Nuts and Bolts of Quantstrat, Part V

This post will be about pre-processing custom indicators in quantstrat–that is, how to add values to your market data that do not arise from the market data itself.

The first four parts of my nuts and bolts of quantstrat were well received. They are even available as a datacamp course. For those that want to catch up to today’s post, I highly recommend the datacamp course.

To motivate this post, the idea is that say you’re using alternative data that isn’t simply derived from a transformation of the market data itself. I.E. you have a proprietary alternative data stream that may predict an asset’s price, you want to employ a cross-sectional ranking system, or any number of things. How do you do this within the context of quantstrat?

The answer is that it’s as simple as binding a new xts to your asset data, as this demonstration will show.

First, let’s get the setup out of the way.

require(quantstrat)
require(PerformanceAnalytics)

initDate="1990-01-01"
from="2003-01-01"
to="2012-12-31"
options(width=70)

options("getSymbols.warning4.0"=FALSE)

currency('USD')
Sys.setenv(TZ="UTC")

symbols <- 'SPY'
suppressMessages(getSymbols(symbols, from=from, to=to, src="yahoo", adjust=TRUE))  

stock(symbols, currency="USD", multiplier=1)

Now, we have our non-derived indicator. In this case, it’s a toy example–the value is 1 if the year is odd (I.E. 2003, 2005, 2007, 2009), and 0 if it’s even. We compute that and simply column-bind (cbind) it to the asset data.

nonDerivedIndicator <- as.numeric(as.character(substr(index(SPY), 1, 4)))%%2 == 1
nonDerivedIndicator <- xts(nonDerivedIndicator, order.by=index(SPY))

SPY <- cbind(SPY, nonDerivedIndicator)
colnames(SPY)[7] = "nonDerivedIndicator"

Next, we just have a very simple strategy–buy a share of SPY on odd years, sell on even years. That is, buy when the nonDerivedIndicator column crosses above 0.5 (from 0 to 1), and sell when the opposite occurs.

strategy.st <- portfolio.st <- account.st <- "nonDerivedData"
rm.strat(strategy.st)
initPortf(portfolio.st, symbols=symbols, initDate=initDate, currency='USD')
initAcct(account.st, portfolios=portfolio.st, initDate=initDate, currency='USD')
initOrders(portfolio.st, initDate=initDate)
strategy(strategy.st, store=TRUE)

add.signal(strategy.st, name = sigThreshold, 
           arguments = list(column = "nonDerivedIndicator", threshold = 0.5, relationship = "gte", cross = TRUE),
           label = "longEntry")

add.signal(strategy.st, name = sigThreshold, 
           arguments = list(column = "nonDerivedIndicator", threshold = 0.5, relationship = "lte", cross = TRUE),
           label = "longExit")


tmp <- applySignals(strategy = strategy.st, mktdata=SPY)


add.rule(strategy.st, name="ruleSignal", 
         arguments=list(sigcol="longEntry", sigval=TRUE, ordertype="market", 
                        orderside="long", replace=FALSE, prefer="Open", orderqty = 1), 
         type="enter", path.dep=TRUE)

add.rule(strategy.st, name="ruleSignal", 
         arguments=list(sigcol="longExit", sigval=TRUE, orderqty="all", 
                        ordertype="market", orderside="long", 
                        replace=FALSE, prefer="Open"), 
         type="exit", path.dep=TRUE)

#apply strategy
t1 <- Sys.time()
out <- applyStrategy(strategy=strategy.st,portfolios=portfolio.st)
t2 <- Sys.time()
print(t2-t1)

#set up analytics
updatePortf(portfolio.st)
dateRange <- time(getPortfolio(portfolio.st)$summary)[-1]
updateAcct(portfolio.st,dateRange)
updateEndEq(account.st)

And the result:

chart.Posn(portfolio.st, 'SPY')

In conclusion, you can create signals based off of any data in quantstrat. Whether that means volatility ratios, fundamental data, cross-sectional ranking, or whatever proprietary alternative data source you may have access to, this very simple process is how you can use quantstrat to add all of those things to your systematic trading backtest research.

Thanks for reading.

Note: I am always interested in full-time opportunities which may benefit from my skills. I have experience in data analytics, asset management, and systematic trading research. If you know of any such opportunities, do not hesitate to contact me on my LinkedIn, found here.

The Problem With Depmix For Online Regime Prediction

This post will be about attempting to use the Depmix package for online state prediction. While the depmix package performs admirably when it comes to describing the states of the past, when used for one-step-ahead prediction, under the assumption that tomorrow’s state will be identical to today’s, the hidden markov model process found within the package does not perform to expectations.

So, to start off, this post was motivated by Michael Halls-Moore, who recently posted some R code about using the depmixS4 library to use hidden markov models. Generally, I am loath to create posts on topics I don’t feel I have an absolutely front-to-back understanding of, but I’m doing this in the hope of learning from others on how to appropriately do online state-space prediction, or “regime switching” detection, as it may be called in more financial parlance.

Here’s Dr. Halls-Moore’s post.

While I’ve seen the usual theory of hidden markov models (that is, it can rain or it can be sunny, but you can only infer the weather judging by the clothes you see people wearing outside your window when you wake up), and have worked with toy examples in MOOCs (Udacity’s self-driving car course deals with them, if I recall correctly–or maybe it was the AI course), at the end of the day, theory is only as good as how well an implementation can work on real data.

For this experiment, I decided to take SPY data since inception, and do a full in-sample “backtest” on the data. That is, given that the HMM algorithm from depmix sees the whole history of returns, with this “god’s eye” view of the data, does the algorithm correctly classify the regimes, if the backtest results are any indication?

Here’s the code to do so, inspired by Dr. Halls-Moore’s.

require(depmixS4)
require(quantmod)
getSymbols('SPY', from = '1990-01-01', src='yahoo', adjust = TRUE)
spyRets <- na.omit(Return.calculate(Ad(SPY)))

set.seed(123)

hmm <- depmix(SPY.Adjusted ~ 1, family = gaussian(), nstates = 3, data=spyRets)
hmmfit <- fit(hmm, verbose = FALSE)
post_probs <- posterior(hmmfit)
post_probs <- xts(post_probs, order.by=index(spyRets))
plot(post_probs$state)
summaryMat <- data.frame(summary(hmmfit))
colnames(summaryMat) <- c("Intercept", "SD")
bullState <- which(summaryMat$Intercept > 0)
bearState <- which(summaryMat$Intercept < 0)

hmmRets <- spyRets * lag(post_probs$state == bullState) - spyRets * lag(post_probs$state == bearState)
charts.PerformanceSummary(hmmRets)
table.AnnualizedReturns(hmmRets)

Essentially, while I did select three states, I noted that anything with an intercept above zero is a bull state, and below zero is a bear state, so essentially, it reduces to two states.

With the result:

table.AnnualizedReturns(hmmRets)
                          SPY.Adjusted
Annualized Return               0.1355
Annualized Std Dev              0.1434
Annualized Sharpe (Rf=0%)       0.9448

So, not particularly terrible. The algorithm works, kind of, sort of, right?

Well, let’s try online prediction now.

require(DoMC)

dailyHMM <- function(data, nPoints) {
  subRets <- data[1:nPoints,]
  hmm <- depmix(SPY.Adjusted ~ 1, family = gaussian(), nstates = 3, data = subRets)
  hmmfit <- fit(hmm, verbose = FALSE)
  post_probs <- posterior(hmmfit)
  summaryMat <- data.frame(summary(hmmfit))
  colnames(summaryMat) <- c("Intercept", "SD")
  bullState <- which(summaryMat$Intercept > 0)
  bearState <- which(summaryMat$Intercept < 0)
  if(last(post_probs$state) %in% bullState) {
    state <- xts(1, order.by=last(index(subRets)))
  } else if (last(post_probs$state) %in% bearState) {
    state <- xts(-1, order.by=last(index(subRets)))
  } else {
    state <- xts(0, order.by=last(index(subRets)))
  }
  colnames(state) <- "State"
  return(state)
}

# took 3 hours in parallel
t1 <- Sys.time()
set.seed(123)
registerDoMC((detectCores() - 1))
states <- foreach(i = 500:nrow(spyRets), .combine=rbind) %dopar% {
  dailyHMM(data = spyRets, nPoints = i)
}
t2 <- Sys.time()
print(t2-t1)

So what I did here was I took an expanding window, starting from 500 days since SPY’s inception, and kept increasing it, by one day at a time. My prediction, was, trivially enough, the most recent day, using a 1 for a bull state, and a -1 for a bear state. I ran this process in parallel (on a linux cluster, because windows’s doParallel library seems to not even know that certain packages are loaded, and it’s more messy), and the first big issue is that this process took about three hours on seven cores for about 23 years of data. Not exactly encouraging, but computing time isn’t expensive these days.

So let’s see if this process actually works.

First, let’s test if the algorithm does what it’s actually supposed to do and use one day of look-ahead bias (that is, the algorithm tells us the state at the end of the day–how correct is it even for that day?).


onlineRets <- spyRets * states 
charts.PerformanceSummary(onlineRets)
table.AnnualizedReturns(onlineRets)

With the result:

> table.AnnualizedReturns(onlineRets)
                          SPY.Adjusted
Annualized Return               0.2216
Annualized Std Dev              0.1934
Annualized Sharpe (Rf=0%)       1.1456

So, allegedly, the algorithm seems to do what it was designed to do, which is to classify a state for a given data set. Now, the most pertinent question: how well do these predictions do even one day ahead? You’d think that state space predictions would be parsimonious from day to day, given the long history, correct?


onlineRets <- spyRets * lag(states)
charts.PerformanceSummary(onlineRets)
table.AnnualizedReturns(onlineRets)

With the result:

> table.AnnualizedReturns(onlineRets)
                          SPY.Adjusted
Annualized Return               0.0172
Annualized Std Dev              0.1939
Annualized Sharpe (Rf=0%)       0.0888

That is, without the lookahead bias, the state space prediction algorithm is atrocious. Why is that?

Well, here’s the plot of the states:

In short, the online hmm algorithm in the depmix package seems to change its mind very easily, with obvious (negative) implications for actual trading strategies.

So, that wraps it up for this post. Essentially, the main message here is this: there’s a vast difference between loading doing descriptive analysis (AKA “where have you been, why did things happen”) vs. predictive analysis (that is, “if I correctly predict the future, I get a positive payoff”). In my opinion, while descriptive statistics have their purpose in terms of explaining why a strategy may have performed how it did, ultimately, we’re always looking for better prediction tools. In this case, depmix, at least in this “out-of-the-box” demonstration does not seem to be the tool for that.

If anyone has had success with using depmix (or other regime-switching algorithm in R) for prediction, I would love to see work that details the procedure taken, as it’s an area I’m looking to expand my toolbox into, but don’t have any particular good leads. Essentially, I’d like to think of this post as me describing my own experiences with the package.

Thanks for reading.

NOTE: On Oct. 5th, I will be in New York City. On Oct. 6th, I will be presenting at The Trading Show on the Programming Wars panel.

NOTE: My current analytics contract is up for review at the end of the year, so I am officially looking for other offers as well. If you have a full-time role which may benefit from the skills you see on my blog, please get in touch with me. My linkedin profile can be found here.

An Introduction to Portfolio Component Conditional Value At Risk

This post will introduce component conditional value at risk mechanics found in PerformanceAnalytics from a paper written by Brian Peterson, Kris Boudt, and Peter Carl. This is a mechanism that is an easy-to-call mechanism for computing component expected shortfall in asset returns as they apply to a portfolio. While the exact mechanics are fairly complex, the upside is that the running time is nearly instantaneous, and this method is a solid tool for including in asset allocation analysis.

For those interested in an in-depth analysis of the intuition of component conditional value at risk, I refer them to the paper written by Brian Peterson, Peter Carl, and Kris Boudt.

Essentially, here’s the idea: all assets in a given portfolio have a marginal contribution to its total conditional value at risk (also known as expected shortfall)–that is, the expected loss when the loss surpasses a certain threshold. For instance, if you want to know your 5% expected shortfall, then it’s the average of the worst 5 returns per 100 days, and so on. For returns using daily resolution, the idea of expected shortfall may sound as though there will never be enough data in a sufficiently fast time frame (on one year or less), the formula for expected shortfall in the PerformanceAnalytics defaults to an approximation calculation using a Cornish-Fisher expansion, which delivers very good results so long as the p-value isn’t too extreme (that is, it works for relatively sane p values such as the 1%-10% range).

Component Conditional Value at Risk has two uses: first off, given no input weights, it uses an equal weight default, which allows it to provide a risk estimate for each individual asset without burdening the researcher to create his or her own correlation/covariance heuristics. Secondly, when provided with a set of weights, the output changes to reflect the contribution of various assets in proportion to those weights. This means that this methodology works very nicely with strategies that exclude assets based on momentum, but need a weighting scheme for the remaining assets. Furthermore, using this methodology also allows an ex-post analysis of risk contribution to see which instrument contributed what to risk.

First, a demonstration of how the mechanism works using the edhec data set. There is no strategy here, just a demonstration of syntax.

require(quantmod)
require(PerformanceAnalytics)

data(edhec)

tmp &lt;- CVaR(edhec, portfolio_method = &quot;component&quot;)

This will assume an equal-weight contribution from all of the funds in the edhec data set.

So tmp is the contribution to expected shortfall from each of the various edhec managers over the entire time period. Here’s the output:

$MES
           [,1]
[1,] 0.03241585

$contribution
 Convertible Arbitrage             CTA Global  Distressed Securities       Emerging Markets  Equity Market Neutral
          0.0074750513          -0.0028125166           0.0039422674           0.0069376579           0.0008077760
          Event Driven Fixed Income Arbitrage           Global Macro      Long/Short Equity       Merger Arbitrage
          0.0037114666           0.0043125937           0.0007173036           0.0036152960           0.0013693293
        Relative Value          Short Selling         Funds of Funds
          0.0037650911          -0.0048178690           0.0033924063 

$pct_contrib_MES
 Convertible Arbitrage             CTA Global  Distressed Securities       Emerging Markets  Equity Market Neutral
            0.23059863            -0.08676361             0.12161541             0.21402052             0.02491917
          Event Driven Fixed Income Arbitrage           Global Macro      Long/Short Equity       Merger Arbitrage
            0.11449542             0.13303965             0.02212817             0.11152864             0.04224258
        Relative Value          Short Selling         Funds of Funds
            0.11614968            -0.14862694             0.10465269

The salient part of this is the percent contribution (the last output). Notice that it can be negative, meaning that certain funds gain when others lose. At least, this was the case over the current data set. These assets diversify a portfolio and actually lower expected shortfall.

&gt; tmp2 &lt;- CVaR(edhec, portfolio_method = &quot;component&quot;, weights = c(rep(.1, 10), rep(0,3)))
&gt; tmp2
$MES
           [,1]
[1,] 0.04017453

$contribution
 Convertible Arbitrage             CTA Global  Distressed Securities       Emerging Markets  Equity Market Neutral
          0.0086198045          -0.0046696862           0.0058778855           0.0109152240           0.0009596620
          Event Driven Fixed Income Arbitrage           Global Macro      Long/Short Equity       Merger Arbitrage
          0.0054824325           0.0050398011           0.0009638502           0.0044568333           0.0025287234
        Relative Value          Short Selling         Funds of Funds
          0.0000000000           0.0000000000           0.0000000000 

$pct_contrib_MES
 Convertible Arbitrage             CTA Global  Distressed Securities       Emerging Markets  Equity Market Neutral
            0.21455894            -0.11623499             0.14630875             0.27169512             0.02388732
          Event Driven Fixed Income Arbitrage           Global Macro      Long/Short Equity       Merger Arbitrage
            0.13646538             0.12544767             0.02399157             0.11093679             0.06294345
        Relative Value          Short Selling         Funds of Funds
            0.00000000             0.00000000             0.00000000

In this case, I equally weighted the first ten managers in the edhec data set, and put zero weight in the last three. Furthermore, we can see what happens when the weights are not equal.

&gt; tmp3 &lt;- CVaR(edhec, portfolio_method = &quot;component&quot;, weights = c(.2, rep(.1, 9), rep(0,3)))
&gt; tmp3
$MES
           [,1]
[1,] 0.04920372

$contribution
 Convertible Arbitrage             CTA Global  Distressed Securities       Emerging Markets  Equity Market Neutral
          0.0187406982          -0.0044391078           0.0057235762           0.0102706768           0.0007710434
          Event Driven Fixed Income Arbitrage           Global Macro      Long/Short Equity       Merger Arbitrage
          0.0051541429           0.0055944367           0.0008028457           0.0044085104           0.0021768951
        Relative Value          Short Selling         Funds of Funds
          0.0000000000           0.0000000000           0.0000000000 

$pct_contrib_MES
 Convertible Arbitrage             CTA Global  Distressed Securities       Emerging Markets  Equity Market Neutral
            0.38087972            -0.09021895             0.11632406             0.20873782             0.01567043
          Event Driven Fixed Income Arbitrage           Global Macro      Long/Short Equity       Merger Arbitrage
            0.10475109             0.11369947             0.01631677             0.08959710             0.04424249
        Relative Value          Short Selling         Funds of Funds
            0.00000000             0.00000000             0.00000000

This time, notice that as the weight increased in the convertible arb manager, so too did his contribution to maximum expected shortfall.

For a future backtest, I would like to make some data requests. I would like to use the universe found in Faber’s Global Asset Allocation book. That said, the simulations in that book go back to 1972, and I was wondering if anyone out there has daily returns for those assets/indices. While some ETFs go back into the early 2000s, there are some that start rather late such as DBC (commodities, early 2006), GLD (gold, early 2004), BWX (foreign bonds, late 2007), and FTY (NAREIT, early 2007). As an eight-year backtest would be a bit short, I was wondering if anyone had data with more history.

One other thing, I will in New York for the trading show, and speaking on the “programming wars” panel on October 6th.

Thanks for reading.

NOTE: While I am currently contracting, I am also looking for a permanent position which can benefit from my skills for when my current contract ends. If you have or are aware of such an opening, I will be happy to speak with you.

How To Compute Turnover With Return.Portfolio in R

This post will demonstrate how to take into account turnover when dealing with returns-based data using PerformanceAnalytics and the Return.Portfolio function in R. It will demonstrate this on a basic strategy on the nine sector SPDRs.

So, first off, this is in response to a question posed by one Robert Wages on the R-SIG-Finance mailing list. While there are many individuals out there with a plethora of questions (many of which can be found to be demonstrated on this blog already), occasionally, there will be an industry veteran, a PhD statistics student from Stanford, or other very intelligent individual that will ask a question on a topic that I haven’t yet touched on this blog, which will prompt a post to demonstrate another technical aspect found in R. This is one of those times.

So, this demonstration will be about computing turnover in returns space using the PerformanceAnalytics package. Simply, outside of the PortfolioAnalytics package, PerformanceAnalytics with its Return.Portfolio function is the go-to R package for portfolio management simulations, as it can take a set of weights, a set of returns, and generate a set of portfolio returns for analysis with the rest of PerformanceAnalytics’s functions.

Again, the strategy is this: take the 9 three-letter sector SPDRs (since there are four-letter ETFs now), and at the end of every month, if the adjusted price is above its 200-day moving average, invest into it. Normalize across all invested sectors (that is, 1/9th if invested into all 9, 100% into 1 if only 1 invested into, 100% cash, denoted with a zero return vector, if no sectors are invested into). It’s a simple, toy strategy, as the strategy isn’t the point of the demonstration.

Here’s the basic setup code:

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

symbols <- c("XLF", "XLK", "XLU", "XLE", "XLP", "XLF", "XLB", "XLV", "XLY")
getSymbols(symbols, src='yahoo', from = '1990-01-01-01')
prices <- list()
for(i in 1:length(symbols)) {
  tmp <- Ad(get(symbols[[i]]))
  prices[[i]] <- tmp
}
prices <- do.call(cbind, prices)

# Our signal is a simple adjusted price over 200 day SMA
signal <- prices > xts(apply(prices, 2, SMA, n = 200), order.by=index(prices))

# equal weight all assets with price above SMA200
returns <- Return.calculate(prices)
weights <- signal/(rowSums(signal)+1e-16)

# With Return.portfolio, need all weights to sum to 1
weights$zeroes <- 1 - rowSums(weights)
returns$zeroes <- 0

monthlyWeights <- na.omit(weights[endpoints(weights, on = 'months'),])
weights <- na.omit(weights)
returns <- na.omit(returns)

So, get the SPDRs, put them together, compute their returns, generate the signal, and create the zero vector, since Return.Portfolio treats weights less than 1 as a withdrawal, and weights above 1 as the addition of more capital (big FYI here).

Now, here’s how to compute turnover:

out <- Return.portfolio(R = returns, weights = monthlyWeights, verbose = TRUE)
beginWeights <- out$BOP.Weight
endWeights <- out$EOP.Weight
txns <- beginWeights - lag(endWeights)
monthlyTO <- xts(rowSums(abs(txns[,1:9])), order.by=index(txns))
plot(monthlyTO)

So, the trick is this: when you call Return.portfolio, use the verbose = TRUE option. This creates several objects, among them returns, BOP.Weight, and EOP.Weight. These stand for Beginning Of Period Weight, and End Of Period Weight.

The way that turnover is computed is simply the difference between how the day’s return moves the allocated portfolio from its previous ending point to where that portfolio actually stands at the beginning of next period. That is, the end of period weight is the beginning of period drift after taking into account the day’s drift/return for that asset. The new beginning of period weight is the end of period weight plus any transacting that would have been done. Thus, in order to find the actual transactions (or turnover), one subtracts the previous end of period weight from the beginning of period weight.

This is what such transactions look like for this strategy.

Something we can do with such data is take a one-year rolling turnover, accomplished with the following code:

yearlyTO <- runSum(monthlyTO, 252)
plot(yearlyTO, main = "running one year turnover")

It looks like this:

This essentially means that one year’s worth of two-way turnover (that is, if selling an entirely invested portfolio is 100% turnover, and buying an entirely new set of assets is another 100%, then two-way turnover is 200%) is around 800% at maximum. That may be pretty high for some people.

Now, here’s the application when you penalize transaction costs at 20 basis points per percentage point traded (that is, it costs 20 cents to transact $100).

txnCosts <- monthlyTO * -.0020
retsWithTxnCosts <- out$returns + txnCosts
compare <- na.omit(cbind(out$returns, retsWithTxnCosts))
colnames(compare) <- c("NoTxnCosts", "TxnCosts20BPs")
charts.PerformanceSummary(compare)
table.AnnualizedReturns(compare)

And the result:


                          NoTxnCosts TxnCosts20BPs
Annualized Return             0.0587        0.0489
Annualized Std Dev            0.1554        0.1553
Annualized Sharpe (Rf=0%)     0.3781        0.3149

So, at 20 basis points on transaction costs, that takes about one percent in returns per year out of this (admittedly, terrible) strategy. This is far from negligible.

So, that is how you actually compute turnover and transaction costs. In this case, the transaction cost model was very simple. However, given that Return.portfolio returns transactions at the individual asset level, one could get as complex as they would like with modeling the transaction costs.

Thanks for reading.

NOTE: I will be giving a lightning talk at R/Finance, so for those attending, you’ll be able to find me there.

Create Amazing Looking Backtests With This One Wrong–I Mean Weird–Trick! (And Some Troubling Logical Invest Results)

This post will outline an easy-to-make mistake in writing vectorized backtests–namely in using a signal obtained at the end of a period to enter (or exit) a position in that same period. The difference in results one obtains is massive.

Today, I saw two separate posts from Alpha Architect and Mike Harris both referencing a paper by Valeriy Zakamulin on the fact that some previous trend-following research by Glabadanidis was done with shoddy results, and that Glabadanidis’s results were only reproducible through instituting lookahead bias.

The following code shows how to reproduce this lookahead bias.

First, the setup of a basic moving average strategy on the S&P 500 index from as far back as Yahoo data will provide.

require(quantmod)
require(xts)
require(TTR)
require(PerformanceAnalytics)

getSymbols('^GSPC', src='yahoo', from = '1900-01-01')
monthlyGSPC <- Ad(GSPC)[endpoints(GSPC, on = 'months')]

# change this line for signal lookback
movAvg <- SMA(monthlyGSPC, 10)

signal <- monthlyGSPC > movAvg
gspcRets <- Return.calculate(monthlyGSPC)

And here is how to institute the lookahead bias.

lookahead <- signal * gspcRets
correct <- lag(signal) * gspcRets

These are the “results”:

compare <- na.omit(cbind(gspcRets, lookahead, correct))
colnames(compare) <- c("S&P 500", "Lookahead", "Correct")
charts.PerformanceSummary(compare)
rbind(table.AnnualizedReturns(compare), maxDrawdown(compare), CalmarRatio(compare))
logRets <- log(cumprod(1+compare))
chart.TimeSeries(logRets, legend.loc='topleft')

Of course, this equity curve is of no use, so here’s one in log scale.

As can be seen, lookahead bias makes a massive difference.

Here are the numerical results:

                            S&P 500  Lookahead   Correct
Annualized Return         0.0740000 0.15550000 0.0695000
Annualized Std Dev        0.1441000 0.09800000 0.1050000
Annualized Sharpe (Rf=0%) 0.5133000 1.58670000 0.6623000
Worst Drawdown            0.5255586 0.08729914 0.2699789
Calmar Ratio              0.1407286 1.78119192 0.2575219

Again, absolutely ridiculous.

Note that when using Return.Portfolio (the function in PerformanceAnalytics), that package will automatically give you the next period’s return, instead of the current one, for your weights. However, for those writing “simple” backtests that can be quickly done using vectorized operations, an off-by-one error can make all the difference between a backtest in the realm of reasonable, and pure nonsense. However, should one wish to test for said nonsense when faced with impossible-to-replicate results, the mechanics demonstrated above are the way to do it.

Now, onto other news: I’d like to thank Gerald M for staying on top of one of the Logical Invest strategies–namely, their simple global market rotation strategy outlined in an article from an earlier blog post.

Up until March 2015 (the date of the blog post), the strategy had performed well. However, after said date?

It has been a complete disaster, which, in hindsight, was evident when I passed it through the hypothesis-driven development framework process I wrote about earlier.

So, while there has been a great deal written about not simply throwing away a strategy because of short-term underperformance, and that anomalies such as momentum and value exist because of career risk due to said short-term underperformance, it’s never a good thing when a strategy creates historically large losses, particularly after being published in such a humble corner of the quantitative financial world.

In any case, this was a post demonstrating some mechanics, and an update on a strategy I blogged about not too long ago.

Thanks for reading.

NOTE: I am always interested in hearing about new opportunities which may benefit from my expertise, and am always happy to network. You can find my LinkedIn profile here.

Are R^2s Useful In Finance? Hypothesis-Driven Development In Reverse

This post will shed light on the values of R^2s behind two rather simplistic strategies — the simple 10 month SMA, and its relative, the 10 month momentum (which is simply a difference of SMAs, as Alpha Architect showed in their book DIY Financial Advisor.

Not too long ago, a friend of mine named Josh asked me a question regarding R^2s in finance. He’s finishing up his PhD in statistics at Stanford, so when people like that ask me questions, I’d like to answer them. His assertion is that in some instances, models that have less than perfect predictive power (EG R^2s of .4, for instance), can still deliver very promising predictions, and that if someone were to have a financial model that was able to explain 40% of the variance of returns, they could happily retire with that model making them very wealthy. Indeed, .4 is a very optimistic outlook (to put it lightly), as this post will show.

In order to illustrate this example, I took two “staple” strategies — buy SPY when its closing monthly price is above its ten month simple moving average, and when its ten month momentum (basically the difference of a ten month moving average and its lag) is positive. While these models are simplistic, they are ubiquitously talked about, and many momentum strategies are an improvement upon these baseline, “out-of-the-box” strategies.

Here’s the code to do that:

require(xts)
require(quantmod)
require(PerformanceAnalytics)
require(TTR)

getSymbols('SPY', from = '1990-01-01', src = 'yahoo')
adjustedPrices <- Ad(SPY)
monthlyAdj <- to.monthly(adjustedPrices, OHLC=TRUE)

spySMA <- SMA(Cl(monthlyAdj), 10)
spyROC <- ROC(Cl(monthlyAdj), 10)
spyRets <- Return.calculate(Cl(monthlyAdj))

smaRatio <- Cl(monthlyAdj)/spySMA - 1
smaSig <- smaRatio > 0
rocSig <- spyROC > 0

smaRets <- lag(smaSig) * spyRets
rocRets <- lag(rocSig) * spyRets

And here are the results:

strats <- na.omit(cbind(smaRets, rocRets, spyRets))
colnames(strats) <- c("SMA10", "MOM10", "BuyHold")
charts.PerformanceSummary(strats, main = "strategies")
rbind(table.AnnualizedReturns(strats), maxDrawdown(strats), CalmarRatio(strats))

                              SMA10     MOM10   BuyHold
Annualized Return         0.0975000 0.1039000 0.0893000
Annualized Std Dev        0.1043000 0.1080000 0.1479000
Annualized Sharpe (Rf=0%) 0.9346000 0.9616000 0.6035000
Worst Drawdown            0.1663487 0.1656176 0.5078482
Calmar Ratio              0.5860332 0.6270657 0.1757849

In short, the SMA10 and the 10-month momentum (aka ROC 10 aka MOM10) both handily outperform the buy and hold, not only in absolute returns, but especially in risk-adjusted returns (Sharpe and Calmar ratios). Again, simplistic analysis, and many models get much more sophisticated than this, but once again, simple, illustrative example using two strategies that outperform a benchmark (over the long term, anyway).

Now, the question is, what was the R^2 of these models? To answer this, I took a rolling five-year window that essentially asked: how well did these quantities (the ratio between the closing price and the moving average – 1, or the ten month momentum) predict the next month’s returns? That is, what proportion of the variance is explained through the monthly returns regressed against the previous month’s signals in numerical form (perhaps not the best framing, as the signal is binary as opposed to continuous which is what is being regressed, but let’s set that aside, again, for the sake of illustration).

Here’s the code to generate the answer.

predictorsAndPredicted <- na.omit(cbind(lag(smaRatio), lag(spyROC), spyRets))
R2s <- list()
for(i in 1:(nrow(predictorsAndPredicted)-59))  { #rolling five-year regression
  subset <- predictorsAndPredicted[i:(i+59),]
  smaLM <- lm(subset[,3]~subset[,1])
  smaR2 <- summary(smaLM)$r.squared
  rocLM <- lm(subset[,3]~subset[,2])
  rocR2 <- summary(rocLM)$r.squared
  R2row <- xts(cbind(smaR2, rocR2), order.by=last(index(subset)))
  R2s[[i]] <- R2row
}
R2s <- do.call(rbind, R2s)
par(mfrow=c(1,1))
colnames(R2s) <- c("SMA", "Momentum")
chart.TimeSeries(R2s, main = "R2s", legend.loc = 'topleft')

And the answer, in pictorial form:

In short, even in the best case scenarios, namely, crises which provide momentum/trend-following/call it what you will its raison d’etre, that is, its risk management appeal, the proportion of variance explained by the actual signal quantities was very small. In the best of times, around 20%. But then again, think about what the R^2 value actually is–it’s the percentage of variance explained by a predictor. If a small set of signals (let alone one) was able to explain the majority of the change in the returns of the S&P 500, or even a not-insignificant portion, such a person would stand to become very wealthy. More to the point, given that two strategies that handily outperform the market have R^2s that are exceptionally low for extended periods of time, it goes to show that holding the R^2 up as some form of statistical holy grail certainly is incorrect in the general sense, and anyone who does so either is painting with too broad a brush, is creating disingenuous arguments, or should simply attempt to understand another field which may not work the way their intuition tells them.

Thanks for reading.

On The Relationship Between the SMA and Momentum

Happy new year. This post will be a quick one covering the relationship between the simple moving average and time series momentum. The implication is that one can potentially derive better time series momentum indicators than the classical one applied in so many papers.

Okay, so the main idea for this post is quite simple:

I’m sure we’re all familiar with classical momentum. That is, the price now compared to the price however long ago (3 months, 10 months, 12 months, etc.). E.G. P(now) – P(10)
And I’m sure everyone is familiar with the simple moving average indicator, as well. E.G. SMA(10).

Well, as it turns out, these two quantities are actually related.

It turns out, if instead of expressing momentum as the difference of two numbers, it is expressed as the sum of returns, it can be written (for a 10 month momentum) as:

MOM_10 = return of this month + return of last month + return of 2 months ago + … + return of 9 months ago, for a total of 10 months in our little example.

This can be written as MOM_10 = (P(0) – P(1)) + (P(1) – P(2)) + … + (P(9) – P(10)). (Each difference within parentheses denotes one month’s worth of returns.)

Which can then be rewritten by associative arithmetic as: (P(0) + P(1) + … + P(9)) – (P(1) + P(2) + … + P(10)).

In other words, momentum — aka the difference between two prices, can be rewritten as the difference between two cumulative sums of prices. And what is a simple moving average? Simply a cumulative sum of prices divided by however many prices summed over.

Here’s some R code to demonstrate.

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

getSymbols('SPY', from = '1990-01-01')
monthlySPY <- Ad(SPY)[endpoints(SPY, on = 'months')]
monthlySPYrets <- Return.calculate(monthlySPY)
#dividing by 10 since that's the moving average period for comparison
signalTSMOM <- (monthlySPY - lag(monthlySPY, 10))/10 
signalDiffMA <- diff(SMA(monthlySPY, 10))

# rounding just 
sum(round(signalTSMOM, 3)==round(signalDiffMA, 3), na.rm=TRUE)

With the resulting number of times these two signals are equal:

[1] 267

In short, every time.

Now, what exactly is the punchline of this little example? Here’s the punchline:

The simple moving average is…fairly simplistic as far as filters go. It works as a pedagogical example, but it has some well known weaknesses regarding lag, windowing effects, and so on.

Here’s a toy example how one can get a different momentum signal by changing the filter.

toyStrat <- monthlySPYrets * lag(signalTSMOM > 0)

emaSignal <- diff(EMA(monthlySPY, 10))
emaStrat <- monthlySPYrets * lag(emaSignal > 0)

comparison <- cbind(toyStrat, emaStrat)
colnames(comparison) <- c("DiffSMA10", "DiffEMA10")
charts.PerformanceSummary(comparison)
table.AnnualizedReturns(comparison)

With the following results:

                          DiffSMA10 DiffEMA10
Annualized Return            0.1051    0.0937
Annualized Std Dev           0.1086    0.1076
Annualized Sharpe (Rf=0%)    0.9680    0.8706

While the difference of EMA10 strategy didn’t do better than the difference of SMA10 (aka standard 10-month momentum), that’s not the point. The point is that the momentum signal is derived from a simple moving average filter, and that by using a different filter, one can still use a momentum type of strategy.

Or, put differently, the main/general takeaway here is that momentum is the slope of a filter, and one can compute momentum in an infinite number of ways depending on the filter used, and can come up with a myriad of different momentum strategies.

Thanks for reading.

NOTE: I am currently contracting in Chicago, and am always open to networking. Contact me at my email at ilya.kipnis@gmail.com or find me on my LinkedIn here.