Moving Averages and the Cost of Lag

What a moving average is, precisely

A simple moving average is the mean of the last n closes. That definition contains everything worth knowing about it. The mean of five closes is $24.60 when those closes are 21, 22, 24, 26 and 30, and it is not a price that anyone is holding — it is a statistic describing a window. Because the window is fixed, each day has the same arithmetic effect: the newest close enters and the oldest leaves, so a day the stock fell hard can still move the average up if the close that dropped out of the window was lower still. This is not a flaw to be worked around; it is the property that makes the line smooth and the reason it cannot lead. The exponential moving average weights recent closes more heavily, which is often described as making it faster. That description is half true. In a 20-period EMA the effective average age of the closes is nine and a half sessions, exactly as it is for the 20-period simple average; what changes is the *distribution* of the weights, which is spread exponentially rather than equally. So the EMA responds sooner to a new move while still being an average, and the cost of that responsiveness is more sensitivity to noise, which is why a fast EMA whipsaws more than a slow SMA of the same lookback. The practical rule is that the difference between the two matters far less than the length of the window: a 200-period average of either kind is slow, and a 5-period average of either kind is mostly noise. The average age of the window is the useful mental model, because it explains both the lag and the use. A 5-day average describes the middle of the last week; a 50-day average describes a point about seven weeks back if it were a single day, because the average age of its closes is about 24 sessions. That is why a 200-day moving average can sit far below price for months in a recovery and still be rising: it is reporting a statistic about last autumn. And it is why the crossover between two averages, which is the most quoted signal in charting, is a slow agreement between two old windows rather than a discovery about the present. A 5-day average, one day apart — Window yesterday: 20, 21, 22, 24, 26 → $22.60 · Window today: 21, 22, 24, 26, 30 → $24.60 ← · What changed: A $20.00 dropped out and a $30.00 went in — the average rose $2.00 while price rose $4.00 · If price falls to $24 tomorrow: 22, 24, 26, 30, 24 → $25.20 — the line still rises ← The average age of a 5-day window is two sessions; of a 20-day window, nine and a half. Multiply nothing, add nothing — the lag is simply the mean of the offsets, which is why a longer window is not "more accurate" but "differently late".

What an average is for, and what it costs

There are three defensible uses and one indefensible one. The defensible uses: as a filter, allowing longs only when price is above a rising average and refusing the rest; as a dynamic reference, a level that trends with the market and that traders watch because everyone else is watching it, which is the same reflexive mechanism that makes any level matter; and as a system component with a known, measurable expectancy — the classic moving-average crossover backtests as a positive-expectancy trend follower across long histories, but only because a small number of very large trends pay for a long series of small whipsaws. All three share the property of being a rule about participation rather than a prediction about direction. The indefensible use is using the line to forecast a turning point, which it structurally cannot do: it is made of prices that have already printed, weighted and averaged. The honest framing of a crossover is "the recent window has become more favourable than the older window", which is a statement about the past. A trend can be twenty per cent along before a 20/50 crossover fires, and in the range that follows, the same crossover will fire eight times and be wrong five of them while costing the spread each time. The cost is not theoretical: each whipsaw is two entries and exits at the spread, so a setting that produces twelve signals a year in a range can lose more to friction than it makes in the trend that eventually arrives. The way to keep the tool honest is to put the regime check before the signal, which TR14 does properly and which can be anticipated here. Crossovers are for trending markets: measure the market first, and only then read the signal. In a market with an efficiency ratio near zero — one that does a lot of work to go nowhere — the correct number of crossover signals to take is none of them, and the correct action is to let the averages cross each other unattended while the position size falls. That is a discipline about context, and it is the difference between a tool and a habit. • Legitimate uses: a trend filter, a dynamic reference level, and a rule-based system with measurable expectancy. • Not a use: forecasting a turn. The line is made of prices that have already printed. • Longer window: slower, smoother, fewer whipsaws; shorter window: sooner, noisier, more of them. • Cost of a whipsaw is two entries and exits at the spread, and a range produces them mechanically. • Regime first, signal second: in a range, the right number of crossover signals to take is none. The most common misuse among learners is drawing three averages at once — 20, 50 and 200 — and reading the alignment as confirmation. Three windows of the same data are not three pieces of evidence; they are one smoothed view of the past at three resolutions, so agreement between them is a restatement rather than corroboration. Before treating a stack of averages as confirmation, ask what independent information the second and third lines add.

Choosing the length is the decision

The lesson usually ends with a piece of folklore — use the fifty and the two hundred — and the folklore is a convention rather than a reason. The length of an average is a parameter, and the parameter decides both what you see and how much you give back. Choosing it is the actual decision, and it deserves the treatment any other rule gets: stated in advance, tested across a range, and justified by something other than having looked best on the chart in front of you. What the length encodes is a single trade-off. A short average tracks price closely and changes its mind often; a long one is slow and stable. Sensitivity to new information and immunity to noise are the same dial turned in opposite directions, so there is no setting that is both responsive and calm. The honest way to choose is to match the average to the holding period you can actually sustain — an average that turns weekly cannot be traded by somebody who only looks at the portfolio on Sundays — rather than to whatever line fits the last move. Then the trap, which is a search pretending to be a choice. Trying lengths from five to two hundred on one price history and keeping the winner is a search over roughly two hundred candidates, and the winner will be the one that best fitted the noise in that particular sample. It will look excellent and it will mean nothing. The remedy is not to refuse parameters; it is to stop treating the peak as the finding. The discipline that makes it honest is to plot the result against the parameter and read the *shape* rather than the high point. If expectancy rises to a plateau across a broad band of lengths, there is likely something real in the idea, because many settings capture it. If it spikes at 37 and collapses at 35 and 40, the peak is noise wearing a number, and the next sample will move it. Reporting the whole curve, including the disappointing ends, is the difference between a test and a hunt for a flattering number. There are adaptives that try to escape the choice — averages that shorten in fast markets and lengthen in quiet ones, Kaufman’s version, the hull form, and the double- and triple-smoothed exponentials that reduce lag by combining averages of averages. Each buys responsiveness at the price of another parameter, and every parameter is another opportunity to fit the past rather than measure it. It is worth trying one, and worth remembering that “adaptive” describes the mathematics, not the honesty: an adaptive rule that was tuned on a history is as fitted as a fixed one. So the defensible answer to “which length” has four parts: the one whose turn rate matches your holding period, tested as a family rather than a point, reported across the range instead of at the peak, and re-tested — never re-tuned — on data the choice was not made from. What it looks like when it is written down is unglamorous and useful: “a 50-day average on daily bars, chosen because the median hold is six weeks, and the 30-to-80 band all showed positive expectancy in the test”. That sentence can be argued with. “It looked best” cannot. • Length trades responsiveness against stability; there is no setting that is both. • Match the turn rate to the holding period you will actually keep. • Plot expectancy against length and believe the plateau, not the spike. • Adaptive averages add parameters rather than removing the choice. The same curve answers a second question: how much the result depends on the parameter at all. A flat curve across a wide band says the idea is robust; a narrow one says the strategy is a claim about a number, and claims about numbers decay.

Sources

• Simple and exponential moving averages, and the lag they carryMurphy, “Technical Analysis of the Financial Markets”
• Moving-average crossovers as a trend-following systemFaber, “A Quantitative Approach to Tactical Asset Allocation”
• Why trend-following pays in trends and bleeds in rangesCovel, “Trend Following”