The Law of Large Numbers October 1, 2018

The Law of Large Numbers

In an earlier article I wrote “Beating the Efficient Market Hypothesis” where I tested for “momentum” in the markets, one of our readers responded by emailing in:

The Law of Large Numbers

Yes, it was by design that I did it in deciles. Also I have more than two hundred stocks per decile per rebalancing period. But why so many?

This is because of the Law of Large Numbers (LLN).

LLN states that given a sufficiently large enough sample size of independent results, the mean of the sample size is roughly equal to the mean of the population.

To put it simply, if I group enough momentum stocks together, their returns on average will be “momentum-like”. There may be outliers such as outperformers or laggards in the group but aggregating their performance will cancel the effects of those out. The more momentum stocks there are in the group, the more “momentum-like” the portfolio will be on average. Variation of individual returns in the portfolio is likely to be lower as well.

You can test this out yourself with a six-sided dice.

Getting Dicey?

Given a normal six-sided dice with the faces labelled from 1 to 6, the theoretical mean of each throw is (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5.

Whether it is luck or skill, our first few throws may give us 1s or 6s, throwing our averages off the theoretical value. As the number of throws increase towards 1,000, the sample mean converges around the theoretical mean at 3.5.

Try doing this in Microsoft Excel with the formula =RANDBETWEEN(1,6).

I have simulated different numbers of throws and produced the results in the table below.

Six-Sided Dice
Number of throws 1 5 10 20 50 100 200 300 500 1000
Sample Mean 6.00 3.60 3.10 3.35 3.44 3.59 3.43 3.37 3.55 3.51
Theoretical Mean 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5
Difference 2.500 0.100 0.400 0.150 0.060 0.090 0.070 0.130 0.054 0.009

We can also capture the idea visually with a graph:
The Law of Large Numbers
Fig 1

Back to the Lab

I started off with “The best performing stocks in the past will continue to be the best performing stocks in the future.”

I consider two hundred as a meaningful enough number for each decile to be averaged to “momentum-like”.

The returns of each decile is almost linear. We can see almost immediately:
a. The top half (Q1 to Q5) outperforms the bottom half
b. The top quintile (Q1 & Q2) outperforms the next four quintiles
c. The top decile (Q1) outperforms the rest of the nine deciles

The Law of Large Numbers
Fig 2

La crème de la crème?

This got another of our readers thinking:

The Law of Large Numbers

Surely the cream of the crop is the best?

I can also rephrase the above email, along with the previous email, as such:

How low can we go? What is the minimum number to make this work?

In the previous sample tested, there were about two thousand stocks during the earlier years to about two thousand eight hundred stocks in the recent years. The top decile of the top decile (10% of 10%) is about 20 to 28 stocks per rebalancing period. I’ve decided to round it up for consistency’s sake and chose the top 30 stocks per rebalance period. The top 50 and 100 are also included for good measure. This has produced rather mixed results.

a. While the Top 30 underperforms Q1 (Top 1% underperforms Top 10%)
b. The Top 30/50/100 still outperforms Q2 (although only slightly in 6m3m’s case)

Top 30 Top 50 Top 100 Q1 Q2
US 6m1m 14.54% 15.03% 15.22% 15.41% 12.59%
US 6m3m 12.12% 13.15% 12.98% 14.28% 11.91%

The Law of Large Numbers
Fig 3

The number of stocks to hold also suggests a trade-off between diversification and practical application.

Going by the decile implies a lower standard deviation of returns. Remember LLN? As the number of stocks held increases, sample mean converges to theoretical mean whereas going by an absolute number (Top 30/50/100) is more digestible for a layman.

Now, if only there is something we can do to have the best of both worlds…

Stay tuned for more next time!

(*Please note that information contained in this article is given for educational purposes only. Past performances are not indicative of future returns. All information and opinions provided in this piece do not constitute as investment advice.)


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