The concept is based on the principles of Jensen’s inequality (named after the Danish mathematician Johan Jensen). Nassim Taleb has dealt with this concept extensively in his book Antifragile (highly recommended reading).

The Generic function for the value of an investment ($X) invested @ a compounded rate of return (R) after N years = $X * (1+R)^N…now this is a convex function for all values of N > 1

Let’s take a 10 year investment horizon.

The value of the $2000 investment @ 10% annualised return would be equal to $2000 * (1.1)^10…This is denoted by Q in the picture below

If we plot the graph for the values of the investments for different annualised returns, the graph will look like this:

Here P is the future value of $2000 after 10 years invested @6% and R is the future value of $2000 after 10 years invested @ 14%.

Now to exemplify the application of Jensen’s inequality here….due to the ‘convex’ nature of the curve, even though the arithmetic average of 6% and 14% = 10%, The value of P+R > 2Q (had the function been liner P+R would be = 2Q)

Now if we make the individual returns more variable around the mean of 10%…..let’s say the individual returns get pulled out to 2% and 18% (the mean is still = 10%), then the Future Value of the $2000 invested @ 2% = S, and the Future Value of the $2000 invested @ 18% = T…..now again because of the convexity, (T-R) will always be greater than (P-S)….hence the gains in future value because of the shift from 14% to 18% (=T-R) will always more than compensate the drop in future value due to the shift from 6% to 2% (=P-S)

Hence, because of the convexity, the more variable the returns of the individual investments are, as long as the arithmetic mean is same between the two portfolios, the future value of Portfolio B (comprising 5 such investments) will always be higher than the future value of Portfolio A (comprising 5 investments of $2K invested @ 10%).

In other words, ‘The function of the mean is always lower than the mean of functions….in case of a convex curve’.

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