Understanding The Link Between Volatility And Compound Returns

All credit to www.cssanalytics.wordpress.com/2012/03/12/understanding-the-link-between-volatility-and-compound-returns/

This is a very clear discussion of a stunningly important concept (emphasis added):

…It seems obvious from the research that volatility is an important factor to consider in portfolio management, but it is necessary to understand why this is the case. The answer lies in the relationship between the geometric return and the arithmetic return. The geometric return is the return that is achieved through reinvestment or compounding, while the arithmetic return is the simple average of the returns. Often the difference between the two is framed in terms of  the impact of positive versus negative returns when you stay invested (compound your wealth). Most traders understand that large losses hurt their accounts more than a gain of the same magnitude would help their account: for example if you lose 33% of your wealth you need to make 50% to get back to even. Therefore avoiding large losses should be more important than seeking gains to maximize the growth of your account.

This insight is part of the rationale behind trend-following and is also the common justification for traders to judiciously use stop losses. It makes for an excellent example of homespun investment folk wisdom that captures the spirit but fails to capture the science. What is less clear to the vast majority of traders is the deleterious impact of a re-investing wealth in a volatile stream of smaller-sized returns. More formally, Cooper http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1664823  expresses this relationship is  by the equality:

(1-x)*(1+x)= 1-x^2

where x is equal to the daily return. Assuming a leverage factor or portfolio size of 100%, when x= 5%, an investor would have one 5% up day followed by a -5% down day.  Instead of breaking even as one would assume, the investor would have lost -.25%.

…The takeaway from these formulas is that risk/variation/volatility is the enemy of compound returns – especially when the average return is low and the risk is high (think 2008!). Assuming returns are moderate to low, volatility has a dominating role in explaining the gap between arithmetic and compound (geometric) returns.

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