# 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


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.

• The Spearman correlation is like the ordinary correlation except that instead of using the data, you use the Ranks of the data. So you would find the correlation between [3,1,2,4] and [2,3,1,4] with the ordinary correlation formula. The original data is not used. – noob2 Mar 14 at 20:51
• Thanks for the answer. But the ranks are all the time the same, the only change is the length of the data vector. Or do I miss something? – NewNY1990 Mar 14 at 21:13
• I added a simple example at the end the spearman correlation would be the same for both examples right? – NewNY1990 Mar 14 at 21:17
• Now I got it. There is a dependence . Thanks – NewNY1990 Mar 14 at 21:34
• Spearman Correlation really aims to address the following question: If my worst HPL occurred on day X, then did also the RTPL did occur on this day X? So its not about the absolute levels of the HPL and RTPL but more about whether their ranks are strongly correlated. An extreme example would be that 3 months ago HPL reports the most profitable P&L whereas your risk model reports the worst P&L on the same day. Independent of how large they were. Just as another viewpoint to gain some understanding of this concept – SI7 Mar 14 at 21:39