ارزیابی خطر بازگشت تجارت کردن در پرتفولیو وام

This paper proposes a methodology to analyse the risk and return of large loan portfolios in a joint setting. I propose a tractable model to obtain the distribution of loan returns from observed interest rates and default frequencies. I follow a sectoral approach that captures the heterogeneous cyclical features of different kinds of loans and yields moments in closed form. I investigate the validity of mean–variance analysis with a value at risk constraint and study its relationship with utility maximisation. Finally, I study the efficiency of corporate and household loan portfolios in an empirical application to the Spanish banking system.

The risk and return of empirical loan portfolios has generally been analysed separately. Chirinko and Guill (1991) is one of the first empirical attempts to model the credit risk associated with loan portfolios. More recently, the Basel II framework has originated the development of many quantitative models to estimate the loss distribution of loan portfolios (see Embrechts et al., 2005 for a textbook review of the literature). Meanwhile, a parallel literature studying the determinants of interest rates has simultaneously grown during the previous years (see e.g. Martín et al., 2007 and Mueller, 2008). However, much less is known about the interaction between risk and profitability. Mean–variance analysis, introduced by Markowitz (1952), is the most popular tool to compare the characteristics of different assets. The theoretical banking literature has extensively used this tool to analyse the investment problem of banks (see Koehn and Santomero, 1980, Rochet, 1992 and Kim and Santomero, 1988, among others). However, these papers typically assume Gaussian returns, despite the well known asymmetric features of loan returns. More recently, Altman (1996) used the mean–variance framework to analyse the risk and return characteristics of loan and bond portfolios. However, his approach relies on ratings and market data. Unfortunately, this information is not available for loans to small and medium enterprises or households, which constitute the largest share of most banks’ portfolios. The purpose of this paper is to model the distribution of loan portfolios explicitly and study the usefulness of mean–variance analysis when the actual features of loans are taken into account. As Chamberlain (1983) shows, mean–variance analysis may still be valid in a non-Gaussian context, as long as the higher order moments are fixed once the mean and variance have been chosen. In this sense, my model generates negatively skewed loan returns but at the same time remains a function of a vector of underlying Gaussian state variables. In this context, I can argue that the means and variances of portfolio returns remain valid measures of profitability and risk, respectively. From an empirical perspective, my goal is to make this investment problem operational in broad loan portfolios, including personal loans as well as corporate loans of firms that have no access to the bond market. In contrast to other applications, the fact that loan portfolio returns are not observable makes the use of mean variance analysis far from trivial. Hence, returns have to be inferred from available information. Unfortunately, in most cases neither the loans themselves nor any other product from their borrowers are traded in any organised market. To overcome this problem, the usual approach is to assess credit risk by grouping loans in buckets of similar characteristics (see Foglia, 2009). Then, it is easy to compare the historical performance of loans with similar features. Following this methodology, I show how to obtain the return distribution from interest rates and historical loan default rates by loan category, which is generally the only information available to bank managers and supervisors when dealing with comprehensive samples of loans. I consider stochastic probabilities of default for each loan category, proxied by default frequencies, and model them as a probit function of an underlying multivariate Gaussian vector of state variables. This approach can be interpreted as a multiasset portfolio extension of the popular Gaussian copula of Li (2000). In this setting, I obtain closed form expressions for the expected returns, variances and covariances between different loans. The covariance matrix of returns does not only depend on the distribution of the probabilities of default, but also on the granularity of the portfolios. I also describe the multivariate distribution of returns and the value at risk (VaR) in closed form. I analyse risk and return jointly by describing the set of feasible portfolios in mean–variance space. The frontier of this set can be easily obtained thanks to the analytical tractability of my framework, which makes mean–variance analysis in loan portfolios (almost) as easy to implement as in traded securities. The investment opportunity set may be narrowed by introducing the minimum capital requirement imposed by the regulator or possibly by an even more stringent rating target. Both conditions 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, Alexander and Baptista, 2006 and Alexander et al., 2007 have previously considered mean–variance analysis with a VaR constraint when returns are elliptical. However, the negative skewness of loan portfolios can cause large biases in the elliptical estimates of the VaR. In addition, this literature does not explain how to estimate the mean and variance when returns are not observable. I also analyse the risk-return tradeoff from a pricing point of view. In this regard, I obtain the relationship between interest rates and the risk of borrowers that ensures absence of arbitrage opportunities, which provides a useful tool to understand loan pricing in the presence of credit risk. Finally, I consider an empirical application to Spanish loans. I use quarterly data from the Spanish credit register, from 1984.Q4 to 2008.Q4. I group corporate loans in four categories based on the economic sector of the borrower, and consider two additional groups for household loans. I propose a dynamic model to deal with the persistence and time varying features of the data. In addition, I exploit the information from banks’ confidential reports about average interest rates for several classes of loans. I use this information to set the interest rates in the loan categories. In particular, I assume homogeneity within each loan class, so that all constituents from a given class have the same probability of default and interest rate. With this data, I compute the multivariate return distributions, the conditional value at risk by loan category and the mean–variance frontiers at different points of the credit risk cycle. Lastly, I investigate the validity of mean–variance analysis when bank managers maximise a constant relative risk aversion utility function. The rest of the paper is organised as follows. I describe the model in Section 2 and introduce mean–variance analysis in Section 3. Then, I develop the pricing model in Section 4. Section 5 presents the results of the empirical application and Section 6 concludes. Proofs and auxiliary results can be found in appendices.

This paper provides an analytical empirical framework to compare the risk and return of loan portfolios in a joint setting. I propose the use of mean–variance asset allocation techniques to carry out this analysis, allowing for a value at risk constraint in order to take into account regulatory requirements. At the same time, my approach captures the empirically observed negative skewness and time varying correlations between different types of loans. Specifically, I analyse the distribution of loan returns with a flexible albeit easily implementable model, which is driven by a vector of Gaussian state variables. This model extends the Gaussian copula approach to deal with different types of assets. In addition, it is designed for applications to comprehensive samples of banks’ loan portfolios, where returns are unobservable and have to be inferred from balance sheet and credit register data. Thanks to the analytical tractability of my methodology, I obtain closed form expressions for the mean vector and covariance matrix of loan returns. I interpret these formulas and show that the variance of returns can be decomposed in diversifiable and non-diversifiable risk, where the diversifiable component can be eliminated by increasing portfolio granularity. I also describe value at risk and the multivariate return distribution in closed form for infinitely granular portfolios. In addition, I illustrate how this model can capture default correlations between different loans. These correlations are introduced by means of the state variables. In this sense, the correlations of the underlying Gaussian distribution can be approximately interpreted as the default correlations under infinite granularity. I develop an asset pricing model to obtain “fair” loan prices that compensate for the presence of credit risk. Specifically, I obtain closed form expressions under absence of arbitrage, in which spreads are increasing functions of the risk-adjusted probabilities of default and decreasing functions of expected recovery rates. In addition, they depend negatively on the covariance between recovery rates and default probabilities. I consider an empirical application in which I study the aggregate loan portfolio of the Spanish banking system. I group corporate loans by the economic sector of the borrower. Similarly, I group loans to households in two categories: consumption loans and mortgages. I estimate a multivariate dynamic model for the loan return distribution of these six categories based on historical interest rates and default frequencies over more than two decades of data. The volatilities that result from my model display GARCH-type features with highly cyclical and persistent patterns, although I observe the largest variations in volatility for corporate loans, especially construction. Construction and services display large correlations, while mortgages tend to have smaller correlations with other kinds of loans. On the mean–variance space, mortgages turn out to be the safest category of loans, although they yield smaller returns than consumption loans. In contrast, corporate loans are generally riskier and more sensitive to recessions. My model also quantifies how the investment opportunity set on the mean–variance space has quickly moved to a riskier area on the 2008 crisis. Finally, I find that mean–variance analysis is empirically consistent with the maximisation of a constant relative risk aversion utility function. An interesting avenue for future research would be to explore the effect of macroeconomic variables such as GDP growth on the mean–variance frontier, extending the idea of Wilson, 1997a and Wilson, 1997b to mean–variance analysis. The integration of asset allocation on loan portfolios and other market investments through a common mean–variance analysis might be greatly helpful as well. It would also be possible and potentially useful to generalise the model in Section 2 to allow for stochastic recovery rates. Lastly, it would be helpful to extend this model to a multiperiod setting.