19Funding, Margin and Capital Adjustments

In 2012 John Hull and Alan White asked in print whether funding is a cost for derivatives desks, and argued that it should not enter a derivative’s price. A practitioner’s reply appeared the same year under the title “Yes, FVA is a cost for derivatives desks”. In January 2014 JPMorgan reported a 1.5 billion dollar loss from implementing a funding valuation adjustment, describing an industry migration towards pricing the cost of funding. Both sides were right about something: the debate turned on whose value is being measured, the firm’s as a whole or its shareholders’, and on what an uncollateralised trade forces a bank to borrow. This chapter builds the adjustments that followed CVA and DVA: the cost of funding uncollateralised positions, of posting initial margin, and of holding capital against counterparty and CVA risk, and stacks them into the price a bank quotes.

19.1 Funding a derivative: the debate

A bank that trades an uncollateralised swap with a client hedges it with an opposite swap under a CSA with the market. When the client’s swap is in the bank’s favour, the hedge is against it: the bank posts variation margin on the hedge and receives nothing from the client. It must borrow that cash at its own funding rate, above the overnight rate paid on collateral. When the client’s swap is against the bank, the bank receives margin on the hedge and can lend it or reduce its borrowing (Figure 19.1).

The FBA overlaps with DVA: both are value the bank captures from its own credit spread on what it owes. Firm-value arguments (Hull and White) say FVA should not affect prices; shareholder-value arguments (Andersen, Duffie and Song) say funding costs transfer value to creditors and a desk must charge them. Conventions differ between banks: some count FCA and FBA symmetrically, some only the cost, some remove the overlap with DVA.

19.2 Funding cost and benefit adjustments

def fca(dee: np.ndarray, times: np.ndarray, spread: float, survival: np.ndarray | None = None) -> float:
    """Funding cost of the discounted positive exposure (trapezoid on the grid)."""
    s = np.ones_like(dee) if survival is None else survival
    f = dee * s
    return float(spread * np.sum(0.5 * (f[1:] + f[:-1]) * np.diff(times)))


def fba(dne: np.ndarray, times: np.ndarray, spread: float, survival: np.ndarray | None = None) -> float:
    return fca(dne, times, spread, survival)

19.3 Initial margin and MVA

Uncleared margin rules (One Quant Book 2, chapter 10) and clearing houses require initial margin, posted by both sides and segregated: it cannot be reused, and the poster earns little on it while funding it at its own rate.

19.4 Capital: counterparty and CVA capital, and KVA

A bank holds capital against the default of its counterparties and against the volatility of its CVA. Regulatory formulas turn netting sets into exposures and exposures into capital.

def ir_addon(trades: list[IrTrade], mpor_days: float | None = None) -> float:
    total = 0.0
    for ccy in sorted({t.currency for t in trades}):
        D = {1: 0.0, 2: 0.0, 3: 0.0}
        for t in trades:
            if t.currency == ccy:
                D[t.bucket()] += (t.delta * t.notional * supervisory_duration(t.start, t.end)
                                  * maturity_factor(t.maturity, mpor_days))
        en2 = (D[1] ** 2 + D[2] ** 2 + D[3] ** 2 + 1.4 * D[1] * D[2] + 1.4 * D[2] * D[3] + 0.6 * D[1] * D[3])
        total += SF_IR * math.sqrt(max(en2, 0.0))
    return total


def fx_addon(pairs: dict[str, float], maturity: float = 1.0, mpor_days: float | None = None) -> float:
    """pairs: net notional (domestic currency, signed) per currency pair; one maturity factor for simplicity."""
    return SF_FX * sum(abs(v) for v in pairs.values()) * maturity_factor(maturity, mpor_days)


def multiplier(v_minus_c: float, addon: float) -> float:
    if addon <= 0:
        return 1.0
    return min(1.0, FLOOR + (1 - FLOOR) * math.exp(v_minus_c / (2 * (1 - FLOOR) * addon)))


def sa_ccr_ead(v: float, addon: float, collateral: float = 0.0, th: float | None = None, mta: float = 0.0,
               nica: float = 0.0) -> float:
    """EAD = alpha (RC + multiplier AddOn); RC unmargined max(V - C, 0), margined max(V - C, TH + MTA - NICA, 0)."""
    rc = max(v - collateral, 0.0) if th is None else max(v - collateral, th + mta - nica, 0.0)
    return ALPHA * (rc + multiplier(v - collateral, addon) * addon)

19.5 The whole stack

The adjustments overlap and interact: DVA and FBA both monetise the bank’s own spread; capital depends on CVA hedges, which reduce CVA capital but need funding; margin reduces CVA and FVA but creates MVA. A desk computes them on common scenarios and quotes their sum, the subject of chapter 20.

19.6 Tutorial: the adjustment stack

19.7 Build: funding, margin and capital

19.8 Exercises

19.9 Problem: The Funding Charge

19.10 Interview questions