Why does the collateral rate specified in a CSA determine the discount curve, and how does this affect derivative valuations?

The collateral rate determines the discount curve because of a fundamental no-arbitrage argument: the cost of funding a derivative position is directly linked to the return on the collateral posted against it. If you can earn the OIS rate on your collateral, then the appropriate discount rate for the derivative's cash flows is the OIS rate.\n\nThe Funding Argument:\n\nConsider a dealer who enters a derivative that will pay $1 million in one year. To hedge the risk of this future payment, the dealer needs to set aside reserves today. The cost of carrying these reserves depends on what they earn:\n\n- If collateral earns the overnight rate (SOFR): the carry cost is SOFR, so discount at SOFR\n- If collateral earns a different rate (say EUR overnight): discount at that rate\n- If no collateral is posted: the dealer funds at their own borrowing rate (higher), increasing the carry cost\n\nMathematical Foundation:\n\nThe present value of a collateralized derivative cash flow C at time T is:\n\nPV = E^Q[C x exp(-integral from 0 to T of r_c(s) ds)]\n\nwhere r_c is the collateral rate. This is the risk-neutral expectation discounted at the instantaneous collateral rate — not LIBOR, not the dealer's funding rate.\n\nImpact on Valuations:\n\n| CSA Collateral Terms | Discount Curve | Typical Spread to OIS |\n|---|---|---|\n| Cash (USD) earning Fed Funds | OIS (SOFR) | 0 bps |\n| Cash (EUR) earning ESTR | EUR OIS | Cross-currency basis |\n| Government bonds | OIS minus haircut adjustment | -5 to -15 bps |\n| No collateral | Dealer funding rate | +50 to +200 bps |\n\nWorked Example:\nStonegate Partners holds a 5-year receive-fixed interest rate swap with $100 million notional. The swap has a positive mark-to-market of $2.3 million. They evaluate the impact of different CSA terms:\n\nScenario A — Cash collateral earning SOFR (standard bilateral CSA):\n- Discount at OIS curve\n- Swap MTM = $2,300,000\n\nScenario B — No CSA (uncollateralized):\n- Discount at dealer funding curve (OIS + 85 bps)\n- Swap MTM = $2,195,000\n\nScenario C — EUR cash collateral (cross-currency CSA):\n- Discount at EUR OIS converted via cross-currency basis\n- Swap MTM = $2,278,000\n\nThe difference between collateralized and uncollateralized valuation ($105,000 in this example) represents the funding valuation adjustment (FVA) — a real economic cost that dealers pass through to clients.\n\nPractical Implications:\n- Identical derivatives can have different values depending on the CSA\n- Moving a portfolio from one CSA to another changes its mark-to-market\n- Clearing houses use a single collateral rate (SOFR for USD), simplifying discounting\n- The choice of discount curve directly affects hedge ratios and Greeks\n\nLearn CSA-dependent pricing in our FRM Part I materials.