# How do I model the diversification benefit of adding something to my portfolio?

## 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?

• If only it were that easy... Backtest only tells you how it would have affected you, not necessarily how it will affect you. 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
• 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
• 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