Default Risk and Sovereign Bond Pricing: Dittmar, Hsu, Roussellet & Simasek (2026)

What this is. The paper’s core results, the model it builds on (an affine no-arbitrage term structure with sovereign default), the estimation method (Extended Kalman Filter maximum likelihood), and the empirical specifications (IV regression, GMM): enough to know what it found and how, without reading all 42 pages. To replicate or extend it, read the full source at https://doi.org/10.1111/jofi.70014 (paywalled).

TL;DR

The ILSBEI differential is a version of the TIPS-Treasury no-arbitrage mispricing studied by Fleckenstein, Longstaff, and Lustig (2014), and this paper extends their liquidity and slow-moving-capital account by adding a credit risk channel. Treating U.S. default risk as nontrivial follows the macrofinance view of U.S. sovereign CDS premia in Chernov, Schmid, and Schneider (2020), applied here to relative bond pricing.

Using monthly U.S. data from June 2005 to December 2020, the paper documents that the spread between the inflation-linked swap (ILS) rate and the breakeven inflation rate (BEI), the ILSBEI differential, is significantly positively related to two measures of U.S. sovereign default risk: growth in Treasury debt held by the public and Euro-denominated five-year CDS spreads. Controlling for liquidity (VIX, LIBOR-OIS, OTR spread, HPW noise), debt growth remains a robust predictor. Treasury debt growth is used as an instrument for CDS to address endogeneity, and instrumented CDS loads significantly on the ILSBEI spread across tenors of two through seven years. A new affine no-arbitrage term structure model estimated by Extended Kalman Filter decomposes the ILSBEI spread into credit and liquidity components: at the 10-year maturity, most of the ILSBEI spread is explained by the credit component outside the financial crisis. The dominant transmission channel is the correlation between inflation dynamics and default probability (hyperinflation upon default), not differential loss-given-default between nominal Treasuries and TIPS.

Core results

Magnitudes and significance are as reported; */**/*** = 10%/5%/1%. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	ILSBEI differential is positively related to Treasury debt growth (G)	Table II, p. 837	G coef 0.013*** (level), 0.025*** (first diff); R^2 = 0.217 (level), 0.113 (FD); debt growth captures ~22% of level variation
R2	Debt growth relationship is robust to liquidity and slow-moving-capital controls (HPW noise, LIBOR-OIS, VIX)	Table III, p. 839	G coef 0.004** (level), 0.018*** (FD) with full liquidity controls; HPW noise also significantly positive (0.053*** level); overall R^2 = 0.808
R3	Postcrisis (2010-2020) results are qualitatively similar; CDS is a stronger predictor than in the full sample	Table IV, p. 840	CDS coef 0.352*** (spec 2); with both G and CDS, CDS coef 0.287*** and G coef 0.002*
R4	Treasury debt growth is a strong instrument for CDS; instrumented CDS is significantly positively related to ILSBEI	Table V Panel B, p. 842	2SLS: instrumented CDS coef on ILSBEI = 1.205** (SE 0.475); on ILS = -2.174***, on nominal TSY = -3.643**; TIPS coefficient insignificant (-0.265)
R5	Effect is robust across maturities 2-7 years; 10-year loading positive but not statistically significant	Table VI, p. 843	Instrumented CDS on ILSBEI: 2y = 1.097** (SE 0.522), 3y = 1.324*** (SE 0.511), 7y = 0.868** (SE 0.341), 10y = 0.385 (SE 0.250)
R6	Term structure model fits the data well (R^2 93-99% for ILS; 99%+ for nominals; ILSBEI R^2 81-91% for maturities 2-7y)	Table VIII, p. 859	ILS RMSE 6-13 bps; Nominal RMSE 6-10 bps; ILSBEI RMSE 8-11 bps (range 7.77-10.92 bps across maturities); model-implied default probabilities peak at ~3% (10y) during 2008-2009 crisis
R7	Credit component of ILSBEI is small at short maturities but large at long maturities; outside the crisis, more than half of the 10-year spread is credit	Table IX / Figure 7, pp. 859-862	Mean credit component (Cred.): 7.57 bps (2y), 10.07 bps (3y), 14.46 bps (5y), 18.16 bps (7y), 23.05 bps (10y); credit share rises from ~46% (2y) to ~78% (10y) on average
R8	Dominant channel is inflation jump upon default (hyperinflation), not differential LGD or default priced in SDF alone	Figure 10 / Section IV.F, pp. 865-866	Comparative statics: inflation-upon-default channel (kappa_delta^pi) contributes 1-10 bps to ILSBEI (95% CI, orange curve); inflation/PD correlation channel contributes ~1 bp at 10y median (~3.5 bps at 97.5th percentile); default-in-SDF channel contributes near zero

Overall (paper’s conclusion). Credit risk can drive persistent deviations between ILS rates and BEI rates that are often attributed solely to liquidity. Where Pflueger and Viceira (2016) attribute the ILSBEI differential to a liquidity premium, this paper finds credit risk is also a nontrivial driver, especially at long maturities. The interaction between inflation dynamics and default is the primary source of differential pricing between nominal Treasuries and TIPS.

Theory / model

The paper builds intuition through a one-period two-equation toy model (Section II, p. 844) before embedding these mechanisms in the full multivariate term structure model (Section III, p. 852).

Toy model (Section II, eq. 1, p. 844). The real pricing kernel and inflation rate are:

$$\log M^*_{t+1} = \bar{M} + \Lambda_\delta \cdot \mathbf{1}\{\delta^{(c)}_{t+1} > 0\}$$ $$\pi_{t+1} = \kappa_0 + \kappa_y \lambda_t + \kappa_\delta \cdot \mathbf{1}\{\delta^{(c)}_{t+1} > 0\}$$

where $\delta^{(c)}_{t+1}$ is a nonnegative default process with conditional jump probability $\lambda_t$, $\Lambda_\delta > 0$ is the price of default risk, $\kappa_0$ governs average inflation, $\kappa_y$ is the loading of inflation dynamics on default probability, and $\kappa_\delta$ is the inflation jump upon default. Under independence (default event is transitory), all bond prices are in closed form.

Bond yield decomposition (eq. 2, p. 845). Any nominal Treasury yield can be decomposed into four parts:

$$\begin{aligned} R^{(n)}_t \;=\; & r^{(n)*}_t & &\text{(a) real risk-free rate} \\ + & \bigl(r^{(n)}_t - r^{(n)*}_t\bigr) & &\text{(b) ILS} \\ + & \bigl(R^{(n)*}_t - r^{(n)*}_t\bigr) & &\text{(c) real credit spread} \\ + & \bigl(R^{(n)}_t - r^{(n)}_t\bigr) - \bigl(R^{(n)*}_t - r^{(n)*}_t\bigr) & &\text{(d) -ILSBEI} \end{aligned}$$

where $*$ denotes real yields and capitalization denotes defaultable bond yields. The ILSBEI spread in component (d) is negative when the nominal credit spread exceeds the real credit spread, that is, when nominal Treasuries are more exposed to default than TIPS. Expanding around small default probability $\lambda_t$, the nominal Treasury yield decomposes as (eqs. 3.a-3.d, pp. 845-846):

$$\begin{aligned} R^{(n)}_t \;\approx\; & -\bar{M} + \lambda_t(1 - e^{\Lambda_\delta}) & &\text{(3.a: risk-free real yield)} \\ + & \kappa_0 + \kappa_y \lambda_t + \lambda_t e^{\Lambda_\delta}(1 - e^{-\kappa_\delta}) & &\text{(3.b: ILS)} \\ + & \lambda_t e^{\Lambda_\delta} \cdot \text{LGD}^* & &\text{(3.c: real credit spread)} \\ + & \lambda_t e^{\Lambda_\delta}(e^{-\kappa_\delta} \cdot \text{LGD} - \text{LGD}^*) & &\text{(3.d: -ILSBEI)} \end{aligned}$$

where $\text{LGD}$ ($\text{LGD}^*$) is the loss given default of a nominal (real) Treasury. Equation (3.d) shows that even equal LGDs produce a nonzero ILSBEI if $\kappa_\delta > 0$, that is, if there is hyperinflation upon default.

Three channels. The model identifies three channels through which default affects ILSBEI (pp. 844, 865):

1. Default priced in the SDF ($\Lambda_\delta$): a price of default risk that lowers riskless real bond yields through the pricing kernel.
2. Inflation jump upon default ($\kappa_\delta^{(\pi)}$): hyperinflation upon default raises the value of inflation swaps relative to nominal Treasuries, widening ILSBEI.
3. Negative correlation between default probability and inflation ($\kappa_y^{(\pi)} < 0$): higher default probability predicts lower current inflation, lowering the ILS rate.

Lucas tree motivation (Section II.D, p. 850). The inflation/default correlations arise naturally in a CRRA representative-agent Lucas tree economy with long-run risk and a central bank following a Taylor rule with coefficient on inflation less than one (passive monetary policy). The passive central bank stance means the Taylor principle is violated: inflation must fall when default probability rises, consistent with the negative $\kappa_y$ in the regression results.

Method

Full term structure model (Section III, pp. 852-855). The paper builds a dynamic discrete-time affine no-arbitrage term structure model following Monfort et al. (2020), using affine-term-structure and gamma-zero-processes as building blocks.

Risk factors (eqs. 5-8, pp. 852-853). Three blocks of state variables:

Riskless real short rate driven by three Gaussian factors $x_t$ (eq. 5):

$$r^{(1)*}_t = \kappa_0^{(r)} + \kappa_x^{(r)\prime} x_t$$

with VAR(1) dynamics (eq. 6):

$$x_t = \Phi_x x_{t-1} + \epsilon_{x,t}, \qquad \epsilon_{x,t} \sim \text{iid}\; N(0, I_3)$$

Credit and liquidity event processes $\delta_t = (\delta^{(c)}_t, \delta^{(l)}_t)$ modelled as gamma-zero processes (eq. 7): for $i \in \{c, l\}$,

$$\delta^{(i)}_t = \sum_{j=1}^{P^{(i)}_t} \xi^{(i)}_{j,t}, \quad \text{where } P^{(i)}_t \mid \lambda^{(i)}_t \sim \text{Poisson}(\lambda^{(i)}_t) \text{ and } \xi^{(i)}_{j,t} \sim \text{Exp}(1/c^{(i)}_\delta)$$

Event intensities driven by nonnegative factors $y_t = (y^{(c)}_{1,t}, y^{(c)}_{2,t}, y^{(l)}_t)$ following VARG dynamics (eq. 8):

$$\begin{aligned} \lambda^{(c)}_t &= \beta^{(c)}_{\lambda,1} y^{(c)}_{1,t} + \beta^{(c)}_{\lambda,2} y^{(c)}_{2,t} \\ \lambda^{(l)}_t &= \beta^{(l)}_\lambda y^{(l)}_t \end{aligned}$$

Pricing kernel (eq. 9, p. 853):

$$\log(M^*_{t+1}) = -r^{(1)*}_t + \Lambda_{x,t}' x_{t+1} + \Lambda_y' y_{t+1} + \Lambda_\delta \delta^{(c)}_{t+1} - \zeta_t$$

where $\Lambda_{x,t} = \Lambda_{0,x} + \Lambda_{1,x} x_t$ are the prices of riskless factor risk, $\Lambda_y$ are the prices of credit/liquidity factor risk, and $\Lambda_\delta$ is the price of credit event risk. The structure- preserving property of the pricing kernel ensures the model belongs to the affine class, so all pricing formulas are in closed form.

Inflation dynamics (eq. 10, p. 854):

$$\pi_t = \kappa_0^{(\pi)} + \kappa_x^{(\pi)\prime} x_t + \kappa_y^{(\pi)\prime} y_t + \kappa_\delta^{(\pi)} \delta^{(c)}_t$$

The expected positive sign on $\kappa_\delta^{(\pi)}$ (hyperinflation upon default) and expected negative sign on $\kappa_y^{(\pi)}$ (lower inflation when default probability is high) are both key model predictions tested against the data.

Estimation (eq. 11, p. 856). The model is cast in state-space form with 23 observable variables $\mathbf{y}_t \in \mathbb{R}^{23}$ (ILS, nominal yields, TIPS, CDS at multiple maturities, OIS, CPI, TIPS liquidity proxy):

$$\mathbf{y}_t = F(x_t, y_t, \delta_t, \theta^Q) + \eta_t, \qquad \eta_t \sim N(0, \Sigma_\eta)$$

where $F(\cdot)$ is a nonlinear closed-form function summarizing the affine pricing equations and $\theta^Q$ is the set of risk-neutral parameters. Since $F(\cdot)$ is nonlinear (due to the gamma-zero processes), the model is estimated by Extended Kalman Filter approximate maximum likelihood (extended-kalman-filter). CDS measurement error standard deviations are allowed to depend on a CDS liquidity proxy (Grischenko and Huang (2013)) to account for mismeasurement.

Empirical specifications

Baseline regression (Tables II-IV, pp. 837-840). OLS in levels and first differences of the five-year ILSBEI spread on measures of default risk:

$$\text{ILSBEI}_t = \alpha + \beta G_t + \epsilon_t \qquad \text{(level)}$$ $$\Delta\text{ILSBEI}_t = \alpha + \beta\, \Delta G_t + \epsilon_t \qquad \text{(first diff)}$$

• $\text{ILSBEI}_t$: five-year inflation-linked swap rate minus breakeven inflation rate at time $t$
• $G_t$: year-over-year log growth in Treasury debt held by the public (level regressions) or monthly log variation (first-difference)
• $\alpha$: intercept
• $\beta$: slope coefficient on default risk proxy
• $\epsilon_t$: error term
• Standard errors: Newey-West with three lags
• Sample: June 2005 to December 2020, monthly

Extended specification adds liquidity controls:

$$\text{ILSBEI}_t = \alpha + \beta_G G_t + \beta_{\text{CDS}} \text{CDS}_t + \beta_{\text{VIX}} \text{VIX}_t + \beta_{\text{OIS}} (L\text{-}OIS)_t + \beta_{\text{OTR}} \text{OTR}_t + \beta_{\text{HPW}} \text{HPW}_t + \epsilon_t$$

• $G_t$: Treasury debt growth (default risk proxy)
• $\text{CDS}_t$: Euro-denominated five-year U.S. sovereign CDS spread
• $\text{VIX}_t$: CBOE Volatility Index (liquidity/risk-aversion control)
• $(L\text{-}OIS)_t$: LIBOR-OIS spread (counterparty risk / liquidity control)
• $\text{OTR}_t$: off-the-run/on-the-run 10-year Treasury spread (liquidity control)
• $\text{HPW}_t$: Hu-Pan-Wang noise measure (slow-moving capital proxy)

Instrumental variables (Tables V-VI, pp. 841-843). Single-stage GMM using $G_t$ (Treasury debt growth) as an instrument for CDS spreads. First stage: regress each liquidity/slow-moving-capital control on $G_t$ to obtain orthogonalized residuals $\text{VIX}^\perp$, $(L\text{-}OIS)^\perp$, $\text{OTR}^\perp$, $\text{HPW}^\perp$; then regress $\text{CDS}_t$ on $G_t$ and these residuals. Second stage: regress $\text{ILSBEI}_t$ (and individually: ILS, nominal TSY, TIPS) on the first-stage predicted $\widehat{\text{CDS}}_t$ and the orthogonalized liquidity controls. Standard errors are Newey-West with three lags, estimated simultaneously via single-stage GMM. Specification repeated at 2y, 3y, 5y, 7y, 10y tenors. Identifying assumption: Treasury debt issuance decisions are exogenous to financial market pricing frictions (p. 841).

Term structure model estimation (Section IV, pp. 855-856). Extended Kalman Filter maximum likelihood on 23 observable monthly series from November 2004 to December 2019 (data availability for TIPS liquidity proxy). Observable variables are the term structures of ILS (2-10y), nominal Treasuries (1-10y), TIPS (implied from BEI), sovereign CDS (5y, 10y), OIS (6m), monthly CPI inflation, and the Grischenko-Huang TIPS liquidity index. The riskless real one-period yield and all bond prices are derived in closed form under the affine structure. Model parameters are estimated jointly including $\kappa_\delta^{(\pi)}$, $\kappa_y^{(\pi)}$, $\Lambda_\delta$, recovery fractions, and the TIPS disindexation rate $\rho^*$.

Comparative statics (Figure 10, Section IV.F, pp. 865-866). Sequential counterfactuals setting $\Lambda_\delta = 0$, $\kappa_\delta^{(\pi)} = 0$, or $\kappa_y^{(\pi)} = 0$ in turn, computing the deviation between fitted and counterfactual yield curves. The median contribution of each channel to ILSBEI is measured in basis points at each maturity, with 95% confidence intervals across sample dates.

Datasets used

Dataset	Role in paper	Wiki page
Gurkaynak, Sack, and Wright (GSW) zero-coupon yields (nominal Treasury and TIPS)	BEI construction; nominal yield curve estimation	no page yet (Federal Reserve Board data, publicly available)
Bloomberg ILS rates (zero-coupon, 2-10y maturities)	ILSBEI spread construction; model estimation target	Bloomberg (licensed)
U.S. Treasury debt held by public (monthly, Federal Reserve/Treasury)	Default risk proxy (G); instrument for CDS	FRED, series available via FRED
Euro-denominated 5-year U.S. Treasury CDS spreads	Alternative default risk measure; model estimation target	Bloomberg / Markit CDS (licensed)
BLS CPI-U (monthly)	Inflation measure for model	BLS (CPI-U), pulled via FRED series CPIAUCSL
VIX (CBOE)	Liquidity/slow-moving capital control	FRED, series VIXCLS
LIBOR-OIS spread	Liquidity control (counterparty risk)	no page yet
Off-the-run / on-the-run 10y Treasury spread (OTR)	Liquidity control	no page yet (derived from GSW and Bloomberg)
HPW noise measure (Hu, Pan & Wang 2013)	Slow-moving capital proxy; TIPS liquidity intensity proxy	no page yet (academic dataset, Jun Pan’s website)
OIS 6-month rate	Short-term riskless nominal yield for model	Bloomberg (licensed)

Sample: monthly, Nov 2004 to Dec 2019 (model estimation); Jun 2005 to Dec 2020 (regressions).

When to read the full paper

Consult the original via https://doi.org/10.1111/jofi.70014 if you are: replicating (code in journal Supporting Information); extending the affine model to other sovereign issuers; studying how monetary policy passivity interacts with default to generate inflation dynamics; auditing a specific coefficient; or tracing the closed-form bond pricing derivations (Internet Appendix Sections I-XII). The locators above point to the exact tables and figures. For “what did this paper find,” the table above is sufficient and is the intended default.

Attribution and rights

Source: peer-reviewed, The Journal of Finance 81(2). (c) 2026 the American Finance Association. This distillation was extracted by an LLM on 2026-05-31 and augmented on 2026-06-01; it is not human-verified or independently reproduced. The paper is paywalled; no verbatim content is reproduced here.

Dittmar, Robert F., Alex Hsu, Guillaume Roussellet, and Peter Simasek. “Default Risk and the Pricing of U.S. Sovereign Bonds.” The Journal of Finance 81, no. 2 (April 2026): 829-869. DOI: 10.1111/jofi.70014. (c) 2026 the American Finance Association. All rights reserved. This page is an extract-only distillation by the Institute for Automated Research: core results summarised; no verbatim text reproduced.