How many stocks should I put in my portfolio ?

TL; DR: Not only one stock. Invest in 5 stocks to reduce your risk by half vs one stock investment (Result without optimizing portfolios, i.e., picking stocks at random and weight them evenly).

A diversified investment

When answering the question in the title of this post, one may address the more general question of diversification, which is allocating investment resources through many assets. The idea is that if one asset, just one I hope, collapses the other assets keep your investment stable, at least.

In simple words “Never put all your eggs in the same basket”.

The legend wants this quote to be a wise piece of advice transmitted by our parents and grandparents for us to confront the world strongly and firmly. Nobody told me that, except my finance professor at the university. Unfortunately, our eyes are not trained to see life through a diversification lens.

An example in another investment context: diversification is the same principle that should make you avoid work in the same field as your partner… or avoid a partner in the same field as you, for the more hardcore of us.

Suppose that you and your partner are working in the same field, say banking. If the industry collapse, both of you will encounter financial difficulties.

Now the most important part of the story: you have no clear incentives to work in the same field since working in any field pays, roughly, the same. However, the risk while working in two different fields can decrease. 

This point is at the heart of diversification and deserves some formalization.

 

Maths behind in an example

The key to understanding diversification is to have a look at the maths behind it. Say we have two assets with their daily returns :

sweet_stock nice_stock
-0.01 0.05
0.01 0.04
0.025 -0.02
0.03 -0.015
-0.005 0.03
0.01 0.025
0.01 -0.005

sweet_stock has a return of 0.01 and a variance around 0,00021. nice_stock has a return of 0.015 and a variance of 0,00078.

This gives you the performance: expected payoff (return) vs risk (variance), in investing in all your wealth in one of these two assets.

Note that I sneakily used the term variance, volatility old sister, to depict risk measurement which can be challenged.

However, if you invest in both stocks, say 50% in one and 50% in the other, the return is 0.0125 (0.5 * 0.01 + 0.5 * 0.015) and variance of 0,0001 (0.5^2*0,00021 + 0.5^2*0,00078 – 2 * 0.5*0.5*0,00029, with the last term the covariance of both stocks. See here).

Diversification doesn’t affect the expected return. On the other hand,  the negative covariance guarantee that the risk of the portfolio is less than the risk of any stocks.

Is this always the case in the real world of stocks?

The experiment

Nothing too original here: we replicate a famous study by Wagner and Lau in 1971. We update the result using SP500 from 2013 to 2018. We also provide all the materials here

To put it in a nutshell, we generate portfolios composed of a number of stocks that will vary. What we want is to know how performances, i.e. return and risk, will vary with their number of stocks. For each number of stocks, we generate many portfolios (like samples) to ensure that results are robust. Stocks in each portfolio are chosen randomly and are represented evenly. 

Note that it is possible to build portfolios in a more optimized fashion using random stocks at each experiment. This will complicate the discussions. Also, evenly designed portfolios make the point about diversification stronger. 

First, let’s see the return part of the experiment :

These are flat. Thus, varying the number of stocks doesn’t affect the return of the portfolios. But to be fair, one may expect some costs when having multiple stocks that have to imputed from the return. These are transaction costs, but also some cognitive costs (costs of headache) due to investigating and researching with stock is worthy.

Experiment results

Now let’s see the risk part of the experiment :

These are decreasing in a nonlinear fashion. Concave to be specific, since the second derivative is negative. This means that increasing the number of stocks decreases the risk… but the decrease when the number of stocks is low is more important than the decrease when the number of stocks is important. In other words, the diversification process end at some point. At 10 stocks, there is no benefit in adding new stock. This is due to the so-called systemic risk, or market risk, which can’t be eliminated by diversification.

 For the sake of science, let’s see the result in the Wagner and Lau 1971 paper :

 

Same shape, 40 years later.

It appears that the example case reflects reality. Mixing stocks reduce the risk for the same level of return.

The take away is that with only 5 stocks, which is not a heroic task, you can reduce the risk of your investment by near half.

References

  • https://en.wikipedia.org/wiki/Variance#Weighted_sum_of_variables
  • https://en.wikipedia.org/wiki/Diversification_(finance)
  • https://www.jstor.org/stable/4470866
  • https://github.com/soufianekhoudmi/optimalportfolioweights/blob/master/naive_random_portfolios.ipynb