EWI163's 0.139 annual growth over cash is the number a trader would want to pursue. The authors attribute it to equal weighting across 163 sub-industries. Their model's new element is the activity clock, which the paper never tests as a signal. Those claims need to stay separate when adapting the idea to a tradable book. Only the clock travels.
The model Platen and Fergusson built
The growth-optimal portfolio supplies the starting point; the authors call it the benchmark. Following Platen (2026b), they assume that the stationary densities of normalized factors and normalized activity maximize surprisal, which they equate with Shannon entropy. That assumption yields a minimal market model (MMM). Under the benchmark-neutral (BN) pricing measure, which uses the benchmark as numéraire, the benchmark follows a time-changed squared Bessel process of dimension four. Its clock is activity time τ. To construct it, take the quadratic variation of the square root of the cash-denominated index, add e^{τ0}, then take the log. Activity measures the clock's speed, its derivative.
The model makes two predictions. The normalized benchmark Y follows a CIR process of dimension four with mean 2. Squared volatility in activity time, 2/Y, therefore has an inverse gamma distribution. Log-returns should follow a Student t with four degrees of freedom. Random activity in calendar time should pull the estimated degrees of freedom slightly below four.
The authors model activity through an exponential moving average of the driving Brownian motion. They interpret its smoothing parameter κ, between 1 and 2, as evidence of traders overreacting. Normalized activity is the scaled squared increment of that average. It follows a dimension-one CIR process, the square of an Ornstein-Uhlenbeck process.
For the empirical work, the authors construct equal-weighted indices (EWIs) using daily Thomson-Reuters GICS data from 67 countries, January 1973 to July 2024. Within each of 11 sectors, 25 industry groups, 74 industries or 163 sub-industries, country indices receive equal weights. The global indices formed this way are equal-weighted again. Turnover incurs a 40bp charge. Measured in US savings-account units, EWI163 grows by 0.139 annually, compared with 0.05 for the MSCI world total return index. Student t fits produce degrees of freedom ranging from 3.15 (EWI1) to 3.48 (MSCI). A likelihood ratio test against the four-parameter symmetric generalized hyperbolic (SGH) density fails to reject the t for every index.
Equal weighting across 163 sub-industries explains EWI163's growth
No activity term enters the EWI recursion. The 8.9-point growth gap concerns naive diversification across 163 sub-industries and 67 country markets, following Platen and Rendek (2012). The authors make that attribution themselves: incorporating increasingly diverse sources of randomness "by applying equal weighting, results in higher average growth rates". Growth increases with finer groupings in Table 5.8: 0.102 for EWI1, 0.124 for EWI11, 0.143 for EWI25, 0.162 for EWI74 and 0.177 for EWI163.
Volatility requires a closer look. EWI163's σ of 0.130 in Table 5.8 matches its normal-fit daily σ of 0.008071 multiplied by √260. Applying the same calculation to EWI25 gives 0.166 from a normal σ of 0.010296. Its Student t scale, however, is just 0.004292. A sample standard deviation that high usually reflects a few enormous days. The authors identify one: a 2020 jump caused "by a single significant price jump of a Venezuelan stock". They say excluding it would bring the degrees of freedom closer to four.
The paper describes EWI163's volatility as approximately equal to the MSCI's. EWI25's 0.166 is considerably higher than the MSCI's 0.133. Dividing normal-fit standard deviation by Student t scale gives 1.52 for the MSCI, 1.60 for EWI1 and 1.59 for EWI11. For EWI25, EWI74 and EWI163, the ratios are 2.40, 2.06 and 1.84 respectively. These are the three indices the paper says contain the 2020 Venezuelan jump.
The prose also conflicts with the table. Growth "over 0.17" and volatility "less than 0.11" are described as attainable for EWI11, EWI25, EWI74 and EWI163. Yet Table 5.8 reports 0.124 growth for EWI11 and 0.166 volatility for EWI25. EWI163 also has different growth figures: 0.139 in Figure 2.1 and 0.177 in Table 5.8. Between those sources, the MSCI gap is 0.043 and the EWI163 gap is 0.038. Both are close to a cash rate, suggesting that Table 5.8 uses USD rather than savings-account units. Section 4.1 expresses index values in US dollars. Section 5 does not say that log-returns were converted to savings-account units. Because the gaps differ, a single cash deflator cannot explain them fully.
Rebalancing occurs at every observation, hence daily. We found no turnover figure. The flat 40bp charge is fixed "for simplicity" and applies even to frontier-market sub-industry indices.
What does the activity clock add to realized volatility?
Building the clock is straightforward. Accumulate squared daily changes in √(index/cash account), add a constant and take logs. Across five decades the result is nearly linear. For the MSCI, the slope is 0.065 and R² is 0.993; for EWI163, the slope is 0.171 and R² is 0.976. The authors choose the starting constant to maximize that R², obtaining τ0 of -3.6 and -6.9. This choice estimates one number per index in-sample. In calendar time, the derivative moves quickly and spikes, including a jump at the 1987 crash.
A trader needs to know what activity adds. Equation (3.15) expresses calendar-time squared volatility as 2a/Y. Rearranging it makes activity equal to squared volatility times Y/2. The sole distinction from realized variance is Y = S/(2e^τ), the index level relative to its clock. An activity timing rule therefore reweights a volatility signal by how stretched the index is. Any adaptation needs to demonstrate that this Y term contributes something absent from a matched realized-volatility rule.
The paper does not run that test.
Instead, it illustrates the activity model with a simulation using κ = 1.97, ā = 0.15 and ξ = 0.0042. The paper calls these estimated parameters of the daily EWI163. We did not find an estimation procedure or standard errors. Its predicted variance-gamma density with one degree of freedom applies to high-frequency returns, and the only tests offered are from cited literature. On the paper's daily sample, VG has the lowest likelihood among the heavy-tailed fits: 49203.09 versus 49519.80 for the Student t on EWI163.
Table 5.10 prints 0.0000566 for EWI1; recalculation gives 2.80
Using the printed log-likelihoods, recompute L = 2(ℓ_SGH − ℓ_t). For the MSCI, 2 × (46994.43 − 46994.36) = 0.14. For EWI11, the result is 0.98. EWI1 gives 2 × (49524.41 − 49523.01) = 2.80. EWI25, EWI74 and EWI163 each give zero to two decimals. For the first three, Table 5.10 instead prints 0.00000298, 0.0000194 and 0.0000566. Those values match -2 ln of a ratio of log-likelihoods rather than likelihoods. For EWI1, 2.80/49524 = 5.65 × 10^-5.
The authors' conclusion of no rejection at 1% still holds: 2.80 is below 6.635. The apparent margin of comfort shrinks. EWI1's 2.80 exceeds the 10% critical value of 2.706. The Student t also lies at ᾱ = 0, on the boundary of the SGH parameter space. At that boundary, the null distribution mixes a point mass at zero and χ²(1) in equal proportions. This halves the p-value to about 0.047. One of the six indices then rejects at 5%. There is a further problem with multiplying likelihoods over days: the paper's own random-activity model implies that daily returns are not i.i.d.
Even an untroubled test would establish only that the marginal density resembles a t. Before fitting, the paper predicts degrees of freedom between three and four. Its reported fitted range extends further: "smaller than 4 and larger than 2.9". EWI1's SGH-implied 2.91 falls below the predicted floor of three. No number accompanies the explanation that activity and jumps account for lower values. Nor can a density test establish whether activity helps with timing.
Our SPY adaptation
We could not trade EWI163 or the MSCI. We substituted SPY, a US equity ETF whose growth optimality has not been established. Our run applies an activity-based allocation overlay derived from the model. It neither reconstructs the EWI indices nor addresses the BN pricing and hedging claims. We estimate quadratic variation from discretized daily squared increments and calibrate over a trailing window.
The overlay first divides SPY by a cash account accruing lagged fed funds, then accumulates squared √ increments. From 401 candidates, it selects the log constant that maximizes R² over 1,260 training days, mirroring the authors' τ0 selection. That constant then stays frozen. We measure activity as the annualized 20-day change in the log clock. The book switches to cash above activity's expanding 90th percentile and re-enters below the 50th, with execution at the next open. Alongside it, we run buy-and-hold and a realized-volatility book using identical hysteresis. The period is January 2020 to July 2024.
The activity book earned 57.02% in total during that window. Its Sharpe was 0.99 and Sortino 1.11. Annualized volatility came to 12.79%, max drawdown to -17.37% and Calmar to 0.61. Every figure here belongs to the activity book. We have no figures for either the realized-volatility control or buy-and-hold.
Our 57.02% total return covers four and a half years in SPY. The paper's 0.139 annual return above cash for EWI163 covers 1973 to 2024. Its 0.139 and 0.177 figures describe a 163-index global portfolio held over 51 years without timing. Differences between our result and theirs tell us nothing about the paper. Our trades execute at the next open, and costs use the broker commission model rather than our intended 3bp one-way charge. With only the activity book reported, this run cannot establish whether Y adds information beyond realized volatility.
Two implementation choices matter more to our result than the theory. The window contains one sharp stress episode, March 2020. Any switching benefit therefore depends on a single event. Residual cash earns zero, although the signal accrues fed funds, penalizing periods out of the market through 2023. A thin or flat outcome here says more about our one-pass build than about Platen and Fergusson.
We would change our view if the activity book beat the realized-volatility control with identical costs. Only that comparison would earn the Y term in 2a/Y its place.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.