a multi-objective multi-period stochastic programming model for public debt management

European Journal of Operational Research 205 (2010) 205–217

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

A multi-objective multi-period stochastic programming modelfor public debt management

Emre Balibek a,b, Murat Köksalan b,*

a Turkish Treasury,1 06510 Ankara, Turkeyb Industrial Eng. Dept., Middle East Technical University, Ankara, Turkey

a r t i c l e i n f o

Article history:Received 10 August 2007Accepted 1 December 2009Available online 6 December 2009

Keywords:OR in governmentMultiple objective programmingRisk analysisStochastic programmingPublic debt management

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.12.001

1 The ideas expressed in this study are only those of* Corresponding author.

E-mail addresses: [email protected] (E. B

a b s t r a c t

While raising debt on behalf of the government, public debt managers need to consider several possiblyconflicting objectives and have to find an appropriate combination for government debt taking intoaccount the uncertainty with regard to the future state of the economy. In this paper, we explicitly con-sider the underlying uncertainties with a complex multi-period stochastic programming model that cap-tures the trade-offs between the objectives. The model is designed to aid the decision makers informulating the debt issuance strategy. We apply an interactive procedure that guides the issuer to iden-tify good strategies and demonstrate this approach for the public debt management problem of Turkey.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Public debt management (PDM) is concerned with meeting the funding requirements of a country that arise from budgetary and otherfinancial liabilities of the government. More specifically, it can be defined as the ‘‘process of establishing and executing a strategy for man-aging the government’s debt to raise the required amount of funding, pursue its cost/risk objectives, and meet any other public debt man-agement goals the government may have set, . . .” (International Monetary Fund – The World Bank, 2003, p. 5). Public debt managers have arange of financial instruments, securities, at their disposal and have to form a specific portfolio, in terms of maturity, currency and interesttypes, that would suit the government’s objectives. Given the exposure of public sector balances and the country’s financial stability topublic debt, the overall structure of public debt portfolio is fundamentally important for a country’s macroeconomic stability. Once a gov-ernment is in a financial problem, i.e. facing difficulties in fulfilling fiscal liabilities or having to pay excessive costs when issuing debt, thishas spill-over effects on the entire economy, as demonstrated by a number of recent macroeconomic crises in several emerging countries.Therefore, the financial liability portfolio of the government must be effectively managed.

Given its importance, the problem of designing the public debt management strategy, in terms of setting the composition of financing,draws attention of both practitioners and academicians from various perspectives. Alesina et al. (1990) elaborate on the choice of maturityof public debt and argue that issuing debt at long maturities that is evenly concentrated in time will boost public confidence and reduceperceived likelihood of confidence crisis about debt default. Missale and Blanchard (1994) claim that government can use the maturity ofdebt to show her commitment to anti-inflationary policies and thus should prefer short-maturity or indexed debt. The tax smoothing ap-proach assumes that the main reason for the government to change taxes is to meet the long-term financing constraint, and the objective isto smooth taxes by choosing the optimal composition of debt with respect to maturity and contingencies. There is uncertainty about mac-roeconomic variables such as public expenditures, tax base, etc. and therefore, the composition of debt matters (Barro, 1995, 2003). Theargument is that debt can serve as a buffer against tax rate changes if the government can issue debt with costs that are lower whennet tax receipts are lower. A good review of theoretical and practical concepts regarding public debt management can be found in Dorn-bush and Draghi (1990) and Leong (1999).

Debt management authorities take a practical point of view and apply concepts and tools derived from those employed by private finan-cial institutions. The simulation models of PDM offices are generally derived from the ‘‘Value at Risk” (VaR) concept widely used by banks

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the authors’ and do not necessarily reflect the views and policies of the Turkish Treasury.

alibek), [email protected] (M. Köksalan).

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206 E. Balibek, M. Köksalan / European Journal of Operational Research 205 (2010) 205–217

and other financial firms. For sovereigns, this approach is modified into ‘‘Cost-at-Risk” or ‘‘Cash-Flow-at-Risk” simulation models by whichthe cost and risk performances of alternative debt management strategies are tested under various macroeconomic simulation scenarios(see Danmarks Nationalbank (2005) and Bergström et al. (2002) for two country examples). Bolder (2003) explains the simulation modelfor debt strategy analysis in Canada. More recently, Bolder and Rubin (2007) try to combine simulation and optimization approaches indebt strategy analysis. Countries also apply other methods like stress testing or scenario analysis to compare different PDM strategies(see International Monetary Fund – The World Bank (2003) and OECD (2005) for discussions on debt management practices of selectedcountries).

The general aim in these applications is to quantify costs and risks associated with policy choices. On the other hand, there is a need forproviding support to decision makers (DM) to find efficient solutions. To the best of our knowledge, this is the first study that explicitlyconsiders multiple objectives and guides the DM towards desirable strategies. In this paper, we model the PDM problem as a multi-periodstochastic program with multiple objectives. We formulate the problem as a deterministic equivalent linear programming model, in whichthe decision variables are the amounts of different types of bonds to be issued, accounting for the cash-flow constraints for the government.We develop our approach as a decision aid tool in analyzing the trade-offs between alternative courses of action. In this context, we identifyefficient solutions based on different preferences and apply an interactive Multi-Criteria Decision Making (MCDM) approach to guide theDMs in developing the debt strategy. We demonstrate how sovereign DMs can experiment with such a tool in a practical setting, drawingon the case of Turkey.

The following section defines the public debt management problem and discusses its main features. We then present our multi-stagestochastic programming (SP) model, developed to guide issuance decisions. Section 4 discusses how we employ the SP model to obtainefficient solutions and use an interactive algorithm by which the DMs can experiment to explore alternatives. The proposed methodsare illustrated in Section 5.

2. Characteristics of the PDM strategy formulation problem

Governments often announce auction schedules or financing programs to publicize the amounts and dates with regard to planned bondissuance schedules. These issuance programs are prepared in line with the with government’s debt management objectives and describethe types of bonds the government is planning to issue to meet the projected financing requirement in a certain period. Early announce-ment of issuance strategies leaves time for market participants, i.e. potential investors to absorb the information revealed and to adjusttheir cash-flow schemes.

The common aim in PDM is to minimize the cost of debt. It will be tax payers who will be paying back the debt and one of the mainobjectives of debt management offices is to find the necessary funds at the lowest possible cost in line with the expectations of citizens.Even though the relevant cost definition may differ for each country, the most common measure of cost in borrowing funds is the interestrate requested by lenders. When a government issues debt, the cost of borrowing is reflected in the national budget in terms of interestexpenditures. For countries that issue debt in foreign currencies, the change in the value of the debt, measured in the local currency,due to fluctuations in the exchange rate also adds to the cost of debt. Debt management offices that engage in frequent secondary marketactivities such as debt buybacks or bond exchanges may also follow the marked-to-market value of their debt portfolios. When costs aredistributed over a number of years, they can be measured in a present value basis. They can also be normalized with respect to a macro-economic magnitude or the size of debt portfolio to allow period-wise comparisons.

A well-known characteristic of financial markets is that there is a trade-off between return and risk. Generally, the higher the returnsfrom an investment, the higher are the associated risks. Considering the fact that an investor’s return on a financial instrument is a cost forthe issuer, the ‘‘risk/return trade-off” concept has its mirror image for the government as the ‘‘cost-risk dilemma”. The recent financial cri-ses, accompanied by the increased volatility of international fund flows and the complexity of financial instruments highlighted the impor-tance of risk-related criteria, in addition to cost, while raising public debt. Most public debt managers are now concerned with the risks andassociated macroeconomic issues as well as cost. The public debt management objective in the United Kingdom, for example, is ‘‘to min-imise, over the long-term, the costs of meeting the Government’s financing needs, taking into account risk, . . .” (HM Treasury, 2007).

An important characteristic of the multi-objective PDM problem is that decisions are made under uncertainty. Debt managers are notfaced with choices that have deterministic outcomes. Their decisions are concerned with future actions of the government and while mak-ing strategy decisions, debt managers are not certain about the future states of nature for the relevant macroeconomic variables. There is adegree of uncertainty associated with the evolution of economic factors such as interest and exchange rates that drive the cost of borrow-ing. The actual outcomes of the decisions made while formulating the issuance strategy are contingent on realizations of macroeconomicvariables that exhibit different types of stochasticity. In fact, it is this uncertainty that raises the need to consider risk objectives.

The major risks public debt managers face are the market risk, which is defined as the risk of an increase in the cost of debt service as aresult of unfavourable movements in market conditions and the liquidity (re-funding) risk that indicates the possibility of having insuffi-cient funds in order to make debt repayments.

The cost and market risk objectives are generally conflicting by their nature, as short-term interest rates are usually lower than longer-term rates. This is also true in an economy where interest rates tend to decline. In such a context, it would be less costly for the governmentto issue short-term debt to make use of lower or declining interest rates. The aim in issuing short-term bills or longer-term variable ratebonds indexed to short-term interest rates is to shorten the interest rate fixing period of the debt stock. This policy will expectedly serve forcost minimization purposes. However, in case of a sudden surge in market interest rates, the cost on a major portion of the government’sdebt will have to increase. This is the market risk in public debt management, which is different from the market risk perceived by inves-tors. Short-term bonds which have short duration are less sensitive to price changes, therefore expose less risk of market value movementsfor investors. However, for a government short duration means a short interest fixing period, i.e. higher risk of volatility in interest costs.Therefore, everything else being the same, ‘‘Short-duration debt (short-term or floating) . . . is usually considered more risky than longduration (long-term, fixed-rate) debt” (OECD, 2005, p. 41).

For countries that have liabilities denominated in foreign currencies, the volatility of exchange rates also constitute a major market risk.The Mexican Crisis at the end of 1994 is partly attributable to the 29 billion United States Dollar (USD) tesobonos maturing in 1995, with 10

Scenario Paths

quarter 1 2 3 4 5 6 7 8 9 10 11 12

210egatsnoisiced

Set Strategy for Year 1

Revise Strategy

Fig. 1. A sample scenario tree.

E. Balibek, M. Köksalan / European Journal of Operational Research 205 (2010) 205–217 207

billion USD payable in the first 3 months, while the country’s foreign reserves stood at a level of 6.3 billion USD (Cassard and Folkerts-Lan-dau, 1997).

Liquidity risk or re-funding/re-financing risk as it is sometimes called is also a critical concern for public debt managers, especially indeveloping countries. This type of risk may arise from the level of cash reserves of the government, for example due to a decline in taxrevenues or from the lenders’ reluctance in renewing their loans. However, in many cases, the level of funds from net revenues availableto governments to service debt is very low when compared to public debt redemptions. On the other hand, market conditions that affectinvestor demand are not controllable by debt managers. Therefore, ‘‘the guiding principle for countries is to spread maturities across thematurity spectrum to avoid bunching of maturity payments” (OECD, 2005, p. 34). In this regard, many countries target total redemptionsover the near term and try to extend maturities to control liquidity risk which is crucial for a government’s reputation. Lenders, other gov-ernment institutions, public employees and in the end all the parties in the economy will be affected from the government’s liquidity crises.

In many cases, there can also be a trade-off between the cost and liquidity risk objectives, since reducing the liquidity risk may requirelong-term borrowing at high costs and/or keeping a certain level of excess cash reserve which also induce a cost for the government. Aim-ing to minimize market risk may dictate to borrow fixed-rate long-term bonds whose repayments accumulate at a certain point in timewhich in turn induces a certain level of liquidity risk.

While formulating the issuance strategies the authorities also need to consider several constraints. To begin with, the amount of fundsraised should not be less than those required by the budget. Governments generally hold a cash account which serves as a buffer to coverunexpected cash needs and this allows borrowing more or less than needed for a certain period of time. However, there are also limitationsto the levels of this account, i.e. governments cannot over or under borrow continuously. Thus, the inter-temporal budgetary and cash ac-count constraints must be satisfied.

The size and efficiency of a country’s financial markets, the government’s ability to access international markets and other macroeco-nomic environmental conditions impose several limitations on the type of securities the debt managers can issue. For under-developed ordeveloping countries, where the level of domestic savings and efficiency of internal financial markets are limited, the main option is to optfor funds from international markets. Some developing countries have a functioning domestic financial market, but they also need to issuedebt in foreign currency to lengthen maturity when domestic lenders prefer shorter maturities. Given a certain instrument set, public debtmanagers should also consider market constraints with regard to the availability of funds and the demand for different types of securities.In a volatile environment, creditors may not be willing to extend long-term loans, and the government’s insistence on lengthening matu-rities may result in a funding-crisis. Institutional investors, individuals, banks might have different preferences which may impose severalconstraints on the size of bonds to be offered to different market segments.

The debt strategy is not a one-off decision. The PDM problem embodies a sequence of decisions that would allow the government’s debtportfolio to adjust to changing environmental conditions. A decision made now for the portfolio structure is subject to revision in the futuredepending on changing outlooks for the macro-economy. Therefore, debt managers should incorporate this flexibility in their decisionmaking processes. They need to consider the effects of the potential for adjusting decisions in the future, since future decisions will be con-tingent on the previous actions and prevailing market conditions.

3. A multi-objective stochastic programming approach

In recent years, (SP) models have been increasingly used to address real life multi-period asset and liability management problems. Aseminal contribution was made by Bradley and Crane (1972) who proposed a multi-stage model for bond portfolio management. Morerecently, Carino et al. (1994) applied SP to the asset–liability management problem of the insurance industry, and Zenios et al. (1998)and Topaloglou et al. (2008) formulated models for a portfolio of fixed income securities. Nielsen and Poulsen (2004) proposed a multi-stage SP model for managing mortgage backed loans. Volosov et al. (2004) developed a two-stage decision model for foreign exchangeexposure management. Grill and Östberg (2003) have applied an optimization approach for debt management. Yu et al. (2003) providea bibliography of SP models in financial optimization. Extensive collections of SP models for financial problems can also be found in Ziembaand Mulvey (1998) and Dupacova et al. (2002).

The general approach of multi-stage SP models in representing uncertainty is forming a scenario tree that reflects the evolution of ran-dom variables in each stage of the decision horizon, by discretizing their joint probability distributions. The simplest approach is to use


historical data regarding random variables without any modelling and claim that future will replicate the past, i.e. sampling from differentpoints in time for generating scenarios. Another method is trying to create scenarios that replicate the empirical characteristics of randomvariables such as the moment matching method of Hoyland and Wallace (2001). A more sophisticated method requires econometric mod-elling such as Boender (1997) and Villaverde (2003) who use vector autoregressive (VAR) time series models for scenario generation.

We formulate the PDM problem as a linear deterministic equivalent model based on a scenario tree representation of macroeconomicfactors that affect the cost of debt. The model has a multi-stage structure that takes into account sequential decisions concerned with debtissuance policies. We assume that at the beginning of each year the government sets a borrowing strategy, which embodies the amount ofbonds to be offered in each month (or quarter) of the following year and revises this strategy annually. We start with a given liability cash-flow scheme (arising from the current debt portfolio) and a set of anticipated scenarios about future states of relevant macroeconomic vari-ables such as the interest and exchange rates. The objective is to specify a sequence of bond issuance decisions at discrete points in time.Since the funding requirements might be different in each quarter, total funds raised through issuance of bonds might also be different.

We present a general n-stage model in which each period is divided into several sub-periods, t. (If the periods correspond to years, sub-peri-ods can be months or quarters.) Decisions are made at the start of each period for the following sub-periods, i.e. issuance decisions are not re-vised in each sub-period, but only at decision stages. The scenarios between decision stages combine to form a sequence of joint realizations fora certain period. These sequences of scenarios are linked at the decision nodes and we have scenario paths covering the entire planning horizon.

The issuance strategy for the sub-periods of the first year is not scenario specific and is set to be implemented across all scenarios. At theend of the first year we have a new liability portfolio and we now have to make a new set of decisions incorporating this new portfoliostructure, thus the updated cash-flow scheme is contingent on the scenario realization in the interim (first year) and the current scenariotree branches. Thus decisions, other than the first stage decision, are path-dependent and we have a SP problem with recourse. Fig. 1 illus-trates the structure of a problem with three periods each divided into four quarters.

One main assumption we make is that the macroeconomic environment is independent of the government’s policy actions with regardto public borrowing. That is, the amount and the type of bonds the government decides to issue do not affect the level of interest rates inthe market. This is a realistic assumption, especially for countries that have deep and liquid bond markets with many issuers and lenders.

Even though the constraints imposed by the characteristics of the economy and the financial markets may differ from one country toanother, the objectives in debt management are similar and the general structures of liabilities are comparable. For example, Turkey andCanada issue the same type of inflation linked bonds. The floating rate notes of the Turkish Treasury are similar to those of Italy. Therefore,the generic model can serve for different country characteristics with small modifications. There can be as many periods as relevant, andthe number of sub-periods in each period can vary. The model can also consist of two dissimilar periods, the first period corresponding toyear one and the second covering all remaining years in the model horizon.

3.1. Notation

The model is based on cash-flow equations that guarantee that the total out-flows of the government match the inflows. We include acash account in our model that would absorb any excess or short borrowing that might occur when certain scenarios are realized since cashout-flows are based on some parameters that are scenario specific. Thus, the debt managers set the amount of each bond to be issued in allsub-periods of the following period considering the possibilities for the level of the financing requirement. If, in some cases, the total debtraised is more (less) than needed, the excess (short) amount is injected into (withdrawn from) the cash account of the government.

Parameters and index sets

T
length of the planning horizon (in terms of sub-periods) N number decision stages (periods) t time index (denoting sub-periods), t = 1, . . . ,T Ti length of period i, i = 1, . . . ,N (in terms of sub-periods) S the scenario set s scenario index, s 2 S ps probability associated with scenario s J1 set of zero-coupon bonds/bills (bonds that pay interest at maturity) J2 set of variable coupon bonds/bills (with interest fixing at the start of each coupon period) J3 set of fixed coupon bonds/bills J set of all bonds ðJ ¼ J1 [ J2 [ J3Þ J0t set of bonds with maturities shorter than t, J0g;t ¼ fj 2 J : t �mj > 0g J0g;t set of type g bonds with maturities shorter than t, J0g;t ¼ fj 2 Ji : t �mj > 0g; g ¼ 1;2;3 mj maturity of instrument j, j 2 J (in terms of sub-periods) cj coupon period of instrument j, j 2 J2 [ J3 (we assume all coupons are semi-annual) ut;j upper bound for the issuance of bond j at time t PSt primary surplus (net non-debt cash-flow) at time t, t = 1, . . . ,T Ys;t;j coupon payment indicator for instrument j ðj 2 J2 [ J3Þ issued at times for time t (Ys;t;j ¼ 1 if instrument j issued at time s pays
coupon at time t)
CB0 starting balance at the government’s cash account nðs; tÞ decision node for scenario s, for period t
The decisions are made at the nodes of the scenario tree, thus nodes are where scenario paths branch. The parameter nðs; tÞ denotes thenode in which the issuance decision is made for time t under scenario s. For all the scenarios in period 1, the decision is made in node 0:

nðs; tÞ ¼ 0; for 8s 2 S and t 6 T1:


The scenario dependent inputs are given below:

Stochastic variables:

rst;j
interest rate prevailing at time t for instrument j under scenario s
rst;c
interest earned on the cash balance of the government at time t under scenario s
est;j
exchange rate prevailing at time t for instrument j under scenario s (es
t;j ¼ 1 for local currency instruments)

Lst
liability payments fixed before the decision horizon for time t under scenario s
The decision variables are defined for each node of the scenario tree:

Decision variables:

Xnðs;tÞt;j

amount of instrument j to be issued in period t under scenario s, decided at decision point nðs; tÞ

Auxiliary variables:

Ist
total net interest paid at time t under scenario s
Dst
total principal (debt) paid at time t under scenario s
Fst
the difference in the market value of maturing foreign currency debt
TCs
total cost for scenario s
Cst
withdrawal from cash account at time t, under scenario s
CBst
level of cash account (cash balance) at time t, under scenario s
Rst
interest revenue on the cash account balance
VR
variable used in the definition of Conditional Cost-at-Risk – equals to VaR at the optimal solution cvs excess cost beyond VaR for scenario s PR variable used in the definition of Conditional Payment-at-Risk (similar to VR) cps excess payment beyond PR for scenario s
ut;j; PSt ; CB0; Lst ; Xnðs;tÞ

t;j ; Ist ; Ds

t ; Fst ; TCs; Cs

t ; CBst ; Rs

t ; VR; cv s; PR; cps are in units of numeraire currency.

3.2. Constraints

The constraints of the model include inter-temporal cash-flow equations that provide for balance between the government’s payments,amortization and interest payments, and revenues. There is also a balance equation for the cash account. We also include constraintsregarding the marketability of the bonds, as there might be bond specific limitations for the amount of issuance due to the structure ofmarket demand. Below are the constraints of our model:

– Total principal paid back at time t, scenario s:

Dst ¼

Xj2J0t

Xnðs;t�mjÞt�mj ;j

8t; s: ð3:1Þ

This equation sums the principal values of all the bonds that mature at time t for a specific scenario s.– Total net interest paid at time t, scenario s:

Ist ¼

Xj2J01;t


rst�mj ;j

�es

t;j

est�mj ;j

þX0j2J02;t

Xt�1

s¼t�mj

Xnðs;sÞs;j Ys;t;jrs

t�cj ;j�

est;j

ess;jþXj2J03;t

Xt�1

s¼t�mj

Xnðs;sÞs;j Ys;t;jrs

s;j �es

t;j

ess;j� Rs

t 8t; s: ð3:2Þ

The interest cash-flow equation for scenario s, consists of the interest paid on maturing zero coupon bonds and the coupons paid for livefixed and floating rate bonds at time t, all adjusted for changes in the underlying exchange rate. The interest paid is computed by mul-tiplying the principal value of a bond by the applicable interest rate, which is fixed at time of issuance for zero and fixed coupon bonds andat the start of coupon period for variable rate notes. The interest earned on the cash account is deducted from the interest payments.

– Appreciation in the value of foreign currency debt at maturity:

Fst ¼

Xj2J0t


est;j

est�mj ;j

� 1

!8t; s: ð3:3Þ

Countries that issue foreign currency denominated debt also have to pay the difference in the market value of debt due to exchange ratefluctuations.

– The cash-flow balance:
Xj2J
Xnðs;tÞt;j þ Cs

t ¼ Dst þ Is

t þ Fst þ Ls

t � PSt 8t; s: ð3:4Þ

The cash-flow balance equation indicates that the total amount of bonds issued at time t (for scenario s) and the amount used from thegovernment’s cash account (Cs

t can take both positive and negative values) should equal the sum of debt repayments, including principaland interest, and the non-debt liabilities of the government, accounting for the primary surplus available for time t.


– Cash account balance:

CBst ¼ CB0 � Cs

t 8s and if t ¼ 1; ð3:5ÞCBs

t ¼ CBst�1 � Cs

t 8s; 8t : t > 2: ð3:6Þ

The cash account balance must be adjusted after each time step, taking into account in and out-flows.– Interest earned from the cash account balance:

Rst ¼ CB0rs

t;c 8s and if t ¼ 1; ð3:7ÞRs

t ¼ CBst�1rs

t;c 8s; 8t : t > 2: ð3:8Þ

The balance at the government’s cash account generates revenue for the government and the amount is based on the outstanding balanceat the end of previous time step

– Non-negativity:

CBst P 0 8t; s; ð3:9Þ

Xnðs;tÞt;j P 0 8t; s; j: ð3:10Þ

We assume that the government does not allow its cash account to deplete. The amount of bonds issued cannot as well be negative (no-buybacks are allowed).

– Marketability:

Xnðs;tÞt;j 6 ut;j 8t; s; j: ð3:11Þ

The marketability constraint accounts for the demand for government bonds.

3.3. Objective functions

We formulate the PDM problem as a three-criteria model, accounting for the objectives of minimizing cost, market risk and liquidityrisk. Other objectives of debt management such as enhancing the investor base or improving market efficiency are hard to formulate insuch a modelling framework.

3.3.1. Expected costIn our model, we only account for the cost of bonds issued during the decision horizon. The payoffs for existing variable rate bonds are

dependent on the scenario realizations; however, since we do not include any buyback facilities in the model, there is no means to changethe costs arising from the current debt stock. Therefore, their contribution to cost and risk measures will be constant for all alternativestrategies. That is, even though their costs might vary from scenario to scenario, as our cost and risk measures rely on expected values,their contribution calculated over the same scenario tree will be the same for all strategies. On the other hand, varying borrowing require-ments due to existing liabilities are considered in the model, i.e. Ls

t is scenario specific. If the model is extended to include buybacks anddebt exchanges that would allow for decisions on changing the structure of the starting debt portfolio, then the cost definition can be wid-ened to include all liabilities including those fixed before time t = 0.

The expected cost can be calculated by multiplying the cost associated in each scenario with the respective probability

Minz1 ¼Xs2S

psTCs: ð3:12Þ

Here, the cost definition ðTCsÞ is to be determined taking into account the relevance of possible alternative measures. A possible formu-lation is provided in Section 5.1.

3.3.2. Market riskWith regard to market risk, we consider measures that preserve LP solvability (see Mansini et al. (2007) for linear risk measures used in

portfolio optimization models) and use the ‘‘Conditional Cost-at-Risk” (CCaR) concept. This is based on the ‘‘Conditional Value-at-Risk”(CVaR) measure, also referred as the ‘‘mean excess loss” or the ‘‘expected shortfall”, that has emerged as an alternative risk measure asa response to the limitations of VaR.

Despite their popularity, ‘‘Value-at-Risk” models, used to obtain ‘‘a measure of the maximum potential change in value of a portfolio offinancial instruments with a given probability over a pre-set horizon” (RiskMetrics, 1996, p. 6), have undesirable mathematical character-istics such as lack of subadditivity (VaR of a portfolio can be larger than the total of that of individual assets). It is difficult to optimize when itis calculated from scenarios (see Pflug, 2000) and does not provide any information about the level of risk if the confidence level is exceeded.

The portfolio’s CVaR is the expected loss given that the loss is greater than (or equal to) its VaR. In other words, it is the expected value of100a% worst costs over the entire scenario set at a given level of a. Pflug (2000) has shown that CVaR possesses the required properties ofcoherent risk measures in the sense identified by Artzner et al. (1999). Rockafellar and Uryasev (2000) show that CVaR can be efficientlyminimized using linear programming in a scenario-based framework. For a government that is concerned with the level of interest costsrather than the value of the debt portfolio, the CVaR measure can be turned into a Conditional Cost-at-Risk (CCaR) metric.

To compute the CCaR value, we define an auxiliary variable, cv s, which takes positive values when a certain level, VR, is exceeded. In theoptimal solution, the VR value equals the associated VaR level for a given a:

Min z2 ¼ VRþ 1aXs2S

ðpscv sÞ ð3:13Þ


s:t: cv s P TCs � VR 8s; ð3:14Þcv s P 0 8s: ð3:15Þ

Depending on the DM’s concerns or preferences with regard to market risk, it is also possible to use other measures such as the worst-case cost. The government might also have a target level for the debt service expenditures, and any deviation above this level due to marketconditions can be a measure of market risk. If high deviations are more important than lower ones, one can assign different weights todifferent levels of excess cost and invent a piece-wise linear objective function.

3.3.3. Liquidity riskWhile the cost and the market risk can be measured in accounting terms, liquidity or re-financing risk, is associated with the actual debt

service or total cash flows of the government For this risk objective, we again adopt a Conditional ‘‘at-Risk” measure:

Min z3 ¼ PRþ 1aXs2S

ðpscpsÞ ð3:16Þ

s:t: cps P Dst þ Is

t þ Lst � PR 8t; s; ð3:17Þ

cps P 0 8s: ð3:18Þ

This Conditional Payment-at-Risk (CPaR) formulation considers the highest debt service level in a single time step for each scenario andaims to minimize the expected value of 100a% of highest payments over the entire scenario set for a given level of a. In other words, theCPaR metric accounts for the expected highest possible payment level under worst-case scenarios. The governments can compare thisamount with their available resources or with the level of funds they would be able to generate in extreme market conditions to manageliquidity risk.

Alternative formulations are also possible depending on the preferences of governments such as a metric that would quantify the var-iability of cash flows through time to serve as a means for smoothing out debt repayments.

4. MCDM approaches to the PDM problem

MCDM literature contains many examples which combine MCDM tools with the financial decision making process. Zopounidis (1999)and Steuer and Na (2003) present extensive bibliographies on the subject showing that methods like multi-objective/goal programming,outranking relations approaches, Analytical Hierarchical Process (AHP) have been applied to the fields of portfolio analysis, financial plan-ning, budgeting, risk analysis, corporate management, etc.

In our framework, the SP model forms the basis on which the DMs explore the trade-offs between different objectives. We experimentwith our model, which has the following form, for possible achievements of the objectives and discuss the results to demonstrate how sov-ereign DMs can utilize these models in making their decisions:

ðPÞ\Min" z ¼ fz1ðxÞ; z2ðxÞ; z3ðxÞgs:t: x 2 X;

where zi; i ¼ 1;2;3 denote the objective functions, x and X represent the decision variables and the feasible set, respectively. We use quo-tation marks since the minimization of a vector is not a well defined operation. When multiple criteria are considered, it is unusual to have asingle solution that is best for all criteria, and typically, one needs to sacrifice in some criteria in order to improve in others. In general, if wehave p criteria, a solution x 2 X is said to be ‘‘efficient” if there does not exist x0 2 X such that ziðx0Þ 6 ziðxÞ for all i, and ziðx0Þ < ziðxÞ for at leastone i. If x 2 X is efficient then its image in the criterion space fz1ðxÞ; z2ðxÞ; . . . ; zpðxÞg is said to be non-dominated.

4.1. Identifying efficient solutions using the PDM model

To help the DMs explore the outcomes of alternative issuance strategies, we obtain a set of non-dominated solutions, i.e. a portion ofthe efficient frontier (E) by utilizing an achievement scalarizing program (see Steuer (1986, pp. 400–405) for a discussion on achieve-ment scalarizing functions). We first identify an ideal point ðz�Þ in the criterion space where each objective attains its respective min-imum by solving variants of the problem with each individual objective separately. These values then become the targets towardswhich feasible solutions are aimed. To accomplish this, we project the ideal reference point onto the non-dominated surface. We em-ploy a weighted Tchebycheff metric to discover the projected point on the surface, which is defined by the criterion vector that has theshortest weighted Tchebycheff distance to the ideal point. This projection is obtained by solving the following achievement scalarizingprogram:

Min bþ eXp

i¼1

ziðxÞ ð4:1Þ

s:t: b P ki ziðxÞ � z�i� �

i ¼ 1; . . . ; p; ð4:2Þx 2 X; ð4:3Þ

where e is a very small positive constant, which guarantees that the solution obtained is non-dominated and the CCaR and CPaR metrics areproperly computed, i.e. in line with Rockafellar and Uryasev (2000). The approach is illustrated in Fig. 2. By changing the values of ki, i.e. theweights assigned to the Tchebycheff distance and solving the above program iteratively, we end up with a set of different points on the effi-cient surface.

z*

An Efficient Solution

Tchebycheff contours

z1

z2

Fig. 2. Illustration of the Tchebycheff program.


4.2. Visual interactive approach of Korhonen and Laakso

In our continuous objective space, it is not practically possible to identify all alternative efficient solutions. Therefore, getting the DMinvolved in the analysis is critical for assessing preferences and exploring distinct alternative solutions.

For example, in the visual interactive approach of Korhonen and Laakso (1986), the DM can interact with the solution process by spec-ifying a reference direction, d ¼ ðd1; . . . ; dpÞ that indicates the objectives to be improved based on a given solution, h ¼ ðh1; . . . ;hpÞ. The DMthen selects a preferred solution from a set of efficient solutions obtained along direction d. This provides an opportunity to explore parts ofthe non-dominated solution set according to the DM’s choices and constitutes a learning environment. The method is based on the solutionof the following achievement scalarizing program:

Min bþ eXp

i¼1

ziðxÞ ð4:4Þ

s:t: b P ki½ziðxÞ � hi � hdi� i ¼ 1; . . . ; p; ð4:5Þx 2 X; ð4:6Þ

where e is a very small positive constant and h is the step size along direction d. The DM is assisted with a graphical display where thechanges in the objective function values are depicted based on different d and h values. Korhonen and Laakso solve the achievement scalar-izing program for h going from 0 to1. The kinks of the objective function value trajectories occur at h values that correspond to basis changesin the solution of the linear program.

5. An application: Examples from the case of Turkey

We illustrate the ideas discussed in the previous sections for the case of sovereign debt management in Turkey using our generic SPmodel. We assume that the government prepares an annual borrowing program at the beginning of each year deciding on the issuancestrategy for the following four quarters. This decision is then to be revised at the beginning of the next year.

Our model covers a period of 3 years, in line with the central government’s Medium Term Fiscal Plan (MTFP). The short-horizon of theTurkish case limits the model’s capabilities in this application, especially in consideration of the liquidity risk objective. The model tends tochoose longer-term bonds that mature beyond the decision horizon when this objective is minimized. However, it was not possible to ob-tain longer-term fiscal assumptions.

As a simplification, we assume that the existing forward liabilities forecast as of that date are scenario-independent ðLst ¼ Lt 8sÞ as well

as the non-debt cash flows of the government ðPSst ¼ PSt 8sÞ. The parameters Lt are based on the available data at the Turkish Treasury web-

site (www.hazine.gov.tr) as of December 2005 and PSt values are in line with the fiscal policy assumptions in the MTFP 2006–2008 detailsof which can be found at the Official Gazette (no. 25863) dated 02.07.2005 (in Turkish). The cash account of the Treasury, held at the Cen-tral Bank of Turkey (CBT), is not remunerated due to the current legal framework. Therefore, the revenues on the cash account, Rs

t , are all setto zero. The starting cash balance is assumed as zero for simplification purposes. The numeraire currency is the Turkish Lira (TRY).

Our model presents a selection of seven different kinds of bonds: four of which are TRY denominated zero-coupon bonds with matu-rities of 3, 6, 12 and 18 months. We also include a 3 year TRY fixed-rate coupon bond (3YTF), a 3 year TRY variable rate coupon bond in-dexed to 6 month Treasury Bill yields (3YTV), and a 3 year USD denominated fixed-rate coupon bond (3YUF). All coupons are semi-annuallyredeemed. The variable rate bond is assumed to be issued with a fixed spread over the prevailing 6-month interest rate. We include twomarketability constraints: we assume that in one quarter the amount of 3-month bills the Treasury can issue is capped at 10 billion TRYand the market availability for the USD denominated bonds is 3 billion USD per quarter, considering the size and preferences of the lendersin those segments of the debt market.

5.1. The SP model

In our application, we evaluate interest costs in accrual terms so that the interest payments can be attributed to the periods they aregenerated. However, for Turkey, the interest charge is not the sole source of cost for the government. There are bonds issued in other

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currencies and any increase in debt repayments, including principal and interest, due to changes in the exchange rates, adds up to the costof debt. Thus, our cost definition covers not only the interest charges to accrue during the planning period but also the change in the marketvalue of foreign currency denominated debt (see Turkish Treasury, 2004, p. 59). Foreign currency linked debt that mature beyond the deci-sion horizon are marked to market value at the end of the model period.

As a result, we include the following accrual cost measure in our model:

– Total accrued cost at T in scenario s:

2 Thecarried

As ¼Xj2J

XT�1

s¼T�mjþ1

Xnðs;sÞs;j

esT;j

ess;j� 1

!þXj2J1

XT�1

s¼T�mjþ1

Xnðs;sÞs;j

esT;j

ess;j

T � smj

rss;j

� �þXj2J2

XT�1

s¼T�mjþ1

Xnðs;sÞs;j ð1� Ys;T;jÞrs

T�cjþ1;j �es

T;j

ess;j� 1cj

þXj2J3

XT�1

s¼T�mjþ1

Xnðs;sÞs;j ð1� Ys;T;jÞrs

s;j �es

T;j

ess;j� 1cj8t; s; ð5:1Þ

where cj is 2, since the regular coupon period for Treasury bonds is assumed as 6 months.

The accrued cost is calculated for bonds and bills that have not yet matured at time T. It consists of the changes in the value of the bondsdue to movements in the exchange rate and the interest that has accumulated on a bond since its issuance (for zero-coupon bonds) or itslast coupon payment (for coupon bonds) up to time T. We compute the accrued interest rate by multiplying the effective interest rate bythe fraction of days that have passed since the last the coupon payment to the coupon period. We then adopt the following cost definition,which is the sum of actual interest payments made in cash and interest costs accrued:

TCs ¼XT

t¼1

Ist þ Fs

t þ As� �: ð5:2Þ

As measures of market and liquidity risks, we use the CCaR and CPaR metrics, respectively. In this application we calculate these valuesat the 90th percentile of their respective distributions (i.e. a ¼ 10%). Thus the model becomes:

ðPÞ\Min" z ¼ fz1; z2; z3gsubject to ð3:1Þ—ð3:18Þ; ð5:1Þ; ð5:2Þ:

5.2. Scenario tree generation

The scenarios for the SP model were generated by a modified version of one of the macroeconomic simulation models of the TurkishTreasury, a VAR time series model containing the short and medium term local interest rate, the USD/TRY parity, the inflation rate (Con-sumer Price Index) and the Treasury’s funding rate in USD denominated issues.2 The VAR approach, as illustrated in the following equationwith l lags, models the co-movement of selected variables as functions of lagged values of their own and others:

Yt ¼ C þXl

i¼1

AiYt�i þ et ; ð5:3Þ

where Yt is a vector of n variables, C is an n � 1 vector of constants, Ai ði ¼ 1; . . . ; lÞ are n � n matrices of coefficients and et is the vector oferror terms with the following properties:

EðetÞ ¼ 0 for all t; ð5:4Þ

Eðes � e0tÞ ¼X for s ¼ t;

0 otherwise;

�ð5:5Þ

where X is the variance/covariance matrix assumed to be positive definite.The parameters of the time series model are estimated based on a monthly data set from 2001 to 2005. We would have preferred to

work with a larger dataset to reduce the impact of estimation errors, however, due to regime shifts in the Turkish economy, data before2001 are highly unstable. The 3 and 12 months interest rates reflect the rates that emerged in Treasury auctions for securities in thosematurities. The inflation rate is the monthly rate of change in the 1994 based Consumer Price Index. The USD/TRY exchange rate is themonthly average calculated over daily figures announced by the CBT. We take the mid point of the official purchase and sales rates ofthe bank. At the end of 2005, annualized interest rates for 3 and 12 months stood at a level of 14.2% and 14.1%, respectively, while theaverage annual interest rate for 1-year USD denominated bonds was about 4.8%. The monthly average value of the USD in December2005 was 1.35 TRY and the annual inflation rate for 2005 was recorded as 7.7%.

We create random scenarios for our stochastic variables by making use of the VAR model via imposing correlated random shocksthrough the error term. The random shocks are achieved by drawing five random variables from the standard normal distribution. Tobe able to create scenarios consistent with the empirical co-movements of our macroeconomic variables, we make use of the Choleskydecomposition of the covariance vector X. Thus, we first find a matrix F such that

F 0 � F ¼ X: ð5:6Þ

parameters of the model cannot be disclosed due to the confidentiality reasons. However, estimation of the model parameters is a straightforward process which can beout by using commercially available econometrics packages.

Table 1Stability results (based on 50 independent replications).

Scenario tree Objective: min cost (billion TRY) Objective: min CCaR (billion TRY)

Average St. Dev. Average St. Dev.

10 � 10 � 10 50.9 4.9 83.5 10.930 � 10 � 10 52.3 3.7 88.7 7.140 � 10 � 10 52.9 3.0 89.6 5.380 � 10 � 10 52.7 2.5 90.7 4.2


We then transform the 5 � 1 vector that contains the standard normal random variables by multiplying it by F and impose the resultingvector to the VAR model as a random shock. The monthly paths created by the model are converted to quarterly figures by taking averagesover 3-month periods. Once we obtain simulated values for the short and 1 year interest rate, we compute the yields for maturities in-be-tween by linear interpolation. For maturities longer than a year, a flat yield curve is assumed. That is, interest rates are not allowed to varywith maturity, but taken constant for maturities over a year. The scenario generation process is implemented on MATLAB 6.5.

5.3. Assessment of scenario tree generation

Since our scenario generation method is stochastic, it generates different scenarios on different runs. We need to ensure that solving theSP model on different trees generated by the same method yields similar optimal values. Kaut and Wallace (2007) focus on this issue anddiscuss the evaluation of the quality of scenario generation methods, defining some minimal requirements. Specifically, they propose twomeasures to test the suitability of a certain generation method for a given SP model: one related with the robustness of the tree generator(stability) and the other regarding the bias it contains.

By what they define as ‘‘in-sample” stability, the authors propose to check that the optimal objective values obtained from differentscenario tree instances are approximately identical. While ‘‘in-sample stability” is concerned with the variability of the optimal objectivefunction value, ‘‘out-of-sample stability” is related with the performance of the optimal solutions in the decision space. In this regard, weneed to test whether solutions obtained on different scenario trees yield similar results when plugged in the real problem. However, this isnot always possible since we may not have full information about the actual distributions that drive our stochastic variables. To ensure thatthe scenario generation method contains no bias, we need to compare the optimal values in the scenario-based problem to that of the trueproblem and see whether or not they are close to each other. This is again impossible in most cases since it requires solving the true prob-lem optimally. As a proxy, Kaut and Wallace (2007) recommend the employment of a larger ‘‘reference tree” that is believed to have a bet-ter representation of the true stochastic process and use the results as a benchmark to test for a possible bias.

To test the ‘‘in-sample stability” of our method, we solved our SP model based on different instances obtained from the same scenariotree generator, trying to optimize our objective functions separately. We have generated 50 independent and identically distributed sce-nario trees and solved our model separately on these to see how the optimal value of each objective function varies due to the stochasticityincluded in our modelling framework. The model is implemented on GAMS 2.0 using CPLEX as the linear programming solver.

Table 1 depicts the averages and standard deviations of optimal cost and market risk values when models are solved on trees of differentsizes. The results are all based on a three-stage model, only the numbers of branches differ. The notation of 10 � 10 � 10 corresponds to athree-stage tree with 10 branches from each node in each stage. Thus, in the final stage there are 1000 branches. The results in Table 1 showthat we need to increase the dimensionality to achieve a significant reduction in the variations of the optimal objective function values.However, that adds to the complexity of the model, i.e. causes an increase in solution times.

The 3000-branch model is solved in the order of 5–10 minutes on a Pentium 4, 728 MB RAM PC when a single objective is minimized.Solution times can be longer when we implement multi-objective approaches (the problem size reaches around 500,000 rows and col-umns). For our illustrative model, we are content with a 3000-branch scenario tree. As regards to measuring the ‘‘bias” in our model,the optimal solutions of the 3000 and 8000 branch trees are comparable.

5.4. Experiments

The figures in Table 1 for the cost measure depict the level of interest expenditures due to issuances within the 3-year planning period,while the values for the CCaR measure indicate the interest expenditures expected under the worst-case scenarios. The table points outthat while the expected cost is estimated to be around 50 billion TRY for a 3 year period, the government can expect to incur up to 90 billionif the market conditions deteriorate. This potential increase is very significant considering that the total level of expenditures in TurkishCentral Government Budget in 2006 was around 178 billion TRY.

We now provide the optimal borrowing strategies generated by our model when each decision criterion is optimized separately. Table 2includes the issuance policy generated for four quarters of year one with respect to each objective. Since decisions for years 2 and 3 arescenario dependent, we only include the bonds to be issued in the first year. As expected, since our VAR model generally generated sce-narios with declining interest rates in line with the macroeconomic environment in Turkey during the period of analysis, the model choosesshort-term or variable rate securities for the first year when expected cost is minimized. Short-term rates are also lower on average. How-ever, as far as the market risk is concerned, funding is raised through a combination of short-term and long-term fixed-rate bonds. Themodel aims to extend maturities to minimize liquidity risk. We acknowledge that this is mainly caused by the short decision horizon inthis application. The similarity in the issuance amounts of USD denominated bonds is due to the marketability constraint we include inthis model for this type of securities.

The model is then used to obtain a set of points on the efficient frontier as depicted in Fig. 3. The graph depicts the ranges of objectivefunction values and shows how the government’s preferences affect these levels. The steep nature of the efficient surface in z3, liquidityrisk, dimension suggests that we can obtain a sizeable improvement in the liquidity risk objective with relatively small sacrifices in thelevels of cost and market risk.

Table 2Optimal borrowing strategy for each objective (initial stage decisions).

Fig. 3. A representation of the efficient frontier (billion TRY).

52 53 54 55 56 57 58 59

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0theta

Cos

t

40

50

60

70

80

90

Mar

ket R

isk

Liq

. Ris

k

Liq. Risk

Cost

Market Risk

Fig. 4. Criterion value trajectories in reducing cost (billion TRY).


Let us now assume that the DMs in our problem would like to explore the areas where the level of liquidity risk is at its minimum toachieve a cost reduction. To this end, we employ the visual interactive approach of Korhonen and Laakso (1986). Fig. 4 displays the effect ofaltering the step size ðhÞ in moving from the point where liquidity risk is at its lowest level in a direction towards the minimum of the costobjective, i.e. d ¼ ð�3:6;8:8;94:0Þ, with k ¼ ð1=3;1=3;1=3Þ. The trajectories indicate that in order to achieve a reduction in cost from itscurrent level, the DM has to admit significant increases in the level of liquidity risk, which means an important degree of concentrationin the government’s repayments, i.e. the piling up of bond redemptions within a single period. The apparent insensitivity of market riskis due to representing both risk types on the same scale. The market risk does, in fact, vary sizeably within its own range.

Let us assume that despite the increase in the level of liquidity risk, the DM likes the solution at h ¼ 0:7 ðz1 ¼ 52:9; z2 ¼ 86:1; z3 ¼ 83:7Þamong the solutions in Fig. 4. Table 3 contains the corresponding issuance strategy, which contains a mixture of short-term and long-term

Table 3Optimal borrowing strategy for the selected solution (initial stage decisions).

Quarter Bond

3m bill (Bil. TRY) 6m bill (Bil. TRY) 12m bill (Bil. TRY) 18m bill (Bil. TRY) 3YTV (Bil. TRY) 3YTF (Bil. TRY) 3YUF (Bil. TRY)

1 10.0 19.8 2.22 40.9 2.23 10.0 43.84 78.0

52

53

54

0 0.1 0.2 0.3 0.4 0.5theta

Cos

t

78

82

86

90

Mar

ket R

isk

Liq

. Ris

k

Liq. Risk

Cost

Market Risk

Fig. 5. Criterion value trajectories in reducing market risk (billion TRY).


bonds for the first year. The short-term securities provide for the chance to renew debt at expectedly lower levels of interest, while 3 yearfixed-rate bonds help the debt management keep liquidity risk at acceptable levels.

Changing the direction (d) and ðhÞ interactively with the DMs will produce different trajectories on which the DMs can analyze andexperiment with their decisions. Let us now suppose the DMs would like to explore solutions on the direction of reducing the market risk.Fig. 5 plots the trajectories assuming that direction change occurs at h ¼ 0:7 in Fig. 4 (h ¼ 0 corresponds to h ¼ 0:7 in the previous figure) inan attempt to reduce the market risk ðz2Þ, for example with d ¼ ð2:7;�4:7;17:3Þ. An analysis of the figure reveals that in return for a onebillion TRY increase in the expected cost of debt, we can obtain savings in the order of three billion TRY in the levels of both risks. A debtmanager who has a lower risk appetite, especially in terms of market risk, may choose to accept this increase in cost to contain the chargeof debt under volatile conditions.

These experiments can be repeated with many different starting points and directions to be explored. Based on the DM preferences, wecan identify various combinations of short- and long-term, fixed and floating rate bonds that will guide the government in meeting its over-all cost and risk objectives.

6. Conclusions

The existing methods employed for public debt strategy analysis rely on enumeration of costs and risks associated with given financingstrategies under various different macroeconomic scenarios. Since, these methods are limited with user-supplied alternatives; they do notguarantee efficient solutions. In this paper, we formulate the PDM strategy problem as a multi-objective stochastic programming model. Inour analysis, we first identify several efficient solutions to understand the ranges of objective functions and to obtain an overall view of thetrade-offs using the developed SP model. We then experiment in different regions of the efficient surface based on DM preferences untilobtaining a satisfying solution using an interactive procedure. The procedure relies on DM involvement and provides a graphical represen-tation of the trajectories of the objective function values. This provides for assessing preferences, exploring distinct alternative solutionsand guiding the DMs in making debt strategy solutions.

Our experiments with the case of Turkey show that this framework can be of practical use in a real setting. In our example, the modelsuggests issuing short-term bonds to minimize expected cost and longer-term fixed-rate securities to decrease the level of market risk asexpected. The DMs can solve the model attaching different weights to the objectives and gain insights about the resulting debt strategycompositions. With the help of such a quantitative tool, the sovereign debt issuers will have the means to see the effects of different riskand cost preferences on the debt issuance policy. Our generic model can serve for different country characteristics with small modifica-tions, i.e. by changing the number of periods, including different types of instruments, incorporating the relevant cost and risk measures.

Experimentation on the model can help the assessment of the DMs’ preferences with regard to associated criteria, which are not onlycrucial in debt management policies, but also in other financial decisions of the government. Formulation of robust debt management strat-egies is critical for preparing a sound government budget which has to account for expected expenditures and potential increases in theirlevel. Governments must meet their payment commitments at all times, and therefore many countries include buffers in their budgets tocope with the effects of fluctuations in the economy. Our PDM model can also be used to assess the level of contingencies in the nationalbudget with regard to debt service and interest costs.

Disclosing mainline results from the modelling work to general public opinion can also help the debt management offices conveytheir strategies to market participants and inform stakeholders, such as tax payers, about the objectives of the public debt managementpolicy. Similar models can also be employed by independent organizations that conduct research on the macroeconomic policies of thegovernment. This will provide them the means to comment on the actions and plans of the government in raising debt on behalf of thecitizens.


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http://www.riksgalden.se

http://www.riksgalden.se

http://www.nationalbanken.dk

http://www.dmo.gov.uk

http://www.hazine.gov.tr

a multi-objective multi-period stochastic programming model for public debt management

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