User:ComovementResearch/sandbox

In finance, the Gerber Statistic (a co-movement measure, part of the Gerber family of co-movement measures), is a robust co-movement tool for covariance matrix estimation which can be used for portfolio construction. It was named after Sander Gerber, a hedge fund manager who developed it in collaboration with several distinguished authors including Professor Harry Markowitz and Professor Philip Earnst. The former introduced Modern Portfolio Theory (MPT), for which he was later awarded a Nobel Memorial Prize in Economic Sciences; see Markowitz model. It can be calculated using vote counts, or by the enhanced Smyth-Broby approach which uses contributions.

Since the Gerber Statistic and the Smyth-Broby enhancement are not affected by extremely large or extremely small co-movements, they are well-suited for active fund managers seeking to understand un-priced risk.

Gerber Statistic Definition
The equation for the Gerber Statistic $${g}_{ij}(t) $$ measures the number of the proportion of simultaneous co-movements in a time series when their amplitudes exceed data-dependent thresholds:

$${g}_{ij}(t)$$ = $$\frac{\sum_{m} (\Delta_{ij}(t_m)_C - \Delta_{ij}(t_m)_D)}{\sum_{m} (\Delta_{ij}(t_m)_C - \Delta_{ij}(t_m)_D) + \Delta_{ij}(t_m)_N))} $$

where $$\{m\}$$ is the set of qualifying co-movements in the lookback window $$\left[t-T,t\right]$$ and $$\left\{\mbox{C,D,N}\right\}\equiv\left\{\mbox{Concordant, Discordant, Neutral}\right\}$$.

And, $$ {ij} $$ are two securities that co-move where the qualifying co-movements $${C,D,N}$$ are given votes in a quantitive ballot so as to determine the nature of the way two assets are connected. The total is carried forward and when all votes have been made, the Gerber statistic is determined. In effect, it represents the ratio of excess concordant counts over discordant counts, to the total number of qualifying counts.

The Smyth-Broby Enhancement
This measure was named after Dr William Smyth and Professor Daniel Broby. It does not replace the Gerber Statistic as it captures different co-movement dynamics. Both measures can be used in combination to understand co-movement.

The equation for the Smyth-Broby enhancement $$\tilde{g}_{ij}(t) $$ of the Gerber Statistic is essentially the same but the inputs are different:

$$\tilde{g}_{ij}(t)$$ = $$\frac{\sum_{m} (\Delta_{ij}(t_m)_C - \Delta_{ij}(t_m)_D)}{\sum_{m} (\Delta_{ij}(t_m)_C - \Delta_{ij}(t_m)_D) + \Delta_{ij}(t_m)_N))} $$

where $$\{m\}$$ is the set of qualifying co-movements in the lookback window $$\left[t-T,t\right]$$ and $$\left\{\mbox{C,D,N}\right\}\equiv\left\{\mbox{Concordant, Discordant, Neutral}\right\}$$.

And, $$ {ij} $$ are two securities that co-move. Where the qualifying co-movements $${C,D,N}$$ are calculated using contributions. It amends the Gerber Statistic in a way that is designed to reward stronger evidence of connection. Contributions represent the geometric mean of the gross standardized returns for two assets in a selected period.

Python code
This repository reproduces the analysis in the paper The Gerber Statistic: A Robust Co-Movement Measure for Portfolio Optimization.