Abstract
The aim of the chapter is to propose the new approach to valuation of individual banks which takes into account the risk of the whole interbank market network. We show that the value of the bank is equal to the value of the call option on the bank’s debt which is the standard step in the valuation theory. However, the underlying value process depends on the possible interbank payments the bank expects to receive from other participants of the interbank market. In this way valuation theory originated to Black and Scholes (J Polit Econ 81:637–653, 1973) is embedded into the systemic framework a la (Cifuentes et al. (2004) Liquidity risk and contagion. London School of Economics) and we are able to prove that the value of a bank should not only depend on its internal financial standing but on the ability of their interbank counterparties to repay their debts. Our model has two unique features. Firstly, we demonstrate how losses originated to the interbank exposures can be reflected into the valuations of the banks even if it is extremely difficult to estimate the default probabilities of the interbank deposits. Secondly, liquidity of the bank and marketability of the bank’s counterbalancing capacity is an outcome of the interbank market equilibrium. We apply the developed theory to study the relationship between the US banking system structure and the valuations of the US banks. We solely use publicly available data: the financial statements of the US banks provided by FDIC and the stock exchange quotes.
DISCLAIMER: The chapter presents views of the author which are not necessarily those of the ECB. Most of the results presented in the chapter were completed when the author was working for Bank Pekao SA, Warsaw (UniCredit Group).
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- 1.
We will recall the following operators “ ∧ ”, “ ∨ ”, “ + ” and “ − ” useful in the notation of equilibrium, i.e. x ∧ y: = min{x, y} and x ∨ y: = max{x, y}, x + : = max{x, 0} and x − : = − min{x, 0}. By ∏ j ∈ J A j we denote the Cartesian product of sets A j indexed by the set J. By \(\bar{\mathbb{N}}: =\{ 1,\ldots ,N\}\) we denote the set of cardinal numbers of banks in the banking system. Symbol “ ⊤ ” denotes transposition. By \(\mathcal{P}\) we denote \(\prod \limits _{i\in \bar{\mathbb{N}}}[0,\bar{{p}}_{i}]\).
- 2.
The risk factors are univariate random variables instead of being multivariate, correlated factors. This assumptions can be easily relaxed in this setting to account for the dependance structure of the banking income.
- 3.
The guarantees should be thought of as MBS issuance of Government National Mortgage Association (Ginnie Mae), Federal Home Loan Mortgage Corporation (Freddie Mac) and Federal National Mortgage Association (Fannie Mae).
- 4.
Here, we apply the standard game-theoretical notation for the “other players” than i putting − i.
- 5.
Symbol : = means “by definition”.
- 6.
For example USD bn. We skip “units” for brevity.
- 7.
- 8.
- 9.
The assumption about independent income processes for banks is very much simplified. In this way, rather lower bound of the interbank contagious losses can be captured – positive correlation of the income processes would probably amplify the losses. Generalization of the model setting to the case of income correlation is straightforward.
- 10.
In this way all moments of I(i) are finite.
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Acknowledgements
The author would like to express his gratitude to the participants of the research seminar in the National Bank of Belgium (September 2009) and of the Hurwicz Workshop in Warsaw (October 2009) and the anonymous referee for very valuable comments on the interbank liquidity part of this chapter. The content of the chapter was essentially improved during the author’s postdoc fellowship at the Fields Institute (Toronto) in 2010.
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Hałaj, G. (2012). Systemic Valuation of Banks: Interbank Equilibrium and Contagion. In: Kranakis, E. (eds) Advances in Network Analysis and its Applications. Mathematics in Industry, vol 18. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30904-5_3
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