Standard capital market theory states that there is a risk-return trade off in equilibrium. The more risk one is willing to take, the higher the return one will be able to get. This relationship has been extensively analysed in the context of liquid assets that trade in organised markets (see e.g. Fama and MacBeth, 1973; Ghysels, Santa-Clara, and Valkanov, 2005). However, much less is known about its implications on the behaviour of banks as risk managers and profit maximisers. Banks aggregate profits have been analysed by Behr et al. (2007) and Hayden, Porath, and von Westernhagen (2007), among others, who find that more specialisation tends to yield higher returns but also a higher level of risk. However, the optimal degree of bank specialisation is not analysed by these papers. In addition, they study the different activities of banks jointly, that is, they are not able to separate market and credit activities. As already explained, the features of liquid assets are well known. Thus, my goal is to focus on the less well understood credit activities of banks.
The risk and return of loans portfolios has generally been analysed separately. On the one hand, the Basel II framework has originated the development of many quantitative models to estimate the loss distribution of loans portfolios (see Embrechts, Frey, and McNeil, 2005, for a textbook review of the literature). On the other hand, a parallel literature that studies the determinants of interest rates has simultaneously grown during the previous years (see e.g. Mart?n, Salas, and Saurina, 2007; Mueller, 2008). Unfortunately, to the best of my knowledge, an specific common framework to analyse the risk and return characteristics of bank’s loans portfolios is still missing in the literature.
In this paper, I develop a flexible although analytically tractable model for loans returns. I introduce conditional default correlation between different loans. Defaults areindependent conditional on the realisation of an underlying multivariate Gaussian vector of state variables. This vector will follow a dynamic process so that it can account for the strong persistence that is usually observed in probabilities of default.1 In this context, I am able to obtain closed form expressions for the expected returns, variances and covariances between different loans. The expected return is a linear combination of the interest rate required by the lender and the expected recovery rate, where the relative importance of each of these factors is determined by the probability of default. As for the covariance matrix of returns, it not only depends on the distribution of the probabilities of default, but also on the granularity of the portfolios.
I explore the asset pricing implications of my model. Following Gourieroux and Monfort (2007), I consider an exponentially affine stochastic discount factor (SDF) in terms of the state variables of the model. Then, I derive the distribution under the risk-neutral measure, and compute the spread over the risk-free rate that banks should require to ensure absence of arbitrage. In terms of portfolio allocation, I apply the mean-variance analysis of Markowitz (1952) to obtain the set of efficient portfolios. In this sense, the properties of the return distribution, and in particular the absence of probability mass at the right tail, make variance a suitable measure of risk in this context. Thus, banks will minimize the variance of their loans portfolio for any given target expected return. Of course, this strategy may be restricted by the minimum capital requirement imposed by the regulator or possibly by a rating target. This requirement can be interpreted as a constraint on the minimum return that the bank must obtain, which technically corresponds to a bound on the admissible Value at Risk (VaR). Sentana (2003) and Alexander and Baptista (2006) have previously considered mean-variance analysis with a VaR constraint when returns are Elliptical. In this sense, I extend their approach into the non-elliptical statistical framework of this paper.
I consider an empirical application to Spanish loans. I use quarterly data from the Spanish credit register, from 1984 to 2008, to calibrate the model for the probabilities of default and obtain the granularity of empirical portfolios. I consider an additional database in which banks inform about the average interest rates for several classes of loans. Since this database does not start until 2003, I extend it backwards by constructing average interest rates from the marginal interest rates reported by banks on new operations, which are available from 1990. I analyse the historical evolution of risk and return. Then, I compare the risk-return efficiency of the two main types of institutions that operate in Spain: commercial and savings banks. I also compare the efficiency of small and large institutions. In both cases, I consider three kinds of loans: corporate loans, consumption loans and mortgages. Finally, I estimate the area in the mean-standard deviation space that yields a 1% excess return over the risk-free rate with 99.9% probability for different periods, and compare the risk-neutral and actual probabilities of default.
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Assessing the risk, return and efficiency of banks’ loans portfolios
