The formulas I actually coded: simple averages, exponential averages, RSI, and a basic return.
01
Price as a time series
A stock does not give you one number. It gives you a sequence. For each trading day I stored a close price P(t). If Monday is t = 1 and the close is 10, then Tuesday might be 11, Wednesday 12, and so on. Almost every beginner rule I read starts from this list. Technical analysis is, at this level, just functions of past prices (Investopedia, n.d.).
I also stored a simple daily return, because later I need to grow a pretend account. The return from yesterday to today is just the percent change:
r(t) = P(t) / P(t − 1) − 1
(1) daily return
If price goes from 10 to 11, r = 0.10. If it goes from 11 to 10, r is a little worse than −10%, because the base is larger.
If the price goes from 10 to 11, r = 0.10, or +10%. If it goes from 11 to 10, r = −0.0909. I used this later when I compared a moving average rule to sitting in the market the whole time.
Simple moving average
The simple moving average (SMA) is the mean of the last n closes. People use it as a smoother. A 20-day SMA is “what the last month of trading days looked like, on average.” A 50-day or 200-day SMA is slower and is often treated as a stand-in for the trend (Fidelity Learning Center, n.d.; StockCharts, n.d.).
Every day in the window has the same weight. Drop the oldest close, add the newest, divide by n. That is the entire indicator.
Worked example with n = 5. Suppose five closes are 10, 11, 12, 11, 13. The Friday SMA is (10 + 11 + 12 + 11 + 13) / 5 = 11.4. That is the entire indicator. There is no hidden step.
Day
Close
5-day SMA
Mon
10
—
Tue
11
—
Wed
12
—
Thu
11
—
Fri
13
11.4
Notice the SMA cannot exist until you have n prices. On Monday through Thursday the cell is empty. In code I skip those days so I do not pretend I had a signal I did not have. That sounds obvious, but it is a common beginner bug: filling the first days with zeros makes the program think the stock is cheap.
SMA-50 stays fixed. Drag to see how a faster average hugs price.
Figure 3. Close (navy), a fast SMA you can retune (teal), and SMA-50 (grey) on my illustrative series. After the mid-sample drop, both averages keep falling for a while even if price starts to bounce. That delay is why SMA rules get in late and get out late.
Figure 4. A 50-day SMA drawn through candlesticks. The average is a smooth line; the candles are the noisy path. Image: Investopedia, “Moving Average (MA).”
Exponential moving average
An exponential moving average (EMA) does the same job as an SMA but gives more weight to recent days. You pick a smoothing factor α (alpha). A common choice is α = 2 / (n + 1), so a 20-day EMA uses α ≈ 0.095. Then:
EMA(t) = α · P(t) + (1 − α) · EMA(t − 1)
(3) exponential moving average
Today is a blend: a little of the new price, a lot of yesterday’s EMA. The memory never fully dies; it just fades.
I did not use EMA as my main test because SMA is easier to show in a table. I still coded it. The practical difference is that EMA reacts a bit faster after a shock, which can help in a clean trend and can also create extra false flips in a choppy market (StockCharts, n.d.).
Crossover rules
A very old trading rule is: buy when a fast average crosses above a slow average, sell when it crosses below. People call the up-cross a “golden cross” and the down-cross a “death cross.” Brock, Lakonishok, and LeBaron (1992) tested simple versions of this on the Dow and reported that some of the rules looked better than a random walk in their sample. That paper is one of the reasons these formulas are still taught. It does not mean they still work after costs, after everyone knows them, or on every stock I pick.
An even simpler rule, which I used in Findings, is: hold the market when price is above the SMA, hold cash when it is below. Faber (2007) used a 10-month SMA on monthly data as a timing overlay. I used a 50-day SMA on daily data because I was working with a short illustrative series, not 40 years of monthly index levels.
Figure 5. A long 200-day SMA. Price spent a long stretch below the line, then recovered through it. Image: Fidelity Learning Center, “Simple Moving Average.”
Relative Strength Index
RSI was described by J. Welles Wilder (1978). It tries to measure how one-sided recent moves have been. First you split each daily change into a gain or a loss. Then you average those over n days (Wilder used n = 14). Relative strength is the ratio of average gain to average loss. RSI rescales that ratio onto 0 to 100.
RS = average gain / average loss
(4) relative strength
If average gain and average loss are equal, RS = 1 and the market has been a coin flip lately.
RSI = 100 − 100 / (1 + RS)
(5) RSI
RS = 1 maps to RSI = 50. Huge gains with almost no losses push RSI toward 100; the opposite toward 0.
If every day is an up day, average loss is near zero and RSI is near 100. If every day is a down day, RSI is near 0. Wilder’s original notes treated RSI above 70 as “overbought” and below 30 as “oversold.” Quantpedia and later timing papers also use RSI in a different way: as a trend filter, for example staying long when a slow RSI is above 50 (Quantpedia, 2023). Those are two different jobs — mean reversion versus trend — using the same number.
Figure 6. 14-day RSI on the same series. Dashed lines at 70 and 30. The middle of the sample spends a lot of time under 30, which matches the price drop in Figure 3. I did not trade every touch of 30. I only plotted it so I could see the formula move.
A thing I had to be careful about: Wilder’s smoothing is recursive, like an EMA of gains and losses. If I used a plain SMA of the last 14 changes, I get a close cousin called Cutler’s RSI. For a student project the difference is small. I used the recursive version so the code matches the usual write-up.
Volatility, in one line
Volatility here just means “how jumpy are the returns.” The version I used is the sample standard deviation of daily r(t), sometimes multiplied by the square root of 252 if I want a yearly number (because there are about 252 trading days in a year):
σ_ann ≈ stdev(r) · √252
(6) annualized volatility
√252 is a calendar trick, not a trading rule. I report it as a warning, not as a signal.
I am not using this as a trading signal. I am using it as a warning. A rule that looks good because it sat out one crash still has to be judged by how much the account bounced around on the days it was in. Findings reports a rough drawdown next to return for that reason.
Why these formulas and not others
I picked SMA, a crossover, and RSI because each one is a short function of past closes, each one has a name in the literature, and I can show the arithmetic on this page without a matrix library. MACD, Bollinger bands, and Kalman filters are the same family with more knobs. I did not add them. Extra knobs would have made the charts look busier without teaching me anything new about the pipeline: read a bar, compute a number, wait until tomorrow to act.
The next section is the software that holds these formulas in the right order so I do not cheat on time.
Glossary I actually used
Bar / OHLCV
One row of open, high, low, close, and volume for a day.
Lookahead
Using a number in the decision that would not have been known yet. Same-close fills are the usual form.
Whipsaw
A rule that buys, then sells, then buys again in a noisy range, paying costs for no trend.
Drawdown
The worst peak-to-trough drop on an equity line, not the same as a single down day.
Long / cash
Either hold the series or hold nothing. I did not short in this study.
Walk-forward
Choose a setting on one stretch of data, then report it on a later stretch you did not tune on.
Sources
Wilder, J. W. (1978). New Concepts in Technical Trading Systems. Trend Research.
Brock, W., Lakonishok, J., & LeBaron, B. (1992). Simple technical trading rules and the stochastic properties of stock returns. Journal of Finance, 47(5), 1731–1764.
Faber, M. (2007). A quantitative approach to tactical asset allocation. Journal of Wealth Management, 9(4), 69–79.
