Self-Fulfilling Runs:
Evidence from the U.S. Life Insurance Industry
∗
Nathan Foley-Fisher
Borghan Narajabad
Stéphane Verani
†June 2019
Abstract
The interaction of worsening fundamentals and strategic complementarities among investors renders identification of self-fulfilling runs challenging. We propose a dynamic model to show how exogenous variation in firms’ liability structures can be exploited to obtain variation in the strength of strategic complementarities. Applying this identification strategy to puttable securities offered by U.S. life insurers, we find that at least 40 percent of the $18 billion run on life insurers by institutional investors during the 2007-08 crisis was amplified by self-fulfilling expectations. Our findings suggest that other contemporaneous runs in shadow banking by institutional investors may have had a self-fulfilling component.
JEL Codes: G01, G22, G23, E44
Keywords: Shadow banking, self-fulfilling runs, life insurance companies, funding
agreement-backed securities
∗All authors are in the Research and Statistics Division of the Federal Reserve Board of Governors.
This paper greatly benefited from comments and suggestions by Ali Hortaçsu (the editor) and three anonymous referees. For providing valuable comments, we also would like to thank, without implicating, Felton Booker, Moshe Buchinsky, Francesca Carapella, Mark Carey, Gabe Chodorow-Reich, Ricardo Correa, Lukasz Drozd, Stefan Gissler, Itay Goldstein, Valentin Haddad, Diana Hancock, Zhiguo He, Sebastian Infante, Anastasia Kartasheva, Todd Keister, Ralph Koijen, Stephen LeRoy, Ralf Meisenzahl, Michael Palumbo, Rodney Ramcharan, Rich Rosen, Larry Schmidt, Amit Seru, Alp Simsek, René Stulz, Gustavo Suarez, Amir Sufi, Luke Taylor, Ted Temzelides, Moto Yogo, and the conference and seminar participants in the Macro Finance Society Workshop 2019, CEPR ESSFM Corporate Finance 2016, NBER SI Corporate Finance 2016, EWFC 2016, IBEFA ASSA 2016, Vienna Macro Cafe 2016, SED 2016, SEM 2015, LAEF CYCLE 2015, EEA 2015, RES 2015, FIRS 2015, WFA 2015, Becker-Friedman Institute Conference on Financial Regulation, Wharton Conference on Liquidity and Financial Crisis 2015, Federal Reserve System Committee on Financial Structure and Regulation 2014, UNSW, U Melbourne, ANU, Deakin University, Federal Reserve Board, Johns Hopkins University, Rice University, the Federal Reserve Banks in St. Louis, Philadelphia, and Atlanta, IMF, University of Bern, NUIM, CBoI, UCSB, and the SNB. We are grateful to Caitlin Briglio, Della Cummings, and Shannon Nitroy for exceptional research
assistance. The views in this paper are solely the responsibility of the authors and should not be
interpreted 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.
Introduction
Institutions and markets that are vulnerable to runs pose a threat to financial stability.
In the traditional model of banking, individual banks fund long-term illiquid assets with
short-term demand deposits, rendering them vulnerable to depositor runs. By contrast, in
shadow banking, financial intermediation is performed by chains of institutions operating
outside of the regulated banking sector (Cetorelli, Mandel & Mollineaux 2012). While
chains of shadow banking institutions facilitate greater risk sharing in the economy, each
link in the chain may be vulnerable to runs, potentially increasing the fragility of the
financial system. Policies designed to address the threat to financial stability from runs
have focused on traditional banks, where the causes of runs have been studied extensively,
but there remains considerable debate among academics and policymakers on the causes
of runs affecting shadow banking. Understanding the mechanisms behind these runs is
vital to address the vulnerabilities of the financial system.
In this paper we study the role of self-fulfilling expectations in shadow bank runs—
that is, when investors run because they expect other investors will run and there are
strategic complementarities. In an empirical setting, we would like to analyze investors’
responses to other investors’ actions. But to study how actions of individuals in a group
are associated with actions of the group requires us to confront the reflection problem
(Manski 1993). The key empirical hurdle to identifying self-fulfilling runs is that investors
may be running in response to common fundamentals.1 Indeed, theory suggests that the
two reasons are connected (Morris & Shin 1998, Goldstein & Pauzner 2005, He & Xiong
2012). Weak fundamentals trigger a run, which is amplified by investors’ self-fulfilling
expectations about other investors’ actions. We refer to the amplification of withdrawals
due to strategic complementarities among investors as the self-fulfilling component of
the run. The interaction between fundamentals and strategic complementarities renders
empirical identification of self-fulfilling runs very challenging (Goldstein 2012).
We tackle this empirical challenge using a strategy based on exogenous variation in
investors’ strategic complementarities. We first develop a dynamic model to show how
1 Fundamentals include changes in investors’ liquidity demand, risk appetite, regulatory constraints,
firms’ liability structures are associated with the degree of strategic complementarity
among investors.2 Intuitively, the larger the amount that investors might withdraw from a
firm, the stronger investors’ strategic complementarity. Our model describes a mechanism
whereby runs may have a self-fulfilling component, in which adverse fundamentals interact
with investors’ expectations about potential future withdrawals, amplifying the initial
adverse fundamental shock. We derive the conditions under which a self-fulfilling run
equilibrium is unique. We show that the prospect of bad fundamentals can trigger a
self-fulfilling run. Even a small probability that fundamentals may be bad in the future,
when combined with a possibility of significant withdrawals by other investors, is enough
for an investor to run today.
The main contribution of our model relative to the existing bank run models, such
as Goldstein & Pauzner (2005) and He & Xiong (2012), is to show how fluctuations
in a firm’s liability structure can act as a coordination device for investor expectations
conditional on the fundamental state. An important implication of the model is a
one-to-one mapping between a measureable quantity, the amount of puttable debt that investors
can exercise, and the degree of strategic complementarities among investors, conditional
on the underlying asset’s fundamentals.3 The model shows how the variation in a debt
issuer’s liability structure can be used to test for strategic complementarities during a
run and structurally estimate the amplification in withdrawals due to the run externality.
We take this identification strategy to the data using contractual features of puttable
liabilities issued by U.S. life insurers to institutional investors. Since the early 2000s,
U.S. life insurers have issued extendible funding agreement-backed notes (XFABN) to
access short-term wholesale funding markets. On predetermined recurring election dates,
2 Several recent papers have offered alternative sources of variation in strategic complementarity.
Chen, Goldstein & Jiang (2010) use the liquidity of investments by U.S. mutual funds as a measure of strategic complementarities among investors in each fund. Hertzberg, Liberti & Paravisini (2011) exploit the 1998 reform of a national public credit registry in Argentina as a natural experiment that revealed investors’ strategic complementarities. And Schmidt, Timmermann & Wermers (2016) use heterogeneity in the costs associated with investing in U.S. money market mutual funds as a proxy for the sophistication of investors in each fund, and thereby measure investors’ strategic complementarity.
3 The models of Goldstein & Pauzner (2005) and He & Xiong (2012) cannot be used to derive a
investors in these XFABN decide whether to extend the maturity of their holding.4 Hence,
XFABN are puttable in the sense that investors have the option not to extend the maturity
of any or all of their holdings. In such cases, the non-extended holdings are converted
into short-term fixed maturity bonds with new security identifiers. This funding structure
is analogous to an asset-backed commercial paper (ABCP) program with full liquidity
guarantees from the issuers. XFABN are designed to appeal to short-term investors,
such as money market funds (MMFs), whose investment decisions may be constrained by
liquidity and concentration requirements.5
We first document that institutional investors ran on U.S. life insurers’ XFABN at the
same time that they ran on the ABCP market (Covitz, Liang & Suarez 2013, Acharya,
Schnabl & Suarez 2013, Schroth, Suarez & Taylor 2014) and the repo market (Gorton
& Metrick 2012, Krishnamurthy, Nagel & Orlov 2014) when fundamentals began to
deteriorate in the summer of 2007. To show this, we collected new data for each XFABN—
including daily amounts outstanding, election dates, and terms for withdrawals—by hand
from individual security prospectuses and Bloomberg corporate action records. At that
time, widespread concerns about financial market liquidity had developed in concert with
the subprime mortgage crisis and declining house prices.
Life insurers are vulnerable to unexpected XFABN withdrawals even though they have
large asset portfolios (Foley-Fisher, Narajabad & Verani 2019). These assets are held
for actuarial and regulatory reasons and are difficult to liquidate. When spare capital
and alternative sources of funding become scarce, unexpected XFABN withdrawals can
put tremendous pressure on an insurer’s liquidity and, in some cases, lead to a failure
even though the insurer remains solvent. Although insolvency is rarely an issue for life
insurers, and insurance liability holders can be reasonably certain they will eventually be
paid, there could be tremendous uncertainty overwhen short-term institutional investors
will get their money back.
Investors in XFABN are sensitive to the timing of their payoffs and to the timing of
other investors’ payoffs because those investors are themselves vulnerable to runs.
Short-term institutional investors—such as MMFs—avoid any asset that might cause their own
investors to withdraw. In the face of growing concerns about the financial system in the
4 For each XFABN, there is a final maturity date beyond which no extensions are possible.
5For example, SEC Regulation 2a-7 generally requires MMFs to hold securities with residual maturity
second half of 2007, institutional investors sensitive to the risk of future illiquidity of life
insurers executed their option to withdraw.
We propose an identification strategy based on variation in strategic complementarity
among investors in the puttable XFABN market. We construct an instrument for
investors’ expectations about other investors’ actions, using the contractual structure of
XFABN. Our instrumental variable (IV) is the maximum fraction of XFABN that could
be withdrawn between an individual XFABN’s consecutive election dates. The intuition
for this instrument follows from the predictions of our theoretical model: If the number
of potential other investors that can run is low (high), there is weak (strong) strategic
complementarity among investors. Differences across each insurer’s XFABN contractual
terms creates variation in the instrument over time and across XFABN and insurers.
Crucially, the election dates are determined when the XFABN were first issued, often
years before the run, and are therefore plausibly exogenous to changes in fundamentals
during the run.
This type of instrumental variable can be used to identify strategic complementarities
in other applications. One possible application is ABCP and sovereign debt, where
short-term debt from the same issuer may mature on different dates and are usually rolled
over. In such cases, there is variation in the amount of the other debt from the same
issuer that is rolled over during the lifespan of one particular debt obligation. Provided
this variation is exogenous to underlying fundamentals, it can be used to tease out the
effect of coordination among investors. A second possible application is credit default
swaps (CDS), where different contractual expiry dates creates variation in the fraction of
“empty” creditors as defined by Bolton & Oehmke (2011). This variation potentially yields
a useful instrumental variable to identify strategic interaction among these creditors,
leading to a deeper understanding of manufactured credit events.
Our IV estimates consistently show that there was a self-fulfilling component to the
run on U.S. life insurers’ XFABN. In addition to the baseline estimates, we implement
a series of robustness tests, including controlling for high-frequency financial conditions,
controlling for group behavior unrelated to expectations, and exploring the sensitivity of
our estimates to variation in the date at which the instrumental variable is calculated. We
also estimate our IV specification including week fixed effects to address the reasonable
life insurance industry as a whole or a common shock to short-term investors’ liquidity
demand.
While our reduced-form IV regression results offer compelling evidence for the
existence of the self-fulfilling component, we can better measure the magnitude of the
self-fulfilling component by taking our dynamic run model to the data. Motivated by the
recent structural empirical work of Schroth et al. (2014) and Egan, Hortaçsu & Matvos
(2017), we use a combination of model calibration and estimation to account for potential
nonlinear relationships between investors’ concerns about fundamental developments and
their concerns about other investors’ withdrawals during the run. Our estimate of the
self-fulfilling component of the run is at least 40 percent of withdrawals.6
The contributions of our paper are fivefold. First, our model shows how an investor’s
decisions depend on beliefs about other investors’ decisions and those beliefs respond
to exogenous shocks to a puttable debt issuer’s liability structure. Second, our
hand-collected data shed light on the connection between U.S. life insurers and shadow banking.
Third, we provide a new empirical strategy, based on our theoretical finding, to identify
strategic complementarities among investors. Fourth, we apply this identification strategy
to our data and find compelling evidence that the run had a self-fulfilling component. And
fifth, we structurally estimate our model, finding that the interaction of concerns about
fundamentals and strategic complementarities among investors was a major factor in the
withdrawal decisions by institutional investors in the shadow banking system during the
financial crisis.
Our evidence of a sizable self-fulfilling component to the run on U.S. life insurers
contributes to a deeper understanding of the vulnerability of shadow banking to runs.
While the market for XFABN is small relative to the ABCP and repo markets, the same
institutional investors participate in all of them. Because their behavior is likely to have
been similar across markets, our study offers evidence that there may have been a
self-fulfilling component to the contemporaneous runs by institutional investors in those larger
markets.
A better understanding of self-fulfilling runs by institutional investors is important
because the traditional methods of dealing with self-fulfilling runs by bank depositors—
that is, through liability insurance and regulatory supervision of assets—are either
6The trigger for the run is nonetheless fundamental concerns, in the absence of which no withdrawals
infeasible or ineffective to cope with runs by institutional investors. Efforts to mitigate
the run risk have been made at some links in the shadow banking intermediation chain by
adapting the traditional methods of dealing with runs. For example, new rules imposed
by the Securities and Exchange Commission (SEC) are intended to reduce the likelihood
of runs on MMFs (Cipriani, Martin, McCabe & Parigi 2014).7 However, the wide range of
liabilities and assets on institutional investors’ balance sheets renders liability insurance
and regulatory supervision impractical for dealing with runs by institutional investors.
Our analysis suggests that careful monitoring of security issuers’ liability structures is
essential for supervisors tasked with financial stability.
The remainder of the paper proceeds as follows. Section 1 presents a model in
which a firm’s liability structure affects its vulnerability to self-fulfilling runs. Section 2
discusses the institutional background to our analysis, including the vulnerability of large
modern U.S. life insurers to relatively small unexpected withdrawals by institutional
investors. Section 3 presents our data and summary statistics on extendible funding
agreement-backed securities. Section 4 discusses the implementation of our test of
strategic complementarities and presents our main reduced-form IV regression results.
Section 5 discusses estimates of the magnitude of the self-fulfilling component based on
structurally estimating the model. Section 6 concludes with some remarks on broader
implications of our findings and suggests some avenues for further study.
1
A model of liability structure and self-fulfilling runs
We analyze the dynamic coordination problem among investors in a firm that issues
puttable debt obligations. Investors in puttable bonds have the right, but not the
obligation, to demand early repayment of the principal. The put option can be exercised
on a number of specified dates, and the fraction of bonds for which investors can withdraw
varies over time because the put options cannot be exercised on the same dates. The
proceeds from issuing the puttable bonds are used to finance a longer-term risky asset.
Our model maps to a wide range of empirical settings, including banks that provide full
liquidity guarantees to ABCP programs set up to finance their loans off-balance sheet
7 SEC 17 CFR Parts 230, 239, 270, 274 and 279. Release No. 33-9616, IA-3879; IC-31166;
FR-84; File No. S7-03-13. See https://www.sec.gov/News/PressRelease/Detail/PressRelease/
and insurance companies that issue funding agreement-backed securities structured as
bonds with embedded put options or commercial paper.8
The main contribution of our model is to show how a firm’s liability structure,
summarized by the fraction of bonds for which investors can exercise their put option,
is a coordination device for investor expectations conditional on the underlying asset’s
fundamental state. The key variable of the model is the fraction of puttable bonds that
give investors the right to exercise their put optionnow relative to the amount of puttable
bonds for which investors may have the right to exercise their put option in the future.
An important implication of the model is to highlight the one-to-one mapping between a
measureable quantity, the fraction of bond with exercisable put options, and the degree
of strategic complementarity among investors.
As in Goldstein & Pauzner (2005) and He & Xiong (2012), self-fulfilling expectations
can be triggered by the prospect of a deterioration in asset fundamentals and lead to a
run.9 Unlike those papers, which assume the bond issuer’s liability structure is fixed,
we show that variation in the issuer’s liability structure can have a significant effect on
investors’ propensity to run. Concerns about deteriorating asset fundamentals can trigger
a self-fulfilling run that becomes more severe when the fraction of bonds with exercisable
put options is high.10
1.1
Environment and investor decisions
Time is continuous and infinite. A continuum of risk-neutral investors discount the future
at rateρ >0. Each investor holds one unit of short-term puttable bonds issued by a firm
to finance a long-term risky asset. The underlying risky asset could be, for example, a
debt obligation issued by a commercial or industrial corporation.
The underlying asset yields a constant coupon streamr >0and matures at a random
date determined by a Poisson process with arrival rateφ >0. The issuer returns one unit
8 See online appendix A for a description of funding agreement-backed commercial paper.
9 In seminal theoretical contributions, Bryant (1980) and Diamond & Dybvig (1983) show that firms
issuing demandable liabilities are potentially vulnerable to swift changes in investors’ beliefs about the actions of other investors. Such a run is in contrast to a fundamental-based run, in which investors decide to withdraw based on a signal they receive about the state of fundamentals as in Chari &
Jagannathan (1988), Jacklin & Bhattacharya (1988) and Allen & Gale (1998). Our theory follows
recent work suggesting that the two reasons are connected (Goldstein 2012).
10In online appendix B, we show how holding the issuer’s liability structure fixed in our model results
of principal to each investor when the asset matures. However, the asset could default
before it matures with Poisson arrival rate π >0. The parameter π >0 summarizes the
asset fundamentals: a greater value of π is associated with a higher default risk, and an
increase in π corresponds to an increase in default risk. The default value of the asset is
constant and denoted by νD <1.
Investors in puttable bonds have the right to demand early repayment of the principal
only on some dates. Puttable bonds can be in one of two statesS ∈ {E, N}. When a bond
is in its exercisable state E, investors receive the right to exercise their put option with
Poisson arrival rateδ. An investor exercising the put option receives1−ω, whereωis an
independent and identically distributed withdrawal cost from a cumulative distribution
function Ω. The withdrawal cost is drawn by each investor every time they receive a
put option and captures the cost of reinvesting the principal elsewhere.11 We assume the
distribution Ω has full support over [0,1] and no mass point. When investors exercise
their put option, their bonds are replaced by new puttable bonds in the same state,
unless the underlying asset is liquidated by the issuer during a run, which we discuss
below. When the bond is in its nonexercisable state N, investors cannot exercise their
put options. Each puttable bond matures with Poisson arrival rate ζ. If their bond
matures, investors receive their principal and their bond is replaced by a new puttable
bond in the same state as the maturing bond.
The issuer’s liability structure is summarized by the aggregate state variablee∈[0,1],
which denotes the fraction of puttable bonds that are in the exercisable state E. We
assume that there is an exogenous shock to e with Poisson arrival rate , which we
refer to as a liability structure shock. We model shocks to e as exogenous from the
investors’ perspective because it is a descriptive assumption of the environment and not
a simplifying assumption. After a liability structure shock, a new fraction of bonds e0
will be in state E, where e0 is drawn from a Beta distribution with parameters α =e·η
and β = (1−e)·η. This assumption implies that, following a liability structure shock,
some bonds will enter state E, while other bonds will remain in their current state, and
yet other bonds will exit state E. Thus, every bond switches state from time to time,
11 The withdrawal costω can also be interpreted, for example, as the opportunity cost of withdrawal
in terms of the liquidity service that an investor derives from holding the bond. In this interpretation, an investor perceiving the bond to be illiquid has a low opportunity cost of withdrawal—i.e., a low value
ofω—and prefers to exercise her put option. We will discuss below a special case of the model whereΩ
and each bond is equally likely to be in state E with probability e0.12
A run occurs only when a positive measure of investors exercise their put options.
During a run, the issuer may be able to roll over its debt by issuing new puttable bonds
with no effect on the issuer’s liability structure. However, the issuer may be forced to
liquidate the underlying asset if it cannot roll over its debt. If the underlying asset is
liquidated during a run, then investors in the puttable bond receive the liquidation value
νL < 1, which may be different from the underlying asset default value νD. A forced
liquidation occurs with Poisson arrival rate θ ·e ·Ωb ≥ 0, where Ωb is the fraction of
investors who are exercising their put option and e·Ωb is the flow of withdrawals.13 If
there is a larger fraction of puttable bonds in state E, or a larger fraction of investors
who are exercising their put option, or both, then the likelihood of a forced liquidation
during a run is higher. We will describe below how the parameter θ captures the degree
of strategic complementarity among investors.
An investor’s required return on one unit of a puttable bond in state S ∈ {E, N} is
equal to the expected increment in her continuation value. Assume that each investor
takes as given the values, V¯ = {V¯E,V¯N}, that other investors derive from holding one
unit of a puttable bond in states E and N, respectively. Moreover, assume for now that
the two value functions are continuous in e.14 The functional equation associated with
holding one unit of a puttable bond is given by:
ρVS e; ¯V = ε· Ee0|ee0·VE e0; ¯V+ (1−e0)VN e0; ¯V−VS (1) +π·(νD−VS)
+r+φ·(1−VS)
+θ·e·Ω(1−V¯E(e))·(νL−Vs)
+ζ· 1−VS+1{S=E}·δ· EΩ
maxVS,1−ω −VS ,
where the arguments of VS are omitted on the right-hand side when they are same as
the arguments on the left-hand side.
The right-hand side term on the first line of equation (1) is the expected change
12 Note thateis a martingale process because
E[e0|e] =e.
13 As we will describe below, the fraction of investors
b
Ωwho are exercising their put option is related
to the distribution of withdrawal costsΩ.
in value caused by variation in the issuer’s liability structure. The second line is the
change in value caused by an asset default shock. The third line is the yield on the
underlying asset plus its payoff at maturity. The fourth line captures the degree of
strategic complementarity among investors, through its effect on the expected liquidation
value during a run. The fifth line is the change in value caused by the bond maturing
plus the change in value if the investor exercises her put option. Naturally, an investor
always chooses to exercise her put option if the value of holding the puttable bond is less
than the principal amount net of the withdrawal cost ω.
There are strategic complementarities only when θ > 0, and the degree of strategic
complementarity depends on the value of e. During a run, the likelihood of a forced
liquidation depends on the flow of withdrawals e·Ωb. The fraction of investors exercising
their put options Ωb = Ω(1−V¯S(e)) is a function of the measure of investors for whom
the withdrawal cost ω is less than 1−V¯S(e). Because investors receive νL < 1 if the
underlying asset is liquidated, an investor is more sensitive to changes in the bond issuer’s
liability structure when other investors’ valuations are lower. Consequently, an investor’s
decision to withdraw is affected by her expectation about other investors’ valuations, and
the degree of strategic complementarity is greater when e is higher.
We formalize this argument in Lemma 1.1, which establishes that an investor’s bond
valuation is uniquely determined by other investors’ bond valuations.
Lemma 1.1 Given V¯, there is a unique pair of investor value functions V ={VE, VN}
that solves equation (1), such that VE(e) ≥ VN(e) for all e ∈ [0,1]. Moreover, these value functions are continuous in e.
Proof Define the operator L onVS for S ∈ {E, N} as:
LVS e; ¯V
= r+φ+π·νD +θ·e·Ω(1−
¯
VE(e))·ν
L+ζ +1{S=E}·δ·EΩ
maxVS,1−ω
ρ+φ+π+θ·e·Ω(1−V¯E(e)) +ζ+ε+1
{S=E}·δ
+ ε·Ee0|e
e0·VE e0; ¯V
+ (1−e0)VN e0; ¯V
ρ+φ+π+θ·e·Ω(1−V¯E(e)) +ζ+ε+1
{S=E}·δ
, (2)
where
ε+1{S=E}·δ
ρ+φ+π+θ·e·Ω(1−V¯E(e)) +ζ+ε+1
{S=E}·δ
<1.
[0,1] to R2
+. The result follows because the fixed point LVS =VS solves equation (1)
and LVE ≥LVN .
An implication of Lemma 1.1 is that whenever θ > 0, an investor is more likely to
exercise her put option when other investors are more likely to do the same. To see this
point, note that the probability that an investor withdraws in state S = E is given by
b
Ω= Ω(1−VE(e)). The value of a bond in state E, V¯E(e), is lower and Ω(1−V¯E(e))
is higher when other investors are more likely to exercise their put options for all levels
ofe. It follows that the solution for LVS =VS is non-increasing inV¯E because the term
Ω(1−V¯E(e))appears in the denominator of the operator Lin equation (2) and ν
L <1.
In addition, the proof of Lemma 1.1 implies that VE(e)≥ VN(e). This means that the
put option is valuable to investors holding bonds in state S=E, as it gives investors an
opportunity to demand the early repayment of their principal net of the withdrawal cost
ω.
1.2
Equilibrium
We now turn to the definition of a symmetric equilibrium. In a symmetric equilibrium, an
investor’s expectation about other investors’ value functions should be consistent with the
value functions implied by the other investors’ optimal withdrawal decisions. Formally, a
symmetric equilibrium consists of a pair of functions V ={VE, VN} such that V solves
equation (1) and V¯ =V. In other words,
LVS(e;V) = VS(e;V) for S ∈ {E, N}, (3)
where L is defined in equation (2). Proposition 1.2 below establishes the conditions
under which a unique symmetric equilibrium exists.
Proposition 1.2 Given that the withdrawal cost distribution Ωdoes not have any mass
point over its support on [0,1], if π > 1r−−νρ
D and θ <
ρ+φ+π+ζ
O+δ+ε , where O is the maximum of the probability density function associated with Ω, then there is a unique pair of value
functions V∗ = {VE∗, VN∗} that solves equation (3). Each value function maps from
[0,1]to [0,1] and VE∗ ≥VN∗
The symmetric equilibrium is unique because the idiosyncratic withdrawal cost
ω plays a similar role as the noise variables in Morris & Shin (1998) and Frankel
& Pauzner (2000). When strategic complementarities are present (θ > 0) and the
underlying asset fundamental is relatively good (π ≤ 1r−−νρ
D) there could be multiple
equilibria if the withdrawal cost distribution has a mass point. For example, in the
special case where (i) ω = 0 for all investors, (ii) the liability structure is constant
at = 0, and (iii) e > r−ρ−π(1−νD)
θ(1−νL) , there is both a no-run equilibrium and a run
equilibrium. In the no-run equilibrium, no investor exercises their put option and
VnoE−run = VnoN−run =
r+π·νD+φ+ζ
ρ+π+φ+ζ > 1. In the run equilibrium all investors exercise their
put options with VE
run =
πνD+r+φ+θ·e·νL+ζ+δ
r+π+φ+θ·e+ζ+δ and V N run =
πνD+r+φ+θ·e·νL+ζ
r+π+φ+θ·e+ζ . Exercising the
put option is rational in the run equilibrium because the fraction of investors that can
exercise their option is large enough to affect other investors’ valuation of the bond, that
is e > r−ρ−π(1−νD)
θ(1−νL) , which means V
N
run < VrunE <1. In both equilibria there are strategic
complementarities among investors, which creates self-fulfilling expectations about other
investors’ decisions. In the no-run equilibrium, investors do not expect other investors to
exercise their put options and, conditional on this expectation, the value of holding the
bond is higher than the payoff they receive from exercising the put option. In the run
equilibrium, investors expect other investors to exercise their put options and the value of
holding the bond is lower than the payoff they receive from exercising the put option. In
the no-run equilibrium all investors are better off because the value of holding the bond
is higher than it is in the run equilibrium, that is VnoE−run = VnoN−run >1 > VrunE > VrunN .
Therefore, strategic complementarities result in Pareto-ranked equilibria, as in Bryant
(1980) and Diamond & Dybvig (1983). As shown in Proposition 1.2, the absence of any
mass point in the withdrawal cost distribution rules out this type of multiplicity.
1.3
Properties of the run equilibrium
Having defined the symmetric equilibrium, we can study the benefits and costs of a higher
e for an investor that holds a puttable bond.15 In general, there are benefits to investors
15 A higheremay also be beneficial to the issuer under some conditions. We do not model the firm’s
issuance of bonds before the run because our focus is on the investors’ decisions to withdraw. That said, it is reasonable to think that the firm may lower its cost of funding by issuing bonds that are more
valuable to its investors. In particular, since a higherecan increase the value of holding the bond when
θis close to zero (no run externality) andπis positive. Consequently, the firm may choose to issue more
from an increase in e, but these benefits may be dominated by a relatively large run externality, resulting in a net-negative effect on the bond value. Proposition 1.2 shows
that VE∗ ≥VN∗
because investors only receive the right to exercise the put option when
the bond is in stateE. Consequently, a higheremeans that a higher fraction of bonds are
more valuable to investors. In addition, because the law of motion for e is a martingale
process, the probability of switching to state E following a liability structure shock is
higher for bonds that are in state N. Therefore, for all bonds, the probability that an
investor receives the right to exercise her put option is increasing in e, which is valuable
when the underlying asset fundamental is relatively weak. Proposition 1.3 shows that in
the absence of strategic complementarities, there is no run externality and there are only
benefits to investors from an increase in e. This means that, all else equal, investors are
less likely to withdraw when a higher fraction of investors can exercise their put options.
Proposition 1.3 If θ = 0 and the conditions of Proposition 1.2 hold, then V∗ =
VE∗, VN∗ is non-decreasing in e.
Proof See Appendix.
When θ > 0, there are strategic complementarities and concerns about fundamentals can trigger a self-fulfilling run. Moreover, strategic complementarities are more severe
when e is higher. In some cases, the run externality can be so large that its negative
effect may dominate the benefits to investors from a highere. The size of the benefits of a
higheredepends on the arrival rate of the liability structure shocks, ε, and the difference
between VE and VN. For a sufficiently low asset liquidation value, ν
L, the difference
between VE and VN is bounded. We can exploit this property to derive a threshold
value forεbelow which the adverse effect of eon the bond value that arise from strategic
complementarities dominates the benefits associated with a higher e. Proposition 1.4
characterizes the sufficient conditions guaranteeing that V∗ is non-increasing in e, which
are necessary conditions to estimate the contribution of strategic complementarities to a
run.
Proposition 1.4 When θ > 0, νL is sufficiently small, ε is not too large, and the
conditions of Proposition 1.2 hold, then V∗ = VE∗, VN∗ is decreasing in e and
∂VS(e; ¯V)
∂θ ≤0.
Proof See Appendix.
In summary, the model highlights the distinction between runs due only to a
deterioration in asset fundamentals, runs that areamplified by self-fulfilling expectations,
and runs that are purely self-fulfilling. When there are no strategic complementarities
(θ = 0) investor withdrawal decisions are not sensitive to e. Investors only withdraw
when the asset fundamentals are relatively bad, π > 1r−−νρ
D, because V
E∗(·;·) < 1 .
Such withdrawals correspond to a pure fundamental run. When there are strategic
complementarities (θ > 0) an investor is more likely to withdraw when eis high, because
she expects other investors to withdraw as well. In that situation, a run may be triggered
by deteriorating underlying asset fundamentals and be amplified by the effect of
self-fulfilling expectations. Lastly, a run can be purely self-self-fulfilling in the absence of any
concerns about the fundamental (π = 0) ifω = 0 for all investors so that the withdrawal
cost distribution has a mass point at zero.16
Propositions 1.3 and 1.4 show how the variation in a debt issuer’s liability structure
can be used to test for strategic complementarities during a run and estimate the
amplification in withdrawals due to the run externality. Proposition 1.3 shows that in
the absence of strategic complementarities, and all else equal, investors are less likely to
withdraw when a higher fraction of investors can exercise their put options. Therefore,
a positive correlation between investor withdrawals and the variation in the fraction
of investors with the right to exercise their put option would be evidence of strategic
complementarities. When there are strategic complementarities, Proposition 1.4 shows
that investors’ withdrawals increase monotonically in the fraction of investors who can
exercise their put options. In addition, the monotone effect of θ on the value of holding
the bond and the probability of withdrawal mean that we can use the model to identifyθ
and estimate the contribution of strategic complementarities. These theoretical findings
suggest that some empirical environments may allow researchers to overcome the key
identification challenge that investor withdrawals during a run can be correlated because
there are strategic complementarities, or because investors are responding to changes in
the underlying fundamentals, or both. In the next section, we will describe one such
institutional setting.
2
Institutional background
The use of institutional funding agreements by U.S. life insurers emerged as a response to
long-run macroeconomic and regulatory changes that affected the industry. Life insurers
traditionally offer insurance to cover either the financial position of dependents in the
event of the death of the main income earner or individuals at risk of outliving their
financial wealth. Under this model, policyholders make regular payments to an insurance
company in exchange for promised transfers from the insurer at a future date. The
promised transfers are long-term illiquid liabilities for insurers, which are backed by
assets funded by the regular payments from policyholders. The assets backing insurance
liabilities need to be low risk and are managed carefully to pay insurance claims as
required. While striving to match the duration and risk of their insurance liabilities,
insurers typically maintain only just enough liquid assets to meet expected claims and
satisfy regulatory constraints, investing as much as possible in higher-yielding illiquid
longer-term assets.
2.1
Rise of life insurers’ non-traditional liabilities
Throughout the middle part of the 20th century, U.S. life insurers enjoyed easy profits,
as high interest rates on safe long-term U.S. Treasuries that were attractive during World
War II were replaced with high interest rates on long-term corporate bonds (Briys & De
Varenne 2001). Soon after, however, pension funds emerged, offering higher returns to
savers and challenging the traditional business model of life insurers. Pension funds could
afford to offer higher returns because they could invest freely in booming equity markets.
Life insurers responded to the threat from pension funds by pursuing more aggressive
investment strategies and offering products with higher (sometimes guaranteed) yields
and greater flexibility to withdraw funds early.
The combination of greater liability run-risk and risky assets resulted in an insurance
crisis in the late 1980s. Many insurers failed, as capital losses on high-risk assets
caused surrender runs by policyholders, intensified by falling credit ratings of insurers
(DeAngelo, DeAngelo & Gilson 1994). Realizing that life insurers had overweighted
their portfolios with risky assets, the National Association of Insurance Commissioners
risk-based capital (RBC) requirements, financial regulation accreditation standards, and an
initiative to codify accounting principles.17 For their part, life insurers redressed the
balance of their portfolios toward safer and more liquid assets.
Insurers’ refocus on safe assets after the crisis of the late 1980s gave rise to a new
problem, as interest rates on safe assets continued the decline they had begun in the
early 1980s. The prospect of persistently low interest rates meant life insurers were
at risk of being unable to deliver the guaranteed returns promised to policyholders
when the expected path of interest rates was higher. This rising interest rate risk led
insurance industry state regulators to adopt new regulations requiring life insurers to hold
higher statutory reserves in connection with term life insurance policies and universal
life insurance policies with secondary guarantees.18 However, higher risk-based capital
requirements necessarily imply a lower return on equity, as larger reserves must be backed
by safe, low-yield assets.19
Life insurers responded to higher capital requirements and falling interest rates by
finding innovative ways to increase their return on equity. One way is to reduce the
risk-based capital requirement by shifting insurance risk off-balance sheet to captive reinsurers
(Koijen & Yogo 2016).20 Another way is to loan out securities to raise cash and fund
a portfolio of longer-term, higher return assets (Foley-Fisher, Narajabad & Verani 2016,
Foley-Fisher, Gissler & Verani 2019). And yet another way is to fund an expansion of the
insurer’s portfolio of high yield assets using funding agreement-backed securities (FABS),
which is part of what the industry terms its “institutional spread business.”21
17 Under the state-based insurance regulation system, each state operates independently to regulate
its own insurance market, typically through a state insurance department. State insurance regulators created the NAIC in 1871 to address the need to coordinate regulation of multistate insurers. The NAIC acts as a forum for the creation of model laws and regulations.
18 NAIC Model Regulation 830 (Regulation XXX) and Actuarial Guideline 38 (Regulation AXXX).
19 The new statutory reserve requirements are typically higher than the reserves that life insurers’
actuarial models suggest will be economically required to back policy liabilities. For context, insurers’ statutory reserves tend to be much higher than reserve requirements for banks under U.S. generally accepted accounting principles (GAAP).
20 Captive reinsurers are onshore and offshore affiliated unauthorized reinsurers that are not licensed
to sell insurance in the same state as the ceding insurer and do not face the same capital regulations as the ceding insurer. Koijen & Yogo (2016) estimate that the regulatory capital reduction from transferring insurance liabilities to captives increased from $11 billion in 2002 to about $324 billion in 2012.
21 Funding Agreement-Backed Notes are sometimes referred to as Guaranteed Investment
2.2
Funding agreement-backed securities
Life insurers issue FABS and invest the proceeds in a portfolio of relatively higher-yield
assets such as mortgages, corporate bonds, and private label ABS, to earn a spread. In a
typical FABS structure, shown in Figure 1, a hypothetical life insurer sells a single funding
agreement to a special purpose vehicle (SPV).22 The SPV funds the funding agreement
by issuing smaller denomination FABS to institutional investors. Importantly, FABS
issuance programs inherit the ratings of the sponsoring insurance company, and investors
are treated pari passu with other insurance obligations, as the funding agreement issued
to the SPV is an insurance liability. This provides FABS investors with seniority over
regular debt holders, and it implies a lower cost of funding for the insurer relative to senior
unsecured debt. For example, this structure allows a life insurer rated AA to “borrow”
at AAA and earn a sizable return by investing the funds in BAA- or lower-rated assets.
A further benefit is that FABS do not increase standard measures of leverage because a
funding agreement is legally an insurance obligation.
The U.S. FABS market grew rapidly during the early 2000s. Figure 2 shows the
end-of-year total amount of FABS outstanding by insurance company. At its peak in 2007,
new issuance reached over $50 billion, with more than $170 billion in notes outstanding,
or about 90 percent of the auto ABS market. It is apparent from Figure 2 that only the
largest highly rated U.S. life insurers issue FABS.
FABS are flexible capital market instruments that may feature different types of
embedded put options to meet demands from various investors, including short-term
investors, such as MMFs. XFABN are a particular type of FABS designed for
short-term investors that give investors the option to extend the maturity of their investment.
XFABN are structured as floating-rate notes with a variable coupon that is the sum of
a benchmark interest rate plus a spread (or margin) that steps up at regular intervals.
The embedded put option in an XFABN allows an insurer to place a medium-term note
with, for example, an MMF that is legally bound to hold short-term debt instruments.23
22 Note that FABS can only be issued by life insurers, as a funding agreement is a type of annuity
product without morbidity or mortality contingency.
23 Some evidence suggests life insurers even went as far as hiring specialist salespeople to place their
2.3
Extendible funding agreement-backed securities
Each XFABN prospectus specifies election dates on which investors may extend the
maturity of their holdings.24 If the investor chooses to extend, the XFABNinitial maturity
date is extended by some pre-specified term, and the option to extend carries over to
the next election date or until the maturity date reaches a pre-specified final maturity
date. Panel A of Figure 3 provides a graphical example of the timeline for XFABN
election decisions. The period over which the XFABN maturity may be extended is
called the election window. An investor in a particular XFABN may choose to extend
it while other investors in the same XFABN may choose not to extend it. Furthermore,
investors’ decisions to withdraw from an XFABN need not be “all or nothing,” as they
can choose to only extend a fraction of their own holdings of that XFABN. On every
election date, the portion of an XFABN that is not extended is converted into a new
zero-coupon note, called a spinoff. Each spinoff is given a new identifier (CUSIP) from
that of the original XFABN. These new securities are no longer eligible for extension and
have a pre-specified fixed maturity. Any remaining portion of the XFABN continues to be
eligible for extension, pays the variable coupon, and retains its original CUSIP identifier.25
Importantly, information about an insurer’s liability structure is public knowledge among
participating institutional investors. Referring to their XFABN program circa 2000, the
then director of new initiatives at Aegon Institutional Markets explains:
“The customers that we sell to are pretty sophisticated. They know exactly
what they’re buying. They are generally investment managers in their own
right. [...] [T]he computer systems have been developed to a point that
everybody knows exactly what options are on each contract. At any point
in time most of our customers know what’s on first and who’s on second.”
(Society of Actuaries 2000).
12-month put business is effectively all that Aegon does. We actually like the business. It’s a perpetual contract. The contract holder can’t get out of the contract unless they give a 12-month notice. Part of risk management is case specific underwriting. Each ticket, as I mentioned before, is pretty large and a lot of risk management needs to happen at the individual sale each time you make the sale” (Society of Actuaries 2000).
24 Typically, holders only notify the XFABN dealer on or around each election date if they want
to extend the maturity of their XFABN (either in part or the entire security). In the event that no notification is made, the security holder is assumed to have elected not to extend the security.
25 XFABN programs are essentially similar to ABCP programs withfull liquidity guarantees from the
A representative example of XFABN terms is provided in online appendix C.26
This $800 million MetLife XFABN is backed by a funding agreement issued by
Metropolitan Life Insurance Company (MLIC), MetLife’s flagship New York-based
insurer, to Metropolitan Life Global Funding I, the Delaware-based SPV issuing all
MetLife FABS backed by a funding agreement from MLIC to date.27 This XFABN
was issued on June 14, 2011 with an initial maturity date of July 6, 2012 (397 days) and
a final maturity date of July 6, 2017 (six years plus one month). Beginning on July 6,
2011, institutional investors have the option to extend the initial maturity by one month
on the sixth calendar day of each month up until June 6, 2016 (397 days before the final
maturity date). The variable coupon rate during the first year is the USD three-month
LIBOR plus a spread of 0.125 percent. In each year thereafter, the spread over the
variable coupon increases to a maximum of 0.25 percent in the last two years before the
final maturity date. An investor choosing not to extend a portion or all of his holding
of this particular XFABN on one of the election dates obtains a spinoff that pays zero
coupon and matures 365 days from this particular election date.
The increasing coupon spread over the variable base rate gives an incentive to investors
to extend the maturity of an XFABN until its final maturity date. Because privately
placed debt instruments such as XFABN have relatively high fixed issuance costs, the
lower (and, in some cases, negative) spread in the first year after the issuance of an
XFABN penalizes investors that withdraw early, while the higher spread in the subsequent
years rewards investors that extended the maturity of the XFABN.
Panel B of Figure 3 illustrates the source of strategic complementarities among
investors in XFABN. On election date t, an investor—say Investor A—must decide
whether to withdraw. If no other XFABN from the same insurer are up for election
between t and t+m, Investor A’s decision to withdraw on election date t should not be
affected by her expectations of other investors’ withdrawal since no one can withdraw
betweent and t+m. However, if Investor A chose to extend her XFABN ont and if two
26 Online appendix C provides the first three pages of an XFABN prospectus specifying the election
dates and relevant conditions; the overall prospectus totals over 900 pages.
27 MetLife typically issues its FABS in Ireland and converts them into Reg S and 144A securities to
other XFABN from the same insurer are up for election betweent andt+m, investors in
these two other XFABN could potentially withdraw between tand t+m and would need
to be paid by the insurer before Investor A. This is depicted by the dotted purple lines.
Consistent with the model in Section 1, on any given election date, investors need
to think about potential withdrawals of other investors between any two election dates
because these investors’ decisions affect their place in the queue of payments. Investors
are likely to be concerned about withdrawals that may occur during the term of their
spinoff, but only if these withdrawals are scheduled to mature before their spinoff comes
due. Withdrawals that need to be paid after an investor’s spinoff matures do not increase
the insurer’s illiquidity risk at the date this investor’s spinoff is due. Consequently, the
amount that can potentially be withdrawn between any two election dates is a shifter
for investors’ expectations about other investors’ withdrawals. In normal times, investors
never exercise their put option and the maturity of XFABN is always extended, allowing
insurers to borrow relatively long term at shorter-term interest rates.
2.4
XFABN investors and runs on life insurers
Investors in XFABN are typically short-term institutional investors, such as money
market funds, that fund themselves by issuing on-demand liabilities to sophisticated
investors (Kacperczyk & Schnabl 2013, Schmidt et al. 2016). To avoid a run on
themselves, money funds are highly sensitive to any possible change in the performance of
their investments (Hanson, Scharfstein & Sunderam 2015). Importantly, these investors
do not withdraw only when they are troubled by issuer insolvency. Their extreme risk
aversion encourages them to exercise their put option even when facing a small amount
of uncertainty about when they might be repaid.28 Online appendix D provides a more
detailed description of XFABN investors’ sensitivity to illiquidity and an analysis of the
28 The possibility of untimely payoffs arises whenever a state regulator intervenes, or the company’s
credit rating is downgraded, or both. State insurance regulators can intervene prior to insolvency if the regulator deems the “company in hazardous financial condition based upon adverse findings in a financial analysis or examination, a market conduct examination, audits, actuarial opinions or analyses, cash flow and liquidity analyses; [...] and, any other finding determined by the commissioner to be hazardous to
the insurer’s policyholders, creditors, or general public.” (https://www.naic.org/cipr_newsletter_
size of the run risk posed by life insurers’ XFABN programs using life insurers cash flows.29
Lastly, the run on U.S. life insurers’ XFABN is not the first time that liquidity
problems arose when life insurers deviated from their traditional business, and it is
unlikely to be the last.30 The experience of XFABN issuers in 2007 and 2008 illustrates
a general principle that short-term institutional investors withdraw when facing even a
small risk of illiquidity. Their run on ABCP in August 2007 (Covitz et al. 2013, Schroth
et al. 2014) and the run on repo in September 2007 (Gorton & Metrick 2012) were early
signals of an impending financial crisis, with widespread illiquidity. Coincident with
those runs, the XFABN market collapsed. In the next section, we give an overview of our
database and describe the run on XFABN that began in the summer of 2007.
3
Data
The main source of data about XFABN is our database of all FABS issued by U.S. life
insurers covering the period beginning when FABS were first introduced in the mid-1990s.
To construct our dataset, we combined information from various market observers and
participants on FABS conduits and their issuance. We then collected data on contractual
terms, outstanding amounts, and ratings for each FABS issue to paint a complete picture
of the market for FABS at any point in time. Finally, we added data on individual
conduits and insurance companies, as well as aggregate information about the insurance
sector and the broader macroeconomy. A more detailed description of our FABS database
29 Foley-Fisher, Narajabad & Verani (2019) explain why comparing life insurers’ nontraditional
liabilities, including XFABN, to the overall size of their balance sheets is not appropriate to assess the size of the run risk.
30The issuance of XFABN is not even the first time thatfunding agreementswere used by life insurers
is provided in online appendix A.
Our data for XFABN were collected by hand from individual security prospectuses and
the Bloomberg corporate action records. We use these sources to construct the universe
of XFABN CUSIP identifiers and pair them with their spinoffs’ CUSIP identifiers. We
thereby obtain a complete panel of those XFABN outstanding, those still eligible for
extensions, and those whose holders elected to spinoff their holdings earlier than the
final maturity date. While our analysis does not require information about XFABN
investors’ holdings, we nevertheless checked that XFABN are not concentrated among
MMFs. On a case-by-case basis, we can observe individual MMF exposure to XFABN
conduits through their SEC Form N-Q and N-CSR filings. For example, in the third
quarter of 2007, Fidelity and JPMorgan held 3.7 percent and 0.5 percent, respectively, of
all outstanding XFABN.
In total, we record 51 XFABN issuances during the period of our analysis, from which
104 individual spinoffs were issued. The average XFABN issuance amount is $480 million,
while the average spinoff amount is $190 million, or roughly 40 percent of their parent
XFABN. About 63 percent of spinoffs mature in a year or less, consistent with an issuance
strategy that targets investment by MMFs.31
Additional summary statistics are separated into three groups in Table 1. In the first
group, we report statistics on the number of XFABN and spinoffs issued by each insurer,
as well as the total number of election dates for each life insurer in our sample period.32
The second group of statistics are calculated across XFABN, providing information on
the number of days between XFABN election dates, together with duration and issuance
amounts in U.S. dollars.33 Lastly, we group by election date a set of summary statistics
for withdrawals by investors, both as fractions and in dollar terms. This group includes
measures of regular FABS maturing that may affect investors’ withdrawal decisions, as
described below.
Figure 4 shows the daily time series of outstanding XFABN (green line) and
31 The median initial maturity at issuance for all XFABN in our sample is about two years, less than
one-quarter of the median duration at issue of the entire sample of FABN (roughly eight years).
32 During the run period we study, three insurers have only a single XFABN outstanding at some
point. These cases account for a small fraction of the total number of observations, and our results are not sensitive to their exclusion.
33 The statistics for the minimum and median number of days between election dates are 28 and
outstanding spinoffs (blue line) from the beginning of 2006 to the end of 2009. In
the twelve months before June 2007, the amount of XFABN outstanding rose by about
$8 billion to about $23 billion, or about 20 percent of total U.S. FABS outstanding. From
August 2007, institutional investors in XFABN began to exercise their put.
The figure contrasts the decline in the amount of XFABN outstanding (green line)
with the fastest possible withdrawal that investors could have made from August 1, 2007
(black line). The gap between these two series shows that, while investors did withdraw
swiftly, the run was not as immediate as it could have been.34 It is therefore reasonable to
expect investors could have formed nontrivial expectations about other investors’ future
actions. The total outstanding amount remained roughly flat throughout the run period
and declined in 2008 as the spinoffs created during the run matured. This second decline
might mislead an observer of insurers’ total liabilities to conclude that investors withdrew
later in 2008. In fact, the run occurred almost a year earlier. The question we address in
the next section is whether the run was amplified by strategic complementarities.
4
Testing strategic complementarities among investors
The sample comprises multiple life insurers indexed by k∈1, . . . , K, each with multiple
XFABN outstanding indexed by i ∈ 1, . . . , Nk. Let t = 1 and t = T denote the first
and last day of the sample period, respectively. For each insurer, we know the amount
Mk
it of each XFABN outstanding on every day t ∈ 1, . . . , T. We also know the set
of idiosyncratic, recurring election dates on which investors have the option to withdraw
their investment. These election dates are represented by the indicator variableLk
it, which
takes the value 1 on election dates and 0 otherwise. Lastly, we know the amount Wjtk
of each XFABN that is withdrawn by investors. The amount withdrawn is by definition
zero on any day that an XFABN does not have an election date i.e.,Lkit = 0. Our sample
is based on the election dates (Lk
it = 1) to study the decisions by investors to withdraw.
Each XFABN i issued by an insurer k contributes Tik = PT
t=1L
k
it observations to the
sample. Our sample contains PK
k=1
PNk i=1T
k
i = 1,119 XFABN-election date observations
from January 1, 2005, to December 31, 2010.
34 We observe two XFABN that were extended in full during the course of the run from June 2007 to
We use these data to construct three key variables to implement our empirical test
of strategic complementarities. First, the variable Dk
it is the withdrawal rate of XFABN
i from insurer k on day t, which is defined as the the amount Wk
it of XFABN i from
insurer k withdrawn on day t divided by the amount Mk
it of XFABN i outstanding at
the beginning of t—i.e., it is the amount outstanding before any withdrawal is made on
day t. It is clear that Dk
it =LkitDkit, as investors in XFABN i can only withdraw on the
predetermined and idiosyncratic election dates indicated by Lkit = 1.
Second, the variable Sk
it+mk i
measures the withdrawal rates for all XFABN from
insurer k other than i that are up for election between XFABN i’s consecutive election
dates t and t + mk
i, where mki is the idiosyncratic number of days between XFABN
i’s election dates.35 This variable is given by Sitk+mk
i =
P
j6=i
Pt+mki
τ=t+1wkjτDkjτ, where
wk
jτ = Mjτk/(
PNk
j=1Mjtk), for τ ∈ {t+ 1, t+ 2, ..., t+mki}. Thus, Sitk+mk i
is a weighted
average of the withdrawal rates for all the XFABN other than i that are up for election
betweeni’s election datest andt+mk
i. Clearly,Sitk+mk
i
= 0whenever an insurer has only
one XFABN or has many XFABN that all have the same set of election dates. Note that
on any given dayt,Dk
it andSitk+mk i
measure withdrawal rates corresponding to a mutually
exclusive set of XFABN from insurerk, respectively.
Third, the variable REk
it+mk i
is the upper bound for Sk
it+mk i
at date t, which is the
maximum amount that may be withdrawn strictly between i’s election dates t and t+
mk
i.36 This is given by REitk+mk
i
= Lk
it
P
j6=i
Pt+mki
τ=t+1wkjτ. As with Sitk+mk
i
, REk
it+mk i
does
not exist when all XFABN have the same election dates and REitk+mk
i is constant when
the election dates of all XFABN j 6=i always fall between the election dates of XFABN
i and there is no withdrawal.
Our test of strategic complementarities among investors seeks to establish that
investors’ expectations about Sitk+mk
i at date t have a positive causal effect on their
withdrawals at date t, measured by Dk
it. Although we do not observe investors’
expectation about future withdrawals, the unobserved EtSitk+mk
i and its realized value
Sk it+mk
i
should be positively correlated under the assumption that investors have rational
expectations.37 The main challenge to establishing a causal relationship is that observed
35 Note thatmk
i = arg maxτ{P
t+τ
x=t+1L
k
ix= 0s.t. Lkit= 1}is constant for each XFABN but may vary
across XFABN from the same insurer.
36 Equivalently, REk
it+mk
i
is the highest possible weighted average withdrawal rate on XFABN other
thanibetween XFABNi’s election datestandt+mki.
37 Our identification strategy does not assume that investors knowSk
it+mk
i
withdrawal rates Sk it+mk
i are a function of unobservable fundamentals that may also be
affecting withdrawals at date t, Dk
it. The potential for unobserved fundamentals leads to
the classic reflection problem (Manski 1993): Investors may be withdrawing as a response
to their expectations of future withdrawals, or to fundamentals developing during the run,
or to both. In the remainder of this section, we show how the structure of XFABN leads
to variation in REk
it+mk i
that can be exploited to identify strategic complementarities
among investors and overcome the reflection problem.
Importantly, our test for a positive causal relationship between Dk
it and expectations
about Sitk+mk
i is not a test of whether self-fulfilling expectations or fundamentals is the
driving force for the run on XFABN. Rather, we test for the existence of strategic
complementarities among investors, which leads to amplification of withdrawals. Hence,
this approach is fully consistent with the application of the global games framework to
understanding runs (Goldstein 2012) and the dynamic debt run models of He & Xiong
(2012) and Section 1. We will quantify the effect of strategic complementarities during
the run on XFABN in Section 5.
4.1
Correlation between present and future withdrawals
Our empirical analysis begins by establishing that there was a positive association between
investors’ decisions to withdraw and their expectations that holders of other XFABN
issued by the same insurer will withdraw in the future. This correlation forms the basis
of our argument that there might have been a self-fulfilling component to the run on U.S.
life insurers. When establishing the correlation, we do our best to control for obvious
economic fundamentals that might be driving the run.
Our main specification is summarized by equation (4):
Ditk =γk+γ1Sitk+mk
i +γ2Q
k it+x
k it
0
β+ξitk , (4)
t∈1, . . . , T ,i∈1, . . . , Nk , and k∈1, . . . , K .
Using the notation developed above for each XFABN (i), issuer (k), and day (t), our
baseline specification allows for a suite of issuer, time, and aggregate controls. Many
to assuming perfect foresight EtSitk+mk
i =Sk
it+mk
i
. We require only that EtSitk+mk
i
and Sk
it+mk
i
of these are summarized by the vector xk
it and described in further detail below. One
control that deserves further discussion is Qk
it, which is constructed as the fraction of all
outstanding fixed maturity FABS (including spinoffs created prior to election datet) that
are scheduled to mature on or before the maturity date t+nk
i ·mki, where nki ≥ 1 and
nk
i ·mki is the maturity of a spinoff created at date t. Intuitively, Qkit controls for the
amount of claims on the insurer that are already ahead of any spinoff created by decision
Dkit.38 The specification also allows for unobserved heterogeneity at the insurer level (γk)
to control for persistent insurer characteristics that could affect their vulnerability to runs
by institutional investors.39
Column 1 of Table 2 reports the results from estimating equation (4) by ordinary least