## 1. Introduction

The 2009/2010 European fiscal crisis has brought renewed global attention to the challenges involved in public debt management. These challenges entail determining the optimal size, the sustainability, the optimal structure, and the risk management of a sovereign debt portfolio. In their intriguing study of benchmark government debt portfolios, Hahm & Kim (2003) assert that the ultimate objective of sovereign debt management is to minimize the long-term cost of debt given the trade-off between expected debt-service costs and the risks associated with various borrowing strategies (meaning the composition of government bond issuance by maturity) to finance borrowing requirements. Optimizing the cost–risk trade-off refers to the level where debt-service costs cannot be reduced further without incurring higher risk, which might result when the distribution of shorter-term yields (or borrowing costs) has a lower mean and a larger standard deviation than those of longer-term yields. This has typically been the case for US yield curves.

To assess alternative sovereign borrowing strategies within a cost–risk analytical framework, the costs and risks associated with a specific strategy have to be defined. Towe *et al.* (2009) focus on two specific cost measures; namely, annual interest payment-to-GDP and nominal stock of debt-to-GDP, and define risk as the maximum increase in cost, given a particular macroeconomic scenario, relative to the baseline. As they note, however, their framework for measuring risk is deterministic rather than stochastic because of the severe data limitations in the low income countries in their sample.

Pick & Anthony (2007) propose a small-scale, reduced-form macroeconomic model and a Nelson–Siegel (1987) type yield curve model. Similar to Towe *et al.* (2009), they define the cost of debt in any given period as the ratio of the sum of all nominal coupon payments to GDP. They argue that the debt–cost ratio provides a better indication of the government’s sustainable investment rule that compares public sector net debt to nominal GDP. As for risk measures, Pick and Anthony (2007) use two statistics: volatility of the debt–cost ratio and debt-service cost-at-risk (CaR), which is defined as the upper 95th percentile of the debt–cost ratio distribution. They use a two-step procedure to estimate the Nelson–Siegel-type model of the term structure of interest rates: they estimate the cross-sectional Nelson–Siegel model for each period in the first step and build the time-series model for the factor sensitivity parameters that determines the level and shape of the yield curve in the second step. The two-step procedure is a practical method that is often used in other applications, such as the strategic asset allocation between stocks and bonds. Nevertheless, its theoretical validity is open to question.

Hahm & Kim (2003) examine methods of determining the optimal benchmark government debt portfolio for Korea using the average debt–service cost ratio, defined as a percentage of the outstanding balance and the volatility (and CaR) of the debt–service costs as measures for the cost and risk of servicing debt. For the yield curve simulations, they use daily Korean Government bond yields from January 1996 to March 2000 and use bootstrapping due to the unavailability of reliable Treasury spot rates.^{1} They find a clear negative cost–risk trade-off among borrowing strategies. Rhee (2005) applies a similar method to analyze Nepal’s management of its public debt.

When applying a cost–risk analytical framework, it is important to keep track of governments’ borrowing needs to pay for interest, to roll over maturing bonds and to finance the primary budget deficit requirements. The temporal paths of the term structure of interest rates play a central role in determining the future debt–service cost of a given borrowing strategy. Despite being a main source of uncertainty, however, few attempts have been made to incorporate more rigorous term structure models of interest rates into a cost–risk model and to examine the effects of theoretical no-arbitrage restrictions on the choice of optimal borrowing strategy. Furthermore, it is not clear whether the negative cost–risk trade-off documented in the published literature still holds in countries where the distributional characteristics of yield curves are more complex than others such as those of the USA.

One purpose of the present paper is to extend the work of Hahm & Kim (2003) by introducing more elaborate term structure simulation models to debt management analysis. This paper focuses on two models: the dynamic Nelson–Siegel (DNS) model proposed by Diebold & Li (2006) and the arbitrage-free Nelson–Siegel (AFNS) model proposed by Christensen *et al.* (2008a,b). The DNS model is a time-series model that relates three latent factors of the term structure of interest rates, which are interpreted as the level, the slope, and the curvature factors in the literature, to the observed spot rates in a state-space model framework. Maximum likelihood (ML) estimates and simulations of future interest rates can be conveniently handled in this framework.^{2}

The AFNS model is a special version of Duffie & Kan (1996) affine term structure models, yet still preserves the parsimonious functional form of the Nelson–Siegel model and can be handled conveniently in the state-space model framework.^{3} The empirical evidence provided by Cha & Kim (2010) using the KTB market data suggests that the superior out-of-sample performance of one model to the other is not conclusive and is sensitive to the choice of sample, such as the inclusion or exclusion of the 2008 financial crisis. Rather than decide whether a theoretical or time-series based model better describes the KTB market, however, the present paper will examine both and compare the effects of different types of term structure models on the determination of optimal sovereign debt strategy.

Cost–risk analytical tools can also be applied to address such issues as borrowing strategy aimed at developing the domestic debt market by rolling over certain portions of maturing debt into new domestic instruments. For example, the Korean Government issues KTB mainly at maturities of 3, 5, 10, and 20 years.^{4} Because large financial institutions tend to hold them to maturity, the liquidity in the market is relatively low. Moreover, this also makes the informational content of the term structure of interest rates, particularly at the shorter end, limited, because the term structure is usually computed from on-the-run bonds of longer maturity. Therefore, another purpose of this paper is to discuss the issue of introducing new debt instruments into the KTB market and to quantify the potential gains of doing so.

This paper is organized as follows. Section 2 introduces the cost–risk analytical tool. The method is parallel to that of Hahm & Kim (2003) and Rhee (2005). Section 3 briefly introduces the DNS and AFNS models. Section 4 presents the ML estimates of these models and compares their performance. Section 4 also presents the results from implementing a cost–risk analytic tool with different term structure models, along with the results from introducing new 1-year zero coupon bonds to the domestic KTB market. Section 5 summarizes and concludes the paper.