Finding Single Name Representatives of S&P 500 Sectors Using Vine Copulas

According to data by Koyfin, the S&P 500 has gained 17.95% this year as of Friday 13th September. However,  four of the 11 S&P 500 sector ETFs have outperformed it namely XLU (utilities, return=26.05%), XLC (Communication Services, 20.20%), XLF (Financials, 19.29%) and XLP (Consumer Staples, 18.27%).

It is interesting to ask if it is possible to create a long-short portfolio of the sector ETFs that would significantly outperform the S&P; I haven’t found any research of that nature, but I will leave it for another day.

For now, I was wondering if single name proxies for the different sectors would outperform the index in the short-term or long-term, giving better expressions for sector-specific plays for investors of either or both horizons. Since sector ETFs are made up of several stocks, it is not farfetched to think that each sector has at least one stock that is strongly correlated with the sector ETF but moves more than the ETF to the upside or downside. If this is the case, then it be better to simply trade the single name rather than the ETF.

Methodology

I searched around for a good way to find out and I stumbled upon an excellent methodology using Vine Copulas by Claudia Czado in her book “Analyzing Dependent Data with Vine Copulas” where she did something similar for the DAX index.

The idea is to use vine copulas to find single name stocks that are proxies for the entire sector. This can be done by gathering the returns for all S&P 500 constituents, grouped by sector then fitting a Vine copula—specifically a C-Vine copula due to its ‘star’ structure (a ‘root’ asset connected to the other assets)—to each sector and find this root (Czado, 2019). The root is the stock such that all pairwise correlations, measured by the Kendall’s tau, between it and other stocks are the highest. A comparison of the root’s performance versus the corresponding sector ETF would sow if it is indeed a better short term and/or long-term play.

In the next section, I give a brief overview of how copulas and vine copulas work. I also share a downloadable mathematical treatise that serves as an introduction to copulas and Vine copulas.

To fit a vine copula, you first need to remove serial dependence for each stock’s returns by fitting a univariate GARCH model. I used GARCH (1,1)—which has been shown to be hard to beat. This removes the serial dependence of each stock’s returns and leaves only the dependence structure among the stocks. R has a very good Vine copula package. You can fork the repo with all the code used in this analysis here: Code and supplementary data on Github.

How Copulas Work

Copulas are a neat innovation, if used with caution! This is how they work. Consider a set of assets. I will use five energy stocks: APA, BKR, CVX, COP and CTRA, using their returns since the beginning of the year to 13th Friday Sept, here are their pair plots, return distributions and Kendall’s tau correlation coefficients.

I am using Kendall’s tau because it measure’s concordance (if A moves up so does B) and discordance (if A moves up, B moves down) and makes no assumptions about the data’s distribution. Pearson’s correlation assumes normality.

They energy stocks are all positively correlated with their Kendall’s tau values all well above 0. You can also see from their pairs plots that they have positive dependencies. With copulas, the idea is to isolate the dependency structure among the assets. Notice by the distribution plots in the diagonals that each asset has it’s own unique marginal distribution. Even if you were to fit Gaussian distributions, they would have different parameters.

It is possible to apply a transformation—the rank transformation—to each of the return series so that they preserve their dependence structure. To do this, take the smallest value in each series and give it the value 1/n, and keep going until you have n/n. This way, all the return series will have values between 0 and 1, i.e. they will be uniformly distributed. It so happens that what you have done is actually create an empirical CDF of each return series, and all CDFs are uniformly distributed. Here is how that looks like.

Notice that the diagonals are uniform distributions and that their Kendall’s tau values are the same—the dependence structure among the variables hasn’t changed after the transformation. The pairs plots also show this dependence structure.

A copula is a joint distribution of these marginals. A theorem called Sklar’s theorem proves that the copula of these uniform marginals is equal to a multivariate distribution of the original asset returns. What this means is that you can find a multivariate distribution for assets that follow different marginal distributions. It is useful in fields where you have to take into account factors that each have their own unique distribution, such as weather or asset returns.

Copulas also allow you to model the dependence structure among assets to account for dynamic dependencies—something which linear correlation cannot do. For example, you can model tail dependencies among different assets, which can be useful in modeling default probabilities for example.

Vine copulas extend this idea further. Using Bayes theorem, you can decompose a joint distribution into conditional distributions.

I can’t talk about copulas without talking about why they made a mess in the Sub-prime bubble. The problem is that the pricing of mortgage baskets was done using Gaussian copulas, basically a Multivariate Normal CDFs on uniform marginals, and this meant the copulas were symmetric and also assumed the tail risk of all the marginals was the same.

Editor’s note: To keep this post from being too long and to allow me to write in LaTeX, which Substack isn’t great at facilitating, I made a separate PDFs that explains in detail the mathematics behind copulas and Vine copulas. Download it below.

Single Name Representatives of S&P 500 Sectors

Energy Sector

Editor’s Note: The network plots show the first tree of the fitted C-Vine models for each sector. The ticker in the middle is the root and the edges are the Kendall’s tau of the root versus other stocks in the sector. The root is the stock with the highest Kendall tau’s with other stocks.

In the energy sector, the root is Occidental Petroleum (OXY) . There is a lot of fuss about this stock because Berkshire Hathaway have owned it since early 2022. However, it hasn’t been doing too well. It is down 12.6% YTD while XLE is up 3.3%. Last week, Barron’s published an article about how Berkshire Hathaway are losing a lot of money on this stock. However, according to Reuters, Berkshire bought 2.57 million additional shares in June this year and are rumored to be looking to buy more as prices drop. Last Friday, Goldman published a report that the stock is a long-term AI stock that will eventually rally, and there are other analysts who think it could rally as much as 36%.

While it is true that this stock has underperformed the sector. You can see from the cumulative returns chart above that up to April, it moved in sync with XLE and even though both the sector ETF and the stock dropped since then OXY dropped by more. This shows that you could have used the stock as a proxy for the ETF if you were bullish the sector in Feb and March, or bearish in January and since April for better returns.

The monthly returns shows that in Jan, XLE dropped by -1.6% but OXY dropped by -4.2%, in May, OXY plummeted by -5.7% while XLE dropped by -0.4%, and in the August and September, XLE dropped by -2.1% and -6.4% while OXY dropped by -6.5% and -10.6%. That gives four months when OXY plummeted by more than the sector ETF.

For three monthly OXY outperformed the sector ETF, but it is only in one of them where it was a directional proxy to the sector. OXY outperformed XLE in February gaining 5.1% versus the sector ETF’s 3.2% gain, in April OXY gained 1.8% while XLE fell by -0.9%, and in June, OXY gained 1.2% while XLE fell by -1.4%.

August saw XLE outperform the sector ETF gaining 2.2% versus OXY’s -3.6% drop.

In terms of risk-return characteristics, XLE is the safer bet with a CAGR of 3.8% and an annualized volatility of 17.3%. OXY has a CAGR of -19.3% and an annualized volatility of 20.9%.

Financial Sector

In financials, the root is Principal Financial Group (PFG). It has gained only 0.8% YTD and is underperforming the sector. Sentiment is extremely bearish on this stock as you can see from this article, but the analysts recommend a buy.

The monthly returns show that PFG outperformed XLE in March( 7.4% vs 4.7%), May (3.6% vs 3.1%) and September(0.8% vs -2.7%). Furthermore, in April and June when the sector ETF was down for the month, PFG was down by more. In March PFG dropped by -8.7% versus -4.3% for XLF, and in May PFG lost -3.6% versus XLF’s -0.9%. These would have been good months to use the proxy if you had the correct directional view on the sector. However, in Jan, Feb, July and August, XLF outperformed the sector proxy.

In terms of risk-reward characteristics, XLF is more attractive with a CAGR of 27.8% and an annualized vol of 12.6%. PFG has a CAGR of 8.0% and an annualized vol of 19.6%

Healthcare

In healthcare, the root is Agilent Technologies (A). The cumulative chart below shows that it is underperforming XLV. It suffered a loss of about 11.1% since May, however, it is up 21.1% in the past 52 weeks versus XLV’s 17.9% in the same period. There are mixed feelings about the stock in terms of sentiment as some think it is bullish and some think that it could suffer more losses.

The monthly returns chart shows that in Feb, March and July, the sector proxy outperformed XLV gaining 5.4%, 5.8% and 8.9% versus XLV’s 3.1%, 2.3% and 2.9%. Furthermore, in April and September when XLV dropped, A dropped by more, losing -5.8% and -4% respectively versus XLV’s 5.1% and -0.7%. These were good times to use the proxy for directional bets on the sector ETF. However, XLV outperformed A in January, May, June and August.

The risk-return chart of the healthcare sector shows that XLV has better risk-return characteristics than the single name proxy with a CAGR of 19.6% and an annualized volatility of 10.4% versus A’s CAGR of -1% and annualized volatility of 26.6%.

Industrials

In industrials, the root is Parker-Hannifin (PH). Here we have a different story with the sector proxy consistently outperforming the sector ETF, XLI, despite some sharp pullbacks. Sentiment is currently quite bullish on the stock. Recently, TD Cowen raised it’s price target for the stock due to how well the company has dealt with challenges in the market. Futhermore, according to the same article, 17/22 brokerages rate the stock as a ‘buy’ and five rate it a ‘hold’—none are bearish the stock.

The monthly returns chart shows that in 6/9 months so far this year, PH would have provided better returns for directional bets on the industrials sector. In January, XLI was unchanged but PH gained 1.3%.

In Feb, July and August, it gained 14.5%, 10.4%and 7%,compared to XLI’s 6.9%, 4.8% and 2.8%. Furthermore, in June and September when XLI dropped by -1% and -0.7%, the single name sector proxy lost -5% and -1.5%.

In April, XLI lost by more, losing -3.6% vs PH’s -2%, but it outperformed the single name proxy in March and May gaining 4.3% and 1.6% versus PH’s 3.7% and -2.2%.

In terms of risk-return, XLI is the safer choice with a CAGR of 24.2% and an annualized volatility of 13.3%, versus PH’s 45.4% CAGR and 26.9% annualized volatility. In this case you get more reward for more risk with the single name proxy.

Communications

In communications, the root is News Corp (NWSA). The cumulative returns chart below shows that it has underperformed the XLC gaining 6% YTD versus XLC’s 20%. Note that XLC has also outperformed the S&P 500. In terms of sentiment, analysts are mixed on the stock. Goldman think it will benefit from AI and is therefore a long-term buy. Newscorp and Open AI signed a content deal in May this year giving Open AI access to Newscorp’s publications and archives.

In five out of the nine months so far this year, NWSA would have been a better proxy for the XLC on directional communications sector plays.

In February, NWSA gained 8.7% compared to XLC’s 4.5%. In May and August, it gained 13.3% and 2.7% compared to XLC’s 6.7% and 1.8%.

In April and September when XLC dropped, NWSA lost -9.5% and -7.7% versus XLC’s -4.8% and -0.6%.

XLC outperformed the single name in January, March and June.

In terms of risk-reward, XLC is the better option giving more returns for less risk. It has a CAGR of 31% and an annualized vol of 16.2%, while NWSA has a CAGR of 11.4% and an annualized volatility of 21.8%.

Consumer Cyclicals

In Consumer Cyclicals, the root is Wynn Resorts(WYNN). It is significantly underperforming XLY, having lost 18% YTD while XLY gained 8%. However, the divergence between the two begun in June when it plummeted below unch and realized all of its YTD losses. The stock crashed 14.6% in June after results from it’s Macau gaming revenue in May were not as strong as expected. Despite the decline, and price target cuts, analysts remain bullish on the stock. 11/14 analysts rate it a ‘strong buy’ and three rate it a ‘hold’, according to this article.

In five out of nine months this year, WYNN was the better expression for a consumer discretionary trade. In Feb, it gained 11% versus XLY’s 7.6% and in May it gained 3.7% versus XLY’s 0.5%.

It dropped more than XLY in March, April and August losing -2.9%, -10.9% and -7.1% versus XLY’s -0.3%, -4.6% and -0.4%.

In June, July and September, XLY gained by 3.8%, 2.8% and 2.8% respectively while the single name dropped by -5.8% and -7.8% in June and July and gained by 2% in September making XLY the better expression of a consumer discretionary trade.

XLY has better risk-reward characteristics with a CAGR of 13.2% versus an annualized volatility of 18.4% while WYNN has a CAGR of -22.5% and an annualized vol of 25.9%.

Consumer Defensive

In Consumer Defensive, the root is Kimberly-Clark (KMB). The cumulative returns chart below shows that KMB has outperformed XLP. It has gained 19% compared to XLP’s 17% due to strong earnings. It is heavily touted as a good momentum stock with more upside having outperformed the Nasdaq this year.

In four out of nine months this year, KMB significantly outperformed XLP which would have made it a better expression for a long consumer staples play. In March, April, June, and August it gained 7.5%, 5.4%, 4.5% and 6.9% versus 3.3% gain, -1.1% drop, and 4.5% and 5.8% gains.

In January, February, June, and July, XLP outperformed KMB; the sector ETF gained 2.1%, 2.4%, 1.6% and 1.7% while KMB gained 0.7%, 0.7%, dropped by -2.4% and gained 2.3% in those months.

In September XLP gained 1.7% while KMB gained 0.3%.

KMB had a CAGR of 30.4% and an annualized vol of 19.1% versus XLP’s CAGR of 25% and an annualized volatility of 9.8%. It gave more returns but with higher risk.

Real Estate

In Real Estate, the root is UDR (UDR). Analyst are bullish on the stock i.e this article, but this is after the stock has outperformed this year gaining 24% while the sector ETF gained 13%. It is also touted as a momentum stock. Scotiabank raised its price target today and analysts the average price target set by analysts has increased by 6.78% as shown below.

The monthly returns show that in January, March, June, August and September, the stock would have been a great proxy for the sector. In January both assets lost -5.9%, in February UDR gained 5.2% while XLRE gained 1.7%, in June UDR gained 6.3% while XLRE gained 2%, in August UDR gained 10.5% while XLRE gained 5.6%, and in September UDR gained 5.7% while XLRE gained 3.7%. In the February, May, and July XLRE outperformed UDR. In February, XLRE gained 2.5% while UDR lost -1.5%, in May XLRE gained 5% while UDR gained 1.4%, and in July XLRE gained 7% while UDR lost -1.6%. In April XLRE lost -8.8% while UDR gained 2.9%.

In terms of risk-reward UDR has a CAGR of 39% and an annualized volatility of 18.9%, while XLRE had a CAGR of 20.1% and an annualized volatility of 16.3%. UDR therefore had a higher return but with more risk.

Technologies

In Technologies, the root is Keysight Technologies (KEYS). KEYS has plummeted by 7% this year while XLK gained 16%. However, since mid-August, the single name has outperformed the sector ETF largely due to strong financials. It is expected to gain further after it boost security standard tests to its vehicle-to-everything (V2X) communication solution, and demonstrated new AI infrastructure and accelerated radio frequency innovations.

In March, July and August this year, Keysight would have been a better proxy for the Technologies sector. It outperformed the XLK during these months gaining 1.3%, 2%

and 9.9% compared to XLK’s corresponding 0.8% gain, -3.3% drop and 0.7% gain.

In five months, January, February, May, June and September, XLK outperformed the technology sector proxy, gaining 5.3%, 4.6%, 6.8%, 7.5% and 0.3% compared to Keysight’s -1.4% drop in January, 0.7% gain in February, -6.6% drop in May, -1.3% drop in June, and -2.7% drop in September.

In April, XLK dropped by -5.9% while Keysight dropped by -5.5%.

In terms of risk-reward characteristics, XLK was the better choice with a CAGR of 26.6% and an annualized volatility of 23.4%. Keysight had a CAGR of -5% and an annualized volatility of 32.1%.

Basic Materials

In Basic Materials, the root is DuPont de Numours (DD). XLB has returned 9% YTD while DD has returned 4% in the same period. However, the stock is up 17% in the past six months which is a higher return than the sector ETF’s 4.5% and the S&P 500 9.5% gains in a similar period. According to this article, the gain driven by higher productivity, more innovations and acquisitions. In May, the stock jumped after it split into three companies and replated the then CEO.

For six months out of nine this year, DD would have been a better proxy for the basic materials sector. It gained more than the sector ETF in the of those months it was the better short and in the other three it was the better long. In January, April and September, DD lost -22.7%, -5.6% and -3.4% compared to XLB’s losses of -3.8%, -4.7% and -1.7% in those months.

In February, March, May DD gained more than the sector ETF returning 11.9%, 10.3% and 13% compared to XLB’s gains of 6.3%, 6.3% and 3.2% in those months.

The three months when XLB was the better expression for a basic materials play were June, July and August. In June XLB shed -3.1% compared to DD’s -2.1%. June and July saw the sector ETF gain 4.2% and 2.3% respectively compared to DD’s 3.9% and 1.1%.

XLB had the better risk-return characteristics with a CAGR of 13.7% and an annualized volatility of 13.7% while DD had a CAGR of 9.5% and an annualized volatility of 28.4%.

Utilities

In Utilities, the root is DTE energy. The cumulative returns plot shows that XLU outperformed DTE YTD. It gained 23% while DTE gained 13%. Despite this, the stock is touted as a strong value play and a great momentum stock due to its strong gains in recent weeks—so much so that redditors believe it will go to the moon.

DTE has not been a good proxy for a utilities play since in only three months out of nine, DTE would have been the better pick. In January, XLU lost -4.5% while DTE lost -5.8%, which would have made it a better utilities short than XLRE. February saw XLU gain 1.1% while DTE gained 2.7% , while in July XLU gained 6.6% compared to DTE’s gain of 8.2% making it the better choice for long utilities. In the remaining six months, XLU outperformed DTE to the upside.

From a risk-reward point of view, XLU was the better of the two. It had a CAGR of 36.2% and an annualized volatility of 15%. DTE didn’t perform too bad with a CAGR of 21.7%. and an annualized volatility of 18.9%.

Final Thoughts

I would say that the results show some promise that using sector representatives to proxy the sector ETF can give better returns on both long and short short-term trades. It seems to work most of the time for most sectors, but it is not a surefire thing. Some work would need to be done to develop this strategy. There is some hindsight bias here and you’d expect the sector proxy to change over time. One way to find a strong sector proxy would be to use a rolling window over a long period and check the evolution of the sector proxies. Then test if the strategy works out-of-sample and when it works or fails. For example, I have compared the monthly returns but it possible that the strategy works when a stock assumes the status of sector proxy and stops when it loses that status. A rolling window estimation approach would catch these changes.

It is also crucial that you come to the trade with a short-term view and an opinion about the sector, but also be aware of the narratives driving the stock. You would also need to check if the stock has lost it’s status of sector representative during the trade.

Such an analyses are computationally intensive and need to run on real-time data which is the kind of thing I will have running on the Quant (h)Edge website once it is up and running.

I’m open to feedback on this piece. Please share it with someone who would find it useful.

Over and out,

Brian

References

Czado, C. (2019). Analyzing Dependent Data with Vine Copulas: A Practical Guide with R. Springer.