Momentum: RSI and MACD

Chapter 5's moving averages follow price. This chapter's indicators try to measure how fast it has been moving, which is a different quantity and a more interpreted one.

RSI, defined

The relative strength index compares the size of recent gains with the size of recent losses.

Over a lookback of n periods, compute the average gain on up days and the average loss on down days, then:

$RS = \frac{\text{average gain}}{\text{average loss}}, \qquad RSI = 100 - \frac{100}{1 + RS}$

The conventional n is 14.

Read what that produces. If average gains equal average losses, $RS = 1$ and $RSI = 50$. If there have been no down periods at all, $RS$ is unbounded and $RSI$ approaches 100. If no up periods, $RSI$ approaches 0.

So RSI is a bounded restatement of the ratio of recent up-moves to recent down-moves, squashed onto a 0–100 scale. The squashing is the whole of what the formula adds — it makes an unbounded ratio comparable across instruments and periods.

What it is not is a measure of whether something is expensive. The name "relative strength" does not refer to strength relative to other stocks, and "overbought" is not a valuation statement. A stock can rise for a year with RSI high throughout, because a persistent rise is precisely what produces a high ratio of gains to losses.

MACD, defined

Moving average convergence divergence is built from chapter 5's exponential averages.

$\text{MACD} = \text{EMA}_{12} - \text{EMA}_{26}$

$\text{Signal} = \text{EMA}_{9}(\text{MACD})$

$\text{Histogram} = \text{MACD} - \text{Signal}$

A fast average minus a slow one, then an average of that difference, then the gap between the two.

What the quantity means. The difference between a short and a long average is a measure of how far recent prices sit above or below older ones — that is, a rate of change expressed through smoothing. When the fast average is above the slow one, recent closes have been higher than older ones.

Everything in MACD is a function of past closes. Three exponential averages stacked on each other, each with the lag chapter 5 described. The stacking does not remove lag; it compounds it, which is why the histogram turns later than price and the signal line later than the MACD.

The property both share

Both are deterministic functions of the closing price series.

Give me the closes and I can compute the RSI and the MACD exactly, with no other input. Which means:

They contain no information the closes did not already contain. They are transformations, not measurements of something new. Chapter 1 made this point generally; here it is concrete.

They cannot know anything the price does not reflect. Earnings, a regulatory decision, a change of management — none enters.

Agreement between them is not corroboration. If RSI and MACD "confirm" each other, that is two functions of the same series agreeing, which is close to guaranteed when price has been moving one way. Treating it as two independent pieces of evidence is a mistake people make constantly, and it is the same error as reading twelve indicators as twelve sources.

The parameters nobody derived

RSI uses 14 periods. MACD uses 12, 26 and 9. The thresholds are 70 and 30.

None of these numbers was derived from a theory of markets. They are conventions that originated with particular authors working with particular data decades ago, and they persist because they are what software ships with.

The research literature's warning applies with full force: results depend on assumptions including parameter choices and in-sample optimisation periods, and a dataset repeatedly searched over such choices will yield settings that look successful by chance.

With RSI there are at least three free parameters — the lookback, the upper threshold, the lower threshold — plus chapter 2's interval. Four dials on one indicator. The number of combinations is large, and finding one that fits a given history is not an achievement.

Working the problem

RSI above 70 described as "overbought".

What the number measures. At $RSI = 70$, $RS = 100/(100-70) - 1 = 2.33$. So the average gain on up periods has been about 2.33 times the average loss on down periods over the last 14 periods. That is all. It is a statement about the recent ratio of up-moves to down-moves, nothing more.

Where 70 comes from. It is a convention, introduced with the indicator and carried forward by convention and by default settings in charting software. It was not derived, and it is not calibrated to any particular instrument, interval or market regime. The same 70 is applied to a government bond and a small-cap share.

What "overbought" asserts. That the ratio having been high implies price is likely to fall. Notice this is a mean-reversion claim — and notice that it sits in direct tension with chapter 3's trend idea, which is a continuation claim. The same subject contains both, and which one a reader invokes tends to depend on what the chart has just done.

What would have to be true for crossing 70 to mean anything:

1. A measured base rate. Across many instruments and long periods, with the rule fixed in advance: after RSI crosses 70, what happens next, and how does that compare with a matched sample where it did not?
2. A result that survives the parameter search. If 70 works but 68 and 72 do not, the finding is about the number rather than the market. A real effect should be robust to small changes in an arbitrary threshold.
3. Out-of-sample confirmation. The replication approach chapter 9 describes — does it hold on data not used to find it?
4. Survival of transaction costs. The literature's standard is that a rule is profitable only if risk-adjusted profits exceed transaction costs. An effect too small to cover costs is not useful even if it is real.

Without those, "overbought above 70" is a convention being reported as a property of the market. The indicator is computing something definite; the threshold and the interpretation are the parts that have not been established.

The point

RSI squashes the ratio of recent average gains to recent average losses onto a 0–100 scale, so 70 means gains have run about 2.33 times losses over the lookback — a statement about recent price, not about value. MACD subtracts a slow exponential average from a fast one and smooths the result, compounding chapter 5's lag through three stacked averages. Both are deterministic functions of the closing prices, so they add no information and their apparent agreement is two transformations of one series moving together. Their parameters — 14, 12, 26, 9, 70, 30 — were conventions rather than derivations, and with four dials on a single indicator, a setting that fits any given history can always be found.