This post will display some robustness results for KDA asset allocation.

Ultimately, the two canary instruments fare much better using the original filter weights in Defensive Asset Allocation than in other variants of the weights for the filter. While this isn’t as worrying (the filter most likely was created that way and paired with those instruments by design), what *is* somewhat more irritating is that the strategy is dependent upon the end-of-month phenomenon, meaning this strategy cannot be simply tranched throughout an entire trading month.

So first off, let’s review the code from last time:

# KDA asset allocation
# KDA stands for Kipnis Defensive Adaptive (Asset Allocation).
# compute strategy statistics
stratStats <- function(rets) {
stats <- rbind(table.AnnualizedReturns(rets), maxDrawdown(rets))
stats[5,] <- stats[1,]/stats[4,]
stats[6,] <- stats[1,]/UlcerIndex(rets)
rownames(stats)[4] <- "Worst Drawdown"
rownames(stats)[5] <- "Calmar Ratio"
rownames(stats)[6] <- "Ulcer Performance Index"
return(stats)
}
# required libraries
require(quantmod)
require(PerformanceAnalytics)
require(tseries)
# symbols
symbols <- c("SPY", "VGK", "EWJ", "EEM", "VNQ", "RWX", "IEF", "TLT", "DBC", "GLD", "VWO", "BND")
# get data
rets <- list()
for(i in 1:length(symbols)) {
returns <- Return.calculate(Ad(get(getSymbols(symbols[i], from = '1990-01-01'))))
colnames(returns) <- symbols[i]
rets[[i]] <- returns
}
rets <- na.omit(do.call(cbind, rets))
# algorithm
KDA <- function(rets, offset = 0, leverageFactor = 1.5, momWeights = c(12, 4, 2, 1)) {
# get monthly endpoints, allow for offsetting ala AllocateSmartly/Newfound Research
ep <- endpoints(rets) + offset
ep[ep < 1] <- 1
ep[ep > nrow(rets)] <- nrow(rets)
ep <- unique(ep)
epDiff <- diff(ep)
if(last(epDiff)==1) { # if the last period only has one observation, remove it
ep <- ep[-length(ep)]
}
# initialize vector holding zeroes for assets
emptyVec <- data.frame(t(rep(0, 10)))
colnames(emptyVec) <- symbols[1:10]
allWts <- list()
# we will use the 13612F filter
for(i in 1:(length(ep)-12)) {
# 12 assets for returns -- 2 of which are our crash protection assets
retSubset <- rets[c((ep[i]+1):ep[(i+12)]),]
epSub <- ep[i:(i+12)]
sixMonths <- rets[(epSub[7]+1):epSub[13],]
threeMonths <- rets[(epSub[10]+1):epSub[13],]
oneMonth <- rets[(epSub[12]+1):epSub[13],]
# computer 13612 fast momentum
moms <- Return.cumulative(oneMonth) * momWeights[1] + Return.cumulative(threeMonths) * momWeights[2] +
Return.cumulative(sixMonths) * momWeights[3] + Return.cumulative(retSubset) * momWeights[4]
assetMoms <- moms[,1:10] # Adaptive Asset Allocation investable universe
cpMoms <- moms[,11:12] # VWO and BND from Defensive Asset Allocation
# find qualifying assets
highRankAssets <- rank(assetMoms) >= 6 # top 5 assets
posReturnAssets <- assetMoms > 0 # positive momentum assets
selectedAssets <- highRankAssets & posReturnAssets # intersection of the above
# perform mean-variance/quadratic optimization
investedAssets <- emptyVec
if(sum(selectedAssets)==0) {
investedAssets <- emptyVec
} else if(sum(selectedAssets)==1) {
investedAssets <- emptyVec + selectedAssets
} else {
idx <- which(selectedAssets)
# use 1-3-6-12 fast correlation average to match with momentum filter
cors <- (cor(oneMonth[,idx]) * momWeights[1] + cor(threeMonths[,idx]) * momWeights[2] +
cor(sixMonths[,idx]) * momWeights[3] + cor(retSubset[,idx]) * momWeights[4])/sum(momWeights)
vols <- StdDev(oneMonth[,idx]) # use last month of data for volatility computation from AAA
covs <- t(vols) %*% vols * cors
# do standard min vol optimization
minVolRets <- t(matrix(rep(1, sum(selectedAssets))))
minVolWt <- portfolio.optim(x=minVolRets, covmat = covs)$pw
names(minVolWt) <- colnames(covs)
investedAssets <- emptyVec
investedAssets[,selectedAssets] <- minVolWt
}
# crash protection -- between aggressive allocation and crash protection allocation
pctAggressive <- mean(cpMoms > 0)
investedAssets <- investedAssets * pctAggressive
pctCp <- 1-pctAggressive
# if IEF momentum is positive, invest all crash protection allocation into it
# otherwise stay in cash for crash allocation
if(assetMoms["IEF"] > 0) {
investedAssets["IEF"] <- investedAssets["IEF"] + pctCp
}
# leverage portfolio if desired in cases when both risk indicator assets have positive momentum
if(pctAggressive == 1) {
investedAssets = investedAssets * leverageFactor
}
# append to list of monthly allocations
wts <- xts(investedAssets, order.by=last(index(retSubset)))
allWts[[i]] <- wts
}
# put all weights together and compute cash allocation
allWts <- do.call(rbind, allWts)
allWts$CASH <- 1-rowSums(allWts)
# add cash returns to universe of investments
investedRets <- rets[,1:10]
investedRets$CASH <- 0
# compute portfolio returns
out <- Return.portfolio(R = investedRets, weights = allWts)
return(list(allWts, out))
}

So, the idea is that we take the basic Adaptive Asset Allocation algorithm, and wrap it in a canary universe from Defensive Asset Allocation (see previous post for links to both), which we use to control capital allocation, ranging from 0 to 1 (or beyond, in cases where leverage applies).

One of the ideas was to test out different permutations of the parameters belonging to the canary filter–a 1, 3, 6, 12 weighted filter focusing on the first month.

There are two interesting variants of this–equal weighting on the filter (both for momentum and the safety assets), and reversing the weights (that is, 1 * 1, 3 * 2, 6 * 4, 12 * 12). Here are the results of that experiment:

# different leverages
KDA_100 <- KDA(rets, leverageFactor = 1)
KDA_EW <- KDA(rets, leverageFactor = 1, momWeights = c(1,1,1,1))
KDA_rev <- KDA(rets, leverageFactor = 1, momWeights = c(1, 2, 4, 12))
# KDA_150 <- KDA(rets, leverageFactor = 1.5)
# KDA_200 <- KDA(rets, leverageFactor = 2)
# compare
compare <- na.omit(cbind(KDA_100[[2]], KDA_EW[[2]], KDA_rev[[2]]))
colnames(compare) <- c("KDA_base", "KDA_EW", "KDA_rev")
charts.PerformanceSummary(compare, colorset = c('black', 'purple', 'gold'),
main = "KDA AA with various momentum weights")
stratStats(compare)
apply.yearly(compare, Return.cumulative)

With the following results:

> stratStats(compare)
KDA_base KDA_EW KDA_rev
Annualized Return 0.10990000 0.0879000 0.0859000
Annualized Std Dev 0.09070000 0.0900000 0.0875000
Annualized Sharpe (Rf=0%) 1.21180000 0.9764000 0.9814000
Worst Drawdown 0.07920363 0.1360625 0.1500333
Calmar Ratio 1.38756275 0.6460266 0.5725396
Ulcer Performance Index 3.96188378 2.4331636 1.8267448
> apply.yearly(compare, Return.cumulative)
KDA_base KDA_EW KDA_rev
2008-12-31 0.15783690 0.101929228 0.08499664
2009-12-31 0.18169281 -0.004707164 0.02403531
2010-12-31 0.17797930 0.283216782 0.27889530
2011-12-30 0.17220203 0.161001680 0.03341651
2012-12-31 0.13030215 0.081280035 0.09736187
2013-12-31 0.12692163 0.120902015 0.09898799
2014-12-31 0.04028492 0.047381890 0.06883301
2015-12-31 -0.01621646 -0.005016891 0.01841095
2016-12-30 0.01253209 0.020960805 0.01580218
2017-12-29 0.15079063 0.148073455 0.18811112
2018-12-31 0.06583962 0.029804042 0.04375225
2019-02-20 0.01689700 0.003934044 0.00962020

So, one mea culpa: after comparing AllocateSmartly, my initial code (which I’ve since edited, most likely owing to getting some logic mixed up when I added functionality to lag the day of month to trade) had some sort of bug in it which gave a slightly better than expected 2015 return. Nevertheless, the results are very similar. What is interesting to note is that in the raging bull market that was essentially from 2010 onwards, the equal weight and reverse weight filters don’t perform too badly, though the reverse weight filter has a massive drawdown in 2011, but in terms of capitalizing in awful markets, the original filter as designed by Keller and TrendXplorer works best, both in terms of making money during the recession, and does better near the market bottom in 2009.

Now that that’s out of the way, the more interesting question is how does the strategy work when not trading at the end of the month? Long story short, the best time to trade it is in the last week of the month. Once the new month rolls around, hands off. If you’re talking about tranching this strategy, then you have about a week’s time to get your positions in, so I’m not sure the actual dollar volume this strategy can manage, as it’s dependent on the month-end effect (I know that one of my former managers–a brilliant man, by all accounts–said that this phenomena no longer existed, but I feel these empirical results refute that assertion in this particular instance). Here are these results:

lagCompare <- list()
for(i in 1:21) {
offRets <- KDA(rets, leverageFactor = 1, offset = i)
tmp <- offRets[[2]]
colnames(tmp) <- paste0("Lag", i)
lagCompare[[i]] <- tmp
}
lagCompare <- do.call(cbind, lagCompare)
lagCompare <- na.omit(cbind(KDA_100[[2]], lagCompare))
colnames(lagCompare)[1] <- "Base"
charts.PerformanceSummary(lagCompare, colorset=c("orange", rep("gray", 21)))
stratStats(lagCompare)

With the results:

> stratStats(lagCompare)
Base Lag1 Lag2 Lag3 Lag4 Lag5 Lag6 Lag7 Lag8
Annualized Return 0.11230000 0.0584000 0.0524000 0.0589000 0.0319000 0.0319000 0.0698000 0.0790000 0.0912000
Annualized Std Dev 0.09100000 0.0919000 0.0926000 0.0945000 0.0975000 0.0957000 0.0943000 0.0934000 0.0923000
Annualized Sharpe (Rf=0%) 1.23480000 0.6357000 0.5654000 0.6229000 0.3270000 0.3328000 0.7405000 0.8460000 0.9879000
Worst Drawdown 0.07920363 0.1055243 0.1269207 0.1292193 0.1303246 0.1546962 0.1290020 0.1495558 0.1227749
Calmar Ratio 1.41786439 0.5534272 0.4128561 0.4558141 0.2447734 0.2062107 0.5410771 0.5282311 0.7428230
Ulcer Performance Index 4.03566328 1.4648618 1.1219982 1.2100649 0.4984094 0.5012318 1.3445786 1.4418132 2.3277271
Lag9 Lag10 Lag11 Lag12 Lag13 Lag14 Lag15 Lag16 Lag17
Annualized Return 0.0854000 0.0863000 0.0785000 0.0732000 0.0690000 0.0862000 0.0999000 0.0967000 0.1006000
Annualized Std Dev 0.0906000 0.0906000 0.0900000 0.0913000 0.0906000 0.0909000 0.0923000 0.0947000 0.0949000
Annualized Sharpe (Rf=0%) 0.9426000 0.9524000 0.8722000 0.8023000 0.7617000 0.9492000 1.0825000 1.0209000 1.0600000
Worst Drawdown 0.1278059 0.1189949 0.1197596 0.1112761 0.1294588 0.1498408 0.1224511 0.1290538 0.1274083
Calmar Ratio 0.6682006 0.7252411 0.6554796 0.6578231 0.5329880 0.5752771 0.8158357 0.7493000 0.7895878
Ulcer Performance Index 2.3120919 2.6415855 2.4441605 1.9248615 1.8096134 2.2378207 2.8753265 2.9092448 3.0703542
Lag18 Lag19 Lag20 Lag21
Annualized Return 0.097100 0.0921000 0.1047000 0.1019000
Annualized Std Dev 0.092900 0.0903000 0.0958000 0.0921000
Annualized Sharpe (Rf=0%) 1.044900 1.0205000 1.0936000 1.1064000
Worst Drawdown 0.100604 0.1032067 0.1161583 0.1517104
Calmar Ratio 0.965170 0.8923835 0.9013561 0.6716747
Ulcer Performance Index 3.263040 2.7159601 3.0758230 3.0414002

Essentially, the trade at the very end of the month is the only one with a Calmar ratio above 1, though the Calmar ratio from lag15 to lag 21 is about .8 or higher, with a Sharpe ratio of 1 or higher. So, there’s definitely a window of when to trade, and when not to–namely, the lag 1 through 5 variations have the worst performances by no small margin. Therefore, I strongly suspect that the 1-3-6-12 filter was designed around the idea of the end-of-month effect, or at least, not stress-tested for different trading days within the month (and given that longer-dated data is only monthly, this is understandable).

Nevertheless, I hope this does answer some people’s questions from the quant finance universe. I know that Corey Hoffstein of Think Newfound (and wow that blog is good from the perspective of properties of trend-following) loves diversifying across every bit of the process, though in this case, I do think there’s something to be said about “diworsification”.

In any case, I do think there are some future research venues for further research here.

Thanks for reading.

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A very interesting piece, thank you, Ilya!

I have two questions/ideas:

1. you suspect that 1-3-6-12 filter is overfit to the end of month, right? But don’t you think that introducing any lag to the system may be the root cause of the problem? Throughout my testing, introducing even a small lag always deteriorates the system (to which extent – matter of specific case). Another idea here is that last+first 3 days of each month are much more profitable then other days. I read several articles on that + I have my own Excel sheet that shows that some days are statistically brilliant (1-3, 16, 29-31), some are neutral, some are awful. So if the system lags and doesn’t enter those best days then it will lose a lot of profit.

2. your investigation on 3 possible filters shows that the faster filter wins, have you tried to change weights, to overweight last month (or last week) even more? Say, 20*M1 + 4*M3 + 2*M6 + M12?

1) The filter definitely seems to be based around exploiting the end of month anomaly for sure. An yes, in robustness testing, inducing lag does deteriorate the results.

2) Have not tried highly over-weighting a back month. But given the evidence for reversed weighting which puts an inordinate amount of weight on the back month as it stands, I’m not sure going to even higher extremes will help much.

Thank you for your comments.

Some extra ideas:

1. I created a spredsheet to check end of month (and other superdays) anomaly, wastly superior (CAR 16,6% vs 7,66% SP500 since 1950) compared to SP500 with lower drawdown (38.6% vs 58.6%) and only 1/3 exposition to the market (in days). And I didn’t use any bond, only cash, otherwise results would be much better (2/3 of bond return for these years).

But! I tried simple VUSTX + SPY momentum with rebalancing based, say on 31,62,83 days and holding for a month. Got some results for EOM rebalancing, tested EOM vicinity, say, if we want to exit SP500, and we know that the best days are from 28th till 3th, then rebalance on 3th, the same is for enter – enter on 28th. Results are mixed, in 2/3 of cases it’s better to rebalance on vicinity days, in 1/3 – on exact month end.

2. The more I test, the more I see that the most robust combination is to calculate momentum for 1 month and hold for 3 months. All other combinations are worse for a wide set of params and assets. I think that the reason may be quaterly rebalances of funds or some other real-life recurrent events. Did you notice such anomaly?

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Hello, Ilya! Staying in cash means investing in money market funds or holding real cash?

Take your pick. It doesn’t change the outcome too much.

Would you please explain in simple words why you are calculating momentum in the way presented in the text? In particular, why is the last month given so much weight?

The book “Quantitative Momentum” says that the last month momentum is not a signal, but noise. And that it has rather the opposite predictive power. This thesis in the book is supported by a reference to a scientific article in a financial journal. If interested, I’ll find it and send it to you. The authors of the book recommend calculating the momentum for the previous 12 or 6 months, ignoring the last month.

I am interested in this question from a practical point of view. Which formula should I use to manage my money?

And I didn’t really figure out when to update the portfolio using the presented method. Should I run the code on the weekend before the last week of the month? Trade on Monday last week of the month? Rebalance the portfolio once a month or quarterly?

The last month being noise is something intrinsic to individual equities, not asset class indices. I also reviewed Alpha Architect’s Quantitative Momentum, so I’m familiar with it.

For momentum on asset class ETFs, you generally don’t do 12-1. For individual equities, to my understanding, you do.

As for this code, it rebalances at the end of the month. And because it’s written by Brian Peterson, the re-allocations take a day to actually set in. That is, if you get your weights on Jan 31’s close, you get your weights locked in by the end of Feb 1.

Thanks, Ilya. I like long backtests (preferably from 1926) for which only monthly data are available. So I never optimised the 13612W filter for end-of-month as compared to other days. Also notices that the 13612W weights (12*M1+4*M3+2*M6+1*M12) implies equal weights for annualized returns. Therefore, I also did not optimise the 13612W weights in our recent (VAA/DAA) models. In our older (EAA/CAA) models we used unweighted returns, like Faber did.

Thanks for the explanation. I also suppose that may be another reason why this algorithm is sensitive to end-of-month effects–it was tested using monthly data, and there may be dynamics that were present before the advent of computers that may or may not still be around today.