## Wednesday, November 2, 2011

### Confusion about Correlation of Investments

Most of us have heard that it is good to hold asset classes with low or negative correlation. The informal explanation for this is that risk is lower because when one asset class, such as stocks, is going down, another asset class, such as bonds, is going up. However, this explanation is misleading.

It is possible for two investments to both be going up over a period of time, but have negative correlation. Consider the following example:

Investment A earns either 2% or 20% each year based on a 50/50 coin toss. Investments B, C, and D do the same. Investment B's return is based on the same coin as A uses. Investment C uses its own independent coin. Investment D does the opposite of A's coin. All 4 investments have an expected compound return of 10.63% (for math geeks, this is 1 less than the square root of 1.02 x 1.20).

Even though the investments all look the same based on their returns, their correlations are different:

A and B are +100% correlated (perfect correlation).
A and C are 0% correlated (uncorrelated)
A and D are -100% correlated (perfect negative correlation).

We can see the effect of correlation by looking at the expected compound return of investing strategies that use half investment A and half of each of investments B, C, and D (assuming yearly rebalancing):

Half A, half B: 10.63%
Half A, half C: 10.82%
Half A, half D: 11%

Because A and B are exactly the same, it's not surprising that a 50/50 mix looks the same as either investment on its own. The expected compound return goes up when we mix independent investments A and C. The return is highest for perfectly negatively correlated investments A and D. In this case, every year one investment returns 2% and the other returns 20% for a blend of 11%. Of course, investments like A and D don't exist in the real world or else you could get a certain return of 11% without any risk.

Getting back to the informal explanation of correlation, it's not the case that when one investment goes up, a negatively-correlated investment must go down. The real explanation relates to how the investments perform relative to their average returns.

When one of the investments returns only 2%, this is a downside surprise, and when it returns 20% we have an upside surprise. Investments A and B always have upside surprises together and downside surprises together. Investments A and C have surprises in the same direction half the time, and A and D always have surprises in opposite directions.

So, correlation has nothing directly to do with whether investments go up or down; it has to do with whether they tend to have surprises in the same direction. If an investment has an expected return of -10% and one year it returns -5%, this is an upside surprise. If another investment has an expected return of 10% and returns 5% one year, this is a downside surprise. Correlation measures the extent to which two investments tend to have surprises in the same direction.

1. An interesting bit you touched on was that perfect negative correlation increased expected returns. I don't understand the math that led to a 10.63% return intuitively, so I'll trust you on that. Can you explain why this is true without using the calculation? Seems like a 50/50 chance of either 2 or 20 would lead to an average compounded return of 11.00%.

2. @Gene: Consider what happens after two years. With the negatively-correlated A and D you would get 11% each year which compounds to 23.21%. With just A on its own the "average case" in the compound sense occurs when you get one year of 2% and one year of 20%, which compounds out to only 22.4%. The 10.63% comes from finding the one-year return that compounds to 22.4% after 2 years.

3. Thanks for that. Does it follow that the less a portfolio's annual returns fluctuate, the higher the compounding will be? That didn't seem intuitive to me, but I suppose it makes sense.

I just tried a more drastic example including a negative return, and it was even more exaggerated. Something to make me think.

4. @Gene: If two portfolios have the same average yearly returns, then the more volatile one will have a lower average compound return. By "average return" I mean just adding up the percentages and dividing by the number of years. By "average compound return", I mean compounding out all the returns and taking the n-th root, where n is the number of years.

5. Do you put any credence in risk-adjusted return? Seems that accepting lower returns by mixing in a few GICs might alleviate some of the penalty of lower compounding we see in highly volatile portfolios.

I'd rather be 100% stocks personally, since mixing in GICs will reduce volatility, but also returns (probably) over the long term.

6. @Gene: The value of risk-adjusted returns depends on how you do the adjusting. The CAPM way is to make investing at the risk-free rate the same as investing in a stock index. This type of risk-free rate is used as a way to measure whether your portfolio is on the efficient frontier. But this has little to do with an investor's risk tolderance. The way that I do risk-adjusted returns for myself is based on maximizing the compound return of my portfolio with the added consideration of avoiding big blowups (black swan events) to the extent that I can. Morningstar have their own version of risk-adjusted returns that is roughly 3 times more punitive to the level of volatility than my approach is. Morningstar claim that their method best matches investor feelings, but I'm not overly concerned about emotional responses from the typical investor in choosing my own portfolio.

7. I agree with your assessment of Morningstar's focus of investor feelings. I'm not quite as well-versed in math as you, but I do prefer a lumpy 15% to a smooth 10%, to paraphrase Buffett. Feelings are very unreliable, given that we feel pain in loss at double the rate of pleasure derived from gains.

I'm at a point where a 7% decline in one day feels horrible, but I also realize I'm being irrational and I don't make any sudden portfolio changes. Our feelings are less powerful if we detach ourselves from them, trying to inject some rationality.

Thanks again for taking the time to discuss this with me.