Dividend Tree 19 January 2009 Commentary, Risk  Email This Post  Print This Post

In an earlier article, I discussed about the importance of quality of stocks in a portfolio rather than number of stocks for risk management. The common notion is that higher number of stocks in a portfolio provides better diversification (i.e. better risk management). In today’s post, I am presenting a probabilistic mathematics to demonstrate the relationship between number of stocks and its impact on diversification.

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, one would like to have all positive. If a portfolio has 10 stocks, one would like to have all positive and so on. At this point, I am not thinking about what would be 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.

Here, my interest 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 1 stock being +ve in 1 stock portfolio, in a 2 stock portfolio, in a 5 stock portfolio, in a 10 stock portfolio, and in a 20 stock portfolio. Furthermore, this probability can the extended to 2 stocks being positive or 3 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) 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 2 positive stocks, three stocks, 4 positive stocks, and 5 positive stocks.

In general, it can be observed 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.

We want to have higher probability for stock to be positive. Let us say we want to have 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.

A Customary Caveat: The results from these plots are good for understanding the relationship among different variables (in this case number of stocks vs. diversification). These may not be used for making portfolio decision. While it correctly explains the fundamentals, it is not designed for portfolio construction or decision making. This is for informative-purpose only.