Introduction to Hypothesis Driven Development — Overview of a Simple Strategy and Indicator Hypotheses

This post will begin to apply a hypothesis-driven development framework (that is, the framework written by Brian Peterson on how to do strategy construction correctly, found here) to a strategy I’ve come across on SeekingAlpha. Namely, Cliff Smith posted about a conservative bond rotation strategy, which makes use of short-term treasuries, long-term treasuries, convertibles, emerging market debt, and high-yield corporate debt–that is, SHY, TLT, CWB, PCY, and JNK. What this post will do is try to put a more formal framework on whether or not this strategy is a valid one to begin with.

One note: For the sake of balancing succinctness for blog consumption and to demonstrate the computational techniques more quickly, I’ll be glossing over background research write-ups for this post/strategy, since it’s yet another take on time-series/cross-sectional momentum, except pared down to something more implementable for individual investors, as opposed to something that requires a massive collection of different instruments for massive, institutional-class portfolios.

Introduction, Overview, Objectives, Constraints, Assumptions, and Hypotheses to be Tested:

Momentum. It has been documented many times. For the sake of brevity, I’ll let readers follow the links if they’re so inclined, but among them are Jegadeesh and Titman’s seminal 1993 paper, Mark Carhart’s 1997 paper, Andreu et. Al (2012), Barroso and Santa-Clara (2013), Ilmanen’s Expected Returns (which covers momentum), and others. This list, of course, is far from exhaustive, but the point stands. Formation periods of several months (up to a year) should predict returns moving forward on some holding period, be it several months, or as is more commonly seen, one month.

Furthermore, momentum applies in two varieties–cross sectional, and time-series. Cross-sectional momentum asserts that assets that outperformed among a group will continue to outperform, while time-series momentum asserts that assets that have risen in price during a formation period will continue to do so for the short-term future.

Cliff Smith’s strategy depends on the latter, effectively, among a group of five bond ETFs. I am not certain of the objective of the strategy (he didn’t mention it), as PCY, JNK, and CWB, while they may be fixed-income in name, possess volatility on the order of equities. I suppose one possible “default” objective would be to achieve an outperforming total return against an equal-weighted benchmark, both rebalanced monthly.

The constraints are that one would need a sufficient amount of capital such that fixed transaction costs are negligible, since the strategy is a single-instrument rotation type, meaning that each month may have two-way turnover of 200% (sell one ETF, buy another). On the other hand, one would assume that the amount of capital deployed is small enough such that execution costs of trading do not materially impact the performance of the strategy. That is to say, moving multiple billions from one of these ETFs to the other is a non-starter. As all returns are computed close-to-close for the sake of simplicity, this creates the implicit assumption that the market impact and execution costs are very small compared to overall returns.

There are two overarching hypotheses to be tested in order to validate the efficacy of this strategy:

1) Time-series momentum: while it has been documented for equities and even industry/country ETFs, it may not have been formally done so yet for fixed-income ETFs, and their corresponding mutual funds. In order to validate this strategy, it should be investigated if the particular instruments it selects adhere to the same phenomena.

2) Cross-sectional momentum: again, while this has been heavily demonstrated in the past with regards to equities, ETFs are fairly new, and of the five mutual funds Cliff Smith selected, the latest one only has data going back to 1997, thus allowing less sophisticated investors to easily access diversified fixed income markets a relatively new innovation.

Essentially, both of these can be tested over a range of parameters (1-24 months).

Another note: with hypothesis-driven strategy development, the backtest is to be *nothing more than a confirmation of all the hypotheses up to that point*. That is, re-optimizing on the backtest itself means overfitting. Any proposed change to a strategy should be done in the form of tested hypotheses, as opposed to running a bunch of backtests and selecting the best trials. Taken another way, this means that every single proposed element of a strategy needs to have some form of strong hypothesis accompanying it, in order to be justified.

So, here are the two hypotheses I tested on the corresponding mutual funds:

symbols <- c("CNSAX", "FAHDX", "VUSTX", "VFISX", "PREMX")
getSymbols(symbols, from='1900-01-01')
prices <- list()
for(symbol in symbols) {
  prices[[symbol]] <- Ad(get(symbol))
prices <-, prices)
colnames(prices) <- substr(colnames(prices), 1, 5)
returns <- na.omit(Return.calculate(prices))

sample <- returns['1997-08/2009-03']
monthRets <- apply.monthly(sample, Return.cumulative)

returnRegression <- function(returns, nMonths) {
  nMonthAverage <- apply(returns, 2, runSum, n = nMonths)
  nMonthAverage <- xts(nMonthAverage, = index(returns))
  nMonthAverage <- na.omit(lag(nMonthAverage))
  returns <- returns[index(nMonthAverage)]
  rankAvg <- t(apply(nMonthAverage, 1, rank))
  rankReturn <- t(apply(returns, 1, rank))
  meltedAverage <- melt(data.frame(nMonthAverage))
  meltedReturns <- melt(data.frame(returns))
  meltedRankAvg <- melt(data.frame(rankAvg))
  meltedRankReturn <- melt(data.frame(rankReturn))
  lmfit <- lm(meltedReturns$value ~ meltedAverage$value - 1)
  rankLmfit <- lm(meltedRankReturn$value ~ meltedRankAvg$value)
  return(rbind(summary(lmfit)$coefficients, summary(rankLmfit)$coefficients))

pvals <- list()
estimates <- list()
rankPs <- list()
rankEstimates <- list()
for(i in 1:24) {
  tmp <- returnRegression(monthRets, nMonths=i)
  pvals[[i]] <- tmp[1,4]
  estimates[[i]] <- tmp[1,1]
  rankPs[[i]] <- tmp[2,4]
  rankEstimates[[i]] <- tmp[2,1]
pvals <-, pvals)
estimates <-, estimates)
rankPs <-, rankPs)
rankEstimates <-, rankEstimates)

Essentially, in this case, I take a pooled regression (that is, take the five instruments and pool them together into one giant vector), and regress the cumulative sum of monthly returns against the next month’s return. Also, I do the same thing as the above, except also using cross-sectional ranks for each month, and performing a rank-rank regression. The sample I used was the five mutual funds (CNSAX, FAHDX, VUSTX, VFISX, and PREMX) since their inception to March 2009, since the data for the final ETF begins in April of 2009, so I set aside the ETF data for out-of-sample backtesting.

Here are the results:

pvals <- list()
estimates <- list()
rankPs <- list()
rankEstimates <- list()
for(i in 1:24) {
  tmp <- returnRegression(monthRets, nMonths=i)
  pvals[[i]] <- tmp[1,4]
  estimates[[i]] <- tmp[1,1]
  rankPs[[i]] <- tmp[2,4]
  rankEstimates[[i]] <- tmp[2,1]
pvals <-, pvals)
estimates <-, estimates)
rankPs <-, rankPs)
rankEstimates <-, rankEstimates)

plot(estimates, type='h', xlab = 'Months regressed on', ylab='momentum coefficient', 
     main='future returns regressed on past momentum')
plot(pvals, type='h', xlab='Months regressed on', ylab='p-value', main='momentum significance')
abline(h=.05, col='green')
abline(h=.1, col='red')

plot(rankEstimates, type='h', xlab='Months regressed on', ylab="Rank coefficient",
     main='future return ranks regressed on past momentum ranks', ylim=c(0,3))
plot(rankPs, type='h', xlab='Months regressed on', ylab='P-values')

Of interest to note is that while much of the momentum literature specifies a reversion effect on time-series momentum at 12 months or greater, all the regression coefficients in this case (even up to 24 months!) proved to be positive, with the very long-term coefficients possessing more statistical significance than the short-term ones. Nevertheless, Cliff Smith’s chosen parameters (the two and four month settings) possess statistical significance at least at the 10% level. However, if one were to be highly conservative in terms of rejecting strategies, that in and of itself may be reason enough to reject this strategy right here.

However, the rank-rank regression (that is, regressing the future month’s cross-sectional rank on the past n month sum cross sectional rank) proved to be statistically significant beyond any doubt, with all p-values being effectively zero. In short, there is extremely strong evidence for cross-sectional momentum among these five assets, which extends out to at least two years. Furthermore, since SHY or VFISX, aka the short-term treasury fund, is among the assets chosen, since it’s a proxy for the risk-free rate, by including it among the cross-sectional rankings, the cross-sectional rankings also implicitly state that in order to be invested into (as this strategy is a top-1 asset rotation strategy), it must outperform the risk-free asset, otherwise, by process of elimination, the strategy will invest into the risk-free asset itself.

In upcoming posts, I’ll look into testing hypotheses on signals and rules.

Lastly, Volatility Made Simple has just released a blog post on the performance of volatility-based strategies for the month of August. Given the massive volatility spike, the dispersion in performance of strategies is quite interesting. I’m happy that in terms of YTD returns, the modified version of my strategy is among the top 10 for the year.

Thanks for reading.

NOTE: while I am currently consulting, I am always open to networking, meeting up (Philadelphia and New York City both work), consulting arrangements, and job discussions. Contact me through my email at, or through my LinkedIn, found here.

I’m Back, A New Harry Long Strategy, And Plans For Hypothesis-Driven Development

I’m back. Anyone that wants to know “what happened at Graham”, I felt there was very little scaffolding/on-boarding, and Graham’s expectations/requirements changed, though I have a reference from my direct boss, an accomplished quantitative director In any case, moving on.

Harry Long (of Houston) recently came out with a new strategy posted on SeekingAlpha, and I’d like to test it for robustness to see if it has merit.

Here’s the link to the post.

So, the rules are fairly simple:

ZIV 15%
SPLV 50%
TMF 10%
UUP 20%
VXX 5%

TMF can be approximated with a 3x leveraged TLT. SPLV is also highly similar to XLP — aka the consumer staples SPY sector. Here’s the equity curve comparison to prove it.

So, let’s test this thing.


getSymbols('XLP', from = '1900-01-01')
getSymbols('TLT', from = '1900-01-01')
getSymbols('UUP', from = '1900-01-01')
download('', destfile='ZIVlong.csv')
download('', destfile = 'VXXlong.csv')
ZIV &lt;- xts(read.zoo('ZIVlong.csv', header=TRUE, sep=','))
VXX &lt;- xts(read.zoo('VXXlong.csv', header=TRUE, sep=','))

symbols &lt;- na.omit(cbind(Return.calculate(Cl(ZIV)), Return.calculate(Ad(XLP)), Return.calculate(Ad(TLT))*3,
                         Return.calculate(Ad(UUP)), Return.calculate(Cl(VXX))))
strat &lt;- Return.portfolio(symbols, weights = c(.15, .5, .1, .2, .05), rebalance_on='years')

Here are the results:

compare &lt;- na.omit(cbind(strat, Return.calculate(Ad(XLP))))
rbind(table.AnnualizedReturns(compare), maxDrawdown(compare), CalmarRatio(compare))

Equity curve (compared against buy and hold XLP)


                          portfolio.returns XLP.Adjusted
Annualized Return                 0.0864000    0.0969000
Annualized Std Dev                0.0804000    0.1442000
Annualized Sharpe (Rf=0%)         1.0747000    0.6720000
Worst Drawdown                    0.1349957    0.3238755
Calmar Ratio                      0.6397665    0.2993100

In short, this strategy definitely offers a lot more bang for your risk in terms of drawdown, and volatility, and so, offers noticeably higher risk/reward tradeoffs. However, it’s not something that beats the returns of instruments in the category of twice its volatility.

Here are the statistics from 2010 onwards.

rbind(table.AnnualizedReturns(compare['2010::']), maxDrawdown(compare['2010::']), CalmarRatio(compare['2010::']))

                          portfolio.returns XLP.Adjusted
Annualized Return                0.12050000    0.1325000
Annualized Std Dev               0.07340000    0.1172000
Annualized Sharpe (Rf=0%)        1.64210000    1.1308000
Worst Drawdown                   0.07382878    0.1194072
Calmar Ratio                     1.63192211    1.1094371

Equity curve:

Definitely a smoother ride, and for bonus points, it seems some of the hedges helped with the recent market dip. Again, while aggregate returns aren’t as high as simply buying and holding XLP, the Sharpe and Calmar ratios do better on a whole.

Now, let’s do some robustness analysis. While I do not know how Harry Long arrived at the individual asset weights he did, what can be tested much more easily is what effect offsetting the rebalancing day has on the performance of the strategy. As this is a strategy rebalanced once a year, it can easily be tested for what effect the rebalancing date has on its performance.

yearlyEp &lt;- endpoints(symbols, on = 'years')
rebalanceDays &lt;- list()
for(i in 0:251) {
  offset &lt;- yearlyEp+i
  offset[offset &gt; nrow(symbols)] &lt;- nrow(symbols)
  offset[offset==0] &lt;- 1
  wts &lt;- matrix(rep(c(.15, .5, .1, .2, .05), length(yearlyEp)), ncol=5, byrow=TRUE)
  wts &lt;- xts(wts,[offset]))
  offsetRets &lt;- Return.portfolio(R = symbols, weights = wts)
  colnames(offsetRets) &lt;- paste0("offset", i)
  rebalanceDays[[i+1]] &lt;- offsetRets
rebalanceDays &lt;-, rebalanceDays)
rebalanceDays &lt;- na.omit(rebalanceDays)
stats &lt;- rbind(table.AnnualizedReturns(rebalanceDays), maxDrawdown(rebalanceDays))
stats[5,] &lt;- stats[1,]/stats[4,]

Here are the plots of return, Sharpe, and Calmar vs. offset.

plot(as.numeric(stats[1,])~c(0:251), type='l', ylab='CAGR', xlab='offset', main='CAGR vs. offset')
plot(as.numeric(stats[3,])~c(0:251), type='l', ylab='Sharpe Ratio', xlab='offset', main='Sharpe vs. offset')
plot(as.numeric(stats[5,])~c(0:251), type='l', ylab='Calmar Ratio', xlab='offset', main='Calmar vs. offset')
plot(as.numeric(stats[4,])~c(0:251), type='l', ylab='Drawdown', xlab='offset', main='Drawdown vs. offset')

In short, this strategy seems to be somewhat dependent upon the rebalancing date, which was left unsaid. Here are the quantiles for the five statistics for the given offsets:

rownames(stats)[5] &lt;- "Calmar"
apply(stats, 1, quantile)
     Annualized Return Annualized Std Dev Annualized Sharpe (Rf=0%) Worst Drawdown    Calmar
0%            0.072500             0.0802                  0.881000      0.1201198 0.4207922
25%           0.081925             0.0827                  0.987625      0.1444921 0.4755600
50%           0.087650             0.0837                  1.037250      0.1559238 0.5364758
75%           0.092000             0.0843                  1.090900      0.1744123 0.6230789
100%          0.105100             0.0867                  1.265900      0.1922916 0.8316698

While the standard deviation seems fairly robust, the Sharpe can decrease by about 33%, the Calmar can get cut in half, and the CAGR can also vary fairly substantially. That said, even using conservative estimates, the Sharpe ratio is fairly solid, and the Calmar outperforms that of XLP in any given variation, but nevertheless, performance can vary.

Is this strategy investible in its current state? Maybe, depending on your standards for rigor. Up to this point, rebalancing sometime in December-early January seems to substantially outperform other rebalance dates. Maybe a Dec/January anomaly effect exists in literature to justify this. However, the article makes no mention of that. Furthermore, the article doesn’t explain how it arrived at the weights it did.

Which brings me to my next topic, namely about a change with this blog going forward. Namely, hypothesis-driven trading system development. While this process doesn’t require complicated math, it does require statistical justification for multiple building blocks of a strategy, and a change in mindset, which a great deal of publicly available trading system ideas either gloss over, or omit entirely. As one of my most important readers praised this blog for “showing how the sausage is made”, this seems to be the next logical step in this progression.

Here’s the reasoning as to why.

It seems that when presenting trading ideas, there are two schools of thought: those that go off of intuition, build a backtest based off of that intuition, and see if it generally lines up with some intuitively expected result–and those that believe in a much more systematic, hypothesis-driven step-by-step framework, justifying as many decisions (ideally every decision) in creating a trading system. The advantage of the former is that it allows for displaying many more ideas in a much shorter timeframe. However, it has several major drawbacks: first off, it hides many concerns about potential overfitting. If what one sees is one final equity curve, there is nothing said about the sensitivity of said equity curve to however many various input parameters, and what other ideas were thrown out along the way. Secondly, without a foundation of strong hypotheses about the economic phenomena exploited, there is no proof that any strategy one comes across won’t simply fail once it’s put into live trading.

And third of all, which I find most important, is that such activities ultimately don’t sufficiently impress the industry’s best practitioners. For instance, Tony Cooper took issue with my replication of Trading The Odds’ volatility trading strategy, namely how data-mined it was (according to him in the comments section), and his objections seem to have been completely borne out by in out-of-sample performance.

So, for those looking for plug-and-crank system ideas, that may still happen every so often if someone sends me something particularly interesting, but there’s going to be some all-new content on this blog.

Thanks for reading.

NOTE: while I am currently consulting, I am always open to networking, meeting up (Philadelphia and New York City both work), consulting arrangements, and job discussions. Contact me through my email at, or through my LinkedIn, found here.

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 <-, 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 <-, 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 <-, 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;-, 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) {

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]

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),
    weights[[i]] &amp;lt;- weight
  weights &amp;lt;-, weights)
  stratRets &amp;lt;- Return.portfolio(R = returns, weights = weights)
  if(returnWeights) {
    return(list(weights, 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")

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")

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")

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.


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 <-, 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),
    weights[[i]] <- weight
  weights <-, weights)
  stratRets <- Return.portfolio(R = returns, weights = weights)

gmr <- logicInvestGMR(returns)

And here’s the performance:

> rbind(table.AnnualizedReturns(gmr), maxDrawdown(gmr), CalmarRatio(gmr))
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.

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.

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 <-, 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),
               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.

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.

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 <-, 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,, 1)))
  #replace cash back into returns
  returns <- cbind(assetRets, cashRets)
  weights <-, 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)

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")
apply.yearly(comparison, Return.cumulative)
stats <- rbind(table.AnnualizedReturns(comparison),
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]), 
      colnames(returnConfig) <- paste(inst1, weightSeq[j], 
                                inst2, 1-weightSeq[j], sep="_")
      count <- count + 1
      configs[[count]] <- returnConfig
  configs <-, 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) {
  #configs$zeroes <- 0 #zeroes for initial periods during calibration
  weights <- t(apply(monthlyModSharpe, 1, findMax))
  weights <- weights*1
  weights <- xts(weights,
  weights[] <- 0
  weights$zeroes <- 1-rowSums(weights)
  configCopy <- configs
  configCopy$zeroes <- 0
  stratRets <- Return.portfolio(R = configCopy, weights = weights)

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[![,4]),]
etfAdj <- 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) {
               maxDrawdown(returns), CalmarRatio(returns)))

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

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


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:


                          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 <-, divReturns)


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 Logical Invest “Hell On Fire” Replication Attempt

This post is about my replication attempt of Logical Invest’s “Hell On Fire” strategy — which is its Universal Investment Strategy using SPXL and TMF (aka the 3x leveraged ETFs). I don’t match their results, but I do come close.

It seems that some people at Logical Invest have caught whiff of some of the work I did in replicating Harry Long’s ideas. First off, for the record, I’ve actually done some work with Harry Long in private, and the strategies we’ve worked on together are definitely better than the strategies he has shared for free, so if you are an institution hoping to vet his track record, I wouldn’t judge it by the very much incomplete frameworks he posts for free.

This post’s strategy is the Logical Invest Universal Investment Strategy leveraged up three times over. Here’s the link to their newest post. Also, I’m happy to see that they think positively of my work.

In any case, my results are worse than those on Logical Invest’s, so if anyone sees a reason for the discrepancy, please let me know.

Here’s the code for the backtest–most of it is old, from my first time analyzing Logical Invest’s strategy.

LogicalInvestUIS <- function(returns, period = 63, modSharpeF = 2.8) {
  returns[] <- 0 #impute any NAs to zero
  configs <- list()
  for(i in 1:11) {
    weightFirst <- (i-1)*.1
    weightSecond <- 1-weightFirst
    config <- Return.portfolio(R = returns, weights=c(weightFirst, weightSecond), rebalance_on = "months")
    configs[[i]] <- config
  configs <-, 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) {
  #configs$zeroes <- 0 #zeroes for initial periods during calibration
  weights <- t(apply(monthlyModSharpe, 1, findMax))
  weights <- weights*1
  weights <- xts(weights,
  weights[] <- 0
  weights$zeroes <- 1-rowSums(weights)
  configCopy <- configs
  configCopy$zeroes <- 0
  stratRets <- Return.portfolio(R = configCopy, weights = weights)
  weightFirst <- apply(monthlyModSharpe, 1, which.max)
  weightFirst <-, weightFirst)
  weightFirst <- (weightFirst-1)*.1
  align <- cbind(weightFirst, stratRets)
  align <- na.locf(align)
  chart.TimeSeries(align[,1], date.format="%Y", ylab=paste("Weight", colnames(returns)[1]), 
                                                           main=paste("Weight", colnames(returns)[1]))

In this case, rather than steps of 5% weights, I used 10% weights after looking at the Logical Invest charts more closely.

Now, let’s look at the instruments.

getSymbols("SPY", from="1990-01-01")

getSymbols("TMF", from="1990-01-01")
TMFrets <- Return.calculate(Ad(TMF))
getSymbols("TLT", from="1990-01-01")
TLTrets <- Return.calculate(Ad(TLT))
tmf3TLT <- merge(TMFrets, 3*TLTrets, join='inner')
discrepancy <- as.numeric(Return.annualized(tmf3TLT[,2]-tmf3TLT[,1]))
tmf3TLT[,2] <- tmf3TLT[,2] - ((1+discrepancy)^(1/252)-1)
modifiedTLT <- 3*TLTrets - ((1+discrepancy)^(1/252)-1)

rets <- merge(3*Return.calculate(Ad(SPY)), modifiedTLT, join='inner')
colnames(rets) <- gsub("\\.[A-z]*", "", colnames(rets))

leveragedReturns <- rets
colnames(leveragedReturns) <- paste("Leveraged", colnames(leveragedReturns), sep="_")
leveragedReturns <- leveragedReturns[-1,]

Again, more of the same that I did from my work analyzing Harry Long’s strategies to get a longer backtest of SPXL and TMF (aka leveraged SPY and TLT).

Now, let’s look at some configurations.

hof <- LogicalInvestUIS(returns = leveragedReturns, period = 63, modSharpeF = 2.8)
hof2 <- LogicalInvestUIS(returns = leveragedReturns, period = 73, modSharpeF = 3)
hof3 <- LogicalInvestUIS(returns = leveragedReturns, period = 84, modSharpeF = 4)
hof4 <- LogicalInvestUIS(returns = leveragedReturns, period = 42, modSharpeF = 1.5)
hof5 <- LogicalInvestUIS(returns = leveragedReturns, period = 63, modSharpeF = 6)
hof6 <- LogicalInvestUIS(returns = leveragedReturns, period = 73, modSharpeF = 2)

hofComparisons <- cbind(hof, hof2, hof3, hof4, hof5, hof6)
colnames(hofComparisons) <- c("d63_F2.8", "d73_F3", "d84_F4", "d42_F1.5", "d63_F6", "d73_F2")
rbind(table.AnnualizedReturns(hofComparisons), maxDrawdown(hofComparisons), CalmarRatio(hofComparisons))

With the following statistics:

> rbind(table.AnnualizedReturns(hofComparisons), maxDrawdown(hofComparisons), CalmarRatio(hofComparisons))
                           d63_F2.8    d73_F3    d84_F4  d42_F1.5    d63_F6    d73_F2
Annualized Return         0.3777000 0.3684000 0.2854000 0.1849000 0.3718000 0.3830000
Annualized Std Dev        0.3406000 0.3103000 0.3010000 0.4032000 0.3155000 0.3383000
Annualized Sharpe (Rf=0%) 1.1091000 1.1872000 0.9483000 0.4585000 1.1785000 1.1323000
Worst Drawdown            0.5619769 0.4675397 0.4882101 0.7274609 0.5757738 0.4529908
Calmar Ratio              0.6721751 0.7879956 0.5845827 0.2541127 0.6457823 0.8455274

It seems that the original 73 day lookback, sharpe F of 2 had the best performance.

Here are the equity curves (log scale because leveraged or volatility strategies look silly at regular scale):

chart.TimeSeries(log(cumprod(1+hofComparisons)), legend.loc="topleft", date.format="%Y",
                 main="Hell On Fire Comparisons", ylab="Value of $1", yaxis = FALSE)
axis(side=2, at=c(0, 1, 2, 3, 4), label=paste0("$", round(exp(c(0, 1, 2, 3, 4)))), las = 1)

In short, sort of upwards from 2002 to the crisis, where all the strategies take a dip, and then continue steadily upwards.

Here are the drawdowns:

dds <- PerformanceAnalytics:::Drawdowns(hofComparisons)
chart.TimeSeries(dds, legend.loc="bottomright", date.format="%Y", main="Drawdowns Hell On Fire Variants", 
                 yaxis=FALSE, ylab="Drawdown", auto.grid=FALSE)
axis(side=2, at=seq(from=0, to=-.7, by = -.1), label=paste0(seq(from=0, to=-.7, by = -.1)*100, "%"), las = 1)

Basically, some regular bumps along the road given the CAGRs (that is, if you’re going to leverage something that has an 8% drawdown on the occasion three times over, it’s going to have a 24% drawdown on those same occasions, if not more), and the massive hit in the crisis when bonds take a hit, and on we go.

In short, this strategy is basically the same as the original strategy, just leveraged up, so for those with the stomach for it, there you go. Of course, Logical Invest is leaving off some details, since I’m not getting a perfect replica. Namely, their returns seem slightly higher, and their drawdowns slightly lower. I suppose that’s par for the course when selling subscriptions and newsletters.

One last thing, which I think people should be aware of–when people report statistics on their strategies, make sure to ask the question as to which frequency. Because here’s a quick little modification, going from daily returns to monthly returns:

> betterStatistics <- apply.monthly(hofComparisons, Return.cumulative)
> rbind(table.AnnualizedReturns(betterStatistics), maxDrawdown(betterStatistics), CalmarRatio(betterStatistics))
                           d63_F2.8    d73_F3    d84_F4  d42_F1.5    d63_F6   d73_F2
Annualized Return         0.3719000 0.3627000 0.2811000 0.1822000 0.3661000 0.377100
Annualized Std Dev        0.3461000 0.3014000 0.2914000 0.3566000 0.3159000 0.336700
Annualized Sharpe (Rf=0%) 1.0746000 1.2036000 0.9646000 0.5109000 1.1589000 1.119900
Worst Drawdown            0.4323102 0.3297927 0.4100792 0.6377512 0.4636949 0.311480
Calmar Ratio              0.8602366 1.0998551 0.6855148 0.2856723 0.7894636 1.210563

While the Sharpe ratios don’t improve too much, the Calmars (aka the return to drawdown) statistics increase dramatically. EG, imagine a month in which there’s a 40% drawdown, but it ends at a new equity high. A monthly return series will sweep that under the rug, or, for my fellow Jewish readers, pass over it. So, be wary.

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.

Introduction to my New IKReporting Package

This post will introduce my up and coming IKReporting package, and functions that compute and plot rolling returns, which are useful to compare recent performance, since simply looking at two complete equity curves may induce sample bias (EG SPY in 2008), which may not reflect the state of the markets going forward.

In any case, the motivation for this package was brought about by one of my readers, who has reminded me in the past of the demand for the in-the-ditches work of pretty performance reports. This package aims to make creating such thing as painless as possible, and I will be updating it rapidly in the near future.

The strategy in use for this post will be Flexible Asset Allocation from my IKTrading package, in order to celebrate the R/Finance lightning talk I’m approved for on FAA, and it’ll be compared to SPY.

Here’s the code:



symbols <- c("XLB", #SPDR Materials sector
             "XLE", #SPDR Energy sector
             "XLF", #SPDR Financial sector
             "XLP", #SPDR Consumer staples sector
             "XLI", #SPDR Industrial sector
             "XLU", #SPDR Utilities sector
             "XLV", #SPDR Healthcare sector
             "XLK", #SPDR Tech sector
             "XLY", #SPDR Consumer discretionary sector
             "RWR", #SPDR Dow Jones REIT ETF

             "EWJ", #iShares Japan
             "EWG", #iShares Germany
             "EWU", #iShares UK
             "EWC", #iShares Canada
             "EWY", #iShares South Korea
             "EWA", #iShares Australia
             "EWH", #iShares Hong Kong
             "EWS", #iShares Singapore
             "IYZ", #iShares U.S. Telecom
             "EZU", #iShares MSCI EMU ETF
             "IYR", #iShares U.S. Real Estate
             "EWT", #iShares Taiwan
             "EWZ", #iShares Brazil
             "EFA", #iShares EAFE
             "IGE", #iShares North American Natural Resources
             "EPP", #iShares Pacific Ex Japan
             "LQD", #iShares Investment Grade Corporate Bonds
             "SHY", #iShares 1-3 year TBonds
             "IEF", #iShares 3-7 year TBonds
             "TLT" #iShares 20+ year Bonds


#SPDR ETFs first, iShares ETFs afterwards
if(!"XLB" %in% ls()) {
  suppressMessages(getSymbols(symbols, from="2003-01-01", src="yahoo", adjust=TRUE))

prices <- list()
for(i in 1:length(symbols)) {
  prices[[i]] <- Cl(get(symbols[i]))
prices <-, prices)
colnames(prices) <- gsub("\\.[A-z]*", "", colnames(prices))

faa <- FAA(prices = prices, riskFreeName = "SHY", bestN = 6, stepCorRank = TRUE)

getSymbols("SPY", from="1990-01-01")

comparison <- merge(faa, Return.calculate(Cl(SPY)), join='inner')
colnames(comparison) <- c("FAA", "SPY")

And now here’s where the new code comes in:

This is a simple function for computing running cumulative returns of a fixed window. It’s a quick three-liner function that can compute the cumulative returns over any fixed period near-instantaneously.

"runCumRets" <- function(R, n = 252) {
  cumRets <- cumprod(1+R)
  rollingCumRets <- cumRets/lag(cumRets, k = n) - 1

So how does this get interesting? Well, with some plotting, of course.

Here’s a function to create a plot of these rolling returns.

"plotCumRets" <- function(R, n = 252, ...) {
  cumRets <- runCumRets(R = R, n = n)
  cumRets <- cumRets[![,1]),]
  chart.TimeSeries(cumRets, legend.loc="topleft", main=paste(n, "day rolling cumulative return"),
                   date.format="%Y", yaxis=FALSE, ylab="Return", auto.grid=FALSE)
  meltedCumRets <-, data.frame(cumRets))
  axisLabels <- pretty(meltedCumRets, n = 10)
  axisLabels <- round(axisLabels, 1)
  axisLabels <- axisLabels[axisLabels > min(axisLabels) & axisLabels < max(axisLabels)]
  axis(side=2, at=axisLabels, label=paste(axisLabels*100, "%"), las=1)

While the computation is done in the first line, the rest of the code is simply to make a prettier plot.

Here’s what the 252-day rolling return comparison looks like.


So here’s the interpretation: assuming that there isn’t too much return degradation in the implementation of the FAA strategy, it essentially delivers most of the upside of SPY while doing a much better job protecting the investor when things hit the fan. Recently, however, seeing as to how the stock market has been on a roar, there’s a slight bit of underperformance over the past several years.

However, let’s look at a longer time horizon — the cumulative return over 756 days.

plotCumRets(comparison, n = 756)

With the following result:

This offers a much clearer picture–essentially, what this states is that over any 756-day period, the strategy has not lost money, ever, unlike SPY, which would have wiped out three years of gains (and then some) at the height of the crisis. More recently, as the stock market is in yet another run-up, there has been some short-term (well, if 756 days can be called short-term) underperformance, namely due to SPY having some historical upward mobility.

On another unrelated topic, some of you (perhaps from Seeking Alpha) may have seen the following image floating around:

This is a strategy I have collaborated with Harry Long from Seeking Alpha on. While I’m under NDA and am not allowed to discuss the exact rules of this particular strategy, I can act as a liaison for those that wish to become a client of ZOMMA, LLC. While the price point is out of the reach of ordinary retail investors (the price point is into the six figures), institutions that are considering licensing one of these indices can begin by sending me an email at I can also set up a phone call.

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:

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 <-, 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) {

weights <- t(apply(monthlyModSharpe, 1, findMax))
weights <- weights*1
weights <- xts(weights,
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))

Which gives the results:

> rbind(table.AnnualizedReturns(stratRets), maxDrawdown(stratRets))
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')
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 <-, 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.