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

Collaboration CNR – Ministry of Economy and Finance

Massimo Bernaschi

Alba Orlando

Marco Papi

Benedetto Piccoli

Davide Vergni

Alessandra Caretta

Paola Fabbri

Davide Iacovoni

Francesco Natale

Stefano Scalera

Antonella Valletta

Istituto Applicazioni del Calcolo:

What is the Public 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

Public Debt Management

The rules of the pact require that

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 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

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

2003 2004 2005Deficit/GDP 2,4 2,9 4,3Debt/GDP 106,2 105,9 106,6

Public debt composition

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

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

CCT Floating Rate Coupon Bond 7 year maturity

BTP €i Floating Capital Coupon Bond is similar to a BTP but its capital is linked to the european inflation growth

Interest Rate

Is the measure, in percentage terms (interests) of the money due by the state in one year to investors that lend

money.

Each Bond has its own interest rate that determines the

corresponding price.

Usually, for long-term loan, the interest rate is high.

[issuance price, coupon] Yearly interest rate

3, 6, 12, 24, 60, 120, 180, 360

INFLATION

Interest Rates Evolution

Historical term structure

How to manage Public Debt

We can manage public debt just acting on the debt composition in terms of

issued securities

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?

“Analisi dei problemi inerenti alla gestione del debito pubblico interno ed al funzionamento dei mercati”.

IAC and Ministry of Economy

Project

Optimization Structure

Stochastic Components

drt = μ(rt, t)+ σ (rt ,t) dBt

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:

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, )

dft(T) = (t, T, )+ σ (t, T, ) dBt

Our model for interest rates

Rates decompositio

n

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

Comparison interpolated ECB and rates (1)

Comparison interpolated ECB and rates (2)

Decomposition Example

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 vectorof gaussian independent increments

U is the diagonalization matrix for the square root of the covariance matrix, , and D is the diagonal matrix associated to

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 independentincrements with only the first three component different from 0

• 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

Second model of fluctuations - CIR

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

The cross-correlation of interest rates could be very different from the historical ones

We control the growth and the convexity of the generated

term structure

We control the simulated cross correlation

Term structure example using PCA

Term structure example using CIR

Macroeconomical model

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

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

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

Macroeconomic variables

ECB official discount rate

Harmonized Index of Consumer Prices (HICP) ex tobacco

Annual Inflation Rate

Euristic model

The inflation evolves according to the rule

where

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

Each change of the ECB rate acts on the probability of s

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

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

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

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