An Example Of Using Probability To Build An Intuition For Correlation

The power of negative correlations is powerful when you see how rebalancing increases your expected compounded return. This isn’t intuitive to a typical, especially retail investor.

I’ve tried to make it easier to understand:

• You Don’t See The Whole Picture (4 min read)
• Your Portfolio Intuition Is Poor (5 min read)
• Lesson From Coin Flip Investing (1 min read)

One of my favorite finance educators recently wrote an absolute must-read thread on this topic.

He creates a model with 2 simplifying features:

• There are only 2 stocks
• They are rebalanced to equal weight

You can use the intuition from this exercise to guide your portfolio thinking more broadly. It’s beautifully done and you should work through it carefully not just for the intuition but the practical knowledge of how to compute an expected return in a compounding context. However, there is a part I struggled with that I want to zoom in on because I’ve never before seen it presented as @10kdiver does it:

He converts probability to an estimate of correlation!

This is really cool. But because I struggled and the learnings of the thread are both important I dual purpose to writing this post.

1. The meta-lesson
   This is the easy one:
   When I read the post, it was easy to nod along thinking “yep, that makes sense…ok, ok, got it”. Except for that, I don’t “got it”. I couldn’t reconstruct the logic on my own on a blank sheet of paper which means I didn’t learn it. Paradoxically, this demonstrates how good @10diver’s explanation was. Extrapolate this paradox to many things you think you learned by reading and you will have internalized a useful life lesson — get your hands dirty to actually learn.
   
2. Diving into the probability math I struggled with.
   Let’s do it…

Zooming In: The Probability Basis For Correlation

Assumptions

Example computation for CAGR (also seen in tweet #4):

CAGR_A = =((1+A_up_size)^(A_prob_up*hold_period)*(1+A_down_size)^(A_prob_down*hold_period))^(1/hold_period)-1

Define the probability space

We are focusing on tweets 6-10 in particular. The summary matrix:

Understanding the boxes:

Start with the logic: “what would the probability space look like if they were perfectly correlated?”

• Top left box = X (This corresponds to both up)
  They would go up together 80% of the time if they were perfectly correlated. We generalize “probability of stocks up together as X”
  
• Top right box = .8-X (This corresponds to B up, A down)
  Since stock B goes up 80% of the time we know its probability of going down is .8-X
  
• Bottom left box = .8 – X  (This corresponds to A up, B down)
  Since stock A goes up 80% of the time we know its probability of going down is .8-X
  
• Bottom right box = X – .6 (This corresponds to both down)
  With one box left it’s easy, we know all the boxes must sum to 100% probability.
  100% – [X + 80% -X + 80% – X] = X – .6

We called the probability of moving up together X. We set the matrix up using the simple case of the stocks being perfectly correlated (ie moving up together 80% of the time). But they don’t need to be perfectly correlated. So now we can find the range of X, a joint probability, that is internally consistent with each stock’s individual probability of going up.

What is the probability range of X ie “how often the stocks move together”?

Upper bound

X is defined as “how often they move up together”. Another way to think of this:  the upper bound of the joint probability is the lower bound of how often either stock goes up.

Let’s change the numbers and pretend stock A goes up 50% of the time and stock B goes up 80% of the time. Then 50% is the upper bound of how often they can both up together. (Stock A is the limiting reagent here, it can’t move up more than 50% of the time). So the minimum of their “up” probabilities represents an upper bound on X.

Back to the original example, the upper bound of how often these stocks move together is 80% because the minimum of either stock’s individual probability of going up is 80%. Mathematically this is

.8 – X > 0 so:

Upper bound of X = 80%

Lower bound

Proceeding with the logic that no box can be negative, the bottom right box cannot be less than 60%. This represents the least co-movement possible given the stocks’ probabilities.

Lower bound of X = 60%

Think of it this way, if there were 10 trials each stock could have 2 down years. If they were maximally correlated the stocks would share the same down 2 down years. If they were minimally correlated they would never go down at the same time. The probability of both stocks going down simultaneously would be zero, but since the 4 down years would be spread out over 10 years, the pair of stocks would only go up simultaneously 60% of the time.

Checkpoint

The probability of the stocks moving together, X, is bounded as:

60% < X < 80%

X is not a correlation. X is a probability. The fact that the stocks can co-move from 60-80% of the time maps to a correlation.

A Key Insight

A zero correlation means 2 variables are independent! If they are independent, the joint probability is a simple product of their individual probabilities.

That’s why the 0 correlation point corresponds to 64%:

X = .8 x .8 = 64%

Loosely Mapping Probability to Correlation

If you’re feeling spry, you can use the probability space and covariance math to compute the actual correlation. But, we can estimate the rough shape of the correlation using zero correlation (statistical independence corresponding to X = 64%, the joint probability of both stocks going up together) as the fulcrum.

Look back at tweet #10 to see the extremes:

At the lowest correlation, corresponding to a co-movement of 60% frequency:

• The correlation is slightly negative. It’s below the 64% independence point.
• The stocks NEVER go down together.
• The stocks move in opposite directions 40% of the time
• When the stocks do move together, it’s up.
• The stocks have a negative correlation despite being up together 60% of the time.

At the highest correlation point, corresponding to 80% frequency of co-movement:

• The stocks go up 80% of the time together
• They go down 20% of the time together
• They never move in opposite directions.
• The magnitude of the max positive correlation is greater than the magnitude of the maximum negative correlation since the independence point is near the lower end of the range.

Rebalancing Benefits Improve As Correlations Fall

The thread heats up again in tweet #17 by identifying the possible values of the portfolio rebalanced to 50/50 at the end of a year.

In tweet #18, those states are weighted by the probabilities to generate expected values of the portfolio, which can finally be used to compute the CAGR of the portfolio if rebalanced annually.

The lower the value of X (the joint probability of the stocks moving up together), the lower the correlation.

The lower the correlation, the higher the expected value of a rebalanced portfolio.

The remainder of the thread speaks for itself:

• When X = 60% (ie, strongly negative correlation), we have:
  • Without re-balancing: $1 –> $5.94
  • With re-balancing: $1 –> $17.85 (>3x as much!), over the same 25 years.
  • Thus, negative correlations + re-balancing can be a powerful combination.
    
• If we do this well, our portfolio can end up getting us a HIGHER return than any single stock in it! We just saw an example with 2 stocks. Each got us only ~7.39%. But a 50/50 re-balanced portfolio of them got us ~12.22%. When I first saw this, I couldn’t believe it!
  
  [Moontower note: in practice, portfolios usually have many names and a variety of weighting schemes. While the intuition is similar the math is more complex and you are now looking at a matrix of pairwise correlations, assets with varying volatilities and therefore different weights in the portfolio]
  
• This is the ESSENCE of diversification. We minimize correlations, so our portfolio nearly always has both risen and fallen stocks. We “cash in” on this gap via re-balancing — ie, we periodically sell over-valued stocks and put the money into under-valued ones.
  
• Negative correlations aren’t strictly necessary. We could use stocks with zero — or even positive — correlation. But the MORE heavily correlated our stocks, the LESS “bang for the buck” we get from re-balancing.

Wrapping Up

The idea that low or negative correlations improve with falling correlations is common knowledge in professional circles. Still, the intuition is elusive. The sheer size of the effect on total CAGR is shocking.

Until @10kdriver’s thread, I hadn’t seen a mapping from probability which is intuitive to correlation which is fuzzy (recall that when the 2 stocks had a negative correlation they still went up together 60% of the time!)

When I read the thread, I found myself nodding along but I needed to walk through it to fully appreciate the math. That’s a useful lesson on its own.

If you found this post helpful, I use another of @10kdiver’s threads to show how we can solve a compounding probability problem using option theory:

Solving A Compounding Riddle With Black-Scholes (13 min read)