Diversification premium difference to benchmark in different time periods

portfolios of different sizes all were below the parabola of the benchmark portfolio. We now turn our focus in the diversification premium difference to benchmark which is the portfolio risk premium minus the benchmark portfolio risk premium.

We will show predicted and bootstrapped diversification premium differences for various portfolio sizes in different time periods. The predicted metric, for portfolios greater than one stock, is based on equation (40) and, after deciding the investment fraction and portfolio size of interest, only needs two inputs, the idiosyncratic variance of a single stock portfolio and number of stocks in the benchmark portfolio. For single stock portfolios, we use equation (41) which only requires the idiosyncratic variance of a single stock portfolio in addition to investment fraction as an input. Idiosyncratic variance of a single stock portfolio is acquired as an output from regression specified by equation (47).

We will show the results for the whole time period extending from July 1926 to June 2018 and for the three subperiods shown in Figure 8. In addition, to demonstrate the effect of fat tails, we will show the results for two additional sets of data from the third time period, January 1973 to June 2018, where we cut the tails of the distribution.

For each of the diversification premium difference to benchmark figures, we will show the portfolio sizes up to 500 or 1000 stocks, depending on the minimum monthly number of stocks for the period. 1000 will be used as the maximum whenever the monthly minimum exceeds that amount. Similarly, as with the figures in the section 5.1, we will plot the lines until the investment fraction is high enough for the first bootstrapped portfolio to exceed 100% loss.

Figure 18 shows the diversification premium difference to benchmark for the whole 92-year time period. Bootstrapping without replacement is implemented by creating 25 000 random portfolios per portfolio size each month. Given that the time period consists of three distinct stock populations as shown by Figure 8 and Figure 9, the predicted value is surprisingly close to bootstrapped realized values.

Figure 18. Diversification premium difference to benchmark between Jul-1926 and Jun-2018. The single stock portfolio prediction, for which the realized value is based on exhaustive data rather than bootstrapped, is particularly accurate. Also, the larger portfolio sizes seem very accurate although small differences are difficult to spot from figure.

In Figure 19 we show the diversification premium difference to benchmark for the NYSE stock era. 75 000 random portfolios are created per portfolio size each month.

Figure 19. Diversification premium difference to benchmark between Jul-1926 and Jul-1962. The era contains a tail event, the Great Depression, and on average large firm sizes compared to forthcoming time periods. Despite the massive tail event, the predicted values still appear to explain the realized diversification premium difference to benchmark rather accurately. On the other hand, when the tail event occurs on the market level, stocks on average “ride on top of the market wave” and therefore may not be that susceptible to the tail event. Because of this “market neutral” nature of the diversification premium, it can be hypothesized to be resilient against market volatility implying diversification may actually be highly beneficial regardless the market moves. This would be against the conventional wisdom that diversification is beneficial until it is most needed during a severe bear market. The reference for the value of diversification should be relative not absolute. Using relative reference, as in our case, is about comparing the outcomes between perfect and less than perfect diversification. Using absolute reference, as in the case of the conventional wisdom, is about assessing the outcome of perfect diversification when the market falls. If

diversification indeed could prevent bear markets, would there be such thing as equity risk or equity risk premium?

In Figure 20 we can see the diversification premium difference to benchmark for the NYSE Amex stock era. 100 000 random portfolios are created per portfolio size each month during this relatively short period of time.

Figure 20. Diversification premium difference to benchmark between Aug-1962 and Dec-1972. Our main interest is the modern era of U.S. stock returns, the 45.5-year period extending from the beginning of the 1973 until June 2018. This period is characterized and dominated by microcap stocks which on average are very volatile and exhibit very large idiosyncratic variance. Another characteristic of the microcaps is fat tailed and positively skewed monthly excess returns. Returns between portfolios loaded with volatile, skewed and fat tailed microcap stocks can differ wildly, which should put our diversification metrics to real test. Fortunately, there are plenty of stocks and monthly excess returns, 2 987 476 in total. The large number of data points ensures relatively accurate predictions. However, regardless the huge data, we still see from Figure 21 that the realized metrics for the intermediate portfolio sizes don’t exactly align with the predictions.

Figure 21. Diversification premium difference to benchmark between Jan-1973 and Jun-2018. Table 5 summarizes the diversification premium difference to benchmark metrics at investment fraction one for different time periods. Predicted and realized values are shown for different portfolio sizes. Predicted and realized values are typically almost identical for portfolio size one. Predictions are fairly accurate also for larger portfolio sizes, but we notice a trend that the absolute value of the realized diversification premium difference to benchmark is typically slightly greater than the absolute value of the predicted metric. We suspect this is mainly due to fat tailed monthly excess returns and will further investigate this hypothesis in section 5.2.2.

Table 5. Summary of predicted vs. realized diversification premium differences to benchmarks.

Diversification premium difference to benchmark [pp]

Jul-1926 to

Jun-2018 Jul-1926 to Jul-1962 Aug-1962 to Dec-1972 Jan-1973 to Jun-2018 Predicted Δ𝐷𝑃_𝑛=1𝐵𝑀 -10.626 -6.799 -5.790 -14.765 Realized Δ𝐷𝑃_𝑛=1𝐵𝑀 -10.682 -6.890 -5.904 -14.783 Predicted Δ𝐷𝑃_𝑛=10𝐵𝑀 -1.059 -0.672 -0.576 -1.474 Realized Δ𝐷𝑃_𝑛=10𝐵𝑀 -1.166 -0.741 -0.634 -1.642 Predicted Δ𝐷𝑃_𝑛=25𝐵𝑀 -0.422 -0.264 -0.229 -0.588 Realized Δ𝐷𝑃_𝑛=25𝐵𝑀 -0.467 -0.288 -0.246 -0.682 Predicted Δ𝐷𝑃_𝑛=100𝐵𝑀 _{-0.103 -0.060 -0.055 -0.145} Realized Δ𝐷𝑃_𝑛=100𝐵𝑀 -0.113 -0.070 -0.055 -0.171

By observing the diversification premium difference to benchmark metrics in the latest time period, from January 1973 to June 2018, we can see the head start that the investor has given to fully diversified benchmark investor by being content with low diversification. Ten-stock and twenty-five-stock portfolios are often considered as sufficiently diversified. Based on Table 5, however, in the absence of stock picking skill, a ten-stock portfolio on average has lost more than 1.6 percentage points in risk premium compared to fully diversified benchmark portfolio. A twenty-five-stock portfolio has lost close to 0.7 percentage points. A difference of 1.6 percentage points is comparable to the cost of an expensive active mutual fund.

In addition to our empirical results, we can apply the concept of diversification premium difference to benchmark to empirical results in Tidmore et al. (2019) study. Tidmore et al., as described in section 2.2, measure the difference in expected (geometric) excess return between a portfolio of selected size and the fully diversified benchmark portfolio and call this difference as average expected excess return. This is the same metric as our diversification premium difference to benchmark. Tidmore et al. find that for a single stock portfolio the expected excess return is -9.9 percentage points. It is not clear whether this is continuously or annually compounded rate, but it does not make a practical difference. Alpha in our regression equation (46) estimates

single stock portfolio’s geometric mean return difference to geometric mean return of a fully diversified benchmark portfolio. The -9.9 percentage points for a single stock portfolio in the Tidmore et al. study is the empirically measured return difference to benchmark and can be considered as approximately equal to alpha in equation (46). We therefore can consider the -9.9 percentage points to represent the alpha in our approximate alpha-based equation (54) for diversification premium difference to benchmark. Table 6 shows the relationship between our predicted (based on the -9.9 percentage points acquired for single stock portfolio) diversification premium difference to benchmark and empirical results from Tidmore et al. study. Equation (54) accurately explains the empirical Tidmore et al. expected excess return for different portfolio sizes.

Table 6. Tidmore et al. results explained by approximate diversification premium difference to benchmarkequation.

Diversification premium difference to benchmark [pp] Portfolio size Prediction based on eq. (54) Tidmore et al. result

5 -2.0 -2.0 10 -1.0 -1.0 15 -0.7 -0.7 30 -0.3 -0.4 50 -0.2 -0.2 100 -0.1 -0.1 200 0.0 -0.1 500 0.0 0.0