### Market Reaction and Portfolio Concentration

I was talking with my coworker the other day about the differences between arithmetic and geometric means. Arithmetic means are the averages that we are familiar with, whereas geometric means come into play in the compounding of returns. For example, let's say one of our stocks went up 2X in Year 1, and 4.5X in Year 2. The geometric return would be 3x ( square root [ 2 * 4.5 ] ). Equivalently, we could have had another stock with the same geometric returns and achieved the same results over two years (3 * 3 = 2 * 4.5).

With that being said, the differences between the two types of means differs greatly and changes with how we construct our portfolios. For the sake of simplification, let's assume that all financial assets consist of numerous fair independent coinflips that occur daily, with a 50% positive return for heads, and a 40% loss for tails. Although we can allocate our portfolio into any number of these assets, we must remain fully allocated and play everyday. What is the optimal strategy?

If we compute the two means, we see they differ greatly. The arithmetic mean is +5% ( [+50 - 40] / 2), whereas the geometric mean is -5.1% ( sqrt[ 1.5 * .6 ] ). To achieve the arithmetic mean, we would divide our bankroll into as many bets as possible, so that our sample means would converge to the population mean. This is the so-called "free lunch" of diversification. To achieve the geometric mean, we would place all of our bankroll on a single bet, and continually parlay the results of those coinflips into more coinflips, so that over a long enough series of days, our returns would approach the geometric mean. So is there ever a scenario where we would not want to diversify?

I got the idea that "it depends" from reading Bill Miller's Q4 2006 Letter to Investors. In it, he explains the reason why his streak of beating the S&P 500 was finally broken, and his mistakes of overly concentrating the portfolio at the wrong time.

If we go back to our earlier example, let us now assume that instead of numerous similar bets, there are different classes of coinflips we can make, and that they are limited in number. Class A coins offer +100% returns for heads, -60% returns for tails, and there are only 2 of them. Class B coins offer +20% returns for heads, -10% returns for tails, and there are 40 of them. Doing the math, we see that Class A has an arithmetic mean of +20% and a geometric mean of -10.6%. Class B has an arithmetic mean of +5% and a geometric mean of +3.9%. The arithmetic means of Class A clearly surpass those of Class B, but there aren't enough of them to properly achieve the population average.

So now the question of diversification / concentration is largely a function of opportunity diversity and becomes a conversation of tradeoffs between diversifying to achieve the geometric return, versus striving to achieve a higher arithmetic return through concentrating in rare "good bets". If your ideas are mostly similar and no ideas are clearly better than others, then diversify as much as possible.

The recent market events have introduced greater variability in the risk / returns profiles of my ideas, which has manifested in some positions being untouched whereas others have become hammered. Given an assumption of unchanged fundamental business value, I think now is the time to concentrate the portfolio, increasing positions in those ideas that seem 70+% undervalued (ie multi-bagger prospects) and exiting positions with only 30-40% undervaluations. Although this will almost necessarily introduce greater volatility in portfolio returns, I am confident that these actions will bear fruit 1-3 years hence.

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