Investment advisors may overestimate the danger of equities for longer-term investors. We analyzed stock market returns for 15 different countries from 1870 to 2020 and located that the optimal equity allocation increases with longer investment horizons.
Optimization models based on one-year returns generally ignore the historical seriality of returns. Therefore, they could overestimate equity risk for longer-term investors. This is particularly true for investors who’re more risk-averse and anxious about inflation risk.
In our previous blog post we shared the findings from our recent Paper that the returns of asset classes don’t evolve completely randomly over time. In fact, a sort of serial dependence exists in numerous asset classes.
While there are significant differences in optimal equity allocations across countries, there is powerful evidence that investors with longer investment horizons would have been higher served with higher equity allocations previously. Of course, it’s not possible to predict how these relationships will evolve in the longer term, but investment professionals should concentrate on these findings when determining the suitable level of risk for a client.
Determine optimal portfolios
Optimal portfolio allocations are determined using a utility function. Utility-based models might be more comprehensive and relevant than defining investor preferences using more common optimization metrics comparable to variance. More specifically, optimal asset class weights are determined that maximize expected utility assuming constant relative risk aversion (CRRA), as given in Equation 1. CRRA is a performance utility function that’s widely utilized in the tutorial literature.
Equation 1.
The evaluation assumes different levels of risk aversion (), assuming that a certain initial amount (e.g., $100) grows over a certain time period (i.e., typically one to 10 years, in one-year increments). More conservative investors with higher risk aversion would correspond to investors with lower risk tolerance. No more money flows are assumed within the evaluation.
Data for the optimizations come from the Jordà-Schularick-Taylor (JST) Macrohistory Database. The JST dataset incorporates data on 48 variables, including real and nominal returns for 18 countries from 1870 to 2020. Historical return data for Ireland and Canada usually are not available, and Germany is excluded as a result of relatively extreme returns within the Twenties and the return gap within the Nineteen Forties. This limits the evaluation to fifteen countries: Australia (AUS), Belgium (BEL), Switzerland (CHE), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), United Kingdom (GBR), Italy (ITA), Japan (JPN), Netherlands (NLD), Norway (NOR), Portugal (PRT), Sweden (SWE), and United States (USA).
The evaluation includes 4 time series variables: inflation rates, bill rates of interest, bond yields, and stock returns, with the optimal allocation between bills, bonds, and stocks determined by maximizing collateralized wealth using Equation 1.
Three different risk aversion levels are assumed: low, medium and high, corresponding to risk aversion levels of 8.0, 2.0 and 0.5 respectively. These in turn roughly correspond to equity allocations of 20%, 50% and 80%, assuming a one-year investment period and ignoring inflation. The actual allocation varies considerably from country to country. Any 12 months of hyperinflation during which inflation exceeds 50% is excluded.
Figure 1 presents the optimal equity allocation for every of the 15 countries for five different investment periods: one, five, 15 and 20 years, assuming a medium risk tolerance (=2), where the optimizations are based on nominal or real wealth growth and using the actual historical return sequence or returns randomly chosen (i.e., extrapolated) from the historical values (assuming 1,000 trials).
Bootstrapping evaluation would capture any skewness or kurtosis within the historical return distribution since it is predicated on the identical returns. However, bootstrapping effectively assumes that the returns are independent and identically distributed (IID), which is consistent with common optimization routines comparable to mean-variance optimization (MVO).
Figure 1. Optimal equity allocations for a moderate level of risk aversion by country and investment period: 1870–2020
Key findings
There are several necessary lessons to be learned from these results. First, there are significant differences in historically optimal equity allocations across countries, even when specializing in the identical time horizon (returns over one 12 months). For example, equity allocations range from 16% equities (for Portugal) to 70% (for the UK) when nominal, actual historical returns.
Second, the typical equity allocation for the one-year period is roughly 50% across all 15 countries, no matter whether wealth is defined in nominal or real terms.
Third, and maybe most notably, while the equity allocations for the optimizations using actual historical return sequences increase over longer asset optimizations, there is no such thing as a change within the optimal allocations for the bootstrapped returns. The equity allocations for the nominal asset optimizations increase to about 70% after 20 years, and the equity allocations for the true asset optimizations increase to about 80% after 20 years, representing annual slopes of 1.3% and 1.5%, respectively. In contrast, the equity allocations for the bootstrapped optimizations are virtually constant (i.e., zero).
This result’s value repeating: the optimal equity allocation is different when using actual historical return data (which have nonzero autocorrelation) than within the bootstrap simulation, where the returns are literally IID.
Figure 2 incorporates the typical equity allocations across the 15 countries for the three different levels of risk aversion, specializing in nominal and real wealth and whether the actual historical sequence of returns is used or whether these are calculated by bootstrapping. Note that the averages in Figure 1 (for the one-, five-, ten-, 15- and 20-year periods) are literally reflected within the leads to the subsequent figure for the respective test.
Annex 2. Optimal equity allocation based on risk tolerance and investment period (years)
Here again, we see that optimal equity allocations are inclined to increase over longer investment horizons when using actual historical return histories, but optimal allocations on a bootstrap basis remain effectively constant over the whole investment horizon.
The impact of the investment horizon based on actual return performance is especially noticeable for probably the most risk-averse investors. For example, the optimal equity allocation for a highly risk-averse investor who focuses on nominal assets and has an investment horizon of 1 12 months could be around 20%. With an investment horizon of 20 years, this value rises to around 50%.
These results display that capturing the historical serial dependence exhibited by market returns can significantly affect the optimal equity allocation. In particular, the optimal equity allocation tends to extend with investment duration using actual historical returns, suggesting that equities turn out to be more attractive to investors than fixed income with longer holding periods.
Even when ERP is eliminated, we discover that equity allocations persist and increase over longer investment horizons, suggesting that even without generating higher returns, equities can provide necessary diversification advantages over the long run.
So what?
The investment horizon and the consequences of serial correlation have to be explicitly taken into consideration when constructing portfolios for investors with longer investment horizons. As the evaluation shows, this is particularly true for more conservative investors who would normally receive smaller equity allocations.
In our next blog post, we’ll explore how allocation to an asset class (commodities) that could seem inefficient from a standard perspective might be efficient when examined more closely.
