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What is the optimal equity allocation for your clients?

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Whats Your Clients Optimal Equity Allocation Scaled.jpg

Investment advisors may be overestimating the risk of equities for long-term investors. We analyze stock market returns for 15 countries from 1870 to 2020 and find that the optimal equity allocation increases as the investment horizon increases.

Optimization models using one-year returns generally ignore the historical sequential dependence of returns and so may naturally overestimate equity risk for long-term investors, especially those who are more risk averse and concerned about inflation risk.

In my last blog post, I paper Asset class returns do not vary completely randomly over time – in fact, some form of sequential dependency exists across different asset classes.

While optimal equity allocations vary widely across countries, there is significant evidence that investors with longer investment horizons have historically fared better with higher allocations to equities. Of course, it is impossible to know how these relationships will develop in the future, but investment professionals should take note of these findings when determining appropriate risk levels for their clients.

Determining the optimal portfolio

The optimal portfolio allocation is determined using a utility function. Utility-based models are more comprehensive and relevant than defining investor preferences using more general optimization metrics such as variance. More specifically, the optimal asset class weights that maximize expected utility given constant relative risk aversion (CRRA) are determined, as shown in Equation 1. CRRA is a power utility function that is widely used in academic literature.

Equation 1.

U(w) = w-y

This analysis assumes different levels of risk aversion (Yeah) assumes that an initial amount of capital (i.e., $100) grows over a period of time (i.e., typically 1 to 10 years, in annual increments). Conservative investors with high risk aversion correspond to investors with low risk tolerance. No additional cash flows are assumed in the analysis.

The data for optimization is Jordà-Schularick-Taylor (JST) macro history databaseThe JST dataset contains data on 48 variables including real and nominal earnings for 18 countries from 1870 to 2020. Historical earnings data are not available for Ireland and Canada, and Germany is excluded given its relatively extreme earnings in the 1920s and earnings gap in the 1940s. This limits the analysis to 15 countries: Australia (AUS), Belgium (BEL), Switzerland (CHE), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great Britain (GBR), Italy (ITA), Japan (JPN), Netherlands (NLD), Norway (NOR), Portugal (PRT), Sweden (SWE), and the United States (USA).

The analysis includes four time series variables: inflation rate, bill yield, bond yield, and stock yield, and the optimal allocation between bills, bonds, and stocks is determined by maximizing certainty equivalent wealth using Equation 1 .

Three levels of risk aversion are assumed: low, medium, and high, corresponding to risk aversion levels of 8.0, 2.0, and 0.5, respectively. These correspond to approximately 20%, 50%, and 80% equity allocations assuming a one-year investment horizon and ignoring inflation. Actual allocations will vary significantly by country. Hyperinflationary years with inflation above 50% are excluded.

Figure 1 shows the optimal equity allocation for each of the 15 countries over five different investment horizons – 1 year, 5 years, 15 years, and 20 years – assuming a moderate risk tolerance.Yeah=2) Here, the optimization is based on nominal or real wealth growth and assumes 1,000 trials using a sequence of actual past returns, or returns randomly selected from past values ​​(i.e., bootstrapped).

Because bootstrap analysis is based on the same returns, it captures any skewness or kurtosis present in the historical return distribution, but bootstrap effectively assumes that returns are independent and identically distributed (IID), consistent with common optimization routines such as mean-variance optimization (MVO).

Figure 1. Optimal equity allocation at moderate levels of risk aversion (by country and investment horizon): 1870-2020

Key Takeaways

These results reveal several important takeaways: First, even when focusing on the same time period (one-year returns), there is considerable variation in the optimal historical equity allocation across countries: for example, when considering nominal and actual historical returns, equity allocations range from 16% equities (Portugal) to 70% equities (UK).

Second, whether wealth is defined in nominal or real terms, the average equity allocation across the 15 countries over a one-year period is around 50%.

Third, and perhaps most notable, is that the stock allocation for the optimization using the actual historical return sequence increases over the longer investment optimization, while the optimal allocation for the bootstrap returns remains unchanged. The stock allocation for the nominal assets optimization increases to about 70% over 20 years, and the stock allocation for the real assets optimization increases to about 80% over 20 years, representing annual slopes of 1.3% and 1.5%, respectively. In contrast, the stock allocation for the bootstrap optimization remains essentially constant (i.e., zero).

This finding is worth repeating: the optimal allocation to stocks differs when using actual historical return data (where autocorrelation is nonzero) from the case of bootstrap simulations where returns are truly IID.

Figure 2 includes the average allocation to equities in 15 countries at three different levels of risk aversion, focusing on nominal and real assets, and depending on whether we use actual historical return ordering or bootstrap. Note that the averages in Figure 1 (for 1-, 5-, 10-, 15-, and 20-year periods) are effectively reflected in the results of the respective tests in the following figures.

Appendix 2. Optimal stock allocation based on risk tolerance and investment horizon (years)

Again, using actual historical return sequences, we find that the optimal equity allocation tends to increase as the investment horizon increases, but the bootstrapped optimal allocation remains essentially constant over the entire investment horizon.

The effect of investment horizon using the sequence of actual returns is particularly pronounced for the most risk-averse investors. For example, the optimal equity allocation for an investor with a high level of risk aversion who focuses on nominal assets and has an investment horizon of one year is around 20%, but increases to around 50% when we assume a 20-year investment horizon.

These results suggest that capturing the historical serial dependence found in market returns can have a significant impact on the optimal allocation to stocks. In particular, when using actual historical returns, the optimal allocation to stocks tends to increase with investment horizon, suggesting that stocks become more attractive than bonds for investors with longer holding periods.

One explanation for the change in optimal equity allocation over time using the sequence of actual historical returns is the existence of a positive equity risk premium (ERP). paperThe CFA Institute Research Foundation regularly brings together leading investment professionals to Discuss new ERP research And share different opinions on the topic.

We find that when ERP is eliminated, allocations to stocks are maintained and increase over longer investment horizons, suggesting that stocks can provide important long-term diversification benefits even if they do not generate higher returns.

so what?

When constructing portfolios for investors with longer investment horizons, the effects of investment horizon and serial correlation must be explicitly considered. As our analysis shows, this is especially true for conservative investors who typically have lower allocations to equities.

In future blog posts we will consider how allocations to asset classes (commodities) that appear inefficient from a traditional perspective can become efficient when considered in a more robust way.

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