# Diversification In the Context of Number of Stocks

*This article originally appeared on The DIV-Net on June 25, 2009*

There is a general perception that higher number of stocks in a portfolio provides better diversification (i.e. better risk management). I am discussing the relationship between number of stocks and its effect on diversification. I am using very simple probability mathematics for an ideal scenario.

I am using a stock being positive or negative as measure of diversification. In an ideal scenario, at a minimum, one would like to have all stocks to be positive relative to the buy price. For example, if a portfolio has 5 stocks, then one would like to have all positive side. If a portfolio has 10 stocks, one would like to have all in positive side and so on. Please note that I am not discussing the value of individual stock or portfolio. It is likely that a positive value in one stock can offset the negative value of other stock.

My interest, here, is to break it down to the best possible scenario, and that is, stocks being +ve or stocks being –ve. I am calculating the probability of “one stock being +ve” in “one stock portfolio”, in a “two stock portfolio”, in a “five stock portfolio”, in a “10 stock portfolio”, and in a “20 stock portfolio”.

Furthermore, I am extending this probability to “two stocks being positive” or “three stocks being positive”. The results are then plotted for graphical presentation and discussion. The chart below shows the plot of probability of stock being positive vs. number of stocks in a portfolio. This chart is read as follows:

- The curve, “1 +ve stock” is the curve (topmost curve, in red color) for probability of one stock being positive for a portfolio with 5 stocks, 10 stocks, 15 stocks, and 20 stocks.
- If a portfolio consists of only one stock, then probability that it will be positive is 0.5 (i.e. 50% chance that it will be positive).
- If a portfolio has 5 stocks, then the probability that at least one will be positive is 0.9.
- If a portfolio has 10 stocks, then the probability that at least one will be positive is 0.95.
- If a portfolio has 20 stocks, then the probability that at least one will be positive is 0.98.
- The chart shows similar curves for probability of “two positive stocks”, three positive stocks, four positive stocks, and five positive stocks.

This chart shows that there is a significant increase in probability of stocks being positive until 5 or 10 stocks in the portfolio. Beyond that the change in probability of stocks being positive is very very small.

What we really want is to have a higher probability for stock to be positive. Let us say we want 0.8 probability, so we look at this chart horizontally at 0.8. We can have that with 5 stocks, 8 stocks, 10 stocks, and 12 stocks (intersection points between curve and full digit stock). As an individual investor what would you do? If the odds are same wouldn’t you try to use lesser number of stocks?

These simple probability curves show that there is an optimum point beyond which more stocks will not have any diversification benefits.

The best examples are DOW Index (with 30 stocks) and S&P500 index (500 stocks). I have superimposed these two indices on the same chart below. Both these index follow each other. If more stocks are supposed to provide diversification benefits, shouldn’t the S&P500 perform better than DOW?

In future post, I will continue my discussion on some reason why quality is more important than quantity of stocks for better diversification.

Diversification is N-dimensional. Your examples shows that the S&P outperforms the DOW–but it is not so obvious from your time scheme. Do a chart of small, med & large cap stocks from 1900 to present. Without small caps you’d be horribly underserved–far and away dwarfing returns of the other caps. Another dimension might be growth-value (good luck on an operational definition to distingish these). Or, How about asset allocation over time–a higher level perspective on investing–that you began to address in another post.

Best,

lma

Regarding quality: I believe diversification originated as a way to minimize the blow of a stock or bond suspending or reducing payment. Ideally, nothing would happen to the [dividend/interest] income stream. If something went wrong, though, income was chopped but not by much.

Daniel,

That’s a good observation. Perhaps, diversification is for income stream (and not capital appreciation?). Thanks for stopping by.

Best Wishes,