# Volatility Harvesting: From Theory to Practice

We’ve all heard the old adage that diversification is the only free lunch in investing, but Paul Bouchey, CFA, debunked that notion at the CFA Institute Wealth Management 2012 conference in Miami. To be clear, Bouchey did not challenge the notion that we can reduce risk without sacrificing return through diversification. He did, however, call out another free lunch — volatility harvesting.

As the recent financial crisis illustrates all too well, volatility causes investors heartburn and tempts them to abandon otherwise prudent long-term investment plans. Bouchey pointed out, however, that volatility actually creates alpha-generating rebalancing opportunities for any core portfolio. One of the criticisms of capitalization-weighted indexing, especially in global indices, is that indexes can become very concentrated as specific securities or sectors outperform other sectors. Concentration reduces diversification and thereby increases volatility, which hurts capital accumulation.

We often forget that growth in wealth approximately equals the arithmetic average return over an investment horizon less half the volatility of the return. This is a fundamental, indisputable mathematical principle. As a result, volatility creates a drag on compound growth rates, and hence wealth accumulation, holding returns constant. In other words, the journey matters. A tough journey creates worse investment outcomes, even if the average return stays the same.

Here is a thought experiment: suppose we flip a coin and you double your money on heads, but lose half your money on tails. The expected return is 25% [i.e., 0.5 × 1.00 + 0.5 × (-0.50)]. Despite this very attractive expected return, this bet will not accumulate wealth over time because the volatility inherent in the coin flip impedes capital growth.

In an almost Zen-like approach to investing, we can turn the notion of volatility drag to our advantage through rebalancing. Suppose you only bet half your capital on each flip of the coin, which is analogous to a 50–50 allocation to the stock market and the risk-free asset. After each flip, you rebalance to restore a 50–50 allocation. In that case, you can expect to grow your wealth by more than 6%. A portion of the difference from the no-growth case represents the rebalancing premium.

In a portfolio context, it implies that a contrarian rebalancing strategy selling outperforming assets and buying underperforming assets avoids portfolio concentration, manages volatility, and increases a portfolio’s growth rate when it would otherwise be impeded by excess volatility. In other words, there is a growth premium for rebalancing that Bouchey refers to as “volatility harvesting.”

Portfolio managers can harvest more volatility through rebalancing as the volatility of individual assets increases. However, if the assets are highly correlated, individual positions will not outgrow other positions, and rebalancing opportunities will be scarce. So, Bouchey notes that emerging market portfolios are replete with opportunities to harvest volatility.

Interestingly, this source of alpha cannot be arbitraged away. The mathematical principle remains even if all investors follow this principle.

This principle lends support for different investment strategies, like naïve 1/n diversification, minimum variance, equally weighted, fundamental weighted, or “diversity” weighted portfolios. A diversity weighted portfolio underweights the positions with the largest market capitalizations and overweights positions with the smallest market capitalizations, but not so much that it makes the portfolio equally weighted. One can imagine an infinite number of algorithms to accomplish this, but Bouchey suggested one that looks slightly more equally weighted than capitalization weighted. So, the diversity adjustment need not be dramatic. It works particularly well for creating country diversity in global portfolios. By definition, however, it introduces some tracking errors.

Matthew Kenigsberg, CFA, who attended Bouchey’s presentation, pointed out that Stephen Hawking would caution us against taking the idea of the rebalancing premium to extremes, in order to avoid being sucked into a financial black hole. Suppose you were to continuously rebalance (e.g., every millisecond) an equally weighted portfolio in which one of the securities goes to zero. What would happen? You would continuously reinvest in the security that goes to zero until it consumed the entire portfolio into a sucking, swirling financial eddy of despair.

In the end, Bouchey encouraged the audience to recognize two things. First, capitalization weighted indexing may maximize diversification, but it does not maximize diversity because positions can become very concentrated. Second, volatility is not the same as risk; it creates exploitable opportunity than cannot be arbitraged away through systematic rebalancing that preserves diversity.

As Stephen states, the journey does matter: “volatility creates a drag on compound growth rates.” This means that wealth accumulation has as much to do with finding diversification opportunities as it does profit opportunities. If the returns of each portfolio constituent are equal, then the more “truly” diversified portfolio will accumulate the most wealth (greatest return) over time.

I discuss this at length throughout my book “Jackass Investing: Don’t do it. Profit from it.” (The Amazon Kindle #1 best-seller in the mutual fund category). I’m pleased to provide complimentary links to two of the book’s chapters:

“Myth #19: Diversification Lowers Returns:” http://bit.ly/Ahj2lc

and

“Myth #20: There is No Free Lunch:” http://bit.ly/vxDo6v

The statistical claims in this article are pure nonsense.

To begin with, in the paragraph beginning “Here is a thought experiment…”, yes, your expected return at every interval is a 25% gain. The expected return over N trials remains (1.25)^N. It is not true in any sense that “this bet will not accumulate wealth over time because the volatility inherent in the coin flip impedes capital growth”. I can’t even fathom the error in statistical/mathematical understanding that led to this claim. It reflects badly on the author and the editorial board.

It’s impossible to make sense of the errors in the next paragraph without understanding what led to the errors in the previous paragraph.

This is excellent. There is also one more candidate advantage to increased volatility, at least during the accumulation phase: there is evidence to suggest that investors making regular contributions ought to prefer higher volatility for a given level of return. This flies in the face of finance theory, but there is some sense to it. Dollar Cost Averaging is not what an investor with a lump sum should prefer, but an accumulator with little choice can accumulate more shares of a volatile asset by virtue of being able to purchase more shares when the extremes below the mean take hold. See http://ddnum.com/articles/dollarcostaveraging.php

Also: yes, there is a potential mistake in reasoning with the claim “this bet will not accumulate wealth over time because the volatility inherent in the coin flip impedes capital growth.” It’s beyond debate, however, that there is a “rebalance bonus,” as William J Bernstein puts it. http://www.efficientfrontier.com/ef/996/rebal.htm

In response to Chris’s comment above…it seems that some people (namely statistics people) have trouble with the coin flipping example. And there have been some critics of the idea that rebalancing can enhance returns. The best articulation of this is in Chambers and Zdanowicz “Limitations of Diversification Return”. They point out correctly that the expected wealth (and expected return) is indeed growing over time. However the expected mean return is not a great descriptor of the central tendency of the final wealth distribution. The median (zero) is much better. Even if you are not a statistics person, its easy to understand that you should expect to have about an even number of heads and tails over any trial…thus, by construction, you should expect no growth in this example. The full article “Volatility Harvesting in Theory and Practice” is coming out in the Winter 2015 issue of Journal of Wealth Management. In it I avoid discussing coin flipping and focus on investments. I hope the new paper addresses the issues raised by critics. Rebalancing can add value and there is also risk of underperforming. But just because there is risk, doesn’t mean its a bad idea to rebalance.

The terminal wealth is the result of a product of independent, equally distributed random variables e.g.

Final Wealth = (R1*R2*R3* … *R1000)*Initial Wealth

where R is your discrete process with mean 1.25 and variance 1.125.

Therefore, unless I’m mistaken, it will follow a log-normal distribution (take the logs of both sides in the expression above – the sum of logs will be approximately normal if we appeal to the central limit theorem). If you look at a histogram of the log of the terminal wealth, this might be more instructive. You can do this in R doing something like:

terminal.wealth = numeric(1000000)

for(i in 1:1000000){

# Each example has 1000 coin tosses

s = sample(c(2,0.5),1000,replace=TRUE)

s = c(1,cumprod(s))

terminal.wealth[i] = tail(s,1)

}

plot(density(log(terminal.wealth)),lwd=3)

lines(seq(-100,100,by=0.01),dnorm(seq(-100,100,by=0.01),

mean=mean(log(terminal.wealth)),sd=sd(log(terminal.wealth))),

col=”red”,lwd=3)

The mean of the log terminal wealth is about zero, hence the median of the log normal variable (the terminal wealth) is exp(0) = 1 i.e. you gain nothing. In the above example with a 1000 coin tosses, the variance of the log terminal wealth is about 480, so the mean of the terminal wealth itself is exp(480/2) = 1.7e+104.

The wiki link to the log normal distribution might also be helpful:

https://en.wikipedia.org/wiki/Log-normal_distribution

I love it when a comment chain has embedded R code in it!

The code above simulates the fully invested case, where you are flipping coins. If you rebalance to half invested, half of the assets in your pocket (risk free), then you need to rerun the above code with the parameters “sample(c(1.5,0.75)” because you can only gain 50% and only loose 25% each period.

Now the distribution is centered on a positive number. Indeed if you calculate the median of terminal.wealth in the fully invested case, the median is 1 (indicating no growth). In the second case, the median is a large number (indicating growth). In both cases, the means are very large numbers, skewed by the outlier cases where there happen to be many more heads than tails.

Well said!

In this simple example, you don’t need simulations. Analytically, the expected growth rate for a large number of trials of the rebalancing case is 0.5*ln(1+0.5)+0.5*ln(1-0.25) = 0.059. That is, 5.9% expected growth each period. For the full investment case, 0.5*ln(1+1.0)+0.5*ln(1-0.5) = 0.

Thank you for the interesting article, it surprises me how practitioners do not often grasp the relationship between compounding return and risk.

BTW when you say “half the volatility of the return”, I guess you mean half the variance of the returns, and also “approximately” is actually equal in the long term (or mathematically).

“We often forget that growth in wealth approximately equals the arithmetic average return over an investment horizon less half the volatility of the return. This is a fundamental, indisputable mathematical principle”

Growth in its simplest meaning must be a ratio between ending and beginning wealth, this end/beginning ratio by definition do not care what happened in between.

The author may be referring to the arithmetic drag, which happen because something that goes up and down by the same ratio (e.g., +-20%) do not end up in its original place. The reason being each time a ratio is applied, the basis of the percentage calculation is changed. The arithmetic mean is a familiar tool but it is not a suitable tool to calculate return. If it must be used, then you will need to take account of the the above error, the volatility drag, which can be approximated by variance/2, such that, (arithmetic mean return) – (variance

of return)/2 ≈ geometric mean return, which is the above described ratio.

A higher variance however will not produce a lower geometric return. If we stick with the begin/end definition of return.

That being said, Shannon has demonstrated how volatility can be harvested systemically using what’s later called CPPI in the 1960s, using a similar coin toss example as described above. It subsequently inspired much study in this area of portfolio selection. In my study of the above subject, I found this site. I hope this may help anyone who stumble on this in a search.