XVA: FVA, DVA And MVA

Textbook option pricing assumes a frictionless world: you can borrow and lend at the same risk-free rate, your counterparty never defaults, and posting collateral is free. A trading desk lives nowhere near that world — its counterparty might go bankrupt owing money, the desk itself might go bankrupt owing money, and every dollar it borrows to fund a hedge costs more than the textbook risk-free rate. XVA is the umbrella name for the price adjustments a desk bolts onto the textbook value to account for these frictions — the derivative's real price is the textbook price plus or minus a stack of XVA terms.

Think of the textbook price as a car's sticker price, quoted for a cash buyer with perfect credit. XVA is everything a real dealership adds or subtracts once an actual person shows up: a discount for paying cash (nobody's credit risk matters), a markup for financing (the dealership's own cost of capital), an adjustment if the dealership itself might not survive to honor the warranty. Different letters, same idea: adjust the sticker price for who's actually standing on each side of the deal.

The main pieces

$$\text{Price} = \text{Textbook value} - \text{CVA} + \text{DVA} \pm \text{FVA} - \text{MVA} .$$

CVA is the expected loss if your counterparty defaults while owing you money — it lowers the price you'll pay them. DVA is the mirror image: the benefit you get if you default while owing them money, since that debt disappears — a strange but standard accounting entry that raises the value on your own books. FVA accounts for hedging an uncollateralized derivative requiring cash borrowed at the desk's own funding rate, above the risk-free rate the textbook formula assumes — that spread, paid over the trade's life, is a real cost. MVA is the cost of funding the initial margin regulation now requires many derivatives to post upfront, money frozen in a margin account for the trade's life.

Worked example 1 — CVA on an uncollateralized swap

A dealer has a 5-year interest rate swap with a small corporate client, no collateral posted either way. The swap's expected positive exposure (what the client would owe the dealer, averaged over paths where it's positive) works out to $2 million over the trade's life. The client's CDS curve implies a 5-year default probability of 4%, with 40 cents on the dollar expected recovery. CVA is approximately expected exposure times default probability times loss-given-default: $2{,}000{,}000 \times 0.04 \times 0.6 = 48{,}000$ — the dealer prices the swap $48,000 cheaper than the textbook mid-market value, purely because the client might not be there to pay at the end.

Worked example 2 — FVA on the same trade

Suppose hedging that swap requires the dealer to fund, on average, $3 million of cash over its life, and the dealer's own unsecured borrowing rate runs 1.2% above the risk-free OIS rate the textbook model assumes. Over an average remaining life of 3 years, the extra funding cost is roughly the funded amount times the rate spread times the years outstanding: $3{,}000{,}000 \times 1.2\% \times 3 = 108{,}000$ — that's FVA in dollars, $108,000: money the desk genuinely pays its own treasury desk to borrow the cash the hedge requires, on top of what the risk-free curve says the trade is worth. Between CVA and FVA, this single swap costs the dealer roughly $156,000 more than the textbook price implies — charged to the client or eaten as a loss.

The exposure profiles feeding CVA and FVA are themselves distributions of possible future values, simulated the same way this plot draws a bell curve — instead of a single "expected exposure" number, a desk runs thousands of simulated paths and averages the positive exposure across all of them, at every future date, before turning that profile into a dollar CVA or FVA charge.

What this means in practice

Before 2008, dealers mostly ignored these adjustments and priced close to textbook value; the crisis made counterparty default and funding costs impossible to ignore, and XVA desks — whose only job is managing this stack of adjustments — became standard at major banks. A single trade's all-in XVA charge can be a meaningful fraction of its notional, and since posting collateral (a CSA) sharply reduces CVA and FVA, XVA is now a direct lever clients can pull by agreeing to post more margin.

Practice

1. If a client agrees to post full collateral (a zero-threshold CSA) on the swap in worked example 1, what happens to its CVA, and roughly why?
2. Why does a bank's own DVA rise when its credit spread widens, even though nothing about the trade itself has changed?