Path: How Compounding Alters Return Distributions

Compounded returns experience “variance drain”. This idea captures the fact that typical result of compounded returns is lower than if you compute arithmetic returns even though the expected value is the same. We mostly care about compounded returns. This describes the situation in which your bet size or allocation is a fixed percent of your wealth, savings, or bankroll.

This is in contrast to keeping your bet size fixed (ie if you invested $10,000 in the stock market every year regardless of your wealth).

The distinction is critical because as humans we experience the path of our investments so we care about the distribution of returns in addition to the expected value.

Let’s back up for moment.

Recapping Intuition

• What land are we in?
  
  • Compounding LandIf you bet 1% of your wealth on a coin flip and win then lose, you are net down money. This is symmetrical. If you lose, then win, still down money.1.01 * .99 = .99 * 1.01
    
  • Additive LandIn additive or non-compounding land we bet a fixed dollar amount regardless of wealth.So if I start with $100 and win a flip, then bet $1 again and lose the flip I’m back to $100. The obvious reason is the $1 I bet when my bankroll increased to $101 is less than 1% of my bankroll.
• The order of win then lose, or lose then win leaves you in the same place in both worlds.The order does not matter if we are consistent about how we size the bet (so long as we are consistent to the style whether it’s fixed dollar or fixed percentage).

So is fixed percentage somehow “bad” in that it opens you up to volatility or variance “drag”? 

Well in the last example we used an alternating paths. Win then lose or vice versa. Let’s look at the case where instead of alternating wins and losses, we trend. Win-win or lose-lose.

• In the additive case, we are either up 2% or down 2%
• In the compounded case we are up 2.01% or down 1.99%

Wait a minute. In the compounded case, we are better off both ways! So the compounded case is not always worse.

The compounded case is better when we trend and worse when we “chop”.

If we bet a fixed percent of our bankroll fair coin toss game we are in compound return land.

Compounding is not “bad”, it just alters the distribution of our terminal wealth

Your net compounded return in the coin-flipping game is negative more often than it’s positive, even though the game has zero expectancy.

So why is the median outcome negative?

It goes back to the trend vs the chop. Compounding likes trending and hates chopping as we saw earlier.

•  Chopping happens more 𝐨𝐟𝐭𝐞𝐧 so you get a negative median
• …but this is balanced by a larger trending bonus due to compounding.

Let’s illustrate.

2 Coin Flips

There are 4 scenarios:

2u (trend)
1u, 1d (chop)
1d, 1u (chop)
2d (trend)

Zoom in on “compounding bonus/drag”:

Observations:

• Chop and trend happen equally.
• The magnitude of the boost/drag is also equal.

3 Coin Flips

There are 8 total outcomes, but again order doesn’t matter. So there are really just 4 outcomes.

The “chops” are bolded. They represent compounding “drag”

Note:

• You drag 75% of the time!
• The larger positive boost magnitudes make up for the frequency.

Now that you have the gist, let’s do 10 flips.

10 Coin Flips

• 65% of the results are chop giving you compounding drag.
• The times you trend though crush your performance if you only bet fixed dollar!

Visualizing “The Chop”

Let’s take a look visually at paths where N=10  to see the “chop”.

Pascal’s Triangle is a quick way to to get the coefficients of a binomial tree. The coefficients represent combinations which are weighted by the probabilities in the binomial expansion.

I enclosed the “chop” or drag paths

100 Coin Flips

• The negative median now becomes very apparent in the “cumulative probability” column.
• The chop occurs in 68% of paths. The median return is -.50% after 100 flips though the expectancy is still zero.
• In additive world if you win 50 $1 bets and lose 50 $1 bets your p/l is zero.
• In compounding world, where you bet 1% each time you are down 50 bps in that scenario.
• The negative median associated with compounding is balanced by better outcomes in the extremes.

Both the maximum and minimum returns in simulations are better than the fixed bet case. This simulation by Justin Czyszczewski (thread) shows just how substantial the improvement is in those less probably trending cases:

Lessons From Compounding Coin Flips

• Your overall expectancy is zero because the common chop balances the rare but heavily compounding trends.
• Paths affect distribution of p/l even if they don’t affect expectancy.

Since we actually experience “path” and all its attendant emotions, it pays to think about the composition of expectancy and returns.