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## Introduction

Value investors try to value an asset based on estimations of future cash flows and invest in those instruments that are most undervalued. However, this approach overlooks correlation between assets and the additional returns that can be captured from rebalancing uncorrelated assets. In my opinion correlation between assets is also a value component and should be taken into account when trying to value an asset.

William Poundstone describes in his book, Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, the "Shannon's Demon". This is a strategy developed by Claude Shannon to make money off a random walk. The strategy invests 50% in cash and 50% in stock and rebalances this portfolio to recover the 50-50 proportion. This strategy will generate a bonus return of about 0.125 * σ ^{2 } if the return of the stock is 0. The σ is the standard deviation of the stock movement. This trading system forces an investor to buy low and sell high, which explains this additional return from randomness.

Let's assume there is an asset A and an asset B. Both have the same volatility, but are perfectly negatively correlated with a correlation of -1. Both also have an expected return that exceeds the risk-free market rate. In this case it is possible to create a portfolio without risk and positive returns by allocating 50% to asset A and 50% to asset B. In addition, the rebalancing forces you to buy low and sell high, which adds additional returns. Unfortunately this theoretical situation doesn't exist in real markets.

In financial markets two things are certain:

- Asset prices change (have volatility) or, in mathematical terms, stock prices have a standard deviation ( σ ) or variance ( σ
^{2 }). - The relative change of two assets are not equal or, in mathematical terms, the correlation between returns of assets is less than 1.

These are the two conditions required to extract returns from randomness by rebalancing. Several articles have been written that describe this mathematical phenomenon. The article Volatility Harvesting: Extracting Return from Randomness, written by Hendrik Jan Witte, describes the mathematics of rebalancing returns in more detail.

The article Volatility Harvesting:Why does diversification and rebalancing create portfolio growth by Paul Bouchey, Vassilii Nemtchinov, Alex Paulsen and David M. Stein describes the importance of diversification and rebalancing of an investment portfolio and why it increases the returns.

## Additional returns of rebalancing

Equal-weighted indices outperform market capitalization weighted indices. For example, the S&P 500 equal-weighted index ETF (RSP) outperforms the S&P 500 index ETF (SPY). Why is this? Is this a market inefficiency? In an efficient market, asset returns have a lognormal distribution. A lognormal distribution has a compounding component 0.5 * σ^{2 }, where σ is the standard deviation. This compounding component is captured when rebalancing a portfolio. There is a positive rebalancing return if the sum of the variations of each asset in a portfolio is bigger than the variation of the total portfolio. The article Equal-weighted strategy: Why it outperforms value-weighted strategies? Theory and evidence, written by Rama Malladi and Frank J. Fabozzi, provides evidence for the outperformance of equal-weighted indices.

However an equal-weight portfolio is not optimal. The returns can be improved by giving higher weights to uncorrelated assets and lower weights to assets with high correlations. Apparently the specific risk profile of an asset can also be a source of alpha.

The volatility alpha of a long positions is always bigger than 0. In theory asset weights can be negative, which would imply a short position. Unfortunately the volatility alpha of a short position is always negative (smaller than 0). As a result, I don't include short positions.

## The optimal portfolio weights

I wrote a computer program that calculates the optimized weights of a set of stocks based on their historical correlations. The program has calculated the optimal weights of a portfolio consisting of about 600 assets as input, including stocks, stock indices, currencies, bond ETFs and commodities. The 10 assets of the optimal portfolio with the highest weights are the following:

- Ishares MBS ETF (MBB) - 87.4%
- Invesco CurrencyShares Japanese Yen (FXY) - 3.1%
- Weight Watchers International, Inc. (WTW) - 0.9%
- Hertz Global Holdings, Inc. (HTZ) - 0.7%
- Career Education Corporation (CECO) - 0.7%
- ACADIA Pharmaceuticals Inc. (ACAD) - 0.7%
- Acadia Healthcare Company, Inc. (ACHC) - 0.7%
- Axcelis Technologies, Inc. (ACLS) - 0.5%
- Pilgrim's Pride Corporation (PPC) - 0.5%
- John B. Sanfilippo & Son, Inc. (JBSS) - 0.5%
- All other assets - 4.3%

The biggest daily drawdown of this portfolio was less than 1% in the period 2008-2018. The volatility of this portfolio is much lower than the average annual 16% volatility of the S&P 500, because of the high allocation to bonds. Returns can be increased using leverage. The theoretical bonus returns or volatility alpha of this optimized portfolio using four times leverage exceeds 10%, with annual volatility that is lower than the volatility of the S&P 500. This is a rebalancing bonus return on top of the returns of the underlying assets. Combining those two would result in a Sharpe Ratio of more than 1.

About 550 out of the 600 assets had a weight of 0.01% or less in the optimal portfolio. This means that, out of the 600 assets, only a small percentage creates volatility alpha. I assume that this percentage will drop further when increasing the number of assets.

Indices don't seem to produce volatility alpha. Rebalancing a portfolio of single stocks will outperform indices in the long term. It could be rational to hedge a portfolio of stocks with a short position in a market capitalization weighted index.

I was surprised that gold was not part of the top 10 constituents. It appears that mortgage-backed securities are a better hedge than gold against the market risk of equities.

## Characteristics of the assets in the optimal portfolio.

I analyzed the 50 assets that were part of the optimized portfolio and tried to find specific characteristics of these assets. Assets that have one or more of the following characteristics seem to generate volatility alpha:

- An asset that benefits from market volatility;
- An asset with large amounts of excess cash;
- An asset with specific regulatory uncertainty or legal risks;
- An asset that is dependent on commodity prices;
- An asset with stable revenues independent of the economic situation;
- An asset that benefits from deflation;
- An asset that has a potential future blockbuster product.

Diversification of a portfolio based on countries or industries created only limited volatility alpha, because correlations tend to be too high.

## Why is rebalancing so difficult?

If rebalancing generates bonus returns, why isn't everybody doing it? Rebalancing is in theory simple, but it is psychologically very difficult to do in reality. You have to buy 50% of the original value if the asset drops 50% in value and you rebalance. If this asset drops 50% four times in a row, you need to invest three times the original investment amount in it. I don't know many investors who will continue to rebalance an asset if it has dropped more than 90%. Instead, most investors will start selling the asset when the price start to drop.

## Conclusion

Rebalancing a portfolio of uncorrelated assets will produce a bonus return. This can be proven mathematically. Based on simulations and calculations the annual bonus return of an optimal-risk diversified portfolio can exceed 10% if it is rebalanced periodically.

**Disclosure:** I am/we are long MBB, FXY, ACHC, WTW, PPC, HTZ. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.