# 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.

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Very enlightening post! I have gone through several backtesting exercises in which I compare only investing in the N instruments with the lowest volatility, volatility weighting, etc. and the low volatility approach outperformed the high volatility strategy in mostly all cases. At first, this did not make sense and actually seemed counter intuitive. The more I dug into the strategies was when I realized that a high volatility strategy had bigger losers more often and the key to surviving and allowing one’s money to compound is avoiding the big losers. Your post is such a clear and concise explanation of this “volatility drag” phenomenon with a real world mathematical example. I really enjoyed this post as well as several other articles on your blog.

Thanks,

Ross

Hi Ross, thank you– your findings coincide with a study done by Connor’s Research that found a higher percentage of winning holding periods for low volatility versus high volatility stocks. I appreciate your kind words.

best

david

Not matter what any research suggests or how you try to rationalize it this is irrelevant stocks (still applies to leveraged ETFs). Stock clearly trade based on price and valuation, therefore if a stock loses 33% due to a change in valuation, it will gain 50% back when that same valuation returns. If you have a time horizon of a couple weeks or more investors and traders are frequently more concerened with the valuation of a stock, not percentage gain or loss it has experience. Although you -30%, 50% analysis is true, stocks have no problem routinely giving you these returns if their valuation justifies it.

Hi Sam, I think what you are suggesting is a valuable insight for many. However, my comment is that you would not have to lose 33% in the first place if you managed volatility (that was likely caused by the index and not the stock) properly. In that position you could post a large and significant gain instead of making your money back. Many value investors suffer tremendous losses during periods where the market corrects, volatility is high, and correlations (systematic risk) is high. Their subsequent returns in the recovery are excellent, but there is no need to suffer from these market dislocations to the same degree if volatility is forecastable.

best

david

David, A nice concise piece that cements many of the points I’ve read in the past but only recently accepted through my own position size testing. But there is one derivation of the fundamental equation that I am yet to solve or see discussed anywhere. Maybe you can help.

The downside volatility, or Lower Partial Moment (LPM), of trades is really the only component that damages a portfolio by causing drawdown and therefore hindering compound growth. Loosely described… Given a set of fixed positive returns, the negative returns control the potential drawdown and the compound growth. Given a fixed set of negative returns, the potential drawdown is defined and the positive returns now control the geometric growth. It appears that the geometric growth is effected by the positive and negative return volatilities independently.

Can you comment on this and maybe even break the fundamental equation down into the two volatility components if appropriate.

Hi Look, I “partially”- excuse the pun- agree with you. ALL volatility hurts geometric returns including only positive returns–as an experiment try compounding gains between 5 and 25% over a long sample and compare the average/arithmetic return to the compound return. That said, downside volatility is comparably more damaging if we are after positive and high compound returns. It is worth discussing LPM and other downside measures in a later post.

best

david

Actually, I think the data on the arithmetic return to high betas is probably due to the size factor more than beta. I constructed the beta portfolios from about the top 2000 market cap non-etfs going back in time, and this will tend to have a lower market cap than the S&P500, which had a higher sample return in this period (62-2010). You can download it at http://www.betaarbitrage.com. In any case, the arithmetic returns across all categories are pretty darn close.

Hi Eric, thank you for the link and some excellent reading on your blog. I agree with your assessment. I was primarily trying to highlight the major disparity between compound and arithmetic returns. It would be interesting to see the compound return of the high beta stocks versus low beta stocks when their volatility is adjusted to be equivalent using for example the short-term ewma on a rolling basis.

best

david

Excellent post! This concept is SO important and yet it is lost on the vast majority of investors. I’ve always thought that an interesting hedge fund strategy would be to short offsetting triple long and triple short ETFs to try to capture the volatility drag. They key would be knowing when to rebalance, which isn’t so cut and dry. Thanks again for the post!

I ask a related question in this well known QuantForum where I referenced this post – perhaps somebody wants to contribute:

http://quant.stackexchange.com/questions/3821/is-arithmetic-return-bias-basis-of-low-vol-anomaly

The geometric-artithmetic return relationship you used is a first order approximation for the case in which the time interval is close to zero. You cannot use it to make general conclusions as you did and such conclusions may not be true.

Apologies for posting such a late reply, and for my ignorance. I’m having difficulty understanding how “volatility drag” is anything more than a mathematical artifact, based on the misunderstanding of arithmetic vs geometric returns. If we measure value in absolute dollars or index points rather than annual percentages, It seems to me that a single security, purchased at $100 and held for 5 years with an ending value of $150, would have a CAGR of 8.45% regardless of how its value fluctuated during that 5 years. Specifically, 2 such securities, each purchased for $100 and sold 5 years later for $150, with the intervening year-end valuations below, would each have a CAGR of 8.45%, despite the first being more volatile than the second:

100, 130, 120, 140, 130, 150

100, 108, 118, 128, 138, 150

Frankly, my understanding of quantitative finance is too weak for me to truly follow the second half of your post where you explain this, so I am fairly certain the misunderstanding is on my end. So I apologize for posting on a forum where I’m not really qualified to participate, but I would really appreciate it if someone could explain this, or point me to an appropriate resource.

Thanks to all of you for making forums like this public, where the rest of us can get a free education!

hi bryan, there is no difference in cagr if you stipulate the cagr in advance- as you alluded to the difference is when you have an established arithmetic return and differing volatility-which leads to a difference in cagr. thanks for the kind words.

best

david

Hi David

Thank you for the post. I fully agree on the gist (volatility drag). However, I think your math is wrong.

Let’s assume:

Base index starts at 50, gains 5 points the first and loses 5 points on the second day, to close the 2nd day exactly net flat. In absolute terms gains equals loss, but that’s irrelevant in this context, what is relevnt instead are the percentage changes. Thus:

end value = 100*(1+(55-50)/50)*(1+(50-55)/55) = 100 * (1.1) * (0.9090)

As you can see the gain (10%) is not the same as the loss (9.1%), and thus your core term

(1-x)(1+x)

does not apply.

However, the correct term (whatever it would look like in any given case) will always conform to the inequality

(1-x)(1+y) < 1.0

so your gist is correct just the same.

Sincerely

Urs

Hi Max–sorry you are wrong in this case. The x refers to the return–if you refer to the paper by Cooper there is a simple example:

“For example

the market goes down by 5% then up by 5%. Then the net result is that the market has gone

to (1-0.05) times (1+0.05) = 0.9975 which is a drop of 0.0025 or 0.25%.”

best

david