Understanding the Link Between Volatility and Compound Returns

It is clear from looking at the current landscape that volatility is rapidly becoming a key focus for asset management. Witness the birth of “low-volatility” ETFs and the popularity of minimum-variance portfolios borne from empirical studies that demonstrate their superior performance to alternative methodologies.  It seems obvious from the research that volatility is an important factor to consider in portfolio management, but it is neccessary 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%.

This phenomenon  is not widely understood (surprisingly even among low-volatility advocates!) and generalizes to the mathematical conclusion that the geometric mean is significantly and directly impacted by the variance of returns. Other things being equal, highly levered or very volatile investments tend to be very hazardous to compound performance. Most investors in 3x leveraged bull or bear etfs- that promise constant arithmetic leverage- have learned this relationship the hard way. Often the returns of both bull and bear etfs lose money over time while the underlying index has positive returns. The mathematics that drive this  relationship are very simple:

Geometric Return= Arithmetic Return -.5*Variance

rearranging this equation we have:

Arithmetic Return-Geometric Return= .5*Variance

In “English” this means that the average return is always greater than the compound return by half of the variation or risk of the asset returns. The greater the risk or variation, the larger the difference between average returns and compound returns. This relationship is a mathematical identity that is true for all time series and is used in the disparate fields of telecom, computer science and biology.  The difference between arithmetic returns and geometric returns is the “volatility drag”. The name implies that compound returns are “dragged down” by high levels of volatility to the point that they can be negative despite a high level of arithmetic returns.  However, Cooper shows that leverage plays a key moderating factor in this relationship. This subject is more complex, and for now lets assume the more typical case where one is fully invested. Here is the derivation of the formula above expressed in volatility terms:

Geometric Return= Leverage x Arithmetic Return -(.5 x Leverage^2 x Volatility^2) x  1/(1+Leverage x Arithmetic Return)

for a leverage of 100%-or a typical situation where one is fully invested-we substitute  100% for “Leverage” and this simplifies the equation to:

Arithmetic Return-Geometric Return (volatility drag)=  (.5 ^2 x Volatility^2) x  1/(1+Arithmetic Return)

since the second term is almost always very close to 1, this simplifies the formula to:

Volatility Drag=  .5*(Volatility)^2

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. Given that most academic research studies focus on compound returns, it should not be surprising that research shows that high-risk stocks underperform low-risk stocks. Apparently many people seem to think that this is an “anomaly” or a great way to find stocks that will have a higher return. Academics are puzzled by this finding that seems to fly in the face of the conventional theory that to increase returns you need to take on more risk. But it is important to note that risk premiums- or the relationship between required higher rates of return for higher risk- are arithmetic. That means that the curious underperformance of high risk stocks/assets versus low risk stocks/assets probably has less to do with a hidden risk factor or behavioral bias, but rather the fact that we are compounding our wealth. Falkenstein   http://falkenblog.blogspot.com/2012/01/is-arithmetic-return-bias-basis-of-low.html  shows that high risk stocks actually have a higher arithmetic return than the market, however the compound return is much lower due to the volatility drag. The “low-volatility anomaly” is generally explained by the mathematical relationship between volatility and compound returns. The residual difference is likely explained by the behavioral bias towards seeking higher returns without regard to the added risk– clearly part of this bias is due to the ignorance of subtleties of the math.