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.