“Matching” portfolios is a technique for generating a reasonable benchmark for
determining the relative performance of a specific equity portfolio and is basedon the work in Ho et al. (2005a). Consider the simplest case of a long-onlymutual fund that has returned 10% in the last year. Has the portfolio donewell? If the average stock in the universe has gone up 50% then, obviously,the portfolio has done poorly. If, on the other hand, the average stock hasgone down 25%, then the portfolio has done remarkably well. In other words,it is impossible to evaluate the performance of a portfolio without consideringthe hypothetical performance of other possible portfolios. In this case, we arecomparing the performance of our portfolio to that of a hypothetical portfoliothat is equal-weighted all the stocks in the universe. But, depending on thecharacteristics of our mutual fund, this may not be a reasonable benchmark.1
Matching portfolios provide a benchmark which matches the characteristics
— sector exposures, capitalization biases, position sizes — of the target port-folio. The portfolio package provides the matching method as a means ofcomputing a matching portfolio. In this article we describe the intuition behindmatching in general, frame a real-world problem in which computing a portfo-lio benchmark is difficult, and show how the matching facility of the portfoliopackage can be used to solve this problem.
To focus ideas, let’s examine a specific portfolio formed on December 31, 2004.
Assay Research2 is a forensic accounting that provides “in-depth insight intofinancial statements, accounting practices and policies, and quality-of-earningsof publicly-traded companies.” Assay maintains a “Focus List” of companies forwhich its concerns are “heightened.” Although Assay does not provide buy/sellrecommendations, most if its customers would expect the stocks on its FocusList to perform poorly going forward.
1More information on the performance measurement problem and possible solutions can
On December 31, 2004, there were 33 companies on Assay’s Focus List. The
list of companies, along with data on a total of 4,000 stocks that were tradingat that time, are available as part of the R portfolio package.
> data(assay)> assay[c(1407, 1873, 1058, 2453, 1833, 1390), c("id", "symbol",+
"name", "country", "currency", "price", "sector", "liq",
"on.fl", "ret.0.3.m", "ret.0.6.m")]
❼ id is an identifier for each security, generally a CUSIP for companies
traded on US exchanges and a SEDOL for companies traded elsewhere.
❼ symbol is a human-readable identifier that is generally the ticker that the
security trades under in its home market; exchange specific informationis sometime appended to it. For example, Newcrest Mining trades underthe ticker “NCM” in Australia, indicated by the “AU” suffix.
❼ name is the name of the company. We will generally use the terms “com-
pany” and “security” interchangeably even though a “company” is a singlelegal entity which often has several different types of securities associatedwith it. In this example, we are only examining the single primary equitysecurity for each company.
❼ country is the ISO code for the country in which the security is traded.
Note that this is not necessarily the same as the country in which the com-pany is headquartered or incorporated. For example, Garmin (GRMN)trades in the US but is incorporated in the Cayman Islands.3
❼ currency is the ISO code for the currency in which the security trades.
❼ price is the latest closing price for the security as of December 31, 2004.
(Not all securities traded on that date.)
❼ sector is the economic sector in which a majority of the company’s busi-
ness takes place. There are 10 sectors in this data including Communica-tions, Cyclicals, Energy, Financials and Technology.
❼ liq is a measure of the typical daily dollar volume of trading in the secu-
rity. We normalized it to be N (0, 1).
❼ assay is a TRUE/FALSE indicator of whether or not the security was on
the Assay Focus List on December 31, 2004. Thirty three companies wereon the list at that time.
❼ ret.0.3.m and ret.0.6.m are the three and six month, respectively, re-
turns for each security, including dividends.
There are no missing observations. The universe of 4,000 companies consists
of large companies and all the AFL stocks, and is restricted to companies thattrade on exchanges in developed markets. For example, we include Japan butnot South Korea, Austria but not Croatia.
Consider a portfolio formed by taking equal-weighted short positions in eachof the Assay Focus List stocks and focusing in the returns for the first threemonths, through March 31, 2005.
> assay$assay.wt <- ifelse(assay$on.fl, -1, NA)> p <- new("portfolioBasic", name = "AFL Portfolio", instant = as.Date("2004-12-31"),+
data = assay, id.var = "symbol", in.var = "assay.wt", type = "relative",
size = "all", ret.var = "ret.0.3.m")
This short-only portfolio returns 7.64% because the average Assay stock
fell in price by this amount during the first quarter of 2005. Making 7% inthree months is rarely a bad thing, but whether or not this counts as goodperformance depends on how other stocks in the universe performed duringthis time period. If the universe consistently outperforms the AFL portfolio,then we could achieve better returns by shorting stocks randomly selected fromthe universe. We might question the utility of subscribing to Assay’s focus listexamine the opportunity cost of such a subscription.
A simple analysis suggests that the AFL portfolio significantly outperforms
the universe. The average stock in the universe of 4,000 was up 1.4%, and theAFL portfolio returned 7.64%. If we shorted 33 randomly selected stocks fromthe universe, we would expect the return to be −1.4%. If we consider this to bea reasonable benchmark, then the AFL portfolio outperformed the universe by9%.
But is the Assay portfolio similar to the rest of the universe? To some extent,
it is. All the securities in the universe are relatively larger capitalization, liquidequities traded on developed market stock exchanges. But the AFL portfoliois also very different since all of its components are US stocks. Is it fair to usea benchmark with international stocks as a comparison for a US-only portfoliolike APL? Probably not.
Another major difference between the AFL portfolio and the universe is that
the AFL is concentrated in a limited number of sectors.
> exposure(p, exp.var = "sector")
The analysts at Assay do not place companies from sectors like Financials,
Energy and Utilities on to their Focus List because they lack the industry knowl-edge to evaluate the financial statements for such companies. Any benchmarkwhich includes securities from such sectors is inappropriate for judging the skillof Assay. After all, if Energy stocks did very well in the first quarter of 2005,a benchmark short portfolio which included them would do very poorly. Assaywould hardly deserve “credit” for this since it is not claiming that stocks in sec-tors which it does not cover will do well. It makes no predictions about howenergy stocks, on average, will do.4
Even if we were to eliminate from the universe both non-US stocks and stocks
in sectors that Assay does not cover, a variety of incompatibilities between theAFL portfolio and possible benchmarks would remain. For example, the averageAssay stock has a liquidity of over 0.5, more than 1/2 a standard deviationgreater than the universe as a whole. The Assay portfolio has almost 40% of itsholdings in Technology stocks, but the universe is only 7% Technology. Whatwe need is a benchmark portfolio that looks like, that “matches,” the Assayportfolio in terms of variables like country, sector, liquidity and so on but whichis otherwise randomly selected from non-Assay stocks in the universe.
The solution to the problem of constructing a benchmark for a portfolio like thatderived from the Assay Focus List is to create a “matching” portfolio, a portfoliowith characteristics as similar as possible to the AFL portfolio without beingidentical to it. For the AFL benchmark, we would like a portfolio with similarcountry and sector breakdown as well as a similar distribution of liquidity. If theAFL portfolio does much better (because the Assay stocks do very poorly) thanthis benchmark, we have evidence that Assay has in fact identified companieswith significant problems. It isn’t just a matter of the AFL doing well relativeto the overall universe because, for example, Energy stocks have risen so muchand the AFL isn’t short any energy stocks.
One way to think about the assessment of the AFL is to consider an analogyto a randomized scientific experiment. Recall that a randomized experiment ortrial begins by selecting a group of subjects to work with. From this population,a group of subjects is randomly selected and to whom is applied the treatment.
The rest of the group receives the control. Since the treatment was appliedrandomly, any differences in the outcome should be the result of the treatmentrather than be caused by systematic differences between the treatment andcontrol groups (Rubin (1974)).
Consider a group of 4000 individuals with a headache. We want to determine
if the treatment of “taking an aspirin” relieves the headache better than the
4Energy was the best performing sector in Q1 2005, with the average stock up over 15%.
control of “taking a placebo.” If we only have, say, 33 aspirins to use for thetest, we should select 33 people at random from the group of 4000 and givethem each an aspirin. The other individuals get the placebo. Afterward, wecan see how the treatment group (having taken the aspirin) compares to thecontrol group (who took the placebo). If, for example, the reported headachepain of the treated group is much lower than that of the control group, we mightconclude that aspirin works.
Imagine that we have a “treatment” which we believe causes stock prices to
fall. We want to test to see if this treatment actually has that effect. The bestway to do so is to run a randomized trial. Select, say, 33 stocks at random fromthe total universe of 4,000 stocks. Apply the treatment to those 33 stocks butnot to the other 3,967. If the price of the 33 treated stocks falls more (or risesless) then the prices of the 3,967 control stocks, we might conclude that thetreatment works.
The problem arises, for both tests of aspirin and tests of Assay, when we
can no longer do random assignment. Imagine that, instead of assigning as-pirin/placebo randomly, 33 of the 4,000 people in our group volunteer to takeit. The problem is that these 33 might be very different from the others. Theymight be all men or mostly old or very fat. Unless we somehow “control” forthis problem, we will not be able to conclude that the treatment, the aspirin,actually caused the decrease in headache pain. Instead, it could just be thatheadaches go away more quickly in old, fat men. Instead of comparing our33 volunteers to everyone else, we need to compare them to a subgroup that
“matches” them. If they are mostly male, old and fat, we should select a con-
trol group of people who took the placebo that is equally male, old and fat. Ifaspirin-takers report less pain in this group, then we might reasonably concludethat — at least within this subpart of the population — aspirin works.
The same intuition lies behind the construction of a matching portfolio. We
need a portfolio that looks like the AFL portfolio in terms of country, sector andliquidity. If the only difference between the AFL and matching portfolio is thatthe former consists of stocks that Assay has “heightened” concerns about whilethe latter consists of similar stocks without such concerns, we may concludethat any differences in their subsequent performance is due to the treatmentreceived.
Now, of course, placement on the Focus List does not cause a stock decline
in the same way that taking an aspirin causes, by hypothesis, headache pain todecrease. The price of a stock does not decrease as an Assay employee typesthe focus list, but the price of stocks on the focus list may decrease when Assaycustomers receive the list and sell or short the stocks on the focus list. Whetherinformation from Assay or conditions internal to a Focus List company causeprice decline is not important. What matters is that one can enter a shortposition before the price declines.
Ultimately, the concerns should be first,
whether or not Assay can predict negative future performance and second, ifone can enter a position before this information is incorporated into the price.
Note: Due to internal changes to the package there may be some inconsistenciesin this and subsequent sections of the document.
We want a matching portfolio which is as similar as possible to the AFL
portfolio but which does not include the same stocks. The matching methodin the portfolio package provides this functionality, with a little help fromthe MatchIt package (Ho et al. (2005b)). Calling this method on a portfo-lio object, p, returns a portfolio of different stocks that share attributes withthe stocks in p. These positions most closely resemble those in p along thedimensions specified in the covariates argument:
> p.m <- matching(p, covariates = c("country", "sector", "liq"))
Original portfolio return: 0.076, with 0 NAs.
Original return relative to matches: 0.029
Original portfolio outperformed 100% of matches.
A quick inspection of the new portfolio’s positions confirms that none of the
> all(!p.m@matches[, 1] %in% p@weights$id)
Having created a matching portfolio using country, sector, and liquidity as
covariates, we would expect p.m and p to have similar exposures to these vari-ables. First, all of the positions in p.m have country USA. This makes sensebecause all AFL stocks, and thus all stocks in p, are US stocks. More inter-estingly, however, the sector exposures of p.m are quite similar to the sectorexposures of p:
> exposure(p, exp.var = "sector")
> exposure(p.m, exp.var = "sector")
The only difference sector-wise between the two portfolios is that p.m has
one more stock in Staples and one less stock in Communications. Finally, theexposure of p to the numeric variable liquidity:
> exposure(p, exp.var = "liq")
is quite close to p.m’s exposure to liquidity:
> exposure(p.m, exp.var = "liq")
Since we matched using more than one covariate, we shouldn’t expect the
matching portfolio’s exposures to the covariates to be exactly the same as thoseof the original portfolio. However, given a large enough universe upon which thematching method can draw, we expect those exposures to be reasonably close.
Now that we’ve run matching on our AFL portfolio and calculated a match,we can examine how the AFL portfolio performed relative to the match. Recallthat the AFL portfolio returned 7.64% during Q1 2005, and that members ofour 4000 stock universe were up 1.4% on average during this period. The AFLportfolio, then, outperformed a randomly selected portfolio of 33 stocks fromour universe by 9%.
The match, however, performed far better than a randomly selected portfo-
The match returned 4.78% during Q1 2005, a far better return than the
-1.4% of a randomly selected portfolio. If we then use the matching portfolioas the AFL portfolio’s benchmark, the AFL portfolio had an excess return of2.86%. While this excess return is lower than the 9% we would calculate using arandomly selected benchmark, it more accurately reflects the excess return forwhich Assay should receive “credit”.
For example, while the average stock in our universe returned 1.4%, the aver-
age US stock returned -3.6%. We could have simply shorted a random collectionof 33 US stocks and walked away with 3.6%. Futhermore, stocks in the Technol-ogy and Staples sectors on average returned -4.7% and -4.5%, respectively. Thematching portfolio, like the AFL portfolio, benefits from having over two-thirdsof its positions in these sectors. Finally, stocks with liquidity values close to0.5, the average liquidity value of AFL stocks, have the same or slightly poorerreturns than the average stock in the universe. The AFL portfolio does notperform better or worse than a random portfolio due to its exposure to higherliquidity stocks.
It is clear that the matching portfolio is a better benchmark for the AFL
portfolio than a randomly selected portfolio, particularly due to the poor averagereturn of US stocks and stocks in the Technology and Staples sectors relative tothe entire universe.
Patrick Burns. Performance measurement via random portfolios, 2004. URL
Daniel Ho, Kosuke Imai, Gary King, and Elizabeth Stuart.
nonparametric preprocessing for parametric causal inference, 2005a. URLhttp://gking.harvard.edu/files/matchp.pdf.
Daniel Ho, Kosuke Imai, Gary King, and Elizabeth Stuart. MatchIt: Non-
parametric Preprocessing for Parametric Casual Inference, 2005b.
http://gking.harvard.edu/matchit. R package version 2.2-5.
Donald B. Rubin. Estimating causal effects of treatments in randomized and
nonrandomized studies. Journal of Educational Psychology, 66:688–701, 1974.
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