I was thinking that since XVA is on uncollaterized exposure, we should be using LIBOR discounting environment. Why don't we do that?
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Libor, besides its name, is not a good proxy for uncollateralised funding. In fact, a large bank I worked for had an uncollateralised funding curve which was a a spread of +100bps to Libor.
The fact that it was a spread to Libor was for legacy reasons and easier to adopt to existing systems. However, a spread of say +115bps to CSA was equally appropriate.
When you trade a collateralised (CSA) versus uncollateralised trade and want to hedge there is an element of how you account for the trades on balance sheet.
If you mark as a (generally fixed) uncollateralised spread to Libor then you will also need to hedge the LIBOR/CSA basis. If you mark as a (generally fixed) uncollateralised spread to CSA then you can effectively ignore the LIBOR part of discounting completely.
A second point is that to provide an effective comparative base every derivative should be first valued with the CSA curve and then the adjustments (CVA/FVA/KVA) measured from that basis, this avoids any kind of double counting, and provides a standardised means of comparison.
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Changing the discounting curve is not related to CVA, but to ColVA. CVA stands for the counterparty risk in uncolletarized trades, but the discounting is related to the CSA terms of collateral remuneration and risk free rate.
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1$\begingroup$ CVA has common terminology of credit valuation adjustment, but it has also been used in a funding context for collateral valuation adjustment where the terms of a CSA are non-standard.. I'm assuming your institution refers to it as ColVA... $\endgroup$– Attack68 ♦Feb 18, 2020 at 19:33