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# Historical Volatility Ratio

This study calculates and displays the Historical Volatility Ratio of the Close Prices.

Let \(C\) be a random variable denoting the Close Price, and let \(C_t\) be the value of the Close Price at Index \(t\). Then we compute the Logarithmic Return (which is defined in the Volatility - Historical study) of \(C\) for \(t > 0\) as follows.

\(\displaystyle{LR_t(C) = \ln\left(\frac{C_t}{C_{t-1}}\right)}\)Let the Inputs **Short Length** and **Long Length** be denoted as \(n_S\) and \(n_L\), respectively, and let \(LR\) be a random variable denoting the Logarithmic Return for \(C\). The Standard Deviations of \(LR\) with **Lengths** \(n_S\) and \(n_L\) at Index \(t\) are denoted as \(\sigma_t\left(LR,n_S\right)\) and \(\sigma_t\left(LR,n_L\right)\), respectively. \(\sigma_t\left(LR,n_S\right)\) is calculated for \(t \geq n_S\), and \(\sigma_t\left(LR,n_L\right)\) is computed for \(t \geq n_L\).

The **Historical Volatility Ratio** at Index \(t\) for the Closing Prices and the given **Lengths** is denoted as \(HVR_t\left(n_S,n_L\right)\), and we compute it in terms of these Standard Deviations for \(t \geq n_L\) as follows.

#### Inputs

#### Spreadsheet

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*Last modified Monday, 26th September, 2022.