We find that the belief that returns are independent over time is inconsistent with historical evidence domestically and internationally for stocks, bonds and alternatives.
These results suggest that investment professionals might have to rethink their portfolio optimization routines – including mean-variance optimization (MVO) – which usually assume returns are random over time.
This article is the primary of a three-part series. Here we offer some context on how returns have historically performed over time. In the next articles we are going to describe what this implies for equity portfolios and portfolios with real assets similar to commodities.
Risk and investment horizon
Many investors and financial advisors consider that the chance of certain asset classes, particularly stocks, decreases over longer investment periods, an effect commonly known as “time diversification.”
In support of this hypothesis, there’s evidence that the distribution of total returns tends to converge over longer investment horizons, as shown in Figure 1, which is predicated on U.S. stock returns from 1872 to 2023.
Exhibition 1. The distribution of cumulative stock returns by investment horizon 1872 to 2023.
A key problem with this angle is that investors shouldn’t deal with total returns. Rather, they need to deal with compound assets. And compound wealth tells a unique story.
Using the identical returns over the identical period, Figure 2 shows how the wealth distribution changes depending on investment horizon, and there is obvious evidence that it’s diverging relatively than converging.
Figure 2. The distribution of composite wealth by investment horizon for an equity investor 1872 to 2023.
In fact, if risk is defined as increasing diversification of assets, the chance of virtually all investments increases over time. This perspective is consistent with option pricing models. While the chance of all investments increases, it can be crucial to notice that the speed of increase may vary over time and this variation has vital implications for investors with longer holding periods.
If the relative risks of investments change depending on the investment horizon, this might suggest that there’s some type of serial dependence, meaning that returns move in a way that will not be completely random.
Previous research suggests that the return on an investment like stocks is comparatively random. This theory is probably best illustrated in Burton Malkiel’s book. However, our research has shown that there’s an autocorrelation.
Our article says: “Investment horizon, serial correlation and higher (age) portfolios“We provide context on autocorrelation, or how past returns relate to future returns. We examine five U.S. yield series – notes, bonds, stocks, commodities and inflation – using historical annual returns from 1872 to 2023 using data from the Jordà-Schularick-Taylor (JST) dataset and the Bank of Canada.
Figure 3 incorporates the coefficients from a series of bizarre least squares (OLS) regressions, where the dependent variable is the actual return for that calendar 12 months, while the returns for the previous five calendar years are included as independent variables.
Historical returns for every asset class are re-centered to have a mean return of zero and a regular deviation of 1 to cut back any impact related to historical differences in returns and risk levels. In other words, the regression is effectively based on the Z-scores of historical time series returns.
Negative coefficients are highlighted in blue because this suggests that an asset’s risk decreases over time, as a positive return is more more likely to be followed by a negative return. Statistically significant positive coefficients, which mean that the chance of the asset increases over time, are highlighted in red.
Figure 3. Regression coefficients for an Ordinary Least Squares (OLS) regression where the dependent variable is the present calendar 12 months for the asset class 1872 to 2023.
Going back to Figure 1, there are several statistically significant coefficients, defined as an ap value below 0.05, suggesting that the historical return series usually are not truly independent over time.
Certain asset classes, similar to bonds, have a history of positive autocorrelation, while other asset classes, similar to stocks, have negative autocorrelation. This suggests that the longer-term risks of owning an asset could change based on the investment horizon. The relative risk of owning stocks, for instance, should decrease in comparison with bonds.
Next, let’s take a look at how the chance of assets can change when taking inflation under consideration. For this evaluation, we estimate the correlation between cumulative asset growth and the cumulative impact of inflation over different investment horizons for a similar 4 asset classes.
In certain forms of optimizations (e.g. a “surplus” or liability-relative optimization), inflation is commonly explicitly taken under consideration. However, a possible problem with inflation is that changes in the costs of products or services usually are not necessarily synchronized with changes in financial markets. In other words: There might be delayed effects.
For example, while financial markets can experience sudden fluctuations in value, inflation tends to have a more latent effect, meaning changes might be delayed and take years to manifest. Focusing on the correlation – or covariance – of inflation with a specific asset class, similar to stocks, over a one-year period can mask potential longer-term effects.
The correlations of the 4 asset classes vary significantly with inflation depending on the investment horizon. For example, over a one-year investment horizon commonly used for MVO assumptions, correlations are relatively low across asset classes, suggesting little potential hedging profit.
However, there are notable increases over a 10-year period that might be at the least partially explained by positive drift for every asset. For example, the correlation between commodities and inflation rises to 0.62.
The significant increase in correlations between banknotes and commodities is especially striking as returns on banknotes and commodities have been significantly lower over the historical period. We will discuss this in a future article. This suggests that the effect will not be simply on account of higher historical returns, but relatively to differences in the way in which asset classes have responded to inflation over time.
The results suggest that there’s a point of serial dependence between the asset classes considered, which could potentially impact optimal portfolio allocations over longer time periods (e.g. 10+ years). We explore this effect further, using sequential historical returns from 1872 to 2023 to look at how the usual deviation of wealth for every asset class has modified across different investment horizons.
The actual historical standard deviations are in comparison with the deviations from a bootstrap simulation, during which the historical returns for the respective asset classes are randomly recombined or bootstrapped.
Bootstrapping is helpful since it preserves the possibly interesting features of time series data. These features keep the means and covariances constant, but change the ordering of the particular historical returns and make them random.
Bootstrapping would capture things like skewness and kurtosis, so the differences in wealth distributions would largely be on account of some type of serial dependence (e.g. the autocorrelations mentioned earlier).
The first 12 months annual standard deviation is adjusted based on the ratio of the long run standard deviation of the ultimate assets to the bootstrap value for investment periods of as much as 10 years. If there have been no form of serial dependence within the historical returns (e.g., autocorrelation), the lines in Figure 4 can be flat, while a falling line would indicate negative autocorrelation and a rising line would indicate positive autocorrelation.
Exhibit 5. Standard deviation for notes, bonds, stocks and commodities for various investment periods 1872 to 2023.
Nominal returns Real returns
This evaluation provides evidence that the chance of assets may vary depending on the investment period, particularly when inflation is taken under consideration. In nominal terms, for instance, that is the usual deviation of the wealth of stocks over longer investment periods, while the usual deviation of notes, bonds and commodities.
However, once we have a look at inflation (Panel B, Figure 5), the usual deviation of commodities falls at concerning the same rate as that of stocks. This is a notable shift and suggests that the perceived efficiency of commodities is more likely to fluctuate dramatically no matter inflation being taken under consideration. We will reveal this in a future article.
In our work, we further extend the evaluation to look at international markets and include the ends in Appendix 2. The international results are just like the US results: wealth distribution tends to diminish relative to bootstrapped values for stocks, while it tends to extend for bonds and bills. This is very important since it suggests that these effects are simply not a US phenomenon.
Overall, there’s notable evidence that asset classes similar to debt, bonds, stocks and commodities exhibit various degrees of serial dependence. This suggests that a portfolio’s optimal allocation may change depending on the investment horizon, which we are going to explore in our next article on this series.
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Image courtesy of Nick Webb. This file is licensed under the License Creative Commons Attribution 2.0 generic License. Circumcised.
