How many times have you heard, or said, “past performance is no guarantee of future performance” or something similar?

SEC rule 482 explicitly contains the language “past performance does not guarantee future results,” and the SEC requires this language in rule 482 advertisements.

While past performance may not predict future performance, we learn a great deal about what is possible, and perhaps about what is likely, by looking at past performance. For example, almost everything we know about the riskiness of stocks and bonds is based on examining what has happened in the past.

The standard deviation of annualized returns on the S&P 500 has averaged about 17.5% during the post-WWII period. Do you have a better way to estimate the volatility in the future? (If you do, please let us know!)

Because for many purposes we have no better guide than history, we study past results despite the knowledge that past results are no guarantee of future results.

In a previous post, we found that over the last 34 years (an arbitrary period that was based on data availability) the covariance matrix of global market sector equity markets looked like the following.

There is no reason to believe that these specific historical correlations are good predictors of the correlations going forward.

For example, during the period, the correlation between the returns to the Japanese market and the North American (US plus Canada, which is dominated by US), was .48. There is no valid reason to believe that that low correlation will continue.

Perhaps there is more reason to believe that looking forward, the expected correlation between any two markets would be around .73, because that is the historical average.

Not Enough Data

Most investors invest for the long term, which is measured in decades. Ideally, we’d want to know what are the long-term correlations likely to be. Unfortunately, there is not enough historical data to state with any degree of confidence.

Here’s an explanation. Suppose we want to know the correlations among markets over a period of ten years. To use historical data, we would need independent ten-year periods. There are only ten such periods in a century.

There’s a statistical rule of thumb that suggests that statistical validity you usually need at least 30 observations of whatever the phenomenon is. So, we’d need 3 centuries of data, at least. Which we don’t have.

Hence, we really don’t have enough data to make a statistically valid historical inference on what long run correlations to expect.

Correlation, Covariance, and Mean-Variance Optimization

We’ve talked about correlation, because it is intuitively more understandable for most people when compared to its close relative, covariance.

The math at the heart of modern portfolio theory is something called quadratic optimization. You don’t have to know what it is or how it works, except that it involves three parameters that we have to estimate.

Those three parameters are:

1. Expected return for each asset (or asset class)
2. Expected volatility (measured as variance of return) for each
3. For each pair of assets, the covariance.

Expected return is fairly straightforward. It is what you expect, over the long run, the return from owning that asset class will be. For US stocks, the long run historical return has been about 10% per year, compounded. (You sometimes see 12%, which is not wrong, but rather represents the arithmetic return instead of the compound return.)

Expected volatility is a bit less intuitive. Volatility is measured as the standard deviation or variance of returns. These two numbers—standard deviation and variance—represent the same underlying information. Standard deviation is the square root of variance. For purposes of the optimization calculations, variance is easier to work with.

Covariance is the least familiar of these three parameters to most people. For the present purposes, it is sufficient to say that covariance is closely enough related to correlation that if you know the correlation, and the volatility, you can get the covariance.

Mean-Variance Optimization

The late University of Chicago economist Harry Markowitz demonstrated in 1952 that given a number of different assets with less than perfectly correlated returns, if we know the expected return and the expected volatility of each asset, and the correlations of return for each pair of assets,[1] for any given level of return (provided that such level of return is greater than or equal to the lowest expected return, and less than or equal to the highest expected return of the various assets) we can calculate the minimum variance portfolio.

When Markowitz did this work in the 1950s, the calculations were laborious and expensive. Now, you can find free online calculators to do them for you.

The Markowitz model provides a mathematical way of finding the portfolios that, for each level of expected return, have the minimum volatility.

Markowitz’ theory produced the famous (and to financial economists beautiful) concept of the efficient frontier.

Efficient Frontier: Examples With Two Assets

To keep the analysis simple, and illustrate the concept of efficient frontier, let’s look at several examples. In each example, we will assume that there are only two assets. We’ll call them “stocks” and “bonds.”

Again, for simplicity, we’ll assume that the expected return on “stocks” is 10% per year with a standard deviation of 20%, and the expected return on “bonds” is 5% per year, with a standard deviation of 10%.

Remember that there’s one more variable: the correlation of returns between the assets. That correlation can run (in theory) from 1 to negative 1. Let’s start with the very long run average correlation of 30%.[2]

What is the Effect of Different Correlations?

Remember that diversification among assets works only if the returns on the assets are less than perfectly correlated. What if the returns are perfectly correlated? There is no diversification benefit. This is illustrated below.

The lack of diversification shows up as an efficient frontier that is perfectly linear. As correlation becomes less than one, the efficient frontier becomes more convex.[3]

Zero Correlation

The following example keeps all the assumptions the same, except that the correlation is zero.

Negative Correlation

What if the two assets were perfectly negatively correlated? (E.g. when stocks go up, bonds go down, and vice versa)?

Introducing the “Riskless” Asset

In the fanciful world of pure academia, there is a concept of the “riskless” asset. This non-existent asset has the almost magical property of delivering positive return (it’s not “expected return” because by assumption the return is certain) and no risk.

This so-called riskless asset is usually assumed (in the US) to be short-term treasury bills.

Here’s how it looks if we add in a “riskless” asset with a return of 4%. You can see it at the lower left.

Now that we’ve developed the concept of the Efficient Frontier, we are ready to introduce the idea of the Capital Market Line. That Line, in theory, will tell us exactly what portfolio of risky assets it is “optimal” to hold.

We’ll address the Capital Market Line in a future post. In the meantime, click here to request one of our free advisor guides.

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[1] Technically the covariance matrix of the returns, but as shown in the previous footnote, if you know the correlations and the standard deviations, you can find the covariance.

[2] Edward McQuarrie, Stocks for the Long Run? Sometimes Yes, Sometimes No, Financial Analysts Journal, 80:1, 2023.

[3] Convexity in this case is a mathematical property that shows up on the graph as a curve. Its practical significance is that for a given level of risk, the return on the portfolios that are part bonds and part stocks is higher than the returns on the straight line connecting the bonds-only and the stocks-only portfolios.