Math

The moving average over a window of length $n$ is the mean of the last $n$ closing levels:

\mathrm{SMA}_n(t) = \frac{1}{n} \sum_{i=0}^{n-1} P(t-i)

The window counts closing levels, that is trading days, not calendar days. The difference is substantial: a year has just under a third fewer trading days than calendar days. The same number means a markedly different window in each counting, and values from other analyses are comparable only if the counting is the same.

Buffer

The state condition with buffers reads separately for entry and exit, with the buffers $b_{\text{in}}$ and $b_{\text{out}}$ as a share of the line:

\text{risk\_on}: \quad P(t) > \mathrm{SMA}_n(t)\,\bigl(1 + b_{\text{in}}\bigr)

\text{risk\_off}: \quad P(t) < \mathrm{SMA}_n(t)\,\bigl(1 - b_{\text{out}}\bigr)

Between the two thresholds the previous state persists. That intermediate zone is the whole point of the buffer: without it the two conditions coincide, and every touch of the line produces a switch. Confirmation days $k$ tighten the condition in time: it has to hold on $k$ consecutive trading days before the state changes.

Why the filter is connected to leverage at all is shown by the volatility drag. A product with daily leverage $L$ tracks $L$ times the daily return. For the mean continuous growth rate $g_L$ with drift $\mu$ and volatility $\sigma$ , approximately:

g_L \approx L\mu - \frac{L^2\sigma^2}{2}

For the unleveraged case $g_1 \approx \mu - \sigma^2/2$ . Substituting leaves, for twofold leverage:

g_2 \approx 2g_1 - \sigma^2

Leverage therefore does not double the growth rate; it additionally subtracts a term that grows with the square of volatility. That is why a leveraged product is cheap in calm rising phases and expensive in turbulent ones, and why a filter that avoids the turbulent phases matters more for leveraged holdings than for unleveraged ones. This is the arithmetic core behind the argument of Gayed and Bilello, who show the same relationship empirically, through volatility above and below the average. The approximation itself is not in their paper; it comes from the literature on leveraged products, among others from Double-Digit Numerics.

What this means for the rule itself follows from the same approximation, because the continuous growth rate adds up over the days. Split the trading days by the state of the rule: on a share $q$ of the days it is out, on the rest it is invested. $\mu_{\text{out}}$ and $\sigma_{\text{out}}$ are the drift and volatility of the index on the days out, $\mu_{\text{in}}$ and $\sigma_{\text{in}}$ on the others, both measured from daily returns and annualised; $r$ is the continuous interest rate the money earns while out. Holding throughout grows at

g_{\text{hold}} \approx q\Bigl(L\mu_{\text{out}} - \frac{L^2\sigma_{\text{out}}^2}{2}\Bigr) + (1-q)\Bigl(L\mu_{\text{in}} - \frac{L^2\sigma_{\text{in}}^2}{2}\Bigr)

The rule swaps the first term for the interest rate:

g_{\text{rule}} \approx q\,r + (1-q)\Bigl(L\mu_{\text{in}} - \frac{L^2\sigma_{\text{in}}^2}{2}\Bigr)

The difference therefore depends only on the days out:

g_{\text{rule}} - g_{\text{hold}} \approx q\Bigl(r - L\mu_{\text{out}} + \frac{L^2\sigma_{\text{out}}^2}{2}\Bigr)

Three things follow. First, the rule lowers the volatility term, averaged over the whole period, from $q\sigma_{\text{out}}^2 + (1-q)\sigma_{\text{in}}^2$ to $(1-q)\sigma_{\text{in}}^2$ . It lowers it by a larger fraction than its time share $q$ only if the days out are more volatile than the others, that is $\sigma_{\text{out}} > \sigma_{\text{in}}$ . Second, it pays for this with the drift of those days: the rule is ahead only if the leveraged product would have grown more slowly than the interest rate on the days out, that is $L\mu_{\text{out}} - L^2\sigma_{\text{out}}^2/2 < r$ . If being out is random, unrelated to both drift and volatility, the days out on average look like the average; then the rule is ahead only if holding itself grows more slowly than the interest rate, and otherwise it trades growth for interest. Third, inside the bracket the volatility term grows with $L^2$ , the drift only with $L$ . Above $L = \mu_{\text{out}}/\sigma_{\text{out}}^2$ the bracket grows with leverage; if the drift on the days out is zero or negative, this holds for any leverage. So the same days out can give the twofold leveraged product a lead where they give the index none. This is the relationship from $g_2$ , shown on the rule itself. Because the days are split by the position held, not by the signal day, the delay after a signal is already contained in the split; trading costs of the switches are missing from the calculation. Whether a moving average is actually out during the volatile phases is not something it tells you; only the data show that.

The calculation in detail

Distribution and signals per year. Both compute with the rule from the box below, day by day across the whole history.

A period runs from January 2000 to January 2020, the next one starts a week later. The calendar year is the unit here: every period of ten, twenty or thirty years that fits into the history is taken; a new one starts every week, and the periods overlap. Per period the average annual return is shown for the rule and for holding the same product in your version. Per series the chart shows the median and alongside it the share of periods in which the rule was ahead of holding.

If the rule exits in March and enters again in June, the year counts two signals. Counted across the whole history with the window length of the standard version: every purchase and every sale is one signal, summed per calendar year, and years without signals stand at zero. Every buffer setting of the switcher, 0, 1, 2 and 3 percent, is a computation of its own. The running year counts up to today and does not go into the yearly average.

The signals per year depend only on the index and the line; costs, taxes, interest and the product price change nothing about them. For the distribution: before the launch the leverage model, while out the interest of the product currency, and taxes, trading costs and an execution lag only if your version sets them.

Day by day

The website applies the rule day by day, each time at the closing price (Close).

What is computed is one buffer $b$ for both directions, one confirmation day and the comparison with $\ge$ ; the separate buffers $b_{\text{in}}$ and $b_{\text{out}}$ and the confirmation days $k$ above are the general form.

The line depends on the state of the previous day. If the index was in, it lies one buffer below the average; if it was out, one buffer above; on the first day with a full line the average itself applies:

L(t) = \begin{cases} \mathrm{SMA}_n(t)\,(1 - b) & \text{previous day in} \\ \mathrm{SMA}_n(t)\,(1 + b) & \text{previous day out} \end{cases}

Whether the money is in the market is decided by comparing index and line: $P(t) \ge L(t)$

When the result changes, that is a crossing (Cross) of the line, and the entry or the exit follows at the same closing price. Line and money follow the same comparison; rounding is done only for the display. Every entry and every exit costs the share $c$ of the value $W$ :

W \leftarrow W\,(1 - c)

While out, the money earns the daily money-market rate on every trading day with a product price. $r$ is the last known rate in percent per year, 2.5 % without a known rate; a negative rate counts as 0:

z = \bigl(1 + \max(r, 0)/100\bigr)^{1/252} - 1

On the entry day the money still earns interest, on the exit day it no longer does. The start decision on the first day with a full line costs nothing and does not count as a trade.

Chapter 11 of 13Recompute