Term Structure in a Heterogeneous Monetary Union: Costain, Nuno & Thomas (2025)

What this is. This is an LLM-distilled skeleton of the paper; read the original to replicate or extend. Not human-verified; not reproduced.

TL;DR

Costain, Nuno, and Thomas build an arbitrage-based affine term structure model (ATSM) for a two-country monetary union (Core = Germany, Periphery = Italy) where the Peripheral sovereign faces rollover-crisis default risk. The model extends Vayanos and Vila (2021) to a multicountry setting with sovereign default risk, building on Duffie and Singleton (1999) for defaultable bond pricing. It decomposes yields into four components: expectations, term premium, expected default loss, and credit risk premium. Calibrated to German and Italian zero-coupon yields from 1999 to 2022, the model finds that the credit risk premium accounts for roughly three-quarters of the long-run Italy-Germany sovereign spread. For the ECB PEPP announcement of March 18, 2020, which drove a ~71 bps fall in Italian 10-year yields, the model attributes about four-fifths of the sovereign spread compression to a reduction in the credit risk premium via a default risk extraction channel, not the conventional duration risk channel identified in Greenwood and Vayanos (2014) and Hamilton and Wu (2012).

Core results

#	Result	Locator	Magnitude as reported
R1	Credit risk premium dominates the long-run IT-DE 10Y sovereign spread	Table II, p. 2419; Figure 5, p. 2420	Model spread = 79 bps; expected default loss = 22 bps (28%); credit risk premium = 62 bps (78%); data spread = 126 bps
R2	Model matches German 10Y term premium and Sharpe ratio	Table II, p. 2419; Figure 4, p. 2418	Model TP = 142 bps vs. data term premium ~125 bps; model Sharpe 0.57, data 0.51
R3	PEPP announcement: 81% of IT yield decline attributed to credit risk premium	Figure 6, p. 2424; Table III, p. 2427	Observed IT 10Y decline = 71 bps (March 18-20, 2020); model matches at calibration; credit risk premium share = 81% across robustness checks
R4	PEPP announcement: small effect on German yields, consistent with asymmetry	Figure 6, p. 2424	German 10Y: roughly -10 bps (model); nonmonotonic, small observed shift; asymmetry explained by default risk channel operating only on Peripheral bonds
R5	Credit risk premium dominance is robust across 12 alternative calibrations	Table III, p. 2427	CRP share: 74% long-run, 81% PEPP impact; low-gamma and low-delta calibrations worsen model fit substantially
R6	Pandemic shock (out-of-sample): model predicts large upward shift in Italian yields	Figure 9, p. 2431	Model-predicted shift = 120-160 bps across maturities; broadly matches observed Feb-to-Mar 2020 data; mainly driven by credit risk premium increase

Overall (paper’s conclusion). The paper proposes a model in which default risk opens a novel default risk extraction channel for central bank asset purchases, allowing large-scale parallel yield curve shifts like those observed in Italy during the Covid-19 pandemic. Asset purchases compress both the credit risk premium (by extracting defaultable bonds from private markets) and the expected default loss (endogenously, by relieving fiscal pressure). The duration risk channel is secondary in the euro-area context. The model identifies that credit risk premia, not expected losses, dominate sovereign spreads, and that the flexible PEPP design substantially enhanced its impact relative to the earlier, rigid APP.

Theory / model

The economy has two countries (Core and Periphery) in continuous time with an infinite horizon. Core issues risk-free bonds; Periphery may partially default on its obligations (Section I, pp. 2395-2408).

Short-rate process. The instantaneous riskless rate $r_t$ follows an Ornstein-Uhlenbeck process (eq. 1, p. 2396):

dr_t = \kappa(\bar{r} - r_t)dt + \sigma dB_t \tag{1}

Bond prices. Conjecture that bond prices are log-affine:

P_t(\tau) = e^{-[A_t(\tau)r_t + C_t(\tau)]}, \quad P_t^*(\tau) = e^{-[A_t^*(\tau)r_t + C_t^*(\tau)]} \tag{8}

where $\tau$ is maturity, $P_t(\tau)$ is the Peripheral bond price, and $P_t^*(\tau)$ is the Core bond price (p. 2398).

Yield decomposition (Proposition 1, p. 2400, eq. 16-17). Peripheral yields $y_t(\tau)$ decompose into four affine components:

y_t(\tau) = \underbrace{\frac{1}{\tau}\mathbb{E}_t\int_0^\tau r_{t+s}ds}_{y_t^{EX}(\tau)} + \underbrace{\frac{1}{\tau}\mathbb{E}_t\int_0^\tau \left\{A_{t+s}(\tau-s)\lambda_{t+s} - \frac{\sigma^2}{2}[A_{t+s}(\tau-s)]^2\right\}ds}_{y_t^{TP}(\tau)} + \underbrace{\frac{1}{\tau}\mathbb{E}_t\int_0^\tau \delta\psi_{t+s}ds}_{y_t^{DL}(\tau)} + \underbrace{\frac{1}{\tau}\mathbb{E}_t\int_0^\tau \xi_{t+s}ds}_{y_t^{CR}(\tau)} \tag{16}

where $\psi_t$ is the Peripheral default arrival rate, $\delta$ is the haircut, and $\xi_t = \gamma\psi_t\delta^2\int_0^\infty X_t(\tau)d\tau$ is the credit risk premium price (eqs. 13, 15, pp. 2399-2400).

Proposition 2 (low default risk, p. 2401). When $\psi \to 0$, term premia are equalized across countries and depend only on the aggregate net bond supply in the monetary union, so Core and Peripheral term premia move symmetrically with ECB purchases.

Proposition 3 (default risk shifts short yields, p. 2402-2403). In the ergodic distribution with constant default rate, the Peripheral yield is:

y_t(\tau) = (\psi\delta + \bar{\xi}) + \frac{(1+\Xi)(1-e^{-\hat\kappa\tau})}{\tau\hat\kappa}r_t + \frac{\int_0^\tau [A(u)(\kappa\bar{r}+\bar\lambda) - \frac{1}{2}\sigma^2[A(u)]^2]du}{\tau} \tag{\text{Prop. 3}}

where $\lim_{\tau\to 0}y_t(\tau) = (1+\Xi)r_t + (\psi\delta + \bar\xi)$, so the default-related constant $\psi\delta + \bar\xi$ shifts the entire Peripheral yield curve in parallel, including the short end, independent of changes in the short-term riskless rate (p. 2403).

Endogenous default (Section II, pp. 2403-2408). Following Corsetti and Dedola (2016), the paper models rollover crises as in Calvo (1988) and Cole and Kehoe (2000). In a rollover crisis (arrival rate $\eta$), the Peripheral government compares the cost of repayment $V_0^R$ (discounted emergency taxation) to the cost of default $V_0^D$ (fixed restructuring cost $\chi$) and defaults if $V_0^R > V_0^D$:

\mathbb{P}(\text{default at time }0|\text{crisis}) \approx \Phi(V_0^R) \tag{25}

The unconditional default rate is $\psi_t = \eta\Phi_t$, where fiscal pressure $F_t$ (eq. 29) aggregates primary deficits, bond redemptions, and central bank remittances. Central bank purchases reduce $F_t$ (via remittances), hence reduce $\psi_t$, reinforcing both duration and default risk extraction (eq. 26-29, pp. 2405-2407).

Method

The paper solves for equilibrium in two stages.

One-factor analytical solution (Section I). Market clearing (eq. 7, p. 2397) combined with the arbitrageurs’ first-order conditions (eqs. 10-13) yields two integral equations for the factor loadings $\lambda_t$ (price of interest-rate risk) and $\xi_t$ (price of default risk):

\lambda_t = \gamma\sigma^2\int_0^\infty\left[(S_t(\tau)-Z_t(\tau))A_t(\tau) + (S_t^*(\tau)-Z_t^*(\tau))A_t^*(\tau)\right]d\tau \tag{14}

\xi_t = \gamma\psi_t\delta^2\int_0^\infty (S_t(\tau)-Z_t(\tau))d\tau \tag{15}

When $\psi_t$ is a deterministic function of time, both sides of (15) are affine in $r_t$, yielding time-varying affine solutions $\lambda_t = \Lambda_t r_t + \bar\lambda_t$ and $\xi_t = \Xi_t r_t + \bar\xi_t$.

Multifactor quantitative model (Section III, pp. 2408-2416). For calibration, the model adds two mean-zero PH demand shifters $\varepsilon_t^h$ and $\varepsilon_t^{h*}$ to capture PH demand fluctuations:

Z_t(\tau) = h(\tau) - \varsigma(\tau)\varepsilon_t^h + \tau\alpha(\tau)(y_t(\tau) - \hat\delta\psi_t) \tag{30}

The factor vector $q_t \equiv [r_t, \varepsilon_t^h, \varepsilon_t^{h*}]^\top$ follows a multivariate Ornstein-Uhlenbeck process:

dq_t = -K(q_t - \bar{r}\mathcal{E}_1)dt + \Sigma dB_t \tag{31}

Bond prices remain log-affine in $q_t$: $P_t(\tau) = e^{-[A_t(\tau)^\top q_t + C_t(\tau)]}$, $P_t^*(\tau) = e^{-[A_t^*(\tau)^\top q_t + C_t^*(\tau)]}$ (eq. 32, p. 2409).

Calibration. Two-step procedure: (i) directly calibrate observable parameters including the OU parameters of $r_t$ (mean $\bar{r} = 1.22\%$, $\kappa = 0.062$, $\sigma = 63$ bps), fiscal variables, and Eurosystem purchase paths from ECB data; (ii) estimate the remaining 10 parameters (Table I, p. 2416) by minimizing a distance criterion over long-run yield moments (1999-2022) and the two-day PEPP announcement window shift. The numerical solution uses finite-difference methods for the time-varying PDE system.

Empirical specifications

Long-run moments target. The distance criterion minimizes the sum of squared deviations between model ergodic distribution and data for (Section III.C, pp. 2413-2414):

• Mean yields on 1m, 1Y, 5Y, 10Y German bonds and 1Y, 5Y, 10Y Italian bonds
• Standard deviations of yields at same maturities (both countries)
• Within-country correlations between 1Y and 10Y yields (Germany and Italy)
• Cross-country correlation of 1Y yields and cross-country correlation of 10Y yields

All yields expressed in annualized percentage points. Sample: January 1999 to December 2022 (monthly zero-coupon yields from Datastream).

PEPP announcement target. The second component of the distance criterion matches the observed yield curve shifts (March 18 to March 20, 2020) for 1m, 1Y, 5Y, 10Y, and 20Y German and Italian bonds. The model compares equilibrium yields under the pre-PEPP and post-PEPP fiscal/purchase scenarios (Section III.C, p. 2414). The scale of the Italian yield shift pins down the elasticity parameter $\theta$ (slope of the default rate with respect to fiscal pressure).

Robustness (Table III, p. 2427). For each of 12 alternative parameterizations (varying $\kappa$, $\sigma$, $\delta$, $\gamma$, $\psi$, $\theta$, $\hat{r}+\phi$, $\sigma_h$, $\alpha_h$, $\zeta=0$ cases), the paper reports model fit (sum of squared deviations), credit risk premium share of the total default compensation in the long run and in the PEPP episode, and mean 10Y yields for Germany and Italy. The benchmark credit risk premium share of 74% (long run) and 81% (PEPP) is robust across all plausible calibrations.

Pandemic out-of-sample test (Figure 9, p. 2431). The model compares average weekly German and Italian zero-coupon yields from February 13-19, 2020 to those from March 12-18, 2020, four weeks later. These observations are not used in estimation and serve as an out-of-sample check.

Datasets used

Dataset	Role in paper	Wiki page
German and Italian zero-coupon sovereign bond yields (Datastream)	Main estimation targets: yields at 1m, 1Y, 5Y, 10Y, 20Y maturities, Jan 1999-Dec 2022; also two-day PEPP announcement window	no page yet
ECB and Eurosystem bond holdings (ECB Data Portal / official ECB publications)	Calibrate net bond supply and Eurosystem purchase paths for APP and PEPP; maturity distribution of Eurosystem holdings as of July 2021	ECB Data Portal
Banco de Espana fiscal projections (in-house debt sustainability model)	Long-run fiscal forecasts for Italy and Germany (primary deficits, debt, interest) extended from two-year-ahead Eurosystem projections	no page yet
Eser et al. (2023) PH demand estimates	Calibrate the fraction of sovereign debt held by preferred-habitat investors (44.2% of net debt for each country)	no page yet
Cruces and Trebesch (2013) haircut evidence	Fix haircut parameter $\delta = 0.25$ consistent with international evidence on sovereign defaults	no page yet

Sample. Monthly data, January 1999 to December 2022. For the PEPP announcement, the estimation window is two trading days (March 18-20, 2020).

When to read the full paper

Read the original if you are:

• Modeling the term structure of sovereign spreads in the euro area and need a tractable affine framework with credit risk.
• Studying how ECB asset purchase programs (APP, PEPP) transmit to yields in heterogeneous monetary unions via the default risk channel.
• Seeking the yield decomposition into expectations, term premium, expected default loss, and credit risk premium (Proposition 1, Table II for numbers).
• Interested in the identification strategy: Germany’s negligible default risk identifies risk aversion from the term premium; Italy adds the spread to identify the long-run default rate and credit risk premium (pp. 2392, 2417-2421).
• Working on fiscal-monetary interactions with rollover crises, building on Calvo (1988) and Cole and Kehoe (2000) in a dynamic ATSM setting.

The key quantitative tables are Table I (estimated parameters), Table II (10Y yield decomposition and Sharpe ratios), Table III (robustness), and Table IV (conditional Sharpe ratios around the PEPP). Figures 4-9 document model fit to long-run moments, the PEPP announcement, and the pandemic shock.

Attribution and rights

Costain, James, Galo Nuno, and Carlos Thomas. “The Term Structure of Interest Rates in a Heterogeneous Monetary Union.” The Journal of Finance 80(4), August 2025, 2389-2434. DOI: 10.1111/jofi.13463.

Published under Wiley VOR terms (paywalled). Rights held by the American Finance Association. This page contains only distilled extracts (tables, equations, and findings with locators). LLM-distilled; not human-verified; not reproduced.