Indicators and the 200-day moving average

01Indicators and rules

A backtest replays history through a rule, day by day, charging the trades it would have made, and reports what the rule would have earned; most of this note is about making that replay honest.

Write P_t for the closing price (or total-return index) on day t. An indicator is a function I_t = f(P_t, P_t−1, …) of prices up to and including the close of t. A rule maps the indicator to an exposure w_t+1 held over the next day. Everything that goes wrong in practice happens in the space between those two sentences: which close the indicator uses, when the order can execute, and what the trade costs. Figure 1 shows the chain; the notation is the one used throughout the series.

02The main families

Most indicators in use belong to five families. Each has a standard definition; the differences between implementations (a simple or exponential average, a close or a typical price, the smoothing of Wilder’s averages) matter less than the lookback and the execution around it.

The trend and momentum families are two views of the same thing: a price above its n-day average has, roughly, risen over the last n/2 days or so, since the average is centred that far back. Time-series momentum, which Market anomalies reviews, is the academic name for the second view.¹ Volatility indicators are the input to Volatility targeting. Wilder introduced the RSI and the ATR,² and the RSI’s mean-reversion logic is a form of the short-term reversal described in Market anomalies. Figure 2 draws four of the families on the same days.

def sma(price: pd.Series, n: int) -> pd.Series:
    """Simple moving average of the last n closes."""
    return price.rolling(n, min_periods=n).mean()


def ema(price: pd.Series, n: int) -> pd.Series:
    """Exponential moving average with span n (alpha = 2 / (n + 1))."""
    return price.ewm(span=n, adjust=False, min_periods=n).mean()


def roc(price: pd.Series, n: int) -> pd.Series:
    """Rate of change over n periods (time-series momentum)."""
    return price / price.shift(n) - 1.0


def rsi(price: pd.Series, n: int = 14) -> pd.Series:
    """Wilder's relative strength index: 100 - 100 / (1 + avg gain / avg loss)."""
    d = price.diff()
    gain = d.clip(lower=0.0).ewm(alpha=1.0 / n, adjust=False, min_periods=n).mean()
    loss = (-d.clip(upper=0.0)).ewm(alpha=1.0 / n, adjust=False, min_periods=n).mean()
    return 100.0 - 100.0 / (1.0 + gain / loss)


def realised_vol(ret: pd.Series, n: int = 21, periods=None) -> pd.Series:
    """Annualised standard deviation of the last n returns. `periods` defaults to the sessions
    per year at each date (`sessions_per_year`), so pre-1952 data is annualised correctly."""
    periods = sessions_per_year(ret.index) if periods is None else periods
    return ret.rolling(n, min_periods=n).std(ddof=1) * np.sqrt(periods)

03How to evaluate an indicator

Indicators are easy to compute and easy to overrate. Six checks separate a result from an artefact; the analysis that follows applies each of them.

1. Look-ahead. A signal computed from the close of day t cannot earn day t’s return. The error is common because in a vectorised backtest it is one missing shift, and it is flattering: on the rule below it adds 8.7 percentage points a year.
2. Execution lag. Even correctly lagged, a rule that trades “at the close” assumes the signal is known in time to send a market-on-close order. Test one day later as well; the difference measures how much the result depends on precise timing.
3. Costs. Charge every switch. A rule that trades a few times a year looks cheap until the switches cluster in volatile, wide-spread markets.
4. Whipsaw. Count round trips that reverse within weeks, and measure what they cost. They are the price of a trend rule, and they are paid in exactly the range-bound markets where the rule adds nothing.
5. Regime dependence. Report results by decade and by episode, not only in aggregate. A century-long average can be one decade’s result in disguise.
6. Multiple testing. The 200-day lookback is famous because it has been examined for decades, which is itself a form of selection. Show the whole neighbourhood of lookbacks, and read Market anomalies on deflating the best of many trials.

04The 200-day rule on a century of data

The rule is the simplest trend rule there is: at each close, hold the US stock market if its total-return index is above its 200-day simple moving average, otherwise hold one-month Treasury bills; trade at that close. The data are the daily CRSP value-weighted US market total return and the one-month T-bill return from the Kenneth R. French Data Library,³ 5 May 1927 to 31 August 2026 after 252 sessions of warm-up (about ten months, since the market then also traded on Saturdays). Sharpe ratios are annualised with the number of sessions actually traded each year. That is the broad US market rather than the S&P 500 or the SPY exchange-traded fund, which has existed only since 1993: a long daily S&P 500 history is not freely licensed. As a check, the 200-day regime computed on the S&P 500 price index (FRED, 12 July 2017 to 31 August 2026)⁴ agreed with the regime on the CRSP series on 98.5% of 2,297 days.

The headline is real but needs reading carefully. The rule earned a higher return than the market with two-thirds of its volatility, so its Sharpe ratio was materially higher (0.73 against 0.46), and its worst drawdown was −34% against −84%. It achieved that by being out of the market 28% of the time and switching 5.7 times a year. Two cautions apply before any of these numbers is quoted. The return advantage comes from the years before 1990 (next section); and the 200-day drawdown is the best, not the typical, result among the lookbacks tested (Figure 4). Trading at the close that produced the signal is also the optimistic case; the next-close row below is the conservative reading.

import sys
from pathlib import Path

import pandas as pd

sys.path.insert(0, str(Path(__file__).resolve().parents[1]))
from fieldnotes import backtest, data, indicators  # noqa: E402

d = data.market_daily()                          # daily total return, T-bill return
price = (1 + d["mkt"]).cumprod()                 # total-return index
signal = indicators.sma_regime(price, 200)       # 1 above the 200-day average, else 0

rule = backtest.run(d["mkt"], signal, d["rf"], lag=1)                       # lag=1: decided at the close of t, held over t + 1
hold = backtest.run(d["mkt"], pd.Series(1.0, index=d.index), d["rf"])

start = d.index[252]                             # 252 sessions of warm-up for every lookback
for name, res in (("buy and hold", hold), ("200-day rule", rule)):
    r = res[res.index >= start]
    print(name, {k: round(float(v), 4) for k, v in
                 backtest.stats(r["ret"], d["rf"][r.index], weight=r["weight"]).items()})

def run(ret: pd.Series, weight: pd.Series, cash: pd.Series | None = None,
        cost: float = 0.0, lag: int = 1) -> pd.DataFrame:
    """Daily P&L of holding `weight` in `ret`, the rest in `cash`.

    lag=1: the weight decided at close t earns the return of t+1 (trade at the signal's close).
    lag=2: trade one period later.  lag=0: LOOK-AHEAD (the signal earns its own day's return).
    cost:  a fraction of capital charged on every unit of exposure traded (0.001 = 10 bp).
    """
    w = weight.shift(lag).fillna(0.0)
    cash = pd.Series(0.0, index=ret.index) if cash is None else cash.reindex(ret.index).fillna(0.0)
    traded = w.diff().abs().fillna(w.abs())
    pnl = w * ret + (1.0 - w) * cash - cost * traded
    return pd.DataFrame({"ret": pnl, "weight": w, "traded": traded})

def sma_regime(price: pd.Series, n: int = 200, band: float = 0.0) -> pd.Series:
    """1 when the close is above its n-day SMA, else 0, decided at each close.

    band > 0 adds hysteresis: switch IN only above sma * (1 + band) and OUT only below
    sma * (1 - band); in between, keep yesterday's state. band = 0 is the plain rule.
    """
    m = sma(price, n)
    up, down = price > m * (1 + band), price < m * (1 - band)
    state = pd.Series(np.nan, index=price.index)
    state[up] = 1.0
    state[down] = 0.0
    state[m.isna()] = 0.0
    return state.ffill().fillna(0.0)

05Where the result comes from

By period

The century-long return advantage belongs to the years before 1990, and mostly to the first quarter-century: from 1927 to 1952 the rule compounded at 11.8% against 7.1%. From 1953 to 1989 it was ahead by 2.2 percentage points, and since 1990 it has returned less than the market (9.2% against 11.0%). What held in every period is the risk profile: a higher Sharpe ratio and a much smaller worst drawdown.

By decade

The rule beat the market in the decades of long bear markets, the 1930s, the 1970s and the 2000s, and lagged it in strong, steady ones: the 1950s, the 1990s, so far the 2020s and, most of all, the 2010s, when it returned 8.0% a year against 13.6%. Its drawdowns were smaller in almost every decade. Anyone who adopted the rule in 2010 on the strength of the previous eighty years would have spent ten years underperforming and been right to ask whether it had stopped working.

In the deepest drawdowns

This is where the rule earns its reputation: in the five deepest market drawdowns its own worst loss over the same span was between 24% and 59% of the market’s, about a quarter to three-fifths. The mechanism is simple: a decline deep enough to matter lasts long enough to take the index below its average, and the rule steps aside for the rest of it. The price is paid later. In three of the five episodes (1929, 2007 and 1973) the rule’s worst loss began only after the market’s low: having gone back in, it rode a later fall (September 1932 to March 1933, April to August 2010 and July to September 1975). In the others it began before the low. Measured from each peak to the market’s trough only, the rule lost between 8% and 26%. The decline of 2007–09 shows the mechanism and its price, step by step.

Whipsaw

The cost of that protection is whipsaw. Of 281 round trips (an exit and the next re-entry), 247 (88%) were out of the market for 60 trading days or less, and on average the market rose 1.2% more than T-bills while those quick round trips were out. The worst are listed below; the largest, from 18 August 1939 to 6 September 1939, missed 7.8% in 15 trading days.

06Lookback, execution and costs

The lookback

Figure 4 repeats the analysis for every lookback from 50 to 300 days in steps of ten. The Sharpe ratio ranges from 0.68 to 0.84 and the annual return from 11.1% to 12.2%; every lookback beats buy and hold on Sharpe ratio, and 200 days is not special. That flatness is reassuring, with a caveat: every lookback shares the same few bear markets, so the 26 results are far from 26 independent confirmations. The worst drawdown, by contrast, jumps around, because it is decided by how a handful of crashes happened to line up with each average. At 200 days it was −34%, the best of the 26 (tied with another lookback); the median across lookbacks was −41%. Quote the median, not the famous number.

Execution and costs

Four things in this table matter. First, look-ahead multiplies the return by 1.75: the error is not a rounding matter. Second, one day of delay costs 0.9 percentage points a year and changes the worst day from −9.2% to −17.4%. The worst-day change is a single date: the index closed below its average on Friday 16 October 1987, so selling at that close avoided Monday’s −17.4%, and selling at Monday’s close took all of it. That one day explains only part of the annual difference; the rest is spread across hundreds of switches, each a day late. Third, one day of delay and 10 bp a switch together remove the return advantage entirely (10.0% against 10.2%), while the Sharpe ratio and drawdown still favour the rule. Fourth, a band cuts the switches by more than half, but both bands cost a little return and deepen the worst drawdown, from −34% to −36% at 1% and to −43% at 2%, so a band is a cost decision with a price of its own.

07What the literature says

The evidence on technical rules is long and divided. Fama and Blume found that simple filter rules on individual stocks did not beat buy and hold once costs were counted.⁵ Brock, Lakonishok and LeBaron tested moving-average and trading-range rules on the Dow Jones index from 1897 to 1986 and found returns after buy signals that standard time-series models could not explain.⁶ Sullivan, Timmermann and White then corrected for the full universe of rules that could have been tried: the best rule remained significant in the original sample but not in the decade that followed.⁷ Lo, Mamaysky and Wang gave chart patterns a statistical footing and found that some carried incremental information,⁸ and Park and Irwin’s survey concluded that the positive results cluster in older samples and weaken with better controls for data snooping and costs.⁹

For asset allocation, Faber popularised a monthly version of this note’s rule, a 10-month average applied to several asset classes, and reported equity-like returns with much smaller drawdowns;¹⁰ Zakamulin examined the real-life performance of moving-average and momentum timing rules and found the out-of-sample advantage smaller and concentrated in bear markets.¹¹ The wider trend-following literature puts the same idea across many markets.^1,12 This note’s result sits comfortably inside that literature: protection in long declines, a price paid in whipsaw and in strong bull markets, and returns that depend on the period.

08Implications

1. Treat a trend indicator as a risk overlay first: its robust property is avoiding the middle of long declines.
2. Lag every signal, test one day later as well, and distrust any result that hinges on a single close.
3. Charge realistic costs per switch, and consider a band or a monthly check if switches cluster.
4. Report the whole neighbourhood of lookbacks and the per-decade table, not a single number.
5. Decide in advance how long the rule may lag the market, remembering the 2010s, before concluding that it has stopped working.