Assessing Sovereign Debt Strategies Under Alternative Term Structure Models

Authors


  • Acknowledgments: Myung-Jig Kim’s participation in this study is supported, in part, by the support from the Hanyang Economic Research Institute. The authors are grateful to Editor Bong-Soo Lee, two anonymous referees, and Hyung-Soo Park for their helpful comments and to Sang-Heon Lee and Joong-Hyun Lee for programming assistance. The authors are also grateful to the Ministry of Strategy and Finance and KIS Pricing for providing us with the data. Any remaining errors are entirely those of the authors.

Corresponding author: Hangyong Lee, 17 Haengdang-dong, Seongdong-gu, College of Economics and Finance, Hanyang University, Seoul 133-791, Korea. Tel: +82-2-2220-1030, Fax: +82-2-2293-1787, email: hl306@hanyang.ac.kr.

Abstract

This paper examines the theoretical restrictions on alternative term structure models in assessing sovereign borrowing strategies. Our approach draws upon Hahm & Kim’s (2003) cost–risk analytic model of sovereign debt management within a mean–variance framework. To explore the effects of different interest rate modeling strategies on government debt portfolio selection, two models are considered; namely, the time series-based dynamic Nelson–Siegel (DNS) model proposed by Diebold & Li (2006) and the DNS model with arbitrage-free restrictions proposed by Christensen et al. (2008a). Using monthly spot rates for 12 maturities of nominal Korea Treasury Bonds (KTB) from September 2000 to November 2008, the present paper finds that a more generic term structure model, such as the DNS model, performs better in terms of smaller out-of-sample root mean squared errors at different forecast horizons. However, looking at the goodness-of-fit, the size of pricing errors and the magnitude of the root mean squared errors suggests that both models are reasonable representations of KTB spot curves. For the actual KTB position as of December 2007, the present paper shows that the 95% cost-at-risk level might be able to trim as much as 5–6% by rebalancing the portfolio. Furthermore, DNS models, both with and without no-arbitrage restrictions, produce a consistent assessment of different strategies. This paper also shows that introducing new short-term domestic debt instruments, such as 1-year zero coupon KTB, would benefit government in terms of lowering both the average debt-service cost and the 95% cost-at-risk. However, it is found that such benefits might dissipate if the issuance weights for such instruments exceed a certain level, which is approximately 4% of the position in the case of Korea.

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.

2. Cost–Risk Analytic Model of Alternative Borrowing Strategies

The debt-service cost of a debt portfolio in any given period depends upon debt issuance requirements and the temporal paths of interest rates over the period. To assess alternative borrowing strategies in such a cost–risk analytical framework, this paper adopts the approach of Hahm & Kim (2003). More specifically, suppose that the debt management horizon (T) is 20 quarters (or years) and the vector of the projected quarterly primary budget deficit Pt is given as follows:

image(1)

Assume that interest and principal payments on maturing bonds are financed by issuing new bonds. Assume further that there are only four debt instruments with maturities, say at 3, 5, 10, and 20 years. Let inline image denote the time-invariant debt strategy, which is the portion of outstanding bonds at different maturities over the horizon under consideration, and Zt denote the vector of the outstanding balance of debt at the end of period t:

image(2)

The notation inline image indicates J-year bond issued at time t − 4J + 1. For a given borrowing strategy w, the government’s total financing requirement for the next year, denoted by Bt+1, can be represented as the sum of the primary budget deficit Pt+1 and the refinancing amount αYt+1, where α ∈ [0, 1] is the fraction of refinancing and Yt+1 is the sum of the principal of maturing debt and interest costs to be paid in period + 1. Assuming 100% of maturing debt principal and interest expenses are refinanced in period + 1 is equivalent to setting α equal to 1. Denoting inline image as the interest rate on the j-year bond issued in period t, the government’s total financing requirement in period t + 1 becomes:

image(3)

where

image(4)

For instance, inline image and inline image in equation (4) represent the amount of maturing 3-year bonds in period t + 1 and that of newly issued 3-year bonds in period t, respectively.

Under a specific time-invariant borrowing strategy w, the amount of new issuance for bonds of four different maturities in period t + 1 can be computed as follows:

image(5)

The temporal paths of total financing requirements and the actual issuance amount under a given borrowing strategy w can be computed by repeating the calculation of equations (3) and (4) for periods + 2, + 3, …, + 20. If this process is repeated, say 1000 times, using a simulated term structure of interest rates for specific borrowing strategy w, one can derive the distribution of the borrowing costs in the terminal year; namely, the distribution of the sum of inline image and inline image when the assessment horizon is 5 years. The mean (μ) of this distribution is regarded as a measure for cost; that is, the expected debt–service cost.5 The corresponding risk can be measured in terms of its standard deviation (σ) or, alternatively, by the 95% significance level of the CaR, which is defined as:

image(6)

If the distribution of inline image is approximately normal, two measures would coincide. When the distribution is not normal, the 95% CaR can be measured by the 95th percentile of the simulated distribution. To determine which borrowing strategy is optimal, one has to define the government’s objective function. Typically, the extent to which the target duration and/or target CaR is achieved is used to define the objective function. As an illustration, this paper considers the minimum CaR criterion as a metric for optimality.

3. Term Structure Models of Treasury Bond Yields

3.1. Dynamic Nelson–Siegel Model

The Diebold & Li (2006) DNS model draws upon the Nelson & Siegel (1987) model, which provides a parsimonious representation of the term structure of interest rates at a particular period. More specifically, the Nelson & Siegel (1987) model fits a nonlinear curve at a particular point in time assuming the following specification:

image(7)

where y(τ) denotes spot rates at different maturities, τ. α, β, γ, and λ in equation (7) are parameters. In particular α, β, and γ are interpreted as the level, slope, and curvature factors of the term structure of interest rates, with factor loadings (or sensitivities) of 1, inline image and inline image respectively.

The dynamic Nelson–Siegel model extends the original Nelson–Siegel model by allowing the first three parameters, α, β and γ, to vary over time. The dynamics of the three factors are usually assumed to follow an autoregressive process. The unobserved factor processes and their association with the observed yield curves constitute a so-called state-space model:

State-space representation of the dynamic Nelson–Siegel model

Measurement equation:

image(8)
image(9)

State equation:

image(10)
image(11)

Denoting inline image, and inline image, equations (8) and (10) can be rewritten as:

image(12)
image(13)

where I(3) denotes a (3 × 3) identity matrix. Assuming that the disturbance terms in both equations are normally distributed, but uncorrelated with each other, this system of equations in (12) and (13) can be estimated by using the ML estimation via a Kalman filter.

3.2. Arbitrage-Free Nelson–Siegel Model

To derive a theoretical term structure model that imposes no-arbitrage conditions but still adheres to the structure of the DNS model, Christensen et al. (2008a,b) also assume three factors and specify the following factor process of Xt under risk neutral probability (Q-measure), which is a restricted version of the more general continuous affine term structure model of interest rates of Duffie & Kan (1996):

image(14)

Wt denotes the Wiener process. Equation (14) can be expressed using compact notation as follows:

image(15)

Assuming that the short rate rt is affine with respect to the unknown factors, that is,

image(16)

where ρ1 is a vector of ones, it is known that the theoretical price of a riskless zero coupon bond with remaining maturity inline image, denoted by Pt(τ), is determined by:

image(17)

where Bt(τ) and Ct(τ)′ denote solutions to particular ordinary differential equations, as described in Duffie & Kan (1996). Spot rates, which are computed as inline image, are then given by:

image(18)

Restricting ρ1 = (1, 1, 0)′ and μQ = 0 and assuming the factor process given by equation (14), Christensen et al. (2008a) show that the affine price in equation (17) can be expressed as follows:

image(19)

Namely, there is a difference between the DNS model in equation (12) and the theoretical price under the no-arbitrage restrictions above by the yield adjustment term, Ct (τ)/τ. Christensen et al. (2008a) also show that the yield adjustment term has the following form:

image(20)

Under the assumption of a diagonal volatility matrix, that is, inline imagej = 1, 2, 3, the yield adjustment term in equation (20) has the following closed solution:

image(21)

To represent AFNS in state-space form, note that the factor process under a risk neutral Q measure as expressed in equation (15) can be rewritten under a physical P measure using the change of measure technique; that is, inline image

image(22)

Therefore, the state-space representation of the AFNS model can be written as follows:

State-space representation of the dynamic Nelson–Siegel model

Measurement equation:

image(23)
image(24)

State equation:

image(25)

In order to make equation (25) operable, it can be discretized as follows:

image(26)
image

Again, the system of equations in (23) and (26) can be estimated by using the ML estimation via a Kalman filter.

4. Empirical Results

4.1. Data

Data used in the paper consist of the outstanding balance of nominal KTB at maturities of 3, 5, 10, and 20 years as of 31 December 2007, which amounts to KRW227.37tn.6 The position is summarized in Table 1. New issuance during 2007 amounted to KRW48.3tn, with the issuance weights for each maturity, respectively, being 22.14, 39.49, 28.25, and 4.89%. The duration of the initial portfolio is 3.84. Setting the outstanding balance as of 31 December 2007 as the initial portfolio, the simulation of borrowing strategies begins from the first quarter of 2008 until the last quarter of 2012. The 2007 issuance weights, wref = (22.1%, 39.5%, 28.3%, 10.1%)′, regarded as the reference borrowing strategy and alternative strategies, are then constructed by allowing each of four issuance weights to change within a ±4% boundary, yielding 489 active strategies.

Table 1.   Summary of 2007 nominal Korea Treasury bonds (KTB) position
Data are from the Ministry of Strategy and Finance. Figures are expressed in trillion KRW. Figures in parentheses are percentages. The magnitude of the new issuance of consumer price index-linked index bonds in 2007, not shown in the table, amounts to KRW1.9tn. aDuration figures are the authors’ calculations.
Maturity of nominal KTBNew issuance during 2007New issuance during 2007Q4As of 31 December 2007
Outstanding balanceDurationa
3 year10.68 (22.1)1.52 (17.7)34.11 (15.0)1.23
5 year19.06 (39.5)3.75 (43.7)98.78 (43.4)2.34
10 year13.63 (28.3)2.72 (31.7)83.90 (36.9)5.69
20 year4.89 (10.1)0.60 (7.0)10.58 (4.7)12.09
Total48.3 (100.0)8.59 (100.0)227.37 (100.0)3.84

There is an additional institutional feature that has to be considered in the quarterly simulation as opposed to the annual simulation conducted by Hahm & Kim (2003). In order to invigorate the KTB market, the Korean Government introduced fungible issues in 2000, meaning that additionally issued KTB have the same terms applied to them as KTB issued within the previous 6-month or 12-month period, although the redemption yield is likely to be different. New 3-year KTB are issued on 10 June and 10 December each year and 5-year KTB are issued on 10 March and 10 September each year. Hence, 3-year KTB issued between 10 June and 10 December are treated the same as those issued on 10 June. New 10-year and 20-year KTB are issued, respectively, on 10 June and 10 December. This institutional feature is incorporated into the cost–risk simulations. When assessing the effects of introducing imaginary 1-year zero coupon KTB into the market, it is assumed that they are issued every quarter without a fungible issue restriction.

To model the term structure of interest rates, monthly KTB spot rates are collected from September 2000 to November 2008 for 12 maturities: 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 5, 7, 10, and 15 years. Spot rates of 20-year KTB bonds are also available, but they are not included in the comparison of the out-of-sample performance of the two-term structure models because the 20-year KTB was issued for the first time only in 2006.7

Descriptive statistics of the data are summarized in Table 2. It reveals that the spot curves are, on average, upward sloping and that the spot rates at the short end are slightly less volatile that those at the long end, in contrast to US spot curves.8 Time-series plots of KTB spot rates are depicted in Figure 1. The figure shows that the sample extends over phases of both decreasing and increasing fluctuation in interest rates. The sudden reversal of the trend during the last 3 months in the plots reflects the lowered target rate by the central bank following the onset of the 2008 financial crisis after the collapse of Lehman Brothers in September of that year.

Table 2.   Descriptive statistics of monthly Korea Treasury bonds (KTB) spot rates
Statistics are based on monthly KTB spot rates collected from September 2000 to November 2008. Source: KIS Pricing.
Maturity in years (τ)MeanMedianStandard deviation
0.250.0450.0450.0076
0.50.0470.0460.0077
0.750.0480.0480.0068
10.0480.0480.0080
1.50.0500.0490.0084
20.0510.0490.0087
2.50.0510.0500.0091
30.0520.0500.0092
50.0540.0520.0100
70.0560.0530.0100
100.0570.0540.0100
150.0560.0530.0100
Figure 1.

 Historical Korea Treasury bonds (KTB) spot rates by maturity.

Data are KTB spot rates for 0.25-, 0.5-, 0.75-, 1-, 1.5-, 2-, 2.5-, 3-, 5-, 7-, 10-, and 15-year maturities from September 2000 to November 2008. Data are obtained from KIS Pricing. The downward shifts in the curves near the end of the sample period reflect the government’s policy reactions to the 2008 financial crisis.

4.2. Maximum Likelihood Estimations of the Two-term Structure Models

Using historical KTB spot rates from September 2000 to November 2008 for 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 5, 7, 10, and 15-year maturities, the DNS and ANFS models are estimated. The state vector and its covariance matrix for the DNS model are initialized by:

image(27)

For the AFNS model, the initial state vector and its covariance matrix are set equal to the conditional mean and the covariance of the factor process:

image(28)

The ML estimates for both models are reported in Table 3. The significance and signs of the parameters for both models are very similar to those reported in the previous published literature. Both models contain 20 parameters, but the sample log likelihood of the DNS model was slightly larger than that of the AFNS model. In contrast, the pricing error, which is computed by taking the average of pricing errors (i.e. the mean of estimated inline image, j = 0.25, 0.5, …, 15) turns out to be very similar, or approximately 6.5 basis points for both models. Table 3 also shows that the DNS model fits 10-year spot rates most closely and 9-month spot rates for the AFNS model, as indicated by the insignificant estimates for the corresponding inline image.

Table 3.   Maximum likelihood estimates of dynamic Nelson–Siegel (DNS) and arbitrage-free Nelson–Siegel (AFNS) models
The sample period is from September 2000 to November 2008. DNS model, equations (12) and (13) estimates: inline imageinline image AFNS model, equations (23)–(25), estimates: inline imageinline image or in discrete form, inline imageinline image Spot rates, inline image are expressed in decimal form in the estimation.
DNS modelAFNS model
ParameterEstimatet-valueParameterEstimatet-value
φ0.976046.385inline image0.14850.995
φ0.901520.312inline image1.15112.180
φ0.70769.215inline image3.02723.245
inline image8.53 × 10−66.037σ110.007622.383
inline image1.46 × 10−56.590σ220.013211.631
inline image9.09 × 10−56.038σ330.032914.667
μ0.06666.898inline image0.07376.032
μ−0.0154−4.263inline image−0.0191−5.156
μ−0.0008−0.241inline image−0.0063−1.662
λ0.534320.526λ0.520225.176
σ2(τ0.25)2.03 × 10−66.709σ2(τ0.25)2.31 × 10−66.739
σ2(τ0.5)4.77 × 10−76.638σ2(τ0.5)5.30 × 10−76.732
σ2(τ0.75)8.70 × 10−100.429σ2(τ0.75)3.81 × 10−300.000
σ2(τ1.0)2.06 × 10−76.532σ2(τ1.0)2.15 × 10−76.663
σ2(τ1.5)3.98 × 10−75.307σ2(τ1.5)4.85 × 10−76.050
σ2(τ2.0)1.66 × 10−74.072σ2(τ2.0)2.28 × 10−75.022
σ2(τ2.5)1.60 × 10−73.091σ2(τ2.5)1.48 × 10−73.463
σ2(τ3.0)3.77 × 10−73.807σ2(τ3.0)3.47 × 10−74.366
σ2(τ5.0)7.93 × 10−75.999σ2(τ5.0)8.67 × 10−76.259
σ2(τ7.0)2.76 × 10−75.728σ2(τ7.0)3.15 × 10−75.237
σ2(τ10.0)2.51 × 10−80.645σ2(τ10.0)1.11 × 10−71.971
σ2(τ15.0)2.56 × 10−66.357σ2(τ15.0)1.14 × 10−65.363
Log-likelihood6538.81Log-likelihood6531.80

To compare the performance of the two models, both 6-month-ahead and 12-month-ahead out-of-sample root mean squared errors (RMSE) are compared. The 6-month-ahead out-of-sample forecasts are made from December 2006 through May 2008 (or December 2007 in the case of the 12-month-ahead out-of-sample forecasts). The models are then recursively estimated using all of the information available at that time when the forecasts are made.

Table 4 compares the RMSE of two models. The magnitudes of the RMSE are, in general, within ranges reported in the literature. The discrepancy of the RMSE between two models might not be significant, but taken literally, the DNS model appears to perform better at almost all maturities and for different predictive horizons.9 This suggests that the DNS model without no-arbitrage restrictions, which moreover is much simpler to implement in practice, can be used as an acceptable approximation of a more theoretically sound affine-class term structure model, such as the AFNS model. In sum, both models may be regarded as a reasonable representation of KTB spot curves.

Table 4.   Out-of-sample performance of DNS and AFNS models: RMSE
Six-month-ahead out-of-sample forecasts are generated from December 2006 through May 2008 (or December 2007 for 12-month-ahead out-of-sample forecasts). For each period, models are re-estimated using all available observations for the time when forecasts are being generated. RMSEs are expressed in basis points. aA smaller RMSE between comparable forecast horizons.
Maturity (τ years)DNS modelAFNS model
6-month-ahead12-month-ahead6-month-ahead12-month-ahead
0.2547.7a63.1a47.7a66.6
0.551.1a61.4a52.165.6
0.7552.9a60.9a54.565.7
153.8a60.6a55.865.7
1.561.6a66.5a63.871.6
262.9a67.2a64.872.1
2.561.2a65.3a62.669.7
358.3a61.2a59.365.1
551.349.1a51.0a51.4
748.344.0a48.0a45.7
1045.6a38.4a47.342.3
1543.2a34.7a53.348.4

4.3. Assessing Alternative Borrowing Strategies with Different Term Structure Models

Different borrowing strategies are evaluated using DNS models, with and without no-arbitrage restrictions, as the main tools for simulating the term structure of KTB spot rates. As mentioned previously, 20-year spot rates are also included in the final estimation of the term structure models, yielding a total of 13 maturities used in the simulations. Given the final estimates as of December 2007, temporal paths for the term structure are simulated by drawing normal variates and using equations (12) and (13). In the case of the DNS model, for example, spot rates at period T + k are simulated recursively from:10

image((13'))
image((12'))

The size of the primary budget deficit from 2008 to 2012 was projected to vary from KRW7.0tn to KRW7.4tn. Hence, the quarterly primary budget deficit (Pt) is computed stochastically to produce figures of between KRW1.75tn and KRW1.85tn on a quarterly basis. The refinancing parameter, α in equation (3) is assumed to be unity. As explained earlier, the simulation horizon is assumed to be 5 years (20 quarters) and the assessment of alternative borrowing strategies is made in the fifth year. Therefore, the distribution of inline image is used to compute the average debt-service costs and risks for specific borrowing strategies.

Panel (a) of Figure 2 depicts the results from using the DNS model with no-arbitrage restrictions (AFNS model). It evaluates all 489 possible strategies (walternative = wref ± 4%) in the cost-risk space, with each point representing a particular borrowing strategy. The average debt-service cost and risk combination for the reference strategy, wref = (22%, 40%, 28%, 10%)′, is located at around the center of the graph, indicated by a circle. The fifth year’s average debt-service cost, risk (standard deviation), and CaR (95th percentile) are, KRW44.994tn, KRW2.086tn and KRW48.407tn. Average debt-service costs range from around KRW42tn to KRW48tn, whereas risks lie between KRW1.9tn and KRW2.3tn.

Figure 2.

 Assessing alternative borrowing strategies: (a) Nelson–Siegel model with no-arbitrage restrictions (AFNS) and (b) Nelson–Siegel model without no-arbitrage restrictions (DNS).

The vertical axis represents the average debt-service cost (expressed in KRW billion) of particular borrowing strategies computed as the sample mean of the distribution of inline image (i.e. borrowing requirements during the fifth year in the simulation). The distribution is based on 1000 simulated paths for the term structure of Korea Treasury bonds (KTB) spot rates. The horizontal axis represents the risk, measured as the standard deviation (σ) of the distribution of borrowing requirements. Reference strategy, wref = (22%, 40%, 28%, 10%)′, is the actual position at the start of the simulation and 489 strategies are defined as walternative = wref ± 4%. Minimum cost-at-risk (CaR) strategy refers to one that yielded the minimum CaR. A cost–risk efficient frontier that shows a negative cost–risk trade-off is obtained by tracking the contour in the south-western region of the graph.

A noticeable feature of Figure 2 is that the negative cost–risk trade-off is not as distinctive as the one reported previously. One can see this through drawing a cost–risk efficient frontier by tracking the contour in the south-western region of the graph. Hahm & Kim (2003) use three daily Korean Government bond yields from January 1996 to March 2000 with 1, 3, and 5-year maturities to derive optimal borrowing strategies and find evidence of a negative cost–risk trade-off.11 They refer to stylized facts on the patterns of the mean and standard deviation of US yield curves to explain their findings: shorter-term yields tend to have smaller means and larger standard deviations compared to longer-term yields. However, historical KTB spot curves do not exhibit this pattern, as illustrated in Table 2; hence resulting in a less distinctive efficient frontier.

Also depicted in Figure 2a is the minimum CaR strategy, or the one that yielded the minimum CaR among all feasible strategies, as indicated by a rectangle.12 The average debt-service cost, risk, and 95% CaR for this minimum CaR strategy are, respectively, KRW42.674, KRW1.989tn, and KRW45.894tn. This amounts to a 5.15% reduction in the average debt-service cost, or a 5.19% saving in the CaR compared with the reference strategy. A minimum CaR strategy is attained by setting the active weights to Δw = (−4%, −3%, +3%, +4%). Greater positions in 10 and 25-year bonds, while reducing those of 3-year and 5-year bonds, might reflect, in part, smaller estimates of inline image at longer maturities in the AFNS model reported in Table 3.

Panel (b) of Figure 2 depicts the results from using the DNS model without no-arbitrage restrictions (DNS model). The average debt-service cost, risk, and 95% CaR for the reference strategy are, respectively, KRW45.369, KRW 2.307tn, and KRW49.311tn, which are greater than the figures obtained from the simulation using the AFNS model. The range of average costs is similar to that using the AFNS model, lying between around KRW43tn and KRW48tn, whereas the risks are generally greater, ranging from KRW2.1tn to KRW2.5tn. The position with active weights Δw = (−4%, −4%, +4%, +4%) relative to the reference strategy produces an average debt-service cost, risk, and 95% CaR of 42.846, 2.221, and 46.480, respectively. This is equivalent to a 5.56% reduction in the average debt-service cost, or a 5.74% saving in CaR compared with the reference strategy, in line with the patterns observed under the ANFS model.

Figure 2a and b shows that the range of variation in average debt-service costs and active weights that attain minimum CaR are very similar, except that the overall risks are higher for cases where no-arbitrage restrictions are not imposed. This suggests that a simpler DNS model might still be a practical option next to a more complex theoretical term structure model such as the AFNS model for assessing borrowing strategies. To quantify the consistency between the two models, suppose that the borrowing strategies of each model are sorted according to their magnitudes of the empirical 95% CaR. The consistency of the strategies between the two models may then be measured by computing the rank correlation, such as Kendall’s tau. This figure turns out to be 0.80, which is quite high, holding important implications for practitioners: although the arbitrage-free Nelson–Siegel-type model is superior from a theoretical point of view, it is difficult to extend such a model to incorporate the interaction between the term structure and macroeconomic variables of interest. Because the DNS model is more flexible in this regard and appears to be reasonably consistent with the theoretical model, its extensions, such as the Diebold et al. (2006) version, might prove helpful both in assessing borrowing strategies and for conducting macroeconomic stress tests.

4.4. Assessing the Benefits of Introducing New Short-term Debt Instruments

The Bank of Korea has been issuing short-term bonds, called monetary stabilization bonds, as one of the instruments it uses for conducting monetary policy. Such bonds with maturities of 1-year or less are issued as zero coupon bonds, while those of longer maturities are issued as coupon bonds. Meanwhile, no zero coupon bonds have been issued as KTB. The proposed framework for assessing alternative borrowing strategies in this paper can be used to examine the effects of introducing new instruments such as zero coupon KTB with maturities of 1-year and less. If the distribution of shorter maturity bonds has a relatively lower mean and larger standard deviation than those of longer maturity bonds, the introduction of these shorter-term bonds might help to reduce the average debt-service costs, but only at the expense of amplifying risk.

Figure 3 illustrates the combination of expected debt-service costs and risks of 2544 strategies, which are constructed from varying weights to 1-year zero coupon KTB from 0% to 5%, while varying those to existing maturities within ±4% from the reference weight. Also depicted in Figure 3 are minimum CaR strategies amongst those that includes at least 1–5% of 1-year zero coupon KTB’s in the position. For both term structure models, there are cases where the introduction of short-term KTB produces a lower average debt-service cost level given a particular level of risk, or a lower risk given a particular level of average debt-service cost (south-western region in the graph from the reference strategy). Strategies denoted by 1yr 1%, 1yr 2%, and 1yr 3% in Figures 3 belong to this case. However, the majority of borrowing strategies that include 1-year zero coupon KTB tended to yield a cost–risk relationship lying farther to the northeast of the reference position. Thus, the trade-off between average debt-service cost and risk does not seem to improve much by introducing short-term debt instruments.

Figure 3.

 Effects of the introduction of 1-year zero coupon Korea Treasury bonds (KTB) on average debt-service cost–risk relationship: (a) Nelson–Siegel model with no-arbitrage restrictions (AFNS) and (b) Nelson–Siegel model without no-arbitrage restrictions (DNS).

The vertical axis represents the average debt-service cost of particular borrowing strategies, computed as the sample mean of the distribution of inline image (i.e. borrowing requirements during the fifth year in the simulation). The distribution is based on 1000 simulated paths for the term structure of KTB spot rates. The horizontal axis represents the risk measured by the standard deviation (σ) (or 95% cost-at-risk [CaR]) of the distribution of borrowing requirements. The reference strategy, wref = (22%, 40%, 28%, 10%)′, is the actual position held at the start of the simulation. A total of 2544 strategies are obtained by varying the weights of 1-year zero coupon KTB from 0% to 5%, while varying those of existing maturities within ±4% of the reference weight. 1yr 1–5% denotes a strategy that yielded the minimum CaR amongst those including at least 1–5% of 1-year zero coupon KTB in the issuance weight. A cost–risk efficient frontier that shows a negative cost–risk trade-off is obtained by tracking the contour in the south-western region of the graph.

Table 5 compares the effects of introducing new short-term debt instruments. Min 1yr 1–5% in the first column was the strategy that yielded the minimum CaR among those including at least 1–5% of 1-year zero coupon KTB in the issuance weight. When the AFNS model is used, the active strategy with issuance weight w1yr3% = (3%, 18%, 36%, 32%, 11%), namely, the minimum CaR 1 year-3% strategy, would reduce the average debt-service cost by 1.82% or 1.77% in terms of the 95% CaR, compared to the reference strategy with an issuance weight of wref = (0%, 22%, 40%, 28%, 10%). The pattern and magnitude of improvement in the CaR made by introducing 1-year zero coupon KTB is similar for the two-term structure models. However, such benefits seem to dissipate when weights for 1-year zero coupon KTB are pushed over 4%.

Table 5.   Effects of different term structure models on the average debt-service cost and 95% CaR
*Artificial 1-year zero coupon Korea Treasury bonds (KTB). †Cost-at-risk, measured as the 95th percentile of the simulated distribution of inline image (i.e. borrowing requirements during the fifth year in the simulation). SD(σ) denotes the standard deviation of the simulated distribution. ‡Percentage change in the cost-at-risk (CaR) relative to the reference strategy (outstanding KTB position as of December 2007). Minimum CaR strategy refers to one that yielded the global minimum CaR among the 2544 total strategies examined. Min 1yr 1–5% denotes a strategy that yielded the minimum CaR amongst those including at least 1–5% of 1-year zero coupon KTB in the issuance weight.
Borrowing strategyIssuance weightsSD(σ)Average debt-service cost95% CaR†Percentage Δ in CaR‡
1 year*3 year5 year10 year20 yearTotal
(a) Dynamic Nelson–Siegel model with no-arbitrage restrictions
 Reference0.220.400.280.101.002.08644.99448.407 
 Minimum CaR0.180.370.310.141.001.98942.67445.894–5.19
 Min 1yr 1%0.010.180.360.310.141.002.00343.07446.269–4.42
 Min 1yr 2%0.020.180.360.300.141.002.02943.52846.877–3.16
 Min 1yr 3%0.030.180.360.320.111.002.09444.17647.552–1.77
 Min 1yr 4%0.040.180.360.320.101.002.10144.77348.167–0.50
 Min 1yr 5%0.050.180.360.300.111.002.,00045.39748.8220.86
(b) Dynamic Nelson–Siegel model without no-arbitrage restrictions
 Reference0.220.400.280.101.002.30745.36949.311 
 Minimum CaR0.180.360.320.141.002.22142.84646.480–5.74
 Min 1yr 1%0.010.180.370.310.131.002.18043.62247.052–4.58
 Min 1yr 2%0.020.180.360.310.131.002.23444.11147.602–3.47
 Min 1yr 3%0.030.180.360.310.121.002.30344.64948.448–1.75
 Min 1yr 4%0.040.180.360.290.131.002.20745.16348.831–0.97
 Min 1yr 5%0.050.180.370.290.111.002.28545.85549.6130.61

5. Summary and Conclusions

A cost–risk analytic model is often constructed to assess alternative sovereign debt financing strategies. In such models, governments issue bonds to pay for interest on existing bonds, to roll over maturing bonds, and to meet new primary budget deficit requirements. Temporal paths for the term structure of interest rates play a central role in the determination of the relationship between future debt-service costs and the risks of specific borrowing strategies. Despite recent advancements in term structure modeling and simulation techniques, the sovereign debt strategy literature has largely overlooked the potential consequences that might arise from using time-series-based term structure models that are not rooted in theory. A time-series model such as the two-step DNS model adopted by Pick & Anthony (2007) would be one of the most complex enough and up-to-date term structure models used in this area in practice. Limited access to historical term structure data further aggravates the situation.

As such, this paper has attempted to examine how much these theoretical restrictions on term structure models affect the process of assessing alternative sovereign borrowing strategies. Our approach builds on Hahm & Kim (2003), who propose a cost–risk analytic model of sovereign debt management within a mean–variance framework. This paper extends their approach by applying recent developments in term structure models; namely, the DNS model proposed by Diebold & Li (2006) and the AFNS model proposed by Christensen et al. (2008a). The DNS model may be regarded as a one-step procedure of a two-step implementation of the DNS model advocated by Pick & Anthony (2007). The AFNS model is a special version of more general theoretical affine-class term structure models, yet preserves the parsimonious functional form of the DNS model. Both the DNS and AFNS models are able to be handled conveniently within a state-space model framework.

Using monthly nominal Korean Treasury spot rates from September 2000 to November 2008 for 12 maturities (0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 5, 7, 10, and 15 years), this paper finds that more generic term structure models such as the DNS model perform better in terms of out-of-sample RMSE at different forecast horizons. However, looking at results for goodness-of-fit, size of pricing errors, and the magnitude of RMSE suggest that both models appear to provide reasonable representations of KTB spot data.

This paper shows that the 95% CaR would have saved as much as 5–6% by rebalancing the actual position held in December 2007. These figures are similar for both models. Furthermore, the DNS models, with and without no-arbitrage restrictions, produced a very similar ranking among the different strategies: the Kendall’s tau statistic between the borrowing strategies of each model that were sorted according to the magnitude of their empirical 95% CaR turned out to be 0.80, implying that the DNS-type models used in practice are not far off from their theoretically consistent counterpart, and, therefore, are regarded as a second-best option for a more complex theoretical term structure model, such as the AFNS model. The DNS-type model carries the additional advantage of being able to extend to relate the term structure with macroeconomic variables, which would help to facilitate macroeconomic stress testing.

The cost–risk analytic model is also used to address such issues as borrowing strategies aimed at developing the domestic debt market by rolling over a certain portion of maturing debt into new short-term domestic instruments, such as 1-year zero coupon KTB. The empirical findings suggest that, regardless of which term structure model is used, it is possible to lower both the average debt-service costs and 95% CaR by as much as approximately 1% if the weighting of 1-year zero coupon KTB is raised to 3%. It is found, however, that regardless of the choice of term structure models, benefits from new short-term instruments seem to dissipate when the weights to such instruments exceed 4%. These findings might also shed some light on the potential usefulness of a DNS-type model to represent the term structure of riskier bonds, such as bank debentures, which would otherwise be a daunting task for practitioners.

Footnotes

  • 1

    From the mid-2000s, three private bond pricing companies launched in the late 1990s started to compile more reliable Korean Treasury spot rates based upon market data.

  • 2

    Diebold & Li (2006) and Diebold et al. (1996) claim that the three latent factor DNS model, or the extended DNS model, that includes macroeconomic variables as additional explanatory variables in the state-space model framework, tend to yield a better out-of-sample performance than those of the Duffie and Kan-type (1996) theoretical affine term structure models.

  • 3

    Ang & Piazzesi (2003) and Ang et al. (2006) propose alternate time-series term structure models that incorporate the no-arbitrage constraints in the conventional vector autoregressive model framework.

  • 4

    Ten-year index bonds and foreign currency denominated Treasury bonds are also issued, but their portions are relatively small.

  • 5

    Because this paper does not build a separate macroeconomic model, the nominal cost of debt, rather than the debt–cost ratio is used as the measure for cost. Another consideration is that the debt burden relative to GDP is moderate in Korea compared to OECD countries, which makes the government’s fiscal rules, such as its sustainable investment rule, a lesser concern.

  • 6

    The outstanding balance in Table 1 reflects the amount of buybacks made during 2007.

  • 7

    In the simulation of borrowing strategies, however, the term structure models are re-estimated using the data, including 20-year spot rates, for which rates prior to 2006 are replaced by fitted values from the cross-sectional term structure model.

  • 8

    For instance, the sample means (standard deviation) of 3-month, 12-month, 3-year, and 10-year US Treasury spot rates for the January 1970–December 2000 period studied by Diebold & Li (2006) were 0.0675 (0.0266), 0.0720 (0.0257), 0.0763 (0.0234), and 0.0805 (0.0214), respectively.

  • 9

    Cha & Kim (2010) examine the effects of the recent crisis on the performance of the two models. Using the November 2004–August 2009 sample to incorporate the structural shift in the KTB term structure as a consequence of the change in the target rate by the central bank, they find that the DNS model still fairs quite well for all maturities for the 6-month-ahead out-of-sample forecasting, whereas it performs better only at relative longer maturities (5–20 years) for 12-month-ahead out-of-sample forecasting.

  • 10

    Alternatively, a bootstrapping method may be used to accommodate the heteroscedasticity in the disturbance terms. The examination of fitted residuals, however, suggests that heteroscedasticity is not a serious problem during the sample period considered in the paper.

  • 11

    The 1-year government bond is issued by the Bank of Korea as a device for conducting monetary policy whereas the 3-year and 5-year bonds are issued by the Ministry of Strategy and Finance. Hence, the yield curve that these authors refer to consists of heterogeneous set of government bonds, which is in contrast to the spot curves of Treasury bonds used in this paper.

  • 12

    Note that the objective function of policy-makers could be more complex than just minimizing cost or CaR. For instance, the target duration of the bond portfolio might be a bigger concern. If so, the weighted average of the minimum cost and duration targets could be used as the objective function. In this case, weighting can be determined according to the risk appetite of policy-makers.

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