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Question

How can I quantify the marginal diversification benefit of adding a particular holding to a portfolio?

Context

In this question, I asked about manually replicating a world-tracker ETF by buying its top N holdings directly. If you feel like your Boglehead is about to have a Boglestroke, please view that question for why I'd want to do such a silly thing! Anyway, I'd like to know what a reasonable minimal value of N is. In other words, how many of the top holdings of the ETF do I have to buy before buying the N+1'th holding is, in some measurable sense, "not worth it"?

Theory

I understand that diversification reduces the variance of expected returns, in the same way that rolling a million dice will basically guarantee a value-per-dice close to 3.5, while rolling a single die has the same expected value but a much bigger variance. Reducing variance is obviously super important for retirement funds.

In the context of investing $T in a portfolio, I could choose to replicate an S&P 500 tracker by buying its top N holdings, or its top N+1 holdings. How do I quantify the marginal diversification benefit of that extra holding? How do people model this? Are there publicly available tools for this?

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    If only it were that easy... Backtest only tells you how it would have affected you, not necessarily how it will affect you.
    – keshlam
    Nov 12, 2023 at 3:15
  • Of course; this can never be underemphasized, which perhaps I have done. Nov 12, 2023 at 12:30
  • I've replaced "backtest" with the more general "model" throughout the question. Nov 12, 2023 at 14:15
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    This is probably beyond the scope of a simple forum post. You'd want to calculate covariance (investopedia.com/ask/answers/041315/…) and correlation coefficients (investopedia.com/terms/c/correlationcoefficient.asp) as part of modern portfolio theory (investopedia.com/terms/m/modernportfoliotheory.asp). Nov 12, 2023 at 15:21
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    And note that the asset that gives you the greatest diversification benefit may not be (and in fact probably is not) the asset that gives you the closest approximation of the index you are trying to replicate. So you'd want to clarify whether the goal is to maximize the benefit of diversification or to minimize the delta between your portfolio and the index. Nov 12, 2023 at 15:23

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In the context of investing $T in a portfolio, I could choose to replicate an S&P 500 tracker by buying its top N holdings, or its top N+1 holdings. How do I quantify the marginal diversification benefit of that extra holding? Can this be backtested?

Yes it can be backtested. You can go back and look at various periods of time and see what impact not having a particular stock from the index would have done you your returns.

The issue will be how much would it cost you to get the proportions correct. The index isn't just one share from 500 companies, you do have to buy them in the correct proportion. You will periodically have to adjust your holdings. Also next year the ranking of the top X shares could change.

In the end you could measure how the decision a decade ago would have worked out. It tells you nothing about what will happen next year. The company at X+3 could have a stellar year. Or the one a X-1 could collapse.

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  • I could always backtest return values, but diversification is less about increasing expected return, and more about reducing variance without sacrificing any risk premium (see the dice example in the question). So I was looking for whether it's possible to backtest the effect of adding a holding to your portfolio on the variance itself, not sampling this or that value from past statistics. Nov 12, 2023 at 12:35

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