How many stocks should you own? The costs and benefits of Diversification

/How many stocks should you own? The costs and benefits of Diversification

How many stocks should you own? The costs and benefits of Diversification

“How many stocks should I own in my portfolio?”

This question is actually quite complex and doesn’t have a clear answer.

To better understand the costs and benefits of more or less stocks in a portfolio we explore the trade-off between diversification and alpha generation.

Here is a high level summary of the situation:

  1. Owning more stocks in a portfolio lowers “idiosyncratic” risk, or risk that can be eliminated through diversification, however…
  2. Owning more stocks dilutes performance of an “alpha” generating process. (e.g., forcing Warren Buffett to hold a 500 stock equal-weighted portfolio would dampen his alpha)

In summary, fewer stocks in a portfolio imply more expected alpha and more idiosyncratic risk; more stocks in a portfolio imply less expected alpha and less idiosyncratic risk.

But what is the optimal trade-off between alpha and idiosyncratic risk? Do we want to own a 1 stock portfolio? A 50 stock portfolio? Or a 1000 stock portfolio?

Understanding Idiosyncratic Risk

Let’s first tackle idiosyncratic risk.

Idiosyncratic risk is the risk we can eliminate via diversification. Idiosyncratic risks are risks that are uncorrelated with other risks. For example, a building catching on fire, a hurricane, a CEO death, etc. We can hold a bunch of companies and these sorts of risks cancel each other out, on average.

Systematic risks are risks we can’t easily diversify, such as risk that the general economy goes into a depression. If people in the economy aren’t spending money, this affects all companies–we can’t hold a bunch of companies and eliminate this risk.

We have a post talking about idiosyncratic risk here. We also highlight how diversification eliminates idiosyncratic risk via the following graphic and accompanying spreadsheet, which I encourage you to investigate at your leisure.

Trade-off between portfolio size and expected alpha_1

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

diversification

The spreadsheet conducts 1000 simulations. In each simulation run, a system can either take 1 bet, 10 bets, or 100 bets–all uncorrelated. As the chart above suggests, one can manage risk by pooling truly uncorrelated bets together. As the number of bets increases, the volatility goes to zero and the expected value becomes the observation.

In the real world stocks don’t have zero correlation–correlations are much higher across equities. To get a better sense of how increasing portfolio size affects portfolio volatility we simply look to academia.

Elton and Gruber have multiple papers and books on the subject. Here is one paper in particular on the subject of portfolio size and portfolio risk:

Table 8 highlights the relationship between portfolio size and risk. Notice that the portfolio risk goes down as the portfolio size increases, however, risk never goes to zero and the benefits to holding a bigger portfolio decline rapidly after n=50.

Effect of Diversification

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

The effect of portfolio size on portfolio volatility can be seen visually in the following chart:

Risk reduction and portfolio size

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

What’s the bottom line?

30-50 stocks seems to be a sweet-spot where an investor eliminates portfolio idiosyncratic volatility, as additional diversification beyond this point does not help reduce volatility in any dramatic way. However, by diversifying beyond 30-50 stocks, we also prevent our portfolio from concentrating on stocks we feel are “undervalued.” In other words, we probably want Warren Buffett to hold 30 or so stocks to ensure he doesn’t completely blow up, but we don’t want to force him to hold more than that, because it is unlikely he has more than 30 good ideas. In effect, he would be “diworsifying.”

How Does Diversification Affect Stock Selection Strategies?

We look at a simple high-performance value factor: EBIT/TEV (a version of enterprise multiple).

There is a clear historical relationship between the level of EBIT/TEV and future stock returns. Cheap stuff earns more than expensive stuff. We are not going to argue whether this extra return earned by cheap stocks is due to mispricing or fundamental risk, but let’s just agree it is an empirical fact in the historical data.

How does concentration affect returns to EBIT/TEV portfolios?

First, a quick description of the study we conduct from 1964-2013.

  • Identify firms above NYSE 40th percentile for market capitalization that have information to calculate enterprise multiples.
    • In today’s terms, this equates to a universe that looks at stocks with a market capitalization of ~$2B and up. 
  • Split firms into portfolios based on EBIT/TEV
    • Top third (33%), which equates to around 279 names in the portfolio
    • Top quintile (20%), which equates to around 167 names in the portfolio
    • Top decile (10%), which equates to around 83 names in the portfolio

What do the results look like?

  • EBIT Cheap Decile = Value-Weight, monthly re-balanced portfolio of cheapest 10% of EBIT/TEV stocks
  • EBIT Cheap Quintile = Value-Weight, monthly re-balanced portfolio of cheapest 20% of EBIT/TEV stocks
  • EBIT Cheap Tercile = Value-Weight, monthly re-balanced portfolio of cheapest 33% of EBIT/TEV stocks
How Does Diversification Affect Stock Selection Strategies

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

More concentration increases compound annual growth rates, but also increases volatility (this makes sense because the larger portfolios are more diversified).

But what is the tradeoff between CAGR and volatility?

Based on Sharpe and Sortino ratios the evidence suggests that concentration is a better risk-adjusted bet. You get more bang for your buck by concentrating your holdings in systems that presumably earn higher risk-adjusted returns.

Here are the invested growth figures:

How Does Diversification Affect Stock Selection Strategies_2

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

What’s Do We Learn?

The simple exercise above highlights the cost benefit relationship between diversification and diworsifying, or diluting the alpha, of a proposed high performing strategy.

What are the implications for portfolio allocations?

Well, if you truly believe in a specific whiz-bang trading strategy, why would you invest in the strategy and allow the manager to hold 100, 200, or 300+ stocks? You would want your “active” manager to run a concentrated portfolio–otherwise you would be shooting them in the foot!

How does this concept play out in the marketplace?

Amazingly, there are very few true active managers. Most active approaches peddled by various investment managers are not active in any sense of the word–they have limited tracking error, hold large diversified portfolios, and don’t really bet on their presumed edge.

Examples:

Smart beta: You like value, so your approach is to hold a 500 stock portfolio that “tilts” towards value? As the analysis shows above, if you like value, BET ON VALUE. Otherwise, you are diworsifying.

Active Mutual funds and Hedge Funds with 100+ stock holdings: You are paying premium fees, and yet, you’re allowing the manager to diworsify their skill away and force them to act more like an index?


  • The views and opinions expressed herein are those of the author and do not necessarily reflect the views of Alpha Architect, its affiliates or its employees. Our full disclosures are available here. Definitions of common statistics used in our analysis are available here (towards the bottom).
  • Join thousands of other readers and subscribe to our blog.
  • This site provides NO information on our value ETFs or our momentum ETFs. Please refer to this site.

Print Friendly, PDF & Email

About the Author:

Wes Gray
After serving as a Captain in the United States Marine Corps, Dr. Gray earned a PhD, and worked as a finance professor at Drexel University. Dr. Gray’s interest in bridging the research gap between academia and industry led him to found Alpha Architect, an asset management that delivers affordable active exposures for tax-sensitive investors. Dr. Gray has published four books and a number of academic articles. Wes is a regular contributor to multiple industry outlets, to include the following: Wall Street Journal, Forbes, ETF.com, and the CFA Institute. Dr. Gray earned an MBA and a PhD in finance from the University of Chicago and graduated magna cum laude with a BS from The Wharton School of the University of Pennsylvania.
  • Chris Scott

    Wes, nice post. This is something I’ve been thinking about recently. How many stocks should you hold to diversify against idiosyncratic risk? The answer to that question is dependent upon the downside risk associated with the universe of stocks from which you are selecting. Thinking about US treasury bills is instructive. When making an investment in US treasury bills, generally you can just purchase a single issue for the entire allocation. There’s very limited downside risk, so there’s no need to diversify amongst multiple issues. While not the same as investing in stocks as there is no idiosyncratic risk, it illustrates that downside risk drives the need for diversification. So going back to stocks, reducing downside risk allows for more concentration. Concentration is important where returns increase monotonically across the cross-sectional returns of an investment factor. Both value and momentum factors exhibit this behavior. If you can reduce the downside risk of the universe you are selecting from, then you can concentrate holdings further, selecting stocks that exhibit deeper value or stronger momentum. Another example of this would be in examining Warren Buffett’s investment history. He has proven that he is better at evaluating downside risk than anyone else on the planet. At times he has invested ~80% of his funds into 5 or 6 stocks. His exceptional ability to evaluate downside risk has, when warranted, allowed him to concentrate his holdings far more than the average investor successfully could. So do you need 50 stocks to minimize idiosyncratic risk? Not if you can reduce downside risk – less would be required. Note
    that if you want to reduce tracking risk, far more stocks are required – which
    is the risk most active managers are trying to diversify away.

  • Chris, great point: you need to think about how many names you need to maintain sufficient downside risk.

    Another point to ponder is how much your idiosyncratic–and/or downside–risk matters in the context of a well-diversified portfolio. Let’s say we concentrate on our 10 best ideas for US Equity, but we have that wrapped up in a portfolio of assets spread across 100’s of asset classes around the globe. The 10-stock portfolio will be pretty volatile as a stand-alone, but in a portfolio context it may not even move the volatility needle.

    Anyway, the point we make, and you are emphasizing, is that thinking about concentration is important if you believe in the value of active management. If you believe markets are perfectly efficient–stick to passive 100%. But if you believe in active, why pay some clown to invest in 100+ stocks with a small “tilt” towards something they think is mispriced. That is insane and overpaying for what you are really trying to buy–the highly active piece of the pie.

  • Doug

    Good examination of the fundamental tradeoff between concentration and diversification. I’ll add three quick points: 1) The ideal level of diversification is based on the magnitude of the alpha. Weaker alpha has less to gain from concentration, so ideal portfolio size is larger. Virtually all managers overestimate their alpha, if you’re discretionary humans tend to overweight the probability of being right, and if you’re systematic your backtests will always be inflated to some degree by data mining bias. To counteract this the ideal portfolio size is probably slightly more diversified than what you initially conclude.

    2) Eton and Gruber are performing their calculation for randomly selected stocks. If you’re picking your highest conviction stocks, chances are they’re definitely more correlated than a random basket. For example you might have 10 airline stocks or a heavy bias towards momentum. In which case you’ll be much less diversified than what the paper suggests. At a certain level of sophistication it’s probably better to manage diversification through industry, country and risk factor exposure limits rather than number of stocks held.

    3) The horizon of the alpha matters as well. If I have two alpha signals with the same magnitude, but one tends to realize in a week and the other takes a quarter, I can achieve much more diversification while still harvesting the same returns with the former. This tends to be why the highest Sharpe strategies are almost universally short-term oriented.

  • All great points.

    1. I guess this would depend on the diversification benefits you already have in the rest of your portfolio. In a well-diversified context, even a small amount of alpha should be captured via conviction.

    2. Potentially true. But even if you do the simulations sector/factor adjusted, the required number of stocks to eliminate the idiosyncratic piece is not much larger than in their study…although, you can be left with higher systematic risk in those cases. In the example of airlines, you would be left with a higher level of systematic risk–regardless of how many airline stocks you own–but you can still eliminate the idiosyncratic risk with 20-30 names.

    3. Yep. But most short-term alpha signals I’ve seen I don’t believe in–market efficiency brain-washing, I know. If they are easy to exploit in the short-run, they probably aren’t real. Most anomalies require some sort of “horizon” risk to be sustainable/exploitable.

    Thanks for the great insights, Doug.

  • Michael Milburn

    Wesley, An issue related to diversification that I find difficult is balancing the tradeoff between risk adjusted return and market exposure. If I’m long, I want to build a system that keeps me exposed to the market but at the same time has cash available when the best risk adjusted opportunities arise. (I’m trading from individual signals instead of sorting a universe and buying something like the top decile at prescribed date, so the timing of buys is unpredictable. Same w/ sells occasionally due to stops).

    I’m a complete hobbyist at this, but it’s clear in the back-testing that having too much cash idle in most markets is simply a return killer. Waiting for the best opportunities is costly over time, but finding that fine line between maintaining exposure, tight buy constraints, and sufficient # of positions can be rewarding. It brings me around to thinking of an approach for liquidation of some positions “before their time / early” to raise cash for new “better” opportunities that arise when I’m 100% invested.

    As I work on this process I see diversification being a bigger issue, because so many of the stocks giving signals are in similar industries/sectors. Tons of biotech/pharmaceuticals and energy/natural gas/pipeline related companies right now. So even though I might be spread across maybe 50 positions, I’m not as diversified as I’d hope I’d be.

  • Michael,

    The questions you are asking are awesome, but very difficult. I’m fairly certain nobody knows the answer to any of them. I’ve got a PhD in finance and all I do is study financial markets–still haven’t cracked the code on timing, selection, and generating great returns with little risk. If you find the answer, please let me know ASAP–You’re hired!

    re diversification: The biggest piece of advice here is to ensure you don’t shackle yourself with mental accounting issues.
    http://www.alphaarchitect.com/blog/2014/09/04/behavioral-bias-bingo-mental-accounting/#.VBD0_fmwLRc
    Remember, your equity portfolio is only one portion of your portfolio–you presumably have a house, human capital, bonds, commodities, cash, and so forth. So maybe your equity portfolio is fairly undiversified, but in the context of your entire portfolio, your overall portfolio risk may not increase that much relative to a situation where you had invested in 300 random stocks…you might want to run this sort of analysis before determining that your equity portfolio is “too concentrated.”

  • Yang Xu

    Wes, i love this post. i am curious to see EBIT top 1 stock performance, just want to see how big the CAGR we can get or it could be a different story, ha.

  • CAGR increases for smaller portfolios. The smallest we’ve ever tested is ~15. The corresponding volatility is much higher, making the risk-adjusted stats look worse than portfolios ~30-50…this is in line with the discussion of idiosyncratic volatility, however.

  • Chris Scott

    Looking at a 1 stock portfolio is meaningless. If any stock ever selected in the backtest files for bankruptcy, the portfolio value goes to 0. As any stock always has some chance of bankruptcy, any backtest that shows a positive return is misleading. Even Warren Buffett, has never considered putting everything into one stock. Here’s his diversification policy from the original Buffett Partnership:

    “We diversify substantially less than most investment operations. We might invest up to 40% of our net worth in a single security under conditions coupling an extremely high probability that our facts and reasoning are correct with a very low probability that anything could drastically change the underlying value of the investment”

    From what I’ve read, he only reached these concentration limits only one time and was generally more diversified than 40% in one stock. Us mere mortals need more diversification.

  • agree. That would be insane.

  • Chris Scott

    One more Buffett quote from 1966 on this topic:

    “There is one thing of which I can assure you. If good performance of the fund is even a minor objective, any portfolio encompassing one hundred stocks (whether the manager is handling one thousand dollars or one billion dollars) is not being operated logically. The addition of the one hundredth stock simply can’t reduce the potential variance in portfolio performance sufficiently to compensate for the negative effect its inclusion has on the overall portfolio expectation.”

  • Saul Wong-Redondo

    One question regarding EBIT: does this consider only the trailing twelve months of EBIT? Intuitively, using something along the lines of the average of the last 5 years or of the last 10 years in the numerator should give better results in terms of separating cheap versus expensive stocks since one-year results can be volatile and investors are likely to be biased to overreact to a single bad or good year. I wonder if this would hold true statistically after going through the data. I.e.: 5-year average EBIT / TEV has better results than 1-year average EBIT / TEV.

  • We do this analysis in our Journal of Portfolio Management paper:

    http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1970693

    Also explore the concept in our Cyclically adjusted valuation metric paper:

    http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2329948

    Long term price ratios, while intuitive, are not better predictors from an empirical perspective, contrary to popular belief…

  • Saul Wong-Redondo

    Excellent, thank you! It is right there in the abstract. This has always been personally one of the main appeals of using balance sheet/stock holder equity metrics such as book-to-market over income/cash flow metrics. It is exciting to see your research covering this part of the puzzle.

    My only other big question although probably too off-topic and harder to answer because of lack of historical data would be about global investing. For example, right now countries such as USA and Germany are expensive on average on EBIT/TEV and have a narrow gap between cheap and expensive stocks. So intuitively, for somebody with a value mentality, it feels scary to be invested in those countries even with a value strategy in place. On the other hand, Russian, Turkey, and China are likely much cheaper. If an investor were to pursue a top down approach seeking cheap EBIT/TEV countries and even considering geopolitical risks such as Germany losing the World War II, governments not respecting property rights, etc, would that lead empirically to better results versus applying the value strategy in a single developed country such as the United States regardless of its overall market valuation?

  • That is definitely a big question. In general, it if feels scary, its probably a good risk-adjusted bet. Value, in our opinion, is fundamentally a behavioral driven phenomenon–people unfairly punish companies/countries for bad news. This is tied to availability bias, representative bias, and a handful of other potential issues.

    So does buying cheap junk nobody else wants a sure winner? Of course not. You’re gonna buy a falling knife along the way–but that’s good–if there wasn’t a fear of falling off the cliff, value investing wouldn’t be that scary and fewer people would sell for no good reason. Sometimes you gotta take the good with the bad and assess the overall odds.

    Regarding country allocation vs security selection strategies in a single country: Our research shows that security selection across all countries is probably the best approach, as a general rule. Buying cheap countries is essentially a noisy way of buying cheap individual securities. But why would you use a noisy proxy when you can simply construct an actual portfolio of cheap names? I’ve never understand the infatuation with “country” investing based on characteristics, but that might just me…

    In the end, just buy cheap junk nobody wants. Empirically, that seems to be a good risk-adjusted bet, at least historically. You’ll get your face ripped off here and there, but that is part of the game.

  • Saul Wong-Redondo

    Thank you again. One great thing about this research is that in addition to the academic value it also has potential to make a big difference in practice for almost anyone who has the discipline to follow the models against intuition or short term market fluctuations.

    I consider myself contrarian and value oriented, and my having fear of the USA is because value investing strategies have worked so well during the last years that I fear complacency and/or performance chasing have pushed prices in value companies especially small companies to the point where on average they temporarily don’t offer an attractive risk premium. But I might be wrong, and I might be more similar to everyone else than I realize. In that case, the risk premium should still be attractive because of this lingering fear. I also have invested in Russia. In that case, I have other kind of fear that is more natural and not contrarian at all: that the US government might impose sanctions forcing US persons to liquidate assets at fire sale prices or that the Russian government might retaliate and not respect property rights. For these fears, under the assumption that most investors share them, I would expect to be well rewarded in terms of the risk premium for those investments.

    Regarding using country as a proxy, the underlying assumption is that when investors are optimistic about a particular country stocks at a certain level of valuation would tend to be of lower quality compared to stocks with the same valuation in a country about which investors are pessimistic about. In other words, if Germany is perceived as low risk now versus Russia perceived as high risk, if I were to build a portfolio of 20 random companies in each country with EBIT/TEV of 20% then the Russian companies would tend to be of much higher quality. The Russian companies have an extra risk premium because Russia is in the news every day with negative headlines. These investor perceptions would change over time. For example, in 10 years, it might be that the emphasis is on Germany perceived as a slow growth country, whereas Russia is perceived as a fast dynamic growth country. Then at that time the German portfolio at the same level of valuation would be of equivalent or higher quality than the Russian portfolio. Something similar perhaps to how you can purchase a much nicer home with $400,000 on average in San Antonio, TX than in New York, NY no matter how good of a bargain hunter you are. Just anecdotally, the reason I started asking myself this question is because my worst performing investment so far has been to purchase a set of small cap Chinese companies in 2007 that had very attractive value metrics (they are still roughly at the same price today than they were in 2007). The most intuitive explanation to this I could find was that I purchased when investors were very optimistic about China so I overpaid even for the cheap looking companies. In contrast, investing in Greece in the middle of the Euro crisis when Greece was in the news much like Russia is today has been extremely rewarding even if I couldn’t go to the most deep value corners of the market because of investment restrictions.

  • Saul Wong-Redondo

    Of course it would be very exciting to see empirical research that proves or disproves this assumption. If there were some research that said “investing in countries with cheap or expensive valuations does not make any difference as long as you invest in cheap companies regardless of country valuation” that would disprove the assumption that Russian companies at a given level of valuation are higher quality than German companies at the moment or at least that the expected returns are higher.

  • we’ve done this research, internally. Stock-selection based country allocations outperform country-selection based allocations.

  • Chris Scott

    Agreed that security selection irregardless of country would be logically superior for construction of a global value portfolio, yet there are no investment products available to retail investors that take this approach (other than maybe the fundamentally weighted funds that don’t have all that much of a value tilt). I’ve never understood why there isn’t more interest in this approach to portfolio construction. For the typical retail investor, using ADRs is about the only option for doing this, which significantly limits the available universe from which to select.

  • working on it…

  • Chris Scott

    Read the SEC ETF registration filings – will be a very welcome addition to my portfolio once they are available.

  • Jake

    While I am inclined to agree with your 30-50 stock conclusion for quantitative portfolios, I think your model substantially understates tail risk and correlation. According to the model, the 100 stock portfolio never has a return less than +4%, which is unrealistic. Moreover, even the 10 stock portfolio never has a return under -10%. I believe the two main issues are that 1) stock returns are not normally distributed or are at least platykurtic and 2) stock returns are not random and in fact correlation increases during large moves both positive and negative, which is precisely when investors count on diversification to protect them. I would also posit that not every stock has the same expected return and standard deviation, which is why these value anomalies work quantitatively but that is a discussion for another time.

  • Jack Vogel, PhD

    Jake, the assumptions we make in that model (for picture 1 above) are that stock returns are independent and normally distributed with a mean of 10% and a standard deviation of 20%. Obviously if you change the mean, standard deviation, distribution (from normal), or independence assumptions, you will probably get different results.

    That is why we continue in the post and create value portfolios throughout time, using either the top third, top fifth, or top decile based on EBIT/TEV. As you can see from the results, these portfolios have at least a 40% drawdown, so there is correlation between the stocks in the portfolio.

  • Jake

    You are 100% right, my apologies. I had read the piece, then started messing around with the model, and apparently forgot about the rest of the paper. Great analysis.

  • Jack Vogel, PhD

    No problem, thanks for the comments.

  • Pingback: Charlie Munger On “Loading Up,” Tracking Error, And Value Investing | EasyHomeSite()