capital value adjustment

Capital value adjustment (KVA) quantifies the cost of holding regulatory capital against a derivatives portfolio over its lifetime. Because regulatory capital must be funded by equity — which demands a higher return than debt — KVA represents the present value of the future cost of this equity funding. KVA is computed by integrating the expected capital profile (ECP) against the cost of capital (the hurdle rate above the risk-free return that shareholders demand). The general formula is KVA = - sum of ECP(t_i) x CC(t_{i-1}, t_i) x (t_i - t_{i-1}), where ECP is the discounted expected future regulatory capital requirement and CC is the capital cost spread. Alternatively, KVA can be framed through the internal rate of return (IRR) condition: the KVA is the amount that, when charged upfront, generates the required return on capital (ROC) across all future periods.

Unlike CVA and FVA, KVA is not an expected future cost per se but rather a profit that is expected to be paid to shareholders and employees. This fundamental difference has led to a debate over whether KVA should be treated as (1) an immediate profit (the ROC hurdle approach), (2) retained earnings released over the transaction lifetime, or (3) a full valuation adjustment managed like CVA and FVA. Most banks currently follow approach (1), releasing KVA profit at inception, which creates misaligned incentives: the ROC is very high in the first year and zero thereafter. The third approach would allow KVA to be transfer priced to an xVA desk and hedged, locking in a stable ROC over time at the cost of variable P&L from KVA hedges.

KVA is the most controversial of the xVA terms because there is genuine debate about whether capital constitutes a “cost” in the same sense as funding or credit losses. Equity dividends are discretionary (unlike bond coupons), so capital does not generate a fixed cost stream. One view holds that KVA is not a direct cost and should remain a “soft” return-on-capital hurdle rather than a formal valuation adjustment. The opposing view argues that investors implicitly require a certain return and will sell shares if it is not achieved, making capital effectively another cost analogous to funding. At the time of Gregory’s writing (2020), no bank had yet made a full accounting adjustment for KVA. KVA is priced more directly into transactions than in the past (hurdles are less “soft”), but it does not form part of the official MTM valuation. Some argue KVA should eventually be treated similarly to FVA, as it is a clear component of the entry price that all market participants require.

KVA has important overlaps with other xVA terms. With CVA, the overlap arises because risk-neutral CVA already prices in the cost of hedging counterparty risk, while KVA prices the capital cost of not hedging it; the combined cost is sometimes expressed as (1 - alpha) x EL + alpha x CVA + beta x KVA, where alpha measures the degree of CVA hedging and beta the capital impact of hedges. With FVA, the overlap arises because regulatory capital can serve as a funding source, reducing either FVA or KVA. With MVA, received initial margin provides capital relief (lowering KVA) while posted margin incurs a funding cost (increasing MVA), requiring joint optimisation.

Key Details

• The expected capital profile (ECP) is the key input to KVA, representing the expected future regulatory capital requirement discounted at the cost of capital
• Capital profiles are driven by counterparty credit risk (CCR) capital, CVA capital, and potentially the leverage ratio (LR) floor
• LR invariance: ECP is computed as the maximum of risk-weighted asset (RWA) capital and LR-implied capital, which is especially important for collateralised trades where non-cash margin cannot reduce the LR exposure
• Discounting ECP at the cost of capital (rather than the risk-free rate) is appropriate when KVA itself is counted towards regulatory capital (reducing the initial KVA charge by approximately 20%)
• KVA is highly volatile: in scenarios where a trade moves in-the-money, capital requirements increase and KVA becomes more negative, whereas out-of-the-money moves cause rapid capital decline
• The partial hedging framework of Kenyon and Green (2014) parametrises the CVA-KVA overlap as (1 - alpha) x EL + alpha x CVA + beta x KVA, where beta > 1 when hedges increase capital (current regime) and beta < 1 when hedges reduce capital (future FRTB-CVA regime)
• No bank had, as of 2020, taken a generic accounting reserve for KVA, though most include some form of hurdle rate in entry pricing
• KVA covers multiple capital charges: CCR capital, CVA capital, market risk capital, and the leverage ratio
• Banks may use “ratio invariance pricing” to ensure that new transactions maintain the bank’s leverage ratio and NSFR at constant levels
• There is an important overlap between KVA and CVA: capital is held as a buffer against potential default, and this buffer reduces as actual accounting CVA losses are taken
• KVA is always a cost (never a benefit to the counterparty), unlike CVA/DVA which are symmetric

Textbook References

The xVA Challenge (Gregory, 2020)

• Section 3.2.3 (p. 57): KVA defines the cost of holding capital (typically regulatory) over the lifetime of the transaction.
• Section 5.2.4 (p. 90): Table 5.1 shows the evolution from “businesses set soft return on capital metrics” (traditional) to “KVA is priced more directly into transactions but with no valuation impact” (current practice).
• Section 5.4.2 (pp. 101—102): The debate on whether capital is a cost. The “capital is not a direct cost” view argues KVA should remain a hurdle; the “capital is a direct cost” view argues KVA should be an accounting adjustment like FVA. No bank has yet made a full KVA accounting adjustment.
• Section 5.4.3 (pp. 102—104): Ratio invariance pricing (Eq. 5.5): new business should be charged to keep the leverage ratio constant. The formula is Delta_R = alpha * Delta_X, where alpha is the desired ratio and Delta_X is the change in the risk component.
• Section 16.3.2 (p. 482): KVA defines the cost of holding capital (typically regulatory) over the lifetime of the transaction.
• Section 19.2 (pp. 565—577): Definition of KVA via the expected capital profile (ECP). The ECP integrates CCR capital, CVA capital, and LR capital. Table 19.1 gives a worked example for a 10-year interest rate swap yielding KVA of -38.8 bps (discounted at cost of capital) or -49.1 bps (undiscounted). LR invariance is illustrated in Figure 19.7 across uncollateralised, CVA-exempt, and collateralised cases.
• Section 19.3.1 (pp. 577—580): KVA is generally not transfer priced to an xVA desk, not fully reflected in clearing prices, and profits are paid out immediately. Figure 19.8 contrasts CVA/FVA (owned by xVA desk, paid out at exit) with KVA (released as profit at entry, not part of exit consideration).
• Section 19.3.2 (pp. 580—584): Three views on KVA management: (1) KVA is profit, (2) KVA is retained earnings, (3) KVA is a valuation adjustment. Capital profiles under different market scenarios (EFV, 5% PFE, 95% PFE) show highly variable ROC (Figure 19.13). Two mitigants: balanced portfolios and hedging KVA.
• Section 19.3.3 (p. 585): Discounting at cost of capital is appropriate if KVA is withheld and counts towards regulatory capital (Albanese et al. 2016, Kjaer 2018). This reduces KVA by approximately 20%.
• Section 19.3.4 (pp. 586—587): Arguments for and against KVA accounting. Benefits include stable ROC, correct incentives, and centralised management. Obstacles include the large one-off adjustment and highly subjective KVA inputs across banks.
• Section 19.4.1 (pp. 587—589): CVA-KVA overlap via the Kenyon-Green framework. Table 19.4 compares credit risk warehousing (EL + KVA), fully hedged (CVA), and partial hedging approaches. Figure 19.15 shows that partial hedging can increase total costs when beta > 1.
• Section 19.4.2 (pp. 589—590): FVA-KVA overlap — regulatory capital can be used as a funding source, reducing either FVA or KVA. This requires coordination between the xVA desk and treasury.

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