Measuring Interest Rate Risk Management by Financial Institutions

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Measuring Interest Rate Risk Management by Financial Institutions Celso Brunetti, Nathan Foley-Fisher, St´ephane Verani 2023-067 Please cite this paper as: Brunetti, Celso, Nathan Foley-Fisher, and St´ephane Verani (2023). “Measuring Interest Rate Risk Management by Financial Institutions,” Finance and Economics Discussion Series 2023-067. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.067. 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.

Measuring Interest Rate Risk Management ∗ by Financial Institutions Celso Brunetti1, Nathan Foley-Fisher1, and Stéphane Verani1 1Federal Reserve Board First version: June 2022; this version: August 2023 Abstract Financial intermediaries manage myriad interest rate risk exposures. Weproposeanewmethodtomeasurefinancialintermediaries’ residualinterestrateriskusinghigh-frequencyfinancialmarketdata. Our method exploits all available high-frequency information and is valid under extremely weak assumptions. Applying the method to U.S. life insurers, we find their interest rate risk management strategies are generally effective. However, life insurers are more sensitive to changes in long-term interest rates than property and casualty insurers. We show that the term premium helps to explain the difference in sensitivities between the two types of insurer. JEL Codes: G20; C58 Keywords: financial institutions; interest rate risk management; highfrequency financial econometrics; subsampling; life insurers. ∗For providing valuable comments, we would like to thank, without implication, Mark Carey, Burcu Duygan-Bump, Peter Hansen, Max Huber, Anastasia Kartasheva, Borghan Narajabad, Andrew Patton, Matt Pritsker, Roberto Renò, Rich Rosen, Oleg Sokolinskiy, Pavel Szerszen and participants in the Society for Economic Measurement Annual Conference 2023, the International Risk Management Conference 2023, and seminarsattheEuropeanCentralBank, St. GallenUniversity, andtheFederalReserve Board. We are grateful to Julia Silbert and Renee Garrow for exceptional research assistance. Theviewsinthispaperaresolelytheauthors’andshouldnotbeinterpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

1 Introduction Financial intermediaries are exposed to interest rate risk. They have multiple sources of exposure arising from cash flow differences across balance sheet components as well as contractual or embedded options with asymmetric payoff characteristics. Although intermediaries have a wide range of asset and liability management tools available to hedge interest rate risk, they do not fully insulate themselves from all potential changes in interest rates for several reasons.1 Financial markets may be incomplete, fully hedging may be prohibited by its cost, and carrying interest rate risk may be a source of earnings.2 Thus, financial intermediaries carry some residual exposure to interest rate risk, which could have significant consequences for financial stability and macroeconomic outcomes in bad states of the world (Holmstrom and Tirole, 1997; Brunnermeier and Sannikov, 2014). In this paper, we propose a new method to measure the time-varying residual interest rate risk exposure of financial intermediaries using minute- 1Riskmanagersatfinancialinstitutionsareexpectedtomonitorandmanageinterest rate exposures at prudent levels, but not fully eliminate the risk. Supervisors provide detailed guidance on management practices and coordinate their standards. See, for example,theOfficeoftheComptrolleroftheCurrency(OCC)RevisedHandbookMarch 2020,theFederalDepositInsuranceCorporation(FDIC)LetteronFinancialInstitution Management of Interest Rate Risk 2010, the Federal Reserve Board (FRB) Supervisory Manual on Interest Rate Risk, the National Association of Insurance Commissioners (NAIC) Risk-Based Capital for Insurers Model Act, the OCC-FDIC-FRB Joint Policy StatementonInterestRateRisk1996,andtheBaselCommitteeonBankingSupervision Guidance on Standards 2014. 2Even an established hedging strategy may be exposed to “basis risk”—that is, it might lose its effectiveness. 2

by-minutefinancialmarketdata. We calculate thedailyrealizedcovariance of high-frequency stock returns for those intermediaries and Treasury security returns. We construct a conditional covariance by projecting out aggregate stock market returns from stock returns and Treasury security returns. We then introduce realized gamma as the ratio of the conditional covariance to the daily realized conditional variance of Treasury security returns. Realized gamma is a daily estimate of the sensitivity of an individual firm’s stock price returns to realized changes in interest rates. We calculate returns at five-minute intervals using every possible fiveminute grid point in a trading day, exploiting all available high-frequency information as described in Zhang, Mykland and Aït-Sahalia (2005). We then propose a new statistical test of the daily residual interest rate risk exposure of financial intermediaries. We conduct statistical inference on the realized gamma estimates by calculating asymptotically valid confidence intervals using subsampling (Politis, Romano and Wolf, 1999). The essence of the subsampling method is to approximate the sampling distribution of the daily realized gamma with the empirical distribution generated by estimating the realized gamma on an exhaustive set of intra-day subsamples.3 Although computationally intensive, the method of subsampling behaves well under extremely weak, easily satisfied 3Our limiting concept is the length of the time interval between two stock price observations going to zero. We provide the main theoretical results for our application in Appendix B. 3

assumptions.4 Our approach to statistical inference is crucial because it is by definition impossible to know everything about each financial intermediary’s proprietary risk management framework. Our new method provides a time-varying measure of residual interest rate risk exposure because it is based on financial intermediaries’ publiclytraded equity values. Those values reflect intermediaries’ exposure to interest rates after they have executed their interest rate risk management strategies. The correlation of equity values with interest rates reveals market participants’ views on the effectiveness of financial intermediaries’ hedging strategies in relation to the changes in the interest rates that occurred. Themeasureisareflectionofthehedgingstrategyconditionalon the actual changes in interest rates. A measure of zero doesn’t necessarily mean that financial intermediaries are fully hedged. That said, intuitively, the stock price of a financial intermediary with fully hedged interest rate riskwouldbeuncorrelatedwithallpossiblechangesininterestrates(Allen, 1993). Note that we will not address the question of why financial intermediaries bear interest rate risk. Importantly, we are not making any normative statement about how much interest rate risk financial intermediaries could or should carry. In particular, our notion of effectiveness does not imply 4Bycontrast,bootstrappingtheconfidenceintervalswouldrequireshowingthetimeseriespropertieswerepreservedwithinsamplesorimposestrongassumptionsaboutthe data generating process. 4

that intermediaries should aim for zero residual interest rate risk. Nor does it imply that market participants think intermediaries should do so. Rather, our measure derives from the compensation for the interest rate risk borne by the ultimate owners of the intermediary, as in Allen (1993). When ownership is obtained through traded equity, the equity market price reflects that compensation. Monitoring residual interest rate risk exposures is an important component in analysts, policymakers, and supervisors’ evaluation of the financial conditions of intermediaries. Interest rate risk exposures are typically included as part of credit rating reports and investment analysis. As part of their financial stability dicussions, central bankers are attuned to the potential effects on their decisions on financial intermediaries, e.g., Brainard(2022). Supervisorsoffinancialinstitutionsexpectregularreports concerning interest rate risk management and exposures. Monitoring is required because interest rates can change swiftly and significantly, with large potential effects. The profitability of entire financial sector industries has been threatened by interest rate exposures. For example, the life insurance industry struggled to cope with the sharp rise in interest rates in the late 1970s and early 1980s, when the Federal Reserve under Chairman Volcker fought inflation (NAIC, 2013). We apply our new method to publicly-listed U.S. life insurers during the period from 2007 to 2022. Interest rate risk management is at the 5

heart of the modern life insurer business model because the duration of life insurers’ insurance liabilities, such as life insurance policies and annuity contracts, is typically much longer than the duration of the assets available in the economy.5 This negative duration gap means that a decrease in the interest rate increases the present value of a life insurer’s fixed-rate liabilities faster than the present value of its fixed income assets, which could lead to insolvency if left unmanaged. The same duration gap also means that persistently low interest rates depress life insurers’ net investmentspreadonnewbusinessandforcesthemtoreinvesttheproceeds from maturing bonds into bonds paying lower coupon rates, which further depresses their overall net investment spread and, in turn, adversely affects their financial condition. In addition, explicit and implicit options on both assets and liabilities contributes to life insurers’ interest rate risk. Because the prospect of insolvency is incompatible with the sale of long-term life and longevity insurance, state insurance regulations, or both, life insurers must credibly manage interest rate risk. We find that life insurer stock prices are largely uncorrelated with longterm (10-year) Treasury interest rates. This suggests that life insurers’ interest rate risk management is effective most of the time. This finding is comforting given some of the largest life insurers in the U.S. have been managing interest rate risk for over a century. However, in some states of 5For example, the duration of a typical life annuity is ten years, while the median corporate bond duration is around 5 years. 6

the world, realized gamma is statistically significant, revealing that after managing their interest rate risk—with liability driven investment, capital structure, and derivatives—life insurers remain exposed to changes in longterm interest rates in some states of the world. We contrast our analysis of life insurance companies with an analysis of publicly-listed property and casualty (P&C) insurance companies. P&C insurers provide an ideal alternative to life insurers because the structure of their business means that they are relatively less exposed to interest rate risk. For example, the vast majority of P&C premiums are renewable every year and, therefore, P&C insurers do not need to actively manage a duration gap between their assets and insurance liabilities. Consistent with this difference in business model, we find that life insurers are more sensitive to changes in long-term interest rates than P&C insurers. We then show that a measure of the term premium—the compensation for the risk associated with holding longer-term bonds—helps to explain the difference between the estimated sensitivities of life insurers and P&C insurers. We use the estimate of the term premium from the term structure model of Adrian, Crump and Moench (2013). We control for the funding cost of life insurers and a measure of the corporate credit return on life insurers’ assets. Our finding likely reflects the outsized importance of longer-termdebtinlifeinsurers’investmentportfolios. Weusetheseresults to illustrate how our measure provides information about the impact that 7

rapidly changing interest rates may have on insurers. Lastly, we show that our finding that life insurers’ interest rate risk management is generally effective is not due to low long-term interest rate volatility. We provide two alternative approaches to address the potential endogeneity between realized gamma and long-term interest rate volatility. Both approaches are based on the exogenous increase in interest rate volatility that occurs on scheduled Federal Open Market Committee (FOMC) meeting days. 1.1 Related literature Our paper connects to three distinct strands of literature. First, our method contributes to the high-frequency financial econometrics literature. Conceptually, our method is an extension of the single-factor realized beta model of Andersen, Bollerslev, Diebold and Wu (2006) and Hansen, Lunde and Voev (2014). We include a second right-hand side variable, that is Treasury security returns, in the estimated regression specification. To the best of our knowledge we are the first to introduce a second righthand side variable. Our computation of asymptotically valid standard errors using the subsampling approach is unusual in the high frequency financial econometrics literature because the approach is conservative and computationally intensive. Our realized gamma estimates do not suffer from bias due to non-synchronous trading—see, for example, Christensen, 8

Kinnebrock and Podolskij (2010) and Barndorff-Nielsen, Hansen, Lunde and Shephard (2011)—since we use index data aggregated at the oneminute frequency. Our choice of five-minute sampling frequency and averaging should also immunize our estimates from market microstructure noise biases. Second, our method relates to—but is distinct from—studies of interest rateriskthatmeasuretheeffectsofrealizedchangesininterestrates. These studies differ from other interest rate risk assessments that use balance sheet information to describe scenarios of potential effects associated with hypothetical changesininterestrates, e.g., Möhlmann(2021). Otherpapers thatstudyactual changesininterestratestendtofocusonbanks. Flannery and James (1984) studies the correlation between bank stock prices and interest rates using a similar regression model and weekly data. English, Van den Heuvel and Zakrajšek (2018) identify the response of bank stock prices to FOMC interest rate shocks. Paul (2022) revisits the findings of English et al. (2018) by decomposing the effect of monetary policy surprises into changes in future expected short-term rates and changes in term premium. Hoffmann, Langfield, Pierobon and Vuillemey (2018) use supervisory bank balance sheet data to estimate interest rate risk and study its determinants in the cross section. Vuillemey (2019) and Begenau, Piazzesi and Schneider (2015) show that banks increase their exposure to interest rate risk using derivatives. Most of these papers study low- 9

frequency data and, in some cases, attempt to identify interest rate shocks. By contrast, we exploit the information in high-frequency data and we use changes in interest rates rather than identified shocks. Third, our paper adds to the extensive literature on risk management of financial institutions—e.g., Froot, Scharfstein and Stein (1993); Froot and Stein (1998). Our method is applicable to any financial intermediary. We chose to focus on the interest rate risk of life insurers, as they have received much less attention than, for example, banks. The theoretical foundation for our application to life insurers comes from recent work studying interest rate risk management at insurance companies (Foley- Fisher, Narajabad and Verani, 2016; Verani and Yu, 2021). In these papers, limited liability insurers manage the ex-ante risk of insolvency due to future movement in the interest rate by choosing an optimal insurance price, asset portfolio, and capital structure. Our method is an ex-post statistical test of the performance of insurers’ ex-ante interest rate risk management strategy. As such, our analysis is closely related to empirical work that measures the residual interest rate risk exposure of insurers using a two-variable regression model of stock prices and low-frequency data (Brewer III, Mondschean and Strahan, 1993; Berends, McMenamin, Plestis and Rosen, 2013; Hartley, Paulson and Rosen, 2016; Ozdagli and Wang, 2019; Sen, 2021; Koijen and Yogo, 2022; Huber, 2022).6 We show in 6Some of these papers use stock prices only as a motivation for subsequent analysis of insurer balance sheet measures of interest rate risk. 10

Appendix A that estimates obtained through low-frequency rolling window ordinary least squares (OLS) regressions are severely biased, inconsistent, and potentially misleading. For example, in contrast to our findings, inference on the OLS estimates suggests that life insurers are sensitive to any movement in long-term interest rates at almost all times in the post-crisis period. The rest of our paper is structured as follows: Section 2 sets out the empirical framework for our estimation and explains how we construct our standard errors using subsampling. Section 3 describes our application to US life insurers, including institutional background and details on the data. We summarize our main findings in section 3.3 and offer some concluding remarks in section 4. 2 Methodology 2.1 A two-variable regression model In this section, we introduce our new method to measure the residual interest rate risk exposure of financial intermediaries. Let 𝑟 be the stock 𝑖𝑗𝑡 returnof financialintermediary𝑖 indexedto minute 𝑗 within day𝑡. Let𝑟 𝑚𝑗𝑡 be the return on aggregate market 𝑚 and 𝑟 be the return on Treasury 𝑦𝑗𝑡 security 𝑦. Our framework is a regression model with two right-hand side variables 11

using minute-by-minute financial market returns: 𝑟 𝑖𝑗𝑡 = 𝛼 𝑡 + 𝛽 𝑡 𝑟 𝑚𝑗𝑡 +𝛾 𝑡 𝑟 𝑦𝑗𝑡 +𝜖 𝑖𝑗𝑡 (1) where {𝛼 , 𝛽 , 𝛾 } are day-specific coefficients estimated using within-day 𝑡 𝑡 𝑡 returns. Our regression with the restriction 𝛾 𝑡 = 0 is well established in the finance literature and is referred to as the one-factor capital asset pricing model (CAPM). In the CAPM regression, the coefficient 𝛽 is 𝑡 interpreted as a dynamic measure of the comovement of individual stock returns with aggregate market or systematic returns.7 We extend the onevariableCAPMregressiontoincludeasecondright-handsidevariable, that is Treasury security returns.8 The time-varying 𝛾 coefficient estimates the sensitivity of an individual 𝑡 firm’s stock price returns to high-frequency realized changes in Treasury security returns. As Treasury security returns are inversely dependent on changes in interest rates, 𝛾 provides an estimate of that firm’s interest rate 𝑡 sensitivity. We label our 𝛾 𝑡 estimates realized gamma because our method 7A full discussion of the extensive literature studying time-varying 𝛽 𝑡 and its determinants is beyond the scope of this paper. See Fama and French (2004) for an overview. 8To be sure, we are not assuming that the right-hand side variables in our regression model are orthogonal. We are estimating the general equilibrium relationship between the three variables in our regression, which is fully consistent with the standard onefactor CAPM and yields an unbiased estimate of 𝛾 𝑡. Other papers that adopt a similar approachincludeFamaandSchwert(1977)andFlanneryandJames(1984). Weexplore the effect of an exogenous increase in long-term Treasury rate volatility in section 3.4. 12

can also be cast in the nonparametric framework of realized variances and covariances (Meddahi, 2002; Barndorff-Nielsen and Shephard, 2004; Andersen, Bollerslev and Meddahi, 2004). Our estimates of daily gammas are based on realized daily variances and covariances after conditioning on the aggregate market returns. We first project out aggregate stock market returns from stock returns and Treasury security returns by running two auxiliary regressions for each day 𝑡: 𝑟 𝑖𝑗𝑡 = 𝛼ˆ 𝑡 1 + 𝛽ˆ 𝑡 1𝑟 𝑚𝑗𝑡 +𝜖ˆ𝑖𝑗𝑡 , (2) 𝑟 𝑦𝑗𝑡 = 𝛼ˆ 𝑡 2 + 𝛽ˆ 𝑡 2𝑟 𝑚𝑗𝑡 +𝜖ˆ𝑦𝑗𝑡 . (3) The residuals from these auxiliary regressions {𝜖ˆ𝑖𝑗𝑡 ,𝜖ˆ𝑦𝑗𝑡 } are, respectively, the within-day conditional stock returns and Treasury security returns. The daily realized covariance of each financial intermediary’s conditional stock returns and Treasury security conditional returns is given by: ∑︁ 𝜈ˆ𝑖,𝑦,𝑡 = 𝜖ˆ𝑖𝑗𝑡 ·𝜖ˆ𝑦𝑗𝑡 . 𝑗 And the daily realized variance of conditional Treasury security returns is given by: ∑︁ 𝜈ˆ𝑦,𝑡 = 𝜖ˆ 𝑦 2 𝑗𝑡 . 𝑗 So we can define realized gamma as the ratio of the conditional covariance 13

to the daily realized conditional variance of Treasury security returns: 𝜈ˆ𝑖,𝑦,𝑡 𝛾 𝑡 = . (4) 𝜈ˆ𝑦,𝑡 The 𝛾 estimates by equation 1 and equation 4 are identical by what 𝑡 is commonly-known as the Frisch-Waugh-Lovell Theorem (Davidson and MacKinnon, 1993, Section 1.4). However, care must be taken with interpretation. The simple regression shown in equation 1 yields a consistent estimate of the ex-post realized gamma coefficient. That said, obtaining asymptotically valid standard errors is not a simple process, as we will describe in Section 2.2. 2.1.1 Addressing market microstructure noise Controlling for market microstructure noise that is prevalent in high frequency financial market data is an important issue (Aït-Sahalia and Yu, 2009). Microstructure noise naturally arises from a variety of features built in to financial market trading, including prices bouncing from bids to asks, variation in the size of trades, adjustment to new information contained in prices, order flow dynamics, and inventory management. Following Aït-Sahalia and Mykland (2009), we address the presence of market microstructure noise without discarding observations from our samples. 14

We employ two well-established techniques to mitigate concerns that market microstructure noise is clouding our ability to construct estimators and draw inference from high-frequency data. First, we calculate returns at five-minute intervals as their use as a benchmark for estimators generally outperforms all alternatives (Liu, Patton and Sheppard, 2015). We use everypossiblefive-minutegridpointinatradingdaytoexploitallavailable high-frequency information given the data structure as described in Zhang et al. (2005).9 Second, we filter all of our returns time series through AR(1) processes estimated separately for each day. That is, we take the raw returns 𝑟 𝑘𝑗𝑡 for 𝑘 ∈ {𝑖,𝑦,𝑚} and estimate 𝑟 𝑘,𝑗,𝑡 = 𝜌+𝜙𝑟 𝑘,𝑗−1,𝑡 +𝜀 𝑘,𝑗,𝑡 for each day 𝑡. We then use the residuals 𝜀 as our returns time series that 𝑘,𝑗,𝑡 has filtered out market microstructure noise. 2.2 Statistical inference A key principle for our new methodology is to impose minimal assumptions about the data generating process. This principle underpins our use of high-frequency data to estimate nonparametrically the time-varying correlation between interest rates and financial intermediaries’ stock prices. Similarly, we follow this principle when we consider what standard errors areappropriateforvalidinference. Wederiveasymptoticallyvalidstandard 9Our approach is identical to the method commonly referred to as “subsampling” in high-frequency financial econometrics. We avoid using the term here to prevent confusion with the concept of “subsampling” that we use to construct asymptotically valid standard errors. 15

errors without imposing undue structure on the time series processes. Our choice of standard errors is a crucial part of our approach to estimate interest rate risk, as the data generating process underpinning our realized gamma estimates is nonstandard. For example, we use rolling five-minute windows to construct our time series of returns. We adopt the subsampling methodology as it is a valid technique in extremely general cases (Politis et al., 1999).10 The basic idea of subsampling in a time series context is to approximate the sampling distributionusingallpossiblesubsetsofthetimeseries. Theorem4.3.1from Politis et al. (1999), which we reproduce in Appendix B for completeness, shows that we can derive asymptotically valid confidence intervals for the daily estimator 𝛾 . In addition, we can draw asymptotically valid 𝑡 inference about the true 𝛾 by exploiting the familiar duality between 𝑡 the construction of confidence intervals for 𝛾 and the construction of 𝑡 hypothesistestsabout𝛾 . Wecantestthenullhypothesisthatourestimate 𝑡 of the daily 𝛾 is statistically different from 0. That is, under the null, 𝑡 financial intermediaries are hedged against interest rate risk as their stock prices are not sensitive to movements in interest rates. Our algorithm for hypothesis testing uses within-day observations to construct subsamples. We follow Politis et al. (1999) and evaluate statistics 10Alternative methodologies based on the bootstrap technique could be devised, but they typically require additional assumptions, such as a finite fourth moment of the model residuals (Paparoditis and Politis, 2009). 16

on an exhaustive set of subsamples of size 𝑏 < 𝑛 that are created from the original daily sample of size 𝑛. We estimate the distribution of this statistic after a suitable normalization for each day in our sample. Note that our limiting concept is that the number of observations in a day approaches infinity. Tobeclear,eachsubsamplecontainsconsecutiveobservationsfrom theoriginaltimeseriessample. Therefore,eachsubsampleofsize𝑏 isdrawn without replacement from the true data generating process. We calculate a confidence interval for each of the daily 𝛾 using subsampling following 𝑡 Politisetal.(1999)undertheassumptionthattheerrorsareasymptotically stationary. Asymptotic stationarity is a weak condition that means, for example, the errors could follow an 𝐴𝑅(1) process with autocorrelation parameter strictly less than 1 and heteroskedastic innovations. The essence of the subsampling method is to approximate the sampling distribution of the (normalized) 𝛾 estimate with the empirical distribution generated by 𝑡 its subsample counterpart. As we have a large daily sample size (roughly speaking, 𝑛 = 390 observations per day), the choice of subsample size (𝑏) should not have a large effect on the empirical distribution of our statistic. Nevertheless, we need to choose the size of our subsamples. We follow the algorithm proposed by Politis et al. (1999) in section 9.3.3. Let 𝑏𝑡 be the subsample size for day 𝑡, which yields a confidence interval {𝐼 𝑏𝑡,𝑙𝑜𝑤 ,𝐼 𝑏𝑡,ℎ𝑖𝑔ℎ }. We construct a discrete grid of possible values 𝑏𝑡 ∈ {𝑏𝑡 , ..., 𝑏𝑡 }. For 𝑠 𝑠𝑚𝑎𝑙𝑙 𝑙𝑎𝑟𝑔𝑒 17

eachsubsamplesize 𝑏𝑡 weconsideraperturbationofsmallinteger 𝑘 around 𝑠 the subsample size and calculate a measure of variation in the confidence interval: 𝑉𝐼 𝑏𝑡 𝑠 ≡ var(cid:0)𝐼 𝑏𝑡 𝑠−𝑘,𝑙𝑜𝑤 ,...,𝐼 𝑏𝑡 𝑠+𝑘,𝑙𝑜𝑤 (cid:1) +var(cid:0)𝐼 𝑏𝑡 𝑠−𝑘,ℎ𝑖𝑔ℎ ,...,𝐼 𝑏𝑡 𝑠+𝑘,ℎ𝑖𝑔ℎ (cid:1) . Finally, we pick the value of 𝑏 that delivers stable confidence intervals for the most number of days in the entire sample: 𝑇 ∑︁ 𝑏 = argmax 𝑏 𝟙(𝑏𝑡∗ = 𝑏) where 𝑏𝑡∗ = argmin 𝑏𝑡 𝑉𝐼 𝑏𝑡 . 𝑠 𝑠 𝑡=0 Having determined the ‘optimal’ subsample size, we construct the empirical distribution of the normalized 𝛾 estimate for each day 𝑡. We 𝑡 use empirical distributions to obtain confidence intervals, which allow us to make inference about the statistical significance of each 𝛾 . With our 𝑡 new methodology in hand, we can turn to a specific application and data. 3 Application to U.S. life insurers 3.1 Institutional background Life insurers play a major role in the financial system, holding $6 trillion in total assets in their general accounts, of which roughly $3 trillion are in 18

corporate and foreign fixed income securities (Federal Reserve release Z.1 table L.116.g). Their overall business model consists of earning a spread between the yield they owe on their insurance liabilities and the yield they earn on the assets backing those liabilities. Life insurers write liabilities that are traditionally long-term, illiquid, and make fixed payments, such as fixed annuities. Life insurers tend to invest their premiums primarily in fixed rate corporate debt, in an effort to match their asset and liability cash flows and illiquidity profile and to offer a competitive return to policy holders. Like other financial intermediaries, life insurers have multiple sources of exposure to interest rate risk. A key underlying reason for their exposure is that the duration of insurance liabilities is typically much longer than the durationofassetsavailableintheeconomy. IntheU.S.,thetypicalduration of life insurance liabilities is 15–20 years (Huber, 2022). By contrast, in most countries, long-term fixed coupon bonds with more than two-year maturity do not exist (Gajek and Ostaszewski, 2004). Even in the U.S., which has the largest corporate bond market in the world, the supply of long-duration corporate bonds paying fixed interest rates is considerably smaller than the size of the life insurance industry (Verani and Yu, 2021). This means that, in practice, it is difficult for life insurers to hedge interest rate risk by investing in assets that have the same duration and greater cashflowvariabilitythantheirinsuranceliabilitiesi.e., theycannotdirectly 19

implement the classical immunization strategy of Redington (1952). Convexity—the effect of changing interest rates on the duration—of life insurer assets and/or liabilities also contributes to interest rate risk. One well-known source of convexity stems from options on financial contracts. For life insurers, the option for corporate bond issuers to call their bonds creates convexity on the asset side of their balance sheet. Likewise, policyholders may have the option of surrendering their life insurance products–perhaps for some cost–that creates convexity on the liability side of the balance sheet. The combination of these options creates a short straddle position for investors in the life insurer, which means they suffer when volatility is high (Babbel and Stricker, 1987).11 A natural way for life insurers to manage their interest rate risk consists of choosing a price for their insurance liabilities, an asset portfolio to back their insurance liabilities, and a capital structure to prevent insolvency along different paths for interest rates (Verani and Yu, 2021). For example, life insurers can hedge interest rate risk by charging a markup on the actuarially fair cost of their insurance products. The present value of the markup adds to the insurer’s ‘net worth’. Net worth allows the insurer to close its duration gap by financing bonds whose present value is greater than the present value of its insurance liabilities. Or, put differently, net worth acts as precautionary savings and helps cushion the effect of interest 11Briys and de Varenne (1997) provide an alternative formulation for the investor straddle position in which insurance liabilities are more convex than assets. 20

rate changes that disproportionately affect the value of the insurance liabilities.12 Large and sophisticated life insurers also manage interest rate risk by adding net-positive duration to their balance sheets synthetically using derivatives (Sen, 2021; Verani and Yu, 2021) and nontraditional lines of business (Foley-Fisher et al., 2016; Foley-Fisher, Narajabad and Verani, 2020), which amounts to using leverage instead of net worth to close the duration gap. For example, life insurers can add positive duration to their balancesheetbyenteringintoalong-termfixed-for-float interestrateswaps or by financing long-term fixed interest rate assets with nontraditional liabilitiessuchasovernightsecuritieslendingcashcollateral(Gissler, Foley- Fisher and Verani, 2019; Foley-Fisher et al., 2016) and funding agreementbacked short-term funding (Foley-Fisher et al., 2020). All these interest rate hedging strategies amount to closing the insurer’s natural negative duration gap by either directly or synthetically financing fixed-maturity assets with short-term floating rate debt. Nevertheless, insurers typically carry residual interest rate risk after they have implemented their hedging strategies. Investors in the insurers— either the policyholders in the case of mutual insurers or shareholders in the case of publicly-listed insurers—provide additional risk-bearing capital and receive compensation for bearing the insurer’s residual interest rate 12Net worth is not to be confused with what the industry calls reserves, which is the value of insurance liabilities. 21

risk (Allen, 1993). When investment takes place through traded equity, the market price for the equity reflects the interest rate risk compensation. One real-world example when the residual interest rate risk carried by life insurers was realized occurred in the early 1980s. At that time, the Federal Reserve sharply increased short-term interest rates amid persistently high inflation. Life insurers’ financial condition deteriorated as policyholders surrendered their claims or took out policy loans in search of higher interest rates on alternative saving vehicles (Briys and de Varenne, 2001). Life insurers responded by rewriting existing business at a loss and selling new products that offered higher-than-current long-term rates (negative spreads) (NAIC, 2013). While locking in huge losses—eroding their net worth—they avoided even greater losses they would have incurred had they sold their fixed income assets at far-below costs given the rise in current rates. The surge in short-term interest rates occurred after a relatively long period of low interest rate volatility, making these sharp rises largely unexpected. The significance of the episode is underscored by subsequent efforts to develop new tools for managing interest rate risk (Doffou, 2005). Adverse scenarios such as the early 1980s create a need for researchers and policymakers to monitor and assess the effects of rising interest rates on life insurers. However, they do not have access to the complete set of balance sheet information needed to precisely identify the effectiveness 22

of life insurers’ interest risk management and their residual interest rate risk. For example, information about the interest rate sensitivity of life insurance liabilities is difficult to gauge, although it is easier in some non- U.S. jurisdictions (Huber, 2022; Möhlmann, 2021; Kirti, 2017; Domanski, Shin and Sushko, 2017). Furthermore, it is hard to incorporate balance sheet information about the interest rate sensitivity of derivative positions, off-balance sheet liabilities (such as those in offshore captive reinsurers), and nontraditional liabilities. To overcome this problem, researchers turned to analyzing the sensitivity of life insurer stock returns to changes in long-term interest rates. An insurer’s equity valuations reflect the market price for its residual interest rate risk, after it has implemented its hedging strategies. That is, the ex-post effectiveness of life insurers’ management of ex-ante interest rate risk.13 To the best of our knowledge, this approach was first adopted by Brewer III et al. (1993). To assess the dynamics of interest rate risk exposure, some papers run OLS on rolling windows of stock returns e.g., Hartley et al. (2016). Although conceptually valid, the OLS implementation can lead to biased estimates in the presence of heteroskedasticity.14 In Appendix A, we show the bias is extremely large 13Here,again,theterm‘effectiveness’shouldnotbetakentoimplythatinvestorsthink insurers should target any particular level of interest rate risk. Rather, it’s investors’ assessment of the effect that actual interest rate changes had on the net worth of the insurer. 14BrewerIII,Carson,Elyasiani,MansurandScott(2007)recognisedthisconcernand allowed for time-varying volatility in a GARCH-M process. 23

by imposing some structure on the data generating processes. We will now applyourpreferredmethodologydescribedinSection2toobtainconsistent estimates without imposing such structure. 3.2 Data All the price data for our empirical application to life insurers come from Refinitiv. The underlying data are timed to the microsecond and recorded from data feeds covering both over-the-counter and exchange traded instruments on more than 500 trading venues and third parties. We useapreprocessedversionoftheunderlyingdataaggregatedbyRefinitivto aminutelyfrequencyusingthelasttradeduringeachminute. Weconstruct the data so as to follow the previous tick method, that is, if there are no transactions during a specific minute, the last transaction is used. The dataset identifier for each dataseries typically combines a ticker with a code indicating the primary trading market. For example, MetLife’s identifier is MET.N as it trades on the New York Stock Exchange. The list of the individual insurer identifiers and their mapping to the life and P&C insurers used in our analysis is provided in Table 1. Column 2 of Table1showstheinsurersincludedineachindex. Ourlistofpublicly-listed life insurers almost completely overlaps with the list of “publicly traded U.S. variable annuity insurers” used by Koijen and Yogo (2022). This is not surprising because virtually all large listed life insurers offer variable 24

annuities contracts at some point in the sample period.15 In addition, our analysis uses Standard and Poor’s S&P500 index as our measure of the aggregatemarket. Theidentifierfortheindexis.SPX.WealsouseRefinitiv evaluated prices for 10-year Treasury securities. The identifier for the series is US10YT=RRPS. Evaluated prices contain information from actual trades, quotes, and other sources within a model-based methodology. We use minutely data for each trading day beginning at 9:30am through 4pm.16 Except for 9:30am, we use closing prices recorded for each minute. For 9:30am, we use the opening price of 9:31am to avoid concerns about jumps following overnight information and trading. We calculate five-minute log returns of all time series using every possible five-minute grid point in a trading day. That is, we calculate the returns 𝑙𝑛(𝑝 ) − 𝑙𝑛(𝑝 ) for each day 𝑡, data series 𝑖, and all 𝑗 ∈ 𝑖,𝑗,𝑡 𝑖,𝑗−5,𝑡 {9.35𝑎𝑚, 9.36𝑎𝑚, ··· , 3.59𝑝𝑚, 4.00𝑝𝑚}. We construct high-frequency price indexes separately for life insurers and P&C insurers, weighting each individual insurer’s intraday market price by its end-of-day market capitalization. We obtain daily data on market capitalization from the Center for Research in Security Prices hosted by Wharton Research Data Services. Figure 1 shows the life insurer index as a red solid line and the P&C insurer index as a dotted blue 15As we will discuss in Section 3.4, this means that it is not possible to attribute the residual interest rate risk exposure to variable annuities. 16We exclude holidays, weekends, emergency closures, and partial trading days. 25

Table 1: Mapping insurance groups to identifiers. This table shows the insurance groups that we use in our empirical application, with their respective NAIC Group codes, and identifiers. Name Code Life/P&C Identifier Ticker Notes AlleghanyGroup 501 P&C Y.N Y AmericanFinancialGroup 84 P&C AFG.N AFG AmericanIntlGroup,Inc. 12 Life/P&C AIG.N AIG Assurant,Inc. 19 Life/P&C AIZ.N AIZ TheAllstateCorporation 8 Life/P&C ALL.N/BK.N ALL/BX IdentifierchangeforLifein2021 AmeripriseFinancial,Inc. 4 Life AMP.N AMP AmericanNationalFinancialGroup 408 Life ANAT.OQ ANAT ApolloGlobalManagement,Inc. 4734 Life ATH.N/APO.N ATH/APO Identifierchangein2022 BrighthouseFinancial,Inc. 4932 Life BHF.OQ BHF BerkshireHathawayInc. 31 P&C BRKb.N BRK.B ChubbLtd. 626 P&C ACE.N/CB.N ACE/CB Identifierchangein2016 CignaHealthGroup 901 Life CI.N CI CincinnatiFinancialCorporation 244 P&C CINF.OQ CINF CNAFinancialCorporation 218 P&C CNA.N CNA CNOFinancialGroup 233 Life CNO.N CNO ErieInsuranceGroup 213 P&C ERIE.OQ EquitableHoldings,Inc. 4965 Life EQH.N EQH FBLFinancialGroupInc. 513 Life FFG.N FFG Ceasedtradingin2021 FidelityandGuarantyLife 4731 Life FGL.N FGL Ceasedtradingin2017 FidelityNationalFinancial,Inc. 670 Life FNF.N FNF GenworthFinancial,Inc. 4011 Life GNW.N GNW HanoverInsuranceGroup,Inc. 88 P&C THG.N THG TheHartfordFin.SvcsGroup,Inc. 91 Life/P&C HIG.N HIG RemoveidentifierfromLifein2018 HoraceMannGroup 300 Life HMN.N HMN KansasCityLifeInsuranceGroup 588 Life KCLI.OQ KCLI Delistedin2015 KemperCorporationGroup 215 P&C KMPR.N KMPR LincolnNationalCorporation 20 Life LNC.N LNC MercuryGeneralGroup 660 P&C MCY.N MCY MarkelCorporationGroup 785 P&C MKL.N MKL MetLife,Inc. 241 Life MET.N MET ManulifeFinancialCorporation 904 Life MFC.TO MFC NationwideCorporationGroup 140 Life NFS.N NFS Ceasedtradingin2008 ThePhoenixCompanies,Inc. 403 Life PNX.N PNX Ceasedtradingin2016 PrimericaGroup 4750 Life PRI.N PRI PrincipalFinancialGroup,Inc. 332 Life PFG.OQ PFG ProtectiveLifeCorporation 458 Life PL.N PL Ceasedtradingin2015 TheProgressiveCorporation 155 P&C PGR.N PGR PrudentialFinancial,Inc. 304 Life PRU.N PRU SelectiveInsuranceGroup 88 P&C THG.N THG SymetraFinancialCorp. 4855 Life SYA.N SYA Ceasedtradingin2016 TheTravelersCompanies,Inc.Group 3548 P&C TRV.N TRV VoyaFinancial,Inc. 4832 Life VOYA.N VOYA W.R.BerkleyCorporation 98 P&C BER.N/WRB.N BER/WRB Identifierchangein2008 26

line. The dotted blue line lies above the red solid line as P&C insurers have generally outperformed life insurers in the post-crisis low interest rate environment. Figure 1: Insurer price indexes. Each line is a weighted average high-frequency price for large publicly-traded insurers listed in Table 1. The weights for each series are the daily market capitalization of insurers. Source: Authors’ calculations based on data from Refinitiv and the Center for Research in Security Prices. Table 2 shows summary statistics for the high-frequency data used in our analysis. Column 1 shows that the first day that data are available is different for each of our variables. The S&P500 Index is earliest available, while our high frequency data on long-term Treasury bond prices (‘US10YT’) begin only in 2007. Our indexes of large life and P&C insurers stock prices also begin in 2007. By construction, our sample ends on October 31, 2022. In addition to the first date available, we report the 27

numberofdays,andthetotalnumberofminutelyfive-minutereturnsinour data. We also report that there are no zero returns in our data, alleviating concerns about downward bias in our estimates due to zero returns (Bandi, Kolokolov, Pirino and Renò, 2020; Kolokolov and Renò, 2023). Across these returns, we report the mean, median, standard deviation, percentiles, and higher-order moments for each time series. Life insurers’ returns have a higher standard deviation than P&C insurers, but the kurtosis of life insurers’ returns is far lower. Table 2: Summary statistics. For each returns series in our sample, the table shows the first observation date, the number of days, the number of five-minute returns, the number of returns equal to zero, as well as the mean, median, standard deviation, percentiles, skewness, and kurtosis. The statistics reported in columns 6 through 10 are multiplied by 1𝑒+4 for legibility. Source: Authors’ calculations based on data from Refinitiv and the Center for Research in Security Prices. Series Firstdate No. days No. obs. #Zeroes Mean Median Std. dev. p25 p75 Skew. Kurt. S&P500 2000-01-03 5,768 2,218,825 0 -0.01 0.05 10.74 -4.06 4.11 -0.04 38.74 US10YT 2007-04-10 3,961 1,524,682 0 0.01 0.00 3.91 -1.67 1.70 0.99 80.66 Life 2007-01-03 3,996 1,533,761 0 -0.01 0.06 16.55 -5.76 5.79 0.44 35.25 P&C 2007-01-03 3,996 1,533,761 0 0.02 0.03 10.44 -4.00 4.02 2.06 147.25 3.3 Results In this section, we apply the methodology laid out in Section 2 to the data described in the previous section. Panel A of Figure 2 shows the daily point estimate of realized gamma for life insurers, and Panel B of the same figure shows the daily point estimate of realized gamma for P&C insurers. 28

Bothpanelsexhibitvolatility, whichisawell-knownfeatureoftime-varying coefficients estimated using realized variances and covariances (Hansen et al.,2014). Nevertheless, lifeinsurers’realizedgammaevidentlyhasahigher level of volatility than P&C insurers’. Figure 2: Daily realized gammas. The panels show daily realized gammas for life insurers and P&C insurers from 2007 through to the end of 2022. Source: Authors’ calculations based on data from Refinitiv and the Center for Research in Security Prices. We obtain confidence intervals from the empirical distributions, which are estimated for each day. Table 3 shows the results from applying the algorithm described in subsection 2.2 to determine the block size. While any block size satisfying the conditions of Theorem B.1 is valid, the ideal block size is the one that produces the most stable i.e., least variable, 29

Table 3: Optimizing block size. Each of columns 2-4 shows the number of days on which the block size (column 1) produces the most stable, i.e. least variable, confidence intervals. The measures of variation used in columns 2 and 4 is the standard deviation, while columns 3 and 5 use the difference between the minimum and the maximum values. The row with the highest count of days reveals the ideal block size for life insurers (columns 2-3) and P&C insurers (columns 4-5). Source: Authors’ calculations based on data from Refinitiv and the Center for Research in Security Prices. Block Life P&C size (%) Std. dev. Min-max Std. dev. Min-max 15 481 475 519 513 20 419 418 381 379 25 992 1000 988 995 30 676 673 702 700 35 597 594 618 615 40 472 480 452 454 45 418 415 395 399 confidence intervals given a small perturbation in the size of the block.17 Each of columns 2-4 shows the number of days on which the block size (column 1) produces the most stable confidence intervals. The measure of variationusedincolumns2and4isthestandarddeviation,whilecolumns3 and 5 use the difference between the minimum and the maximum values. The row with the highest count of days reveals the ideal block size for life insurers (columns 2-3) and P&C insurers (columns 4-5). For both life insurers and P&C insurers, the optimal block size is 25 percent of the daily observations, corresponding to about 100 consecutive observations in each 17Note that there is no reason to expect variation across grid points to follow a monotonic function or have a global optimum (Politis et al., 1999). 30

block and about 300 points in the empirical distribution. These relatively large values alleviate concerns about the power of the test. To smooth out the volatility in both the point estimates of realized gammas and the confidence intervals, we calculate time-series averages. The smoothed series are simply easier to read. We construct averages using a rolling window of two months. Panels A and B in Figure 3 show the smoothed time series. The red horizontal lines represent the sample means of the respective series. The smoothed time series reveal that realized gamma for life insurers is statistically significant on only 1,261 days, equivalent to roughly 32 percent of the sample. Realized gamma is always negative whenever it is statistically significant, which means that life insurers would benefit from higher long-term interest rates. For the majority of our sample, life insurer stock prices are uncorrelated with long-term interest rates. This suggests that life insurers’ interest rate risk management is effective most of the time. These results should not be interpreted as a normative assessment of life insurers’ interest rate risk management, neither by us nor by equity market participants. The measure is a reflection of how actual changes in interest rates affected—or did not affect—equity investors in life insurers, who expect compensation for bearing interest rate risk. Thetimeseriesalsorevealthatlifeinsurersaremoresensitivetointerest rate changes than P&C insurers. In contrast to life insurers, realized 31

Figure 3: Smoothed daily realized gammas. The panels show daily realized gammas averaged using a rolling window of two months for life insurers and P&C insurers from 2007 through to the end of 2022. The shaded region in both panels represents the 90 percent confidence intervals for each daily estimate. The underlying data are shown in Figure 2. The red horizontal lines represent the sample means of the respective series. A negative realized gamma means that insurers would benefit from higher long-term interest rates. Source: Authors’ calculations based on data from Refinitiv and the Center for Research in Security Prices. 32

gamma for P&C insurers is significant on only 705 days (about 18 percent of the sample). Like life insurers, realized gamma for P&C insurers is always negative whenever it is statistically significant. This finding could be interpreted as evidence that P&C insurers carry less residual interest rate risk, or as evidence that life insurers are exposed to different kinds of interest rate risk. In the next section, we offer some support for the latter interpretation by analyzing individual components of long-term interest rates. 3.4 Analysis 3.4.1 When is life insurer hedging not effective? In this section, we study macroeconomic variables during periods when realized gamma is statistically significant. Our findings help to explain why lifeinsurers’realizedgammaismoreoftenstatisticallysignificantthatP&C insurers’ realized gamma. While we offer an interpretation of our findings, we do not claim causal identification, as we recognize that long-term yields are a general equilibrium outcome of supply and demand (Schneider, 2022). Life insurers’ interest rate sensitivity is potentially endogenous to their demandforcompensationtoholdlonger-termdebtand, aswenotedearlier, life insurers are important investors in the long-term debt market. All the analysis in this section is conducted at a daily frequency out of necessity. 33

In an ideal empirical experiment, we would use intraday data to analyze the force(s) behind the results described in the previous section. However, we do not know of any high-frequency measures of the variables described below. We focus on three key variables based on Verani and Yu (2021), who showed that the relative cost of hedging interest rate risk is determined by the long-term investment grade bond spread relative to life insurers’ cost of funding. As measures of the return on life insurers’ long-term assets, we use the term premium and Moody’s Baa-Aaa seasoned corporate spread.18 We use the term structure model of Adrian et al. (2013) to decompose longterm yields and obtain an estimate of the term premium. While the term premium contributes to the slope of the yield curve, it is more specifically the component that compensates investors for holding longer-term debt instead of rolling over short-term debt. In addition to these measures of asset returns, we use the ICE BoA Single-A U.S. corporate index optionadjusted spread as a proxy for life insurers’ average cost of funding because life insurers are rated around A. Summary statistics for all the variables used in this analysis are provided in Appendix C. We construct a binary variable that takes the value 1 if the estimated 18The corporate bonds used to construct this spread all have at least 20 years of maturity. The yield on Aaa-rated corporate bonds with at least 20 years of maturity is a quasi-risk free benchmark. Under state insurance regulation, corporate bonds rated by Moody’s to be Baa or higher are designated as NAIC 1 and uniformly attract the lowest statutory risk-based capital charge. 34

realized gamma (𝛾 ) is statistically significant on day 𝑡 for insurer type 𝑖𝑡 𝑖 ∈ {Life, P&C}, and takes the value 0 otherwise. We then estimate a linear probability model using as independent variables the term premium (𝑇𝑃 ) estimates from Adrian et al. (2013), the Moody’s Baa-Aaa seasoned 𝑡 corporate spread (𝐵𝑎𝑎 − 𝐴𝑎𝑎 ), and a measure of the funding cost of life 𝑡 insurance companies (𝐹𝐶 ) that is the ICE BoA Single-A U.S. corporate 𝑡 index option-adjusted spread. In more technical terms, we estimate: 𝑃 (cid:98) (𝟙(𝛾 𝑖𝑡 < 0)|𝑇𝑃 𝑡 ,𝐵𝑎𝑎 − 𝐴𝑎𝑎 𝑡 ,𝐹𝐶 𝑡 ) = 𝛼 𝑖 + 𝛽 𝑖 1𝑇𝑃 𝑡 + 𝛽 𝑖 2𝐵𝑎𝑎 − 𝐴𝑎𝑎 𝑡 + 𝛽 𝑖 3𝐹𝐶 𝑡 where 𝑃 (cid:98) (𝟙(𝛾 𝑖𝑡 < 0)|𝑇𝑃 𝑡 ,𝐵𝑎𝑎− 𝐴𝑎𝑎 𝑡 ,𝐹𝐶 𝑡 ) is the predicted probability that 𝛾 𝑖𝑡 < 0 given 𝑇𝑃 𝑡 , 𝐵𝑎𝑎 − 𝐴𝑎𝑎 𝑡 , 𝐹𝐶 𝑡 , and a linear functional form. The results are shown in Table 4 where we report the coefficient estimatesandstandarderrorsinparentheses. Column1showsthebivariate relationship between the term premium and the statistical significance of realized gamma for life insurers. Columns 2-4 provide the main result under a range of standard error estimates, as indicated at the bottom of the table. HC are heteroskedasticity consistent standard errors, HAC are heteroscedasticity and autocorrelation consistent standard errors, and NW are Newey-West standard errors. The dependent variable in column 5 is a binary variable for statistical significance of P&C insurers’ realized gamma. This column acts as a placebo test of the main result for life insurers: The 35

key variables we focus on for life insurers are not statistically important for P&C insurers, consistent with our prior expectations. Noting again that the results are not causal, the estimates nevertheless suggest that there is a strong economic relationship between the variables, in addition to the statistical significance indicated in the table. We use as a benchmark for the economic effects the 32 percent unconditional probability that realized gamma for life insurers is statistically significant (see Table 6 in Appendix C). A one standard deviation increase in the term premium, which compensates investors for holding longer-term debt, reduces the probability that realized gamma for life insurers is statistically significant by about 17 percentage points—equivalent to about half of the unconditional probability that realized gamma for life insurers is statistically significant. A one standard deviation increase in Moody’s Baa- Aaaseasonedcorporatespreadreducestheprobabilitythatrealizedgamma for life insurers is statistically significant by about 24 percentage points. And a one standard deviation increase in the ICE BoA Single-A U.S. corporate index option-adjusted spread raises the probability that realized gamma for life insurers is statistically significant by about 28 percentage points. Our analysis provides support for the view that a flattening yield curve can drive realized gamma below zero. This can be seen, for example, around September 2019 when short-term interest rates rose and the 10- 36

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year Treasury yield fell. Similarly, our findings chime with the broad consensusthatanupwardshiftoftheentireyieldcurveisgenerallygoodfor life insurers. For example, realized gamma remained statistically close to zero during the rapid rise in short-term interest rates that occurred as the Federal Reserve tightened monetary policy in 2022. Our measure suggests that market participants focused on the positive effect on life insurers’ profitability from rising long-term interest rates and widening spreads on long-term investment grade bonds. In summary, our realized gamma measure of stock price sensitivity to long-term interest rates serves as a useful barometer for market sentiment about the effectiveness of insurers’ interest rate risk management. 3.4.2 Is realized gamma low due to interest rate volatility? Column 6 of Table 4 shows that the daily realized volatility of 10-year Treasury security returns is not correlated with the statistical significance of realized gamma. This finding should be intuitive, as we are estimating realized gamma conditional on intraday 10-year Treasury security returns, but is important to emphasize: It means life insurers’ interest rate risk management is generally effective not as a consequence of generally low interest rate volatility. In this section, we provide further evidence for this key result. We provide additional tests as we recognize the potential endogeneity 38

of long-term interest rate volatility and realized gamma. Life insurers are important investors in the long-term debt market, as we noted above. Their willingness to lend at long terms may simultaneously affect their own sensitivity to long-term interest rates and long-term interest rate volatility. We address the potential endogeneity with a source of plausibly exogenous variation in long-term interest rate volatility. Scheduled Federal Open Market Committee (FOMC) meeting days are a well-known source of volatility in interest rates, that is sometimes used as a exogenous source of variation (Rigobon and Sack, 2004; Foley-Fisher and Guimaraes, 2013).19 FOMC meeting days are exogenous to the supply-side variables that give rise to endogeneity concern in our setting. We exploit this source of exogenous variation in two ways. First, we use scheduled FOMC meeting days as an instrumental variable (IV) to obtain exogenous variation in long-term interest rate volatility. Second, we test the difference in means between realized gamma on scheduled FOMC meeting days and on other days when long-term interest rate volatility is lower. Our first test using an IV is reported in columns 7 and 8 of Table 4, where we show the results from estimating a two-stage least squares regression specification. The IV for the endogenous interest rate volatility variable (𝜎10𝑦𝑡 ) is a dummy variable (𝐹𝑂𝑀𝐶 ) that takes the value 1 on 𝑡 𝑡 19Note that monetary policy shocks are the root cause of the exogenous increase in interest rate volatility, but we do not need to identify the size of those shocks to implement our tests. 39

days when the FOMC holds a scheduled meeting and the value 0 otherwise. The first stage, reported in column 7, shows that the FOMC variable is a strong instrument for 𝜎10𝑦𝑡. The coefficient estimate is highly statistically 𝑡 significant and positive, consistent with rising interest rate volatility on days with scheduled FOMC meetings. The F-statistic for the first stage regressionis277.4, indicatingastrongIV.Thesecondstage, whichincludes the fitted values from the first stage as a right-hand side variable to replace 𝜎10𝑦𝑡, is reported in column 8. The coefficient on long-term interest rate 𝑡 volatility remains statistically insignificant. Our second test addresses two limitations of our IV approach: (1) other right-hand side variables in our specification may be invalid as instrumentsinthefirststage,and(2)ourleft-handsidevariableisadummy variable for the statistical significance of realized gamma. We focus on the statistical property of the average difference in realized gammas between high-volatility days when the FOMC has its scheduled meetings and lowvolatility days just before the FOMC meetings. Specifically, in our data sample we have 125 FOMC meeting days from May 2007 to December 2022. We pair these days with two alternative low-volatility samples: (1) the days that are one day before the scheduled FOMC meetings, and (2) the days that are one week before the scheduled FOMC meetings. Our null hypothesis (𝐻 ) is that the average difference between the paired high- 0 40

volatility realized gammas and low volatility realized gammas is zero.20 We implement this test using the sub-sampling approach, which does not require making strong assumptions about the unknown distribution of realized gammas or estimating the sample mean variances. All that is required to obtain an asymptotically valid test is that the sampling distribution of the difference in paired realized gammas converges to some unknowndistributionandthateachpair ofrealizedgammasisindependent and identically distributed. The former is an extremely weak condition and the latter is natural as we estimate realized gamma using the ratio of daily realized covariances. The results are reported in Table 5. Columns 1 and 2 show the 99-percent confidence intervals obtained by sub-sampling for the average difference in paired realized gammas for life insurers and P&C insurers, respectively. The test rejects 𝐻 when the confidence intervals do not 0 contain zero. The first row of the table shows the results when the FOMC meeting days are paired with one-day earlier days. The second row of the table shows the results from pairing FOMC meeting days with oneweek earlier days. In both rows, the confidence intervals contain zero and we cannot reject the null hypothesis that the paired realized gammas are the same. For comparison, column 3 reports the confidence intervals from testing the difference in 10-year Treasury security realized volatility 20Inadditiontocalculatingthepaireddifference,wealsotestedthedifferencebetween the average realized gamma on scheduled FOMC meeting days and non-meeting days. 41

between paired days. Column 3 shows that there was a statistically significant increase in 𝜎10𝑦𝑡, as should be expected. 𝑡 Table 5: Comparing realized gammas on days with high and low interest rate volatility. We test the statistical significance of the average differenceinrealizedgammasbetweenhigh-volatilitydayswhentheFOMC has its scheduled meetings and low-volatility days just before the FOMC meetings. Column 1 reports the test of life insurer gammas. Column 2 reports the test of P&C insurer gammas. Column 3 reports the test of daily realized volatility of 10-year Treasury security returns, multiplied by 1𝑒+4 for legibility. The first row pairs the scheduled FOMC meeting days with one-day earlier days. The second row pairs the scheduled FOMC meeting days with one-week earlier days. The sub-sampled confidence intervals are calculated using 10,000 combinations of 15 paired dates. Source: Authors’ calculations based on data from Refinitiv, the Center for Research in Security Prices and the St Louis Fed’s FRASER database. 99% confidence interval Life insurers P&C insurers 10yr Treasury FOMC days vs. 1 day before [-0.072, 0.102] [-0.027, 0.086] [0.23, 0.606] FOMC days vs. 7 days before [-0.063, 0.096] [-0.038, 0.074] [0.271, 0.629] In summary, the additional tests we implemented to address the potential endogeneity of realized gamma and 𝜎10𝑦𝑡 underscore that low 𝑡 interest rate volatility is not the reason for our finding that life insurers’ interest rate risk hedging is generally effective i.e., that realized gamma is generally statistically insignificant. 42

4 Concluding remarks In this paper, we introduced a new method to measure the time-varying residual interest rate risk of financial intermediaries after they have executed their risk management strategies. Our estimates are daily partial correlations obtained using a nonparametric approach on high-frequency financial market data. We then showed how to conduct statistical inference on our estimates by calculating confidence intervals that are asymptotically valid under extremely weak conditions. Our method can be adapted to include additional variables in the regression model that underpins our framework. Another potential future extension would be to allow for ‘jumps’ when estimating realized variances and covariances (Andersen, Bollerslev and Diebold, 2007). Ourmeasurecanbeusedtoevaluatetheinterestrateriskvulnerabilities of any financial intermediary with high-frequency stock prices almost in real time, which is a useful tool for market analysts, supervisors, and policymakers. We applied our method to life insurers, whose exposure to interest rate risk has received less attention than, for example, banks. In doing so, we offered an alternative to the biased and inconsistent lowfrequency rolling window OLS regressions that are prevalent in the existing literature. We find that life insurers are generally well-hedged against longterminterestratemovements. Thatsaid,theyaremoresensitivetochanges 43

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Appendix for online publication A How large is the rolling window bias? In this appendix, we demonstrate the size of the bias from estimating the two-variable regression model on a rolling window of daily data for insurance companies. We start from the specification: 𝑟 𝑖,𝑡 = 𝛼+ 𝛽𝑟 𝑚,𝑡 +𝛾𝑟 𝑦10,𝑡 +𝜖 𝑖,𝑡 where 𝑟 is the stock price return on the index of life insurers (described 𝑖,𝑡 in section 3.2) on day 𝑡, 𝑟 is the return on the benchmark S&P500, and 𝑚,𝑡 𝑟 isthereturnonthe10-yearTreasurysecurity. The 𝛾 coefficientinthis 𝑦10,𝑡 specification is termed rolling gamma and is a low-frequency counterpart to the realized gamma described in section 2 of the main paper. Selecting the size for the rolling window is typically framed as a tradeoff between (i) including more data to reduce standard errors and (ii) being forced to assume the parameter is stable within the window (Robertson, 2018). We follow the standard approach in the empirical literature estimating interest rate risk for life insurers, and assume a rolling window of two years (Sen, 2021; Huber, 2022). Figure 4 shows the time series of rolling gammas. The shaded region indicates the heteroskedasticity-corrected 90 percent confidence interval for 50

Figure 4: Rolling window regression results. The black line shows the rolling gamma estimates using end-of-day data and a twoyear rolling window. The shaded region indicates the heteroskedasticitycorrected90percentconfidenceintervalfortheestimates. Source: Authors’ calculations based on data from Refinitiv, the Center for Research in Security Prices. the estimates. There are two main takeaways from the figure. First, the estimatesarealmostalwaysstatisticallysignificantinthepost-crisisperiod. This finding led researchers to conclude that life insurers’ risk management became less effective in the aftermath of the GFC and spurred a research agenda to understand the cause of this regime switch—e.g., Sen (2021); Koijen and Yogo (2022); Huber (2022). Second, there are large “jumps” in the time series corresponding to periods of market volatility, such as the beginning of the financial crisis (2008) and the pandemic (2020). Jumps in the time series hint at a problem of time-varying conditional 51

volatility in the underlying data. Figure 5 shows the problem by plotting the square of the residuals (𝑟 𝑖,𝑡 −𝑟ˆ𝑖,𝑡 )2. Volatility clustering, which is clearly presentinourdata,isalong-knownempiricalfeatureoffinancialtimeseries (Bollerslev, Chou and Kroner, 1992). Figure 5: Squared residuals from rolling regression. Source: Authors’ calculations based on data from Refinitiv, the Center for Research in Security Prices. The potential effects of conditional heteroskedasticity for OLS regressions are well known. In some applications, such as when the primary concernisestimatingtheconditionalmean,acommonviewisthatinference can be made using the standard corrections proposed by White (1980) or Newey and West (1987). However, as Hamilton (2008) points out, misspecifying the errors will produce inefficient estimates and incorrect 52

inference. The specific case of the rolling window OLS estimator was studied by Cai and Juhl (2021), who showed that a bias can exist even asymptotically with well-behaved errors. Intuitively, the rolling window OLS estimates are weighted averages of the time-varying parameter and the weights depend on the time-varying volatility. The asymptotic bias arises when the two time series (parameter and volatility) are correlated. In simulations, the rolling window OLS estimates are often unstable and the bias can be substantial (Robertson, 2018). One solution to the problem is to assume some structure for the variance processes. By explicitly modeling the heteroscedasticity in the variance-covariance matrix, we address the bias in the time series of parameter estimates and gain efficiency. A typical approach in financial econometrics is to appeal to autoregressive conditional heteroskedasticity (ARCH) models (Bollerslev, Engle and Nelson, 1994). This class of flexible models and its wide range of extensions are straightforward to implement in off-the-shelf statistical packages. In practice, the generalized ARCH, or ‘GARCH’,modelthatallowsforgreaterserialdependenceintheerrorterm is an extremely common choice. The conditional variance of the process for a GARCH(𝑟, 𝑝) is given by: 𝑉𝑎𝑟(𝜖 𝑡 |Ω𝑡−1 ) = ℎ 𝑡 = 𝑎 0 +𝑎 1 𝜖 𝑡 2 −1 +𝑎 2 𝜖 𝑡 2 −2 +···+𝑎 𝑝 𝜖 𝑡 2 −𝑝 + 𝑏 ℎ +𝑏 ℎ +···+𝑏 ℎ . 1 𝑡−1 2 𝑡−2 𝑟 𝑡−𝑟 53

As an exercise to gauge the size of the rolling window OLS bias in the estimates reported in Figure 4, we assume that our three time series of daily returns follow a multivariate GARCH(1,1) process. We specify the joint process: 𝑟 𝑖,𝑡 = 𝛼 𝑖 +𝑢 𝑖,𝑡 𝑟 𝑚,𝑡 = 𝛼 𝑚 +𝑢 𝑚,𝑡 𝑟 𝑦10,𝑡 = 𝛼 𝑦10 +𝑢 𝑦10,𝑡 so    𝑟  𝛼 (cid:169) 𝑖,𝑡 (cid:170) (cid:169) 𝑖 (cid:170)   (cid:173) (cid:174) (cid:173) (cid:174)   (cid:173) (cid:174) (cid:173) (cid:174)   𝐸  (cid:173) (cid:173) 𝑟 𝑚,𝑡 (cid:174) (cid:174) | Ω𝑡−1 = (cid:173) (cid:173) 𝛼 𝑚 (cid:174) (cid:174)   (cid:173) (cid:174) (cid:173) (cid:174)   (cid:173) (cid:174) (cid:173) (cid:174)    (cid:173) 𝑟 (cid:174)  (cid:173) 𝛼 (cid:174)  𝑦10,𝑡  𝑦10 (cid:171) (cid:172)  (cid:171) (cid:172) and      𝑟   𝑢  𝜎2 𝜎 𝜎  (cid:169) 𝑖,𝑡 (cid:170)   (cid:169) 𝑖,𝑡 (cid:170)  (cid:169) 11,𝑡 12,𝑡 13,𝑡 (cid:170) (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) (cid:174)     (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) (cid:174)     𝑉𝑎𝑟   (cid:173) (cid:173) 𝑟 𝑚,𝑡 (cid:174) (cid:174) | Ω𝑡−1  =𝑉𝑎𝑟   (cid:173) (cid:173) 𝑢 𝑚,𝑡 (cid:174) (cid:174) | Ω𝑡−1  = (cid:173) (cid:173) 𝜎 21,𝑡 𝜎 2 2 2,𝑡 𝜎 23,𝑡 (cid:174) (cid:174) . (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) (cid:174)     (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) (cid:174)      (cid:173) 𝑟 (cid:174)   (cid:173) 𝑢 (cid:174)  (cid:173) 𝜎 𝜎 𝜎2 (cid:174)  𝑦10,𝑡   𝑦10,𝑡  31,𝑡 32,𝑡 33,𝑡 (cid:171) (cid:172)  (cid:171) (cid:172)  (cid:171) (cid:172) Thelastmatrix—knownasthedynamicconditionalvariance-covariance 54

matrix—can be used to form the dynamic conditional ratio: 𝜎 𝛾𝐷𝐶 = 13,𝑡 , 𝑡 𝜎2 33,𝑡 which is obtained after specifying GARCH(1,1) processes for each second moment. We call the ratio 𝛾𝐷𝐶 the dynamic conditional gamma, following 𝑡 theliteraturethatusesthesametechniquetoestimatedynamicconditional betas (Engle, 2016). Figure 6 compares the different estimates of gamma. The blue dotted line shows the rolling gamma estimates, while the green solid line shows the dynamic conditional gamma estimates. The difference between the two estimates is particularly striking during periods of high volatility, such as the financial crisis and the global pandemic, revealing that the rolling gamma is highly biased and misleading. For completeness, we include the realized gamma estimates as the brown dashed line in the figure. The relative proximity of the realized gamma estimates and the dynamic conditional gamma estimates during those periods of stress is a reassuring signthatbothapproachesaresolvingtheunderlyingproblemofconditional heteroscedasticity. Note that dynamic conditional gamma and realized gamma use completely different data and approaches to address the same underlying problem. Although dynamic conditional gamma and realized gamma deliver 55

Figure 6: Comparing rolling gamma, dynamic conditional gamma, and realized gamma. The figure shows three different estimatesofthesensitivityoflifeinsurers’stockpricestochangesininterest rates. Source: Authors’ calculations based on data from Refinitiv, the Center for Research in Security Prices. 56

similar parameter estimates, they are not the same. In particular, the empirical approach that underpins the dynamic conditional gamma is known to suffer from substantial limitations (Caporin and McAleer, 2013). As a stated data representation—rather than derived model—the dynamic conditional gamma has no moments or desirable asymptotic properties. It serves our purposes as a diagnostic tool that reveals a huge bias in rolling gamma. But to avoid reliance on the imposed structure and—most importantly—to conduct valid inference, we strongly prefer the empirical approach that uses realized variances and covariances in our paper. B Hypothesis testing using the subsampling method In each day 𝑡 ∈ {1,...,𝑇}, we estimate 𝛾 𝑡 using the following linear regression model 𝑟˜𝑖,𝑗,𝑠 = 𝛼 𝑡 + 𝛽 𝑡 𝑟˜𝑚,𝑗,𝑠 +𝛾 𝑡 𝑟˜𝑦10,𝑗,𝑠 +𝜖 𝑖,𝑗,𝑠 onasampleof𝑛 = 388observationscorrespondingtoeachofthe388trading minutes for day 𝑡 between 9:31am and 3:59pm indexed by 𝑠. We calculate a confidence interval for each of the daily 𝛾 using subsampling following 𝑡 Politis et al. (1999) under the assumption that the errors are asymptotically 57

stationary. Asymptotic stationarity means that, for example, the errors could follow an AR(1) process with autocorrelation parameter strictly less than one and heteroskedastic innovations. To simplify the exposition of subsampling, we rewrite the linear regression model in matrix form as y = Xβ +ϵ, where y and ϵ are 𝑛 × 1 vectors, β is a 𝑝 × 1 vector which includes 𝛾 𝑡 as an element and X is an 𝑛×𝑝 matrix of five-minute returns and a constant. The estimator of β based on X and y is given by β ˆ ≡ (X′X)−1X′y. For any 𝑏 < 𝑛 such that 𝑏 > 𝑝, define the subvectors and submatrices y𝑏,𝑠 ≡ (𝑦 𝑠 ,...,𝑦 𝑠+𝑏−1 )′ , ϵ𝑏,𝑠 ≡ (𝜖 𝑠 ,...,𝜖 𝑠+𝑏−1 )′ and (5) x′ x′ (cid:169) 𝑠 (cid:170) (cid:169) 1 (cid:170) (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) . (cid:174) (cid:173) . (cid:174) X𝑏,𝑠 ≡ (cid:173) (cid:173) . . (cid:174) (cid:174) , where X ≡ (cid:173) (cid:173) . . (cid:174) (cid:174) (6) (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) (cid:174) (cid:173) x′ (cid:174) (cid:173) x′ (cid:174) 𝑠+𝑏−1 𝑛 (cid:171) (cid:172) (cid:171) (cid:172) The estimator of β based on X𝑏,𝑠 and y𝑏,𝑠 is given by β ˆ 𝑛,𝑏,𝑠 ≡ (X′ 𝑏,𝑠 X𝑏,𝑠 )−1X′ 𝑏,𝑠 y𝑏,𝑠 . Denote by 𝐽 (𝑃) the sampling distribution of the normalized statistic 𝑏 58

√ 𝑏(β ˆ 𝑛,𝑏,𝑠 −β), where 𝑃 istheprobabilitylawgoverningtheestimatorβ ˆ 𝑛,𝑏,𝑠 , which is unknown. For any Borel set 𝐴 ∈ R𝑝, let √ 𝐽 𝑏 (𝐴,𝑃) = 𝑃𝑟𝑜𝑏 𝑃 { 𝑏(β ˆ 𝑛,𝑏,𝑠 −β) ∈ 𝐴}. The approximation to 𝐽 (𝐴,𝑃) is defined by 𝑛 𝑛−𝑏+1 √ 1 ∑︁ 𝐿 𝑛,𝑏 (𝐴) = 1{ 𝑏(β ˆ 𝑛,𝑏,𝑠 −β ˆ) ∈ 𝐴}. 𝑛−𝑏 +1 𝑠=1 Therefore, subsampling consist of evaluating a statistics on an exhaustive set of subsamples of size 𝑏 < 𝑛 that are created from the original sample of √ size 𝑛 and estimating the distribution of this statistics normalized by 𝑏. As should be clear, each subsample contains consecutive observations from the original time series sample. Therefore, each subsample is drawn from the true data generating process. In what follows we summarize the main result from subsampling related to the estimation of a daily 𝛾 using intra-day time series observation. Note 𝑡 that our limiting concept is that the number of equally spaced intraday returns approaches infinity. We refer the readers to Politis et al. (1999) for details and proofs. Assumption 1 There exists a limiting law 𝐽(𝑃) such that 1. 𝐽 𝑛 (𝑃) converges weakly to 𝐽(𝑃) as 𝑛 → ∞. This means that for 59

any Borel set 𝐴 whose boundary has mass zero under 𝐽(𝑃), we have 𝐽 𝑛 (𝐴,𝑃) → 𝐽(𝐴,𝑃) as 𝑛 → ∞. 2. For every Borel set 𝐴 whose boundary has mass zero under 𝐽(𝑃) and for any index sequence {𝑠 𝑏 }, we have 𝐽 𝑏,𝑠 (𝐴,𝑃) → 𝐽(𝐴) as 𝑏 → ∞. 𝑏 Theorem B.1 (Politis et al. (1999) Theorem 4.3.1) Let {(x𝑠 ,𝜖 𝑠 )} be a sequence of random vectors defined on a common probability space. Denote the mixing coefficients for the {(x𝑠 ,𝜖 𝑠 )} sequence by 𝛼(·). Define 𝑠+𝑘−1 1 ∑︁ 𝑇 𝑘,𝑠 ≡ √ x𝑎ϵ𝑎 , 𝑉 𝑘,𝑠 ≡ 𝐶𝑜𝑣(𝑇 𝑘,𝑠 ) , and 𝑀 𝑘,𝑠 ≡ 𝐸(X′ 𝑘,𝑠 X𝑘,𝑠 /𝑘). 𝑘 𝑎=𝑠 Assume the following conditions hold. For some 𝛿 > 0, • 𝐸(x𝑠 𝜖 𝑠 ) = 0 for all 𝑠, • 𝐸|x𝑠,𝑗 𝜖 𝑠 |2+2𝛿 ≤ Δ 1 for all 𝑠 and all 1 ≤ 𝑗 ≤ 𝑝, • 𝐸|x𝑠,𝑗 |4+2𝛿 ≤ Δ 2 for all 𝑠 and all 1 ≤ 𝑗 ≤ 𝑝, • 𝑉 𝑘,𝑠 →𝑉 > 0 uniformly in 𝑠 as 𝑘 → ∞, • 𝑀 𝑘,𝑠 → 𝑀 > 0 uniformly in 𝑠 as 𝑘 → ∞, • 𝐶(4) ≡ (cid:205)∞ 𝑘=1 (𝑘 +1)2𝛼 4+ 𝛿 𝛿(𝑘) ≤ 𝐾. Furthermore, assume that 𝑏/𝑛 → 0 and 𝑏 → ∞ as 𝑛 → ∞. Letting 𝐽(𝑃) = 𝑁(0,𝑀−1𝑉𝑀−1). Then: 60

i. 𝐿 𝑛,𝑏 (𝐴) → 𝐽(𝐴,𝑃) in probability for each Borel set A whose boundary has mass zero under 𝐽(𝑃). ii. Let 𝑍 be a random vector with L(𝑍) = 𝐽(𝑍). For a norm ||cot|| on R𝑘 , define univariate distributions 𝐿 𝑛,||cot|| and 𝐽 ||cot|| (𝑃) in the following way: 𝑛−𝑏+1 √ 1 ∑︁ 𝐿 𝑛,𝑏,||cot|| (𝑥) = 𝑛−𝑏 +1 1{|| 𝑏(β ˆ 𝑛,𝑏,𝑠 −β ˆ||) ≤ 𝑥} 𝑠=1 𝐽 (𝑥,𝑃) = 𝑃𝑟𝑜𝑏{||𝑍|| ≤ 𝑥}. ||cot|| For 𝛼 ∈ (0,1), let 𝑐 𝑛,𝑏,||·|| (1−𝛼) = inf{𝑥 : 𝐿 𝑛,𝑏,||·|| (𝑥) ≥ 1−𝛼}. Correspondingly, define 𝑐 (1−𝛼,𝑃) = inf{𝑥 : 𝐽 (𝑥,𝑃) ≥ 1−𝛼}. ||·|| ||·|| If 𝐽 (·,𝑃) is continuous at 𝑐 (1−𝛼,𝑃) then ||·|| ||·|| √ 𝑃𝑟𝑜𝑏 𝑃 {|| 𝑏(β ˆ 𝑛,𝑏,𝑠 −β ˆ|| ≤ 𝑐 𝑛,𝑏,||·|| (1−𝛼)} → 1−𝛼 as 𝑛 → ∞. 61

Thus, the asymptotic coverage probability under 𝑃 of the region √ {β : || 𝑏(β −β ˆ|| ≤ 𝑐 𝑛,𝑏,||·|| (1−𝛼)} is the nominal level 1−𝛼. Theorem B.1 shows that we can derive asymptotically valid confidence intervals for the daily estimator β ˆ using 𝐿 𝑛,𝑏 (𝐴) because it is a consistent estimator of 𝐽(𝐴,𝑃). By exploiting the usual duality between the construction of a confidence interval for 𝛾 and the construction of a 𝑡 hypothesis test about 𝛾 , subsampling allows us to make asymptotically 𝑡 valid inference about the true 𝛾 . In our application, we wish to test the 𝑡 null hypothesis that the daily 𝛾 is statistically different from 0. Under the 𝑡 null, insurers are hedged against interest rate risk as their stock price is not sensitive to movement in the ten-year treasury rate. If the value of the estimated daily 𝛾 falls outside the daily confidence interval, we reject the 𝑡 null hypothesis on that day. Subsamplingisnotaswellknownasthebootstrapmethodineconomics and finance, which warrants a cursory comparison—see Politis et al. (1999) fortextbook-lengthtreatment. Themostrelevantbootstrapmethodforour time series application is the so-called Moving Blocks Bootstrap (MBB). As with subsampling, MBB breaks down the original time series to smaller blocks of consecutive observations, which preserves the serial correlation 62

structure within each block. Practically, the main difference is that MBB draws samples with replacement from the blocks and connects the sampled blocks together to form a bootstrap sample of size 𝑛. Therefore, by construction, MBB imposes the assumption that blocks of an arbitrary size 𝑏 are uncorrelated. This assumption about the unknown data generating process is rather strong and likely to be violated in our application. From a technical point view, the bootstrap method requires that the distribution of the statistic of interest be locally smooth as a function of the unknown model. Establishing this result, even if it is indeed true, would be nontrivial. With subsampling, we do not need to make these assumptions or verify the smoothness of the distribution under the true model to draw asymptotically valid inferences. All that is required is that our normalized statistic has a limit distribution under the true model. 63

C Summary statistics for Section 3.4 Table 6: Summary statistics. This table reports summary statistics for the variables used to analyze the determinants of the significance of realized gamma. 𝛾 is realized gamma for insurer type 𝑖 ∈ {Life, P&C}. 𝑖,𝑡 The binary variable 𝟙(𝛾 𝑖,𝑡 <0) takes the value 1 when realized gamma for insurer type 𝑖 is statistically significant and 0 otherwise. 𝑇𝑃 is the term 𝑡 premium estimate from Adrian et al. (2013), 𝐵𝑎𝑎 − 𝐴𝑎𝑎 is the Moody’s 𝑡 Baa-Aaa seasoned corporate spread, and 𝐹𝐶 is the ICE BoA Single-A US 𝑡 Life corporate index option-adjusted spread. 𝜎 is the realized volatility of 𝑡 10yt the intraday returns of life insurers. 𝜎 is the realized volatility of the 𝑡 Life 10yt intraday returns on 10-year Treasury. The statistics for 𝜎 and 𝜎 𝑡 𝑡 are multiplied by 1e+4 for legibility. Source: Authors’ calculations based on data from Refinitiv, the Center for Research in Security Prices, FRED, and Adrian et al. (2013). Variable No. obs. Mean Median Std. Dev. p25 p75 𝛾 3,901 -0.19 -0.16 0.31 -0.34 -0.02 Life ,𝑡 𝟙(𝛾 Life <0) 3,923 0.32 0 0.47 0 1 ,𝑡 𝛾 3,901 -0.12 -0.09 0.22 -0.21 0.01 𝑃&𝐶,𝑡 𝟙(𝛾 P&C <0) 3,923 0.18 0 0.38 0 0 ,𝑡 𝑇𝑃 3,896 0.54 0.30 1.10 -0.33 1.51 𝑡 𝐵𝑎𝑎 − 𝐴𝑎𝑎 3,897 1.08 0.96 0.47 0.82 1.19 𝑡 𝐹𝐶 3,921 1.49 1.17 1.01 0.94 1.64 𝑡 10yt 𝜎 3,923 0.24 0.16 0.31 0.10 0.28 𝑡 64

D Data citations • Refinitiv • Center for Research in Security Prices, CRSP 1925 US Indices Database,WhartonResearchDataServices,http://www.whartonwrds. com/datasets/crsp/ • FRED API, accessed using third party R software package fredr, https://fred.stlouisfed.org/docs/api/fred/ – ‘BAA’ — Moody’s, Moody’s Seasoned Baa Corporate Bond Yield [BAA], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/BAA – ‘AAA’ — Moody’s, Moody’s Seasoned Aaa Corporate Bond Yield [AAA], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/AAA 65

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