Optimal monetary policy involving loan rates setting and default rates CASELLINA S.* - UBERTI M.** (*) Banca d’Italia, Rome, Italy, (**) Department of Statistics and Applied Mathematics, University of Turin, Italy, Workshop MDEF 2008 Urbino, September 2008

1.Motivation: macroeconomics and bank decisions 2.Aim 3.Empirical evidence 4.The perspective of Commercial Bank: the model 5.Data and results 6.References OUTLINE

Recently, a lot of studies on the analysis of interest rates dynamics have been carried out according to: the monetary policy decisions of the Central Bank because as a consequence of these decisions, the Central Bank – directly controls the short-term interest rate – and indirectly the long-term rates with the primary goal to limit the fluctuations of the main economic variables. Many proposals are made to extend Taylor rule (1993) by including in the optimal reaction function other variables than those originally introduced MOTIVATION: macroeconomics and Central Bank decisions 1

MOTIVATION: macroeconomics and bank decisions 1-1 <<That some loans are not repaid is central. A theory of m.p. which pays no attention to bankruptcy and default is like Hamlet without the Prince of Denmark and is likely to lead to drastically erroneous policies (as during East Asia crisis). A central function of banks is to determine who is likely to default, and in doing, so banks determine the supply of loans.>> J. Stiglitz – B. Greenwald Towards a new paradigm in monetary economics (2003)

Moreover, In recent years, the increasing interest in financial stability has encouraged the analysis of the links between the macroeconomic environment and the soundness of the banking system The goal is to assess to what extent macroeconomy affects banks’ performance and whether, in turn, banks’ reaction further affects the macroeconomy reinforcing cyclical fluctuations (CREDIT CRUNCH) However, the change in banks’ behaviour through the business cycle is not explicitly modelled. Few studies assess the role that macroeconomy uncertainty plays in determining banks’ behaviour MOTIVATION: macroeconomics and bank decisions 1- 2

The current state of the art is unsatisfactory for two reasons: - no attempt is made in order to model how banks’ management varies in changing macroeconomic environments - the effect of the uncertainty regarding future macroeconomic conditions is typically neglected MOTIVATION: macroeconomics and bank decisions 1 - 3

From a new perspective the links between the dynamics of short- term and long-term interest rates are analysed: through the determination of the loan rates the Commercial Bank reactions to the economic cycle perturbation are examined closely, according to the monetary policy decisions of Central Bank based on Taylor type rules the advantage of this approach is to include into the monetary policy model the default rate usually not involved, though it has a remarkable significance for the banks AIM 2

EMPIRICAL EVIDENCE 3

THE PERSPECTIVE OF COMMERCIAL BANK: THE MODEL 4 NOTATION DOUBFOUL LOANS D t the financing total amount of that at any time goes to default (i.e. doubfoul loans) DR t =D t /L t the default rate that can be considered as a risk measure of the portfolio credits of the Commercial Bank or: log(DR t )=log(D t )-log(L t ) as an alternative measure of risk

INVESTMENTS L t the total amount of the investments of the bank towards the customers at time t (loans, financing,...) r t the short-term interest rate of bank re-financing at any date; (it is directly controlled by the Central Bank with equilibrium value r) i t =r t +d t the lending interest rate (long term interest rate) where d t is the applied spread; the Commercial Bank wishes to maintain it at a fixed level d P t =(i t -r t )L t the total profit of the Commercial Bank in the absence of default NOTATION THE PERSPECTIVE OF COMMERCIAL BANK: THE MODEL 4

SUPPLY SIDE TAYLOR RULE ASSUMPTIONS CENTRAL BANK

ASSUMPTIONS D t and L t as functions of macroeconomic variables the interest rate i t and the output-gap y t INVESTMENTS log(L t )=L t =b 0 t+b 1 Ey t+1 +b 2 (Ei t+1 -E  t+1 ) where b 0 >0 is the equilibrium growth rate, b 1 >0, b 2 <0 Ey t+1 =E(y t+1 |Ω t ) and Ey t+1 =E(y t+1 |Ω t ) DOUBFOUL LOANS log(D t )=D t =a 0 t+a 1 y t-1 +a 2 (i t-1 -  t-1 ) where a 0 >0, a 1 0 if y t <0 (i.e. slackening cyclical phase) D t ↑ if i t ↑ D t ↑ in equilibrium for y t =0 and i t -  t =const.=i*=r+d D e t =a 0 t+a 2 i* a 0 = D e t - D e t-1 a 1 =( ∂D t /∂y t )/D t

ASSUMPTIONS INTERTEMPORAL OBJECTIVE FUNCTION FOR COMMERCIAL BANK The Commercial Bank preferences depend on the deviations from the spread target d and from the equilibrium value D of the default rate U t = λ(d t -d) 2 +(1-λ)( DR t -D) 2 if λ ↑ the bank is less risk adverse λ(d t -d) 2 is the profit component (1-λ)(DR t -D) 2 is the risk component then the control variable d t must 1. maintain the bank spread near to the target d 2. control the risk, i.e. the default rate

DEFAULT RATE D t /L t log(DR t )=DR t =a 1 y t-1 +a 2 (i t-1 -  t-1 )-b 1 Ey t+1 -b 2 (Ei t+1 -E  t+1 ) in equilibrium for y t =0 and i t -  t =const.=i* DR e =(a 2 -b 2 )i* if d ↑ DR t ↑ a risk adverse Commercial Bank would maintain the default rate level at the equilibrium value D=DR e ASSUMPTIONS

THE MODEL 4 quadratic dynamic o.p. where 0< ß<1 is the intertemporal discount factor; π t and y t denote inflation and output gap at date t, respectively and π is the inflation target for the Central Bank; –whose two constraints depict a closed economy (IS-LM curve and Philips curve) the fourth one is the monetary policy function that minimize a Central Bank loss function (Casellina and Uberti, 2008)

DATA AND RESULTS 5 It can be looked at as an extension of Taylor type rules.. The proposed model is calibrated by the VAR approach with respect to historical quarterly date series from 1990 to 2007 of Italy Dennis’s (2005) algorithms are adapted to solve the program since they allow the constraints to be written in a structural form rather than in a state-space form. The optimal policy function for the commercial bank with quadratic intertemporal utility function as in the proposed program is

Fig.1 Reactions of real short and long-term interest rates Fig.3 Dynamic responses of the control variable (d t ) and the default rate (DR t ) Fig.2 Dynamic responses of long-term interest rate (i t ) and the output gap (y t ) if the real economy is in an expansive phase (y t >0) or the inflation rate is greater than the equilibrium level ( π t -π>0), - then the default rate decreases since the fund demand increases and the Commercial Bank can raise the spread (d t ). nevertheless, in this context, the Central Bank reacts - increasing the short rate (r t ) and, as a consequence, - the default rate increases so that the Commercial Bank must limit a rise in spread.

REFERENCES 6 Campbell, J., Shiller, R. (1991) - The Review of Economic Studies 58(3). Casellina, S., Uberti, M. (2008). - Computational Economics, in press. Christensen, A. M., & Nielsen, H.B. (2005). -. FRU Working Papers No. 2005/01 Dennis, R. (2005). - Journal of Economic Dynamics & Control 28, Gerlach, P., Kristen (2003).- ECB Working Paper series No. 258 McCallum, B. (2005). - Federal Reserve of Richmond Economic Quarterly 91(4) Stiglitz, J., Greenwald, B. (2003). - Towards a New Paradigm in Monetary Economics Svensson L. (1997), - European Economic Review 41(6),