The SMA cross: two averages, and the day they swap places

The Bollinger Bands post put a margin around one moving average. This post uses two averages and no margin at all. The rule is as old as charting: average the price over a short window and a long one, and the day the short average moves above the long one, the trend is up; the day it moves below, the trend is down. It is the rule the backtesting post ran with 20 and 100 days, and the one whose flips the indicators post counted. Here it gets the full treatment on SPY’s closes from 2019 to 2023: what a cross is, the four times the classic pair crossed, and what the price had already done by the time each cross arrived.

What a cross is

A simple moving average over $n$ days is the mean of the last $n$ closes, recomputed each day. Achelis starts with one average and the price itself: buy when the price closes above its average, sell when it closes below. Murphy prefers two averages, which he calls the double crossover method: “a buy signal is produced when the shorter average crosses above the longer”, and he names the pairs traders use, 5 and 20 days, 10 and 50. The stock-market pair is 50 and 200 days, and the StockCharts page gives the two crosses their trading-floor names: the short average rising through the long one is a golden cross, falling through it a death cross.

S, the short average, which does the crossingthe day the sign of the gap changes

\begin{aligned} \key{S_t} &= \tfrac{1}{50}\textstyle\sum_{i=0}^{49} P_{t-i}, \qquad L_t = \tfrac{1}{200}\textstyle\sum_{i=0}^{199} P_{t-i} \\[10pt] \bad{\text{cross}} &\iff \operatorname{sign}(\key{S_t} - L_t) \ne \operatorname{sign}(\key{S_{t-1}} - L_{t-1}) \end{aligned}

The formula has no threshold in it. The two averages are in dollars, but the rule only asks which is bigger, so the price level drops out and the same rule reads a $40 stock and a $400 one. On 2020-03-30 SPY’s 50-day average was 274.22 and its 200-day was 273.70, the short one just ahead. The next day the 50-day was $\key{272.90}$ and the 200-day 273.58: the short average had fallen through the long one by 68 cents, and 2020-03-31 was a death cross.

A cross is a comparison of two lagging numbers, so it arrives after the move it names. The 50-day average needs fifty days of new prices before the old ones stop dominating it. The sign flips when enough of the fall has been averaged in, not when the fall begins.

The four crosses on SPY

Five years of SPY gave the 50/200 pair four crosses. For each, the table shows the close that day and how far the price had already travelled from its most recent extreme, the high before a death cross or the low before a golden one:

Date	Cross	Close that day	The price had already moved	From
2020-03-31	death	235.74	−23.4%	the high of 2020-02-19
2020-07-06	golden	291.25	+42.8%	the low of 2020-03-23
2022-03-16	death	408.87	−8.8%	the high of 2022-01-03
2023-01-26	golden	386.06	+14.0%	the low of 2022-10-12

The 2020 pair is the extreme case. The death cross came six trading days after the low, with the fall already over: SPY had lost 23.4% from its February high and had begun to recover. The golden cross came three months later, after the price had risen 42.8% from the low. A rule that sold on the first cross and bought on the second sold near the bottom and bought back higher than it sold. The 2022 pair was gentler, the death cross arriving 8.8% below the high and the golden cross 14.0% above the low, because that fall was slower and the averages had time to follow it. Achelis says what the rule is for in one sentence: it “is not intended to get you in at the exact bottom nor out at the exact top”, but to keep you “in line with the security’s price trend by buying shortly after the security’s price bottoms and selling shortly after it tops”. In 2020 “shortly after” was three months.

Faster pairs arrive sooner and flip more

The lag is set by the windows, and shortening them has a cost. Murphy states the trade-off plainly: a shorter average “has the advantage of giving trend signals earlier in the move”, and “the more sensitive the average, the earlier the signals will be”, but it “generates more false signals”, the whipsaws, and more trades with their commissions. Counting the crosses of his three pairs on the same five years of SPY:

Pair of windows	Crosses, 2019 to 2023	About one every
5 and 20 days	60	month
10 and 50 days	28	two months
50 and 200 days	4	fifteen months

Sixty crosses in five years is a rule that changes its mind every month, and most of those are the short average dipping across the long one and coming straight back. Four crosses is a rule that needs a year and a quarter of patience and, as the table above showed, pays for it in how late each cross arrives. There is no pair that is both early and quiet; the dial runs from one fault to the other. The backtesting post found the same thing from the other side, by searching window pairs for the best past return and watching the best pair fail to hold out of sample.

Where does a pair come from, then? Achelis offers a rule of thumb, half the market’s cycle length plus one day, so a 40-day peak-to-peak cycle wants a 21-day average, and then undercuts it with his own experiment: with hindsight he found the best monthly average for the Dow over 1970 to 1993 was 43 months, and “the key is to find a moving average that will be consistently profitable”, which hindsight cannot tell you. The honest answer is that 50 and 200 are conventions, used because they are used, and a cross is informative mostly because many people watch the same one.

What the tests say

The cross is the one technical rule with a serious academic record, and Kirkpatrick and Dahlquist make a point of it. Brock, Lakonishok and LeBaron tested moving-average crossover rules on ninety years of the Dow Jones Industrials in 1992, with methods built to reduce the biases of earlier tests, and found the signals had “intrinsic value”; the result was argued over, later studies on other countries’ markets “generally” found the same, and Kirkpatrick and Dahlquist count it among the better arguments against the random-walk view of prices. Their own verdict is two-sided. Crossover signals have been “successful but with substantial drawdowns in capital in sideways markets because of the many unprofitable small trades”, and “a moving average requires a trend for a crossover to be profitable”. The method works, when it works, by losing a majority of small trades and catching the one large move; it asks for “the stomach and discipline to continue with the small losses”. Their conclusion is that the most sensible use of a moving average is to say whether a trend exists at all, with entries then timed by something else, which is a more modest job than the names golden cross and death cross suggest. SPY’s four crosses fit that description: two caught the long 2020 to 2021 rise and the 2022 fall, and the first, the death cross of March 2020, was a small loss a week after the low.

Traps

People think	Actually
A golden cross says the price is about to rise	it says the price has risen enough for the short average to catch up; in 2020 that was 42.8% after the low
A faster pair fixes the lag	it shortens the lag and multiplies the crosses: sixty for 5 and 20 against four for 50 and 200
The cross is a prediction	it is a description of two averages; any forecast in it comes from the trend continuing
A rule that loses most of its trades is broken	Kirkpatrick and Dahlquist say that is the design: many small losses in sideways markets, paid for by the one trend it catches
50 and 200 are the right windows	they are the usual windows; Achelis found 43 months best for the Dow with hindsight, and says hindsight is the problem
The rule needs a threshold	it does not; the two averages are compared directly, which is why it reads any price level

The code

The snippet reproduces the death cross of 2020-03-31 and the list of all four crosses. Variables:

Variable	Holds
`px`	SPY’s adjusted close, one row per trading day
`s50`, `s200`	the 50-day and 200-day averages, one value per day once enough days exist
`side`	+1 on days the short average is above the long one, −1 below
`cross`	the days `side` changed: +1 is a golden cross, −1 a death cross

import numpy as np
import pandas as pd

url = "https://raw.githubusercontent.com/ankurtrapasiya/examples/main/data/prices.csv"
px = pd.read_csv(url, index_col="date", parse_dates=True)["SPY"]

s50, s200 = px.rolling(50).mean(), px.rolling(200).mean()
print(f"2020-03-31: 50-day {s50['2020-03-31']:.2f}  200-day {s200['2020-03-31']:.2f}")

side = np.sign(s50 - s200).dropna()                # which average is on top, each day
cross = side[side.diff().fillna(0) != 0]           # the days that changed
for day, s in cross.items():
    print(day.date(), "golden" if s > 0 else "death", f"close {px[day]:.2f}")

It prints a 50-day average of 272.90 and a 200-day average of 273.58 on 2020-03-31, then the four crosses: a death cross on 2020-03-31 and 2022-03-16, a golden cross on 2020-07-06 and 2023-01-26. .diff() is the pandas idiom for “today’s value minus yesterday’s”, so a non-zero difference in the sign is a cross.

The script behind the real numbers

The tables and the chart come from examples/sma-cross/sma_cross.py, which runs on the blog’s five tickers. To run it yourself, clone the examples repo: its README has a three-command Docker setup, or run uv run sma-cross/sma_cross.py.

Summary

Idea	In one line	On SPY, 2019 to 2023
A cross is a change of sign	the short average overtakes the long one, or falls behind it	272.90 under 273.58 on 2020-03-31
It arrives after the move	both averages lag, so the cross confirms a trend that already happened	death cross 23.4% below the high, golden cross 42.8% above the low
Speed costs flips	shorter windows cross sooner and far more often	60 crosses for 5 and 20, 4 for 50 and 200
The windows are conventions	no pair is both early and quiet; a pair found with hindsight is fitted to the past	Achelis’s best Dow average with hindsight: 43 months

References

John J. Murphy, Technical Analysis of the Financial Markets (New York Institute of Finance, 1999). Chapter 9, “Moving Averages”, for the double crossover method, the 5/20 and 10/50 pairs, and the trade-off between early signals and whipsaws.
Steven B. Achelis, Technical Analysis from A to Z, 2nd edition (McGraw-Hill, 2001). The “Moving Averages” entry for the price-over-average rule, what a moving-average system is for, the cycle rule of thumb and the hindsight warning.
Charles D. Kirkpatrick and Julie R. Dahlquist, Technical Analysis: The Complete Resource for Financial Market Technicians (FT Press, 2007). Chapter 14, “Moving Averages”, for the crossover as a buy and sell signal, whipsaws in a flat trend, the drawdown verdict, trend determination as the most sensible use, and the summary of the academic tests.
William Brock, Josef Lakonishok and Blake LeBaron, Simple Technical Trading Rules and the Stochastic Properties of Stock Returns, Journal of Finance 47(5), 1992. The test of moving-average rules on the Dow that Kirkpatrick and Dahlquist summarise.
StockCharts ChartSchool, Moving Averages: Simple and Exponential. The simple-average formula, the golden and death cross names, and the whipsaw warning for trendless markets.
Yves Hilpisch, Python for Finance, 2nd edition (O’Reilly, 2018). Chapter 15 for the crossover rule in pandas and the warning that a window pair chosen on past data is prone to overfitting.