Taking Advantage of Correlation and Portfolio Diversification

Don’t put all of your eggs in one basket. Don’t bet the house on Red 9. These are common refrains that we all intuitively understand: if you drop that basket then all of your eggs will break, if you’re going all in on a single number then more often than not, you’re going to get wiped out. Good advice, right?

If you think so (and hopefully you do), then why would you treat your portfolio any differently? Just like with your eggs or if you’ve ever played roulette then you know it’s a good idea to spread your risk across many different baskets and bets, but how does this work with your investments? Let’s take a moment to discuss portfolio diversification.

Using Correlation in a Portfolio

Portfolio diversification is the practice of spreading your investments across many different assets so that your exposure to any single asset is limited. This basic theme will likely appear often on this blog so let’s briefly touch on the theory and then look at the results.

Below I’ve assembled 14 indices that broadly represent the universe of investable financial assets (stocks, Treasuries, and bonds) as well as a simple allocation strategy that’s 60% stock and 40% bonds. Specifically, the allocation is composed as follows:

• 20% S&P 500
• 10% MSCI EAFE (i.e. International Developed)
• 5% MSCI Emerging Markets
• 7.5% Russell 1000 Value
• 7.5% Russell 1000 Growth
• 5% S&P Mid-Cap
• 5% S&P Small-Cap
• 40% Barclays Aggregate Bond Index

Below is the correlation matrix for these 14 indices. In short, correlation measures the degree to which two assets move together. A high correlation (i.e. close to 1) means that two assets are more likely to move up or down together while a lower correlation (i.e. close to 0) indicates that the assets don’t typically move together. A negative correlation means that the assets move opposite of one another. For example, if assets A and B have a negative correlation and asset A moves up then asset B in likely to move down.

We can see this action at play when we examine the correlation matrix. The S&P 500 and Barclays Aggregate Bond Index have a correlation very close to 0 meaning that they don’t move together much at all. On the other hand, the correlation between the S&P 500 and REITS is about .75 meaning that if the S&P is up/down on a given day, then REITS are likely also up/down.

	Large Cap	Intl. Dev	Emerging	Value	Growth	Mid Cap	Small Cap	REITs	Agg. Bonds	Corp. Bonds	ST Treasury	Int. Treasury	LT Treaury	TIPS	Allocation
Large Cap	1
Intl. Dev	0.89	1
Emerging	0.79	0.88	1
Value	0.97	0.86	0.77	1
Growth	0.97	0.85	0.78	0.9	1
Mid Cap	0.94	0.84	0.79	0.94	0.93	1
Small Cap	0.52	0.44	0.38	0.55	0.49	0.57	1
REITs	0.75	0.71	0.64	0.77	0.71	0.76	0.45	1
Agg. Bonds	0.01	0.14	0.14	0.02	0.01	0	-0.06	0.31	1
Corp. Bonds	0.31	0.39	0.36	0.32	0.28	0.3	0.13	0.45	0.83	1
ST Treasury	-0.32	-0.22	-0.19	-0.31	-0.33	-0.36	-0.24	-0.12	0.59	0.28	1
Int. Treasury	-0.31	-0.21	-0.18	-0.32	-0.29	-0.33	-0.24	-0.01	0.86	0.53	0.71	1
LT Treaury	-0.32	-0.24	-0.22	-0.32	-0.31	-0.34	-0.24	-0.03	0.78	0.49	0.5	0.9	1
TIPS	0.15	0.26	0.3	0.14	0.17	0.16	0.01	0.34	0.79	0.66	0.46	0.68	0.57	1
Allocation	0.96	0.93	0.86	0.94	0.93	0.94	0.62	0.8	0.2	0.46	-0.21	-0.16	-0.19	0.3	1

Correlation and Portfolio Risk

So how can we use the correlation matrix to our advantage? Basically, we can use it to potentially reduce the risk in our portfolio. If we combine the S&P 500 and Long-term (LT) Treasuries in a portfolio then we end up with a portfolio with less risk, but a similar level of return.

Let’s take a totally abstract example to see how this works in practice. Suppose we have two assets, A and B, and we split the portfolio equally between them (50% in asset A and 50% in asset B). Furthermore, let’s assume that the correlation between the two is ‘0’ (like the S&P and Barclays Agg. from our correlation matrix). Finally, let’s assume that the risk (i.e. the volatility) of asset A is 20% and the risk of asset B is 30%. How does this change the risk of our portfolio? Well, we can use the following formula to figure that out! If you don’t like formulas, then don’t worry about it; I’ve added it for completeness and you can still get the intuition without it.

So, what are these variables?

• W_A = Weight of Asset A in the portfolio (50% in our case)
• W_B = Weight of Asset B
• σ_A = Volatility of Asset A (20% in our case)
• σ_B = Volatility of Asset B (30% in our case)
• p_A,B = Correlation of Assets A and B taken from the correlation matrix (0 in our case)
• σ_Portfolio = Volatility of the portfolio (50% asset A and 50% asset B)

If we plug all of this in, we get the following:

What does this tell us? If we split our portfolio 50:50 between assets A and B then we find that the risk of our portfolio is 18%. Recall that the risk of assets A and B were 20% and 30% respectively. But because the correlation between the two is ‘0’, then once we put them in a portfolio together the risk of the portfolio is now less than each of Assets A and B individually. Pretty cool right?!

What happens if the correlation changes? Below is a graph that depicts the same assets A and B mixed in a portfolio 50:50. The X-axis shows varying levels of correlation ranging from -.5 to 1. The Y-axis depicts the volatility of the resultant portfolio. I’ve marked the point for the portfolio we just built with correlation ‘0’ and volatility 18%.

What do we observe by looking at this graph? We see that as the correlation between the two assets increases (moving left to right along the X-axis) then the volatility of the resultant portfolio increases as well; the reverse is also true.

Let’s run through another quick example with our indices and look at some results. Below is the correlation matrix between the S&P 500, Barclays Agg., and Allocation strategy. Looking at the correlation matrix we again see that the correlation between the S&P 500 and Barclays Agg. is approximately 0; however, we also see that the correlation between the S&P and Allocation model is still pretty strong at ~.96 while the correlation between the Barclays Agg. and Allocation model is mild at about .2.

	S.P.500	Barclays.Agg..Bonds	Allocation
S.P.500	1
Barclays.Agg..Bonds	0.01	1
Allocation	0.96	0.2	1

How does this observation translate into risk and performance? Below we apply the same formula from above to calculate the risk of each the S&P, Barclays Agg and Allocation model. We see that the risk of the Allocation model is greatly reduced as compared to the S&P due to the help of correlation and the inherent lower risk of the Barclays Agg.

	S.P.500	Barclays.Agg..Bonds	Allocation
Annualized Standard Deviation	0.143	0.038	0.095

Finally, let’s look at the performance of each of these portfolios and compare the performance over the past 15 years.

The wealth plot above shows the relative performance of investing $1 in the S&P, Agg and Allocation model since 2004. A couple of things stand out: $1 invested in the S&P would have yielded approximately $2.70 by 2020, a dollar invested in the Allocation model would have yielded ~$2.25 while the Barclays Agg. would have yielded about $1.91. We also observe that during periods of high market volatility (2008 and more recently 2020) the Allocation portfolio declines by less than the S&P. In fact, for the first 10 years of our little study, the Allocation model actually outperformed the S&P and was eclipsed only as recently as 2014.

So, what is our take way from this exercise? Having many types of assets in a portfolio can be used to both guard and grow your wealth. By taking advantage of correlation you can lower the risk of your portfolio, guard against market declines and still provide suitable upside appreciation when markets are up.

Thanks for reading!

Aric Lux.