# Discounting with Imperfect Collateral

## TL;DR

This paper develops a new framework for discounting derivatives with imperfect collateral, incorporating collateral liquidity value adjustments and optimization under liquidity constraints.

## Contribution

It introduces a derivative financing rate that accounts for collateral imperfections and proposes a linear programming approach for collateral optimization.

## Key findings

- LVA can be significant for long-duration swap portfolios.
- The model effectively captures the impact of collateral imperfections on discounting.
- Optimization under LCR constraints improves collateral management strategies.

## Abstract

Cash collateral is perfect in that it provides simultaneous counterparty credit risk protection and derivatives funding. Securities are imperfect collateral, because of collateral segregation or differences in CSA haircuts and repo haircuts. Moreover, the collateral rate term structure is not observable in the repo market, for derivatives netting sets are perpetual while repo tenors are typically in months. This article synthesizes these effects into a derivative financing rate that replaces the risk-free discount rate. A break-even repo formulae is employed to supply non-observable collateral rates, enabling collateral liquidity value adjustment (LVA) to be computed. A linear programming problem of maximizing LVA under liquidity coverage ratio (LCR) constraint is formulated as a core algorithm of collateral optimization. Numerical examples show that LVA could be sizable for long average duration, deep in or out of the money swap portfolios.

---
Source: https://tomesphere.com/paper/1702.04053