Maximum of Average vs. Average of Maxima
The maximum of an average cannot exceed the average of the components’ separate maxima. Because components can peak at different times, an index and its individual stocks can show different declines from their respective highs.
THE IDEA
The advent of the financial television channel CNBC presented plenty of benefits to the financial community but it also allowed a collection of extrovert practitioners long on theories to voice them in a few minutes of television time. One often sees respectable people making ludicrous (but smart-sounding) statements about properties of the stock market. Among these are statements that blatantly violate the laws of probability. One summer during which I was assiduous at the health club, I often heard statements such as “the real market is only 10% off the highs while the average stock is close to 40% off its highs,” which is intended to be indicative of deep troubles or anomalies—some harbinger of bear markets.
There is no incompatibility between the fact that the average stock is down 40% from the highs while the average of all stocks (that is, the market) is down 10% from its own highs. One must consider that the stocks did not all reach their highs at the same time. Given that stocks are not 100% correlated, stock A might reach its maximum in January, stock B might reach its maximum in April, but the average of the two stocks A and B might reach its maximum at some time in February. Furthermore, in the event of negatively correlated stocks, if stock A is at its maximum when stock B is at its minimum, then they could both be down 40% from their maximum when the stock market is at its highs! At any time, the average cannot exceed the average of the components’ separate maxima. Their peaks need not coincide; this comparison does not establish a universal ordering of volatility.
When to use it
Checking whether an index and its component stocks peaked at different times before interpreting their different drawdowns as a bear market signal.
Evaluating the performance of a group of projects where each project peaks at different times.
Comparing aggregate economic indicators (e.g., GDP) with sector-specific data to identify potential misleading trends.
In sports, comparing team performance to individual player performances over a season to understand apparent discrepancies.
FROM THEORY TO EVERYDAY LIFE
In practice
CNBC Stock Market Anecdote
One summer at a health club, a statement was heard: 'the real market is only 10% off the highs while the average stock is close to 40% off its highs.' This appears contradictory but is perfectly consistent because stocks do not all peak at the same time. Because those peaks need not coincide, the market index can be down only 10% while individual stocks are down 40% from their respective highs.
A MOMENT TO REFLECT
Put it to work.
When looking at aggregated data, why is it important to consider the timing of individual components?
How can the principle of maximum of averages affect portfolio diversification and risk assessment?
In what other fields might this statistical principle lead to misinterpretation of indicators?
CONTINUE THE CONVERSATION
Source texts
- Fooled by Randomness The Hidden Role of Chance in the Markets and in Life (Nassim Nicholas Taleb)