An Introoduction 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 <- CVaR(edhec, portfolio_method = "component")

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.

> tmp2 <- CVaR(edhec, portfolio_method = "component", weights = c(rep(.1, 10), rep(0,3)))
> 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.

> tmp3 <- CVaR(edhec, portfolio_method = "component", weights = c(.2, rep(.1, 9), rep(0,3)))
> 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.

A Return.Portfolio Wrapper to Automate Harry Long Seeking Alpha Backtests

This post will cover a function to simplify creating Harry Long type rebalancing strategies from SeekingAlpha for interested readers. As Harry Long has stated, most, if not all of his strategies are more for demonstrative purposes rather than actual recommended investments.

So, since Harry Long has been posting some more articles on Seeknig Alpha, I’ve had a reader or two ask me to analyze his strategies (again). Instead of doing that, however, I’ll simply put this tool here, which is a wrapper that automates the acquisition of data and simulates portfolio rebalancing with one line of code.

Here’s the tool.

require(quantmod)
require(PerformanceAnalytics)
require(downloader)

LongSeeker <- function(symbols, weights, rebalance_on = "years", 
                       displayStats = TRUE, outputReturns = FALSE) {
  getSymbols(symbols, src='yahoo', from = '1990-01-01')
  prices <- list()
  for(i in 1:length(symbols)) {
    if(symbols[i] == "ZIV") {
      download("https://www.dropbox.com/s/jk3ortdyru4sg4n/ZIVlong.TXT", destfile="ziv.txt")
      ziv <- xts(read.zoo("ziv.txt", header=TRUE, sep=",", format="%Y-%m-%d"))
      prices[[i]] <- Cl(ziv)
    } else if (symbols[i] == "VXX") {
      download("https://dl.dropboxusercontent.com/s/950x55x7jtm9x2q/VXXlong.TXT", 
               destfile="vxx.txt")
      vxx <- xts(read.zoo("vxx.txt", header=TRUE, sep=",", format="%Y-%m-%d"))
      prices[[i]] <- Cl(vxx)
    }
    else {
      prices[[i]] <- Ad(get(symbols[i]))
    }
  }
  prices <- do.call(cbind, prices)
  prices <- na.locf(prices)
  returns <- na.omit(Return.calculate(prices))
  
  returns$zeroes <- 0
  weights <- c(weights, 1-sum(weights))
  stratReturns <- Return.portfolio(R = returns, weights = weights, rebalance_on = rebalance_on)
  
  if(displayStats) {
    stats <- rbind(table.AnnualizedReturns(stratReturns), maxDrawdown(stratReturns), CalmarRatio(stratReturns))
    rownames(stats)[4] <- "Max Drawdown"
    print(stats)
    charts.PerformanceSummary(stratReturns)
  }
  
  if(outputReturns) {
    return(stratReturns)
  }
} 

It fetches the data for you (usually from Yahoo, but a big thank you to Mr. Helumth Vollmeier in the case of ZIV and VXX), and has the option of either simply displaying an equity curve and some statistics (CAGR, annualized standard dev, Sharpe, max Drawdown, Calmar), or giving you the return stream as an output if you wish to do more analysis in R.

Here’s an example of simply getting the statistics, with an 80% XLP/SPLV (they’re more or less interchangeable) and 20% TMF (aka 60% TLT, so an 80/60 portfolio), from one of Harry Long’s articles.

LongSeeker(c("XLP", "TLT"), c(.8, .6))

Statistics:


                          portfolio.returns
Annualized Return                 0.1321000
Annualized Std Dev                0.1122000
Annualized Sharpe (Rf=0%)         1.1782000
Max Drawdown                      0.2330366
Calmar Ratio                      0.5670285

Equity curve:

Nothing out of the ordinary of what we might expect from a balanced equity/bonds portfolio. Generally does well, has its largest drawdown in the financial crisis, and some other bumps in the road, but overall, I’d think a fairly vanilla “set it and forget it” sort of thing.

And here would be the way to get the stream of individual daily returns, assuming you wanted to rebalance these two instruments weekly, instead of yearly (as is the default).

tmp <- LongSeeker(c("XLP", "TLT"), c(.8, .6), rebalance_on="weeks",
                    displayStats = FALSE, outputReturns = TRUE)

And now let’s get some statistics.

table.AnnualizedReturns(tmp)
maxDrawdown(tmp)
CalmarRatio(tmp)

Which give:

> table.AnnualizedReturns(tmp)
                          portfolio.returns
Annualized Return                    0.1328
Annualized Std Dev                   0.1137
Annualized Sharpe (Rf=0%)            1.1681
> maxDrawdown(tmp)
[1] 0.2216417
> CalmarRatio(tmp)
             portfolio.returns
Calmar Ratio         0.5990087

Turns out, moving the rebalancing from annually to weekly didn’t have much of an effect here (besides give a bunch of money to your broker, if you factored in transaction costs, which this doesn’t).

So, that’s how this tool works. The results, of course, begin from the latest instrument’s inception. The trick, in my opinion, is to try and find proxy substitutes with longer histories for newer ETFs that are simply leveraged ETFs, such as using a 60% weight in TLT with an 80% weight in XLP instead of a 20% weight in TMF with 80% allocation in SPLV.

For instance, here are some proxies:

SPXL = XLP
SPXL/UPRO = SPY * 3
TMF = TLT * 3

That said, I’ve worked with Harry Long before, and he develops more sophisticated strategies behind the scenes, so I’d recommend that SeekingAlpha readers take his publicly released strategies as concept demonstrations, as opposed to fully-fledged investment ideas, and contact Mr. Long himself about more customized, private solutions for investment institutions if you are so interested.

Thanks for reading.

NOTE: I am currently in the northeast. While I am currently contracting, I am interested in networking with individuals or firms with regards to potential collaboration opportunities.

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.

A Book Review of ReSolve Asset Management’s Adaptive Asset Allocation

This review will review the “Adaptive Asset Allocation: Dynamic Global Portfolios to Profit in Good Times – and Bad” book by the people at ReSolve Asset Management. Overall, this book is a definite must-read for those who have never been exposed to the ideas within it. However, when it comes to a solution that can be fully replicated, this book is lacking.

Okay, it’s been a while since I reviewed my last book, DIY Financial Advisor, from the awesome people at Alpha Architect. This book in my opinion, is set up in a similar sort of format.

This is the structure of the book, and my reviews along with it:

Part 1: Why in the heck you actually need to have a diversified portfolio, and why a diversified portfolio is a good thing. In a world in which there is so much emphasis put on single-security performance, this is certainly something that absolutely must be stated for those not familiar with portfolio theory. It highlights the example of two people–one from Abbott Labs, and one from Enron, who had so much of their savings concentrated in their company’s stock. Mr. Abbott got hit hard and changed his outlook on how to save for retirement, and Mr. Enron was never heard from again. Long story short: a diversified portfolio is good, and a properly diversified portfolio can offset one asset’s zigs with another asset’s zags. This is the key to establishing a stream of returns that will help meet financial goals. Basically, this is your common sense story (humans love being told stories) so as to motivate you to read the rest of the book. It does its job, though for someone like me, it’s more akin to a big “wait for it, wait for it…and there’s the reason why we should read on, as expected”.

Part 2: Something not often brought up in many corners of the quant world (because it’s real life boring stuff) is the importance not only of average returns, but *when* those returns are achieved. Namely, imagine your everyday saver. At the beginning of their careers, they’re taking home less salary and have less money in their retirement portfolio (or speculation portfolio, but the book uses retirement portfolio). As they get into middle age and closer to retirement, they have a lot more money in said retirement portfolio. Thus, strong returns are most vital when there is more cash available *to* the portfolio, and the difference between mediocre returns at the beginning and strong returns at the end of one’s working life as opposed to vice versa is *astronomical* and cannot be understated. Furthermore, once *in* retirement, strong returns in the early years matter far more than returns in the later years once money has been withdrawn out of the portfolio (though I’d hope that a portfolio’s returns can be so strong that one can simply “live off the interest”). Or, put more intuitively: when you have $10,000 in your portfolio, a 20% drawdown doesn’t exactly hurt because you can make more money and put more into your retirement account. But when you’re 62 and have $500,000 and suddenly lose 30% of everything, well, that’s massive. How much an investor wants to avoid such a scenario cannot be understated. Warren Buffett once said that if you can’t bear to lose 50% of everything, you shouldn’t be in stocks. I really like this part of the book because it shows just how dangerous the ideas of “a 50% drawdown is unavoidable” and other “stay invested for the long haul” refrains are. Essentially, this part of the book makes a resounding statement that any financial adviser keeping his or her clients invested in equities when they’re near retirement age is doing something not very advisable, to put it lightly. In my opinion, those who advise pension funds should especially keep this section of the book in mind, since for some people, the long-term may be coming to an end, and what matters is not only steady returns, but to make sure the strategy doesn’t fall off a cliff and destroy decades of hard-earned savings.

Part 3: This part is also one that is a very important read. First off, it lays out in clear terms that the long-term forward-looking valuations for equities are at rock bottom. That is, the expected forward 15-year returns are very low, using approximately 75 years of evidence. Currently, according to the book, equity valuations imply a *negative* 15-year forward return. However, one thing I *will* take issue with is that while forward-looking long-term returns for equities may be very low, if one believed this chart and only invested in the stock market when forecast 15-year returns were above the long term average, one would have missed out on both the 2003-2007 bull runs, *and* the one since 2009 that’s just about over. So, while the book makes a strong case for caution, readers should also take the chart with a grain of salt in my opinion. However, another aspect of portfolio construction that this book covers is how to construct a robust (assets for any economic environment) and coherent (asset classes balanced in number) universe for implementation with any asset allocation algorithm. I think this bears repeating: universe selection is an extremely important topic in the discussion of asset allocation, yet there is very little discussion about it. Most research/topics simply take some “conventional universe”, such as “all stocks on the NYSE”, or “all the stocks in the S&P 500”, or “the entire set of the 50-60 most liquid futures” without consideration for robustness and coherence. This book is the first source I’ve seen that actually puts this topic under a magnifying glass besides “finger in the air pick and choose”.

Part 4: and here’s where I level my main criticism at this book. For those that have read “Adaptive Asset Allocation: A Primer”, this section of the book is basically one giant copy and paste. It’s all one large buildup to “momentum rank + min-variance optimization”. All well and good, until there’s very little detail beyond the basics as to how the minimum variance portfolio was constructed. Namely, what exactly is the minimum variance algorithm in use? Is it one of the poor variants susceptible to numerical instability inherent in inverting sample covariance matrices? Or is it a heuristic like David Varadi’s minimum variance and minimum correlation algorithm? The one feeling I absolutely could not shake was that this book had a perfect opportunity to present a robust approach to minimum variance, and instead, it’s long on concept, short on details. While the theory of “maximize return for unit risk” is all well and good, the actual algorithm to implement that theory into practice is not trivial, with the solutions taught to undergrads and master’s students having some well-known weaknesses. On top of this, one thing that got hammered into my head in the past was that ranking *also* had a weakness at the inclusion/exclusion point. E.G. if, out of ten assets, the fifth asset had a momentum of say, 10.9%, and the sixth asset had a momentum of 10.8%, how are we so sure the fifth is so much better? And while I realize that this book was ultimately meant to be a primer, in my opinion, it would have been a no-objections five-star if there were an appendix that actually went into some detail on how to go from the simple concepts and included a small numerical example of some algorithms that may address the well-known weaknesses. This doesn’t mean Greek/mathematical jargon. Just an appendix that acknowledged that not every reader is someone only picking up his first or second book about systematic investing, and that some of us are familiar with the “whys” and are more interested in the “hows”. Furthermore, I’d really love to know where the authors of this book got their data to back-date some of these ETFs into the 90s.

Part 5: some more formal research on topics already covered in the rest of the book–namely a section about how many independent bets one can take as the number of assets grow, if I remember it correctly. Long story short? You *easily* get the most bang for your buck among disparate asset classes, such as treasuries of various duration, commodities, developed vs. emerging equities, and so on, as opposed to trying to pick among stocks in the same asset class (though there’s some potential for alpha there…just…a lot less than you imagine). So in case the idea of asset class selection, not stock selection wasn’t beaten into the reader’s head before this point, this part should do the trick. The other research paper is something I briefly skimmed over which went into more depth about volatility and retirement portfolios, though I felt that the book covered this topic earlier on to a sufficient degree by building up the intuition using very understandable scenarios.

So that’s the review of the book. Overall, it’s a very solid piece of writing, and as far as establishing the *why*, it does an absolutely superb job. For those that aren’t familiar with the concepts in this book, this is definitely a must-read, and ASAP.

However, for those familiar with most of the concepts and looking for a detailed “how” procedure, this book does not deliver as much as I would have liked. And I realize that while it’s a bad idea to publish secret sauce, I bought this book in the hope of being exposed to a new algorithm presented in the understandable and intuitive language that the rest of the book was written in, and was left wanting.

Still, that by no means diminishes the impact of the rest of the book. For those who are more likely to be its target audience, it’s a 5/5. For those that wanted some specifics, it still has its gem on universe construction.

Overall, I rate it a 4/5.

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.