Which investment would you prefer to invest in to diversify your existing stock allocation?

Asset A with an expected:

- 3% annualized return
- 3.5% annualized standard deviation
- 0.00 correlation with your existing investment

Asset B with an expected:

- -5% annualized return
- > 50% annualized standard deviation
- 0.00 correlation with your existing investment

Easy question right? Perhaps not.

Asset B may actually improve long-term returns and reduce risk at the portfolio level, whereas an investment in Asset A may just be a drag on performance. This post will walk through an example, outline the math behind the results, and hypothesize how an investor may want to think about this phenomenon. Going one step further, I will outline how this may, in part, explain the

low volatility anomaly (one example being that low-volatility stocks have produced higher risk-adjusted returns than high-beta stocks in most markets studied).

**Example **

The chart below outlines the performance of an investment in the S&P 500 (using actual monthly S&P 500 returns) going back to the SPY ETF inception (I was being lazy), as well as a monthly rebalanced allocation consisting of a 90% weight to the S&P 500 and a 10% weight to::

__Asset A:__ which reverts monthly from negative to positive performance (-0.75%, +1.26%, -0.75%, +1.26%), compounding to a 3% annualized return at a 3.5% standard deviation; in the chart below this assumes price action of $100 to $99 to $100 to $99... but with an incremental 3% return built in.
__Asset B:__ which reverts monthly from negative to positive performance (-15.4%, +17.2%, -15.4%, +17.2%), compounding to a -5% annualized return at a 56.5% standard deviation; in the chart below this assumes price action of $100 to $85 to $100 to $85... but with an incremental -5% negative return built in.

Despite the 8% annualized outperformance of asset class A vs B that compounded to a 100% gain in asset A and a 70% decline in asset B, the 90% stock / 10% allocation to asset B results in a combined portfolio with higher returns, a higher sharpe ratio, and a lower drawdown relative to an allocation to the positive returning asset A (the benefit of the allocation to asset A was the reduced standard deviation as that allocation reduced risk asset exposure by 10%).

We see below how much asset B benefits the portfolio when both both asset A and B are on equal return footing.

**What gives?**
William Bernstein wrote a great piece on the topic back in 1997 titled

The Rebalancing Bonus. The whole article is worth a read, but he shares the following formula that calculates the "bonus" received by rebalancing across asset classes.

**RB**_{1,2
}= X_{1}X_{2} {SD_{1}SD_{2} (1 - CC) + (SD_{1}
– SD_{2})^{2 }/ 2}

RB = rebalancing bonus

X1 = allocation weight to asset 1

X2 = allocation weight to asset 2

SD1 = standard deviation to asset 1

SD2 = standard deviation to asset 2

CC = correlation coefficient

Before you complain too much about the

math, I'll walk through the equation from left to right (assuming "all else equal") with the applicable points to the example above in bold:

- A more balanced blend (i.e. a closer to 50/50 weighting) will provide a greater rebalancing bonus
**A higher standard deviation of either asset class will result in a greater rebalancing bonus as an investment gets "more bang for your buck" **
- A smaller correlation (or negative correlation) will result in a greater rebalancing bonus
**A larger difference in the standard deviation of the two asset classes will result in a greater rebalancing bonus**

In the case of a higher volatility solution, bullets #2 and #4 both result in a higher rebalancing bonus and higher return (all else equal). In my example, the rebalancing bonus went from roughly 10 bps given a 10% allocation to asset A to 150 bps given a similar 10% allocation to asset B.

Asset A: RB = 90% x 10% [~15% x 3% (1-0) + (~15% - 3%)^2/2] = ~10 bps

Asset B: RB = 90% x 10% [~15% x 56% (1-0) + (~15% - 56%)^2/2] = ~150 bps

Given the 10% allocation, the return differential of 8% (the 3% return for asset A less the -5% return for asset B) x the 10% weight is only 80 bps, significantly lower than the benefit of the rebalancing bonus. In this example, in order for the two blended portfolios to have a similar return, the return gap between asset A and B must be closer to 14%.

**Implications**

From an asset allocation perspective, the above has a number of implications.

For one, it certainly makes the case for an allocation to strategies uncorrelated to an existing portfolio that have higher levels of expected volatility. In my opinion, the most obvious strategy that is largely under-allocated to is managed futures and to a lesser extent certain hedge fund styles and (until the last 10 years when everyone piled in)... commodities. I would also note that return expectations for the traditional diversifier, core bonds, are quite low and volatility of those returns are anticipated to remain low (thus a rebalancing bonus near 0), thus an allocation to a high volatility diversifier only needs low (or potentially negative expected) returns to make sense.

Another potential implication of the positive impact higher volatility strategies have on the rebalancing bonus is that this may partially explain the low volatility anomaly seen within asset classes, such as within

stocks and

bonds. When viewed in isolation, the outperformance of less volatile asset classes seems like an anomaly, but when viewed within a broader portfolio construct it makes sense that these lower volatility investments may need a higher expected return to draw in investors.

Using the rebalancing bonus formula and the following inputs which go back to the December 1990 inception of the S&P 500 High Beta and S&P 500 Low Volatility indices, we get the following rebalancing bonuses despite the higher correlation of the S&P 500 High Beta index with the S&P 500 index:

High Beta: RB = 50% x 10% [14.4% x 28.2% (1-0.89)) + (14.4% - 28.2%)^2/2] = 35 bps

Low Volatility: RB = 50% x 10% [14.4% x 11.0% (1-0.75)) + (14.4% - 11.0%)^2/2] = 11 bps

Pretty close to the 38 bps and 12 bp rebalancing bonus they provided in reality. In this instance, that 26 bp differential makes up ~25% of the excess performance a 50/50 S&P 500 / low volatility blend over that time frame.