Options on Interbank Rates and Implied Disaster Risk

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Options on Interbank Rates and Implied Disaster Risk Hitesh Doshi, Hyung Joo Kim, Sang Byung Seo 2023-054 Please cite this paper as: Doshi, Hitesh, Hyung Joo Kim, and Sang Byung Seo (2025). “Options on Interbank Rates and Implied Disaster Risk,” Finance and Economics Discussion Series 2023-054r1. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.054r1. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

* Options on Interbank Rates and Implied Disaster Risk Hitesh Doshi(cid:132) Hyung Joo Kim(cid:133) Sang Byung Seo§ July 9, 2025 Abstract Theidentificationofdisasterriskhasremainedasignificantchallengeduetotherarityofmacroeconomic disasters. We show that the interbank market can help characterize the time variation in disaster risk. We propose a risk-based model in which macroeconomic disasters are likely to coincide with interbank market failure. Using interbank rates and their options, we estimate our model via MLE and filter the short-run and long-run components of disaster risk. Our estimation results are independent of the stock market and serve as an external validity test of rare disaster models, which are typically calibrated to match stock moments. *We thank George Pennacchi (the editor) and the anonymous referee for their valuable comments and suggestions. We are also grateful to Caio Almeida, Hui Chen, Mathieu Fournier, Kris Jacobs, Mete Kilic, Praveen Kumar, Ivan Shaliastovich, Anders Trolle, Jessica Wachter, Nancy Xu, and seminar participants at the North American Summer Meeting of the Econometric Society, European Finance Association Annual Meeting, NorthernFinanceAssociationConference, FinancialManagementAssociationAnnualMeeting, and University of Houston for helpful comments. We gratefully acknowledge the financial support from the CanadianDerivativesInstitute. Theanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicate concurrence by the Federal Reserve Board or other members of its staff. An earlier version of this manuscript was circulated under the title “What Interbank Rates Tell Us About Time-Varying Disaster Risk.” (cid:132)Bauer College of Business, University of Houston; hdoshi@bauer.uh.edu (cid:133)Federal Reserve Board of Governors; hyungjoo.kim@frb.gov §Wisconsin School of Business, University of Wisconsin-Madison; sang.seo@wisc.edu

1 Introduction Severe macroeconomic contractions are often accompanied by widespread financial sector distress. Historical events such as the Great Depression illustrate how disruptions in the banking systemcantriggeroramplifyeconomiccollapse(Bernanke,1983). Morerecentempiricalwork supports this view, showing that defaults in the financial sector, unlike those in non-financial sectors, are closely tied to macroeconomic downturns (Allen, Bali, and Tang, 2012; Giesecke, Longstaff, Schaefer, and Strebulaev, 2014). The association between macroeconomic and financial crises is particularly strong for extreme episodes: as noted by Reinhart and Rogoff (2013), virtually all consumption disasters documented by Barro and Ursu´a (2008) were coupled with systemic banking crises. In this paper, we exploit this empirical regularity to propose a new method for estimating the risk of rare economic disasters. Macro-finance models featuring disaster risk have become a popular framework for capturing macroeconomic tail risk and explaining prominent asset pricing puzzles.1 Yet, the rare disaster literature faces a major empirical challenge. Disasters are often referred to as “dark matter” because of their rarity and the obscurity of their source. This criticism points to the fact that it is difficult to reliably estimate the parameters associated with disasters, based on the historical consumption time series.2 Estimating timevarying disaster risk is even more difficult; Cochrane (2017) describes time-varying disaster probabilities as “dark energy,” unless they can be independently anchored to external data. We address these limitations by using interbank rates and their options. Our key assump- 1In order to describe an extreme and rare economic downturn, the literature defines macroeconomic disasters as a severe drop in real consumption per capita (e.g., Barro, 2006; Barro and Ursu´a, 2008; Gabaix, 2012; Wachter, 2013) or as a significant reduction in total factor productivity (e.g., Gourio, 2012). 2Inlinewiththis,Chen,Dou,andKogan(2024)raiseconcernsaboutoverfittingin-sampledata: ifamodel overfits, its implications become highly sensitive to small perturbations in disaster parameters. 1

tion is that macroeconomic disasters are likely to coincide with interbank market failure – a view supported by historical data. This assumption provides a direct link between disaster risk and observed prices in the interbank market. We show that interbank rates and their options allow us to characterize the time series evolution of disaster risk, with the aid of a model. For our analysis, we adopt a risk-based model where the nominal pricing kernel is affine and is comprised of Brownian and Poisson shocks to five state variables of the economy: real consumption growth, mean consumption growth, expected inflation, short-run disaster risk, and long-run disaster risk. This framework allows us to derive the expressions for three distinct interest rates: (i) interbank rate (LIBOR), (ii) risk-free rate (proxied by the OIS rate), and (iii) government bill rate (Treasury rate). Within our model, risk-free lending always pays back the promised amount at the end of its maturity, whereas interbank lending is potentially subject to partial defaults in the event of a disaster. Consequently, their interest rate difference, the so-called LIBOR-OIS spread, is highly informative about disaster risk, directly reflecting the two disaster-related state variables. We also show that the spread between the interbank rate and the Treasury rate, commonly referred to as the TED spread, can serve as a useful indicator of disaster risk. However, its signal may be noisier than the LIBOR-OIS spread due to the complexity arising from the Treasury convenience yield. Following recent studies on safe/liquid assets, we assume that government bills may carry a convenience yield in our model, causing the Treasury rate to be, on average, lower than the risk-free rate. In addition to the two interest rate spreads, we take advantage of interest rate caps and swaptions. We approach these financial instruments from a new angle. Prior studies simply 2

consider them as option contracts on future benchmark interest rates, often overlooking the aspect that the underlying rates are not default-free. In contrast, we view them as options on future interbank rates: a cap consists of a series of caplets, each of which is a call option on the LIBOR; a swaption provides its holder the right to enter into an interest rate swap that exchanges fixed coupons with floating coupons based on the LIBOR. Naturally, these instruments reflect the possibility of interbank market failure and hence that of economic disasters. We believe that this is an important distinction because, as witnessed during the 2008 financial crisis, interbank rates can significantly deviate from default-free interest rates. We estimate the model parameters via maximum likelihood estimation using the data from February 2002 to December 2019. As a result, we obtain parameter estimates whose signs and magnitudes are economically sensible. The model-implied time series for interest rates, caps, and swaptions mimic their data counterparts reasonably well. Our results provide additional guidelines that can help discipline the calibration of rare disaster models. An important advantage of our estimation is that it is possible to extract the short-run and long-run components of latent disaster risk through a filtering approach. Specifically, we adopt the unscented Kalman filter because caps and swaption prices are nonlinear functions of the latent and observable state variables.3 From a sensitivity analysis, we discover that the short-run component of disaster risk is mainly identified by the LIBOR-OIS spread with a short maturity. In contrast, the long-run component of disaster risk is primarily filtered through caps and swaptions whose payoffs are contingent on future interbank rates over a long horizon. Importantly, the forward-looking information from interest rate options plays a 3The unscented Kalman filter has been widely used in filtering latent variables from nonlinear financial asset prices. See, for instance, Filipovi´c and Trolle (2013), Christoffersen, Dorion, Jacobs, and Karoui (2014), Park (2016), Doshi, Elkamhi, and Ornthanalai (2018), and Bakshi, Crosby, Gao, and Hansen (2023). 3

crucial role in estimating the whole dynamics of disaster risk. Additionally, based on the filtered time series of the short-run and long-run components of disasterrisk,weverifythetestableimplicationsthatdisasterriskisassociatedwithconditional moments and returns in the equity market. We find that higher disaster risk is generally linked to lower stock market valuations, in terms of the price-dividend and price-earnings ratios. Intuitively, the short-run component of disaster risk significantly explains implied variance, expected realized variance, and the variance risk premium over the next month, as wellasshort-maturityout-of-the-moneyputoptionprices. Thelong-runcomponent, however, becomes more relevant for longer-horizon moments. Shocks to disaster risk are also negatively associated with prominent equity return factors, such as market, size, and liquidity, and have stronger negative effects on returns for cyclical industries. Overall, our results demonstrate that the interbank market can potentially be useful for overcoming the criticisms of rare disaster models, which are typically calibrated to match stock data. Our estimation is independent of equity market moments and, thus, serves as an external/out-of-sample validity test of the models. The estimation results suggest that disaster risk is significant in magnitude and in variation, strongly supporting macro-finance models with the rare disaster mechanism. Our paper relates to several strands of the literature.4 We contribute to the rare disaster literature by estimating the time-varying risk of economic disasters. Granted, we are not the first to attempt to quantify the time series variation in disaster risk. For example, Berkman, Jacobsen, and Lee (2011) proxy the perceived disaster probability by a crisis severity index, 4IntheInternetAppendix,weprovidedetailedcomparisonswithrelatedpapersintableformat,highlighting our modeling innovations and empirical contributions. 4

constructed based on the number of international political crises. Manela and Moreira (2017) create a text-based disaster concern measure, called news implied volatility (NVIX), using the words in front-page articles of the Wall Street Journal.5 Rather than proposing another index that potentially correlates with disaster risk, our goal is to directly estimate the risk of consumption disasters under the assumption that consumption disasters are likely to coincide with interbank market failure. A key advantage of our framework is that it is possible to exploit the information contained in interbank rates and their options, which allows us to separately extract the short-run and long-run components of disaster risk. Our estimation relies on the pricing data on interest rate caps and swaptions. Prior studies, including Longstaff, Santa-Clara, and Schwartz (2001), Han (2007), and Trolle and Schwartz (2009), mainly concentrate on the relative pricing of caps and swaptions. However, the literature has paid little attention to what these financial instruments imply about the aggregate economy or other financial markets. We explore the economic content of caps and swaptions by focusing on the fact that their payoffs are contingent on future interbank rates. We confirm that interbank rate options indeed contain valuable information about the risk of economic disasters. The findings of this paper also potentially relate to the growing literature on the role of financial intermediaries in asset pricing. He and Krishnamurthy (2013) and Brunnermeier and Sannikov (2014) argue that intermediaries, rather than households, act as marginal investors and, therefore, their financial constraints serve as key drivers of market risk premia. Adrian, 5Related,BollerslevandTodorov(2011a),BollerslevandTodorov(2011b),BollerslevandTodorov(2014), and Andersen, Fusari, and Todorov (2015) estimate and study the risk of rare events in the equity market using an essentially model-free approach based on high-frequency index time series and/or short maturity out-of-the-money index options. Bakshi, Gao, and Xue (2023) define a disaster as an event in which the equity market declines over a certain threshold and estimate its probability using S&P 500 options. 5

Etula, and Muir (2014) and He, Kelly, and Manela (2017) empirically support this theory by showing that intermediary-induced factors outperform traditional risk factors in explaining asset returns in various markets. Although our analysis mainly concerns economic disasters, the pricing kernel we adopt has properties that are isomorphic to those of an intermediarybased pricing kernel: when the interbank market is hit by a shock (which is modeled as a shocktodisasterriskinourframework), thepricingkernelrespondsandamplifiesriskpremia. Our results hint that the disaster-based explanation and the intermediary-based explanation of asset markets might share common microfoundations. The rest of this paper proceeds as follows. Section 2 describes the model. Section 3 describes the data as well as explains how we estimate the model and extract time-varying disaster risk. Section 4 reports the estimation results and discusses their implications. Section 5 concludes. 2 Model 2.1 Risk of disasters and interbank rates How can we characterize the time-varying risk of economic disasters? In this section, we illustrate how interbank rates can help. Let C denote aggregate real consumption, which t follows an affine jump diffusion process: dC t = µ dt+σ dB + (cid:0) eZC,t −1 (cid:1) dN , t C C,t t C t− 6

where B is a standard Brownian motion. The occurrence of consumption disasters is spec- C,t ified by a Poisson process N with a negative jump size random variable Z : when a cont C,t sumption disaster occurs (dN t = 1), consumption falls from C t− to C t− eZC,t. Given this setup, our main objective is to properly estimate the time series of the intensity process for N . We t assume that N has stochastic intensity λ whose dynamics are described by t t (cid:112) dλ = κ (ξ −λ )dt+σ λ dB , t λ t t λ t λ,t ¯ (cid:112) dξ = κ (ξ −ξ )dt+σ ξ dB , t ξ t ξ t ξ,t where ξ is the time-varying mean of λ . Simply put, λ and ξ represent the short-run and t t t t long-run components of disaster risk. In the absence of a disaster (dN = 0), the consumption t dynamics simply reduce to a Geometric Brownian motion with time-varying mean µ and t constant volatility σ , where C dµ = κ (µ¯−µ )dt+σ dB . t µ t µ µ,t That is, mean consumption growth in normal-times µ reverts to its long-run mean µ¯ at the t rate of κ , while experiencing random Brownian shocks dB with volatility coefficient σ . µ µ,t µ We study the time series evolution of λ and ξ by linking consumption disasters to the t t bankingsector. AsevidentintheexampleoftheGreatDepression, thecollapseofthebanking sector is closely related to severe economic downturns. In fact, Reinhart and Rogoff (2013) point out that virtually all consumption disasters documented by Barro and Ursu´a (2008) were accompanied by severe systemic banking crises. Motivated by this insight, we assume 7

that consumption disasters are likely to coincide with extreme rare events where the interbank market fails and market participants suffer significant losses. More formally, let L denote the time-t face value of interbank lending. We assume that t in the event of a consumption disaster, the interbank market fails with probability p¯, which results in a loss to the lenders:     L t− eZL,t with probability p¯if a disaster occurs at time t, L = t    L t− otherwise. That is, the face value, or the expected principal payment from interbank lending, reduces from L t− to L t− eZL,t if the interbank market fails.6 By defining I t as a Bernoulli random variable with success probability p¯, we can express the dynamics of L as t dL t = (cid:0) eZL,tIt −1 (cid:1) dN . (1) t L t− Here, wedonotmodelthebehaviorsofbanksnorthestructureoftheinterbankmarket, which can potentially generate consumption disasters and interbank market failure in an endogenous fashion. Moreover, we are agnostic about any causal link: whether interbank market failure leads to consumption disasters or vice versa, and through which mechanism they are related. Since our focus is on empirically characterizing time-varying disaster risk, it suffices that these two types of extreme events are likely to coincide with each other. Under this simple setup, it is intuitive that the spread between the interbank rate and the risk-free rate contains important information about disaster risk. The pricing relation implies 6The reduced-form way in which we model risky interbank lending is reminiscent of fractional-loss-atdefault models in the spirit of Duffie and Singleton (1999) and Duffie (2005). 8

that the τ-maturity zero-coupon interbank rate can be derived as (cid:20) (cid:21) 1 M L y(τ) = − logE t+τ · t+τ , (2) i,t τ t M L t t where Mt+τ represents the nominal pricing kernel whose existence is guaranteed under no Mt arbitrage. While interbank lending contracts are potentially subject to partial defaults, riskfree lending always pays back the promised amount at the end of their maturity. The τmaturity zero-coupon risk-free rate is calculated as (cid:20) (cid:21) 1 M y(τ) = − logE t+τ ·1 . (3) f,t τ t M t Comparing equations (2) and (3) suggests that changes in disaster risk have a direct effect on the interest rate spread y(τ)−y(τ). When disaster risk rises, the expectation of the future i,t f,t payoff Lt+τ decreases, which pushes the interbank rate upward. However, this effect is missing Lt for the risk-free rate as the future payoff from a default-free bond is, by definition, always 1. As a result, the spread between the two interest rates widens when disaster risk increases. These equations also suggest that fluctuations in disaster risk or in other potential risk factorscanhaveindirecteffectsontheinterbankrateandtherisk-freeratethroughthepricing kernel. However, if risk factors only impact the pricing kernel but not the future payoff Lt+τ, Lt they will move the interbank rate with the same degree as the risk-free rate, leaving the spread between the two unchanged. As clear in equation (1), the distribution Lt+τ only depends on Lt instantaneous disaster risk λ and its time-varying mean ξ , and this suggests that risk factors t t that are orthogonal to λ and ξ will have no impact on the interest rate spread y(τ)−y(τ). In t t i,t f,t 9

Section 2.3, we show that this spread indeed depends only on λ and ξ in our fairly flexible t t setup, confirming this intuition. 2.2 Nominal pricing kernel Equations (2) and (3) make it clear that we need a nominal pricing kernel Mt+τ to derive Mt the expressions for zero-coupon yields. Rather than deriving it from investors’ preferences as in typical equilibrium models, we take a reduced-form approach and directly specify the nominal pricing kernel. To this end, we first establish the state variables in the economy. In addition to real consumption (C ), mean consumption growth (µ ), and the short-run/longt t run components of disaster risk (λ , ξ ), we take one more variable that is typically used to t t capture the state of the nominal economy: expected inflation. This is particularly relevant, as we examine the term structure of nominal interest rates. In our model, the expected inflation process q solves the following stochastic differential equation: t dq = κ (q¯−q )dt+σ dB +Z dN , t q t q q,t q,t t where B is another standard Brownian motion. Motivated by Tsai (2015), we assume that q,t q is also subject to the same Poisson process N and that the jump size random variable Z t t q,t is, on average, positive.7 We specify our pricing kernel in a general form so that it can be directly estimated using data. We assume that Brownian shocks (dB , dB , dB , dB , dB ) are independent C,t µ,t λ,t ξ,t q,t 7Since jumps in consumption and expected inflation always occur together through N , our model cannot t generate inflation spikes that are not accompanied by disasters. In fact, Tsai (2015) adopts an additional Poisson process to capture such independent inflation spikes. Given our focus on studying disaster risk, we choose a more parsimonious setup and abstract away from this additional feature. 10

of one another and of a Poisson shock (dN ). These six independent shocks to the five state t variables (C , µ , λ , ξ , q ) are priced and hence constitute the nominal pricing kernel: t t t t t dM t (cid:112) (cid:112) = −r dt−θ dB −θ dB −θ λ dB −θ ξ dB −θ dB t C C,t µ µ,t λ t λ,t ξ t ξ,t q q,t M t− + (cid:0) eθNZC,t −1 (cid:1) dN −λ E(cid:2) eθNZC,t −1 (cid:3) dt, (4) t t where (θ ,θ ,θ ,θ ,θ ,θ ) indicate the market prices of risk and r represents the instanta- C µ λ ξ q N t neous nominal risk-free rate.8 To preserve the affine structure of our setup, we represent the short rate r as a linear function of the state variables, similar to Ang and Piazzesi (2003) t and Joslin, Le, and Singleton (2013): r = δ +δ λ +δ ξ +δ µ +δ q . (5) t 0 λ t ξ t µ t q t The pricing kernel fully characterizes the risk-neutral measure, as the Radon-Nikodym deriva- (cid:82)t tiveprocessoftherisk-neutralmeasurewithrespecttothephysicalmeasureequalsM r ds. t 0 s In Appendix A.1, we derive the risk-neutral dynamics of the underlying processes using Girsanov’s theorem. Based on the pricing kernel specified in equation (4), we finally calculate the expressions for y(τ) and y(τ) in equations (2) and (3). We show that both interbank rates and Treasury i,t f,t 8Thejumpterm (cid:0) eθNZC,t −1 (cid:1) dN t inthepricingkernelcapturestheideathatthereisaveryhighmarginal utility state, which we call a disaster. A common econometric challenge in the disaster literature is that it is very difficult to separately identify the market price of risk θ and the size distribution of disasters Z , N C,t as we do not observe any disaster realizations in the post-war sample. In our estimation, we set Z to C,t the empirical distribution of real consumption disasters compiled by Barro and Ursu´a (2008). This not only helps identify θ but also allows us to interpret the estimated intensity of the Poisson process N as the N t probability/frequency of consumption disasters. 11

rates are linear in state variables λ , ξ , µ , and q : t t t t 1 (cid:104) (cid:105) y(τ) = − a (τ)+b (τ)λ +b (τ)ξ +b (τ)µ +b (τ)q , (6) i,t τ i i,λ t i,ξ t µ t q t 1 (cid:104) (cid:105) y(τ) = − a (τ)+b (τ)λ +b (τ)ξ +b (τ)µ +b (τ)q , (7) f,t τ f f,λ t f,ξ t µ t q t where deterministic functions a , a , b , b , b , b , b and b solve the ordinary differential i f i,λ f,λ i,ξ f,ξ µ q equations in Appendix A.2. 2.3 Interest rate spreads One important aspect when comparing equations (6) and (7) is that the loadings on mean consumption growth µ and expected inflation q are identical for both types of interest rates. t t Therefore, for a given maturity τ, the difference between y(τ) and y(τ) becomes a function i,t f,t of disaster risk (λ , ξ ) alone, consistent with the intuition developed in Section 2.1. In our t t analysis, the data counterparts of the interbank rate and the risk-free rate are the London interbank offered rate (LIBOR) and the overnight index swap (OIS) rate, respectively, after being adjusted for compounding frequencies.9 Thus, we can see that the spread between the (continuously compounded) LIBOR and OIS rate, the so-called LIBOR-OIS spread, can be 9Rigorouslyspeaking,theOISratesarenotriskfree,astheyarederivedfromovernightindexswapsbased ontheeffectivefederalfundsrate. Giventhatthisistherateatwhichbanksborrow/lendtheirreservesonan uncollateralized basis, the OIS rates can essentially be seen as a “continually-refreshed” overnight interbank rate (Hull and White, 2015). Despite this, the OIS rates are widely used as a primary proxy for risk-free interest rates, especially in derivative markets. While the federal funds seller faces overnight credit risk, this is often considered very small, though not negligible, given the short maturity (and the fact that the buyer is another bank). Another contributing factor is that the effective federal funds rate is directly influenced by the Federal Reserve and has not deviated significantly from the target range since the 2000s. 12

expressed by 1 (cid:104) (cid:105) LOIS(τ) = − (cid:0) a (τ)−a (τ) (cid:1) + (cid:0) b (τ)−b (τ) (cid:1) λ + (cid:0) b (τ)−b (τ) (cid:1) ξ . t τ i f i,λ f,λ t i,ξ f,ξ t Our view that the LIBOR-OIS spread directly reflects disaster risk is novel, but not inconsistent with the existing view on the source of interbank risk. Prior studies typically decompose interbank risk (and hence the LIBOR-OIS spread) into a liquidity component and a pure credit component.10 We do not make such a distinction; we study the possibility of extreme tail events implied by the total risk of the interbank market, regardless of where it originates from. Modeling these market phenomena separately is beyond the scope of our paper. Instead, we simply capture the risk of such an economic downturn using disaster risk. We also note that the fluctuations in the LIBOR-OIS spread are purely systematic in our framework, as they are driven by the risk of economic disasters. One might argue that this spread can also be influenced by bank-specific shocks. Granted, the LIBOR is calculated empirically using input data from only about a dozen contributor banks. However, they are large and systemically important banks; significant shocks to these banks cannot be purely idiosyncratic, considering their critical roles in the intricately intertwined banking system. It is worth highlighting that we do not assume the yield on Treasury securities (simply, Treasury rate) to be our benchmark interest rate for riskless discounting; we choose the OIS rate instead, which is a conventional market proxy for the risk-free interest rate. The recent literature has documented that Treasury securities carry a convenience yield, derived from their special role as safe/liquid assets (e.g., Krishnamurthy and Vissing-Jorgensen, 2012). 10See, for example, Michaud and Upper (2008), Taylor and Williams (2009), Acharya and Skeie (2011), Filipovi´c and Trolle (2013), and McAndrews, Sarkar, and Wang (2017). 13

This implies that the Treasury rate can go even below the true risk-free rate. Consistently, we define the τ-maturity Treasury rate y(τ) as g,t y(τ) = y(τ) −y(τ), g,t f,t x,t where y(τ) represents the Treasury convenience yield, typically proxied by the OIS-Treasury x,t spread. Infact, theOIS-Treasuryspreadisonaveragepositiveinthedata, supportingthepresence of a convenience yield from holding Treasury securities. Nevertheless, the data reveals that this spread occasionally turns negative, most notably during the COVID-19 crisis (e.g., He, Nagel, and Song, 2022). To capture these properties in reduced form, we assume that the Treasury convenience yield is driven by an instantaneous convenience rate x , which follows t a mean-reverting Gaussian process: dx = κ (x¯−x )dt+σ dB . t x t x x,t The standard Brownian motion B is assumed to be independent of other Brownian and x,t Poisson shocks. Under this setup, the τ-maturity convenience yield y(τ) is derived as x,t 1 (cid:104) (cid:105) 1 (cid:104) (cid:105) y x (τ ,t ) = τ logE t e (cid:82) t t+τxudu = − τ a x (τ)+b x (τ)x t , (8) where the expressions for the deterministic functions a and b are provided in Appendix A.2. x x With the Treasury convenience, the spread between the LIBOR and the Treasury rate, 14

the so-called TED spread, becomes a function of (λ , ξ ) as well as x : t t t 1 (cid:104) (cid:105) TED(τ) = − (cid:0) a (τ)−a (τ)+a (τ) (cid:1) + (cid:0) b (τ)−b (τ) (cid:1) λ + (cid:0) b (τ)−b (τ) (cid:1) ξ +b (τ)x . t τ i f x i,λ f,λ t i,ξ f,ξ t x t This suggests that while the TED spread is still a meaningful indicator of disaster risk, its signal is relatively less clear than the LIBOR-OIS spread due to the added complexity introduced by the convenience factor x . t 2.4 Options on interbank rates In this section, we introduce derivative contracts called caps and swaptions whose payoffs depend on future interbank rates. When discussing their payoffs and pricing, it is convenient to introduce the following notation: (cid:104) (cid:105) P (t,t+τ) = exp −τ ·y(τ) . i i,t In other words, P (t,t + τ) represents the time-t value of $1 zero-coupon interbank lending i maturing at time t + τ. Additionally, we need the expression for the LIBOR in the model. Recall that the LIBOR is a simple interest rate and, therefore, is not exactly the same as the continuously compounded rate y(τ). We can convert between the two interest rates using the i,t following relation: 1 (cid:104) (cid:105) 1 (cid:104) (cid:16) (cid:17) (cid:105) y(τ) = log 1+τLIBOR(τ) , or equivalently, LIBOR(τ) = exp τ ·y(τ) −1 . i,t τ t t τ i,t 15

An interest rate cap consists of a series of caplets that mature every 6 months.11 Specifically, the first caplet matures 6 months from today, and the last caplet matures 6 months prior to the cap maturity date. Let T denote the time to maturity of a cap from today (time t), and K denote its strike interest rate. For notational convenience, we define ∆ = 0.5 and t = t+∆j. j The j-th caplet provides the holder of the cap with the right, not the obligation, to borrow a dollar at the rate of K between times t and t . If the future 6-month LIBOR j j+1 at time t is higher than the strike K, this caplet is exercised and the holder borrows a j dollar at the lower-than-the-fair interest rate. That is, the payoff from exercising the caplet (cid:16) (cid:17) is ∆ × LIBOR(0.5) −K . This payoff occurs at time t because the interest payment is tj j+1 made at the end of the borrowing period. (cid:0) (cid:1) Since the cap is a collection of a m = T −1 number of caplets, the time-t cap value is T ∆ calculated as (cid:34) (cid:35) (cid:88) mT (cid:18) (cid:90) tj+1 (cid:19)(cid:20) (cid:16) (cid:17) (cid:21)+ V(T)(t,K) = EQ exp − r ds ∆× LIBOR(0.5) −K , (9) cap t s tj t j=1 whereQrepresentstherisk-neutralmeasure(seeAppendixA.1). InAppendixA.3, wedemonstrate that equation (9) can be computed using the transform analysis of Duffie, Pan, and Singleton (2000). An interest rate swaption grants the holder the right to enter into an interest rate swap (IRS). There are two types of swaptions. When exercised, a payer swaption delivers an IRS 11More precisely, a conventional cap contract traded in the market is a collection of caplets that mature every 3 months. As documented by Longstaff, Santa-Clara, and Schwartz (2001), assuming semi-annually spaced caplets for computational convenience is innocuous, generating a negligible difference when it comes to Black-implied volatilities. 16

where the holder pays the fixed leg and receives the LIBOR-based floating leg (a payer IRS); a receiver swaption delivers an IRS where the holder receives the fixed leg and pays the LIBOR-based floating leg (a receiver IRS). We now define T as the time to maturity of payer and receiver swaptions. Let K denote their strike interest rate. The tenor of the IRS at the ¯ maturity of the swaptions is denoted as T. Under this notation, our swaptions of interest are ¯ often referred to as T-into-T swaptions. ¯ The payer swaption is exercised if the future T-maturity swap rate at time t+T is larger than the strike K. In this case, the holder enters into a payer IRS contract and makes a profit by exchanging the fixed rate K (which is lower than the fair swap rate) for the floating rate. The pricing relation implies that the time-t payer swaption value is calculated by (cid:34) (cid:35) (cid:18) (cid:90) t+T (cid:19)(cid:20) (cid:21)+ V(T,T¯)(t,K) = EQ exp − r ds V(T¯) (t+T,K)−V(T¯)(t+T,K) , (10) payer t s floating fixed t wherethevalueofthepayerIRSdeliveredatoptionexpiry,ifexercised,issimplythedifference between its floating leg V(T¯) (t+T,K) and its fixed leg V(T¯)(t+T,K). Similarly, the receiver floating fixed ¯ swaption is exercised if the future T-maturity swap rate at time t + T is smaller than the strike K. The time-t receiver swaption value is expressed as (cid:34) (cid:35) (cid:18) (cid:90) t+T (cid:19)(cid:20) (cid:21)+ V(T,T¯)(t,K) = EQ exp − r ds V(T¯)(t+T,K)−V(T¯) (t+T,K) . (11) receiver t s fixed floating t Traditionally, the industry norm was to calculate both the floating and fixed legs of an IRS based on LIBOR-swap discounting. However, the global financial crisis cast doubt on the validity of LIBOR-swap discounting. Going through the Lehman failure and the subsequent 17

credit crunch that led to a sharp rise in LIBOR, the market migrated to OIS discounting, which is now the new norm for derivative pricing (Hull and White, 2015). While our sample period includes both pre-crisis and post-crisis periods, we believe that OIS discounting is theoretically more suitable, especially given that most derivative trades require margins and are heavily collateralized. However, when it comes to tractability, there is a significant benefit of using LIBOR-swap discounting: it always puts the floating leg at par because the floating coupons and their discount rates are both based on future LIBOR rates. As a result, the IRS pricing reduces to computing the fixed leg V(T¯)(t+T,K), which is equivalent to the value of a coupon bond fixed with the (annualized) coupon rate of K:   T¯/∆ V fi ( x T¯ ed )(t+T,K) = ∆K (cid:88) P i (t+T,t+T +j∆)+P i (t+T,t+T +T ¯ ). (12) j=1 For analytical tractability, we opt for the simplifying assumption that the future IRS contract, delivered at option expiry, is valued using LIBOR-swap discounting.12 Then, equation (12) suggests that interest rate swaptions can essentially be viewed as options on a coupon bond. Due to coupon payments, the expression for the coupon bond price contains multiple terms, which makes it impossible to calculate the expectations in equations (10) and (11) using the semi-analytic approach of Duffie, Pan, and Singleton (2000). We adopt the stochastic duration method implemented by Trolle and Schwartz (2009), which enables us to accurately approximate the price of a coupon bond option by a constant multiplication of the 12Under OIS discounting, the floating leg of a LIBOR-based IRS is no longer at par and is worth more than1. Consequently,V(T¯) (t+T,K)remainsintheswaptionpricingformula. Thistermmakesitvirtually floating impossibletoobtainasemi-analyticalsolution. HandlingthefixedlegV(T¯)(t+T,K)alonealreadyrequiresan fixed approximationcalledthestochasticdurationmethodduetonumerouscouponterms; addingfloatingcoupons further complicates the pricing formula to an extreme level. 18

price of a zero-coupon bond option (see, e.g., Wei, 1997; Munk, 1999). We provide a detailed description of the swaption pricing procedure in Appendix A.3. 3 Estimation 3.1 Data Ourdatasampleconsistsofthefollowingvariables: interbankrates, OISrates, Treasuryrates, Black-impliedvolatilitiesforcapsandswaptions, expectedinflation, andrealconsumptionper capita. All variables are sampled at the monthly frequency at the end of each month, from February 2002 to December 2019. Interbank rates consist of 3-, 6-, and 12-month LIBOR rates as well as 2-, 3-, and 5-year swap rates, all of which are downloaded from Bloomberg.13 We also collect OIS rates from Bloomberg and Treasury rates from FRED for the same maturities. To make the data comparable with the model-implied interest rates, we need to convert the three types of interest rate term structures into continuously compounded zero curves. To do so, we use linear interpolation to construct par curves with maturities ranging from 6 months to 5 years with 6-month intervals. From these interpolated par curves, we extract smooth forward rate curves and, in turn, zero-coupon yield curves via the Nelson and Siegel (1987) parameterization. Summary statistics for these interest rates are reported in Panels A, B, and C of Table 1. Note that our analysis relies on interest rates with maturities of up to 5 years, despite their 13On March 5, 2021, the ICE Benchmark Administration Limited announced the complete discontinuation of the publication of USD LIBOR effective June 30, 2023. The crucial role of USD LIBOR as a benchmark interest rate will be largely taken over by the secured overnight financing rate (SOFR). See Jermann (2019), Jermann (2020a), and Klingler and Syrstad (2021) for further discussion about the transition from USD LIBOR to SOFR. 19

Mean SD Min Median Max AR1 Panel A: Treasury rate 3M 1.32 1.51 0.00 0.95 5.13 0.993 6M 1.43 1.53 0.03 1.01 5.17 0.994 1Y 1.55 1.48 0.11 1.23 5.14 0.993 2Y 1.74 1.40 0.15 1.37 5.09 0.988 3Y 2.00 1.31 0.32 1.61 5.06 0.984 5Y 2.44 1.20 0.57 2.21 5.03 0.976 Panel B: OIS rate 3M 1.45 1.59 0.08 1.00 5.35 0.995 6M 1.48 1.59 0.08 1.04 5.41 0.995 1Y 1.59 1.58 0.10 1.17 5.50 0.993 2Y 1.80 1.53 0.12 1.35 5.45 0.989 3Y 2.07 1.47 0.23 1.61 5.44 0.986 5Y 2.49 1.40 0.48 2.17 5.47 0.981 Panel C: Interbank rate 3M 1.71 1.60 0.22 1.18 5.58 0.992 6M 1.85 1.54 0.32 1.34 5.51 0.993 1Y 2.07 1.42 0.54 1.64 5.56 0.991 2Y 2.10 1.46 0.39 1.66 5.55 0.988 3Y 2.31 1.44 0.44 1.89 5.55 0.985 5Y 2.76 1.34 0.75 2.56 5.56 0.979 Panel D: Cap IV 2Y 49.65 25.10 12.18 50.52 102.70 0.969 3Y 46.41 21.56 13.43 46.30 93.22 0.964 4Y 43.43 18.63 13.99 43.03 82.59 0.958 5Y 40.26 15.97 14.26 39.80 77.47 0.954 Panel E: Swaption IV 1-into-4 36.57 12.88 13.20 37.20 75.22 0.924 2-into-3 34.88 12.01 14.10 34.09 70.75 0.932 3-into-2 33.47 11.67 14.55 32.28 67.86 0.936 4-into-1 32.16 11.35 14.60 30.50 69.33 0.931 Table 1: Summary statistics. This table reports the mean, standard deviation, minimum, median, maximum,andfirst-orderautoregressivecoefficient(AR1)ofcontinuously-compoundedzero-couponTreasuryrates (PanelA),OISrates(PanelB),andinterbankrates(PanelC),aswellascapandswaptionimpliedvolatilities (Panels D and E). All values are expressed in percentage terms, except for the AR1, which is unitless. availability of up to 30 years. This is our intentional choice, as long-term interest rates are potentially subject to some frictions that are beyond our model. In particular, since the 2008 financial crisis, the 30-year swap rate has maintained a lower level than the 30-year Treasury rate, which poses a puzzle. This phenomenon is not just seen in the 30-year maturity: we frequently observe a negative swap spread for any long-term maturity between 5 years and 30 years. Klingler and Sundaresan (2019) argue that negative swap spreads reflect underfunded 20

pension plans’ increased demand for duration hedging with long-term swaps. Even without such explicit demand effects, Jermann (2020b) shows that negative swap spreads can be justified via limits to arbitrage associated with frictions for holding long-term Treasuries. Rather than incorporating extra frictions into our model, we keep our model simple and focus on short- and mid-term interest rates. We download the pricing data on caps and swaptions from Bloomberg.14 Caps and swaptions are typically quoted in terms of Black-implied volatilities. For each market price, the corresponding Black-implied volatility is found by backsolving the volatility term in the Black (1976) model. Caps and swaptions in our sample are at-the-money forward (ATMF), meaning that the strike price of each option is equal to the current forward price of the underlying. Specifically, the strike price of a T-maturity ATMF cap is the T-maturity swap rate. The ¯ strike price of a T-into-T ATMF swaption is the forward swap rate between the option expiry ¯ (T years from today) and the underlying swap expiry (T + T years from today). Panels D and E of Table 1 present summary statistics for cap and swaption implied volatilities. The data on expected inflation is from the Blue Chip Economic Indicators survey. This dataset provides the forecasts of inflation for the current calendar year and the next calendar year. For each month, we calculate a proxy for the 1-year-ahead expected inflation by calculating the weighted average between the two forecasts. Lastly, the monthly time series of real consumption per capita is downloaded from FRED. 14There are multiple sources for caps and swaptions data on Bloomberg. We mainly use the sources “CMPN” for caps and “BBIR” for swaptions, which provide the Black-implied volatilities calculated based on LIBOR-swap discounting. 21

3.2 Unscented Kalman filter and maximum likelihood estimation From the data described in Section 3.1, we filter the time series of the latent processes (λ , ξ , t t µ , x ) using the unscented Kalman filter and estimate the model parameters via maximum t t likelihood estimation (MLE). In this section, we detail our model estimation procedure. Estimating our model is computationally challenging. In each iteration of the MLE procedure, we need to evaluate the log-likelihood function. Due to cap/swaption pricing, this requires numerically solving the system of complex-valued ordinary differential equations of Duffie, Pan, and Singleton (2000) several times. Furthermore, we have a large number of parameters to be estimated. Evidently, estimating all of these parameters all at once in a single MLE is highly time-consuming. To alleviate the computational burden, we first separately estimate the parameters κ , q¯, q andσ , whichgovernthenormal-timedynamicsofexpectedinflation. Thisispossiblebecause q a consumption disaster is absent during our sample period, implying that the observed time series of expected inflation was completely driven by these three parameters. Specifically, κ , q q¯, and σ are estimated by maximizing the log-likelihood of the expected inflation time series, q conditional on no disasters. We also set µ¯ to be 1.05%, which is the average real consumption growth during our sample period. We further reduce the dimension of our parameter space by putting a restriction on the value of δ . Taking expectations on both sides of equation (5) results in: 0 E[r] = δ +δ E[λ]+δ E[ξ]+δ E[µ]+δ E[q]. 0 λ ξ µ q Here, we proxy the unconditional mean of the short rate (E[r]) by the average 1-week OIS 22

rate (Eˆ [r]) from Bloomberg. Then, the value of δ is given by 0 δ = Eˆ [r]−δ ξ ¯ −δ ξ ¯ −δ µ¯−δ Eˆ [q], 0 λ ξ µ q where Eˆ [q] is the average expected inflation during our sample period. ¯ Moreover, we set ξ to be 2.86%, which is the average probability of consumption disasters acrossOECDcountries, accordingtoBarroandUrsu´a(2008). Wealsoconstructtheempirical distributions of Z and Z by compiling consumption declines and inflation rates during C,t q,t historical consumption disasters from the Barro-Ursua dataset.15 Lastly, we assume Z = L,t Z : themoreseveretheinterbankmarketfailure,themoreseveretheconsumptiondisaster.16 C,t Asaresult,weareleftwith20parameterstobeestimatedwithinthemainMLEprocedure: Θ = [κ ,σ ,σ ,κ ,σ ,κ ,σ ,κ ,σ ,x¯,p¯,δ ,δ ,δ ,δ ,θ ,θ ,θ ,θ ,θ ], µ µ C ξ ξ λ λ x x λ ξ µ q λ ξ µ q N besides three extra parameters concerning measurement errors (described below). We derive the likelihood function L under the assumption that we observe the following: (i) LIBOR-OIS spreads and TED spreads with 3-, 6-, 12-month maturities, (ii) interbank rates, OIS rates, and Treasury rates with 3-, 6-, 12-month, 2-, 3-, 5-year maturities, (iii) 2-, 3-, 4-, 5-year cap implied volatilities, (iv) 1-into-4, 2-into-3, 3-into-2, 4-into-1 swaption implied volatilities, and (v) real consumption growth. For notational simplicity, we let Y denote the vector of all these t 15The Barro-Ursua dataset contains a few extreme hyperinflation events, such as the hyperinflation of Weimar Germany in the 1920s when the inflation rate exceeded 3,000%. Given that the number of historical consumption disasters is only 89, such extreme outliers completely dominate the moment generating function of Z . Thus, we exclude the observations that fall more than 3 times the interquartile range above the third q,t quartile. No observations fall more than 3 times the interquartile range below the first quartile. 16We can relax this assumption by setting Z = kZ and estimating k directly. However, in this case, L,t C,t the coefficient k is not well identified separately from p¯; an increase (decrease) in k can be compensated by a decrease (increase) in p¯or vice versa. 23

observations at time t. To construct the likelihood function L, it suffices to derive the transition density of Y . t Specifically, we define L as the likelihood of observing Y conditional on Y : t t t−∆t L = P(Y |Y ;Θ), t t t−∆t where ∆t = 1/12 represents monthly time intervals between observations. Not only does this transition density depend on the observable state variable q , but it also relies on the four t latent variables λ , ξ , µ , and x . t t t t In order to solve this filtering problem, we specify the state equation and the measurement equation in the state-space representation of our model. The state equation describes the dynamics of a latent state vector S = [λ ,ξ ,µ ,x ]⊤. There are two ways to map the t t t t t continuous-time dynamics of S into the discrete-time state equation. The first approach apt plies the Euler discretization to λ , ξ , µ , and x and then finds the discrete-time relation t t t t between S and S . In contrast, the second approach finds the exact relation between S t t−∆t t and S without any approximation and then discretizes the resulting relation. We adopt t−∆t the latter approach following Chen and Scott (2003), as it better captures the square-root diffusions of the first two latent processes. Consequently, we arrive at the following linear state equation: S = η +ΨS +ϵ , where E (cid:2) ϵ ϵ⊤ (cid:3) = Ω . (13) t t−∆t t t−∆t t t t−∆t We provide the expressions for four-dimensional vector η, 4 × 4 matrix Ψ, and 4 × 4 timevaryingcovariancematrixΩintheInternetAppendix. Sinceϵ isnon-normal, weapproximate t it by a normal distribution with the same covariance matrix. Prior studies document that 24

this approximation is innocuous.17 Now, we turn to the measurement equation. We assume that Y is observed with a vector t of measurement errors e : t Y = h(S ,q )+e , where E (cid:2) e e⊤ (cid:3) = Q. t t t t t−∆t t t Note that h is a vector-valued function of the state variables, which generates the model counterparts of the data. The mean-zero random vector e is normally distributed with t covariance matrix Q. For parsimony, all measurement errors are assumed to be independent of one another, making the off-diagonal entries of Q zero. As for the diagonal entries, we assume that the standard deviations of the measurement errors are identical for the interest rate spreads (σ ), for the interest rates (σ ), and for the Black-implied volatilities (σ ). SP ITR OPT The measurement equation clearly suggests that the linear Kalman filter cannot be used in our estimation. This is because h is not a linear function: cap/swaption prices as well as their Black-implied volatilities are nonlinear in the state variables. Therefore, we apply the unscented Kalman filter, proposed by Wan and Van Der Merwe (2000). The Internet Appendix provides a detailed description of how the unscented Kalman filter is implemented under our setup. IneachiterationoftheMLEprocedure, weobtainnotonlythetimeseriesoftheestimated (cid:110) (cid:111) ˆ ˆ latent variables λ , ξ , µˆ , xˆ , but also the time series of the transition densities {L }. Let t t t t t {t }n denote monthly-spaced points in time when the data time series are observed. Then, k k=1 17See, for example, Duan and Simonato (1999), Duffee (1999), Chen and Scott (2003), Trolle and Schwartz (2009), and Filipovi´c and Trolle (2013). 25

the log-likelihood function for the entire observations can finally be expressed as n (cid:88) logL = logL . t k k=1 We determine the parameter estimates by maximizing this log-likelihood function. 4 Estimation results and implications 4.1 Parameter estimates Table 2 reports the values of our parameter estimates, along with their robust standard errors in parentheses. First of all, The estimated dynamics of consumption growth and expected inflation are economically sensible. During normal periods without disasters, consumption growth has a long-run mean (µ¯) of 1.05% and a standard deviation (σ ) of 0.81%, broadly C consistent with the first two moments of the observed consumption time series in our sample. The process of mean consumption growth µ is not only persistent with κ = 0.7343 but also t µ quite volatile with σ = 0.37%. The parameter values for expected inflation are also in line µ with its data time series: the long-run mean (q¯) is 2.12%, the conditional volatility (σ ) is q 0.65%, and the monthly autocorrelation (e−κq∆t) is 0.96.18 Our main focus is on the model parameters that are associated with the dynamics of disaster risk. In our model, disaster risk has a two-factor structure with λ and ξ . While both t t processes are estimated to be highly persistent with low mean reversion speed (κ = 0.4373 λ 18While expected inflation plays a crucial role in capturing the level of interest rates in our model, its omission is not impossible given that our model is reduced-form. In the Internet Appendix, we consider a nested case of our model without expected inflation and show that it is rejected based on a likelihood ratio test. 26

κ σ σ µ¯ µ µ C Consumption growth Est. 0.7343 0.0037 0.0081 0.0105 (SE) (0.0152) (0.0002) (0.0005) κ σ q¯ q q Expected inflation Est. 0.5239 0.0065 0.0212 (SE) (0.3153) (0.0011) (0.0028) κ σ ξ¯ ξ ξ Est. 0.0446 0.0204 0.0286 (SE) (0.0071) (0.0002) Disaster risk κ σ p¯ λ λ Est. 0.4373 0.1560 0.9035 (SE) (0.0130) (0.0039) (0.0295) κ σ x¯ x x Convenience yield Est. 0.8162 0.0017 0.0021 (SE) (0.1915) (0.0003) (0.0004) δ δ δ δ δ λ ξ q µ 0 Short rate Est. -0.1443 -2.1572 0.4594 2.1835 0.0476 (SE) (0.0026) (0.1160) (0.0251) (0.0356) θ θ θ θ θ λ ξ q µ N Market price of risk Est. -0.1355 -0.6377 -3.6234 0.2696 -0.1075 (SE) (0.0031) (0.0589) (0.0870) (0.0139) (0.0061) σ σ σ SP ITR OPT Measurement errors Est. 0.0014 0.0020 0.0742 (SE) (0.0001) (0.0001) (0.0032) Table 2: Parameter estimates. This table reports the values of the model parameters estimated through the unscented Kalman filter/MLE procedure. The model consists of the following five state variables: real consumption (C ), mean consumption growth (µ ), expected inflation (q ), instantaneous disaster intensity t t t (λ ), and the long-run mean of disaster intensity (ξ ). We also report the dynamics of the instantaneous t t Treasury convenience rate (x ), the factor loadings on the short rate, the market prices of risk, and the t standard deviations of measurement errors. Reported together in parentheses are robust standard errors. and κ = 0.0446), λ is relatively less persistent than ξ . This is intuitive because λ cap- ξ t t t tures the short-run component of disaster risk, whereas ξ captures the long-run component. t Consistent with this interpretation, λ exhibits a higher conditional volatility than ξ (i.e., t t σ > σ ). In Section 4.3, we further characterize the time variation in disaster risk by exam- λ ξ ining the filtered time series of λ and ξ . The conditional probability of interbank market t t failure given the occurrence of a disaster (p¯) is estimated as 0.9035, suggesting that the two types of extreme events are indeed likely to coincide. We also report the estimated dynamics of the instantaneous convenience rate x . Relative t to λ and ξ , the process for x is much less persistent with κ = 0.8162, which corresponds t t t x 27

to a monthly autocorrelation of 0.93. The long-run mean (x¯) is 0.21%, and the conditional volatility (σ ) is 0.17%. Given that the average 3-month Treasury rate is only about 1.32% x over our sample period, the relative magnitude of the Treasury convenience yield is not small. The estimated coefficients δ , δ , δ , and δ in Table 2 show how the four state variables λ , λ ξ q µ t ξ , q , and µ affect the nominal short rate in our estimated model. We first observe that the t t t factor loading on expected inflation is positive. This conforms to the economic intuition that expected inflation rises with nominal interest rates. While expected inflation is an important factor, empirical evidence suggests that a significant portion of the variation in the nominal short rate is still attributable to the variation in the real short rate (Haubrich, Pennacchi, and Ritchken, 2012). The table reports that the factor loading is positive for µ but negative for t λ and ξ . This is in accordance with the precautionary savings motive. Investors are inclined t t to save more to secure against future uncertainty when mean consumption growth declines or disaster risk rises, driving down real interest rates. Additionally reported are the market prices of risk. The coefficients θ , θ , and θ are neg- λ ξ q ative, as investors dislike both horizons of disaster risk and inflation risk: investors’ marginal utility, or equivalently, the pricing kernel in equation (4), rises when positive realizations of dB , dB , and dB raise λ , ξ , and q , respectively. Similarly, Poisson shocks are also λ,t ξ,t q,t t t t priced negatively with θ < 0. Since eθNZC,t is larger than 1, the pricing kernel jumps up N when a disaster occurs (dN = 1). In contrast, the coefficient θ is positive, as dB is a good t µ µ,t economic shock that increases consumption growth. All in all, we conclude that the signs and magnitudes of the model parameters from our estimation align well with general economic intuition. 28

4.2 Model fit to interest rates, caps, and swaptions We now investigate whether our estimated model is capable of producing a reasonable fit to the market data on interest rates, caps, and swaptions. Figure 1 first examines short-term interest rates by plotting the time series of the 3-month Treasury rate (Panel A), OIS rate (Panel B), interbank rate (Panel C), LIBOR-OIS spread (Panel D), TED spread (Panel E), and Treasury convenience yield (Panel F) in the data (solid blue lines) and in the model (dashed red lines). Panel A: Treasury rate Panel B: OIS rate Panel C: Interbank rate Panel D: LIBOR-OIS spread Panel E: TED spread Panel F: Convenience yield Figure 1: 3-month interest rates and their spreads in the data and in the model. This figure plots the time series of the 3-month Treasury rate (Panel A), OIS rate (Panel B), interbank rate (Panel C), LIBOR-OIS spread (Panel D), TED spread (Panel E), and Treasury convenience yield (Panel F) in the data and in the model, from February 2002 to December 2019. The solid blue lines represent the data, and the dashed red lines represent the model. All interest rates are calculated using continuously compounded zero ratesandareexpressedinpercentageterms. Datasource: BloombergFinanceLP,BloombergTerminals,and Federal Reserve Bank of St. Louis. Panel A of Figure 1 shows that at the beginning of our data sample, the Treasury rate enteredasteadydownwardtrendupuntil2004. Duringthistime, theFederalReservelowered the federal funds rate from 6% to 1%, as investors faced high economic and financial uncertainty stemming from the 2001 recession, September 11 attacks, and Afghanistan War. This 29

period of expansionary monetary policy was followed by a 3-year period of contractionary monetary policy due to a housing market bubble and high inflation, causing the Treasury rate to gradually increase. From 2007, the Treasury rate began to rapidly decline again when the economy was hit by the subprime mortgage crisis and eventually reached a level close to zero. Since then, the Treasury rate maintained a very low level until the Federal Reserve started raising interest rates in 2015. The 3-month OIS rate in Panel B and the 3-month interbank rate in Panel C also exhibit similartimeseriespatternscomparedtothe3-monthTreasuryrate,althoughtheirmagnitudes are generally larger. However, a distinctive pattern is observed around September 2008 when Lehman Brothers filed for bankruptcy. In contrast to the other two rates, the interbank rate sharply increased, reflecting a serious risk of a potential systemic meltdown in financial markets. This event is more noticeable from the time series of the LIBOR-OIS spread in Panel D and that of the TED spread in Panel E. While the LIBOR-OIS spread stayed at a high level between 2007 and 2009 during the Great Recession period, an exceptionally high value of roughly 2.4% was seen in September 2008. Fluctuations in the TED spread are more magnified. As can be seen in Panel F, the 3-month Treasury convenience yield shot up to about 1% and remained high as investors hoarded Treasury securities (flight-tosafety/liquidity). As a result, the TED spread, which is the sum of the LIBOR-OIS spread and the Treasury convenience yield, sharply went up, even beyond 3%. FromthesixpanelsofFigure1, wefindthatourmodeleffectivelyaccountsforthepatterns of the three types of interest rates as well as their spreads. The dashed red lines that represent the model-implied time series closely resemble the solid blue lines that represent the data time series, peaking and dipping at similar points in time. This holds true for longer-maturity 30

Panel A: 2-year Treasury rate Panel B: 3-year Treasury rate Panel C: 5-year Treasury rate Panel D: 2-year OIS rate Panel E: 3-year OIS rate Panel F: 5-year OIS rate Panel G: 2-year interbank rate Panel H: 3-year interbank rate Panel I: 5-year interbank rate Figure 2: Interest rates in the data and in the model. This figure depicts the time series of the Treasury rates (Panels A, B, and C), OIS rates (Panels D, E, and F), and interbank rates (Panels G, H, and I) with 2-, 3-, and 5-year maturities in the data and in the model, from February 2002 to December 2019. Thesolidbluelinesrepresentthedata,andthedashedredlinesrepresentthemodel. Allvaluesareexpressed in percentage terms. Data source: Bloomberg Finance LP, Bloomberg Terminals, and Federal Reserve Bank of St. Louis. interest rates as well. In Figure 2, the model-implied Treasury rates, OIS rates, and interbank rates with 2-, 3-, and 5-year maturities closely track their data counterparts. More formally, Panel A of Table 3 reports three measures of empirical performance: mean error (ME), mean absolute error (MAE), and mean relative error (MRE). The pricing errors for interest rates, while slightly increasing with maturity, are small across the board. For instance, the MAEs range from 0.05% (6 months) to 0.25% (5 years), confirming that the model captures the interest rate data fairly well. We now turn to caps and swaptions. Panels A, B, and C of Figure 3 present the time 31

Panel A: Mean pricing errors 3M 6M 1Y 2Y 3Y 5Y ME 0.0003 0.0002 -0.0001 -0.0014 -0.0009 0.0013 Interest rates MAE 0.0011 0.0005 0.0010 0.0017 0.0016 0.0025 MRE 0.0190 0.0135 -0.0081 -0.0721 -0.0374 0.0532 2Y 3Y 4Y 5Y ME -0.0369 0.0023 0.0146 0.0094 Cap IVs MAE 0.1293 0.0805 0.0532 0.0466 MRE -0.0743 0.0049 0.0335 0.0234 1-into-4 2-into-3 3-into-2 4-into-1 ME 0.0634 0.0553 0.0405 0.0184 Swaption IVs MAE 0.0702 0.0593 0.0516 0.0489 MRE 0.1734 0.1585 0.1211 0.0573 Panel B: Regressions of data on model-implied values 3M 6M 1Y 2Y 3Y 5Y Coef. 0.98 1.00 1.01 1.04 1.02 0.98 Interest rates t-stat [193.74] [455.85] [230.11] [143.57] [111.10] [74.76] R2 (%) 99.34 99.80 99.35 98.92 98.05 95.42 2Y 3Y 4Y 5Y Coef. 0.74 0.84 0.87 0.83 Cap IVs t-stat [13.91] [19.22] [28.95] [30.36] R2 (%) 71.67 81.57 89.07 91.58 1-into-4 2-into-3 3-into-2 4-into-1 Coef. 0.82 0.84 0.83 0.77 Swaption IVs t-stat [15.32] [17.87] [17.08] [13.88] R2 (%) 79.73 87.74 87.86 84.83 Table 3: Pricing errors. This table evaluates the empirical performance of our model. Panel A reports threemeasuresofmeanpricingerrors: meanerror(ME),meanabsoluteerror(MAE),andmeanrelativeerror (MRE), averaged by security type (interest rates, cap implied volatilities, and swaption implied volatilities) and maturity. Panel B presents regressions of each data series on its corresponding model-implied series and reports the resulting slope coefficients and R2 values. The t-statistics, shown in brackets, are computed using the Newey and West (1987) method with four lags. series of the Black-implied volatilities for the 2-, 3-, and 5-year caps, while Panels D, E, and F present those for the 1-into-4, 2-into-3, and 4-into-1 swaptions. In each panel, the solid blue line denotes the data, and the dashed red line denotes the model. The figure shows that our model’s fit to the data is decent but not perfect. As reported in Panel A of Table 3, the MAEs of cap and swaption implied volatilities range between 5% and 8%, except for the 2-year cap, which has an MAE of 13%. It is not surprising that our model cannot accurately match every cap or swaption. Prior studies document that it is challenging to jointly account for the pricing of caps and swaptions in crisis periods, even with flexible statistical models featuring 32

several latent processes (Longstaff, Santa-Clara, and Schwartz, 2001; Han, 2007; Trolle and Schwartz, 2009).19 Our objective is not to fit the data perfectly. Rather, we attempt to characterize time-varying disaster risk by exploiting the information contained in interbank rates and their options. Instead of adding more factors to improve the fit, we choose to keep our model simple and parsimonious. For our purposes, we believe that our simple economic model does a reasonably good job of capturing the data overall.20 Panel A: 2-year cap Panel B: 3-year cap Panel C: 5-year cap Panel D: 1-into-4 swaption Panel E: 2-into-3 swaption Panel F: 4-into-1 swaption Figure 3: Black-implied volatilities in the data and in the model. This figure presents the time series of the Black-implied volatilities for the 2-, 3-, and 5-year caps (Panels A, B, and C) and those for the 1-into-4, 2-into-3, and 4-into-1 swaptions (Panels D, E, and F) in the data and in the model. The sample period is from February 2002 to December 2019. The solid blue lines represent the data, and the dashed red linesrepresentthemodel. Allvaluesareexpressedinpercentageterms. Datasource: BloombergFinanceLP, Bloomberg Terminals. Panel B of Table 3 further assesses model fit by regressing each data series on its corresponding model-implied series (see, e.g., Bakshi, Crosby, Gao, and Hansen, 2023). The resulting slope coefficients are reported alongside the R2 values. If the model accurately cap- 19Related, Bakshi, Crosby, Gao, and Hansen (2023) explain Treasury yields and option prices by adopting several latent variables, including both spanned and unspanned factors. 20DirectlyinterpretingthemagnitudeandthetimeseriesvariationofBlack-impliedvolatilitiesforcapsand swaptions may not be as intuitive because they represent yield volatilities, rather than bond price volatilities. In the Internet Appendix, we examine our model fit when Black-implied volatilities are converted into their equivalent bond price volatilities. 33

tures the time-series variation in the data, the estimated slope coefficients should be close to one and the regression should yield a high R2. The panel shows that slope coefficients for interest rates are indeed close to one, ranging from 0.98 to 1.04, and are statistically significant with extremely high R2 values between 95.42% and 99.80%. For cap and swaption implied volatilities, the model yields somewhat lower but still substantial slope coefficients ranging from 0.74 to 0.87, with R2 values between 71.67% and 91.58%. Taken together, these results demonstrate that our estimated model provides a good fit to the data. 4.3 Characterizing time-varying disaster risk Our estimation procedure enables us to characterize the time variation in disaster risk via the unscented Kalman filter. Figure 4 presents the filtered time series of the short-run disaster risk component λ (solid blue line) and the long-run disaster risk component ξ (dashed red t t line), along with their summary statistics.21 The figure clearly shows that the instantaneous disaster risk λ is much more volatile than t its time-varying mean ξ . Since λ is highly persistent, it sometimes significantly deviates t t from its mean value of 2.86% for extended periods of time. While λ hovered around a low t level below 1% between 2002 and 2006, it abruptly increased to an extremely high level at the onset of the subprime mortgage crisis in 2007. During the subsequent 2-year period of severe economic downturns and financial market turmoil, λ persisted at levels above 4%. t When Lehman Brothers declared bankruptcy in September 2008, λ reached its highest value, t exceeding 11%. After the crisis, the level of λ came back to a normal level, but we still can t see some small peaks that are associated with economic and financial uncertainty, such as 21The filtered time series and summary statistics of the other two latent variables x and µ are reported t t in the Internet Appendix. 34

Mean SD Min 5th 50th 95th Max λ 1.56 1.65 0.10 0.19 1.07 4.32 11.13 ξ 2.91 0.64 1.46 1.89 3.05 3.73 3.88 Figure 4: Implied time-varying disaster risk. This figure displays the filtered time series of the shortrun disaster risk component λ (solid blue line) and the long-run disaster risk component ξ (dashed red line) t t from February 2002 to December 2019. Reported together are the mean, standard deviation, minimum, 5th percentile, median, 95th percentile, and maximum values for each time series. All values are expressed in percentage terms. in the European sovereign debt crisis.22 The standard deviation of λ is 1.65% during our t sample period. While λ captures a fast-moving component of disaster risk, ξ captures a slow-moving t t component. The filtered time series in Figure 4 reveal that ξ is considerably less volatile than t λ (with a standard deviation of 0.64% versus 1.65%) and moves slowly without deviating too t much from its mean value. Moreover, we observe that ξ is far more persistent than λ : once t t it is hit by a large positive shock, it takes a long time for it to mean-revert back to its previous level. For instance, the level of ξ remained elevated during the post-Great Recession period. t This is in sharp contrast with the behavior of λ , whose level quickly dropped even before the t 22TheInternetAppendixextendsthedatatoDecember2020andexamineshowdisasterriskevolvedduring the COVID-19 pandemic crisis. It shows that the short-run component λ jumped up during the peak of the t crisis, exceeding the long-run component ξ , and then abruptly declined in the following months. t 35

crisis was over. Which aspect of the data makes it possible for us to characterize the time variation in λ and ξ , as discussed above? To understand how time-varying disaster risk is identified t t through our model, we conduct a sensitivity analysis. Figure 5 shows how the 3-, 6-, 12month LIBOR-OIS spreads (Panel A) and the Black-implied volatilities for caps (Panel B) and swaptions (Panel C) change when we vary λ or ξ from the 5th percentile to the 95th t t percentile of its filtered values. In each panel, the solid blue lines describe the sensitivity with respect to λ while fixing ξ at the median, whereas the dashed red lines describe the t t sensitivity with respect to ξ while fixing λ at the median. Expected inflation q and mean t t t consumption growth µ are set at their median values in both cases. Lastly, the black dot in t the middle of each bar represents the model value when λ and ξ are both at their median t t values. Panel A: SensitivityofLIBOR-OIS Panel B: Sensitivity of caps Panel C: Sensitivity of swaptions Figure 5: Comparative statics. This figure shows how the 3-, 6-, 12-month LIBOR-OIS spreads (Panel A) and the Black-implied volatilities for caps (Panel B) and swaptions (Panel C) change when λ or ξ varies t t from the 5th percentile to the 95th percentile of its filtered values. In each panel, the solid blue lines describe the sensitivity with respect to λ while fixing ξ at the median, whereas the dashed red lines describe the t t sensitivity with respect to ξ while fixing λ at the median. Expected inflation q and mean consumption t t t growthµ aresetattheirmedianvaluesinbothcases. Theblackdotinthemiddleofeachbarrepresentsthe t model value when λ and ξ are both at their median values. All values are expressed in percentage terms. t t Panel A of Figure 5 reveals that the short-run component of disaster risk λ is mainly t 36

identified by the LIBOR-OIS spreads. While the LIBOR-OIS spreads increase with both λ t and ξ , they are much more sensitive to λ . For example, the 3-month LIBOR-OIS spread t t moves substantially, ranging from 0.1% to 0.9%, when λ varies between the 5th and 95th t percentiles. In contrast, the 3-month LIBOR-OIS spread barely changes with respect to ξ , t as can be seen in the panel. These results are intuitive. The 3-month interbank rate is higher than the 3-month OIS ratebecauseitisfurtherinfluencedbytheriskofadisasterhappeningovera3-monthhorizon. Therefore, their gap, the 3-month LIBOR-OIS spread, is mostly sensitive to the short-run component of disaster risk. For the same reason, ξ plays a more noticeable role if a longer t horizon is considered: from the panel, we find that the longest LIBOR-OIS spread with a 12-month maturity is more responsive to changes in ξ , compared to the 3-month LIBOR-OIS t spread. However, the effect of ξ is still minuscule across all maturities, relative to the effect t of λ . This confirms that the time series of the LIBOR-OIS spreads are the major channel t through which the time variation of λ is identified. t Although the long-run component of disaster risk ξ has little impact on the LIBOR-OIS t spreads, it has a large impact on interbank rate options. In Panels B and C of Figure 5, we discover completely opposite patterns. The Black-implied volatilities in both panels are highly sensitive to changes in ξ , whereas they are relatively less sensitive to changes in λ . t t The payoffs of caps and swaptions depend on future interbank rates over a long horizon ranging from 1 to 5 years. Hence, the pricing data on caps and swaptions play a critical role in characterizing the time variation of ξ . t Note that these model exercises are predicated on the assumption that both λ and ξ t t are time-varying. In the Internet Appendix, we estimate two nested cases of our model: (i) 37

constant long-run mean of disaster risk (i.e., no variation in ξ ) and (ii) constant disaster t risk (i.e., no variation in λ ). Based on the likelihood ratio test, the two nested models are t rejected, highlighting the significance of incorporating both time-varying λ and ξ . t t In sum, our analysis demonstrates that both the short-run and long-run components of disaster risk are important model ingredients and are well identified using the data on interbank rates and their options.23 It is worth emphasizing that the benefit of using interbank rate options is not limited to identifying the long-run component: the forward-looking information from caps and swaptions helps us accurately estimate the overall dynamics of disaster risk. 4.4 Implications for the equity market Rare disaster models are often criticized as macro-finance models with “dark matter.” In order to explain the high equity premium and volatility in the postwar period, these models need to rely on the possibility of extremely bad events and its substantial time variation. However, it is not possible to measure disaster risk directly from the data, nor statistically test it with meaningful power, due to the rare nature of such events. This dark matter criticism raises some concerns about how disaster risk models are calibrated: in typical variable disaster risk models, the dynamics of disaster risk are calibrated to match some key equity market moments, such as the equity premium and the market volatility. However, Chen, Dou, and Kogan (2024) point out that a model with economic dark matter is likely to be fragile due to the lack of internal refutability and poor out-of-sample performance. Addressing this criticism is challenging: as discussed by Cochrane (2017), it 23In addition, the Internet Appendix provides a discussion of how the short-run and long-run components of disaster risk differently affect the instantaneous bond variance. 38

requires either independently anchoring time-varying disaster risk to some data or reconciling multiple asset classes under one consistent assumption about disaster risk. Our analysis highlights that the interbank market can potentially be useful for addressing the dark matter criticism. We make a plausible assumption that consumption disasters are likely to coincide with interbank market failure. This identification assumption makes it possible to extract time-varying disaster risk that is manifested in LIBOR-OIS spreads as well as caps and swaptions. Since our estimation does not depend on equity market moments, our results can serve as an external/out-of-sample validity test of disaster risk models for the equity market. The parameter estimates from Section 4.1 suggest that the estimated disaster dynamics are fairly close to those implied by Seo and Wachter (2018), whose calibration is primarily based on the equity market. Overall, our finding that disaster risk is significant in size and in variation strongly supports a disaster-based explanation of various asset pricing puzzles (Gabaix, 2012; Wachter, 2013). An additional advantage of our approach is that we obtain the past time series of the short-run and long-run components of disaster risk, namely λ and ξ . These time series t t provide an extra basis for testing the implications of disaster risk for the equity market. First of all, Panel A of Table 4 considers the following conditional moments that are associated with the equity market: the price-dividend ratio (log P/D), price-earnings ratio (log P/E), implied variance (IV), expected realized variance (ERV), and variance risk premium (VRP).24 Since these conditional moments are functions of disaster risk in variable disaster risk models, one testable implication is that their time series variations should be explained by disaster-related state variables. 24The price-dividend and price-earnings ratios are downloaded from Robert Shiller’s website. The three variance-related variables are downloaded from Hao Zhou’s website. 39

Panel A: Valuation ratios and variance-related variables (1) (2) (3) (4) (5) log P/D log P/E IV ERV VRP Coef. -0.68 -0.53 0.72 0.64 0.37 λ t t-stat [-5.13] [-4.18] [5.54] [3.54] [2.81] Coef. -0.48 -0.20 0.02 0.06 -0.05 ξ t t-stat [-7.25] [-2.63] [0.31] [1.01] [-0.80] Adj R2 (%) 68.69 31.87 52.30 41.29 13.83 Panel B: Out-of-the-money put option prices (1) (2) (3) (4) (5) 1 month 3 months 6 months 9 months 12 months Coef. 0.70 0.68 0.65 0.58 0.39 λ t t-stat [5.38] [5.78] [5.34] [5.18] [4.62] Coef. 0.04 0.11 0.21 0.31 0.31 ξ t t-stat [0.62] [1.46] [2.67] [3.88] [4.02] Adj R2 (%) 49.28 47.21 46.52 43.05 24.49 Table 4: Disaster risk and conditional equity moments. This table reports the results of contemporaneous time series regressions that examine the relation between the filtered disaster risk variables (λ and ξ ) t t and conditional equity market moments. Since λ and ξ are correlated, we take the orthogonal component t t of λ with respect to ξ . In both panels, we standardize the independent and dependent variables. In Panel t t A, the dependent variables are the log price-dividend ratio (log P/D), log price-earnings ratio (log P/E), implied variance (IV), expected realized variance (ERV), and the variance risk premium (VRP). In Panel B, the dependent variables are the prices of S&P 500 put options with 90% moneyness, normalized by the underlying index level. We consider options with maturities of 1, 3, 6, 9, and 12 months. The t-statistics, shown in brackets, are computed using the Newey and West (1987) method with four lags. Hence, we run contemporaneous linear regressions of the five conditional equity market moments at time t on λ and ξ . We standardize both independent and dependent variables t t in our regressions to facilitate the interpretation of slope coefficients. In columns (1) and (2) of Panel A, we document that the valuation ratios fall when λ and ξ rise. For instance, a t t one standard deviation increase in λ leads to a 0.68 standard deviation drop in the log pricet dividend ratio. A one standard deviation increase in ξ leads to a 0.48 standard deviation t drop in the log price-dividend ratio. These negative relations are statistically significant with high t-statistics. These results are consistent with economic intuition as well as empirical evidence: stock market valuations are low in bad economic times with high disaster risk. In columns (3), (4), and (5), we examine the relation between disaster risk and each of 40

the three variance-related variables. Note that the implied variance and the expected realized variance measure the risk-neutral and physical expectations of future 1-month equity market variance, respectively. The variance risk premium, calculated as their difference, captures compensation for taking variance risk over a 1-month horizon. We expect positive relations from our regression because higher disaster risk results in higher variance risk as well as higher compensation for variance risk. In line with this, we find a strong positive relation between the short-run component of disaster risk λ and each variance-related variable. However, we t find that the impact of the long-run component ξ is insignificant. This is not surprising, since t the variance-related variables are based on a very short horizon, namely, a month. Panel B of Table 4 considers out-of-the-money put option prices as additional conditional moments since they are particularly informative about tail events. At each point in time, we obtain the prices of S&P 500 put options with 90% moneyness, normalized by the underlying index price.25 This normalization allows us to compare option prices at different points in time by removing the effect of the index level. Columns (1)–(5) report the results from contemporaneous linear regressions of the normalized prices of put options with 1-, 3-, 6-, 9-, and 12-month maturities on the time series of λ and ξ . Column (1) of Panel B demonstrates t t that λ significantly and positively explains the 1-month put option price while ξ has no t t statistically significant impact. This is consistent with the results for the 1-month-ahead implied variance in Panel A. Comparing the five columns in Panel B, we can see that λ is t significant across all maturities. In the case of ξ , its economic magnitude and statistical t 25We download options data from OptionMetrics. Since we do not observe options with fixed moneyness or a constant maturity every day, we use a regression-based interpolation of implied volatilities with respect to moneyness and maturity by adopting the methodology of Seo and Wachter (2019). The price of an option with a specific moneyness and a specific maturity is, then, calculated by plugging the interpolated implied volatility into the Black-Scholes formula. 41

significance gradually increase as option maturity increases. As a result, the effect of ξ t becomessignificantformaturitieslongerthanorequalto6months. Theseresultsaresensible: the long-run component of disaster risk ξ better explains long-horizon equity moments, such t as the price-dividend ratio, price-earnings ratio, and longer-term option prices. Panel A: Factor portfolios (1) (2) (3) (4) (5) MktRf SMB HML MOM LIQ Coef. -0.22 -0.09 0.12 0.09 -0.14 ∆λ t t-stat [-4.98] [-2.20] [1.36] [1.12] [-1.27] Coef. -0.35 -0.27 -0.15 0.17 -0.30 ∆ξ t t-stat [-5.33] [-4.95] [-1.80] [2.04] [-3.10] Adj R2 (%) 14.64 6.92 3.50 2.91 9.62 Panel B: Fama-French 10 industry portfolios (1) (2) (3) (4) (5) Durbl Manuf HiTec Telcm Shops Coef. -0.15 -0.23 -0.21 -0.23 -0.16 ∆λ t t-stat [-4.62] [-4.12] [-5.89] [-6.00] [-4.99] Coef. -0.34 -0.33 -0.38 -0.28 -0.27 ∆ξ t t-stat [-7.10] [-6.09] [-4.92] [-3.75] [-3.96] Adj R2 (%) 12.41 14.19 16.31 11.00 8.74 (6) (7) (8) (9) (10) Enrgy Hlth NoDur Utils Other Coef. -0.19 -0.18 -0.13 -0.25 -0.16 ∆λ t t-stat [-3.96] [-3.06] [-2.15] [-3.92] [-3.21] Coef. -0.29 -0.19 -0.15 -0.09 -0.33 ∆ξ t t-stat [-5.13] [-2.95] [-2.05] [-0.78] [-5.10] Adj R2 (%) 9.99 5.65 3.14 5.91 11.67 Table 5: Shocks to disaster risk and equity returns. Thistablereportstheresultsofcontemporaneous time series regressions that examine the relation between changes in the filtered disaster risk variables (λ t and ξ ) and equity returns. In both panels, we standardize the independent and dependent variables. In t Panel A, the dependent variables are the excess market return (MktRf), size factor return (SMB), book-tomarketfactorreturn(HML),momentumfactorreturn(MOM),andliquidityfactorreturn(LIQ).InPanelB, the dependent variables are the excess returns on the 10 Fama-French industry portfolios. These industries include consumer durables (Durbl), manufacturing (Manuf), business equipment (HiTec), telecommunication (Telcm), retail (Shops), energy (Enrgy), healthcare (Hlth), consumer non-durables (NoDur), utilities (Utils), and others (Other). The t-statistics, shown in brackets, are computed using the Newey and West (1987) method with four lags. So far, we have examined whether disaster risk can explain the variations in conditional equity market moments. Another testable implication of disaster risk for the equity market 42

is that equity returns should be negatively associated with shocks to λ and ξ . Panel A of t t Table 5 reports the results from regressing the market, size, value, momentum, and liquidity factors on changes in λ and ξ .26 We find that changes in both the short-run and long-run t t components of disaster risk are indeed negatively related to the market, size, and liquidity factors. This implies that λ and ξ capture more than just aggregate market risk. t t Lastly, Panel B of Table 5 investigates how shocks to disaster risk affect contemporaneous excess returns on the Fama-French 10 industry portfolios. Intuitively, some industries are less sensitive to disaster risk than others. For example, industries that focus on consumer staples, such as food and utilities, are likely to be less exposed to disaster risk. This intuition is confirmed in Panel B: the relations between changes in disaster risk and industry returns showrelativelysmallert-statisticsforindustriesincludingconsumernon-durablesandutilities (columns (8)-(9)). In contrast, for industries that are more business-cycle sensitive, such as consumer durables, manufacturing, business equipment, telecommunication, and retail (columns (1)-(5)), we obtain more significant negative relations. 5 Conclusion While prior studies on rare disasters calibrate time-varying disaster risk, accurately characterizing it from the data remains a considerable challenge. This is extremely difficult because disasters are nearly unobservable events in the post-war sample. In order to tackle this issue, our paper ties time-varying probabilities of disasters to an independent source of data: interbank rates and their options. 26The market, size, and value factors follow Fama and French (1993), and the momentum factor follows Fama and French (2012). These time series are downloaded from Kenneth French’s website. The liquidity risk factor follows Pastor and Stambaugh (2003) and is downloaded from Robert Stambaugh’s website. 43

Our approach relies on the assumption that macroeconomic disasters are likely to coincide with interbank market failure. This link allows us to derive the model-implied interest rate spreads and option prices on interbank rates as functions of the short-run and long-run components of disaster risk. We show that these data are particularly sensitive to disaster risk, which enables us to reliably infer not only the level of disaster probabilities but also their dynamics. The estimation results suggest that disaster risk is significant in size and in variation, strongly upholding the validity of macro-finance models with the rare disaster mechanism. Using the filtered time series of the short-run and long-run components of disaster risk, we confirm that the implications of these models for equity moments and equity returns are consistent with empirical evidence. Overall, our analysis highlights that interest rates and their options can be valuable tools for deepening our understanding of extreme economic tail risk. 44

Appendix A Model derivations A.1 Risk-neutral dynamics Given the assumption about the pricing kernel in equation (4), the Brownian shocks under the risk-neutral measure are written as follows: dBQ = dB +θ dt, C,t C,t C dBQ = dB +θ dt, µ,t µ,t µ (cid:112) dBQ = dB +θ λ dt, λ,t λ,t λ t (cid:112) dBQ = dB +θ ξ dt, ξ,t ξ,t ξ t dBQ = dB +θ dt. q,t q,t q Moreover, under the risk-neutral measure, the disaster intensity and the moment generating function (MGF) of the jump size distributions are represented as λQ = λ Φ (θ ,0,0), t t Z N Φ (u +θ ,u ,u ) EQ(cid:2) eu1ZC+u2ZL+u3Zq (cid:3) = Z 1 N 2 3 , Φ (θ ,0,0) Z N 45

where Φ (u ,u ,u ) = E(cid:2) eu1ZC+u2ZL+u3Zq (cid:3) is the MGF of (Z ,Z ,Z ) under the physical Z 1 2 3 C L q measure. This leads to the following risk-neutral dynamics of the underlying processes, dC t = µQdt+σ dBQ + (cid:0) eZC,t −1 (cid:1) dN , C C C C,t t t dµ = κQ(µ¯Q −q )dt+σ dBQ t µ t µ µ,t (cid:112) dλ = κQ(νξ −λ )dt+σ λ dBQ , t λ t t λ t λ,t dξ = κQ(ξ ¯Q −ξ )dt+σ (cid:112) ξ dBQ, t ξ t ξ t ξ,t dq = κQ(q¯Q −q )dt+σ dBQ +Z dN , t q t q q,t q,t t where µQ = µ − θ σ , κQ = κ , µ¯Q = κµµ¯−θµσµ, κQ = κ + σ θ , ν = κ λ, κQ = C C C C µ µ κQ µ λ λ λ λ κQ λ ξ κ +σ θ , ξ ¯Q = κ ξ ξ¯ , κQ = κ , and q¯Q = κqq¯−θqσq. ξ ξ ξ κQ ξ q q κQ q A.2 Zero-coupon yields The time-t value of $1 zero-coupon interbank lending maturing at time t+τ is written as (cid:20) (cid:21) L P i (t,t+τ) = EQ t e− (cid:82) t t+τrudu L t+τ . t By multiplying both sides of the above equation with e− (cid:82) 0 truduL t , we obtain the following martingale: (cid:104) (cid:105) e− (cid:82) 0 truduP i (t,t+τ)L t = EQ t e− (cid:82) 0 t+τruduL t+τ . We conjecture that the price of a zero-coupon interbank lending is expressed as P (t,t+τ) = exp(a (τ)+b (τ)λ +b (τ)ξ +b (τ)µ +b (τ)q ), i i i,λ t i,ξ t i,µ t i,q t 46

and its stochastic differential equation is given as (cid:18) dP i,t = b (τ)κQ(νξ −λ )+b (τ)κQ(ξ ¯Q −ξ )+b (τ)κQ(µ¯Q −µ )+b (τ)κQ(q¯Q −q ) P i,λ λ t t i,ξ ξ t i,µ µ t i,q q t i,t− 1 1 1 1 + b (τ)2σ2λ + b (τ)2σ2ξ + b (τ)2σ2 + b (τ)2σ2 2 i,λ λ t 2 i,ξ ξ t 2 i,µ µ 2 i,q q (cid:19) −(a′(τ)+b′ (τ)λ +b′ (τ)ξ +b′ (τ)µ +b′ (τ)q ) dt i i,λ t i,ξ t i,µ t i,q t (cid:112) (cid:112) +b (τ)σ λ dBQ +b (τ)σ ξ dBQ +b (τ)σ dBQ +b (τ)σ dBQ i,λ λ t λ,t i,ξ ξ t ξ,t i,µ µ µ,t i,q q q,t (cid:0) (cid:1) + ebi,q(τ)Zq,t −1 dN . t (cid:16) (cid:17) Since e− (cid:82) 0 truduP i,t (τ)L t is a martingale, the sum of the drift and the jump compensator shouldbezero. Thismartingalepropertyprovidesthesystemofordinarydifferentialequations (ODEs) for a , b , b , b , and b as follows: i i,λ i,ξ i,µ i,q 1 1 a′(τ) = −δ +b (τ)κQξ ¯Q +b (τ)κQµ¯Q +b (τ)κQq¯Q + b (τ)2σ2 + b (τ)2σ2, i 0 i,ξ ξ i,µ µ i,q q 2 i,µ µ 2 i,q q 1 b′ (τ) = −δ −b (τ)κQ + b (τ)2σ2 i,λ λ i,λ λ 2 i,λ λ + p¯Φ (θ ,1,b (τ))+(1−p¯)Φ (θ ,0,b (τ))−Φ (θ ,0,0), Z N i,q Z N i,q Z N 1 b′ (τ) = −δ +b (τ)κQν −b (τ)κQ + b (τ)2σ2, i,ξ ξ i,λ λ i,ξ ξ 2 i,ξ ξ b′ (τ) = −δ −b (τ)κQ, i,µ µ i,µ µ b′ (τ) = −δ −b (τ)κQ, i,q q i,q q with the initial condition: a (0) = b (0) = b (0) = b (0) = b (0) = 0. i i,λ i,ξ i,µ i,q 47

Similarly, the price of $1 zero-coupon risk-free lending is given by (cid:104) (cid:105) P f (t,t+τ) = EQ t e− (cid:82) t t+τrudu ·1 , and we conjecture the price has the form of P (t,t+τ) = exp(a (τ)+b (τ)λ +b (τ)ξ +b (τ)µ +b (τ)q ). f f f,λ t f,ξ t f,µ t f,q t (cid:16) (cid:17) Since e− (cid:82) 0 truduP f,t (τ) isamartingale, thesumofthedriftandthejumpcompensatorshould bezero. ThismartingalepropertyprovidesthefollowingsystemofODEsfora , b , b , b , f f,λ f,ξ f,µ and b : f,q 1 1 a′ (τ) = −δ +b (τ)κQξ ¯Q +b (τ)κQµ¯Q +b (τ)κQq¯Q + b (τ)2σ2 + b (τ)2σ2, f 0 f,ξ ξ f,µ µ f,q q 2 f,µ µ 2 f,q q 1 b′ (τ) = −δ −b (τ)κQ + b (τ)2σ2 +Φ (θ ,0,b (τ))−Φ (θ ,0,0), f,λ λ f,λ λ 2 f,λ λ Z N f,q Z N 1 b′ (τ) = −δ +b (τ)κQν −b (τ)κQ + b (τ)2σ2, f,ξ ξ f,λ λ f,ξ ξ 2 f,ξ ξ b′ (τ) = −δ −b (τ)κQ, f,µ µ f,µ µ b′ (τ) = −δ −b (τ)κQ, f,q q f,q q with the initial conditions: a (0) = b (0) = b (0) = b (0) = b (0) = 0. Note that b (τ) f f,λ f,ξ f,µ f,q i,µ and b (τ) are identical, and so are b (τ) and b (τ), as their ODEs are identical with the f,µ i,q f,q same initial conditions. Therefore, we simply denote them as b (τ) and b (τ), respectively. µ q Now we turn to the Treasury rate. The stochastic differential equation for x implies that t 48

for any u ≥ t, (cid:90) u−t (cid:0) (cid:1) x = x e−κx(u−t) +x¯ 1−e−κx(u−t) + σ e−κx(u−t−s)dB . u t x x,s 0 (cid:16) (cid:17) (cid:104) (cid:105) Clearly, x u is Gaussian and so is (cid:82) t t+τ x u du . Hence, the expression for E t e (cid:82) t t+τxudu in equation (8) can be derived using the moment generating function of a normal distribution. This leads us to the following deterministic functions a and b shown in equation (8): x x (cid:18) σ2 (cid:19)(cid:20) 1−e−κxτ (cid:21) σ2 (cid:20) 1−e−κxτ(cid:21)2 a (τ) = x¯+ x −τ + x , x 2κ2 κ 4κ κ x x x x (cid:18) 1−e−κxτ(cid:19) b (τ) = − . x κ x Based on the identity y(τ) = y(τ) −y(τ), the τ-maturity Treasury rate y(τ) is obtained by g,t f,t x,t g,t 1 (cid:104) (cid:105) y(τ) = − a (τ)−a (τ)+b (τ)λ +b (τ)ξ +b (τ)µ +b (τ)q −b (τ)x . g,t τ f x f,λ t f,ξ t µ t q t x t A.3 Cap and swaption pricing Before deriving the expressions for the prices of caps and swaptions, we first find the pricing formula for a put option on P (T ,T ) with a strike price K. Following Duffie, Pan, and i 0 1 Singleton (2000), Collin-Dufresne and Goldstein (2003), and Trolle and Schwartz (2009), we compute the put option price by using the transform analysis: (cid:104) (cid:105) P(t,T 0 ,T 1 ,K) = EQ t e− (cid:82) t T0rsds(K −P i (T 0 ,T 1 ))1 {Pi(T0,T1)<K} = KG (logK)−G (logK), 0,1 1,1 49

where ψ(a,t,T ,T ) 1 (cid:90) ∞ Im[ψ(a+iub,t,T ,T )e−iuy] 0 1 0 1 G (y) = − du, a,b 2 π u 0 (cid:20) (cid:18) (cid:90) T0 (cid:19) (cid:21) ψ(u,t,T ,T ) = EQ exp − r ds eulog(Pi(T0,T1)) . 0 1 t s t Note that the function ψ solves the complex-valued ODEs of Duffie, Pan, and Singleton (2000), and the operator Im[·] represents the imaginary part of a complex number. The time-t cap price is given by equation (9). Under the approximation e − (cid:82) t t j j+1rsds ≈ P (t ,t ), we re-express the cap pricing formula as i j j+1 (cid:34) (cid:35) (cid:88) mT (cid:18) (cid:90) tj (cid:19) (cid:20) 1 (cid:21)+ V(T)(t,K) ≈ EQ exp − r ds P (t ,t ) −1−∆×K cap t s i j j+1 P (t ,t ) t i j j+1 j=1 (cid:88) mT (cid:18) 1 (cid:19) = (1+∆×K) P t,t ,t , . j j+1 1+∆×K j=1 AsdiscussedinSection2.4, wepriceswaptionsbyadoptingthestochasticdurationmethod suggestedbyWei(1997),Munk(1999),andTrolleandSchwartz(2009). Applyingthismethod to equation (10) results in: (cid:34) (cid:35) (cid:18) (cid:90) t+T (cid:19)(cid:20) (cid:21)+ V(T,T¯)(t,K) = EQ exp − r ds 1−V(T¯)(t+T,K) payer t s fixed t ≈ ζP(t,T,t+D(t,T,T ¯ ),ζ−1), where ζ = (cid:80)T l ¯ = / 1 ∆Y l Pi(t,t+T+l∆) . The stochastic duration D(t,T,T ¯ ), or simply D(t), is defined Pi(t,t+D(t,T,T¯)) 50

as a quantity that satisfies: b (D(t))2σ2λ +b (D(t))2σ2ξ +b (D(t))2σ2 +b (D(t))2σ2 i,λ λ t i,ξ ξ t q q µ µ +[Φ(θ ,0,2b (D(t)))−2Φ(θ ,0,b (D(t)))+Φ(θ ,0,0)]λ N q N q N t  2  2  2 T¯/∆ T¯/∆ T¯/∆ (cid:88) (cid:88) (cid:88) =  w j b i,λ (T +j∆) σ λ 2λ t + w j b i,ξ (T +j∆) σ ξ 2ξ t + w j b q (T +j∆) σ q 2 j=1 j=1 j=1  T¯/∆ 2  T¯/∆(cid:34) (cid:88) (cid:88) + w j b µ (T +j∆) σ µ 2 + w j 2Φ(θ N ,0,2b q (T +j∆))−2w j Φ(θ N ,0,b q (T +j∆)) j=1 j=1  (cid:35) (cid:88) + 2w j w k Φ(θ N ,0,b q (T +j∆)+b q (T +k∆)) +Φ(θ N ,0,0)λ t , k>j where w = YjPi(t,t+T+j∆) , Y = ∆×K for j = 1, 2, ··· , T¯ −1, and Y = 1+∆×K. j (cid:80)T l ¯ = / 1 ∆Y l Pi(t,t+T+l∆) j ∆ T¯/∆ References Acharya, Viral V., and David Skeie, 2011, A model of liquidity hoarding and term premia in inter-bank markets, Journal of Monetary Economics 58, 436–447. Adrian, Tobias, Erkko Etula, and Tyler Muir, 2014, Financial intermediaries and the cross-section of asset returns, Journal of Finance 69, 2557–2596. Allen, Linda, Turan G. Bali, and Yi Tang, 2012, Does systemic risk in the financial sector predict future economic downturns?, Review of Financial Studies 25, 3000–3036. Andersen, Torben G., Nicola Fusari, and Viktor Todorov, 2015, The risk premia embedded in index options, Journal of Financial Economics 117, 558–584. Ang, Andrew, and Monika Piazzesi, 2003, A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables, Journal of Monetary Economics 50, 745–787. Bakshi, Gurdip, John Crosby, Xiaohui Gao, and Jorge W. Hansen, 2023, Treasury option returns and models with unspanned risks, Journal of Financial Economics 150, 103736. Bakshi, Gurdip, Xiaohui Gao, and Jinming Xue, 2023, Recovery with applications to forecasting equity disaster probability and testing the spanning hypothesis in the treasury market, Journal of Financial and Quantitative Analysis pp. 1–35. Barro, Robert J., 2006, Rare disasters and asset markets in the twentieth century, Quarterly Journal of Economics 121, 823–866. 51

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