrow=nrow(prices), ncol=ncol(prices), NA), order.by=index(prices))
positions[prices > upperQ & lag(prices) < upperQ] <- 1 #cross up
positions[prices < lowerQ & lag(prices) > lowerQ] <- -1 #cross down
positions <- na.locf(positions)
positions[is.na(positions)] <- 0

colnames(positions) <- colnames(prices)
return(positions)
}

#find our positions, add them up
d60 <- pctChannelPosition(assetPrices)
d120 <- pctChannelPosition(assetPrices, dayLookback = 120)
d180 <- pctChannelPosition(assetPrices, dayLookback = 180)
d252 <- pctChannelPosition(assetPrices, dayLookback = 252)
compositePosition <- (d60 + d120 + d180 + d252)/4

compositeMonths <- compositePosition[endpoints(compositePosition, on="months"),]

returns <- Return.calculate(prices)
monthlySD20 <- xts(sapply(returns[,-5], runSD, n=20), order.by=index(prices))[index(compositeMonths),]
weight <- compositeMonths*1/monthlySD20
weight <- abs(weight)/rowSums(abs(weight))
weight[compositeMonths < 0 | is.na(weight)] <- 0
weight\$CASH <- 1-rowSums(weight)

#not actually equal weight--more like composite weight, going with
#Michael Kapler's terminology here
ewWeight <- abs(compositeMonths)/rowSums(abs(compositeMonths))
ewWeight[compositeMonths < 0 | is.na(ewWeight)] <- 0
ewWeight\$CASH <- 1-rowSums(ewWeight)

rpRets <- Return.portfolio(R = returns, weights = weight)
ewRets <- Return.portfolio(R = returns, weights = ewWeight)
```

Essentially, with runquantile, you need to give it the “trim” argument, and then manually append the leading NAs, and then manually turn it into an xts object, which is annoying. One would think that the author of this package would take care of these quality-of-life issues, but no. In any case, there are two strategies at play here–one being the percentile channel risk parity strategy, and the other what Michael Kapler calls “channel equal weight”, which actually *isn’t* an equal weight strategy, since the composite parameter values may take the values (-1, -.5, 0, .5, and 1–with a possibility for .75 or .25 early on when some of the lookback channels still say 0 instead of only 1 or -1), but simply, the weights without taking into account volatility at all, but I’m sticking with Michael Kapler’s terminology to be consistent. That said, I don’t personally use Michael Kapler’s SIT package due to the vast differences in syntax between it and the usual R code I’m used to. However, your mileage may vary.

In any case, here’s the updated performance:

```both <- cbind(rpRets, ewRets)
colnames(both) <- c("RiskParity", "Equal Weight")
charts.PerformanceSummary(both)
rbind(table.AnnualizedReturns(both), maxDrawdown(both))
apply.yearly(both, Return.cumulative)
```

Which gives us the following output:

```> rbind(table.AnnualizedReturns(both), maxDrawdown(both))
RiskParity Equal Weight
Annualized Return         0.09380000    0.1021000
Annualized Std Dev        0.06320000    0.0851000
Annualized Sharpe (Rf=0%) 1.48430000    1.1989000
Worst Drawdown            0.06894391    0.1150246

> apply.yearly(both, Return.cumulative)
RiskParity Equal Weight
2006-12-29 0.08352255   0.07678321
2007-12-31 0.05412147   0.06475540
2008-12-31 0.10663085   0.12212063
2009-12-31 0.11920721   0.19093131
2010-12-31 0.13756460   0.14594317
2011-12-30 0.11744706   0.08707801
2012-12-31 0.07730896   0.06085295
2013-12-31 0.06733187   0.08174173
2014-12-31 0.06435030   0.07357458
2015-02-17 0.01428705   0.01568372
```

In short, the more naive weighting scheme delivers slightly higher returns but pays dearly for those marginal returns with downside risk.

Here are the equity curves:

So, there you have it. The results David Varadi obtained are legitimate. But nevertheless, I hope this demonstrates how easy it is for the small details to make material differences.

Thanks for reading.

NOTE: I am a freelance consultant in quantitative analysis on topics related to this blog. If you have contract or full time roles available for proprietary research that could benefit from my skills, please contact me through my LinkedIn here.

# An Attempt At Replicating David Varadi’s Percentile Channels Strategy

This post will detail an attempt at replicating David Varadi’s percentile channels strategy. As I’m only able to obtain data back to mid 2006, the exact statistics will not be identical. However, of the performance I do have, it is similar (but not identical) to the corresponding performance presented by David Varadi.

First off, before beginning this post, I’d like to issue a small mea culpa regarding the last post. It turns out that Yahoo’s data, once it gets into single digit dollar prices, is of questionable accuracy, and thus, results from the late 90s on mutual funds with prices falling into those ranges are questionable, as a result. As I am an independent blogger, and also make it a policy of readers being able to replicate all of my analysis, I am constrained by free data sources, and sometimes, the questionable quality of that data may materially affect results. So, if it’s one of your strategies replicated on this blog, and you find contention with my results, I would be more than happy to work with the data used to generate the original results, corroborate the results, and be certain that any differences in results from using lower-quality, publicly-available data stem from that alone. Generally, I find it surprising that a company as large as Yahoo can have such gaping data quality issues in certain aspects, but I’m happy that I was able to replicate the general thrust of QTS very closely.

This replication of David Varadi’s strategy, however, is not one such case–mainly because the data for DBC does not extend back very far (it was in inception only in 2006, and the data used by David Varadi’s programmer was obtained from Bloomberg, which I have no access to), and furthermore, I’m not certain if my methods are absolutely identical. Nevertheless, the strategy in and of itself is solid.

The way the strategy works is like this (to my interpretation of David Varadi’s post and communication with his other programmer). Given four securities (LQD, DBC, VTI, ICF), and a cash security (SHY), do the following:

Find the running the n-day quantile of an upper and lower percentile. Anything above the upper percentile gets a score of 1, anything lower gets a score of -1. Leave the rest as NA (that is, anything between the bounds).

Subset these quantities on their monthly endpoints. Any value between channels (NA) takes the quantity of the last value. (In short, na.locf). Any initial NAs become zero.

Do this with a 60-day, 120-day, 180-day, and 252-day setting at 25th and 75th percentiles. Add these four tables up (their dimensions are the number of monthly endpoints by the number of securities) and divide by the number of parameter settings (in this case, 4 for 60, 120, 180, 252) to obtain a composite position.

Next, obtain a running 20-day standard deviation of the returns (not prices!), and subset it for the same indices as the composite positions. Take the inverse of these volatility scores, and multiply it by the composite positions to get an inverse volatility position. Take its absolute value (some positions may be negative, remember), and normalize. In the beginning, there may be some zero-across-all-assets positions, or other NAs due to lack of data (EG if a monthly endpoint occurs before enough data to compute a 20-day standard deviation, there will be a row of NAs), which will be dealt with. Keep all positions with a positive composite position (that is, scores of .5 or 1, discard all scores of zero or lower), and reinvest the remainder into the cash asset (SHY, in our case). Those are the final positions used to generate the returns.

This is how it looks like in code.

This is the code for obtaining the data (from Yahoo finance) and separating it into cash and non-cash data.

```require(quantmod)
require(caTools)
require(PerformanceAnalytics)
require(TTR)
getSymbols(c("LQD", "DBC", "VTI", "ICF", "SHY"), from="1990-01-01")

prices <- cbind(Ad(LQD), Ad(DBC), Ad(VTI), Ad(ICF), Ad(SHY))
prices <- prices[!is.na(prices[,2]),]
returns <- Return.calculate(prices)
cashPrices <- prices[, 5]
assetPrices <- prices[, -5]
```

This is the function for computing the percentile channel positions for a given parameter setting. Unfortunately, it is not instantaneous due to R’s rollapply function paying a price in speed for generality. While the package caTools has a runquantile function, as of the time of this writing, I have found differences between its output and runMedian in TTR, so I’ll have to get in touch with the package’s author.

```pctChannelPosition <- function(prices, rebal_on=c("months", "quarters"),
dayLookback = 60,
lowerPct = .25, upperPct = .75) {

upperQ <- rollapply(prices, width=dayLookback, quantile, probs=upperPct)
lowerQ <- rollapply(prices, width=dayLookback, quantile, probs=lowerPct)
positions <- xts(matrix(nrow=nrow(prices), ncol=ncol(prices), NA), order.by=index(prices))
positions[prices > upperQ] <- 1
positions[prices < lowerQ] <- -1

ep <- endpoints(positions, on = rebal_on[1])
positions <- positions[ep,]
positions <- na.locf(positions)
positions[is.na(positions)] <- 0

colnames(positions) <- colnames(prices)
return(positions)
}
```

The way this function works is simple: computes a running quantile using rollapply, and then scores anything with price above its 75th percentile as 1, and anything below the 25th percentile as -1, in accordance with David Varadi’s post.

It then subsets these quantities on months (quarters is also possible–or for that matter, other values, but the spirit of the strategy seems to be months or quarters), and imputes any NAs with the last known observation, or zero, if it is an initial NA before any position is found. Something I have found over the course of writing this and the QTS strategy is that one need not bother implementing a looping mechanism to allocate positions monthly if there isn’t a correlation matrix based on daily data involved every month, and it makes the code more readable.

Next, we find our composite position.

```#find our positions, add them up
d60 <- pctChannelPosition(assetPrices)
d120 <- pctChannelPosition(assetPrices, dayLookback = 120)
d180 <- pctChannelPosition(assetPrices, dayLookback = 180)
d252 <- pctChannelPosition(assetPrices, dayLookback = 252)
compositePosition <- (d60 + d120 + d180 + d252)/4
```

Next, find the running volatility for the assets, and subset them to the same time period (in this case months) as our composite position. In David Varadi’s example, the parameter is a 20-day lookback.

```#find 20-day rolling standard deviations, subset them on identical indices
#to the percentile channel monthly positions
sd20 <- xts(sapply(returns[,-5], runSD, n=20), order.by=index(assetPrices))
monthlySDs <- sd20[index(compositePosition)]
```

Next, perform the following steps: find the inverse volatility of these quantities, multiply by the composite position score, take the absolute value, and keep any position for which the composite position is greater than zero (or technically speaking, has positive signage). Due to some initial NA rows due to a lack of data (either not enough days to compute a running volatility, or no positive positions yet), those will simply be imputed to zero. Reinvest the remainder in cash.

```#compute inverse volatilities
inverseVols <- 1/monthlySDs

#multiply inverse volatilities by composite positions
invVolPos <- inverseVols*compositePosition

#take absolute values of inverse volatility multiplied by position
absInvVolPos <- abs(invVolPos)

#normalize the above quantities
normalizedAbsInvVols <- absInvVolPos/rowSums(absInvVolPos)

#keep only positions with positive composite positions (remove zeroes/negative)
nonCashPos <- normalizedAbsInvVols * sign(compositePosition > 0)
nonCashPos[is.na(nonCashPos)] <- 0 #no positions before we have enough data

#add cash position which is complement of non-cash position
finalPos <- nonCashPos
finalPos\$cashPos <- 1-rowSums(finalPos)
```

And finally, the punchline, how does this strategy perform?

```#compute returns
stratRets <- Return.portfolio(R = returns, weights = finalPos)
charts.PerformanceSummary(stratRets)
stats <- rbind(table.AnnualizedReturns(stratRets), maxDrawdown(stratRets))
rownames(stats)[4] <- "Worst Drawdown"
stats
```

Like this:

```> stats
portfolio.returns
Annualized Return                0.10070000
Annualized Std Dev               0.06880000
Annualized Sharpe (Rf=0%)        1.46530000
Worst Drawdown                   0.07449537
```

With the following equity curve:

The statistics are visibly worse than David Varadi’s 10% vs. 11.1% CAGR, 6.9% annualized standard deviation vs. 5.72%, 7.45% max drawdown vs. 5.5%, and derived statistics (EG MAR). However, my data starts far later, and 1995-1996 seemed to be phenomenal for this strategy. Here are the cumulative returns for the data I have:

```> apply.yearly(stratRets, Return.cumulative)
portfolio.returns
2006-12-29        0.11155069
2007-12-31        0.07574266
2008-12-31        0.16921233
2009-12-31        0.14600008
2010-12-31        0.12996371
2011-12-30        0.06092018
2012-12-31        0.07306617
2013-12-31        0.06303612
2014-12-31        0.05967415
2015-02-13        0.01715446
```

I see a major discrepancy between my returns and David’s returns in 2011, but beyond that, the results seem to be somewhere close in the pattern of yearly returns. Whether my methodology is incorrect (I think I followed the procedure to the best of my understanding, but of course, if someone sees a mistake in my code, please let me know), or whether it’s the result of using Yahoo’s questionable quality data, I am uncertain.

However, in my opinion, that doesn’t take away from the validity of the strategy as a whole. With a mid-1 Sharpe ratio on a monthly rebalancing scale, and steady new equity highs, I feel that this is a result worth sharing–even if not directly corroborated (yet, hopefully).

One last note–some of the readers on David Varadi’s blog have cried foul due to their inability to come close to his results. Since I’ve come close, I feel that the results are valid, and since I’m using different data, my results are not identical. However, if anyone has questions about my process, feel free to leave questions and/or comments.

Thanks for reading.

NOTE: I am a freelance consultant in quantitative analysis on topics related to this blog. If you have contract or full time roles available for proprietary research that could benefit from my skills, please contact me through my LinkedIn here.

# The Quarterly Tactical Strategy (aka QTS)

This post introduces the Quarterly Tactical Strategy, introduced by Cliff Smith on a Seeking Alpha article. It presents a variation on the typical dual-momentum strategy that only trades over once a quarter, yet delivers a seemingly solid risk/return profile. The article leaves off a protracted period of unimpressive performance at the turn of the millennium, however.

First off, due to the imprecision of the English language, I received some help from TrendXplorer in implementing this strategy. Those who are fans of Amibroker are highly encouraged to visit his blog.

In any case, this strategy is fairly simple:

Take a group of securities (in this case, 8 mutual funds), and do the following:

Rank a long momentum (105 days) and a short momentum (20 days), and invest in the security with the highest composite rank, with ties going to the long momentum (that is, .501*longRank + .499*shortRank, for instance). If the security with the highest composite rank is greater than its three month SMA, invest in that security, otherwise, hold cash.

There are two critical points that must be made here:

1) The three-month SMA is *not* a 63-day SMA. It is precisely a three-point SMA up to that point on the monthly endpoints of that security.
2) Unlike in flexible asset allocation or elastic asset allocation, the cash asset is not treated as a formal asset.

Let’s look at the code. Here’s the data–which are adjusted-data mutual fund data (although with a quarterly turnover, the frequent trading constraint of not trading out of the security is satisfied, though I’m not sure how dividends are treated–that is, whether a retail investor would actually realize these returns less a hopefully tiny transaction cost through their brokers–aka hopefully not too much more than \$1 per transaction):

```require(quantmod)
require(PerformanceAnalytics)
require(TTR)

#get our data from yahoo, use adjusted prices
symbols <- c("NAESX", #small cap
"PREMX", #emerging bond
"VEIEX", #emerging markets
"VFICX", #intermediate investment grade
"VFIIX", #GNMA mortgage
"VFINX", #S&P 500 index
"VGSIX", #MSCI REIT
"VGTSX", #total intl stock idx
"VUSTX") #long term treasury (cash)

getSymbols(symbols, from="1990-01-01")
prices <- list()
for(i in 1:length(symbols)) {
prices[[i]] <- Ad(get(symbols[i]))
}
prices <- do.call(cbind, prices)
colnames(prices) <- gsub("\\.[A-z]*", "", colnames(prices))

#define our cash asset and keep track of which column it is
cashAsset <- "VUSTX"
cashCol <- grep(cashAsset, colnames(prices))

#start our data off on the security with the least data (VGSIX in this case)
prices <- prices[!is.na(prices[,7]),]

#cash is not a formal asset in our ranking
cashPrices <- prices[, cashCol]
prices <- prices[, -cashCol]
```

Nothing anybody hasn’t seen before up to this point. Get data, start it off at most recent inception mutual fund, separate the cash prices, moving along.

What follows is a rather rough implementation of QTS, not wrapped up in any sort of function that others can plug and play with (though I hope I made the code readable enough for others to tinker with).

Let’s define parameters and compute momentum.

```#define our parameters
nShort <- 20
nLong <- 105
nMonthSMA <- 3

#compute momentums
rocShort <- prices/lag(prices, nShort) - 1
rocLong <- prices/lag(prices, nLong) - 1
```

Now comes some endpoints functionality (or, more colloquially, magic) that the xts library provides. It’s what allows people to get work done in R much faster than in other programming languages.

```#take the endpoints of quarter start/end
quarterlyEps <- endpoints(prices, on="quarters")
monthlyEps <- endpoints(prices, on = "months")

#take the prices at quarterly endpoints
quarterlyPrices <- prices[quarterlyEps,]

#short momentum at quarterly endpoints (20 day)
rocShortQtrs <- rocShort[quarterlyEps,]

#long momentum at quarterly endpoints (105 day)
rocLongQtrs <- rocLong[quarterlyEps,]
```

In short, get the quarterly endpoints (and monthly, we need those for the monthly SMA which you’ll see shortly) and subset our momentum computations on those quarterly endpoints. Now let’s get the total rank for those subset-on-quarters momentum computations.

```#rank short momentum, best highest rank
rocSrank <- t(apply(rocShortQtrs, 1, rank))

#rank long momentum, best highest rank
rocLrank <- t(apply(rocLongQtrs, 1, rank))

#total rank, long slightly higher than short, sum them
totalRank <- 1.01*rocLrank + rocSrank

#function that takes 100% position in highest ranked security
maxRank <- function(rankRow) {
return(rankRow==max(rankRow))
}

#apply above function to our quarterly ranks every quarter
rankPos <- t(apply(totalRank, 1, maxRank))
```

So as you can see, I rank the momentum computations by row, take a weighted sum (in slight favor of the long momentum), and then simply take the security with the highest rank at every period, giving me one 1 in every row and 0s otherwise.

Now let’s do the other end of what determines position, which is the SMA filter. In this case, we need monthly data points for our three-month SMA, and then subset it to quarters to be on the same timescale as the quarterly ranks.

```#SMA of securities, only use monthly endpoints
#subset to quarters
#then filter
monthlyPrices <- prices[monthlyEps,]
monthlySMAs <- xts(apply(monthlyPrices, 2, SMA, n=nMonthSMA), order.by=index(monthlyPrices))
quarterlySMAs <- monthlySMAs[index(quarterlyPrices),]
smaFilter <- quarterlyPrices > quarterlySMAs
```

Now let’s put it together to get our final positions. Our cash position is simply one if we don’t have a single investment in the time period, zero else.

```finalPos <- rankPos*smaFilter
finalPos <- finalPos[!is.na(rocLongQtrs[,1]),]
cash <- xts(1-rowSums(finalPos), order.by=index(finalPos))
finalPos <- merge(finalPos, cash, join='inner')
```

Now we can finally compute our strategy returns.

```prices <- merge(prices, cashPrices, join='inner')
returns <- Return.calculate(prices)
stratRets <- Return.portfolio(returns, finalPos)
table.AnnualizedReturns(stratRets)
maxDrawdown(stratRets)
charts.PerformanceSummary(stratRets)
plot(log(cumprod(1+stratRets)))
```

So what do things look like?

Like this:

```> table.AnnualizedReturns(stratRets)
portfolio.returns
Annualized Return                    0.1899
Annualized Std Dev                   0.1619
Annualized Sharpe (Rf=0%)            1.1730
> maxDrawdown(stratRets)
[1] 0.1927991
```

And since the first equity curve doesn’t give much of an indication in the early years, I’ll take Tony Cooper’s (of Double Digit Numerics) advice and show the log equity curve as well.

In short, from 1997 through 2002, this strategy seemed to be going nowhere, and then took off. As I was able to get this backtest going back to 1997, it makes me wonder why it was only started in 2003 for the SeekingAlpha article, since even with 1997-2002 thrown in, this strategy’s risk/reward profile still looks fairly solid. CAR about 1 (slightly less, but that’s okay, for something that turns over so infrequently, and in so few securities!), and a Sharpe higher than 1. Certainly better than what the market itself offered over the same period of time for retail investors. Perhaps Cliff Smith himself could chime in regarding his choice of time frame.

In any case, Cliff Smith marketed the strategy as having a higher than 28% CAGR, and his article was published on August 15, 2014, and started from 2003. Let’s see if we can replicate those results.

```stratRets <- stratRets["2002-12-31::2014-08-15"]
table.AnnualizedReturns(stratRets)
maxDrawdown(stratRets)
charts.PerformanceSummary(stratRets)
plot(log(cumprod(1+stratRets)))
```

Which results in this:

```> table.AnnualizedReturns(stratRets)
portfolio.returns
Annualized Return                    0.2862
Annualized Std Dev                   0.1734
Annualized Sharpe (Rf=0%)            1.6499
> maxDrawdown(stratRets)
[1] 0.1911616
```

A far improved risk/return profile without 1997-2002 (or the out-of-sample period after Cliff Smith’s publishing date). Here are the two equity curves in-sample.

In short, the results look better, and the SeekingAlpha article’s results are validated.

Now, let’s look at the out-of-sample periods on their own.

```stratRets <- Return.portfolio(returns, finalPos)
earlyOOS <- stratRets["::2002-12-31"]
table.AnnualizedReturn(earlyOOS)
maxDrawdown(earlyOOS)
charts.PerformanceSummary(earlyOOS)
```

Here are the results:

```> table.AnnualizedReturns(earlyOOS)
portfolio.returns
Annualized Return                    0.0321
Annualized Std Dev                   0.1378
Annualized Sharpe (Rf=0%)            0.2327
> maxDrawdown(earlyOOS)
[1] 0.1927991
```

And with the corresponding equity curve (which does not need a log-scale this time).

In short, it basically did nothing for an entire five years. That’s rough, and I definitely don’t like the fact that it was left off of the SeekingAlpha article, as anytime I can extend a backtest further back than a strategy’s original author and then find skeletons in the closet (as happened for each and every one of Harry Long’s strategies), it sets off red flags on this end, so I’m hoping that there’s some good explanation for leaving off 1997-2002 that I’m simply failing to mention.

Lastly, let’s look at the out-of-sample performance.

```lateOOS <- stratRets["2014-08-15::"]
charts.PerformanceSummary(lateOOS)
table.AnnualizedReturns(lateOOS)
maxDrawdown(lateOOS)
```

With the following results:

```> table.AnnualizedReturns(lateOOS)
portfolio.returns
Annualized Return                    0.0752
Annualized Std Dev                   0.1426
Annualized Sharpe (Rf=0%)            0.5277
> maxDrawdown(lateOOS)
[1] 0.1381713
```

And the following equity curve:

Basically, while it’s ugly, it made new equity highs over only two more transactions (and in such a small sample size, anything can happen), so I’ll put this one down as a small, ugly win, but a win nevertheless.

If anyone has any questions or comments about this strategy, I’d love to see them, as this is basically a first-pass replica. To Mr. Cliff Smith’s credit, the results check out, and when the worst thing one can say about a strategy is that it had a period of a flat performance (aka when the market crested at the end of the Clinton administration right before the dot-com burst), well, that’s not the worst thing in the world.

More replications (including one requested by several readers) will be upcoming.

Thanks for reading.

NOTE: I am a freelance consultant in quantitative analysis on topics related to this blog. If you have contract or full time roles available for proprietary research that could benefit from my skills, please contact me through my LinkedIn here.