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**Bollinger Bands**

Bollinger bands measure the volatility of stock price. It works by defining an upper and a lower bands between which stock price is most likely to fluctuate. The upper and lower bands are calculated through equation

BBU = SMA + K * SD

BBL = SMA - K * SD

SMA is the simple moving average of the last N periods. Investors conventionally use 20 periods. SD is the standard deviation^{*} of the close price of corresponding periods. K typically equals to 2.

Within specific periods of interest, prices are considered high at the upper band and low at the lower band. However, the upper and lower bands should not be used to signal turning points, as prices may move along upper or lower bands, or breakout above or below the bands.

^{*} divide by N rather than N - 1 because in this case N generates an unbiased estimate of the true population variance.

Bollinger bands width (BBW) measures the width of bollinger bands on a normalized basis.

BBW = (BBU - BBL) / SMA

BBU is the upper band. BBL is the lower band. SMA is the simple moving average.

BBW has the following distribution as of May 2018.

We calculated quantile ranking of BBW for each stock comparing with industry, sector, and market peers.

Bollinger bands percentage (BBP, or BB %b) measures the location of current price relative to BBU and BBL. It is calculated as

BBP = (close price - BBL) / (BBU - BBL) * 100%

Note that BBP can be a negative number or larger than 100%. When BBP equals to 50%, price is closed half way between BBL and BBU.

**Bollinger Bands Constraint**

In Katelynn's report, we use bollinger bands constraint (BBC) to quantify the squeeze (or breakout) level of the BBW of current period, comparing with previous several periods. BBC is calculated as

BBC = BBW_{current} / (BBW_{wavg} * (1 + CV))

BBW_{current} is the BBW of current period.

CV is the coefficient of variation of BBW of the last N periods.

BBW_{wavg} is weighted average BBW of the last N periods. In weekly mode, we let N = 5

BBW_{wavg} = (BBW_{p1} * W_{p1} + ... + BBW_{pN} * W_{pN})/100

SUM(W_{p1} + W_{p2} + ... + W_{pN})=100

where pn represents n periods before current period. The weights for pn is calculated as

W_{pn} = (N - n + 1) * (100/SUM(1 + 2 + ... + N))

When N = 5, the weights array is

(5 * 100/15, 4 * 100/15, ... , 1 * 100/15) or (33.35, 26.67, 20.00, 13.34, 6.67)

The basic idea is to give more weights to periods closer to current period. Divide by (1 + CV) cancels out the confounding effect of background variation.

BBC has the following distribution as of May 2018

BBC less than 1 indicates a squeeze (i.e. the BB becomes narrower than before). BBC greater than 1 indicates a breakouts (i.e. the BB becomes wider than before). Usually, in weekly mode, BBC less than 0.8 or greater than 1.1 indicates a significant change in band widths. Investors can also rely on the performance monitor to determine how significant a squeeze (or breakouts) is comparing with peers. Generally, quantile ranking above 95% indicates significant breakouts, below 5% indicates significant squeeze.