metadata: authors: 0: author: Moretti,Matías
1: author: Pandolfi,Lorenzo
2: author: Schmukler,Sergio L.
3: author: Villegas Bauer,Germán
4: author: Williams,Tomás
count: World
docna: 0: docna: Inelastic Demand Meets Optimal Supply of
            Risky Sovereign Bonds
docty: Policy Research Working Paper
abstracts: cdata!: This paper presents evidence of
            inelastic demand in the market for risky sovereign bonds and
            examines its interplay with government policies. The
            methodology combines bond-level evidence with a structural
            model featuring endogenous bond issuances and default risk.
            Empirically, the paper exploits monthly changes in the
            composition of a major bond index to identify flow shocks
            that shift the available bond supply and are unrelated to
            country fundamentals. The paper finds that a 1 percentage
            point reduction in the available supply increases bond
            prices by 33 basis points. Although exogenous, these shocks
            might influence government policies and expected bond
            payoffs. The paper identifies a structural demand elasticity
            by feeding the estimated price reactions into a sovereign
            debt model that isolates endogenous government responses.
            These responses account for a third of the estimated price
            reactions. By penalizing additional borrowing, inelastic
            demand acts as a commitment device that reduces default risk.
display_title: Inelastic Demand Meets Optimal
            Supply of Risky Sovereign Bonds
listing_relative_url: /research/2024/03/34288578/inelastic-demand-meets-optimal-supply-risky-sovereign-bonds
topicv3: Government Bond Markets m1326721 335,Capital Markets m1326967 332,Debt Management m1327211 451,Debt and Fiscal Sustainability m1327677 456,Financial Institutions m1505015 3797
publishtoextweb_dt: 2024-03-26T00:00:00Z
dois: 10.1596/1813-9450-10735
no_of_pages: 53
author_keys: 0: author_key: NA
1: author_key: NA
2: author_key: 142794
3: author_key: NA
4: author_key: NA
model_extractions: Policy Research Working Paper 10735 Inelastic Demand Meets Optimal Supply of Risky Sovereign Bonds

# _Matías Moretti_

_Lorenzo Pandolfi Sergio L. Schmukler Germán Villegas Bauer Tomás Williams_ Development Economics Development Research Group March 2024
Policy Research Working Paper 10735

# **Abstract**

This paper presents evidence of inelastic demand in the market for risky sovereign bonds and examines its interplay with government policies. The methodology combines bond-level evidence with a structural model featuring endogenous bond issuances and default risk. Empirically, the paper exploits monthly changes in the composition of a major bond index to identify flow shocks that shift the available bond supply and are unrelated to country fundamentals. The paper finds that a 1 percentage point reduction in the available supply increases bond prices by 33 basis points. Although exogenous, these shocks might influence government policies and expected bond payoffs. The paper identifies a structural demand elasticity by feeding the estimated price reactions into a sovereign debt model that isolates endogenous government responses. These responses account for a third of the estimated price reactions. By penalizing additional borrowing, inelastic demand acts as a commitment device that reduces default risk.

This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at matias.moretti@rochester.edu; lorenzo.pandolfi@unina.it; sschmukler@worldbank.org; gvillegasbauer@imf.org; tomaswilliams@email.gwu.edu.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Inelastic Demand Meets Optimal Supply of Risky Sovereign Bonds**<sup>∗</sup>

## **Mat ́ıas Moretti**

## **Lorenzo Pandolfi**

University of Rochester University of Naples Federico II and CSEF

**Sergio L. Schmukler Germ ́an Villegas Bauer** World Bank International Monetary Fund

**Tom ́as Williams** George Washington University

**Keywords:** emerging markets bond index, inelastic financial markets, institutional investors, international capital markets, small open economies, sovereign debt **JEL Codes:** F34, F41, G11, G15

> ∗We thank Walker Ray and the participants at a presentation held at the ASSA Annual Meetings for helpful feedback. We are grateful to Patricio Yunis for research assistance and Antonn Park for copy editing. The World Bank Chile Research and Development Center and Knowledge for Change Program (KCP) and George Washington University Facilitating Fund (GWU UFF) provided financial support for this paper. Lorenzo Pandolfi gratefully acknowledges financial support from the Unicredit Foundation. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the IMF or the World Bank, those of their Executive Directors, or the governments they represent. Moretti: matias.moretti@rochester.edu. Pandolfi: lorenzo.pandolfi@unina.it. Schmukler: sschmukler@worldbank.org. Villegas Bauer: gvillegasbauer@imf.org. Williams: tomaswilliams@email.gwu.edu.
# **1 Introduction**

Governments in emerging economies heavily depend on bonds issued in liquid international capital markets for their overall financing. The behavior of investors in these markets is thus crucial to understanding governments’ borrowing costs, default risk, and optimal debt management. Standard sovereign debt models often assume that investor demand is perfectly elastic, implying that investors are willing to lend any amount governments request at the risk-free rate plus a default risk premium. This assumption on investor behavior contrasts with a body of recent work for other asset markets that allows for a richer investor demand structure, typically involving an inelastic or downward-sloping demand (Koijen and Yogo, 2019; Gabaix and Koijen, 2021; Vayanos and Vila, 2021; Gourinchas et al., 2022; Greenwood et al., 2023).

In this paper, we present novel evidence of downward-sloping demand curves in risky sovereign bond markets and analyze their impact on governments’ optimal debt policies. In the context of risky sovereign bonds, estimating a demand elasticity is challenging for two main reasons. Ideally, one would like to identify shocks to the available bond supply that are unrelated to country fundamentals, estimate price reactions around them, and map those effects into an elasticity. However, such exogenous shocks are rare for sovereign bonds. Moreover, even if one were able to identify those shocks, governments could respond to them by adjusting future issuances or their default likelihood. Thus, the estimated price reaction might not just capture a downward-sloping demand curve but also changes in bonds’ expected payoffs.

We overcome these challenges by combining a novel identification strategy with a structural model, which allows us to isolate the endogenous responses of governments. We first estimate price reactions to well-identified shocks to the available bond supply, using monthly changes in the composition of the largest index for emerging economies bonds. To avoid endogeneity concerns, we exploit only the variation generated by the issuance or retirement of bonds from _other_ countries in the index. We find that bond prices significantly react to these shocks in the high frequency, even when they are orthogonal to country fundamentals. Our estimates imply an inverse price demand elasticity of _−_ 0 _._ 33, which we refer to as a reduced-form elasticity, as it does not account for endogenous government responses to the identified shocks. We identify the structural elasticity by indirect inference. We formulate a quantitative sovereign debt model that characterizes governments’ optimal debt and default policies as a function of observables, and we discipline it based on our reduced-form estimates. Through 1
counterfactuals, we can isolate the endogenous responses of governments to shocks and identify the structural elasticity. Our findings show that over one-third of our reduced-form elasticity is explained by endogenous government responses that decrease default risk. Last, we use our calibrated model to analyze the implications of facing an inelastic bond demand. We find that downward-sloping demand curves act as a commitment device that limits governments’ debt issuances and reduces default risk.

We start our analysis with a simple framework to guide our identification strategy. This setup features heterogeneous investors who differ in how they allocate their funds across risky assets. Specifically, they exhibit differences in their levels of activism and passivism. We define passive demand as the portion of investors’ holdings aimed at replicating the composition of the index they follow. This demand is perfectly inelastic and shifts with changes in the index. For any asset in fixed supply, an increase in the passive demand implies a leftward shift in the “effective supply,” namely the quantity available to active investors. If this shift is exogenous, and under the assumption that asset payoffs are fixed, one can use that variation to examine whether demand curves for active investors slope downward (Pandolfi and Williams, 2019; Pavlova and Sikorskaya, 2022). Nevertheless, if expected payoffs endogenously respond to changes in the effective supply, any observed price variation resulting from the shift could over or underestimate the demand elasticity.

On the empirical front, we identify exogenous shifts in a country’s effective supply of sovereign bonds by using monthly rebalancings in the J.P. Morgan Emerging Markets Bond Index Global Diversified (EMBIGD), the most widely tracked index by institutional investors for U.S. dollar-denominated sovereign bonds issued by emerging economies. Changes in the composition of this index affect the effective bond supply because they lead to similar rebalancings in the portfolios of passive investors. Due to potential tracking error costs, passive investors tend not to deviate from the index. Given the EMBIGD’s popularity, these rebalancings can have market-wide effects.

We derive a measure of flows implied by rebalancings (FIR) by combining the assets passively tracking the EMBIGD with the index’s monthly rebalancings. Qualifying new bond issuances are incorporated into the EMBIGD each month, while maturing bonds are removed. These frequent adjustments lead to changes in country weights within the index, generating passive funds flows. To avoid endogeneity issues, we construct an instrument that exploits changes in the FIR generated by the issuance or retirement of bonds from other countries in the index. As such, these changes are orthogonal to a country’s own fundamentals. In addition, we focus on changes in the face amount of the FIR (as opposed to market value) to 2
exclude changes in index composition triggered by endogenous changes in bond prices.

Our analysis reveals that a higher FIR leads to higher bond prices. On average, a 1 percentage point (p.p.) increase in the FIR corresponds to a 33 basis point increase in bond prices. These estimates imply a reduced-form inverse demand elasticity of _−_ 0 _._ 33. We find that these price reactions vary across countries with different levels of default risk. Specifically, for countries with higher default risk, a 1 p.p. FIR inflow can result in up to a 46 basis point increase in bond prices. In contrast, for safer countries, the estimates are close to zero and statistically not significant.

On the quantitative front, we formulate a sovereign debt model where the government has limited commitment and can endogenously default on its debt obligations. Standard models of this nature typically assume a perfectly elastic demand for sovereign bonds, with changes in bond prices driven solely by variations in default risk (Arellano, 2008; Chatterjee and Eyigungor, 2012). We extend these models using a richer demand structure that includes both active and passive investors and a downward-sloping demand curve for active investors. In our model, an exogenous increase in the passive demand that reduces the effective supply affects the bond price through two interconnected mechanisms. First, because the active demand is downward sloping, the implied reduction in the effective supply leads to a higher bond price. Second, this higher bond price lowers financing costs, reducing the government’s incentives to default. This, in turn, increases the expected bond repayment and further raises the price investors are willing to pay for it.

We discipline the model using our empirical estimates. Introducing passive and active investors allows us to replicate our empirical exercise, and we calibrate the model to match the estimated reduced-form elasticity. We then use the model to back out a structural elasticity, isolating changes in governments’ policies and default risk. To this end, we fix the expected bond payoffs and examine how an exogenous shift in the effective supply affects the price that active investors are willing to pay. Our findings reveal that endogenous changes in default risk account for nearly a third of the reduced-form elasticity. Moreover, we find that the larger the persistence of the supply-shifting shock, the greater the impact of changes in default risk on the overall price response.

More broadly, these results underscore the importance of accounting for issuers’ endogenous responses and the resulting changes in the expected repayment of assets. These factors must be considered to avoid significant biases in estimating demand elasticities. Our supply-shifting shock instrument is inherently more temporary than other instruments used in the literature, such as index additions or deletions or index methodological recompositions. Still, we find 3
that the bias can represent about one-third of the total price response.

Our model allows us to examine the impact of a downward-sloping demand on the optimal debt and default policies of governments. In the presence of an inelastic demand, we observe lower default risk and higher bond prices compared to a scenario with a perfectly elastic demand and similar debt levels. This outcome is not driven by investors’ preferences for holding the debt (which could lead to a convenience yield) but rather by the inelastic demand serving as a commitment device for the government. The mechanism behind it is as follows: With a downward-sloping demand, issuing an additional unit of debt decreases bond prices even if the default risk remains fixed. As a result, the government finds issuing large amounts of debt too costly and opts not to do so. An inelastic demand thus limits the maximum amount of debt the government is willing to issue. We find that this limit leads to a quantitative reduction in default risk and an increase in bond prices.

Our findings contribute to several strands of literature. First, we contribute to a longstanding literature using index rebalancings to estimate asset price reactions, demand elasticities, and changes in investors’ portfolios across different asset classes (Harris and Gurel, 1986; Shleifer, 1986; Greenwood, 2005; Hau et al., 2010; Chang et al., 2014; Raddatz et al., 2017; Pandolfi and Williams, 2019; Pavlova and Sikorskaya, 2022).<sup>1</sup> Our contribution lies in showing that demand curves slope downward in one of the most relevant markets for government financing in emerging economies: the international U.S. dollar bond market.

Another key contribution of our work is showing that, even in response to exogenous supply-shifting shocks, part of the price reaction is attributable to changes in the asset’s expected repayment and the rest to the inelastic component of demand. Our analysis can be applied to any asset, beyond sovereign bonds, whose future cash flows or payoffs are affected by movements in the effective supply. As such, it can be extended to a vast literature that uses exogenous shifts in the effective supply as an instrument to estimate demand elasticities. Typical examples are sovereign and corporate bonds and equities from both developed and emerging economies.

Second, a growing literature on inelastic financial markets emphasizes the role of the demand side in explaining asset prices across various financial markets (Koijen and Yogo, 2019; Gabaix and Koijen, 2021; Vayanos and Vila, 2021). Taking as given expected asset payoffs, this literature analyzes how an inelastic demand affects the pricing of risk-free U.S. Treasuries (Krishnamurthy and Vissing-Jorgensen, 2012; Greenwood et al., 2015; Mian et al., 2022; Jiang

> 1Beyond index rebalancings, Droste et al. (2023) use high-frequency U.S. Treasury auctions to estimate the effect of demand shocks on Treasury yields.

4
et al., 2021b) and international financial assets (Koijen and Yogo, 2020; Gourinchas et al., 2022; Greenwood et al., 2023).<sup>2</sup> Similar to our study, Choi et al. (2022) analyze the effects of a downward-sloping demand on the optimal issuance of safe government bonds. In contrast, we focus on the interplay between a downward-sloping demand curve, default risk, and the provision of risky bonds. We show that the demand elasticity interacts with default risk and influences a government’s supply of risky bonds. Failing to account for this endogenous link can lead to biases when estimating demand elasticities.

Third, our paper also connects to a body of work examining how changes in the investor base of government debt impact bond yields (Warnock and Warnock, 2009; Dell’Erba et al., 2013; Peiris, 2013; Arslanalp and Poghosyan, 2016; Ahmed and Rebucci, 2022). A closely related paper (Fang et al., 2022) develops a demand system to quantify how changes in the composition of investors (domestic versus foreign, banks versus non-banks) affect government bond yields in international markets. In this paper, we exploit exogenous changes in the composition of the investor base (passive versus active funds) to provide evidence of downwardsloping demand curves for risky sovereign bonds.

Fourth, our paper relates to a large literature on quantitative sovereign debt models. Our framework extends standard models (Aguiar and Gopinath, 2006; Arellano, 2008; Chatterjee and Eyigungor, 2012) by incorporating two different investor types (active and passive) and introducing a downward-sloping demand. This richer structure allows us to discipline the model using our reduced-form estimates.<sup>3</sup> We then use the model to isolate the role of default risk behind those estimates and to back out the structural demand elasticity.

In our analysis, we are agnostic about the mechanisms behind the downward-sloping demand. Previous work by Borri and Verdelhan (2010), Lizarazo (2013), Pouzo and Presno (2016), and Arellano et al. (2017) analyze sovereign debt models with risk-averse investors. In their models, investors’ downward-sloping demand is a by-product of their risk aversion. In other words, investors are inelastic only because they must be compensated for each additional unit of risky debt they hold. However, several different mechanisms can generate a downward-sloping demand. For example, it can be explained by regulatory limitations, such as a Value-at-Risk (VaR) constraint (as in Miranda-Agrippino and Rey, 2020), by investors’ buy-and-hold strategies (which can be rationalized by a taste for simplicity or

> 2A related literature focuses on U.S. and international corporate bond markets (Dathan and Davydenko, 2020; Bretscher et al., 2022; Calomiris et al., 2022; Kubitza, 2023).

> 3In this regard, our paper connects with recent work by Costain et al. (2022), who introduce endogenous default risk into a Vayanos-Vila preferred habitat model to analyze the term structure of interest rates in the European Monetary Union.

5
agency frictions), or by fixed-share mandates specifying how investors should allocate their funds across assets (as in Gabaix and Koijen, 2021). Our setup relies on a flexible demand structure that can accommodate any of these potential drivers. Our aim is not to uncover the causes of investors’ inelastic behavior but rather to examine its aggregate implications.

The rest of the paper is structured as follows. Section 2 introduces a simple framework to guide our analysis. Section 3 presents the empirical analysis, including details on the institutional setup of EMBI indexes, data sources, the identification strategy, and results. Section 4 formulates a sovereign debt model with endogenous default and inelastic investors. Section 5 presents the quantitative analysis, and Section 6 concludes.

# **2 Demand Elasticity for Risky Bonds**

To guide our analysis, we introduce a simple framework featuring heterogeneous investors who differ in how they allocate their funds across risky assets. This stylized framework illustrates how, under some assumptions, exogenous shifts in the demand of certain investors can be used to estimate a reduced-form price demand elasticity. In doing that, we show how endogenous changes in assets’ expected repayment can influence the estimated elasticity. Although we focus on the case of risky bonds, our framework can be applied to any risky asset (e.g., equities).

## **2.1 Model**

Investors are heterogeneous in how they allocate their funds across risky assets. Let _j_ = _{_ 1 _, ..., J}_ denote the investor. As in Gabaix and Koijen (2021), we assume that each investor _j_ has a mandate or rule that specifies how they should allocate their funds across _i_ = _{_ 1 _, ..., N }_ risky bonds. To tightly link the model with our empirical analysis, we further assume that investors track the composition of a benchmark index _I_ and differ in how actively or passively they do so. Let **_w_** _t_ = � _wt_<sup>1</sup><sup>_, ..., w_</sup> _t_<sup>_N_</sup> � denote the vector of time-varying index weights for each constituent bond of index _I_ . Markets are competitive, and investors take asset prices as given.

We define _x_<sup>_i_</sup> _jt_<sup>=</sup> _qWt_<sup>_iB_</sup> _jtj_<sup>_i_</sup> _t_<sup>astheshareofwealththatinvestor</sup><sup>_j_investsinbond</sup><sup>_i_attime</sup> _t_ . The term _qt_<sup>_i_denotestheunitpriceofbond</sup><sup>_i_,</sup><sup>_B_</sup> _j,t_<sup>_i_denotestheend-of-periodholdingsof</sup> investor _j_ in bond _i_ , and _Wj,t_ denotes their wealth. The share _x_<sup>_i_</sup> _jt_<sup>isgivenbythefollowing</sup> exogenous mandate: 6
The term _θj_ parameterizes the degree of activeness or passiveness of investor _j_ . Purely passive investors can be characterized by _θj_ = 0, indicating that their portfolio simply replicates the benchmark index _I_ . Conversely, active and semi-active investors are those with _θj ∈_ (0 _,_ 1], which captures the fraction of their portfolio that is not linked to index _I_ . Within their active allocation, investors apportion a fixed fraction, _ξj_<sup>_i_,oftheirwealthtobond</sup><sup>_i_and</sup> a varying component determined by Λ _jπ_ ˆ _i,t_ � _rt_<sup>_i_</sup> +1�, where Λ _j >_ 0 parameterizes their demand elasticity and _π_ ˆ _i,t_ is an arbitrary function of the next-period excess return of bond _i_ , _rt_<sup>_i_</sup> +1<sup>.</sup> ˆ For instance, if _πi,t_ � _rt_<sup>_i_</sup> +1� = E _t_ � _rt_<sup>_i_</sup> +1�, investors allocate a higher share of their wealth to bonds with higher expected excess returns.

As we show next, the reduced-form mandate in Equation (1) allows us to introduce an aggregate demand elasticity that can be parameterized by **Λ** _≡{_ Λ1 _, ..,_ Λ _J }_ . While this mandate can have different microfoundations (as shown in Appendix ), we take it as given for our analysis. Our goal is not to explain the reasons behind the inelastic demand for risky bonds but rather to examine its implications.<sup>4</sup> After adding up all the individual demands, we can write the market-clearing condition as follows: where _Bt_<sup>_i_is the bond supply, and</sup> _A_<sup> ̃</sup><sup>_i_</sup> _t_<sup>_≡_�</sup> _j_<sup>_Wj,tθj_</sup> � _ξj_<sup>_ie_Λ</sup><sup>_j_ˆ</sup><sup>_πi,t_</sup> � and _T_<sup> ̃</sup> _t_<sup>_i_(</sup><sup>_w_</sup> _t_<sup>_i_)</sup><sup>_≡_�</sup> _j_<sup>_Wj,t_(1</sup><sup>_−θj_)</sup><sup>_w_</sup> _t_<sup>_i_</sup> denote the market-value active and passive demands, respectively. The passive demand, _T_ ̃ _t_<sup>_i_(</sup><sup>_w_</sup> _t_<sup>_i_),istheportionofinvestors’holdingsaimedatreplicatingtheindexcompositionthey</sup> follow. It captures the holdings of both semi- and fully passive investors. We explicitly write _T_ ̃ _t_<sup>_i_(</sup><sup>_w_</sup> _t_<sup>_i_)asafunctionof</sup><sup>_w_</sup> _t_<sup>_i_toemphasizeitsdependenceontheindexweights.Fortherest</sup> of the analysis, it is useful to rewrite this market-clearing condition as _Bt_<sup>_i_=</sup><sup>_Ai_</sup> _t_<sup>+</sup><sup>_T_</sup> _t_<sup>_i_</sup> � _wt_<sup>_i_</sup> �, where _A_<sup>_i_</sup> _t_<sup>and</sup><sup>_T_</sup> _t_<sup>_i_</sup> � _wt_<sup>_i_</sup> � denote the face-value active and passive demands.<sup>5</sup> For any bond _i_ in fixed supply, an increase in the passive demand implies a decrease in the supply of bonds available to active investors (i.e., a leftward shift in the effective or residual supply). If this increase is exogenous, one can use that variation to analyze whether the demand curves for active investors slope downward. Figure 1 illustrates this point. If the active demand is fully elastic, then an exogenous shift in _Tt_<sup>_i_(</sup><sup>_w_</sup> _t_<sup>_i_)shouldnotaffectbond</sup> prices (Panel (a)). Conversely, bond prices should react to this shift if the active demand slopes downward (Panel (b)).

> 4As argued by Gabaix and Koijen (2021): “While identifying the exact reasons for low market elasticity is interesting, this question has a large number of plausible answers. Fortunately, it is possible to write a framework in a way that is relatively independent to the exact source of low elasticity [...].”

> 5In Section 2.2, we impose additional structure to obtain closed-form expressions for _Ait_<sup>and</sup><sup>_T_</sup> _t_<sup>_i_</sup> � _wt_<sup>_i_</sup> �.

7
Figure 1 <!-- Start of picture text --> Index rebalancing and the demand elasticity<br>(a) Perfectly elastic demand (b) Inelastic demand<br>Effective Effective<br>Price Price<br>Supply Supply<br>q 1 i<br>q 0 i = q 1 i<br>A i<br>q 0 i A i<br>B i −T 1 i B i −T 0 i Quantity B i −T 1 i B i −T 0 i Quantity<br><!-- End of picture text --> Note: The figure depicts a decrease in the effective supply driven by an increase in _T_<sup>_i_</sup> . Panel (a) considers the case when the residual demand is fully elastic and Panel (b) when it is price sensitive.

Based on this graphical intuition, one could exploit exogenous changes in index weights _wt_<sup>_i_tocomputeaninstrumentforshiftsinthepassivedemand,∆</sup><sup>_T_</sup> _t_<sup>_i≡T_</sup> _t_<sup>_i_</sup> +1<sup>(</sup><sup>_w_</sup> _t_<sup>_i_</sup> +1<sup>)</sup><sup>_−T_</sup> _t_<sup>_i_(</sup><sup>_w_</sup> _t_<sup>_i_),</sup> and estimate high-frequency bond price responses around those shifts. For a given ∆ _Tt_<sup>_i_,we</sup> can then estimate the following reduced-form inverse demand elasticity: Equation (3) and the use of changes in index composition as a supply-shifting instrument is a standard practice in the literature to estimate demand elasticities in equity markets. Shleifer (1986) was the first to use index additions to the S&P 500 as an exogenous instrument for ∆ _Tt_<sup>_i_toanalyzewhetherdemandcurvesforequitiesslopedownward.Morerecently,</sup> Pavlova and Sikorskaya (2022) employ a similar strategy and a regression discontinuity design on Russell equity indexes. For the case of bonds (sovereign or corporate), exploiting index changes to construct ∆ _Tt_<sup>_i_hasadditionalchallenges.Inmanycases,changesinindexweights</sup> coincide with large bond issuances that tend to reflect information about the issuer’s own fundamentals. We address this issue using a novel instrument in Section 3.

# **2.2 Endogenous Issuer Responses and Bond Payoffs**

To directly map Equation (3) into a structural elasticity, _η_<sup>_i_</sup> , we need to assume that the intrinsic value of the asset is unaffected by ∆ _Tt_<sup>_i_.However,exogenousshiftsintheeffective</sup> supply might influence issuers’ policies and expected asset payoffs. For example, in the case 8
of risky bonds, a positive ∆ _Tt_<sup>_i_thatleadstoahigherbondpriceimpliesalowerborrowing</sup> cost, potentially affecting the issuer’s default likelihood or its (future) bond issuances. Even if the current supply _Bt_<sup>_i_remainsfixed,theseendogenousresponsescanaffecttheexpected</sup> payoff from holding the bond and, therefore, its price.

Price responses to exogenous shifts in the effective supply can thus capture both an inelastic demand component and endogenous changes in expected payoffs. To decompose these effects, we put more structure behind investor demand, allowing us to derive a closedˆ E _t_ <u>(</u> _rt_<sup>_i_</sup> <u>+1</u><sup><u>)</u></sup> form solution for the price. We assume that _πi,t_ ( _rt_<sup>_i_</sup> +1<sup>) =</sup> V _t_ ( _rt_<sup>_i_</sup> +1<sup>)sothattheactivedemand</sup> is a function of the bond’s expected excess return and its variance (the Sharpe ratio).<sup>6</sup> We define _R_<sup>_i_</sup> _t_ +1<sup>asthenext-periodrepaymentperunitofthebondsothat</sup><sup>_r_</sup> _t_<sup>_i_</sup> +1<sup>_≡_</sup> _Rq_<sup>_i_</sup> _<u>tt</u>_ <u>+1</u><sup>_i_</sup> _− rf_ , where _rf_ denotes the risk-free rate.<sup>7</sup> Based on these definitions and the market-clearing condition in Equation (2), we can write the equilibrium bond price as E _t_ <u>(</u> _R_<sup>_i_</sup> _t_ <u>+1</u><sup><u>)</u></sup> The term _r_<sup>_f_</sup> captures the price under perfectly elastic investors, which is a function of the expected next-period repayment. On the other hand, the Ψ<sup>_i_</sup> _t_<sup>functioncapturesthe</sup> demand’s downward-sloping nature and is given by <u>1</u> The term _κ_<sup>_i_</sup> _t_<sup>(</sup><sup>**Λ**)</sup><sup>_≡_</sup> <u>�</u> _j_<sup>Λ</sup><sup>_jWj,tθjξ_</sup> _j_<sup>_i_characterizesthedegreeofinelasticityinthemarketfor</sup> bond _i_ . When _κ_<sup>_i_</sup> _t_<sup>(</sup><sup>**Λ**) = 0,thedemandisperfectlyelasticandthepriceonlydependsonthe</sup> expected repayment. The _Bt_<sup>_i−T_</sup> _t_<sup>_i_componentiswhatwehavereferredtoastheresidual</sup> supply, and _A_<sup> ̄</sup><sup>_i_</sup> _t_<sup>capturestheinelasticportionoftheactivedemand,whichdependsonthe</sup> fixed component of investors’ mandates, _ξj_<sup>_i_.SeeAppendixforthedetailsandderivations.</sup> From Equation (5), one can decompose the reduced-form elasticity into two components: the structural elasticity, _η_<sup>_i_</sup> , and endogenous changes in expected repayment, _α_<sup>_i_</sup> . In particular,

> 6This is a similar specification to the one in Gabaix and Koijen (2021), which is a function of expected excess returns and a shock to tastes or perceptions of risk. As shown in our quantitative analysis, this specification allows us to capture a demand elasticity that differs across countries with different levels of default risk, which is consistent with our empirical findings.

> 7The repayment function depends on the expected default and next-period issuances. For a short-term (one-period) risky bond, it is given by _R_<sup>_i_</sup> _t_ +1<sup>= 1</sup><sup>_−di_</sup> _t_ +1<sup>,where</sup><sup>_d_= 1denotesadefault.Forlong-termbonds,</sup> it also depends on next-period issuances as they affect the next-period bond price. In Section 4, we analyze this function in detail.

9
<!-- Start of picture text --> Figure 2<br>Endogenous changes in expected payoffs<br>(a) Higher expected payoffs (b) Lower expected payoffs<br>Effective Effective<br>Price Price<br>Supply Supply<br>q 1 i<br>A i 1<br>q 0 i qq 10 ii A i 0<br>A i 0 A i 1<br>B i −T 1 i B i −T 0 i Quantity B i −T 1 i B i −T 0 i Quantity<br><!-- End of picture text --> Note: The figure depicts a reduction in the effective supply driven by an increase in _T_<sup>_i_</sup> . Panel (a) considers a case in which the expected asset payoffs increase after the effective supply decreases, while Panel (b) shows the opposite case.

we can extend Equation (3) as follows: Figure 2 provides a graphical illustration. If a positive ∆ _Tt_<sup>_i_raises the next-period expected</sup> repayment (i.e., _α_<sup>_i_</sup> _<_ 0), investors will be willing to pay a higher price for any given _Bt_<sup>_i_,</sup> leading to an upward shift in the active demand (Panel (a)). Failing to account for this effect might lead to the conclusion that the demand curve is steeper (more inelastic) than it truly is. Conversely, if a positive ∆ _Tt_<sup>_i_lowersthenext-periodexpectedrepayment(i.e.,</sup><sup>_αi>_0),it</sup> would cause the active demand to shift downward (Panel (b)). This shift might lead to the demand curve being estimated as flatter (more elastic) than it truly is.

Since bond prices and payoffs are jointly determined, it is challenging to disentangle the effects on bond prices due to the downward-sloping demand from those resulting from changes in expected payoffs. To formally quantify each mechanism separately, one would need a structural model in which bond prices, the supply of the bond, and its payoffs are endogenous outcomes. Put differently, to quantify _α_<sup>_i_</sup> , we must first understand how ∆ _Tt_<sup>_i_</sup> affects issuers’ policies (debt issuances and default). One could argue that more persistent changes in _Tt_<sup>_i_wouldlikelyhavealargerimpactontheissuers’policies,potentiallymaking</sup> the absolute value of _α_<sup>_i_</sup> larger. Conversely, a more transitory shock could lead to a smaller _α_<sup>_i_</sup> in absolute terms. In any case, one must consider both effects.

In the next section, we construct a novel instrument for ∆ _Tt_<sup>_i_,basedonmonthlyindex</sup> 10
rebalancings for a major sovereign bond index for emerging economies. We estimate bond price reactions to these rebalancings and map the reactions into a reduced-form elasticity. In Section 4, we formulate a structural model to back out the structural elasticity.

# **3 Empirical Analysis**

## **3.1 Index Rebalancings as Passive Demand Shocks**

In this section, we exploit monthly rebalancings in the EMBIGD to identify exogenous shifts in the available bond supply for active investors (the effective supply). The EMBIGD tracks the performance of emerging market sovereign and quasi-sovereign bonds in U.S. dollars issued in international markets.<sup>8</sup> Unlike other indexes that use a traditional market capitalization-based weighting scheme, the EMBIGD restricts the weights of countries with above-average debt outstanding by including only a fraction of their face amount of debt outstanding.<sup>9</sup> Among bond indexes for emerging economies, the EMBIGD is the most widely tracked, followed by funds with combined assets under management (AUM) of around US$300 billion in 2018 (Appendix Figure 2).<sup>10</sup> Rebalancings in the EMBIGD index, triggered by bond inclusions and exclusions, occur on the last U.S. business day of each month. J.P. Morgan announces these updates through a report detailing the updated index composition. Consequently, passive investors tracking the index adjust their portfolios by buying or selling bonds to match the new index weights.

Following Pandolfi and Williams (2019), we construct the flows implied by the rebalancings (FIR) measure for each country at each rebalancing date. The FIR quantitatively measures the relative change in passive demand for a country’s sovereign bonds resulting from index rebalancing. A 1% FIR can be interpreted as a 1% reduction in the available bond supply in

> 8The index includes bonds with a maturity of at least 2 _._ 5 years and a face amount outstanding of at least US$500 million. To be classified as an emerging economy, a country’s gross national income (GNI) per capita must be below an Index Income Ceiling (IIC) for three consecutive years. The IIC is defined by J.P. Morgan and adjusted every year by the growth rate of the World GNI per capita, Atlas method (current US$), provided by the World Bank. Bonds in the index must settle internationally and have accessible and verifiable bid and ask prices. Once included, they can remain in the index until 12 months before maturity. Local law instruments are not eligible.

> 9The J.P. Morgan Emerging Markets Bond Index Global (EMBIG) has the same bond inclusion criteria as the EMBIGD. The only difference between them is that while the EMBIG uses a market capitalization weighting scheme, the EMBIGD modifies this scheme to limit the weights of countries with above-average debt outstanding. Appendix Figure 1 plots the EMBIG country weights of both the EMBIG (a more traditional market-based index) and EMBIGD versions for December 2018. Appendix describes the rules that the EMBIGD uses to compute the weights of the instruments included in the index.

> 10Appendix Figures 3 and 4 show the high preponderance of U.S. dollar-denominated sovereign debt issued by emerging economies in international markets.

11
the market. More precisely, the FIR measure is constructed as follows: The numerator captures the change in passive demand implied by the index rebalancing, and is defined as ∆ _T_<sup> ̃</sup> _c,t ≡_ ( _wc,t − wc,t_<sup>_BH_)</sup><sup>_At_.Itmeasurestheamountoffundsthat,onagiven</sup> rebalancing date, enter or leave a country due to the rebalancing in the portfolio of passive investors tracking index _I_ . For convenience, we normalize ∆ _T_<sup> ̃</sup> _c,t_ by the market value of the bonds available to active investors, _qc,t−_ 1 _Bc,t−_ 1 _− wc,t−_ 1 _At−_ 1.

The first term in the parentheses, _wc,t_ , is the benchmark weight for country _c_ , at time _t_ , in index _I_ . It is defined as _wc,t ≡_<sup>_<u>qc,tB</u>_</sup> _qt_<sup>_c_</sup> _I_<sup>_<u>,</u>_</sup> _t_<sup>_tfc,t_</sup> , where _qc,tBc,t_ denotes the market value of bonds from country _c_ at time _t_ . _qc,t_ denotes the price and _Bc,t_ denotes the face amount outstanding. _fc,t_ is the face amount share of country _c_ ’s bonds in the index, which depends on the country’s amount of outstanding debt rather than on market values.<sup>11</sup> The denominator of _wc,t_ denotes the market value of index _I_ , calculated as _qt_ times _It_ , where _qt_ is the unit price of the index and _It_ is the number of available index units. That is, it captures the relative market capitalization of country _c_ ’s sovereign bonds included in _I_ . The second term in the parentheses, _wc,t_<sup>_BH_,isdefinedas</sup><sup>_w_</sup> _c,t_<sup>_BH_</sup> _≡ wc,t−_ 1 _<u>qcq,tt/q/qct−,t−</u>_ 1 1<sup>.12Itcapturesthe“buy-and-hold</sup> weight,” defined as the weight country _c_ would have had at time _t_ if the index composition had remained unchanged.<sup>13</sup> _At_ in ∆ _T_<sup> ̃</sup> _c,t_ represents the AUM of investors passively tracking the EMBIGD.

Although index changes drive the FIR, this measure might not be entirely exogenous to a country’s fundamentals, for two reasons. First, the FIR is affected by countries’ sovereign bond issuances. When a country issues new bonds that become part of the index, its weight increases, leading to a higher FIR. Second, even for countries whose _Bc,t_ and share _fc,t_ remain constant, the FIR can be mechanically correlated to present or past bond price changes. Note that, generally,<sup>_<u>∂FIR</u>_</sup> _∂qc,t_<sup>= 0and</sup><sup>_∂FIR_</sup> _∂qc,t−_<sup>_c_</sup> 1<sup>_<u>,t</u>_= 0.ThiscanbeseenfromEquation(7),where</sup> the former derivative will not be equal to zero given the effect of current prices through the numerator. In turn, the latter derivative might not be equal to zero due to the effects of past 11To preserve diversification, J.P. Morgan applies a scheme that entails a cap to the weight of countries with greater-than-average sovereign bond markets, for whom the diversification coefficient is therefore smaller than one, _fc,t <_ 1. In contrast to the EMBIGD, for purely market capitalization-weighted indexes (such as the EMBI Global), _fc,t_ = 1 _, ∀c,t_ .

> 12This buy-and-hold weight is computed as if no bonds had entered or exited the index at time _t_ .

> 13Note that _wc,tBH_<sup>=</sup> _qc,tfcq,ttI−t_ 1 _−B_ 1 _c,t−_ 1 . Absent any change in the index composition (i.e., inclusions or exclusions of new bonds or countries), if the price of a country’s sovereign bonds increases more than that of other countries in the index, the weight of that country in the index increases. Nevertheless, investors do not need to rebalance their portfolios as the “buy-and-hold weight” coincides with the new weight in the index, _wc,t_ .

12
prices through the denominator. Given that we aim to isolate the impact of passive demand shocks on bond prices, the potential endogeneity of the FIR could bias our estimates.

We address the potential challenge of the FIR endogeneity in two ways. First, for each rebalancing event, we consider only countries whose amount outstanding of bonds, _Bc,t_ , does not change relative to the previous period. In other words, we focus only on countries that experience no new issuances, bond repurchases, or the removal of bonds from the index due to maturity on the given date. The rationale behind this approach is, for each country, to isolate and examine the impact of index changes driven by changes in other countries’ standings within the index at each specific point in time. Second, we exploit the fact that the EMBIGD’s weighting scheme is based on the face amount of outstanding bonds. This is important as it allows us to net out the variation potentially correlated with current or past bond price changes.

In particular, we construct an instrument for the FIR based on a synthetic index in which country weights are only a function of the diversified face amount outstanding of bonds included in the index, not on market values, _wc,t_<sup>_FA≡_</sup> <u>�</u> _<u>fcc</u>_<sup>_f_</sup> _<u>,t</u>_<sup>_c,t_</sup> _Bc_<sup>_B_</sup> _<u>,t</u>_<sup>_c,t_.Wethencomputethe</sup> fractional change in the synthetic index: Focusing on countries whose debt outstanding in the index remains unchanged ( _Bc,t_ = _Bc,t−_ 1), the instrument becomes By instrumenting the FIR with _Zc,t_ , we can isolate the variation in the FIR that is solely attributable to changes in the outstanding amount of bonds from other countries. These changes are a result of fluctuations in the relative size of other countries’ sovereign bond markets or alterations in the diversification coefficient, _fc,t_ . Because _fc,t_ is not a function of bond prices, and because we only consider countries where _Bc,t_ is fixed,<sup>_∂Z_</sup> _∂qc,t_<sup>_c,t_=</sup> _∂q∂Zc,tc−,t_ 1<sup>= 0.</sup> We then use the _Zc,t_ instrument to estimate the effect of exogenous demand changes induced by passive flows on sovereign bond prices. We take advantage of the specific timing of the rebalancings: index changes always occur on the last business day of each month. For each rebalancing date, we can therefore identify pre- and post-rebalancing periods and estimate the FIR’s effect (instrumented by _Zc,t_ ) on bond prices.

We adopt an instrumented difference-in-differences (DDIV) design and estimate the 13
following main specification using two-stage least squares (2SLS): where _qi,t,h_ is the price of bond _i_ at rebalancing date _t_ , _h_ trading days before or after the rebalancing information is confirmed. For example, _h_ = 1 indicates the first trading after J.P. Morgan releases the EMBIGD’s new composition. This happens during the trading hours on the last business day of each month, meaning that _h_ = 1 falls on this day. For each rebalancing date _t_ , we consider a symmetric _h_ -day window around it. _θc_ ( _i_ ) _,t_ are country-month fixed effects, and _θb_ ( _i_ ) _,t_ are bond characteristics-month fixed effects, including maturity, rating, and bond type (sovereign or quasi-sovereign). _FIR_<sup>�</sup> _c_ ( _i_ ) _,t_ represents the flows implied by the rebalancing, instrumented with the percentage change in the theoretical index weights, _Zc,t_ . We obtain _FIR_<sup>�</sup> _c_ ( _i_ ) _,t_ by regressing _FIRc,t_ on _Zc,t_ (first stage). 1 _h∈Post_ is an indicator function equal to 1 in the _h_ days after the rebalancing and equal to 0 in the _h_ days before. **X** _i,t_ is a vector of monthly bond controls, including the bond’s face amount and (beginning-of-month) spread. Our coefficient of interest is _β_ , which captures the FIR’s effect on bond prices. In particular, how much the average log price change (or percentage change) around the rebalancing day varies with a 1 p.p. exogenous increase in _FIRc_ ( _i_ ) _,t_ . Our main specification replaces the country-month fixed effects, bond characteristics-month fixed effects, and bond controls with bond-month fixed effects.

We also estimate a leads and lags regression in which the instrumented FIR is interacted with trading-day dummies. This analysis allows us to both explore the dynamic effect of the FIR and test for parallel trends before the rebalancing.

# **3.2 Data and Summary Statistics**

We collect data from different sources to compute the FIR and our instrument. Most of the variables used in the analysis come directly from J.P. Morgan. However, one variable is not straightforward to measure: the AUM of funds that passively track the EMBIGD, _At_ . While J.P. Morgan provides data on the amount of assets benchmarked against their indexes, it does not distinguish between passive and active funds. Additionally, even if these data were available, many active funds might passively manage a significant share of their portfolios, as highlighted by Pavlova and Sikorskaya (2022).

To compute _At_ , we start with J.P. Morgan data on assets tracking the EMBIGD, which we then adjust based on an estimate of the share of passive funds. The estimation of this share involves the following steps. We retrieve data from Morningstar on the asset holdings 14
Table 1 Summary statistics

|Variable|Mean|Std. dev.|25th pctl|75th pctl|Min|Max|
|---|---|---|---|---|---|---|
|Log(price)|4.64|0.13|4.59|4.68|3.07|5.19|
|Instrumented FIR (%)|-0.16|0.22|-0.34|0.00|-0.70|0.25|
|Stripped spread (bps)<br>|278|286|129|357|0|4,904|
|EIR duration (%)<br>|6.35|3.91|3.48|7.71|-0.03|19.08|
|Average life (years)<br>|9.6|8.9|4.0|9.9|1.0|99.8|
|Face amount (billion U.S. dollars)<br>|1.3|0.8|0.7|1.6|0.5|7.0|
|CDS (bps)|298|694|104|285|42|6,171|

Note: This table displays summary statistics for the main variables in the analysis. _Stripped Spread_ is the difference between a bond yield-to-maturity and the corresponding point on the U.S. Treasury spot curve, where the value of collateralized flows are “stripped” from the bond. _EIR Duration_ measures the sensitivity of dirty prices to parallel shifts of the U.S. interest rates, expressed as the percentage change of dirty price if all U.S. interest rates change by 100 basis points. _Average Life_ is the weighted average period until principal repayment, and _CDS_ denotes the five-year credit default swap spread of USD-denominated sovereign bonds. Sources: Bloomberg, Datastream, J.P. Morgan Markets, Morningstar Direct, and authors’ calculations.

of funds benchmarked against the EMBIGD and EMBI Global Core for 2016–2017.<sup>14,15</sup> For each fund, we compute their _Passive Share_ = 100 _− Active Share_ , where _Active Share_ is the measure developed by Cremers and Petajisto (2009). We first estimate this variable at the country level, which is the level of the FIR measure.<sup>16</sup> This allows us to separate, even for active funds, the fraction of a fund’s portfolio that might be passive or active, consistent with the model in Section 2. We then compute the average _Passive Share_ weighted by each fund’s AUM. With this strategy, we obtain an estimated passive fund share of 50%.<sup>17</sup> We calculate _At_ by adjusting the AUM tracking the EMBIGD index, using a rescaling factor of 50%, thus obtaining the estimated passive funds tracking the index we use to compute the FIR.<sup>18</sup> We gather data on individual bond prices from Datastream and obtain several bond characteristics (maturity and duration, among others) directly from J.P. Morgan Markets. To clean our dataset, we drop extreme values of daily returns, stripped spreads, and _Zc,t_ .<sup>19</sup> We drop stripped spreads below 0 or above 5 _,_ 000 basis points as well as observations below

> 14The EMBI Global Core uses the same diversification methodology as the EMBIGD to calculate the bond weights, as described in Appendix . The criteria for including bonds in the EMBI Global Core is the same as that for the EMBIGD (and the EMBI Global), except the minimum face amount of the bonds must be US$1 billion and the maturity required to be maintained in the index is of at least one year.

> 15The data sample periods utilized in the paper are determined by data access constraints.

> 16We compute the _Active Share_ at the country level by using the country weights in the index and in the funds’ portfolios rather than bond weights. For the portfolios, we only assign bonds to a given country if they are included in the EMBIGD. Specifically, a country’s weight in a portfolio is determined by adding together the weights of all bonds from that country that are included in the EMBIGD.

> 17Appendix Table 1 provides results using alternative shares of passive funds used to construct the FIR measure. Although our quantitative estimates change slightly, the qualitative implications remain the same.

> 18For comparison, we construct _Active Share_ at the bond level, obtaining a value-weighted average of 72%. Cremers and Petajisto (2009) show an average value-weighted _Active Share_ that fluctuates between 55% and 80%.

> 19Stripped spread is defined in the notes of Table 1.

15
<!-- Start of picture text --> 1<br>_ 9° °<br>SS 5 @ °<br>~~a PO, 74io 0<br>no]N ° 0 alo gies.Do a<br>o °° Sauer °<br>3 ° ¢ Boe<br>® 5 8 ° ote Be<br>©e ;<br>A<br>32 A 0 1 2<br>Residualized Z (%)<br><!-- End of picture text --> nm
# **3.3 Results**

Table 2 reports the results of our baseline estimation using a five-day window around each rebalancing event (i.e., _h ∈_ [ _−_ 5 _,_ 5]).<sup>21</sup> Our coefficient of interest, _β_ , is always positive and statistically significant in the different specifications. The estimate in our preferred specification, with bond-rebalancing month fixed effects, implies that a 1 p.p. increase in the FIR increases bond returns by 0 _._ 29 p.p.

One potential concern with these results is that bonds receiving a larger or smaller FIR during the rebalancings are on different price trends even before the rebalancing date. To show that this is not the case, we estimate a specification with leads and lags, where the instrumented FIR is interacted with a dummy for each of the trading days around the rebalancing event: where 1 _h_ are dummy variables equal to 1 for the _h_ trading day in our [ _−_ 5 : +5] estimation window and 0 otherwise.

The estimated _βh_ coefficients are reported in Figure 4. On the initial four of the five trading days before the index rebalancing, changes in the FIR are not associated with systematic differences in bond prices. Instead, in the trading days after the event, the coefficient increases, becomes positive and significant, and eventually stabilizes below 0 _._ 4 by the end of our estimation window. We do observe a slight anticipation in the day before the index rebalancing, which is not uncommon in these setups. For example, this is consistent with the patterns of portfolio rebalancings by different institutional investors highlighted in Escobar et al. (2021), who show that institutional investors could move in the day before the actual index rebalancing event. In the last column of Table 2, we show the estimates based on our main specification of Equation (10) but after excluding the trading day before the index rebalancing. This leads to an estimate of 0 _._ 33, which we take as our baseline since it does not contain any anticipation effect in the pre-period.

One related concern is the potential for increased anticipation throughout the month. Between the middle and end of every month, J.P. Morgan releases preliminary estimates about end-of-month face amounts, market values, and bond weights. While it is conceivable that active investors traded on this information before the actual index rebalancing date, our data do not support this behavior. Normally, if a significant number of investors were

> 21Appendix Table 2 shows that our results are robust to alternative windows around the rebalancing events. The results are also robust to excluding quasi-sovereign bonds from the analysis (Appendix Table 3).

17
Table 2 Log price and FIR

|Dependent variable: log price<br>Symmetric window:|[-5:+5]||Excl. h=-1|
|---|---|---|---|
|FIR<br>-4.014 ***<br>-0.170<br>1.252 **<br>(0.583)<br>(0.740)<br>(0.572)||||
|FIR X Post<br>0.286 ***<br>0.286 ***<br>0.286 ***|0.287 ***|0.286 ***|0.330 ***|
|(0.091)<br>(0.091)<br>(0.092)|(0.092)|(0.090)|(0.098)|
|Post<br>0.001 **<br>0.001 **<br>0.001 **|0.001 **|0.001 **|0.001 **|
|(0.000)<br>(0.000)<br>(0.000)|(0.000)|(0.000)|(0.000)|
|Bond FE<br>Yes<br>Yes<br>Yes|Yes|No|No|
|Month FE<br>No<br>Yes<br>No|No|No|No|
|Bond char.-month FE<br>No<br>No<br>Yes|Yes|No|No|
|Country-month FE<br>No<br>No<br>No|Yes|No|No|
|Bond-month FE<br>No<br>No<br>No|No|Yes|Yes|
|Bond controls<br>No<br>No<br>No|Yes|No|No|
|Observations<br>107,138<br>107,138<br>107,138|107,098<br>|107,138|96,424|
|N. of bonds<br>738<br>738<br>738|738|738|738|
|N. of countries<br>68<br>68<br>68|68|68|68|
|N. of clusters<br>1,618<br>1,618<br>1,618|1,617|1,618|1,618|
|F (frst stage)<br>1,000<br>693<br>1,008|1,958|2,017|2,023|
|Note:<br>This table presents 2SLS estimates of log bond prices<br>instrumented by _Z_ (Equation (9)), around rebalancing dates.|on the FIR<br> The frst- an|measure (E<br>d second-sta|quation (7)),<br>ge equations|
|are described in Equation (10). The estimations use a symmetri<br>an indicator variable (equal to 1 for the fve trading days after reb<br>efects are dummy variables equal to 1 for each rebalancing mont<br>are fxed efects that interact maturity, ratings, and bond type <br>constructed by dividing a bond’s time to maturity into four difer<br>medium (5–10 years), long (10–20 years), and very long (20+ y<br>Moody’s. Bond type diferentiates sovereign from quasi-sovereign<br>the estimation includes the log face amount and log stripped sprea<br>drops the trading day before rebalancing. Standard errors are cl<br>the sample period is 2016–2018. *, **, and *** denote statistic<br>level, respectively.|c fve-trading-<br>alancing, and <br>h (0 otherwise<br> fxed efects. <br>ent categories<br>ears). Ratings<br> bonds. Bond<br>d of the bond. <br>ustered at the<br>ally signifcan|day window, <br> 0 otherwise)<br>), and bond c<br> Maturity fx<br>: short (less <br> from each b<br> controls ind<br> The last colu<br> country-mo<br>t at the 10%|with _Post_ as<br>. Month fxed<br>haracteristics<br>ed efects are<br>than 5 years),<br>ond are from<br>icate whether<br>mn’s analysis<br>nth level, and<br>, 5%, and 1%|

anticipating the index rebalancing, we would expect to observe pre-trends in bond prices before the actual event. However, our analysis reveals no correlation between the FIR and bond returns in the week leading up to the rebalancing (with the only exception being the day before the event). Finally, if part of the rebalancing-driven inflows were to occur before the event, our FIR measure would overestimate them at the index rebalancing date. This, in 18
turn, implies that our estimates can be understood as a lower bound.<sup>22</sup> <!-- Start of picture text --> Figure 4<br>Leads and lags coefficients<br>.6<br>.4<br>.2<br>0<br>-.2<br>-5 -4 -3 -2 -1 +1 +2 +3 +4 +5<br>Days from rebalancing = h<br>Coefficient of FIR (Instrumented) C.I. (95%)<br><!-- End of picture text --> Note: This figure presents leads and lags coefficients from a 2SLS estimation of bond log prices on a set of trading-day dummies around each rebalancing event, using the same 2SLS procedure as in Table 2. The estimation includes bond characteristics-month fixed effects (maturity, rating, and bond type). The shaded area indicates the rebalancing on the month’s last business day, with _h_ = +1 for returns on that day and _h_ = _−_ 1 for returns on the preceding business day. The vertical red lines show a 95% confidence interval for each horizon. Standard errors are clustered at the country-month level.

The documented effects are heterogeneous across bonds with varying levels of default risk. To show this heterogeneity, we divide our sample into high- and low-spread bonds, those above and below the median spread in our sample, respectively. We estimate Equation (10) for each of these subsamples and report the results in Table 3. The table shows that the price of high-spread bonds is more sensitive to rebalancing shocks, with a 1 p.p. increase in the FIR associated with a 0 _._ 46 p.p. increase in bond returns. In contrast, for low-spread bonds, the effect is quantitatively close to zero and not statistically significant.<sup>23</sup>

# **3.4 Reduced-Form Demand Elasticities**

We can map the estimated reactions of bond prices to a reduced-form demand elasticity, as shown by rewriting Equation (3) to reflect our FIR measure: _η_ ˆ = ( _−_ )<sup>∆lo</sup> _FIR_<sup><u>g(</u></sup> _c,t_<sup>_<u>q</u>_</sup> _<u>t</u>_<sup>_i_</sup><sup><u>)</u>.Thisaligns</sup> 22Appendix Table 1 shows how our estimates change as we proportionally decrease the FIR measure (due to a lower share of passive funds). These results could serve as guidance for what might happen if the FIR measure were lower due to some investors’ portfolio rebalancings being anticipated.

> 23Appendix Table 4 divides bonds into three groups according to their spreads. We find that bond prices are positively associated with the FIR for both high (above 302 basis points) and medium (between 158 and 302 basis points) spread bonds. Instead, for low spread bonds (below 158 basis points), the relationship is statistically insignificant. Additionally, the estimated coefficient increases with the risk profile of the bonds.

19
Table 3 Log price and FIR: Spread heterogeneity

|Dependent varia|ble: log <br>Hi|price<br>gh spread|Low s|pread|
|---|---|---|---|---|
|FIR X Post|0.461|***<br>0.462|***<br>0.100|0.100|
||(0.136)|(0.135)|(0.088)|(0.088)|
|Bond FE|Yes|No|Yes|No|
|Month FE|Yes|No|Yes|No|
|Bond-month FE|No|Yes|No|Yes|
|Observations|53,364|53,364|53,774|53,774|
|N. of bonds|497|497|498|498|
|N. of countries|62|62|51|51|
|N. of clusters|1,249|1,249|895|895|
|F (frst stage)|594|2,333|451|963|

Note: This table presents 2SLS estimates of bond log prices on the FIR measure, instrumented by _Z_ , across rebalancing dates. The sample is divided into high-spread bonds in Columns 1 and 2, above the median stripped spread, and low-spread bonds in Columns 3 and 4, below the median. The sample period, the construction of five-day window, and the 2SLS procedure are identical to those described in Table 2. The estimation includes coefficients for _Post_ and _FIR_ but are not reported for brevity. Standard errors are clustered at the country-month level. *, **, and *** denote statistically significant at the 10%, 5%, and 1% level, respectively.

precisely with the role of our _β_ coefficient in Equation (10). Based on the estimates in Table 2 (last column), the inverse demand elasticity is _−_ 0 _._ 33, implying a demand elasticity of _−_ 3. These estimates are in the ballpark of other estimates in the literature for other financial markets and assets (Appendix Figure 5).

As shown in Section 2, the previous estimates do not capture a structural elasticity because the default risk and expected payoffs of bonds may be affected by the shocks. Thus, part of the documented price reaction could be driven by these endogenous responses and not by a downward-sloping demand curve (as shown in Equation (6)). One way to assess the magnitude of these endogenous effects is to quantify how changes in the FIR affect a country’s default risk. To this end, we use credit default swaps (CDS) as a proxy for default risk. We find that increases in the FIR tend to decrease CDS spreads (see Appendix Table 5). The estimates imply that, for the median CDS spread in the sample (about 190 basis points), a 1 p.p. increase in the FIR decreases CDS spreads by 1 _._ 5 basis points.

20
Using this estimate, and based on a simple back-of-the-envelope calculation, we can quantify the effects of changes in default risk on our reduced-form elasticity. For this, we first convert changes in CDS spreads into changes in bond prices by calculating the median duration of the bonds in our sample and then multiplying it by the estimated effect of the FIR on CDS spreads. Given a median duration for bonds of 5 _._ 5 years, the estimated decrease in CDS spreads increases bond prices by almost 1 _._ 5 _×_ 5 _._ 5 = 8 _._ 25 basis points. This analysis thus suggests that changes in default risk could account for around 25% of our reduced-form elasticity (i.e., 8 _._ 25 _/_ 33).

Although informative, these estimates should be taken with caution as bond prices and CDS spreads are determined jointly. Given that the FIR measure captures shocks to the demand of passive investors, it is possible that shocks are correlated across both bond and CDS markets. Thus, it might be the case that a fall in CDS spreads is not capturing a lower default risk but rather an increase in CDS demand. In the next section, we build a quantitative model that allows us to quantify the role of endogenous movements in default risk behind the documented price responses.

# **4 Optimal Supply of Risky Sovereign Bonds**

We next formulate a quantitative sovereign debt model to study the impact of a downwardsloping demand on a government’s supply of risky bonds. We use the model to disentangle the mechanisms behind our reduced-form demand elasticity and to back out a structural elasticity that isolates the endogenous response of default risk. The model features a riskaverse government that issues long-term debt in international debt markets. The government has limited commitment and can endogenously default on its debt obligations. To tightly link the model with our empirical analysis, we introduce two types of investors (active and passive) and a rich demand structure, allowing us to capture a downward-sloping demand for government bonds.

## **4.1 Issuer Problem**

We consider a small open economy with incomplete markets and limited commitment. Output is exogenous and follows a continuous Markov process with a transition function _fy_ ( _y_<sup>_′_</sup> _| y_ ). Preferences of the representative consumer are given by

21
where _β_ is the discount factor, _ct_ denotes consumption, and the function _u_ ( _._ ) is strictly increasing and concave.

An infinite-lived, risk-averse government issues long-term bonds in international markets to smooth consumption. Let _B_ denote the beginning-of-period stock of government debt. Each unit of _B_ matures in the next period with probability _λ_ . If a bond does not mature (and the government does not default), it pays a coupon _ν_ . Let _d_ = _{_ 0 _,_ 1 _}_ denote the default policy, where _d_ = 1 indicates a default. Default leads to a temporary exclusion from international debt markets and an exogenous output loss, _φ_ ( _y_ ).

International markets are competitive and populated by heterogeneous investors who differ in their degree of activeness and passiveness. Let _A_<sup>_′_</sup> ( _._ ) and _T_<sup>_′_</sup> ( _._ ) denote the active and passive demand functions, respectively. As in Section 2, we assume that the active demand is a function of a bond’s expected return and volatility. The passive demand is perfectly inelastic and given by _T_<sup>_′_</sup> = _T_ ( _τ, B_<sup>_′_</sup> ). That is, it depends on the end-of-period stock of bonds, _B_<sup>_′_</sup> , and on some time-varying index weight, _τ_ . For tractability, we assume that _τ_ is exogenous and follows a continuous Markov process with a transition function _fτ_ ( _τ_<sup>_′_</sup> _| τ_ ). Given an end-of-period bond supply _B_<sup>_′_</sup> , the market-clearing condition can we written as _B_<sup>_′_</sup> = _A_<sup>_′_</sup> ( _τ, B_<sup>_′_</sup> ) + _T_ ( _τ, B_<sup>_′_</sup> ).

The state space can be summarized by the _n_ -tuple ( _h, B, s_ ), where _h_ captures the government’s current default status and _s_ = ( _y, τ_ ) the exogenous states. Under these assumptions, and for a given default status _h_ and choice of _B_<sup>_′_</sup> , the resource constraint of the economy can be written as where _q_ ( _y, τ, B_<sup>_′_</sup> ) denotes the price of a unit of debt, _B_<sup>_′_</sup> _−_ (1 _− λ_ ) _B_ are new bond issuances, and ( _λ_ + (1 _− λ_ ) _ν_ ) _B_ are current debt services.

# **4.2 The Government’s Recursive Problem**

The government is benevolent and chooses _{d, B_<sup>_′_</sup> _}_ to maximize Equation (12), subject to the resource constraint in Equation (13). If the government is not in default, its value function is given by 22
where _V_<sup>_r_</sup> ( _._ ) denotes the value function in case of repayment and _V_<sup>_d_</sup> ( _._ ) denotes the default value. If the government chooses to repay, then its value function is given by the following Bellman equation: If the government defaults, it is excluded from debt markets and cannot issue new debt. The government exits a default with probability _θ_ , with no recovery value. We further assume that the demand from passive investors is zero while the government is in default. Under these assumptions, the value function in case of default is given by

# **4.3 Bond Pricing Kernel**

Foreign lenders are competitive and discount payoffs at the risk-free rate. Based on the analysis in Section 2, given an exogenous state _{y, τ }_ and a choice of _B_<sup>_′_</sup> , the bond price function faced by the government is given by where _β_<sup>_⋆_</sup> _≡_ 1 _/_ (1 + _rf_ ), _R_<sup>_′_</sup> ( _._ ) _≡R_ ( _y_<sup>_′_</sup> _, τ_<sup>_′_</sup> _, B_<sup>_′_</sup> ) denotes the next-period repayment function, and Ψ( _y, τ, B_<sup>_′_</sup> ) captures the downward-sloping component of the active demand.<sup>24</sup> In turn, the next-period repayment function is given by where _d_ ( _y_<sup>_′_</sup> _, τ_<sup>_′_</sup> _, B_<sup>_′_</sup> ) is the next-period default choice and _q_ ( _y_<sup>_′_</sup> _, τ_<sup>_′_</sup> _, B_<sup>_′′_</sup> ) denotes the next-period bond price, which is a function of next-period exogenous states, _{y_<sup>_′_</sup> _, τ_<sup>_′_</sup> _}_ , and the next-period debt policy, _B_<sup>_′′_</sup> _≡ B_<sup>_′_</sup> ( _y_<sup>_′_</sup> _, τ_<sup>_′_</sup> _, B_<sup>_′_</sup> ).

From Equations (17) and (18), it is clear that the bond price decreases with the expected default probability. Specifically, a larger _B_<sup>_′_</sup> (weakly) increases the default risk (conditional on a level of output), and thus _q_ ( _y, τ, B_<sup>_′_</sup> ) (weakly) decreases in _B_<sup>_′_</sup> . As for Ψ ( _y, τ, B_<sup>_′_</sup> ), we only assume for now that<sup>_∂_</sup> _∂_<sup>Ψ</sup> _A_<sup><u>(</u></sup><sup>_<u>′y,</u>_</sup> ( _τ,B_<sup>_τ,B′′_</sup> )<sup><u>)</u></sup><sup>_≤_0.Thus,thistermintroducesanothermechanismfor</sup> the bond price to be decreasing in _B_<sup>_′_</sup> : the downward-sloping demand of active investors.

> 24The case where Ψ( _y, τ, B′_ ) = 1 for all _{y, τ, B′}_ captures the perfectly elastic case. In this instance, the bond price is only a function of the expected next-period repayment.

23
# **4.4 Demand and Supply Elasticities**

Introducing passive and active investors allows us to replicate our empirical exercise. That is, we can exploit exogenous movements in _τ_ , analyze their effects on bond prices, and estimate a demand elasticity. This is straightforward to do in the model as we can directly shock _τ_ while keeping the country’s fundamentals fixed. Let ∆ _T_<sup>_′_</sup> _≡T_ ( _τ_ 1 _, B_<sup>_′_</sup> ) _−T_ ( _τ_ 0 _, B_<sup>_′_</sup> ) denote an exogenous shift in the passive demand.<sup>25</sup> Given ∆ _T_<sup>_′_</sup> , and by means of simulations, we can compute the same reduced-form elasticity within the model as in our empirical analysis. We denote this elasticity as _η_ ˆ = ( _−_ )<sup>∆</sup> ∆<sup>_<u>q</u>_</sup> _T_<sup><u>(</u></sup><sup>_.′_</sup><sup><u>)</u></sup> _<u>Bq</u>_<sup>_′_</sup> _−_ ( _._ ) _<u>T</u>_<sup>_′_.</sup> Through its effects on bond prices, changes in _τ_ affect the government’s value function _V_<sup>_r_</sup> ( _._ ) (as shown in Equation (15)) and thus influence its debt and default policies, _B_<sup>_′_</sup> ( _._ ) and _d_ ( _._ ), respectively. Changes in these policies in turn impact the bond’s expected payoff and its price (as shown in Equations (17) and (18)). Therefore, part of the price reaction captured in _η_ ˆ is reflecting these endogenous responses and not a downward-sloping demand component.

We use the model to decompose _η_ ˆ into a structural demand elasticity _η_ and changes in expected risk and repayment _α_ . Using the model formulas, we can derive an analogous expression to the one in Equation (6): By keeping the repayment function _R_<sup>_′_</sup> ( _._ ) fixed, we can isolate the part of the reduced-form elasticity that is explained by changes in default risk.

The model has important implications for the optimal supply of risky bonds. In our setup, the government internalizes the effects of changes in _B_<sup>_′_</sup> on _q_ ( _y, τ, B_<sup>_′_</sup> ) through both changes in its default probability and the downward-sloping demand component. More formally, let<sup>_<u>q</u>_</sup><sup><u>(</u></sup><sup>_._</sup><sup><u>)</u></sup> _ε ≡_<sup>_∂_</sup> _∂_<sup>lo</sup> log<sup><u>g</u></sup> _B_<sup>_<u>′</u>_denotethe(inverse)supplyelasticity,whichcanbeexpressedas</sup> The first term on the right-hand side captures the elasticity of the expected repayment function with respect to the bond supply. This elasticity is typically negative because a larger _B_<sup>_′_</sup> increases default risk and reduces the expected bond payoff. The second term captures the additional decline in the bond price due to the downward-sloping demand. Given that the government internalizes both effects, a more inelastic demand affects its optimal bond

> 25Note that index weights change but the (end-of-period) bond supply remains fixed.

24
supply. In the next section, we use a calibrated version of the model to quantify the effects of inelastic demand on both a government’s bond supply and its default policy.

# **5 Quantitative Analysis**

## **5.1 Calibration**

We calibrate the model at a quarterly frequency using data on Argentina, a benchmark case commonly studied in the quantitative sovereign debt literature. The calibration follows a two-step procedure. We first fix a subset of parameters to standard values in the literature or based on historical Argentine data. We then internally calibrate the remaining parameters to match relevant moments for Argentine spreads and other business cycle statistics.

In terms of functional forms and stochastic processes, we assume that the government has CRRA preferences: _u_ ( _c_ ) =<sup>_<u>c</u>_</sup> 1<sup>1</sup> _−_<sup>_−_</sup> _γ_<sup>_γ_,where</sup><sup>_γ_denotestheriskaversion.Outputfollowsan</sup> AR(1) process given by log ( _y_<sup>_′_</sup> ) = _ρy_ log ( _y_ ) + _ε_<sup>_′_</sup> _y_<sup>,with</sup><sup>_ε′_</sup> _y_<sup>_∼N_(0</sup><sup>_, σy_).Ifthegovernment</sup> defaults, output costs are governed by a quadratic loss function _φ_ ( _y_ ) = max � _d_ 0 _y_ + _d_ 1 _y_<sup>2</sup> _,_ 0�. For _d_ 0 _<_ 0 and _d_ 1 _>_ 0, the output cost is zero whenever 0 _≤ y ≤−_<sup>_<u>d</u>_</sup> _d_<sup><u>0</u></sup> 1<sup>andrisesmore</sup> than proportionally with _y_ when _y > −_<sup>_<u>d</u>_</sup> _d_<sup><u>0</u></sup> 1<sup>.Thislossfunctionisidenticaltotheoneused</sup> in Chatterjee and Eyigungor (2012) and allows us to closely match the sovereign spreads observed in the data. As for the demand of passive investors, we assume that it is proportional to the (end-of-period) amount of bonds outstanding. Specifically, _T_<sup>_′_</sup> = _T_ ( _τ, B_<sup>_′_</sup> ) = _τ × B_<sup>_′_</sup> . We let _τ_ follow an AR(1) process given by log( _τ_<sup>_′_</sup> ) = (1 _− ρτ_ ) log( _τ_<sup>_⋆_</sup> ) + _ρτ_ log( _τ_ ) + _ε_<sup>_′_</sup> _τ_<sup>,where</sup> _ε_<sup>_′_</sup> _τ_<sup>_∼N_(0</sup><sup>_, στ_).</sup> Based on the analysis in Section 2, we consider the following functional form for the downward-slopping Ψ( _._ ) term: where _κ_ 0 _≥_ 0 captures the “slope” of the demand function and _A_<sup> ̄</sup> denotes the average holdings of active investors (as determined by the fixed component of their mandates, _ξj_<sup>_i_).</sup> For tractability, we assume time-invariant values for both _κ_ 0 and _A_<sup> ̄</sup> .<sup>26</sup> This specification introduces a wedge only for risky bonds (i.e., those with V _s′|s_ ( _R_<sup>_′_</sup> ( _._ )) _>_ 0). As we show next, it allows us to capture the two key features of our empirical analysis: a downward-sloping demand for active investors and a demand elasticity that increases (in magnitude) with

> 26As shown in Section 2, these terms could be, in principle, time-varying functions. We also use an exponential specification purely for computational reasons: to avoid having a negative price.

25
Table 4 Calibration of the model

||Panel a: Fixed parameters<br>||Pa<br>|nel b: Calibrated paramete<br>|rs<br>|
|---|---|---|---|---|---|
|Param.|Description|Value|Param.|Description|Value|
|_γ_|Risk aversion|2_._00|_β_|Discount rate|0_._949|
|_r_|Risk-free interest rate|0_._01| ̄_d_0|Default cost—level|_−_0_._24|
|_λ_|Debt maturity|0_._05| ̄_d_1|Default cost—curvature|0_._29|
|_z_|Debt services|0_._03|_κ_0|Slope parameter|60_._0|
|_θ_|Reentry probability|0_._0385| ̄<br>_A_|Active investors demand|0_._526|
|_ρ_y|Output, autocorrelation|0_._93||||
|_σ_y|Output, shock volatility|0_._02||||
|_τ _<sup>_⋆_</sup>|Share of passive demand|0_._123||||
|_ρτ_|FIR, autocorrelation|0_._66||||
|_ητ_|FIR, shock volatility|0_._02||||

default risk (i.e., with a larger return variance). Note that when _B_<sup>_′_</sup> _−T_<sup>_′_</sup> _− A ≥_<sup> ̄</sup> 0, Ψ( _._ ) _≤_ 1. We thus view Ψ( _._ ) as an “inconvenience” yield, that is, a premium demanded by investors as compensation for holding the bond, in excess of its default risk.<sup>27</sup> Table 4 lists the calibrated parameters. For the subset of fixed parameters (Panel a), we set _γ_ = 2, which is a standard value for risk aversion in the literature. We also set a quarterly risk-free rate of _rf_ = 0 _._ 01, in line with the average real risk-free rate observed in the United States. The probability of re-entering international markets is set to _θ_ = 0 _._ 0385, implying an average exclusion duration of 6.5 years. We set _λ_ = 0 _._ 05 to target a debt maturity of 5 years and _ν_ = 0 _._ 03 to match Argentina’s average debt services. The parameters for the endowment process, _ρy_ and _σy_ , are based on log-linearly detrended quarterly real GDP data of Argentina. All these parameters are taken from Morelli and Moretti (2023). Last, we set _τ_<sup>_⋆_</sup> to match the average share of Argentina’s external debt tracked by passive investors, and calibrate _ρτ_ and _στ_ to match the persistence and volatility of our FIR measure.

We internally calibrate the remaining parameters (Table 4, Panel b). We jointly calibrate the default cost level and curvature, _{d_ 0 _, d_ 1 _}_ , together with the government’s discount factor _β_ , to target Argentina’s average ratio of (external) debt to GDP, average spread, and volatility of spreads.<sup>28</sup> Additionally, we calibrate _κ_ 0 to match the estimated (inverse) reduced-form ˆ demand elasticity, _η_ = _−_ 0 _._ 33. Last, we set _A_<sup> ̄</sup> to match the average holdings of active investors. That is, _A_<sup> ̄</sup> = _B_<sup> ̄</sup> _− T_<sup> ̄</sup> , where _B_<sup> ̄</sup> denotes the average debt stock and _T_<sup> ̄</sup> denotes passive investors’

> 27For low values of _B′_ , when _B′ −T ′ − A_  ̄ _<_ 0, Ψ( _._ ) _≥_ 1. However, when _B_<sup>_′_</sup> is small, the variance of the repayment function tends to be small (due to the low default risk), and hence Ψ( _._ ) is typically close to one in these cases.

> 4

> 28Annualized spreads are computed as _SP_ = 1+ _i_ 1+( _y,τ,Brf_<sup>_′_</sup> <u>)</u> _−_ 1, where _i_ ( _y, τ, B_<sup>_′_</sup> ) is the internal quarterly � � return rate, which is the value of _i_ ( _._ ) that solves _q_ ( _y, τ, B_<sup>_′_</sup> ) = _λ_<sup><u>[</u></sup><sup>_λ_</sup> +<sup><u>+(</u></sup> _i_ ( _y,τ,B_<sup>1</sup><sup>_−λ_</sup><sup><u>)</u></sup><sup>_ν′_</sup> )<sup><u>]</u>.</sup>

26
Table 5 Targeted moments

|Target|Description|Data|Model|
|---|---|---|---|
|E[_SP_]|Bond spreads|472bp|466bp|
|_σ_(_SP_)|Volatility of spreads|200bp|148bp|
|E[_D/Y_]|Debt to output|55%|62%|
|E[Ψ]|Inconvenience yield|1_._0|1_._006|
|ˆ_η_|Reduced-form elasticity|_−_0_._33|_−_0_._31|

Note: The table reports the moments targeted in the calibration and their model counterpart.

Figure 5 Default set and bond prices <!-- Start of picture text -->
1.15<br>1.10<br>1.05<br>1.00<br>0.95<br>0.90<br>0.85<br>0.00 0.25 0.50 0.75 1.00<br><!-- End of picture text -->

<!-- Start of picture text -->
1.0<br>0.8<br>0.6<br>0.4<br>Small y<br>Medium y<br>0.2<br>0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66<br>B'<br>q(.)<br><!-- End of picture text -->

Note: Panel (a) shows the default policy for different combinations of _B_<sup>_′_</sup> and _y_ . The black area depicts combinations of _B_<sup>_′_</sup> and _y_ such that default probability is zero. Lighter colors indicate a higher default probability. In Panel (b), the solid lines show the bond pricing kernel _q_ ( _y, τ, B_<sup>_′_</sup> ) for different values of _B_<sup>_′_</sup> and for two values of output. The dashed lines show the bond price under a perfectly elastic demand, taking as given the same bond and repayment policies as in our baseline model (i.e., _q_ ( _._ ) _/_ Ψ( _._ )).

average holdings. Given Equation (21), this is equivalent to targeting an average Ψ( _._ ) of one. The introduction of Ψ( _._ ) thus only affects the sensitivity of the pricing kernel to changes in _B_<sup>_′_</sup> around the _{B,_<sup> ̄</sup> _T_<sup> ̄</sup> _, A}_<sup> ̄</sup> point.

Figure 5 depicts the default set and the bond price function _q_ ( _._ ) for different values of _B_<sup>_′_</sup> and _y_ . Panel (a) shows that the government defaults in states with high debt and low output. Panel (b) shows that, as a consequence, the bond price is decreasing in _B_<sup>_′_</sup> and increasing in _y_ . The dashed lines in Panel (b) show the bond prices under a counterfactual in which we take the baseline _B_<sup>_′_</sup> ( _._ ) policy but assume that the demand is perfectly elastic (i.e., it shows the _q_ ( _._ ) _/_ Ψ( _._ ) function). At lower _B_<sup>_′_</sup> levels, where default risk is minimal, bond prices remain largely unaffected by the downward-sloping demand. However, as _B_<sup>_′_</sup> increases, increased return volatility decreases Ψ, subsequently lowering the bond price _q_ .

27
<!-- Start of picture text --> Figure 6<br>Disentangling the demand elasticity<br>(a) As a function of B ′ (b) As a function of spreads<br>0.0 0.0<br>-0.1 -0.1<br>-0.2 -0.2<br>-0.3 -0.3<br>Reduced-form Elasticity Reduced-form Elasticity<br>Structural Elasticity Structural Elasticity<br>Changes in Expected Repayment Changes in Expected Repayment<br>-0.4 -0.4<br>0.52 0.54 0.56 0.58 0.60 0.62 0.64 3.5 4.0 4.5 5.0 5.5 6.0<br>B' Spreads, pp<br>Elasticity Elasticity<br><!-- End of picture text --> Note: The figure shows the reduced-form inverse demand elasticity _η_ ˆ (black lines) and the structural one _η_ (blue lines). The vertical differences between the two lines (represented by the red lines) capture the endogenous changes in bonds’ expected repayment, _α_ . Panel (a) shows the results as a function of _B_<sup>_′_</sup> , while Panel (b) shows the results as a function of annualized bond spreads.

# **5.2 Decomposing the Reduced-form Demand Elasticity**

We formally disentangle the different channels through which changes in _T_ affect bond prices. As shown in Equation (19), index rebalancing affects bond prices through two mechanisms: (i) the (inverse) structural demand elasticity of active investors, _η_ , and (ii) changes in expected repayment, _α_ . Using the calibrated model, we can isolate the effects driven by changes in expected repayment to properly identify the structural demand elasticity.

Figure 6 decomposes the channels outlined in Equation (19). The black line shows the reduced-form (inverse) demand elasticity _η_ ˆ, while the blue line depicts the model-implied structural elasticity, _η_ . The vertical differences between these two curves (red lines) indicate the portion of the reduced-form elasticity attributable to endogenous changes in the repayment ˆ function, _α_ . We find that the magnitude of _η_ is always higher than that of _η_ . This is consistent with our empirical analysis based on CDS spreads (described in Section 3). The difference can be substantial, particularly for larger values of _B_<sup>_′_</sup> and for higher bond spreads.

The first column of Table 6 shows the unconditional average for both the reduced-form ˆ elasticity, _η_ , and the structural elasticity, _η_ . On average, the structural elasticity accounts for less than two-thirds of the reduced-form elasticity. The rest is explained by endogenous changes in the expected repayment, _α_ . Despite the fundamentals ( _y, B_<sup>_′_</sup> ) being fixed and the ∆ _T_<sup>_′_</sup> shocks being exogenous, the reduced-form elasticity significantly differs from the structural one.

The magnitudes of the documented biases critically depend on the persistence of the 28
Table 6 Persistence of shocks and demand elasticity

|Moment|Baseline|Lower persistence|Higher persistence|
|---|---|---|---|
|Reduced-form elasticity, ˆ_η_|-0.31|-0.28|-0.33|
|Structural elasticity, _η_|-0.19|-0.21|-0.17|
|Ratio, _η/_ˆ_η_|61%|74%|50%|

ˆ Note: The table compares the reduced-form inverse demand elasticity _η_ with the structural one _η_ . The “Baseline” column shows the elasticities under our baseline calibration. In the “Lower persistence” case, we decrease the persistence of the _{τ }_ process by setting _ρτ_ = 0 _._ 50. The “Higher persistence” column shows the results for _ρτ_ = 0 _._ 80.

shock. The last two columns of Table 6 compare the reduced-form and structural elasticities for different persistence values for the _{τ }_ process (i.e., _ρτ_ ). When the shock is more (less) persistent, a smaller (larger) share of the total price response is explained by the inelastic component of the investors’ demand. This is because more permanent shocks lead to larger changes in government policies and thus in the bond’s expected repayment function.

Overall, our analysis highlights the importance of accounting for issuers’ endogenous responses to an exogenous (supply-shifting) shock and the resulting changes in assets’ expected repayment. Neglecting these factors can introduce significant biases into the estimated demand elasticity, particularly if the shock is persistent. As argued in Section 3, our FIR measure is inherently more temporary than other supply-shifting instruments used in the literature, such as index additions or deletions. However, even in that case, the bias can represent over one-third of the reduced-form elasticity.

# **5.3 Implications of a Downward-sloping Demand**

As shown in Equation (20), in determining its optimal debt policy, the government internalizes not only the effects of a higher _B_<sup>_′_</sup> on _q_ ( _._ ) through changes in its default probability but also its effects through the inelastic demand. This section quantifies the implications of a downward-sloping demand on government policies and their subsequent effects on bond prices. To this end, we compare our baseline model with a downward-sloping demand with an alternative scenario where investors are perfectly elastic.

Table 7 reports a set of targeted and untargeted moments for our baseline model (with _κ>_ ) and for an alternative case with a perfectly elastic demand ( _κ_ 0). All the other model parameters remain the same. Despite similar values of debt, we find that the default frequency and average spreads are _lower_ relative to the perfectly elastic case when facing an inelastic demand.

Two factors explain the lower default rate and bond spreads. First, the government debt 29
Table 7 Comparison with perfectly elastic case

|Moment|Description|Baseline|Perfectly elastic|
|---|---|---|---|
|E(_SP_)|Bond spreads|466bp|831bp|
|_σ_(_SP_)|Volatility of spreads|148bp|446bp|
|E(_B/y_)|Debt to output|62%|59%|
|E(_d_)|Default frequency|3_._9%|4_._43%|
|_σ_(_B_)_/σ_(_y_)|Standard deviation of debt, relative to output|1.88|2.32|
|_ρ_(_SP, y_)|Correlation between spreads and output|-0.71|-0.53|

Note: The table compares a set of key moments between our baseline model with inelastic investors and a counterfactual scenario in which investors are perfectly elastic ( _κ_ 0 = 0).

policy is significantly affected by a downward-sloping demand. Panel (a) of Figure 7 shows the optimal debt policy _B_<sup>_′_</sup> ( _y, τ, B_ ) in our baseline model and in the perfectly elastic case. For large values of _B_ (in states where V ( _R_<sup>_′_</sup> ( _._ )) is high), an additional unit of _B_<sup>_′_</sup> reduces the bond price _q_ ( _._ ) due to both higher default risk and investors’ inelastic behavior. As a result, the government does not find it optimal to issue large amounts of debt because it is too costly to do so. An inelastic demand thus introduces a limit to the maximum amount of debt that a government is willing to issue.

<!-- Start of picture text --> Figure 7<br>Implications on policies and prices<br>(a) Bond policy (b) Bond price<br>0.650<br>1.1<br>0.625<br>1.0<br>0.600<br>0.9<br>0.575<br>Baseline 0.8 Baseline<br>0.550 Perfectly Elastic Perfectly Elastic<br>0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66<br>B B'<br>B'(.) q(.)<br><!-- End of picture text --> Note: The figure presents bond policies and prices for our baseline model with inelastic investors (black solid lines) and in an alternative model in which investors are perfectly elastic (blue dotted lines).

Second, these changes in the optimal bond policy have important effects on the pricing of bonds (Figure 7, Panel (b)). For small values of _B_<sup>_′_</sup> (low default risk), _q_ ( _._ ) is actually _higher_ than under the perfectly elastic case. As shown in Figure 8 (Panel (a)), this larger bond price is not driven by a convenience yield because, by construction, Ψ( _._ ) is typically smaller than one. Instead, the higher bond price is explained by a lower default risk (Panel (b)), which is 30
Figure 8 Inconvenience yield and default risk <!-- Start of picture text --> (a) Ψ( y, τ, B ′ ) (b) Default risk, annualized<br>1.00 50<br>Baseline<br>Perfectly Elastic<br>40<br>0.98<br>30<br>0.96<br>20<br>0.94<br>10<br>0.92 Baseline<br>0<br>0.54 0.57 0.60 0.63 0.66 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66<br>B' B'<br>Psi<br>Default Risk, pp<br><!-- End of picture text --> Note: Panel (a) depicts the Ψ( _y, τ, B_<sup>_′_</sup> ) function for different values of _B_<sup>_′_</sup> . Panel (b) shows the annualized default risk in our baseline model (black solid lines) and in a counterfactual in which investors are perfectly elastic (blue dotted lines).

a direct consequence of the government’s lower incentives to issue large values of _B_<sup>_′_</sup> .

To summarize, an inelastic demand diminishes a government’s incentives to issue additional units of debt, acting as a commitment device that reduces default risk and increases bond prices.

# **6 Conclusion**

In this paper, we present evidence of downward-sloping demand curves in risky sovereign debt markets and analyze their implications for the optimal supply of sovereign bonds. Our approach combines evidence from high-frequency bond-level price reactions to well-identified shocks with a structural model featuring endogenous debt issuances and default risk. This methodology allows us to isolate endogenous changes in default risk behind the estimated price reactions and to back out a structural demand elasticity. Empirically, we find that a 1 p.p. exogenous reduction in the effective bond supply leads to a 33 basis point increase in bond prices. Our structural model reveals that over one-third of this response is due to endogenous changes in the expected repayment of bonds. Moreover, we find that the inelastic demand influences and shapes the governmental policies on optimal debt and default. By diminishing the government’s incentives to issue additional units of debt, an inelastic demand acts as a commitment device that reduces default risk and borrowing costs.

Our results highlight the importance of considering issuers’ endogenous responses and the resulting changes in expected asset payoffs. Failing to account for these responses can introduce significant biases when estimating demand elasticities, particularly for risky assets.

31
Our paper can lead to further research along several dimensions. For example, given the model predictions, it would be interesting to empirically study the impact of inelastic demand on government debt issuances. More importantly, our framework can be extended to other assets and markets, notably equity and corporate bonds. The endogenous responses that we emphasize in this paper can be applied to other issuers’ of risky assets, which we leave for future research.

32