Funding capital and margin valuation adjustments

Definition Fix a base curve with deterministic discount factors $D(0,t)$, the curve on which the clean value $V^{\text{clean}}$ is computed, and a horizon $T$ in years. Let $F_t \ge 0$ be a supplied unsecured funding balance in currency units, the cash that must be borrowed to hold and hedge the position, excluding initial margin, and let $f_t$ be its incremental funding spread over the base rate, annualized with continuous accrual. Assume that $F$ and $f$ are deterministic, or at least fixed in advance, so that neither depends on the value being computed, and that $f_t F_t$ is integrable on $[0, T]$. Over a short interval $[t, t + dt]$ the extra interest is $f_t F_t\,dt$, and discounting and adding these payments gives the funding valuation adjustment of this restricted linear ledger,

A rate per year times a balance in currency units times a time in years gives currency units, and discounting keeps them. For balances held over buckets of length $\Delta_i$ with the extra interest paid at the end dates $t_i$, the integral becomes the sum of $D(0,t_i)\,f_i\,F_i\,\Delta_i$. In this ledger FVA is reported as a cost and subtracted. A negative balance, a cash surplus invested at a lending spread, gives a funding benefit, and when borrowing and lending spreads differ, the integrand becomes $f^{b}_t F_t^{+} - f^{l}_t F_t^{-}$ with $x^{-} = \max(-x, 0)$. The formula charges the spread up to $T$ in every state. Models in which a default ends the funding weight each date by the probability that no default has occurred yet. Any collateral remuneration that is already netted in the spread must not be charged again as a separate cost.

Definition Let $I_t \ge 0$ be a supplied balance of initial margin that we post, in currency units, segregated so that it cannot fund anything else, and let $m_t$ be the incremental spread over the base rate at which it is financed, net of any remuneration the margin earns. With the same discount factors and an integrable product $m_t I_t$, the margin valuation adjustment is

FVA and MVA are separate costs only when the funding balance $F_t$ excludes the margin, so that the two integrals finance disjoint amounts. The adjustment depends on the spread as well as on the balance. When the margin is financed at the base rate, $m_t = 0$, the MVA is zero although the margin is still posted. The margin balance itself comes from the margin model and the portfolio, and here it is supplied rather than modeled.

Definition Let $K_t \ge 0$ be the supplied capital that must be held against the position, in currency units, let $h_t$ be a hurdle rate, the annual return on that capital that a stated policy requires, and let $c_t$ be the annual rate that the capital already earns. The capital valuation adjustment of this simplified policy is

The capital requirement and the hurdle rate are inputs of an economic, regulatory or accounting policy. Risk-neutral pricing does not determine the hurdle rate, so the KVA is a charge under the stated policy and not a no-arbitrage price. Other definitions in use, for example ones that discount at the base rate plus the hurdle rate or let the capital depend on simulated future exposures, give different numbers.

Definition Let $\tau_b$ be our own default time and $\tau_c$ the counterparty’s, with $\mathbb{Q}(\tau_b = \tau_c) = 0$ under a pricing measure $\mathbb{Q}$ with the money-market account $B$ as numeraire and $B_0 = 1$, and let $R_b$ and $R_c$ be the two recovery fractions. Use the risk-free closeout convention of Credit valuation adjustment and wrong way risk, with net claim $X_t = V^{\text{close}}_t - C_t$ after usable collateral. If we default first, a negative net claim is a debt of ours of which only the fraction $R_b$ is paid, so our gain is $(1 - R_b) X^{-}_{\tau_b}$. The debt valuation adjustment is

The bilateral CVA uses the event $A_c = \{\tau_c \le T,\ \tau_c < \tau_b\}$ and the loss $(1 - R_c) X^{+}_{\tau_c}$ in the same way. Under the risk-free closeout convention, with the payment and collateral assumptions of the unilateral case, the value of the netting set including both defaults is the default-free value minus the bilateral CVA plus the DVA. The first-to-default indicators matter, because the first default ends the contract and no second default loss can follow it. A different closeout convention changes both terms.

Definition XVA is the collective name for the valuation adjustments applied to a clean value, among them CVA, DVA, FVA, MVA and KVA. Suppose each adjustment is the discounted value of its own payment stream, the streams are disjoint, all of them are computed for the same party with the same base curve, measure and closeout conventions, and the balances that generate the funding, margin and capital payments are supplied rather than solved for. Linearity of discounted sums then gives the restricted ledger

with CVA, FVA, MVA and KVA reported as costs and DVA as a benefit. Other texts report some terms as signed numbers, so the sign of a term has to be read from its ledger. Outside these hypotheses the additive formula can fail, and a split into named terms is not unique. Suppose the unsecured balance depends on the value being computed, for example $F_t = V_t - C_t - H_t$, with reusable collateral $C_t$ and the part $H_t$ that is financed through the hedge instruments themselves, as in a repo, and suppose that borrowing and lending spreads differ. The funding cost rate

then depends on $V_t$ itself. The value therefore solves a nonlinear equation, which in a stochastic model is a backward stochastic differential equation. The values of a buyer and a seller can differ, and that solution has no universal decomposition into named additive adjustments.

Definition Double counting occurs when two named adjustments charge, or credit, the same payment in the same states. The ledger adds correctly only when every balance is partitioned into parts that each carry one financing rate and one name. For example, if a reported funding balance $F^{\text{incl}}_t = F_t + I_t$ includes the initial margin, then for any spread $f$,

so adding a full MVA to an FVA computed on $F^{\text{incl}}$ finances the margin twice. Other overlaps are harder to see. A lender’s spread can contain compensation for loss if we default, and then a funding benefit and the DVA can describe the same gain in our own default states. Whether they overlap depends on which liabilities and closeout cash flows the valuation includes, not on the labels. Adding a standalone unilateral CVA and a standalone DVA without first-to-default indicators charges a second default after the first default has ended the contract. Collateral remuneration that is already netted in a spread must not be charged again.