Simple three asset class example
In our experiment we will concentrate on a three asset class universe based on US equities, US 7-10 year Treasuries and global commodities. Based on monthly data since 1992 the asset classes have had the following distribution statistics.
The quick observer will realise that bonds have generated a fantastic risk-adjusted return while commodities have delivered the worst. In the middle we find equities.
Below the correlation matrix is shown and here you get an idea of why commodities have been added to asset allocation portfolios. The asset class exhibits very low correlation to other major asset classes.
Commodities monthly mean return of 33 basis points translate into to just below 4% expected annualised return which makes sense as it is slightly above the long run global inflation trend.
It was the wild upswing in commodity prices leading up to the financial crisis together with the low correlation that sparked the euphoria about commodities in asset allocation. Investors seemed to forgot the long period from 1975-2002 when global commodity prices where almost flat. The expectations for the long-run return in commodities are critical, but we will get back to that.
Monte Carlo simulation shows no gain
Based on the monthly statistics since 1992 we run a Monte Carlo simulation creating 500 synthetic return streams for the three asset classes based on their return distribution. We then multiply each run's return matrix by the Cholesky decomposition of the correlation matrix, so we get the approximately right correlation structure for each simulation.
In the end we create two portfolios with equal weight. The first portfolio comprises only of equities and bonds, while the second one contains all asset classes. We then calculate the annualised Sharpe Ratio for each portfolio across all simulations. The density plot below shows the distribution in the Sharpe Ratio. The black line is the equities-bonds portfolio and the red line represents the portfolio with all asset classes.
Instead of only relying on the historical returns we can observe, the Monte Carlo simulation provides us with a much more rich interpretation of the benefits from adding commodities. The exact structure in the observed asset class returns in the period 1992-2015 is a random event. The simulation draws from the distributions so we get a better picture of the inherent variance that exists across the asset classes.
As the density plot shows we are not improving our Sharpe Ratio by adding commodities to the portfolio. The t-statistic between the two samples is 16 so the difference in mean Sharpe Ratio is significant at the 5% level.
How much does expectations mean?
To get an idea of the sensitivity we calculate the annualised Sharpe Ratio based on new expectations for the correlation matrix (see table below). We have changed commodities to have negative correlation with equities and bonds.
As the new density plot below shows it does not change the overall conclusion. In fact the Sharpe Ratio gets worse for the portfolio with commodities.
If we change the expected monthly return to that of bonds (55 basis points) and adjust the volatility down to equities (414 basis points) while keeping the adjusted correlation matrix we used in the prior example, then things get better but it is still not tilting in favour of including commodities in the portfolio.
The crux of the matter is that commodities have a bad Sharpe Ratio and thus do not add meaningful value to the asset allocation portfolio. In addition the low long-term correlation between bonds and equities does the substantial part of the diversification work.
If we changed the correlation between equities and bonds to 0.5 then our previous analyses would turn out differently. In other words, the hurdle for commodities would be lower as the the negative correlation then would add substantially more to the portfolio volatility.
The critique one can aim at this analysis is that we are using long-term stable correlation assumptions despite we know the correlation structure is conditional and the correlation sign between equities and bonds has changed multiple times the last 100 years.
This leads to another interesting driver behind the rise of commodities in asset allocation. It was the very low conditional correlation in the period 1975-2002 that led to commodities assumed benefits. However, the conditional correlation changed dramatically over the financial crisis with correlations going up (commodities behaved like equities and thus not adding diversification).
Another valid observation about commodities and asset allocation is that Bridgewater Associates, one of the largest alternative asset managers, has done extensive research into long-term price series of asset classes and found that commodities add value. To my knowledge they have not change their mind on commodities.
The research into commodities in relation to asset allocation will likely continue and here we have provided some addition to the discussion.