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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_at_iac.cnr.it

Collaboration CNR Ministry of Economy and

Finance

Istituto Applicazioni del Calcolo

- Massimo Bernaschi
- Alba Orlando
- Marco Papi
- Benedetto Piccoli
- Davide Vergni

- Alessandra Caretta
- Paola Fabbri
- Davide Iacovoni
- Francesco Natale
- Stefano Scalera
- Antonella Valletta

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 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

Public Debt ManagementItalian 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

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 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

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

Interest Rate

Is the measure, in percentage terms (interests)

of the money due by the state in one year to

investors that lend money.

issuance price, coupon

Yearly interest rate

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

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

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?

Optimization Structure

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) s (rt ,t) dBt

dft(T) ?(t, T, ?) s (t, T, ?) dBt

A model for the evolution of short term rates

corresponds to a specific functional form for

µ(rt, t) and s (rt ,t). A model for the term

structure evolution corresponds to a specific

functional form for ?(t, T, ?) and s (t, T, ?)

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

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 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 ?

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

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

Term structure example using PCA

Term structure example using CIR

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

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

? 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

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

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