# Trading off Return and Correlation

Abstract: It’s easy to evaluate a trade-off between risk and return, but much more challenging to understand a trade-off between correlation and return, particularly as the number of securities in a portfolio increases. Investors often underestimate the significance correlation plays in portfolio performance, and underweight lower return, low correlation assets as a result. Correlation can be, and often is, more important than return.

In an ideal world, we’d be able to reduce all investment decisions to a few simple variables. Unfortunately, we do not live in an ideal world. Investing can be a difficult and confusing game. Yet the goal has long been to make it as simple as practical.

In 1952, Harry Markowitz introduced an optimization method that has become the foundation of Modern Portfolio Theory. His mean-variance model, while flawed, is nevertheless useful. It relies upon three significant inputs: measures of return, risk, and correlation. Each plays a critical role in portfolio performance, yet investors often prioritize return and risk over correlation when evaluating potential investments.

If return alone mattered, selecting investments would be straightforward. All else equal, higher returns are better.

Investment selection becomes a bit murkier when we factor in risk. Although investments with higher risk-adjusted returns may be more efficient, they are not necessarily more appropriate. After all, an asset with a high risk-adjusted return can still produce a low absolute return if its risk is also low.

To fully capture the value of high risk-adjusted return assets, investors must have the ability to leverage their investments. Fortunately, most institutional investors these days do have the ability to employ at least a limited amount of leverage by using futures and other derivatives instead of fully funded securities. Let’s suppose then that we do have the ability to leverage. Are risk-adjusted returns then the best measure of investment value?

No. Dividing expected return by expected volatility gives you an ideal measure for evaluating assets in a world where you can only invest in a single asset. It does not, however, give you an ideal measure in a world where you can invest in multiple assets. This is because it ignores a fundamental component of portfolio risk: correlation.

Of Markowitz’s three significant measures, correlation is the most difficult to grasp. Although it is conceptually simple, measuring the degree to which two securities move together, it is practically complicated. While risk and return depend only upon the behavior of a single asset, correlation depends on the behavior of multiple assets. Its computation complexity grows exponentially as assets are added to portfolios because it requires one to estimate the dependency between each pair of securities.

It’s easy to evaluate a trade-off between risk and return, but much more challenging to understand a trade-off between correlation and return, particularly as the number of securities in a portfolio increases. Investors often underestimate the significance correlation plays in portfolio performance, and underweight lower return, low correlation assets as a result. Correlation can be, and often is, more important than return.

Consider, for example, a world in which all assets have the same risk and return, differing only by their correlations with one another. Since all assets have the same return, all unleveraged, long-only portfolios will also have that same return. Such portfolios may have different volatilities, however, depending upon the correlations between their component securities.

In this world, we can form more efficient portfolios only by reducing portfolio standard deviations. There are two basic ways to do this. We can swap securities for less correlated securities or we can add more securities, provided that the newly added securities are less than perfectly correlated with the portfolio.

We can estimate the relative impact of each approach by noting the conditions under which they produce portfolios with identical risk. Suppose we form two equally-weighted portfolios that each consist of assets that are identically correlated with each other asset in the portfolio. So, Portfolio 1 consists of assets that are each X correlated with each other and Portfolio 2 consists of assets that are each Y correlated with each other. We can then determine the number of assets that each portfolio would need to hold to produce an identical level of risk. Portfolios consisting of highly correlated assets will require a greater number of assets than portfolios consisting of less correlated assets. A portfolio of four securities, each 0.6 correlated with each other, for example, will produce the same expected volatility as a portfolio of two securities, each 0.4 correlated with each other.

Using a portfolio of uncorrelated assets as one of the two portfolios can illustrate just how profound a role correlation can play in portfolio performance. Suppose our first portfolio consists of two uncorrelated securities. How large would a portfolio of 0.5 correlated securities need to be to produce the same level of risk?

Don’t just read the answer. Think about this for a second. Come up with an estimate. Got one? Is it 5? 10? 100? More?

The answer is that, in this world, where assets all have the same risk and return, you can’t create a portfolio of 0.5 correlated securities that is as efficient as the two-uncorrelated asset portfolio. You would literally need an infinite number of securities to do so (proof).

That is borderline mind blowing. Under these conditions, two uncorrelated assets are as effective as an infinite number of 0.5 correlated assets.

This is relevant because asset classes are relatively highly correlated with each other. In fact, most asset classes are simply repackaging the same underlying risks in slightly different ways.

To illustrate this we can create a custom benchmark for several of the most commonly employed asset classes using a three-factor model consisting of two risk premiums, global equity risk and US government bond term structure risk, and one risk factor, currency risk. We first regress asset class returns on factor returns to estimate each asset class’s factor exposure. We then derive benchmark returns using these exposures, which allows us observe the correlation between each asset class and its corresponding benchmark.

Most asset classes are highly correlated with their benchmark. The equity indexes are all at least 0.9 correlated. Most of the fixed income indexes are at least 0.8 correlated. The residual, unexplained risk tends to be small and often produces negative or only modestly positive information ratios, thereby providing minimal diversification benefits with little to no return enhancement.

This suggests that, regardless of how you allocate your portfolio among traditional asset classes, you are unlikely to receive substantial efficiency gains beyond those offered by a simple portfolio consisting of broad equity and fixed income exposure. In the long run, debates over the “right” splits between large cap and small cap equity, investment grade and high yield credit, and foreign versus US real estate are largely a waste of time.

To be fair, our hypothetical world was unrealistic in that we required every asset to have the same expected risk and return. If we relax these constraints and allow assets in different portfolios to have different returns, we can evaluate trade-offs between correlation and return.

In this case, we create two portfolios of the same size and determine the security return that would be required of each to produce an identical risk-adjusted portfolio return.

Under these circumstances a portfolio of two 0.5 correlated assets can match the performance of a portfolio of two uncorrelated assets, but only by including assets with returns that are 22% higher. That may not sound like a lot, but it is material, particularly in today’s low interest rate world.

The required return increase grows with each step up in portfolio size. A portfolio of seven 0.5 correlated assets would require double the return of a portfolio of seven uncorrelated assets to produce the same risk-adjusted return.

The major takeaway of all this is that you shouldn’t immediately dismiss low correlation assets simply because they offer lower expected returns. This is particularly true for investors that have the ability to employ leverage.

The analysis also underscores the value that hedge funds can provide, at least in theory. Hedge funds can go long and short, employ leverage, and trade across a diverse array of markets and asset classes. This provides them with the potential to harvest a much larger universe of uncorrelated risk premiums than are available to long only vehicles. Unfortunately, most hedge funds fail to do this. Those that do, however, can offer tremendous value to their investors even without generating alpha.