Rebalancing lets you scrape wealth from randomness.

*A boring portfolio*

Consider this chart of four fictitious stocks. (created in Excel using a random number generator.) Each stock meanders randomly about a fixed price, plus or minus 10%.

In this article we will consider how we might use these four equities together in a portfolio. A typical approach would be to distribute 1/4 of our wealth in each one of these stocks and hold the investment. Unfortunately, with these (flat) equities, that approach would not make money.

We can, however, turn this portfolio into a wealth generating machine using a simple procedure called rebalancing.

Here’s how:

Initially invest 25% of our portfolio in each stock. Then, at the end of each trading period “rebalance” by selling some of the profits we made in the stocks that went up and purchasing more of the stocks that went down so that we are again at 25% in each equity… That’s it.

How well does it work? Check out the next chart, Figure 2. If we start with an investment of $10 distributed across the four stocks, the investment grows 100% to $20 after 300 trading periods.

*Why does this work?*

To understand why this works, let’s take a look at a simpler example. Imagine a portfolio of only two equities. Stock B’s price remains constant at $1, while Stock A’s price alternates between $1.00 and $1.10 each day. See Figure 3.

That sort of pattern does not exist in a real market, but it is a useful example to help explain how rebalancing works. (This example is borrowed from a paper by Kalai and Vempala.)

Now, let’s carry on with the strategy in detail. Suppose we have $1 to invest, and we start on the first day by investing 50 cents in each equity. On the second day, stock A rises 10% to $1.10, and our 50 cents there becomes 55. The 50 cents in stock B remains constant at 50. At the end of the second day, our portfolio is worth $1.05 overall. *Whoo hoo!*

We rebalance by transferring 2.5 cents (our profit) from B to A, so that we now have 52.5 cents in each.

On the third day stock A drops 9.1% to $1.00 and our holdings in A drop to 47.72 cents. Fortunately our 52.5 cent holdings in B offset that loss, so we’re still above our original $1 investment at $1.0022.

Each day subsequently the portfolio alternately goes up 5% and drops 4.55%. By the end of the 11th day we’ve accumulated a 1% profit. This increase continues forever.

By rebalancing each day we are effectively reducing our bet on A when it is high (and about to drop), and increasing it when it is low (and about to rise).

*Another way to look at it.*

Another way to view this is to consider that the “true” value of stock A is $1.05 (this is the average value). It may vary up or down but it always reverts back towards the true value. Rebalancing automatically adjusts our bets to take advantage of this “reversion to the mean.”

How does it work with a random stock? The explanation is the same. When we happen to be above the true value of an equity, we automatically take money off the table, and when below we add to the bet.

**How well does it work in practice?**

Daily rebalancing is not feasible because transaction costs would overwhelm any benefit from rebalancing. However the approach adds value even when it is applied less frequently. Rebalancing is especially useful in portfolios with volatile uncorrelated holdings.

**Related writings**

- The Rebalancing Bonus by William J. Bernstein: Addresses this in a more formal way.
- Universal Portfolios by Thomas Cover: An algorithm for finding allocations using constantly rebalanced portfolios with theoretical guarantees.
- Efficient Algorithms for Universal Portfolios by Kalai and Vempala: How to find Cover’s allocations in a tractable way using randomized algorithms.

keywords: portfolio rebalancing, wealth management

related posts: Information Theory, Technical Analysis

Mija

August 11, 2012

Wow impressive explanation of the concept.

Jeremy Roseberry

January 19, 2013

Very Interesting. I am assuming that you would be able to adjust this idea slightly for a long-short portfolio?

Aaron

April 1, 2013

Would you mind uploading your excel spreadsheet that you used to do this experiment?

Tucker Balch

April 8, 2013

That’s a fair request. I’m kinda bogged now now and not able to do it right away :-).

Alexander Mamonov

January 14, 2014

I done spreadsheet with examples from this article, you can view it here https://docs.google.com/spreadsheet/ccc?key=0AlxSoik7QcQndFhvaVpoMjN3dlluQVRhR2VkdjRsX0E#gid=2 .

First example in spreadsheet is “An artificial portfolio helps explain how rebalancing works.”. Second example is “A constantly rebalanced portfolio of 2 random stocks grows in value”. And the last one example is constantly rebalanced portfolio of 2 more realistic random stocks.