Stochastic models for interest rates in the Optimization of Public Debt

1. Stochastic models for interest rates in the Optimization of Public Debt Davide Vergni Istituto per le applicazioni del Calcolo “Mauro Picone” Consiglio Nazionale delle Ricerche Viale del Policlinico, 137 – 00161 Roma – Italy http://www.iac.cnr.it/ E-mail: d.vergni@iac.cnr.it

2. Massimo Bernaschi Alba Orlando Marco Papi Benedetto Piccoli Davide Vergni Alessandra Caretta Paola Fabbri Davide Iacovoni Francesco Natale Stefano Scalera Antonella Valletta CollaborationCNR – Ministry of Economy and Finance Istituto Applicazioni del Calcolo:

3. What is thePublic Debt? Public Debt The compound of the yearly budget deficit in the history • DEFICIT: • Primary Budget Surplus:is the difference between revenues (mostly taxes) and expenditures (mostly salaries). It can be influenced by political orientation: social expenses, investment, selling state's property • Interest over the Debt: expenses for the passive interest on the past debt. It depends on the debt composition and can be modified by optimizing the debt composition

4. Public Debt Management The Growth and Stability Pact (GSP), subscribed by the countries of the European Union (EU) in Maastricht, defines sound and disciplined public finances as an essential condition for strong and sustainable growth with improved employment creation The rules of the pact require that The budget deficit has to be below 3% of Gross Domestic Product The total Debt has to be less than 60% of the GDP Gross Domestic Product: the total output of the economy (PIL) Now the rule are less severe, because they take into account the economic cycle

5. Public Debt Management:Italian situation 1250 billion Euros:Total amount of Italian government stock 277 billion Euros: Bonds expiring in next year This is a very difficult situation. The only lucky fact is that the interest rate are low. With this mass of debt the use of an optimization strategy that reduces only few percentual point in the new issuance, lead to a remarkable money savings A reduction of the 0.4% on the new issuance leads to over than 1 billion euros of money savings

6. Public debt composition BOT, CTZ Zero Coupon Bond 3, 6, 12 and 24 months maturity BTP Fixed Rate Coupon Bond 3, 5, 10, 15 and 30 years maturity CCTFloating Rate Coupon Bond 7 year maturity BTP €iFloating Capital Coupon Bond is similar to a BTP but its capital is linked to the european inflation growth The Italian Public Debt are payed mostly selling different securities (nearly 82% of the total debt). The Italian Treasury regularly issued five different securities: BOT, CTZ, BTP, BTP €i and CCT. The expenses for interest payments on Public Debt are about 15% of the Italian Budget Deficit

7. Interest Rate Is the measure, in percentage terms (interests) of the money due by the state in one year to investors that lend money. Yearly interest rate [issuance price, coupon] Each Bond has its own interest rate that determines the corresponding price. Usually, for long-term loan, the interest rate is high. 3, 6, 12, 24, 60, 120, 180, 360 INFLATION

8. Interest Rates Evolution

9. Historical term structure

10. How to manage Public Debt We can manage public debt just acting on the debt composition in terms of issued securities IAC and Ministry of Economy Project “Analisi dei problemi inerenti alla gestione del debito pubblico interno ed al funzionamento dei mercati”. Debt Management (portfolio composition) can be seen as a constraint optimization problem Fixing a time-window (typically 5 years) what is the optimal debt composition which minimize the debt fulfilling in the meantime all the istitutional and market constraints?

11. Optimization Structure

12. Stochastic Components The most important stochastic elements of the problem are • Primary Budget Surplus: linked to economic policy and macroeconomic factors. It is difficult to modelize. • Evolution of the interest rates: modeled by using of stochastic differential equations like: drt=μ(rt,t)+ σ (rt,t) dBt dft(T)=(t, T, )+ σ (t, T, ) dBt A model for the evolution of short term rates corresponds to a specific functional form for μ(rt,t) and σ (rt,t). A model for the term structure evolution corresponds to a specific functional form for (t, T, )and σ (t, T, )

13. Our model for interest rates All rates are strongly correlated to the official discounted rate determines by the European Central Bank (ECB). Therefore we can think that each rate could be decomposed in a term proportional to ECB and in a term ortoghonal to the ECB Rates decomposition

14. Comparison interpolated ECB and rates (1)

15. Comparison interpolated ECB and rates (2)

16. Decomposition Example

17. First model of fluctuations - PCA For the generation of orthogonal fluctuation we considered a simple multivariate brownian motion We do not use the correlated components of the stochastic terms where Z are a nine component vector of gaussian independent increments but we just use three principal components of the random noise which give 98% of the total variance where z are a nine compoment vector of gaussian independent increments with only the first three component different from 0 U is the diagonalization matrix for the square root of the covariance matrix, , and D is the diagonal matrix associated to 

18. Second model of fluctuations - CIR • Another possibility for the generation of orthogonal fluctuation is by the use of a multivariate extension of the classical model for the short term rate by Cox-Ingersoll-Ross (CIR-1985): • are constant verifying the condition • The settings of the model parameters is by the maximum likelihood applied to the discrete evolution equation

19. Validation for the term structure Our goal is not to forecast rates evolution, but to generate "reasonable" scenario of rates evolution The term structure of interest rates could be very different from the historical ones We control the growth and the convexity of the generated term structure The cross-correlation of interest rates could be very different from the historical ones We control the simulated cross correlation

20. Term structure example using PCA

21. Term structure example using CIR

22. Macroeconomical model • It is a completely interacting model • the inflation modifies the monetary policy of the ECB, • the ECB policy, on the other hand, modify the inflation Basic Model: ECB - Inflation The goal is to capture the link between the inflation and the monetary policy adopted by the ECB. Moreover we are also interested in understanding how the intervention of the ECB reflects on the interest rates evolution in the euro area The principal economic ingredients are: The goal of the ecb is to maintain the inflation around 2% The real Short interest rate has to be positive

23. Macroeconomic variables ECB official discount rate Harmonized Index of Consumer Prices (HICP) ex tobacco Annual Inflation Rate

24. Euristic model The inflation evolves according to the rule  is distributed as the historical absolute value of the inflation increments s could be 1 or -1 according to a certain probability The ECB rate evolves according to the rule where Each change of the ECB rate acts on the probability of s

25. Non linear model We use coupled maps with stochastic element At difference with the previous model now  is a random variable: Where and are binomial random variables whose value can be 0 or 1, with a probability that depends on the value of . K and are constant values obtained by the calibration of the model, f is a non linear function and z is a gaussian random variable

26. Building a complete model Macroeconomic Factors Official discount rate, Inflation Primary Budget Surplus, Gross Domestic Product Microeconomic Factors Interest rates macro-micro economic model A total interacting model involving all the macro and microeconomic factors

27. Building a complete model Economic Cycle Variable Macroeconomic Factor Interest Rates A hierarchical model: each component involves homogeneous quantities, using variables of higher level as quasi-parameters. The economic cycle variable is a non-observable quantity

28. Present State of the Project • The software prototype is complete and running at the Ministry of Economy • All components have been validated on real data • At present the scenario generator implement two different ecb-inflation model and two different interest rates model. Open problems • Improve the interest rate models. • Build a macroeconomic model • Improve the cost-risk analysis