Assessing the risk, return and efficiency of banks' loans portfolios

Assessing the risk, return and efficiency of banks’ loans portfolios ∗ Javier Menc´ıa Bank of Spain June 2008 Preliminary and Incomplete

Abstract This paper develops a dynamic model to assess the risk and profitability of loans portfolios. I obtain their risk premia and derive the risk-neutral measure for an exponentially affine stochastic discount factor. I employ mean-variance analysis with a VaR constraint to assess efficiency. Then I compare Spanish institutions in an empirical application, where small institutions seem to be less efficient than large ones on aggregate terms, while commercial and savings banks perform better on their respective traditional markets. Finally, I find increasing discrepancies between riskneutral and actual default probabilities since June 2007 and discuss their possible sources. Keywords: Credit risk, Probability of default, Asset Pricing, Mean-Variance allocation, Stochastic Discount Factor, Value at Risk. JEL: G21, G12, G11, C32, D81, G28.

∗

This paper is the sole responsibility of its author. The views represented here do not necessarily reflect those of the Bank of Spain. Thanks are due to Alfredo Mart´ın, for his valuable suggestions as well as for help with the interest rate database. Of course, the usual caveat applies. Address for correspondence: Alcalá 48, E-28014 Madrid, Spain, tel: +34 91 338 5414, fax: +34 91 338 6102.

1

Introduction Standard capital market theory states that there is a risk-return tradeoff in equilib-

rium. 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´ın, 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 are independent 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 1

Some authors even find the presence of a unit root in the logit/probit transforms of default rates (see e.g. Boss, 2002). I prefer not to impose a unit root to ensure the existence of the unconditional probability of default.

1

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

2

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. The rest of the paper is organised as follows. Section 2 describes the model for the loan return distribution, derives closed form expressions for the moments and discusses the interpretation of its components. I discuss the asset pricing implications of my model in Section 3 and introduce the mean-variance analysis in Section 4. Section 5 presents the results of the empirical application. Finally, Section 6 concludes. Proofs and auxiliary results can be found in appendices.

2

The model

2.1

Loan characterisation

Consider a portfolio with K types of loans. These types of loans may be interpreted as belonging to different economic sectors, or just as a means of classifying borrowers of different characteristics (e.g.: corporates vs. households). In each of these groups, there are Nkt loans, for k = 1, · · · , K. At time t, I denote the volume of loan i from group k as Lkit , while rkt will be the net interest rate required by the lender for each of these loan types. Interest rates are set at t − 1. As for Lkit , I introduce a time subindex just because loans may enter a bank’s portfolio at different moments. Otherwise, these values may remain constant over time for a particular loan until a substantial portion of it is amortised or the loan is fully repaid. I now turn to modelling borrowers’ credit risk. A loan’s default is driven by a binary variable Dkit that takes a value of 1 if i defaults at time t and 0 otherwise. In case of default, the borrower will obtain a recovery rate δikt , which is a proportion of the total amount that is owed. I will assume that the number of defaults in each group has a binomial distribution conditional on the probabilities of default, which are given by πkt = Φ [xkt ] ,

(1)

for k = 1, · · · , K, where Φ(·) is the standard normal cdf, and xt = (x1t , x2t , · · · , xKt )0 is a random vector of conditionally Gaussian state variables. For the sake of concreteness, 3

I will express this distribution as xt |It−1 ; θ ∼ N [µt (θ), Σt (θ)],

(2)

where It−1 denotes information known at t − 1, θ is a vector of p free parameters, and µt (θ) and Σt (θ) are, respectively, the conditional mean and covariance matrix of xt , i.e. µt (θ) = E[xt |It−1 ; θ], Σt (θ) = V [xt |It−1 ; θ].

(3)

I introduce conditional time information to allow for time series autocorrelation and possibly time varying volatility in probabilities of default. This structure also allows for the presence of exogenous explanatory variables. Interestingly, despite the flexibility of this framework, which may nest many complicated structures as particular cases, it is still possible to derive general closed form expressions for the conditional moments of (1): Proposition 1 Consider a loan with probability of default (1). Then, µkt E[πkt |It−1 ; θ] = πkt|t−1 = Φ √ , 1 + σkkt E[πkt πjt |It−1 ; θ] = ωkjt|t−1 " = Φ2

µ µjt σkjt √ kt ,p ,p 1 + σkkt 1 + σjjt (1 + σkkt )(1 + σjjt )

(4)

# (5)

where µkt is the k-th component of µt (θ), σkjt is the k −j element of Σt (θ), Φ(·) is the cdf of the standard normal distribution, and Φ2 (·, ·, ρ) denotes the cdf of a bivariate normal distribution with zero means, unit variances and correlation ρ. This model can be interpreted as a conditionally independent default model, since defaults are independent conditional on πkt , for k = 1, · · · , K. However, it introduces default correlation because πkt is not known. In fact, it is not even observable at t0 > t unless Nkt grows to infinity.2 From Proposition 1 it is straightforward to show that the conditional probability of default given information known at t−1 is Pr(Dkit = 1|It−1 ; θ) = πkt|t−1 . With this in mind, I can express the conditional default correlation as follows. 2

Specifically, it can be shown that πkt = limNkt →∞ empirical application.

4

P

i

Dikt /Nkt . I will exploit this feature in the

Proposition 2 Consider a loan with probability of default (1). Then cor(Dkit , Djit |It−1 ; θ) = q

ωkjt|t−1 − πkt|t−1 πjt|t−1

,

2 2 ) )(πjt|t−1 − πjt|t−1 (πkt|t−1 − πkt|t−1

where πkt|t−1 and ωkjt|t−1 are defined in (4) and (5). Finally, I will assume that the recovery rates, δikt , are independent of πkt for all k and t.3 Recovery rates can either be taken as fixed values or modelled with a probability distribution. For the sake of concreteness, I will denote the mean of δikt as δk0 .

2.2

Return distribution

I can express the value of the portfolio at time t − 1 as the sum of the outstanding debt at this period, i.e. Vt−1 =

K X

Vkt−1 ,

i=1

where

Nkt−1

Vkt−1 =

X

Ljkt−1 ,

(6)

j=1

for k = 1, · · · , K. According to the model described in Section 2.1, one period later Vkt will be given by Nkt−1

Nkt−1

Vkt = (1 + rkt )

X

Ljkt−1 (1 − Djkt ) +

X

δjkt Ljkt−1 Djkt .

(7)

j=1

j=1

Intuitively, each loan will either pay its interest or default, in which case the lender will only receive the recovery rate times the outstanding amount. I am making some simplifying assumptions in order to make the model operational. Specifically, I assume that, when a loan defaults, the recovery rate is obtained within the period in which the default occurs. In any case, the recovery rate may also be interpreted as the residual value of a loan at the moment of its default, including any future expected payments that can be obtained from the borrower by litigation. 3

This assumption can be relaxed within this framework. However, I will not explore this extension because it is very difficult to calibrate in practice due to lack of sufficient data about recovery rates.

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From (6) and (7), it is straightforward to write the return or yield generated by loans from group k as: ykt =

Vkt − Vkt−1 , Vkt−1 PNkt−1

= rkt −

j=1

(1 + rkt − δjkt )Ljkt−1 Djkt , ¯ kt−1 Nkt−1 L

where ¯ kt−1 = L

(8)

Nkt−1

1

X

Nkt−1

Ljkt−1 .

j=1

Using Proposition 1, it can be shown that the expected value of (8) is given by E(ykt |It−1 ; θ) = rkt (1 − πkt|t−1 ) − (1 − δk0 )πkt|t−1 .

(9)

Hence, the expected return obtained by the lender may in practice differ from the required interest rate rkt under the presence of credit risk. Specifically, (9) is the convex combination of the interest rate obtained in normal conditions and (minus) the expected loss given default, where the weights are determined by the respective probabilities of these events. Now I turn to the question of choosing a measure that is informative about the risk of (8). Notice that, since δjkt will always be smaller than the gross interest rate, the distribution of (8) is bounded on its right tail by the maximum yield that can be obtained, which is rkt . Hence, risk is always undesirable in this setting since it implies receiving a smaller return than the require rate. This is an interesting difference with equity return distributions, which have probability mass on the whole real line, because then positive extreme returns may also occur. Nevertheless, the variance is still a popular measure of risk in these distributions, even though it is well known that a high variance may not necessarily be undesirable if it is due to high unexpected returns (Pe˜ naranda, 2007). Such a concern, though, does not apply in this paper, because there is no right tail. Hence, I can adequately assess the risk of the loans portfolios by means of their covariance matrix, which is fully characterised in the following result: Proposition 3 Consider K portfolios of loans whose returns are given by (8) for k =

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1, · · · , K. Then, V (ykt |It−1 ; θ) = (1/gkt ) E (1 + rkt − δikt )2 |It−1 ; θ πkt|t−1 − (1 + rkt − δk0 )2 ωkkt|t−1 2 +(1 + rkt − δk0 )2 (ωkkt|t−1 − πkt|t−1 ),

(10)

and cov(ykt , yjt |It−1 ; θ) = (1 + rkt − δk0 )(1 + rjt − δj0 ) ×(ωkjt|t−1 − πkt|t−1 πjt|t−1 ), for k 6= j, where

¯2 N2 L kt−1 . gkt = PNkt−1 kt−1 2 L i=1 ikt−1

(11)

(12)

Therefore, both the variances and covariances depend on the moments of recovery rates and probabilities of default. In the case of the variances, (10) is the sum of two components. The first one is proportional to the reciprocal of (12), which can be interpreted as a granularity parameter. For any finite number of loans, (12) will always be finite. However, if I let Nkt−1 grow to infinity while the size of the loans remains bounded, i.e. sup{Likt−1 , 0 ≤ i ≤ Nkt−1 } ≤ Lk+ < ∞, then it can be shown that gkt will tend to infinity and the first term in (10) will disappear. Thus, this parameter increases as granularity grows to infinity. In consequence, the first component in (10) is diversifiable risk, since it can be reduced by just increasing the number of borrowers in the portfolio. For illustrative purposes, I now consider a simple example with two kinds of loans, such that (2) can be expressed as: √ √ −1 x1t Φ (0.05) √1.01 0.01 ρ 0.001 √ xt = ∼ iid N ; , x2t Φ−1 (0.025) 1.1 ρ 0.001 0.1

(13)

For the sake of brevity, I will refer to these portfolios as portfolios 1 and 2. It is not clear which of these to portfolios is safer. On the one hand, the first type of asset has a expected probability of default of 5%, while this value is 2.5% for the second type of assets. On the other hand, the behaviour of loans 2 will be more volatile, since the variance of x2t is ten times larger than the variance of x1t . Figure 1 shows the effect of 7

granularity on the standard deviations of portfolios 1 and 2. Both portfolios have high standard deviations for low granularity, because the smaller the granularity, the higher the idiosyncratic risk. However, portfolio 1 is riskier for low granularity due to its higher unconditional probability of default. Introducing more loans in each portfolio can reduce their risk. Nevertheless, this reduction is smaller for portfolio 2, since the high standard deviation of x2t dominates over the larger expected probability of default of portfolio 1 as the portfolios become more granular. For infinite granularity, only the undiversifiable risk remains. Figure 2 shows the effect of the underlying correlation parameter ρ on the correlation between the returns of portfolios 1 and 2. For very small granularity, the correlation is approximately zero regardless of ρ, because most of the risk is idiosyncratic in this context. For more granular portfolios, ρ becomes a more relevant parameter. Eventually, ρ approximately corresponds to the true correlation for fully diversified portfolios (g = ∞).

3

Asset pricing implications So far, I have treated the interest rates charged by lenders as fixed values. Nevertheless,

in a competitive economy banks should demand an interest rate that compensates for the risk of borrowers’ defaults. To formalise this issue, I introduce an asset pricing framework by modelling the stochastic discount factor (SDF). The SDF is a very useful instrument to obtain prices under absence of arbitrage. For instance, consider an asset with payoff Zt at t. Then, its price at t − 1 can be obtained from the pricing formula: pt−1 = Et−1 [Mt−1,t Zt ] .

(14)

where Mt−1,t is the SDF. The SDF can be related to the marginal rate of substitution between t − 1 and t in equilibrium models (see e.g. Gourieroux and Monfort, 2007). Since the market is incomplete in a discrete time model such as this one, there exists a multiplicity of SDF’s that are compatible with the valuation formula (14). Hence, to operationalise this result I assume a priori an exponentially affine form for the SDF, Mt−1,t = ν0t exp(ν 01t xt ),

(15)

where ν0t and ν 1t are, respectively, a positive scalar and a K × 1 vector that may depend on information known at t − 1. The exponential structure ensures that the SDF is always 8

positive, which in turn guarantees absence of arbitrage opportunities (see e.g. Cochrane, 2001, chapter 4). In addition, this specification corresponds to the Esscher transform used in insurance (see Esscher, 1932), as well as in derivative pricing models (Bertholon, Monfort, and Pegoraro, 2003; León, Menc´ıa, and Sentana, 2007). As Gourieroux and Monfort (2007) argue, the SDF of several important equilibrium models, such as the consumption based CAPM, can be expressed with this structure. In this case, (15) depends on the vector of state variables (2), which is the main determinant of risk in this setting, through K + 1 unknown coefficients. However, I can find their relationship with the interest rates that banks must demand under absence of arbitrage by imposing that the K types of loans and a risk-free asset are correctly priced: 1 = Et−1 [Mt−1,t (1 + rt )] ,

(16)

1 = Et−1 [Mt−1,t (1 + ykt )] , for k = 1, · · · , K

(17)

where rt is the net interest rate of the risk-free asset. Notice that all the prices are normalised to unity because I am working with returns. Using the properties of the log-normal distribution, it is straightforward to show that exp [−ν 01t µt (θ) − (1/2)ν 01t Σt (θ)ν 1t ] ν0t = 1 + rt

(18)

ensures (16). To obtain the remaining coefficients, I will use the following result: Proposition 4 Consider the stochastic discount factor (15). Then Et−1 [Mt−1,t ykt ] =

Q Q rkt (1 − πkt|t−1 ) − (1 − δk0 )πkt|t−1

1 + rt

(19)

where ykt is given in (8), Q πkt|t−1

=Φ

µkt + ν 01t σ •kt √ 1 + σkkt

.

(20)

and σ •kt is the k-th column of (3). Notice the close similarity between (19) and (9). Intuitively, (20) can be interpreted as a risk adjusted probability of default, since the original value under the real measure (4) is modified by introducing the covariance vector of xkt with the whole vector of state variables xt . As shown in Figure 2, the correlations between the elements of xt are essentially those of loan returns under infinite granularity. Hence, a positive covariance 9

Q between ykt and, say, yit increases πkt|t−1 with respect to πkt|t−1 as long as the ν1t,i > 0.

In fact, (19) can be interpreted as the (discounted) expected value of ykt under the riskneutral probability measure that is consistent with the SDF (15), which will be denoted as Q. Interestingly, the distribution of xt can be fully characterised under this measure: Proposition 5 The distribution of xt under the risk-neutral measure Q is multivariate Gaussian with mean vector µQ t (θ) = µt (θ) + Σt (θ)ν 1t

(21)

and covariance matrix Σt (θ). Therefore, as in the multivariate extensions of the model of Black and Scholes (1973), I only have to modify the mean of xt to change the measure, whereas the covariance matrix remains unaltered. Once I have done this change, I could price any asset by just computing its discounted expected price. For example, pt−1 =

1 E Q [Zt ] . 1 + rt t−1

is equivalent to (14). This device can be used to price any combination of the original assets, as well as derivatives or collateralised debt obligations (CDO’s). By introducing (19) in (17), I can express the spread over the risk-free rate, skt = rkt − rt , as skt = (1 + rt − δk0 )

Q πkt|t−1 Q 1 − πkt|t−1

(22)

for k = 1, · · · , K. In this expression, the spread is driven by the product of two components. The first one is the difference between the gross risk-free rate and the recovery rate in case of default. Hence, the higher the recovery rate, the smaller the spread will be. Eventually, if δk0 = 1 + rt , then the spread would be zero because there would be no risk in investing in this asset. The second term depends on the risk-adjusted probability of default. The higher this probability is, the larger the spread. In consequence, spreads are set using a combination of data-based information and preference-based factors. To begin with, they depend on variables from the actual probability measure, such as the recovery rates or the parameters of the actual probability of default. All these parameters are specific of each category of loans, except for σ •kt , 10

which captures the systematic component of the credit risk of loans from a given type k. Q However, the impact of σ •kt on πt|t−1 depends on ν 1t . Hence, these coefficients reflect the

consensus of the market about how to map systematic credit risk into prices. As already mentioned, they are closely related to the utility preferences of the agents, and in particular to their marginal rate of intertemporal substitution. In addition, other determinants of spreads that are not directly related to credit risk may also enter this model through ν 1t . For instance, liquidity risk might be one such example.

4

Mean-Variance frontier with a VaR constraint Let ω t denote a vector containing the proportion of credit that a particular bank

allocates to each of the K different types of loans. This vector is chosen at time t − 1. Hence, ω 0t yt , where yt = (y1t , y2t , · · · , yKt )0 , constitutes the return of the credit portfolio of this bank at t. The expected return and variance will be given by ω 0t E[yt |It−1 ; θ] and ω 0t V [yt |It−1 ; θ]ω t , respectively. I will assume that the bank would like to minimise the variance of its portfolio for every expected return µ0 , subject to the following restriction on the Value at Risk of this portfolio ω 0t yt > τ0 .

(23)

Hence, the bank would like to ensure a minimum return τ0 . This restriction can be due to the minimum capital required by the regulator. For instance, if the bank obtains a return below τ0 it may be forced to raise additional capital. Alternatively, if the bank is targeting a high rating, it may need to satisfy an even more restrictive minimum capital requirement. In general, though, it cannot be ensured that (23) will hold with probability 1. Hence, I need to set the minimum probability 1 − α with which (23) should hold. Notice that the higher this confidence level is the more I will restrict the set of admissible portfolios. In sum, this maximisation problem can be expressed in the following terms:

min ω 0t V [yt |It−1 ; θ]ω t , ωt

11

(24)

such that ω 0t E[yt |It−1 ; θ] = µ0 , Pr[ω 0t yt > τ0 |It−1 ; θ] ≥ 1 − α,

(25) (26)

ω 0t ι = 1,

(27)

k = 1, · · · , K,

(28)

and 0 ≤ ωtk ≤ 1,

where ι is a vector of K ones. Notice that I have introduced (27) and (28) to ensure that the bank can neither short sell nor leverage its investments. For the moment, I am assuming that there is no risk-free rate, but I will relax this assumption below. Following Sentana (2003), this problem can be decomposed in two simpler ones. First, the mean-variance frontier can be computed regardless of restriction (26), that is, by minimising (24) subject to (25), (27) and (28). This frontier can be easily obtained by numerical optimisation, using the closed form expressions of Section 2.2 for the expected return and covariance matrix. If I allow for the presence of a risk-free asset, I do not need to impose (27). As a consequence, it can be shown that then the mean-standard deviation frontier frontier is just a straight line that is tangent to the frontier without risk-free asset and intercepts the mean axis on the risk-free rate. Secondly, the region described by the points such that (26) is satisfied can be analysed separately by Monte Carlo simulation techniques (see Appendix C). Then, the final solution will be obtained from the intersection of these two regions. Following with the example of section 2.2, I show in Figure 3 the effect of different correlations between portfolios 1 and 2 on the mean-standard deviation frontier. I assume infinite granularity and interest rates r1 = 4.5% and r2 = 5%. Since there are only two types of loans in this example, both portfolios will be located in the frontier regardless of ρ. The Figure shows that the frontier is more convex when the correlation is small. Thus, it is easier to obtain diversification gains by combining several portfolios when the loans have small correlations. However, these benefits become increasingly smaller as correlation increases. For instance, if the correlation is close to 1, then any combination of portfolios 1 and 2 is approximately located on the straight line that lies between these two assets. 12

If a bank is able to devise a credit policy that yields returns on the mean-variance frontier, then I will say that this bank is fully efficient since it has exploited all the scope for diversification of its investment technology. In addition, if the mean-variance frontier of a bank a contains the mean-variance frontier of another bank b, then I can say that the technology of bank a is superior to the technology of bank b. Intuitively, the technology of bank a would then have a greater scope for diversification, since it would contain the most efficient situation that bank b can achieve as a particular case. In practice, banks will not generally be able to achieve the mean-variance frontier, since their lending business is constrained by several restrictions. To begin with, loans are highly illiquid assets that cannot be easily traded in the market. Hence, it will only be in the long run that banks will be able to approach the efficient frontier. In addition, the granularity of the loans portfolios may also have an impact on the degree of efficiency that the banks can obtain. In this sense, I can define a “feasible” frontier, taking the granularity of the banks’ portfolios as given, and an infeasible one that assumes that banks can actually get infinitely granular portfolios.

5

Empirical application I now consider an empirical application to the Spanish banking system. In particular,

I consider three types of loans: corporate, consumption loans and mortgages. There is considerable evidence about differences between loans from different economic sectors (see Jiménez and Menc´ıa, 2007). However, I will use this coarser aggregation because I do not have data about the discrepancies between interest rates across economic sectors. I will consider two sets of comparisons between credit institutions. I will first compare commercial with savings banks, and afterwards small vs. large institutions. By estimating a joint model for these pairs of institutions, I can obtain a global efficient frontier for the whole banking system as well as individual frontiers for each type of banks. Hence, I work in practice with K = 6 types of loans, that is, corporate, consumption loans and mortgages of the two types of lenders that I consider in each analysis. I use data from the Spanish credit register to calibrate the risk elements of the model. This database has information about every loan with a volume above e6,000. Since this threshold is very small, I can safely assume that the data is representative of the whole 13

banking system. There are two variables in the database that are of particular interest for this paper.4 In particular, I will obtain the volumes of the existing loans, Lkit , from the outstanding amount of each loan that is reported in the credit register. In addition, the database also has extremely useful information about the situation of each loan. It indicates which loans are overdue, and how long they have been in this situation. Based on this information, I have computed quarterly series from 1984.Q4 to 2008.Q1 of the default frequencies of each type of loans. The default frequency is defined as the ratio of the number of loans that defaulted during a particular quarter, divided by the total number of loans. Following the definition given by the Basel II framework, I assume that a loan is in default if it has been overdue for more than 90 days. I will use default frequencies as proxies of πkt , which I model with (2).5 To do so, I transform them with the inverse of the standard normal cdf. I partition xt as xt = (x0at , x0bt )0 , where xat and xbt are three dimensional vectors that correspond to the first and second type of institutions, respectively.6 I consider the following dynamics: xat = δ a +

d X

diag(αai )xat−i + β a fat + diag(γ a )at ,

(29)

diag(αbi )xbt−i + β b fbt + diag(γ b )bt ,

(30)

i=1

xbt = δ b +

d X i=1

where fat and fbt are two latent factors such that p 2 2 )(1 − ϕ2 ) (1 − ϕ ρ ϕa fat−1 1 − ϕ fat a a b p , I ∼N , ϕb fbt−1 fbt t−1 ρ (1 − ϕ2 )(1 − ϕ2 ) 1 − ϕ2 a

b

b

while 1t and 2t are independent Gaussian vectors. Hence, the correlation structure in this model is driven by a linear factor model whose latent factors are allowed to have time series autocorrelation. A similar structure has been previously considered by Jiménez and Menc´ıa (2007). This model provides a reasonable equilibrium between flexibility in the time series and cross-sectional correlation structures with parsimony in terms of the number of parameters employed. 4

See Jiménez and Saurina (2004), Jiménez, Salas, and Saurina (2006) for a thorough description of the database. 5 This estimation approach relies on the property that πkt can be recovered from default rates if Nkt is sufficiently large (see footnote 2). This is a reasonable approximation in this case because I am considering the whole loans portfolio of Spanish banks. Hence, I have about one million loans on average in each group. 6 That is, either commercial and savings banks or small and large institutions.

14

Unfortunately, the Spanish credit register does not include data about the interest rates of the loans in the database. However, I can use the interest rates reported by banks for corporate loans, consumption loans and mortgages. Banks declared the average interest rate for each type of loans from 2003.Q1 to the end of the sample. Before 2003.Q1, though, they only informed about marginal interest rates, that is, the interest rates of the new operations. Thus, to obtain a consistent database for the whole sample period, I have estimated the age of each loan in the database at each quarter prior to 2003.Q1. With this information, I have constructed series of average interest rates by considering the appropriate marginal rates for each cohort of loans, and weighting them by the proportion of the total credit in the economy that they represent. This procedure yields a consistent time series of interest rates from 1990.Q1 to 2008.Q1. As for recovery rates, I assume that they are constant. Specifically, I set them at 0.65, 0.75 and 0.85 for corporate loans, consumption loans and mortgages, respectively. These values are consistent with the results reported by Spanish banks to the QIS5.7

5.1

Commercial vs. savings banks

Figure 4 shows the historical evolution of default frequencies for commercial and savings banks. Commercial banks seem to have experienced lower default frequencies in corporate loans than savings banks. In contrast, this Figure shows that the consumption loans of savings banks have performed better, while the results for mortgages are quite similar. I have estimated the parameters of (29) and (30), using commercial and savings banks as types a and b of institutions, respectively. As for the autoregressive structure, I consider the first order lag as well as a seasonal component. The estimation has been carried out by maximum likelihood. Table 1 shows that there is strong time series and cross-sectional correlation between the different kinds of loans, since the coefficients of the first lag and the factor loadings on fat and fbt are highly significant in all cases except for mortgages, which seem to display a more idiosyncratic behaviour. In addition, the latent factors are strongly persistent and very highly correlated. Hence, common shocks tend to have lasting effects on the evolution of risk. Based on these estimates, I consider the mean-variance frontier. Table 2 shows the 7

Fifth Quantitative Impact Survey.

15

characteristics of the empirical portfolios. It can be observed in Table 2a that commercial banks have traditionally specialised on corporate loans, while the exposures of savings banks on mortgages have traditionally been higher. Nevertheless, their portfolios are now much more similar than they used to be a decade ago. Table 2b shows that granularity is higher than 104 in all cases. Hence, according to Figure 2, the diversifiable risk due to lack of granularity will be negligible in this application. Nevertheless, corporate loans are generally less granular than either consumption loans or mortgages. Figure 5 shows the historical evolution of the means and standard deviations of the returns generated by the loans portfolios of commercial and savings banks. The two top panels, in which I compare net returns with standard deviations, show that not only the risk of the portfolios has decreased since the early 1990’s, but also their profitability. During the first quarter of 2008, though, this tendency has started to change. For instance, we can observe a significant increase in the standard deviation of commercial banks with respect to 2007. The evolution of returns has been driven to a large extent by the interest rates targeted by the central bank. To eliminate this effect, I consider returns in excess of the threemonth average interbank rate in the bottom panels of Figure 5. It can be noticed that excess returns, which indicate the margin obtained by banks with respect to the interbank market, have also been highly correlated with risk: they were very high during the Spanish recession of 1993, and they achieved their minimum during the last years of the sample. Again, during the first quarter of 2008, excess returns have slightly increased in unison with the rise in standard deviations. On the whole, there seems to be a risk-return tradeoff in the performance of loans from a historical perspective. Finally, I obtain the mean-standard deviation frontiers of these institutions in Figure 6. I consider three different dates: March 2003, December 2005 and March 2008. Each panel includes the frontier above which a 1% excess return will be obtained with 99.9% probability.8 The straight lines correspond to the frontiers when a risk-free asset is included, assuming that its interest rate is the EONIA swap rate. Of course, this is an imperfect proxy of of the risk-free rate. Nevertheless, it seems to be less affected by the liquidity premium that the interbank rate is being charged since 2007. The results show that, on aggregate terms, commercial and savings banks face a fairly similar risk-return 8

I choose this value for illustrative purposes. As already mentioned, the minimum return that a bank should select will depend on its target rating and on regulatory restrictions.

16

tradeoff since their positions on the mean-standard deviation space are very similar. However, this does not mean that the components of their portfolios perform similarly. In fact, the corporate loans of commercial banks are generally more efficient than those of savings banks. That is, the first ones obtain a higher return with less risk. Mortgages have a similar behaviour for both types of institutions and consumption loans seem to be relatively riskier for commercial banks, although they are somewhat more profitable as well. Due to the better performance of commercial banks corporate loans, their frontier is more efficient for low target returns when there is no risk-free asset. However, the lower risk of the consumption loans of savings banks makes their mean-variance frontier more efficient for higher target returns. Due to this feature, when I include the risk-free asset the slope of the frontier of savings banks is generally higher than the one of commercial banks, specially in the last period. In addition, the three panels show that the margins obtained by credit institutions have decreased over the previous years. Thus, a 1% excess return would not require commercial nor savings banks to rebalance their portfolio in 2003, since both of them are situated well above this lower bound. In contrast, they would have needed to concentrate on consumption loans in 2005 or 2008 to achieve this goal. Of course, this would not be feasible in practice, but it nevertheless provides an interesting insight about the high growth of consumption loans during the last years.

5.2

Small vs. Large institutions

In this subsection, I consider small and large institutions as two distinct types of creditors, regardless of whether they are commercial or savings banks. To construct these groups, I sort banks by their size, defined by their total volume of credit, and divide them in two groups. Thus, each group approximately represents half the total amount of credit granted in Spain. Figure 7 shows the evolution of their default frequencies. In general, large institutions seem to have safer borrowers for the three kinds of loans that I consider. However, Figure 7 only has information about risk. Thus, it is only by comparing risk and return jointly that I can decide whether large institutions are more efficient than small ones or vice versa. I have estimated the parameters of (29) and (30) using the default frequencies of Figure 7. Again, I consider a first order lag as well as a seasonal component. The parameter

17

estimates, reported in Table 2, show similar results to those of Table 1. The factor loadings are highly significant in all cases except for mortgages and the latent factors are strongly persistent and very highly correlated. I represent the mean-standard deviation frontiers in Figure 8. As in Figure 6, I also represent the area that yields a 1% excess return with 99.9% probability. Large banks have obtained a slightly larger expected return with a smaller standard deviation in 2003 and 2005. In addition, their lending policy also seems to have been more efficient in these two years, since their mean-standard deviation frontier beats the one of small institutions by achieving a smaller standard deviation for every possible expected excess return. This is mainly due to the better performance of the corporate and consumption loans of large banks. However, the risk of the corporate loans portfolio of large institutions has increased in 2008. It is also possible to observe that the frontiers with the risk-free asset also have a higher slope for large banks. In addition, although Table 4b provides similar granularity for the two types of institutions, this is not the case in reality. I am just analysing the aggregate features of institutions in this exercise. However, from an individual point of view, the size of large banks allows them to have much more granular portfolios or, in other words, less idiosyncratic risk, than small lenders.

5.3

Actual and risk-neutral measures

I have used (22) to obtain the one period ahead predicted risk-neutral default probabilities of corporate loans that are consistent with the interest rate spreads that have been charged by banks.9 Figure 9 (a) shows their evolution since 2003 when I use the EONIA swap as the risk-free asset. For the sake of comparison, I have also plotted the evolution of actual predicted probabilities of default. Risk-neutral probabilities decreased until the middle of 2006, where they were very close or even below the actual default rates. Then, they remained stable until June 2007, increasing again after this period. This change of tendency is of course due to the deterioration of the economy since the middle of 2007. Looking at this issue in more detail, I showed in Section 3 that the difference between risk-neutral and actual probabilities depends on the pricing kernel (15), which in turn can be interpreted as a marginal rate of intertemporal substitution between different periods. 9

An analogous exercise for consumption loans and mortgages yields qualitatively similar results. It is available upon request.

18

Hence, the periods where the difference between risk-neutral and actual default probabilities increases are periods in which a higher risk premium is demanded in the market. Intertemporal consumption models relate this effect to the desire of consumers to soften their consumption path over time. From this perspective, their marginal rate of intertemporal substitution would decrease in bad times to compensate for their decreasing wealth as the economy worsens. However, other more subjective factors, such as the beliefs of the market about the smaller liquidity of loans may also play an important role on this differential. Hence, the increasing discrepancies between risk-neutral and actual expected defaults since June 2007 are probably due to a combination of these two reasons. One remaining important issue is whether the choice of the risk-free asset is reasonable. In this sense, I show in Figure 9 (b) the effect of using the middle reference rate of the Eurosystem as the safe asset. This rate is probably closer to an ideal risk-free rate, since the swap rate may still be affected by some risk and liquidity premia. On the other hand, one of the main disadvantages of the reference rate is that it is not directly tradable. Anyway, I can still observe that the evolution of risk-neutral default probabilities is not affected by this change, although their levels are shifted upwards.

6

Conclusions This paper provides an analytical model to compare the risk and return of loans

portfolios in a joint framework. I propose the use of the mean-variance asset allocation techniques to carry out this analysis, where I allow for a Value at Risk constraint in order to take into account regulatory requirements. The behaviour of the return yielded by loans portfolios is described by means of a flexible dynamic model, which is driven by a vector of Gaussian state variables. In this sense, I obtain closed form expressions for the first two moments of the distribution of 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. 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. 19

I explore the asset pricing implications of the model. I assume an exponentially affine stochastic discount factor. Based on this assumption, I can show that the distribution of the state variables under the risk-neutral measure is another Gaussian distribution with the same covariance matrix but different means. This framework is a powerful valuation tool that can be used to price any portfolio of the original assets, including derivatives and collateralised debt obligations. It can also be used to obtain the spreads over the risk-free rate that banks must require to ensure absence of arbitrage. In this sense, I find that spreads are increasing functions of the risk-adjusted probabilities of default and decreasing functions of recovery rates. In the mean-variance space, the mean-variance frontier represents the set of most efficient portfolios that banks can achieve. Hence, the distance of a bank to its frontier contains information about the scope for improvement that this bank could obtain by changing the allocation of its credit portfolio. I illustrate these features in an empirical application. The dynamic features of the model are crucial to capture the strong time series persistence that is observed in default rates. I find that there seems indeed to be a risk-return tradeoff in historical terms. Periods of high risk (low) usually coincide with high (low) expected returns and, interestingly, large (small) margins with respect to the risk-free interest rate. When I compare the mean-variance frontiers of commercial and savings banks, I find that the lending policy of commercial banks tends to be more efficient for small margin activities because these institutions are more specialised in corporate loans. In contrast, savings banks are more efficient for loans with high margins because of the better performance of their consumption loans. In terms of size, it seems that small banks are generally less efficient than large ones, although this differential is not completely stable over time. I also observe a decreasing trend in the margin that banks are able to obtain with respect to the risk-free rate. In this sense, I compute the area that ensures a 1% excess return over the risk-free rate with 99.9% probability for three different periods: 2003.Q1, 2005.Q4 and 2008.Q1. It turns out that the margins obtained by Spanish institutions have been smaller than this bound in the last two periods. Finally, I can compute from the model the risk-neutral and actual probabilities of default. The differences between these two measures have increased since the middle of 2007, a fact that can be related to the worse perceptions about the current economic situation.

20

An interesting avenue for future research would be to estimate the coefficients of the pricing kernel for the empirical application that I consider. It would also be interesting to explore the effect of macroeconomic variables on the mean-variance frontier, extending the idea of Jiménez and Menc´ıa (2007) into an analysis of risk and return.

21

References Alexander, G. J. and A. M. Baptista (2006). Does the Basle Capital Accord reduce bank fragility? An assessment of the value-at-risk approach. Journal of Monetary Economics 53, 1631–1660. Behr, A., A. Kamp, C. Memmel, and A. Pfingsten (2007). Diversification and the banks’ risk-return-characteristics - Evidence from loan portfolios of German banks. Deutsche Bundesbank Discussion Paper No. 05/2007. Bertholon, H., A. Monfort, and F. Pegoraro (2003). Pricing and inference with mixtures of conditionally normal processes. Mimeo CREST. Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 637–655. Boss, M. (2002). A macroeconomic credit risk model for stress testing the Austrian credit portfolio. Financial Stability Report of the Austrian National Bank 4, 64–82. Cochrane, J. H. (2001). Asset pricing. Princeton: Princeton University Press. Embrechts, P., R. Frey, and A. J. McNeil (2005). Quantitative risk management: concepts, techniques and tools. Princeton: Princeton University Press. Esscher, F. (1932). On the probability function in the collective theory of risk. Skandinavisk Aktuariedskrift 15, 165–195. Fama, E. F. and J. D. MacBeth (1973). Risk, return and equilibrium: empirical tests. Journal of Political Economy 81, 607–636. Ghysels, E., P. Santa-Clara, and R. Valkanov (2005). There is a risk-return trade-off after all. Journal of Financial Economics 76, 509–548. Gourieroux, C. and A. Monfort (2007). Econometric specification of stochastic discount factor models. Journal of Econometrics 136, 509–530.

22

Hayden, E., D. Porath, and N. von Westernhagen (2007). Does diversification improve the performance of German banks? Evidence from individual bank loan portfolios. Journal of Financial Services Research 32, 123–140. Jiménez, G. and J. Menc´ıa (2007). Modeling the distribution of credit losses with observable and latent factors. Banco de Espa˜ na Working Paper 0709. Jiménez, G., V. Salas, and J. Saurina (2006). Determinants of collateral. Journal of Financial Economics 81, 255–281. Jiménez, G. and J. Saurina (2004). Collateral, type of lender and relationship banking as determinants of credit risk. Journal of Banking and Finance 28, 2191–2212. León, A., J. Menc´ıa, and E. Sentana (2007). Parametric properties of semi-nonparametric distributions, with applications to option valuation. Forthcoming Journal of Business and Economic Statistics. Markowitz, H. M. (1952). Portfolio selection. Journal of Finance 7, 77–91. Mart´ın, V. Salas, and J. Saurina (2007). A test of the law of one price in retail banking. Journal of Money, Credit and Banking 39, 2021–2040. Mueller, P. (2008). Credit spreads and real activity. mimeo, Columbia Business School. Pe˜ naranda, F. (2007). Portfolio choice beyond the traditional approach. Forthcoming Revista de Econom´ıa Financiera. Sentana, E. (2003). Mean-Variance portfolio allocation with a value at risk constraint. Revista de Econom´ıa Financiera 1, 4–14.

23

A

Auxiliary results

Proposition 6 Let Φ(·) be the cdf of the standard normal distribution. Then, if z ∼ N (0, 1), E [Φ (a + bz)] = Φ

a √ 1 + b2

.

(A1)

Proposition 7 Let Φ(·) be the cdf of the standard normal distribution. Then, if f, z1 , z2 are independent standard normal variables, ! " 2 # 2 Y Y a2 bi a1 p ,p ; (A2) E Φ (ai + bi f + ci zi ) = Φ2 p 2 2 2 2 2 2 1 + b + c + c + c 1 + b 1 + b 1 2 i=1 1 2 i i i=1 where Φ2 (p1 , p2 | ρ) is the cdf of a bivariate normal distribution with zero means, unit variances and correlation ρ. Proposition 8 Let Φ(·) be the cdf of the standard normal distribution. Then, if x ∼ N (µ, σ 2 ), E[exp(ax)Φ(x)] = exp

B

µ + aσ 2 1 2 2 a σ + µa Φ √ 2 1 + σ2

(A3)

Proofs of propositions

B.1

Proposition 1

Let us first consider (4). Since xt satisfies (2), I can always express πkt as πkt = Φ(µkt +

√

σkkt εkt ),

(B4)

where εkt is a standard normal variable. Then, I can easily obtain (4) from Proposition 6. As for (5), I can use the properties of the Gaussian distribution to express πjt as   s 2 σkkt σjjt − σkjt σkjt πjt = Φ µjt + √ εkt + εjt  , (B5) σkkt σkkt where εjt is independent of εkt . Hence, from Proposition 7, I can show that the expected value of the product of (B4) and (B5) satisfies the required result.

24

B.2

Proposition 2

As usual, I can express cor(Dkit , Djit |It−1 ) as E(Dkit Djit |It−1 ) − E(Dkit |It−1 )E(Djit |It−1 ) cor(Dkit , Djit |It−1 ) = q 2 2 |It−1 ) − E 2 (Dkit |It−1 )][E(Djit |It−1 ) − E 2 (Djit |It−1 )] [E(Dkit E(Dkit Djit |It−1 ) − πkt|t−1 πjt|t−1 = q 2 2 ) )(πjt|t−1 − πjt|t−1 (πkt|t−1 − πkt|t−1 I can easily compute the remaining cross moment by exploiting the conditional independence Dkit and Djit given πkt and πjt : E(Dkit Djit |It−1 ) = E[E(Dkit |πkt , πjt , It−1 )E(Djit |πkt , πjt , It−1 )|It−1 ], = E[πkt πjt |It−1 ] = ωkjt|t−1 .

B.3

Proposition 3

By the law of iterated expectations, I have that V [ykt |It−1 ; θ] = E [V [ykt |πkt , It−1 ; θ]|It−1 ; θ] + V [E[ykt |πkt , It−1 ; θ]|It−1 ; θ] ,

(B6)

where E[ykt |πkt , It−1 ; θ] = rkt − (1 + rkt − δk0 )πkt , 2 V [ykt |πkt , It−1 ; θ] = (1/gkt ) E (1 + rkt − δikt )2 πkt − (1 + rkt − δk0 )2 πkt . Hence, (10) follows directly from introducing the results of Proposition 1 in (B6). Similarly, I can also exploit the law of iterated expectations to express cov[ykt , yjt |It−1 ; θ] as cov[ykt , yjt |It−1 ; θ] = E [cov[ykt , yjt |πkt , πjt , It−1 ; θ]|It−1 ; θ] +cov [E[ykt |πkt , πjt , It−1 ; θ], E[yjt |πkt , πjt , It−1 ; θ]|It−1 ; θ] , where cov[ykt , yjt |πkt , πjt , It−1 ; θ] = 0 due to the conditional independence property of the model, and cov [E[ykt |πkt , πjt , It−1 ; θ], E[yjt |πkt , πjt , It−1 ; θ]|It−1 ; θ] = (1 + rkt − δk0 )(1 + rjt − δj0 ) ×cov(πkt , πjt |It−1 ; θ), which yields (11). 25

B.4

Proposition 4

Using (8), I can express (19) as PNkt Et−1 [Mt−1,t ykt ] = rkt Et−1 [Mt−1,t ] −

j=1 (1

+ rkt − δk0 )Ljkt Et−1 [Mt−1,t Djkt ] , ¯ kt Nt L

(B7)

where Et−1 [Mt−1,t ] = 1/(1+rt ) by virtue of (16). In addition, I can use the law of iterated expectations to show that Et−1 [Mt−1,t Djkt ] = Et−1 [Mt−1,t E(Djkt |xt )] = Et−1 [Mt−1,t Φ(xt )] = ν0t Et−1 [exp(ν 01t xt )πkt ]

(B8)

I can use the conditional properties of multivariate Gaussian distributions to rewrite (B8) as ν 01t σ •kt 0 (xkt − µkt ) + ξkt Φ(xkt ) , Et−1 [Mt−1,t Djkt ] = ν0t Et−1 exp ν 1t µt (θ) + σkkt where (ν 01t Σt (θ)ν 1t )σkkt − (ν 01 σ •kt )2 ξkt ∼ N 0, σkkt is independent of xkt . Hence, ν 01t σ •kt (ν 01t Σt (θ)ν 1t )σkkt − (ν 01t σ •kt )2 0 Et−1 [Mt−1,t Djkt ] = ν0t exp ν 1t µt (θ) − µkt + σkkt 2σkkt 0 ν 1t σ •kt xkt Φ(xkt ) . ×Et−1 exp σkkt I can now use Proposition 8, tho show that µkt + ν 01t σ •kt (ν 01t Σt (θ)ν 1t ) 0 √ Et−1 [Mt−1,t Djkt ] = ν0t exp ν 1t µt (θ) + Φ . 2 1 + σkkt

(B9)

Finally, if I introduce (B9) in (B7), I can obtain the required result.

B.5

Proposition 5

The risk-neutral probability density function of xt can be expressed as f Q (xt ) = (1 + rt )Mt−1,t f (xt ),

(B10)

where f (xt ) is the density of xt under the actual measure: p f (xt ) = exp[−(1/2)(xt − µt (θ))0 Σ−1 t (θ)(xt − µt (θ))]/ 2π|Σt (θ)|.

(B11)

If I introduce (15) and (B11) in (B10), it is straightforward to show the required result when I substitute ν0t for (18). 26

B.6

Proposition 6

I can rewrite (A1) as Z Z ∞ Φ (a + bz) φ(z)dz =

∞

φ (s) ds φ(z)dz,

−∞

−∞

a+bz

Z

−∞

where φ(·) is the pdf of the standard normal distribution. By means of the change of variable t = s − bz, I can obtain Z

∞

Z

a+bz

Z

∞

Z

a

φ (s) φ(x)dsdx = −∞

−∞

φ (t + bz) φ(z)dtdz. −∞

−∞

It is straightforward to show that φ (t + bz) φ(z) is the pdf of the bivariate normal distribution:

z t

∼N

0 0

1 −b ; . −b 1 + b2

This proves the result.

B.7

Proposition 7

By the law of iterated expectations, I can write " 2 # " 2 # Y Y E Φ (ai + bi f + ci zi ) = E E [Φ (ai + bi f + ci zi )| f ] . i=1

(B12)

i=1

Using Proposition 6, I can express (B12) as # " 2 " 2 Y Y E E [ Φ (ai + bi f + ci zi )| f ] = E Φ i=1

!# ai + b i f p 1 + c2i i=1 " # ai +bi f Z ∞Y Z √ 2 2 1+c i = φ (si ) dsi φ(f )df −∞ i=1

(B13)

−∞

p By means of the changes of variables ti = si 1 + c2i − bi f , for i = 1, 2, I rewrite (B13) as ! ! Z ∞ Z a1 Z a2 t1 + b1 f t2 + b2 f φ p φ p φ(f )dt1 dt2 df 1 + c21 1 + c22 −∞ −∞ −∞ It can be shown that this is the integral of the pdf of the following trivariate distribution:       f 0 1 −b1 −b2  t1  ∼ N  0  ;  −b1 1 + b21 + c21  . b1 b2 2 2 t2 0 −b2 b1 b2 1 + b2 + c 2

27

B.8

Proposition 8

I can rewrite (A3) as exp[−(x − µ)2 /(2σ 2 )] √ E[exp(ax)Φ(x)] = exp(ax)Φ(x) dx 2πσ 2 Z 1 2 2 exp[−(x − (µ + aσ 2 ))2 /(2σ 2 )] √ a σ + µa dx. = exp Φ(x) 2 2πσ 2 Z

If I consider the change of variable y = (x − (µ + aσ 2 ))/σ, I can obtain Z 1 2 2 a σ + µa Φ(µ + aσ 2 + σy)φ(y)dy E[exp(ax)Φ(x)] = exp 2 1 2 2 = exp a σ + µa E[Φ(µ + aσ 2 + σy)]. 2 Then, if I use Proposition 6 I can obtain the required result.

C

Value at Risk constraint For each possible variance σ02 , I estimate the points on the mean-variance space for

which (26) holds with equality by solving the following problem:

min ω 0t E[yt |It−1 ; θ] ωt

such that ω 0t V [yt |It−1 ; θ]ω t = σ02 , Pr[ω 0t yt > τ0 |It−1 ; θ] ≥ 1 − α, where Pr[ω 0t yt > τ0 |It−1 ; θ] is estimated for each ω t by computing the proportion of times that M replications of the DGP of ω 0t yt given It−1 is above τ0 . I use M = 100000 in the empirical application.

28

Table 1 Model of the probabilities of default of commercial and savings banks Parameter First order lag: α1

Type of Loan

Comm. Banks

Savings Banks

Corporate Consumption Mortgages

0.79∗∗ 0.27∗∗ 0.94∗∗

0.66∗∗ 0.18∗∗ 0.95∗∗


5.30∗∗ 17.81∗∗ 0.94

5.98∗∗ 17.04∗∗ 0.41

Latent factor loadings: β

Latent factor time series structure ϕa 0.97∗∗ ϕb 0.98∗∗ ρ 0.73∗∗

Notes: Two asterisks indicate significance at the 5% level, while one asterisk denotes significance at the 10% level. Prior to estimation, the dependent variables have been multiplied by 100.

29

Table 2 Characteristics of the aggregate loans portfolios of commercial and savings banks

(a) Portfolio allocations (%)

1993 1997 2001 2005 2008

Commercial Banks Corporate Consumption Mortgages 66 17 17 52 14 34 45 14 41 42 16 42 44 17 39

Savings banks Corporate Consumption Mortgages 39 23 38 28 16 56 30 15 55 32 15 53 36 16 48

(b) Granularity (×10−4 )

1993 1997 2001 2005 2008

Commercial Banks Corporate Consumption Mortgages 3 36 28 5 58 71 9 87 107 8 92 123 9 109 131

30

Corporate 1 2 5 7 9

Savings banks Consumption Mortgages 44 52 58 116 81 164 98 200 108 219

Table 3 Model of the probabilities of default of small and large institutions Parameter First order lag: α1

Type of Loan

Small

Large


0.89∗∗ 0.05 0.92∗∗

0.84∗∗ 0.10 0.95∗∗


1.49∗∗ 16.75∗∗ 2.33

5.45∗∗ 23.66∗∗ 1.48

Latent factor loadings: β

Latent factor time series structure ϕa 0.98∗∗ ϕb 0.98∗∗ ρ 0.65∗∗

Notes: Two asterisks indicate significance at the 5% level, while one asterisk denotes significance at the 10% level. Prior to estimation, the dependent variables have been multiplied by 100.

31

Table 4 Characteristics of the aggregate loans portfolios of small and large institutions

(a) Portfolio allocations (%)

1993 1997 2001 2005 2008

Corporate 56 41 38 36 39

Small Consumption 20 18 18 18 18

Mortgages 24 41 44 46 43

Corporate 52 40 35 37 39

Large Consumption 23 14 12 14 16

Mortgages 25 45 53 50 45

Large Consumption 46 56 74 88 99

Mortgages 37 94 156 176 186

(b) Granularity (×10−4 )

1993 1997 2001 2005 2008

Corporate 2 3 7 7 9

Small Consumption 40 68 102 113 130

Mortgages 36 83 128 160 179

32

Corporate 2 3 8 9 10

Figure 1: Standard deviations of two illustrative portfolios

1.2 Loans type 1 Loans type 2 1

0.8

0.6

0.4

0.2

0 0 10

1

10

2

10 Granularity Parameter

3

10

4

10

Notes: Example based on two types of loans whose defaults are conditionally independent given the evolution of their respective probabilities of default π1t = Φ(x1t ) and π2t = Φ(x2t ), where √ √ −1 x1t Φ (0.05) √1.01 0.01 ρ 0.001 √ xt = ∼ iid N ; . x2t Φ−1 (0.025) 1.1 ρ 0.001 0.1

33

Figure 2: Correlation between two illustrative portfolios.

1 Granularity=1 Granularity=10 Granularity=100 Granularity=1000 Granularity=∞

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 ρ

0.2

0.4

0.6

0.8

1

Notes: Example based on two types of loans whose defaults are conditionally independent given the evolution of their respective probabilities of default π1t = Φ(x1t ) and π2t = Φ(x2t ), where √ √ −1 x1t Φ (0.05) √1.01 0.01 ρ 0.001 √ xt = ∼ iid N ; . x2t Φ−1 (0.025) 1.1 ρ 0.001 0.1

34

Figure 3: Mean-standard deviation frontier between two illustrative portfolios.

4.9 Portfolio 2

ρ=0 ρ=.4 ρ=.8

4.8 4.7

Mean

4.6 4.5 4.4 4.3 4.2 4.1

Portfolio 1 0.05

0.1 Standard Deviation

0.15

Notes: Infinite granularity. Two types of loans whose defaults are conditionally independent given the evolution of their respective probabilities of default π1t = Φ(x1t ) and π2t = Φ(x2t ), where √ √ −1 x1t Φ (0.05) √1.01 0.01 ρ 0.001 √ xt = ∼ iid N ; . x2t Φ−1 (0.025) 1.1 ρ 0.001 0.1 Interest rates are r1 = 4.5% and r2 = 5% for the first and second types of loans, respectively.

35

Figure 4: Evolution of default frequencies for commercial and savings banks (a) Corporate Corporate loans 3.5 Comm. Banks Savings Banks

3 2.5 2 1.5 1 0.5

0 Dec84 Sep87 Jun90 Mar93 Dec95 Sep98 May01 Feb04 Nov06

(b) Consumption Consumption loans 3.5 Comm. Banks Savings Banks

3 2.5 2 1.5 1 0.5


(c) Mortgages Mortgages 3.5 Comm. Banks Savings Banks

3 2.5 2 1.5 1 0.5


36

37

µ0

µ0−r

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

4

6

8

10

12

14

16

0.1

199712 199712

σ0

0.15

199512

199112 199112

0.2

199312

199912

199912

0.05

200803 200803 200712 200712

200512 200512

200112

200112

COMM. BANKS SAVINGS BANKS

0.1

199712

199712

σ0

199512

199112

0.15

199112

0.2

199312

(c) Risk vs. Excess Return (1991-2008)

0.05

200512 200512

199912 200112 199912 200112 200803 200803 200712 200712


(a) Risk vs. Return (1991-2008)

199512

199312

199512

199312

0.25

0.25

200712

200803

0.03

200512 200512


0 0.025

0.5

1

1.5

2

2.5

3

200712

200803

0.035 σ0

200312 200312

200112

0.04

200712

0.035 σ0

200112

200803

200312 200312

0.04

(d) Risk vs. Excess Return (2001-2008)

0.03

200512 200512

200712

200803


3 0.025

3.5

4

4.5

5

5.5

6

6.5

7

(b) Risk vs. Return (2001-2008)

Figure 5: Evolution of the risk and profitability of commercial and savings banks

µ0 µ0−r

200112

200112

0.045

0.045

Figure 6: Mean-standard deviation frontiers of commercial and savings banks (a) June 2003 Mean−Variance Frontier 200306 consb

4 conss.b.

3.5

Joint Comm. Banks Savings banks

3

Pr(w′ yt−r=1%)=99.9%

µ0−r

2.5 2

mortmort s.b. b

COMM. B. BANKS SAVINGS

1.5

corpb corps.b.

1 0.5 0 0

0.02

0.04

0.06

σ0

0.08

0.1

0.12

(b) December 2005 Mean−Variance Frontier 200512 4


3.5

Pr(w′ yt−r=1%)=99.9%

3 µ0−r

2.5 conss.b.

cons

b

2 1.5 1

morts.b. mort b

COMM. BANKS SAVINGS B. corpb

0.5

corp

s.b.

0 0

0.02

0.04

0.06

σ0

0.08

0.1

0.12

(c)Mean−Variance March Frontier 2008200803 4


3.5

Pr(w′ yt−r=1%)=99.9%

3 µ0−r

2.5 2 consb

1.5 conss.b.

1 0.5 0 0

mort mortb s.b.

0.02

COMM. SAVINGS B. BANKS corpb corps.b.

0.04

0.06

σ0

0.08

0.1

0.12

Notes: Excess returns with respect to the EONIA swap rate. The straight lines correspond to the frontiers with a risk-free asset. “corp”, “cons” and “mort” denote corporate loans, consumption loans and mortgages, respectively. The subindex “b” refers to commercial banks while “s.b.” is used for savings banks. 100,000 simulations have been used to estimate the area that yields a 1% excess return with 99.9% probability.

38

Figure 7: Evolution of default frequencies for small and large credit institutions (a) Corporate loans 3.5 Small Big

3 2.5 2 1.5 1 0.5 0 Dec84

Sep87

Jun90

Mar93

Dec95

Sep98

May01

Feb04

Nov06

(b) Consumption loans 3.5 Small Big

3 2.5 2 1.5 1 0.5 0 Dec84

Sep87

Jun90

Mar93

Dec95

Sep98

May01

Feb04

Nov06

(c) Mortgages 3.5 Small Big

3 2.5 2 1.5 1 0.5 0 Dec84

Sep87

Jun90

Mar93

Dec95

39

Sep98

May01

Feb04

Nov06

Figure 8: Mean-standard deviation frontiers of small and large credit institutions (a) June 2003 Mean−Variance Frontier 200306 cons

big

4 3.5

Joint Small Big

conssmall

Pr(w′ yt−r=1%)=99.9%

3 µ0−r

2.5 mort mortbig small

2

BIGSMALL

1.5

corpsmall corp big

1 0.5 0 0

0.02

0.04

0.06

σ0

0.08

0.1

0.12

(b) December 2005 Mean−Variance Frontier 200512 4

Joint Small Big

3.5

Pr(w′ yt−r=1%)=99.9%

3 µ0−r

2.5

consbig cons

2

small

1.5 1

mort BIG mort small

SMALL

big

corp

0.5 0 0

corp

big

0.02

0.04

0.06

small

σ0

0.08

0.1

0.12

(c)Mean−Variance March Frontier 2008200803 4

Joint Small Big

3.5

Pr(w′ yt−r=1%)=99.9%

3 µ0−r

2.5 2 consbig

1.5 1 0.5 0 0

conssmall mort mortbig small

0.02

BIG SMALL

0.04

corpbig corp small

0.06

σ0

0.08

0.1

0.12

Notes: Excess returns with respect to the EONIA swap rate. The straight lines correspond to the frontiers with a risk-free asset. “corp”, “cons” and “mort” denote corporate loans, consumption loans and mortgages, respectively. 100,000 simulations have been used to estimate the area that yields a 1% excess return with 99.9% probability.

40

41

0 Aug03 Feb04 Sep04 Mar05 Oct05 May06 Nov06 Jun07 Dec07

Small (actual) Small (risk−neutral) Big (actual) Big (risk−neutral)


0.5

0.5

1.5

1

Comm. Banks (actual) Comm. Banks (risk−neutral) Savings Banks (actual) Savings Banks (risk−neutral)

1

1.5


Small (actual) Small (risk−neutral) Big (actual) Big (risk−neutral)

(b) Risk-free rate proxied by the marginal middle reference rate in the Eurosystem


0.5

0.5

1.5

1

Comm. Banks (actual) Comm. Banks (risk−neutral) Savings Banks (actual) Savings Banks (risk−neutral)

(a) Risk-free rate proxied by the EONIA swap rate

1

1.5

Q Figure 9: Actual (πt|t−1 × 100) and risk-neutral (πt|t−1 × 100) estimates of the probabilities of default of corporate loans