Wealth Transfers, Indifference Pricing, and XVA Compression Schemes

Abstract

Since the 2008–2009 financial crisis, banks have introduced a family of XVA metrics to quantify the cost of counterparty risk and of its capital and funding implications: the credit/debt valuation adjustment (CVA and DVA), the costs of funding variation margin (FVA) and initial margin (MVA), and the capital valuation adjustment (KVA). We revisit from a wealth conservation and wealth transfer perspective at the incremental trade level the cost-of-capital XVA approach developed at the level of the balance sheet of the bank in [10]. Trade incremental XVAs reflect the wealth transfers triggered by the deals due to the incompleteness of counterparty risk. XVA-inclusive trading strategies achieve a given hurdle rate to shareholders in the conservative limit case that no new trades occur. XVAs represent a switch of paradigm in derivative management, from hedging to balance sheet optimization. This is illustrated by a review of possible applications of the XVA metrics.

This research has been conducted with the support of the Research Initiative “Modélisation des Marchés actions, obligations et dérivés” financed by HSBC France under the aegis of the Europlace Institute of Finance. The views and opinions expressed in this presentation are those of the author alone and do not necessarily reflect the views or policies of HSBC Investment Bank, its subsidiaries or affiliates. The research of Marc Chataigner is co-supported by a public grant as part of investissement d’avenir project, reference ANR-11-LABX-0056-LLH LabEx LMH.

Appendix: Connections with the Modigliani–Miller Theory

The Modigliani–Miller celebrated invariance result is in fact not one but several related propositions, developed in a series of papers going back to the seminal [39] paper. These propositions are different facets of the broad statement that the funding and capital structure policies of a firm are irrelevant to the profitability of its investment decisions. See e.g. [14, 36, 42] for various discussions and surveys. We emphasize that we do not need or use such result (or any negative form of it) in our paper, but there are interesting connections to it, which we develop in this section.

1.1 Modigliani–Miller Irrelevance, No Arbitrage, and Completeness

Modigliani–Miller (MM) irrelevance, as we put it for brevity hereafter, was initially understood by its authors as a pure arbitrage result. They even saw this understanding as their main contribution with respect to various precedents, notably [43]’s law of conservation of investment value (see Sect. 2.3). So, quoting the footnote page 271 of [39]:

See, for example, Williams [21, esp. pp. 72–73]; David Durand [3]; and W. A. Morton [15]. None of these writers describe in any detail the mechanism which is supposed to keep the average cost of capital constant under changes in capital structure. They seem, however, to be visualizing the equilibrating mechanism in terms of switches by investors between stocks and bonds as the yields of each get out of line with their ‘riskiness.’ This is an argument quite different from the pure arbitrage mechanism underlying our proof, and the difference is crucial.

But, thirty years later, judging by the footnote page 99 in [36], the view of Miller on their result had evolved:

“For other, and in some respects, more general proofs of our capital structure proposition, see among others, Stiglitz (1974) for a general equilibrium proof showing that individual wealth and consumption opportunities are unaffected by capital structures; See Hirshleifer (1965) and (1966) for a state preference, complete-markets proof; Duffie and Shafer (1986) for extensions to some cases of incomplete markets”

Non-arbitrage and completeness are intersecting but non-inclusive notions. Hence, implicitly, in Miller’s own view, MM invariance does not hold in general in incomplete markets (even assuming no arbitrage opportunities). As a matter of fact, we can read page 197 of [28]:

“When there are derivative securities and markets are incomplete the financial decisions of the firm have generally real effects”

and page 9 of [25]:

“As to the effect of financial policy on shareholders, we point out that, generically, shareholders find the span of incomplete markets a binding constraint. This yields the obvious conclusion that shareholders are not indifferent to the financial policy of the firm if it can change the span of markets (which is typically the case in incomplete markets). We provide a trivial example of the impact of financial innovation by the firm. DeMarzo (1986) has gone beyond this and such earlier work as Stiglitz (1974), however, in showing that shareholders are indifferent to the trading of existing securities by firms. Anything the firm can do by trading securities, agents can undo by trading securities on their own account. Indeed, any change of security trading strategy by the firm can be accomodated within a new equilibrium that preserves consumption allocations. Hellwig (1981) distinguishes situations in which this is not the case, such as limited short sales.”

Regarding MM irrelevance or not in incomplete markets (including some of the references that appear in the above quotations and other less closely related ones): [14, 31, 32, 38] deal with the impact of the default riskiness of the firm; [13, 37] discuss the special case of banks, notably from the angle of the bias introduced by government repayment guarantees for bank demand deposits; [22] tests empirically MM irrelevance for banks, concluding to MM offsets of the order of half what they should be if MM irrelevance would fully hold.

1.2 The XVA Case

A bit like with limited short sales in [32], a (seemingly overlooked) situation where shareholders may “find the span of incomplete markets a binding constraint” is when market completion or, at least, the kind of completion that would be required for MM invariance to hold, is legally forbidden. This may seem a narrow situation but it is precisely the XVA case, which is also at the crossing between market incompleteness and the presence of derivatives pointed out as the MM ‘non irrelevance case’ in [28]. The contra-assets and contra-liabilities that emerge endogenously from the impact of counterparty risk on the derivative portfolio of a bank (cf. Definition 2.7) cannot be “undone” by shareholders, because jump-to-default risk cannot be replicated by a bank: This is practically impossible in the case of contra-assets, for lack of available or sufficiently liquid hedging instruments (such as CDS contracts with rapidly varying notional on corporate names that would be required for replicating CVA exposures at client defaults); It is even more problematic in the case of contra-liabilities, because a bank cannot sell CDS protection on itself (this is forbidden by law) and it has a limited ability in buying back its own debt (as, despite the few somehow provocative statements in [37], a bank is an intrinsically leveraged entity).

As a consequence, MM irrelevance is expected to break down in the XVA setup. In fact, as seen in the main body in the paper, cost of funding and cost of capital are material to banks and need be reflected in entry prices for ensuring shareholder indifference to the trades.

More precisely, the XVA setup is a case where a firm’s valuation is invariant to funding strategies and, still, investment decisions are not. The point here is a bit subtle. Saying that the value of a company is independent of financing strategies does not imply that investment decisions do not depend on financing strategies. There two numbers we can look at: the value of the equity $\mathrm{E}$ and the sum of the value of equity and debt, $\mathrm{E}+\mathrm{D}.$ Equity holders will naturally seek to optimize $\mathrm{E}$ and will accept an investment opportunity if $\varDelta \mathrm{E}$ is positive. Williams’ law implies that equity plus debt, $\mathrm{E} + \mathrm{D}$, stays invariant under a certain financial transaction. But this does not imply in general that shareholders are indifferent to the transaction: Shareholders are indifferent if $\varDelta \mathrm{E} = 0$, not if $\varDelta (\mathrm{E}+\mathrm{D}) = 0$. To go from Williams’ wealth conservation law to MM irrelevance, we have to assume complete markets or, at least, the availability of certain trades to shareholders. Namely, assuming shareholders can and do change financing strategy, then, even if we start with $\varDelta \mathrm{E} < 0$ for a given transaction (but $\varDelta (\mathrm{E}+\mathrm{D}) = 0$), we may conclude that equity shareholders are actually indifferent as there exists a change in financing strategy for which $\varDelta \mathrm{E} = 0$. However, in the XVA case, the bank cannot freely buy back its own debt, so such a change is not possible and only Williams’ wealth conservation law remains, whereas MM irrelevance breaks down: See Sect. 4 for illustration in a pedagogical static setup.