Stochastic models for interest rates in the Optimization of Public Debt

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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
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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

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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

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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
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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
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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
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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
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Interest Rates Evolution
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Historical term structure
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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?
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Optimization Structure
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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, ?)
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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
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Comparison interpolated ECB and rates (1)
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Comparison interpolated ECB and rates (2)
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Decomposition Example
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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 ?
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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

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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
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Term structure example using PCA
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Term structure example using CIR
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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
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Macroeconomic variables
ECB official discount rate
Harmonized Index of Consumer Prices (HICP) ex
tobacco
Annual Inflation Rate
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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
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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
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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
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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
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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