Rebalancing Frequency Affects Portfolio Construction

An interesting question which seems unresolved in the investment world: Does investment time horizon affect portfolio construction? If you have longer to invest should you favor riskier assets?

This is essentially the time diversification debate, and it’s a complicated one. Too complicated to get deeply into now. But the question of if investment time affects investment decisions reminds me of another question with rebalancing .

Should rebalancing frequency affect portfolio construction?

Some may think the rebalancing period doesn’t matter when constructing a portfolio to maximize geometric returns, but it absolutely does.  Let’s go back to the Shannon’s demon example.

The Coin Flips

As a reminder from the prior post this is the game:

• Heads, up 50%
• Tails, down 33.33%

Here the arithmetic return is 8.33%, and the geometric return is zero.

All we’re going to focus on is the peak, the maximum geometric return, at each rebalance interval.  From our prior game, the peak was found in the middle, 50/50. A ridge runs out straight along the top of the surface, always at 50/50 no matter how many repetitions before rebalancing.  It’s a symmetrical shape at all rebalancing periods.

However, the reason the shape always peaks at 50/50 is because the coin flip example has the same geometric return as cash. Zero.  If the coinflip game provides a slightly positive geometric return, this changes.

A Moving Peak

Let’s change the coin flip to only lose 32% on tails.  This changes the stats of the coin flip to:

• Heads, up 50%
• Tails, down 32%

Here the arithmetic return is 9%, and the geometric return is 0.009955%.

So now we have a slightly positive geometric return.  This means the proper ratio for the game rebalanced each round is near 56%,¹ which is where we find the absolute peak in the overall surface. 

But as repetitions increase, the peaks slide to the right.  You can see this as the peaks of the arc indicating equal geometric return move towards 100% as repetitions increase.

This is most apparent when viewing straight on from the top.

Here is the location of the peak over increasing rebalancing frequencies. 

So you can see, the rebalancing period does affect your portfolio selection when trying to maximize the long term return. 

If we flipped to a negative geometric return on the coin flip, the opposite occurs. We should reduce the exposure to the risky coin as rebalancing frequency increases.

Holding Time Does Affect The Construction of A Rebalanced Portfolio

In a world of pure randomness, with the goal of maximizing long term return, the rebalancing period should affect your portfolio composition as long as the components have different geometric returns.

Now this point doesn’t come close to solving the time diversification debate.  It’s much more complicated than this and examines more than just the peak return. But from what I can tell, the effects of rebalancing don’t often get included into the discussion. And at that level, “holding period” should matter.

From a practical purpose beyond coin flips how do we deal with this?

Short answer: use inputs so that you don’t have too. Where possible try and build portfolios with return, volatility and correlation estimated at the planned rebalancing frequency. If the frequencies of the inputs match the rebalance period, you are rebalancing on the first “period”.

If you’re going to rebalance daily, make sure you build the portfolio with daily data. For monthly rebalancing, use data describing the monthly properties. This way you don’t have to worry about how the optimal portfolio morphs through different rebalancing periods.  You can just rebalance every period.

1- .5 / .32 – .5/.5 = 56.25%. From the Kelly Criteria formula of a coin flip that doesn’t lose everything bet. Investing Games – (breakingthemarket.com)

2-In my portfolio construction posts, I only partially did this. The correlation input just assumes monthly correlations equal daily correlations, which isn’t necessarily correct, and the extrapolation from daily standard deviation to monthly is technically only an approximation because it doesn’t include the return in the calculation.