RDP 9210: Contingent Claim Analysis of Risk-Based Capital Standards for Banks 1. Introduction

Risk-based capital standards are being implemented in Australia and many other countries under the guidance of the Basle-based Bank for International Settlements. These standards require each bank's capital to exceed some proportion of an adjusted asset base. This base is calculated by applying fractional risk weights to the dollar value of various types of assets and converting off-balance-sheet exposures to “credit equivalents”. By more directly linking bank capital to the riskiness of bank portfolios, the new standards may give regulators better control over the safety of banking systems. For example, Kim and Santomero (1988) show that risk-based standards can be designed so that the probability of bank failure is bounded by any particular level desired by regulators. Under traditional capital requirements, in contrast, required capital is not linked explicitly to banks' chosen level of asset risk and the probability of failure may be higher than is desirable.

However, the mean-variance framework used by Kim and Santomero has received some criticism. Keeley and Furlong (1990) argue that the model ignores the fact that limitations to shareholder liability, combined with some degree of imperfectly priced government backing of bank liabilities in most countries,[1] cause the economic value of bank liabilities to be sensitive to bank risk-taking. This sensitivity violates one of the fundamental assumptions of the mean-variance model, making it inappropriate for studying bank decision making. To address this issue, several recent papers (Marcus (1984) and Furlong and Keeley (1989), for example) use contingent claim techniques to model the value of a bank and study the bank's portfolio choice problem.

In this paper we use a contingent claim framework to examine risk-based capital standards. Like Kim and Santomero we derive standards that bound the probability of bank insolvency or failure; we refer to these as failure-probability or “FP” rules. We also consider standards designed to limit the contingent liability arising from the government's commitment to protect depositors; we refer to these as liability-value or “LV” rules.[2] Although these two regulatory goals are related, they are not identical. For example, an FP rule treats all bank insolvencies as equally costly, whereas an LV rule considers the size of the expenditure required to protect depositors in each potential insolvency. Hence a policy designed to achieve one goal may differ from a policy designed to achieve the other.

The paper has two main objectives. The first is to characterise FP and LV capital standards and compare bank behaviour under each type of rule. We are particularly interested in whether banks are more likely to choose high-risk, high-capital positions under one rule than under another. The second main objective is to study the feasibility of constructing standards similar to the BIS standards – specifying risk weights to be applied to different classes of assets – that come close to achieving the goals studied here. The analysis is similar to that of Kendall and Levonian (1992), although that paper develops the theoretical model within the context of an explicit deposit insurance system in which banks pay insurance premiums. This paper generalises the results to the case of a possibly implicit deposit guarantee, thus making the model more broadly applicable. The assumptions regarding the riskiness of the environment, and the precise regulatory objectives under which the numerical results are derived, differ substantially from the Kendall and Levonian paper, and additional results are presented and discussed.

We take as a starting point Merton's (1977) model of a bank with government guaranteed deposits, which follows from Merton's (1974) more general model of financially-levered firms. Bank failure, taken here as synonymous with insolvency, occurs if the value of the bank's deposits at the end of the period exceeds the value of its assets. If the bank fails, a government-backed deposit guarantor ensures that depositors are reimbursed in full. The payout by the deposit guarantor is either zero or the difference between deposits and assets, whichever is larger; hence the deposit guarantee can be modelled as an implicit put option on bank assets. Within this model, we derive the guarantor's liability per dollar of bank deposits and an upper bound on the probability of bank failure, expressing both as functions of the bank's capital and the riskiness of its assets.

In order to study bank behaviour we assume that bank management chooses asset risk and capital to maximise the value of equity inclusive of the implicit put option created by the deposit guarantee and net of contributed capital, subject to the constraint that the capital requirement be met. The model suggests that under an LV rule banks will choose low-risk, low-capital portfolios, while under an FP rule they are likely to choose higher risk and higher capital.

We derive a simple version of the BIS/Basle standards by assuming banks can purchase only riskless assets and one type of risky asset. Risk weights are calculated to yield “best fit” approximations to the LV and FP rules. Under a fairly wide range of assumptions, both approximations yield a risk weight for risky assets that is above (in many cases well above) the maximum 100 percent weight specified by the BIS. The FP approximation yields a weight for riskless assets that is close to the zero weight assigned by the Basle standards; under the LV approximation, however, the weight is substantially less than zero.

The paper is organised as follows. Section 2 presents the model of the value of a bank and the corresponding value of the deposit guarantor's liability, and obtains an expression for an upper bound on the probability of failure. Section 3 presents the LV and FP rules and discusses bank behaviour under each. Section 4 derives a simple approximation to each capital standard and compares the results to the Basle standards. The paper is summarised in Section 5, and some directions for future research are discussed.


In many countries depositor protection is provided by government or central bank guarantee, while in others the protection is provided through a deposit insurance scheme. In either case, the ultimate backing of the national government is usually implicit if not explicit. [1]

Related papers include Sharpe (1978), Ronn and Verma (1989), and Duan, Moreau and Sealey (1991). Sharpe introduces the idea of designing risk-based capital standards to hold constant the value of the government liability. Ronn and Verma use the contingent claim model to derive estimates of the capital infusion several banks would have to make in order to achieve that goal. Duan, Moreau and Sealey consider FP and LV rules in the context of risk-based deposit insurance pricing. Holding the riskiness of assets constant across banks, they derive combinations of the capital ratio and the deposit insurance premium that yield a given constant probability of failure, and also derive combinations that yield a constant value of deposit insurance per dollar of deposits. In contrast, we allow the riskiness of assets to vary across banks, and derive combinations of asset risk and the capital ratio that yield constant failure probabilities and values of the deposit guarantee per dollar of deposits. [2]