Interpreting Spearman Correlation Between HPL and RTPL Ranks
Summary
The document asks how to interpret the Spearman correlation used in the Fundamental Review of the Trading Book to compare hypothetical P&L (HPL) with risk-theoretical P&L (RTPL). The stated procedure ranks each series in ascending order, then applies the usual correlation formula to the paired ranks. The examples show that the calculation uses rank observations aligned by date, rather than the original P&L amounts.
Spearman correlation measures whether the two series move together in rank order: it is the Pearson correlation of their ranks. The P&L values determine those ranks, so changes in the original amounts matter when they alter the ordering; the size of changes beyond that ordering does not directly affect the coefficient. The example includes a tied HPL value, so tie handling matters and the displayed ranks should be checked against the applicable FRTB convention. The text asks for clarification but does not provide an answer or settle that convention.
Key ideas
- Spearman correlation is calculated as the correlation between the ranked HPL and RTPL observations.
- Each HPL rank must be paired with the RTPL rank from the same date.
- The coefficient reflects rank ordering rather than the magnitudes of the original P&L values.
- Tied P&L observations require a consistent ranking convention, which the document does not establish.
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Full text
# FRTB Spearman correlation coefficient definition
# FRTB Spearman correlation coefficient definition
I am just writing my thesis and would like to understand the spearman correlation coefficient definition within the FRTB.
Somehow it is not clear from the definition. The reason what I don't understand is, that when I look at the definition I just rank my PnLs and then I calculate the covariance and the standard deviation, But it doent matter whether PnLs are changing or do misunderstand something? Here the definitions:
- As a first step, the bank must rank each P&L for HPL and RTPL vector in ascending order, so that the lowest value receives a rank of 1.
- After ranking, the bank must calculate the Spearman correlation coefficient applying the following formula:
$r_s = \frac{cov(R_{HPL}, R_{RTPL})}{\sigma_{R_{HPL}}* \sigma_{R_{RTPL}}}$
with
(1) $\sigma_{R_{HPL}} = \sqrt{\frac{\sum_{i}^{250}(R_{HPL_{i}}-\mu_{R_{HPL}})^2}{249}}$
(2) $\sigma_{R_{RTPL}} = \sqrt{\frac{\sum_{i}^{250}(R_{RTPL_{i}}-\mu_{R_{RTPL}})^2}{249}}$
(3) $cov(R_{HPL}, R_{RTPL} = \frac{\sum_{i}^{250}(R_{HPL_{i}}-\mu_{R_{HPL}})(R_{RTPL_{i}}-\mu_{R_{RTPL}})}{249}$
where
- $i$ = the index that denotes the observation in the time series of ranks
- $𝑅_{𝐻𝑃𝐿_𝑖}$ = the ‘i-th’ observation of the time series of ranks $𝑅_{𝐻𝑃𝐿}$
- $\mu_{R_{HPL}}$ = the mean of the time series of ranks $𝑅_{𝐻𝑃𝐿}$
- $𝑅_{RTPL_𝑖}$ = the ‘i-th’ observation of the time series of ranks $𝑅_{RTPL}$
- $\mu_{R_{RTPL}}$ = the mean of the time series of ranks $𝑅_{RTPL}$
An Example would be:
```
Date HPL Rank(HPL) RTPL Rank(RTPL)
3/10/20 100.4 3 89.2 2
3/11/20 80.3 1 94.2 3
3/12/20 110.2 2 80.2 1
3/13/20 112.3 4 99.3 4
```
Additional data:
```
Date HPL Rank(HPL) RTPL Rank(RTPL)
3/10/20 3 3 2 2
3/11/20 1 1 3 3
3/12/20 2 2 1 1
3/13/20 3 4 4 4
```
But as said not clear what to calculate. Taking the ranks it makes no sense taking the PnLs I would have not using the ranks.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.