In Search of Habitat: A First Look at Insurers’

Government Bond Portfolios

Xuanjuan Chen, Zhenzhen Sun, Tong Yao, and Tong Yu∗

October 2012

∗Chen is from Shanghai University of Finance and Economics. Email: xuanjuanchen@gmail.com. Sunis from School of Business, Siena College. Email: zsun@siena.edu. Yao is from Henry B. Tippie Collegeof Business, University of Iowa. Email: tong-yao@uiowa.edu. Yu is from College of Business and Adminis-tration, University of Rhode Island. Email: tongyu@uri.edu. We appreciate the comments from LawrenceHe, Richard Phillips, Dave Simon, Joe Zou, David Bates, Ashish Tiwari, Canlin Li, Michael Gallmeyer,and seminar participants at the FMA meetings, the Financial Intermediation Research Society meetings,the Summer Institute of Finance conference, the American Risk and Insurance Association Meetings, theWestern Risk and Insurance Association meetings, Brock University, City University of Hong Kong, ChineseUniversity of Hong Kong, Shanghai University of Finance and Economics, Northern Illinois University, andUniversity of Iowa. All errors are our own.

In Search of Habitat: A First Look at Insurers’ Government Bond

Portfolios

Abstract

We perform microeconomic level analysis on the preferred-habitat behavior in the govern-

ment bond portfolios of insurance firms, a major group of bond market investors. Insurers’

aggregate government bond portfolios have stable exposure to interest rate factors and lim-

ited sensitivity to term structure changes. Individual insurers’ portfolio risk exposures are

also stable yet widely dispersed, suggesting large heterogeneity in the habitat preference

across insurers. To understand such patterns we consider two forms of habitat behavior

that are nested in a rational dynamic portfolio model—a liability habitat driven by the need

to immunize the interest rate risk of liability, and a horizon habitat due to the preference

for holding securities with riskfree returns at the investment horizon. Consistent with the

liability habitat effect, we find that insurers’ portfolio risk exposures are strongly related to

their liability characteristics, including the level and maturity of claim liabilities. Liability

concerns also dampen portfolio response to term structure changes. The evidence on the

horizon habitat is somewhat mixed; and if any, it suggests that insurers’ investment horizons

tend to be relatively short. The findings highlight the importance of a liability channel for

inelastic demand in the government bond market.

1 Introduction

The preferred habitat hypothesis is one of the earliest theories on the term structure of

interest rates. Its origin can be traced to Modigliani and Sutch (1966) in an analysis of

the early-1960 Treasury endeavor dubbed “Operation Twist.” Under this hypothesis, the

rigidity of investor demand for bonds at specific maturities affects the shape of the interest

rate curve, hence there is room for the government to fine-tune the term structure by changing

the net supply of bonds across maturities. Interest in this hypothesis surged around Federal

Reserve’s second round (QE2) of “quantitative easing” program in 2010. A growing number

of studies have used the concept of preferred habitat to understand the impact of demand

and supply shifts in the bond market and to evaluate monetary policies.1

Despite the relevance of preferred habitat to macroeconomic analysis and monetary poli-

cies, so far there is little evidence on issues that speak to the microeconomic foundation of

the theory. These issues include, for example, what types of bond investors exhibit preferred

habitat? How important are the habitat components in their portfolios? And what causes

their inelastic demand for specific maturities? In part due to data constraints, we have not

had empirical answers to such questions.

Answers to these questions are also important for assessing a perceived drawback of the

hypothesis—i.e., it relies on somewhat arbitrary maturity preferences of investors, implying

arbitrage opportunities in a market that is well regarded as efficient. Recent theoretical ad-

vances in the literature have provided a more rational view of preferred habitat. Vayanos and

1For example, using the UK pension reform of 2004 and the US Treasury’s buyback program of 2000-2002,Greenwood and Vayanos (2010a) demonstrate that shifts to clientele demand and bond supply affect termstructure movements. Greenwood, Hanson, and Stein (2010) present evidence that the maturity structure ofcorporate debt varies in a way that complements the maturity structure change of government bonds becausefirms behave as macro liquidity providers. Hamilton and Wu (2012) show that when short-term interestrate is at the zero lower bound, monetary policy can affect the term structure by changing the maturityof government bonds held by investors. Krishnamurthy and Vissing-Jorgensen (2010, 2011) find that theFederal Reserve’s purchase of long-term Treasuries and other long-term bonds in the 2008-2011 period havesignificant impact on the term structure as well as on yields of mortgage-backed securities. Swanson (2011)uses high frequency data to reevaluate the effectiveness of “Operation Twist,” the event that motivated theoriginal analysis of Modigliani and Sutch (1966). Li and Wei (2012) incorporate the supply factors intoan arbitrage-free term structure model and estimate that the Federal Reserve quantitative easing programshave a combined impact of 100 basis points on the ten-year Treasury yield.

1

Vila (2009) show that the interaction between arbitrageurs and habitat investors can result

in an arbitrage-free term structure. Studies by as Campbell and Viceira (2001), Watcher

(2003), Liu (2007), and Detemple and Rindisbacher (2010), show that a risk-averse investor’s

optimal portfolio has a rational habitat component related to her investment horizon. Fi-

nally, there have been anecdotal observations that habitat-like behavior may arise when

institutional investors such as insurers and pension funds hold long-maturity securities to

hedge to risk of their long-term liabilities. So far, these theoretical predictions and anecdotal

conjectures have yet to meet the data.

This study provides empirical analysis to address several key microeconomic-level ques-

tions regarding the preferred habitat hypothesis. Our analysis takes advantage of the com-

prehensive portfolio data for an important group of institutional investors in the government

bond market—insurance firms. Besides information about their insurance operations, in-

surers are required by regulation to report details of their investment portfolios each year.

Such data are rarely available from any other type of major players, such as pension funds,

in the bond market. The unique data enable us to gauge quantitatively the habitat-like

preference in investors’ government bond portfolios, and evaluate the factors driving their

habitat behavior.

The habitat preference of sophisticated institutional investors is unlikely driven by pure

cognitive biases. Therefore, when looking for likely causes of insurers’ habitat behavior our

attention is on the rational ones—i.e., the need to hedge interest rate risk of liabilities and

the incentive to hold securities deemed risk-free at given investment horizons. To provide

a conceptual framework for empirical analysis, we introduce a simple model of dynamic

portfolio choice that nests these two sources of habitat. In the model, preferred habitat

shows up as two hedging components of an optimal portfolio. The first component immunizes

the interest rate risk of liabilities, while the second component hedges the interest risk with

respect to the investment horizon. For the namesake reason they are referred to as the

liability habitat and horizon habitat, respectively. These components represent inelastic

bond demand in that they are unresponsive to the interest rate conditions. The liability

2

habitat depends on the level and maturity structure of liabilities, but is independent of the

degree of risk aversion (as long as investors are risk averse). By contrast, the importance of

horizon habitat in a portfolio increases with risk aversion.2

Viewing habitat as interest rate risk hedging components of a portfolio leads to a “dif-

ferent” perspective on how habitat should be measured in the portfolio data. While the

general form of habitat is the inelasticity, or stability, of bond holdings at certain maturities,

from the interest rate risk hedging perspective maturity does not perfectly characterizes the

“location” of habitat. Because interest rates are correlated across maturities, the interest

rate risk at any given maturity can be hedged using bonds at other maturities.3 In this

context, a more robust view of habitat is the stability of portfolio exposure to a few rela-

tively orthogonal risk factors in the term structure. Accordingly, in empirical analysis we

look at insurers’ portfolio choices at various maturities as well as three portfolio duration

measures that intuitively capture insurers’ exposure to the major term structure factors, i.e.,

the level, slope, and curvature. These portfolio duration measures are developed under the

Nelson-Siegel term structure model (e.g., Diebold and Li 2006; Diebold, Ji, and Li 2006),

with the level duration equivalent to the Macaulay duration.

We perform analysis on the government bond portfolios of 1378 property and casualty

(“PC”) insurers and 520 life insurers for the period from 1998 to 2009. The aggregate

portfolios of PC insurers and life insurers display tale-telling signs of habitat. First, consistent

with their respective liability characteristics—short-dated property and casualty claims and

long-dated life policies—PC insurers heavily load on short-maturity bonds while life insurers

spread their holdings across maturities. Second, the aggregate portfolio durations fluctuate

2Interestingly, between the two forms of habitat, the liability habitat appears more prominent—an institu-tion must first allocate part of the portfolio to fully immunize its liability, while the horizon habitat appearsonly in the remaining part of the portfolio. However, from a macroeconomic point of view, the liabilityhabitat of institutional investors may be related to the horizon habitat of individuals, because institutionalinvestors’ liabilities could be traced to individuals’ consumption and investment decisions. For example,the maturities of pension liabilities are related to employees’ retirement horizons, and the maturities of lifeinsurance claims are determined by policyholders’ life expectancies.

3Exogenous supply shocks could also affect insurers’ bond holdings at specific maturities. For example,from September 2001 to January 2006 the U.S. Treasury did not issue any new 30-year bonds. This suggestsa further reason for caution when interpreting portfolio responses at the individual maturity level.

3

in a tight range around the means. For example, the interest rate level duration of PC (life)

insurers’ has a standard deviation of 0.49 (0.93) around a mean of 5.56 (10.04) years. The

slope and curvature durations are in even more confined ranges. Third, despite the large

market swings in the 12-year sample period, insurers’ aggregate government bond portfolios

do not appear to be very sensitive to the term structure changes.

Interestingly, the relative stable portfolio characteristics at the aggregate level are the

result of largely heterogeneous portfolio choices by individual insurers. Across PC (life) in-

surers, the level duration varies with a 10th-90th percentile range of 2.38 to 7.70 (2.98 to

11.41) years and a standard deviation of 2.21 (3.53) years. The slope and curvature durations

(i.e., portfolio duration with respect to the slope and curvature factors), as well as portfo-

lio weights around various maturities, also vary widely in the cross-section. Furthermore,

individual insurers’ portfolio characteristics are quite stable over time despite the large cross-

sectional dispersion. Insurers with high portfolio durations in a given year continue to have

high durations for at least five subsequent years. These patterns suggest a possibility that

individual insurers have highly heterogenous but also highly persistent habitat preferences.

What may drive the cross-sectional differences in insurers’ habitat preferences? We follow

the implication of the model to investigate the liability and horizon effects. There is clear ev-

idence for the liability habitat. Across insurers, portfolio level durations are strongly related

to the joint effect of the maturity of claim liabilities and the level of liability (measured by

the liability ratio LTV, the claims-related liability divided by total invested assets). Analy-

sis on the portfolio weights reveals a similar pattern: insurers with higher claim maturities

and higher levels of liability put higher weights on long maturity bonds. This evidence is

consistent with the empirical finding that financial institutions actively manage interest rate

risk exposures (e.g., Schrand and Unal, 1998; Erhemjamts and Phillips, 2012).

There is also some evidence for the horizon habitat effect. The importance of the horizon

habitat in a portfolio increases with risk aversion. We use five firm characteristics to proxy

for risk aversion, all based on the difficulty of obtaining external financing—an insurer’s

corporate form (stock vs. mutual), affiliation with a parent group or holding company, div-

4

idend payment status, firm age, and capital adequacy.4 Firms belonging to parent groups

or holding companies, no-dividend firms, younger firms, and firms with lower capital ad-

equacy ratios tend to have lower interest rate level durations and put higher weights on

short maturities. Based on our model, such a negative relation between risk aversion and

portfolio duration can be construed as that insurers have an investment horizon shorter than

the duration of the “non-habitat” part of the portfolio, i.e., the component responding to

market conditions. However, mutual insurers tend to have higher portfolio durations and

hold more long-term bonds than stock insurers, inconsistent with the hypothesized effect of

risk aversion.

Finally, we examine individual insurers’ portfolio reaction to term structure changes. In

order to get a sense on insurers’ interest rate strategies, we look at the response coefficients

obtained from regressing individual insurers’ portfolio durations and weights onto the factors.

Across insurers, the portfolio response coefficients have small means and a very wide range

of variation, indicating that if insurers have very different reactions to the common term

structure they face. Without taking a view on how insurers should optimally respond to term

structure changes, we examine whether there is habitat behavior in such responses by looking

at the absolute response coefficients. We find a clear liability habitat effect—insurers with

higher liability ratios LTV tend to have lower absolute response coefficients at the portfolio

duration level, significantly so for PC insurers’ duration response to the curvature factor and

for life insurers’ duration response to all three factors. At the portfolio weight level, we find

that a higher liability ratio tends to reduce insurers’ absolute response coefficients at short

maturities. There is also some evidence that risk aversion reduces insurers’ absolute response

coefficients at both the portfolio duration level and the portfolio weight level. However, such

evidence is not consistent across all the risk aversion proxies we examine.

Overall, the findings of this study confirm the existence of habitat-like preference in

4These risk aversion proxies follow the corporate risk management literature, which suggests that thefinancing constraint or convex external financing cost is an important determinant of firms’ risk aversion ininvestment and hedging decisions. Mutual insurers, unaffiliated insurers, firms without dividend payments,young firms, and firms with inadequate capital have higher or more convex external financing costs thanstock insurers, affiliated insurers, dividend-paying firms, seasoned firms, and firms with adequate capital.

5

insurers’ government bond portfolios. Thus, they offer microeconomic-level support to a key

assumption of the preferred habitat theory. Between the two sources of habitat investigated,

there is relatively strong evidence for a liability channel, while the evidence for the horizon

channel is somewhat mixed.5 Our analysis on the demand side of government bonds sets it

apart from, but complements, a growing macroeconomic and macrofinance literature that

examines the supply side issues in this market, such as Greenwood and Vayanos (2010b),

Krishnamurthy and Vissing-Jorgensen (2011), Hamilton and Wu (2012), and Li and Wei

(2012).

Our paper is further related to several recent studies that examine the investment strate-

gies and performance of investors in the government bond market. Ferson, Henry, and Kisgen

(2006) evaluate the performance of government-bond mutual funds using a stochastic dis-

count factor approach that addresses the interim trading bias in performance measurement.

Huang and Wang (2010) use the holdings data to evaluate market timing strategies of gov-

ernment bond mutual funds, while Moneta (2012) use bond holdings to analyze the security

selection activities of fixed income funds including government bond funds. A challenge in

this literature is that interest rate risk factors affect portfolio values and risks in a highly

nonlinear way, making it difficult to apply the linear factor approach popularly used for an-

alyzing equity funds. Further, fixed income strategies have more complicated risk dynamics,

reducing the power of the traditional regression approach based on fund returns. We address

these difficulties by combining the detailed portfolio data with the Nelson-Siegel term struc-

ture framework. The portfolio data enable us to take accurate snapshots of portfolio risk

exposure, while the close-form Nelson-Siegel portfolio duration measures conveniently char-

acterize portfolio risk in terms of major term structure factors and deal with nonlinearities

in a way friendly to empirical analysis.

The remaining of the paper is organized as follows. Section 2 introduces the dynamic

portfolio choice model that nests the liability habitat and horizon habitat. Section 3 discusses

5However, our analysis is not a direct test of the horizon habitat models proposed in the literature, whichare cast under the setting of individuals’ investment and consumption decisions. As noted earlier, institutions’liabilities are related to individuals’ investment and consumption decisions. Thus at the macroeconomic level,institutions’ liability habitat can be potentially reconciled with individuals’ horizon habitat.

6

the data and empirical methodology. Section 4 provides the empirical results. Section 5

concludes.

2 A Tale of Two Habitats: The Model

To provide a conceptual framework for empirical analysis, we first introduce a dynamic

portfolio model that nests two forms of preferred habitat. The horizon habitat arises because

of the preference of a risk-averse investor for securities offering safe returns at her investment

horizon. The liability habitat is due to the need to hedge interest rate risk of her liability.

2.1 Horizon Habitat

We start with a model of only the horizon habitat. Consider an investor with initial wealth

W0 at time 0, whose objective is to maximize expected utility from wealth at time H. There

is no intermediate consumption. At any given time t, there are always M bonds available for

trading. These bonds do not have default risk, and are priced according to a general term

structure of stochastic interest rates. Let Rmt be the one-period gross return from time t-1

to t for bond m. For convenience let the first bond be the one-period riskfree bond. We

assume that the remaining M-1 bonds are non-redundant in the sense that the M-1 by M-1

covariance matrix for the return Rmt (m=2,..., M) has full rank. After one bond matures it

can be replaced by any other non-redundant bond. Market completeness is not required.

Let ωmt be the portfolio weight on bond m at time t. The investor has a power utility

function with a relative risk aversion coefficient of γ. Thus, the optimization problem at

time 0 is:

MaxE0(W 1−γH1− γ

) (1)

subject to the budget constraint:

Wt+1 = WtRpt+1

where Rpt+1 =∑M

m=1 ωmtRmt+1 is the portfolio return. Iterating over the budget constraint

we have WH = WtΠHτ=t+1Rpτ . The value function of the above dynamic programming prob-

7

lem, or the indirect utility function Jt, can be expressed as:

Jt = Et(Jt+1) = W1−γt Et

(ΠHτ=t+1Rpτ

)1−γ1− γ

(2)

To illustrate the horizon habitat we provide an expression for the optimal portfolio weights

based on a “change of numéraire” procedure in the spirit of Detemple and Rindisbacher

(2010) and log-linearization following Campbell and Viceira (1999) and Campbell, Chan,

and Viceira (2003). Appendix A.1 shows that the optimal portfolio weight has the following

form:

ωt =1

γΩ−1(Etrt+1 − rft+1ι+

1

2V) +

γ − 1γ

Ω−1Cov(rt+1, rht+1) +1− γγ

Ω−1Cov(rt+1, xt+1)

(3)

where ωt is a vector of optimal weights for the M-1 risky bonds. Et(rt+1), V and Ω are the

expected return vector, variance vector, and covariance matrix of their log returns. rft+1 is

the log risk free rate. ι is a unit vector. rht+1 is the return of a zero-coupon bond maturing

at time H. We do not require this bond to be among the M bonds available for trading, as

long as its prices are observed or can be synthetically constructed from the prices of other

bonds. rpτ is the log portfolio return and xt+1 =∑H

τ=t+2(rpτ − rfτ ) summarizes the future

portfolio “risk premium” – log portfolio return in excess of the log return to the maturity-H

bond.

The optimal portfolio weight in Equation (3) consists of three terms. The first term is

a static mean-variance component. The second and third terms are hedging components.

The second term hedges against the interest rate risk of the maturity-H bond, and the third

term hedges against future changes in “risk premium.” The horizon habitat is represented

by the second term, which is a demand for securities that can hedge the interest rate risk at

maturity H, i.e., the investment horizon.

To gain further intuition, consider the relation of the three terms with the risk aversion

coefficient γ. Under log utility, i.e., γ = 1, only the first component remains and the two

hedging components disappears. This results in the well-known “myopic portfolio”. On the

other hand, as γ →∞, the myopic component converges to zero, and (γ−1)/γ and (1−γ)/γ

8

in the two hedging components converge to 1 and -1 respectively. On appearance both

hedging components do not disappear. However, in the risk premium hedging component

xt+1 represents future portfolio risk premiums. If the entire portfolio converges to a single

position in the H-maturity bond, xt+1 converges to zero, and so does the entire risk premium

hedging component.6 Therefore, as risk aversion increases, the importance of the interest

rate hedging component increases, and the portfolio weight on the H-maturity bond reaches

1 in the limit.

The horizon habitat effect has been derived previously in various portfolio problems.

Watcher (2003) provides a proof that in complete market and with infinite risk aversion,

the optimal portfolio is a zero-coupon bond maturing at the investment horizon. Liu (2007)

obtains a similar result for incomplete market under quadratic term structure. Based on log-

linearization, Campbell and Viceira (2001) shows that with infinite horizon and intermediate

consumptions, the optimal portfolio converges to a console bond as risk aversion increases.

Lioui and Poncet (2001) and Detemple and Rindisbacher (2010), using the martingale ap-

proach, show that the horizon habitat originates from the interest rate hedging component

of the optimal portfolio. We show that the “change of numéraire” procedure can be applied

in the dynamic programming approach to deliver the same intuition.

2.2 Liability Habitat

We now include the liability habitat. Suppose the investor faces a liability of amount L

maturing on time K, with 1 < K ≤ H. Without loss of generality, we assume the existence

of a buy-and-hold portfolio at time 0 based on the M bonds available, which delivers a riskless

payoff of $1 at time K and zero at any other time. Let ωLmt denote the time-t weight of this

portfolio on bond m. If out of the M zero-coupon bonds there is one with maturity K, then

a feasible portfolio is to put a 100% weight on this bond and zero weights on all other bonds.

Otherwise, it is suffice to assume the existence of a portfolio mimicking the payoff of this

6The convergence of xt+1 to zero can be verified by solving ωt and xt backward, starting from time H-1.With infinite risk aversion, the utility loss due to any risk exposure dominates the utility gain from anyexpected return. Thus the optimal investment has to make the terminal wealth WH riskless.

9

bond.

The investor maximizes the same expected utility function as in (1), with the modified

budget constraint:

MaxE0(W 1−γH1− γ

) (4)

subject to the following budget constraint. For t 6= K,

Wt+1 = WtRpt+1

and for t=K,

Wt+1 = (Wt − L)Rpt+1

Appendix A.2 shows that the optimal portfolio for this problem has two components.

The first component uses the completely immunizes the fluctuation of the present value of

the liability, and the second component is the optimal portfolio without liability, i.e., the

solution to (1).

The details of the optimal portfolio are as follows. Let Bt be the time-t value of the

buy-and-hold portfolio that delivers a safe $1 payoff at time K. At time 0, start holding

this portfolio at the amount of LB0. This position is held without rebalancing until time

K, at which point the portfolio is liquidated to completely pay off the liability. Thus at

any time before K, the value of this position is LBt. This is the immunization component

of the portfolio. The remaining value of the time-t wealth, Wt − LBt, is allocated to bond

m according to the weight ωmt, which is the optimal weight in the problem of (1). Let

αt = LBt/Wt. The weight for bond m in the entire portfolio is thus ω∗mt = αtω

Lmt+(1−αt)ωmt.

After time K, αt = 0 and the portfolio weight goes back to the optimal weight for the problem

(1), i.e., ω∗mt = ωmt. Appendix A.2 provides further discussion on three extensions of the

basic model here.

The liability habitat is represented by the portfolio component αtωLmt. The purpose of

this component is to completely neutralize the interest rate risk of the liability. This form

of hedging is known as complete immunization or cash flow matching in the asset-liability

management practice. In the static optimization setting, there is a question whether complete

10

immunization or some less stringent form of hedging is better. As it turns out, complete

immunization is optimal in the dynamic portfolio problem considered here. Intuitively, this

is because immunization is costless measured by the marginal utility of the investor, which is

in turn due to that in the optimal portfolio without liability, the investor is already indifferent

between holding the maturity-K bond (or the mimicking portfolio ωLt ) and holding any other

bonds.

It is further interesting to note that unlike that of the horizon habitat, the magnitude of

the liability habitat is independent of the level of risk aversion as long as the investor is risk

averse. Indeed, one can verify that the liability habitat exists even under the log utility, i.e.,

when γ = 1 (see further discussion in Appendix A.2). Therefore, even though the liability

habitat can be viewed as ultimately resulting from individuals’ investment horizon effect, its

impact on the bond demand, hence the impact on the term structure of interest rates, may

be quite different from that of the horizon habitat.

2.3 Model Implications

Combining the form of optimal portfolio with liability with the log-linear solution (3) for

the portfolio weights without liability, we have the following log-linear representation of the

optimal portfolio:

ω∗t = αtωLt + (1− αt)

γ − 1γ

ωHt + (1− αt)(1

γωO1t +

1− γγ

ωO2t ) (5)

where ωLt is the vector of ωLmt. ω

Ht = Ω

−1Cov(rt+1, rht+1). ωO1t = Ω

−1(Etrt+1− rft+1ι+ 12V)

and ωO2t = Ω−1Cov(rt+1, xt+1). The first component in the above expression is the liability

habitat, the second one is the horizon habitat, and the third one is the “opportunistic”

component, i.e., the myopic component plus the risk premium hedging component. An

immediate issue to note is that the liability habitat component ωLt matches the risk of

liability but does not necessarily have the same maturity as the liability. The same can be

said about the horizon habitat component ωHt . Thus, these two forms of habitat are better

characterized by their effect on the interest rate risk of the portfolio, rather than by their

effect on the maturities of portfolio holdings.

11

An important feature of the fixed income market is that a few systematic term structure

factors affect the term structure of interest rates—for example, the well-known level, slope,

and curvature factors (e.g., Litterman and Scheinkman 1991). In a general case, a small

number of factors do not span the interest rate space, but a portfolio’s interest rate risk

can be summarized by the sensitivities of portfolio value to these factors. In addition,

these factors serve as state variables for the Markov processes of interest rates. That is, they

summarize the time varying investment opportunities and thus affect the portfolio decisions.7

Consider an example where the interest rate level risk is measured by the Macaulay

duration (which will be generalized for other risk factors). The duration of the optimal

portfolio described by (5) have the following duration expression:

Dt = αtDLt + (1− α)

γ − 1γ

DHt + (1− α)(

1

γDO1t +

1− γγ

DO2t

)(6)

where DLt , DHt , D

O1t and D

O2t are the durations of the sub-portfolios with weights ω

Lt , ω

Ht ,

ωO1t , and ωO2t respectively. We can generalize this duration expression for other interest

rate factors and draw the following empirically testable implications. A special case is that

when αt is 1, Dt = DLt . This is the so-called immunization under which institutions match

durations of their bond portfolios and liabilities.

More generally, the first implication is that the portfolio duration is positively related to

the joint effect of liability duration and level of liability, i.e., αtDLt .

The second empirical implication is on the horizon habitat effect, which depends on the

risk aversion γ as well as DHt . In the context of the Macaulay duration, DHt can be directly

interpreted as the investment horizon. However, the investment horizon is not directly

observed.8 Our empirical strategy is to identify proxies for insurers’ risk aversion, and detect

7A caveat is that yield curve factors may not fully characterize time varying investment opportunities.For example, there might exist unspanned stochastic volatility (e.g., Collin-Dufresne and Goldstein 2002).Cochrane and Piazzesi (2005) identify a forward-rate factor that predicts bond returns but is not spannedby conventional yield curve factors.

8The concept of investment horizon of a financial institution is somewhat complicated. Financial institu-tions such as insurers are expected to survive a long time, therefore they potentially have long investmenthorizons. However, corporate executives and investment managers at these institutions may have muchshorter expected tenures and their performance may be evaluated at even shorter periods. Thus, insurers’effective investment horizons depend on the importance of the agency effect.

12

the presence of the horizon habitat via the relation between observed portfolio duration and

risk aversion. As suggested by Equation (6), the relation between risk aversion γ and the

portfolio duration Dt takes the following form:

Dt = At + (1− αt)γ − 1γ

(DHt −DOt ) (7)

where At is the duration component unrelated to γ, and DOt = D

O1t + D

O2t is the duration

of the opportunistic component. Thus the second implication is as follows. After controlling

for liability (1-αt), if the investment horizon duration DHt is higher than the opportunistic

component duration DOt , then portfolio duration Dt increases with risk aversion; however if

DHt is lower than DOt , Dt decreases with risk aversion.

Finally, consider the role of interest rate risk factors as state variables. There are reasons

to assume that both the liability and investment horizon are relatively exogenous to the

interest rate changes. By contrast, investors may actively adjust the portfolio weights in the

opportunistic component in anticipation of changing risk and expected return as suggested

by the state variables. Thus, in terms of portfolio durations, DLt and DHt are relatively

exogenous, while DO1t and DO2t may actively respond to factor changes.

9 Then, based on (6)

we can further characterize the sensitivity of portfolio duration to a factor ft as:

∂Dt∂ft

= (1− αt)(

1

γ

∂DOt∂ft

− ∂DO2t

∂ft

)(8)

Thus the direction of the impact of αt and γ on∂Dt∂ft

depends on the signs of∂DOt∂ft

and∂DO2t∂ft

.

Without taking a view on what should be their correct signs, we can look at the sensitivity

of Dt as the absolute value of∂Dt∂ft

. It is clear from Equation (8) that αt negatively affects the

absolute value of ∂Dt∂ft

. The effect of γ requires a further note. The term DO2t is the duration

of ωO2t = Ω−1Cov(rt+1, xt+1), which, as explained earlier, converges to zero as γ increases.

Thus∂DO2t∂ft

decreases in absolute value with γ. Hence the third implication: liability (αt) and

risk aversion (γ) reduce the sensitivity of portfolio duration to interest rate factor changes.

9Even when the liability and investment horizon are completely exogenous to factors, factors may stillhave a passive effect on the durations of the liability habitat and horizon habitat, because the durationmeasures are functions of interest rates. We control for such a passive effect in empirical analysis.

13

3 Data and Methodology

3.1 Data and Sample

We use two main datasets on insurance firms. The first is the Schedule D data from the

National Association of Insurance Commissioners (NAIC). NAIC compiles annual regulatory

filings by insurers on their securities holdings and trades in the form known as the “Schedule

D” . Reported securities include stocks, preferred stocks, and bonds. For bonds, the Schedule

D data have detailed information on bond holding by each insurer at the end of each year and

record of each bond transaction occurred during that year (the source of the Mergent FISD

bond transaction data). In addition, the Schedule D data provide basic bond information

such as issuer type, maturity, coupon, yield, and price.

For the government bond sample we start with all straight U.S. treasury bonds and

agency bonds reported in the Schedule D data. These government bonds are further classi-

fied into two categories: 1) Issuer Obligations, which are direct obligations of the government

and government agencies that are backed by the full faith and credit of the United States

government, and 2) Single Class Mortgage-Backed/Asset-Backed Securities, which are pass-

through certificates and other securitized loans issued by the United States government that

are exempt pursuant to the determination of the Valuation of Securities Task Force. We only

keep the issuer obligation type and exclude the mortgage/asset-based securities. Further, we

exclude bonds with special characteristics such as bonds with credit enhancements, convert-

ible bonds, Yankee, Canadian, foreign currency bonds, globally-offered bonds, redeemable,

putable, callable, perpetual, or exchangeable bonds, and preferred securities. We also re-

quire the bonds to have non-missing coupon rate and positive face value. A few bonds with

apparently incorrect CUSIPs are also removed.

Insurers hold other securities and instruments that are subject to interest rate risk, for

example, corporate bonds, municipal bonds, mortgage backed securities, and interest rate

derivatives. Quite possibly, insurers use these alternative instruments in addition to govern-

ment bonds to manage their interest rate risk exposure. We focus on government bonds for

14

the following reasons. First, our primary objective is to examine the inelastic demand for

government bonds per se, rather than insurers’ complete interest rate risk exposure. The

inelasticity of government bond demand is an important issue from a monetary policy per-

spective. The Federal Reserve’s second Large Asset Purchase Program (i.e., QE2) solely

focuses on buying the long-term government bonds. The absence of inelasticity in long-term

government bond demand would call into question the rationale of this endeavor. Second,

other assets such as corporate bonds and mortgage-backed securities carry additional risk

other than the interest rate risk (e.g., default risk and prepayment risk). To quantify such ad-

ditional risks and determine their correlation with the interest rate will necessarily introduce

a substantial amount of further complexity.

The second dataset used in this study is INFOPRO, also from NAIC. INFOPRO pro-

vides insurers’ demographic information and financial statements following the insurance

regulatory requirements rather than in the Compustat format. From the INFOPRO data we

select all property and casualty (PC) insurance companies and life insurance companies. We

exclude pure reinsurance firms (by requiring firms to have non-zero direct underwriting pre-

miums) and a few small insurers with total assets below $5 million. The INFOPRO dataset

has financial information on insurers as well as insurance groups and holding companies that

many insurers belong to. We exclude the group-level and holding company-level firms.

The sample period is from 1998 to 2009. After following the data processing steps

described above, we additionally require an insurer to have at least 5 years during which it

holds a minimum of 5 straight government bonds. This ensures that insurers in our sample

have sufficiently long time series data for analysis. In our final sample, there are 1,378 unique

PC insurers and 520 unique life insurers.

Table 1 reports the summary statistics on asset characteristics of insurers in the sample.

Panel A is for PC insurers. The number of PC insurers in each year varies between 870 (in

1998) and 1,044 (in 2006). Their average total assets increase over time, from $800 million

in 1998 to $1,222 million in 2009. The table further reports the following asset items as

fractions of total assets: i) invested assets, ii) value of bond and stock holdings, iii) value of

15

bond holding, and iv) value of government bond holdings. For P&C insurers, on average,

invested assets account for 85% of total assets, with the majority investments in stocks and

bonds (74% of total assets), and prominently in bonds (63% of the total assets). Notably,

22% of total assets are government bond holdings.

Panel B is for life insurers. Life insurers are much larger, with average total assets of

$7,519 million (vs. $971 million for PC firms). Relative to PC insurers, life insurers have

more invested assets (93% of total assets) and invest more in bonds (71% of total assets).

Interestingly, the fraction of government bond holdings is lower for life insurers than for PC

insurers (16% vs 22%).

3.2 Liability and Risk Aversion Measures

To investigate the liability habitat and horizon habitat effects, we construct several variables

for insurers’ liabilities and risk aversion properties.

The first is a liability ratio LTV, the ratio of liability to total investment value. LTV

serves as a proxy for αt in the optimal portfolio weight expression (5). Insurers’ financial

liabilities typically are small, and their liabilities are mainly operating liabilities due to insur-

ance claims. To construct LTV, we identify claim-related liabilities from insurers’ financial

statements titled “Liabilities, Surplus, and Other Funds,” from the INFOPRO data. For PC

insurers, claim liabilities are the sum of 1) Losses, 2) Reinsurance payable on paid losses and

loss adjustment expenses, and 3) Loss adjustment expenses. For life insurers, claim liabilities

are the sum of 1) Aggregate reserve for life contracts, 2) Aggregate reserve for accident and

healthcare contracts, 3) Liabilities for deposit-type contracts, and 4) Contract claims. The

denominator of LTV, i.e., total investment value, is the data item “Invested Assets” from

the financial statements.

Second, we construct proxies for the maturity of insurers’ claim liabilities. For PC in-

surers, losses already claimed but unpaid represent the main type of long-term operating

liabilities. Loss payments are made throughout a few years after initial claims. The NAIC

Schedule P dataset provides annually cumulative loss payments for losses incurred in prior

16

10 year in each calendar year, based on which we project a 10-year period loss payment

cash flows for claims in a given year, following a “chain ladder” method popular in the in-

surance industry. We consider three measures of claim durations corresponding to the the

level, negative slope, and curvature factors of the yield curve. The calculation of the three

ClaimDur measures is consistent with the three NS durations for insurers’ bond portfolios.

The data are from insurers’ Schedule P filing with NAIC. Further details of this procedure

are available upon request.

Life insurers are not required to disclose claim information similar to that provided in

the Schedule P filings by PC insurers. However, life insurers often engage in other types

of insurance underwriting, such as accidental and healthcare insurance. These alternative

types of insurance tend to have much shorter claim durations. Based on this observation,

we construct a variable PctLife as a proxy for life insurers’ claim maturity. PctLife is the

percentage of life insurance premiums in total premiums collected.

Next, we introduce proxies for risk aversion. So far there is no well-established way of

measuring the risk preference of financial intermediaries such as insurers. Our risk aversion

proxies are motivated by the economic theory of corporate risk management. As pointed

out by Froot, Scharfstein, and Stein (1993) and several other studies, a prominent reason for

firms to engage in financial hedging and risk management is the existence of convex external

financing cost or financing constraints. Following this literature, firms with higher external

financing costs will also act more averse to investment portfolio risk. Based on this financing

constraint effect, we construct the following five proxies for an insurer’s risk aversion:10

• MUTUAL: a dummy variable with a value of one for a mutual firm and zero for a

stock company. Stock companies tend to have easier access to the capital market than

mutual insurers (Cummins and Doherty, 2002; Harrington and Niehaus, 2002).

• INDEP: a dummy variable with a value of one for an insurer not affiliated with any10A popular financing constraint proxy not used in this study is firm size (see Rauh, 2006; Almeida and

Campello, 2007; Li and Zhang 2010). Firm size has a highly positive correlation with LTV (0.44) in oursample. Thus it is unclear whether the relation between firm size and insurers’ portfolio durations is due tofinancing constraint or liability.

17

parent group or holding company, and zero otherwise. Parent groups and holding

companies can reduce external financing costs of subsidiaries and provide additional

risk sharing across subsidiaries. Independent insurers do not enjoy such benefit (see,

e.g., Lamont, 1997; Stein, 1997; Kahn and Winton, 2004).

• NODIV: a dummy variable with a value of one for an insurer not paying any dividend

(including dividends either to policyholders or to stockholders), and zero otherwise.

Dividend paying status is a popular measure of financing constrant (see, e.g., Almeida

and Campello, 2007; Li and Zhang, 2010).

• YOUNG: the negative of the logarithm of firm age. Following prior studies, e.g.,

Fee, Hardlock, and Pierce (2009) and Rauh (2006), younger firms typically are more

financially constrained.

• LOWCAP: the negative of the logarithm of an insurer’s capital adequacy ratio, which

is the total adjusted capital relative to the authorized control capital level (from the

INFOPRO data). Ellul, Jotikasthira, and Lundblad (2011) find that insurers with

lower capital adequacy ratios are more likely to conduct asset fire sales. Shim (2010)

shows that undercapitalized insurers are under pressure to increase capital to avoid

regulatory costs.

Panel A of Table 2 provides summary statistics on insurer characteristics. For PC in-

surers, the average liability ratio LTV is 38%. The average length of the claim duration

ClaimDur is 2.3 years. 30% of PC insurers are mutual firms and roughly one-third are not

affiliated with a parent group or holding company. The liability ratio of life insurers is much

higher, at 67%. Life insurers have a bit more than half of business in the life insurance

sector (60%), with the rest in health and annuity business. 13% of life insurers are mutual

companies and 33% are independent firms.

Panel B of Table 2 reports the correlations among firm characteristics. We use the time-

averaged variable values to compute their cross-sectional correlations. Firms with higher

liability ratios tend to have higher claim maturities, having less likelihood of paying divi-

18

dends, and having lower capital adequacy ratio. Younger firms tend to more often be stock

firms and non-dividend payers. Additionally, among PC firms, non-affiliated firms tend to

more often be mutual firms and non-dividend payers, and firms with lower capital ade-

quacy ratios tend to have higher claim maturities. Among life insurers, dividend payers and

older firms tend to have higher claim maturities, and mutual life insurers are more frequent

dividend payers.

3.3 Interest Rate Factors and Nelson-Siegel Durations

The level, slope, and curvature are considered the most important features of the yield curve

and represent three most important interest rate risk factors. In this paper, we use the

Nelson and Siegel (1987) model to capture these three factors and characterize the yield

curve. The Nelson-Siegel model fits the cross-section of term structure of zero-coupon yields

using a parsimonious polynomial-exponential function. Specifically, we follow the Diebold

and Li (2006) version of the model:

yt(n) = β0t + β1t1− e−nλtnλt

+ β2t

(1− e−nλtnλt

− e−nλt)

(9)

where yt(n) is the continuously-compounded time-t zero-coupon yield for maturity n. β0t,

β1t, β2t, and λt are time-varying parameters. Diebold and Li (2006) show that the three

parameters β0t, β1t, and β2t intuitively can be related to the level, the negative of slope, and

the curvature of the term structure, and show that the correlations of these three parameters

with conventionally defined level, negative slope, and curvature factors are very high. For

this reason we simply refer to β0t, β1t, and β2t as the level, slope, and curvature factors. In

addition, λt determines the location of the curvature top.

When estimating the model, we follow Diebold and Li (2006) to fix the value of λt to a

constant of 0.0609, which exogenously specifies the curvature top at 30 months. This enables

us to estimate β0t, β1t, and β2t using OLS. The zero-coupon yields y(n)t used in estimation

are for the 30 consecutive maturities from 1 to 30 years. We run the regression during each

sample year using the year-end yields. The yield data are obtained from Gürkaynak, Sack,

19

and Wright (2007).11

The first four plots in Figure 1 are the yield curves at four selected years in our sample

period, representing different conditions of the financial market: from shortly after the LTCM

crisis (1998), after the burst of the internet bubble (2002), at the height of the real estate

bubble (2006), and in the midst of the financial crisis (2009). The yield curve is pretty flat in

1998 but becomes steep with low short rates in 2002 in response to a loose monetary policy.

The curve turns flat again in 2006 but becomes steep again in 2009 after the QE1. The fitted

Nelson-Siegel yield curves are also plotted. Overall, the fitted curve stays quite close to the

actual curve, with slightly more deviations at the long end of maturity than the short end.12

The last three plots in Figure 1 are the time series of level, slope, and curvature factors.

Consistent with the term structure variations observed in Figure 1, the three factors exhibit

relatively large swings during the sample period. Note that β1 is the negative of slope thus

a low (negative) value of β1 indicates a positively steep slope.

An advantage of the Nelson-Siegel model framework is that the sensitivities (i.e, partial

derivatives) of bond price to the three term structure factors β0t, β1t, and β2t have close-form

expressions and can be easily computed. Based on bond price sensitivities one can further

derive sensitivity measures of bond portfolio value to term structure factors, in a way similar

to the conventional notion of bond duration. The generalized portfolio duration measures

under the Nelson-Siegel model are derived in Willner (1996) and Diebold, Ji, and Li (2006).

They are referred to as the NS durations—specifically, the level duration, slope duration,

and curvature duration. To derive these durations, note that the price of a bond can be

11Gürkaynak, Sack, and Wright (2007) combine US Treasury bond price data from CRSP and Fed-eral Reserve Bank of New York to provide daily time series of the zero-coupon yields at various ma-turities. Their data are updated quarterly and made available at a Federal Reserve Board website(http://www.federalreserve.gov/pubs/feds/2006/200628/feds200628.xls).

12Such deviations may be due to the “second hump” in the yield curve. The Nelson-Siegel-Svensson modelis designed to capture this feature with a second curvature factor. However, as Gürkaynak, Sack, and Wright(2007) show, there is a parameter instability issue when fitting the Nelson-Siegel-Svensson curve. Plus, thesecond hump is not very significant during our sample period. Therefore we stay with the three-factorNelson-Siegel model.

20

expressed as:

Pt =N∑i=1

Cie−niyt(ni) (10)

where N is the number of payments remaining for the bond. Ci is the i-th remaining pay-

ment (coupon and principal combined) of the bond. ni is the time to the due date of the

payment. Let the weight of the present value of the i-th payment in the bond price be

vit = Cie−niyt(ni)/Pt. The three NS durations for the bond are defined as:

DLt = −∂Pt/Pt∂β0t

=N∑i=1

vitni (11)

DSt = −∂Pt/Pt∂β1t

=N∑i=1

vit(1− e−niλt)/λt (12)

DCt = −∂Pt/Pt∂β2t

=N∑i=1

vit[(1− e−niλt)/λt − nie−niλt

](13)

By construction, the level duration DL is equivalent to the Macaulay duration.

The NS durations for a bond portfolio are the weighted averages of the NS durations of

the bonds held in the portfolio:

DURLt =M∑j=1

wj,tDLj,t, DURSt =M∑j=1

wj,tDSj,t, DURCt =M∑j=1

wj,tDCj,t

where wj,t is the portfolio weight on bond j.

DURL (DURS, DURC) measures the average response of an investor’s government bond

holding to the change of the level (slope, curvature) factor. A greater DURL (DURS, DURC)

suggests a greater sensitivity of the investor’s bond holding to the level (slope, curvature)

changes in interest rates.

3.4 Portfolio Weights Across Maturity Bins

As discussed earlier, if habitat is due to interest rate hedging, then inelasticity of portfolio

durations is a better way to characterize habitat than inelasticity of portfolio weights at

certain maturities. Nonetheless, if in addition to the liability habitat and horizon habitat,

21

there are other factors contributing to insurers’ habitat behavior, then it is still desirable to

look at the inelasticity of portfolio weights.

For this purpose, we calculate the cash flow weights of a portfolio in the following way.

First, we separate each cash flow of a bond (i.e., coupons and principals) according to its own

maturity. We create 10 maturity bins across the 30 year spectrum, with each bin covering

three years of maturity. For example, bin #1 spans zero to three year maturities, bin #2

includes maturities greater than three years and up to six years (inclusive), and so forth.

The 10th bin covers the maturities greater than 27 years and up to 30 years (inclusive). A

very small number of bonds in our sample have cash flows maturing beyond 30 years. We

create an additional bin #11 to include such cash flows. For an insurer’s portfolio in a given

year, we group all the cash flows into the maturity bins. The portfolio weight of cash flows

in a maturity bin is the total amount of cash flows in the bin divided to the total amount of

cash flows in the portfolio.

Our portfolio weight calculation is based on the amount of cash flows regardless of the

present value of cash flows. This weight definition has the advantage of being exogenous to

term structure changes. Consider, alternatively, if we calculate weights based on the present

value of cash flows. This will result in weights endogenous to the term structure since the

interest rates are used to discount the cash flows. As a consequence such portfolio weights

may change in response to interest rate changes even when there is no trading, confounding

inference on portfolio response to the term structure changes.

4 Empirical Results

4.1 Insurers’ Aggregate Portfolios

In macroeconomic or monetary policy analysis, there is an interest in the aggregate behavior

of insurers in the bond market. Accordingly, we start with an analysis on their aggregate

portfolios. The aggregate portfolio treats all the government bond holdings of PC insurers

(and life insurers, separately) as one single portfolio. We compute the three portfolio duration

22

measures for the aggregate portfolios, as well as the portfolios’ cash flow weights in the 11

maturity bins described in Section 3.4.

Table 3 reports the time series distribution of the aggregate portfolio durations and

aggregate portfolio weights. The average interest level duration for PC insurers’ aggregate

portfolio is 5.56 years, while that for life insurers’ is 10.04 years. The duration difference

between PC and life insurers is consistent with their different claim liability maturities—

Property and casualty policies are short-term contracts subject to annual renewal and claims

are paid off fairly quickly, while life policies are long-term contracts and claims are distributed

through a long period of time. In addition, life insurers’ aggregate portfolio has a higher

slope duration and a higher curvature duration than those of PC insurers’, although the

differences are not as large as that in the level duration. The portfolio weights at various

maturities exhibit a consistent pattern. PC insurers put a heavy weight of 58% in the two

maturity bins at the short end, and the weights drop off quickly as the maturity extends.

By contrast, life insurers tend to spread out the weights across maturities. They have 41%

of weights in maturities beyond 15 years. There are two interesting spikes in the weight

distribution, one around the 10-year (bin #4) and another around the 30-year (bin #10),

the maturities with most frequent new Treasury issues.

Another noted pattern is the low time series standard deviations of the portfolio dura-

tions. For example, the standard deviation of the level duration is 0.49 year for PC insurers

and 0.93 year for life insurers, suggesting that insurers manage to keep the interest rate

level risk exposure in a tight range around a target (5.56 and 10.04 years). The standard

deviations of the slope and curvature durations are even smaller: they are around 2% of the

mean for the slope duration and around 4% of the mean for the curvature duration, for both

types of insurers. The fluctuations of the portfolio weights are also pretty narrow ranged at

maturities where there is critical mass of weights. An exception is the large weight variation

in bin #10 that includes the 30-year maturity. An closer inspection of the data shows large

fluctuations around the period when the Treasury stopped issuing new 30-year bonds (from

August 2001 to January 2006). PC insurers’ 30-year bond holdings drop quickly in 2002 and

23

recover in 2006. Life insurers react initially by increasing long-term agency bond holdings,

before dropping their 30-year bond holdings in 2004.

To see how the aggregate portfolios respond to term structure changes, we plot the time

series of the aggregate portfolio durations against the corresponding factors in Figure 2. A

relatively visible pattern is the opposite-direction movements of the level duration and the

level factor for the aggregate life insurer. The slope durations and curvature durations for

both types of insurers are rather flat relative to the fluctuations of the respective factors.

Table 4 confirms this visual pattern by regressions. Specifically, we regress the time series

of aggregate portfolio durations onto the corresponding factors, and obtain the response

coefficients. The aggregate life insurer has a significantly negative duration response to the

level factor. The aggregate PC insurer also spots a negative duration response to the slope

factor, which is statistically significant but the magnitude of the coefficient suggests a small

economic impact. The duration response coefficients in all other regressions are insignificant.

Figure 3 plots the aggregate portfolio weights in the four representative years: 1998,

2002, 2006, and 2009. As noted earlier, the yield curves in 1998 and 2006 are relatively flat

with relatively high levels, while the yield curves in 2002 and 2009 are steep with relatively

low levels. The aggregate PC portfolio does not seem to change much across the four years,

especially for the first two maturity bins where they hold around 60% of total weights. The

aggregate life portfolio weights have more variations. However it is hard to see any common

weight patterns between the pair of years with similar yield curves, i.e., between 1998 and

2006, or between 2002 and 2009. Table 4 further reports the coefficients of regressing the

portfolio weights onto the factors. For the aggregate PC insurer, the only three significant

coefficients are all at maturities above 12 years, thus having small economic impact for the

aggregate PC portfolio. The aggregate life portfolio significantly reduces the weights in the

middle (9-15 year range) and increase the weights in both the short end (0-6 year range)

and the long end (24-27 year range) in response to a rising interest rate level, a somewhat

intriguing way to reduce the level duration. There are a few sporadically significant coeffi-

cients for the aggregate life portfolio’s weight responses to the slope and curvature factors.

24

However they do not appear to be map into any common-sense interest rate strategy.13

Overall, with the exception of life insurers’ duration reduction response to interest rate

level increase, insurers’ aggregate portfolios do not appear aggressively responding to interest

rate factor changes. Combined with the narrow range of fluctuations of portfolio durations

shown in Table 3, such evidence is indicative of the influence of habitat. Of course, due to

a relatively short sample period, it is difficult to develop powerful statistical tests to locate

the habitat in the time series data of the aggregate portfolio. Therefore in the subsequent

analysis, we turn to the cross-section of insurers.

4.2 Portfolio Differences Across Insurers

Table 5 reports the cross-sectional distribution of individual insurers’ portfolio attributes.

The attributes include the three portfolio duration measures as well as the portfolio weights

across the 11 maturity bins. The mean interest rate level duration is 4.78 years for PC

insurers and 6.65 years for life insurers, smaller than the durations of the respective aggregate

portfolios reported in Table 3. This suggests a difference in portfolio duration between

small and large insurers—large insurers, which influence the aggregate portfolio more than

influencing the cross-sectional statistics, tend to higher durations than small insurers.

The small-large differences also help understand a few statistics reported in Table 5. For

example, the cross-sectional means of both PC and life insurers’ portfolio weights concentrate

more on the short end of the maturities, notably the first two bins (i.e., below six years).

The median weights for longer maturities become zero despite positive means, suggesting

that smaller insurers hold much more short-term bonds and much less long-term bonds.

An even more important pattern to note is the large cross-sectional standard deviations of

portfolio durations and portfolio weights relative to the means of the corresponding portfolio

statistics, and further, small time series standard deviations relative to the cross-sectional

standard deviations of the same portfolio characteristics. The cross-sectional standard devi-

13Interest rate changes can induce a passive change in portfolio durations even when portfolio holdings areconstant. We control for this passive effect and still find similar patterns for the aggregate portfolios. Thedetails of controlling for the passive effect are described in Section 4.5.

25

ations are calculated across insurers in each year and then averaged over time, while the time

series standard deviations are estimated for each insurer and then averaged across insurers.

For PC (life) insurers, the cross-sectional standard deviations of the level duration is 1.34

(3.52) relative to the mean of 3.47 (6.65). By contrast, the time series standard deviation

for the level duration is smaller, at 1.34 (1.73). The cross-sectional standard deviations

for the slope and curvature durations are tighter around mean than that of the level dura-

tion. Yet their time series standard deviations are even smaller. A comparison between the

cross-sectional distribution and time series variation of the portfolio weights reveals a similar

pattern. This suggests large heterogeneity in portfolio choices across insurers and simultane-

ously quite stable portfolio choices by individual insurers over time. If such a pattern is due

to habitat, it suggests that habitat is highly dispersed across insurers and yet highly stable

over time for a given insurer.

The persistent cross-sectional portfolio differences are further illustrated in Table 6 and

Figure 4. Table 6 reports the results of Fama-MacBeth cross-sectional regressions, with a

portfolio characteristic (duration or weight) in year t as the dependent variable and the

portfolio characteristic in a lagged year t-k as the explanatory variable, with up to five year

of lag. In Figure 4, we rank insurers into quintiles based on a portfolio characteristic in

year t, and plot the average ranks in the subsequent five years for each quintile. Both sets

of analysis suggest that the three portfolio duration measures are highly persistent insurer

characteristics. The persistence of portfolio weights is somewhat weaker, confirming a caution

against relying on portfolio weights to detect habitat.

4.3 Evidence on Liability Habitat

What drives the persistent cross-sectional differences in insurers’ portfolio choices? We follow

the implications of our model outlined in Section 2.3 to investigate the effect of liability and

risk aversion.

One implication of the model is that the level duration of an insurer’s portfolio is in-

fluenced by the joint effect of maturity of liability. We test this implication by performing

26

cross-sectional regressions. The dependent variable is the time series average of an insurers’

interest rate level duration DURL. In separate regressions, the explanatory variable include

the proxy for liability maturity—as described in Section 3.2, ClaimDur for PC insurers and

PctLife for life insurers, the liability ratio LTV, and the product of LTV with the maturity

proxies. The explanatory variables ClaimDur, PctLife, and LTV are the time series means for

each insurer. Finally, we also include the other two portfolio durations, DURS and DURC,

as dependent variables. However, note that ClaimDur by construction is the level duration

of the liability and we do not have a prediction on how DURS and DURC are related to

ClaimDur.

The results reported in Table 7 show that ClaimDur and PctLife have significantly posi-

tive coefficients in explaining the portfolio level duration DURL. ClaimDur does not explain

the slope duration or curvature duration for PC insurers. However, PctLife has significantly

positive relations with DURS and DURC. Possibly, this suggests that the claim liabilities of

life insurance and those of other insurance business (e.g., healthcare insurance) have quite

different exposures to the slope and curvature risk. In addition, the interactions of LTV with

ClaimDur and PctLife also have significantly positive coefficients for DURL. The interaction

of LTV and PctLife are also significant in explaining DURS and DURC. Finally, since Table

2 shows a sizeable correlation of LTV with both ClaimDur and PctLife, we check if the

liability ratio itself has any explanatory power on the portfolio durations. It does not for PC

insurers but does for life insurers.14

In a panel regression perspective, the relation between the dependent and explanatory

variables has a time series dimension and a cross-sectional dimension. The former is termed

the “within effect” while the latter is known as the “between effect.” The cross-sectional

regression approach we use captures the between effect. We have also performed the panel

regression that capture the “within effect” using demeaned dependent and explanatory vari-

ables (i.e., subtracting insurer-specific means from a time-varying variable). We find that

14For robustness, we perform multivariate regressions that simultaneously include ClaimDur (PctLife),LTV, and LTV ∗ ClaimDur (LTV ∗ PctLife). The result shows the significant positive coefficients onClaimDurPctLife and ClaimDur continue to hold.

27

the “within effect” tends to be insignificant. This suggests that the relation between liability

maturity and portfolio durations is largely a cross-sectional effect. This is consistent with

a pattern of cross-sectionally dispersed but stable-over-time habitat preferences by insurers.

Given this nature of insurers’ habitat, in all subsequent analysis we continue to focus on the

between effect by performing cross-sectional regressions.

Finally, we perform cross-sectional regressions using the portfolio weights in various ma-

turity bins as the dependent variables. The results show that PC insurers with higher

ClaimDur appear to shift weights from the first five bins (0 to 15 years) to the bins with

maturity beyond 15 years. Life insurers with higher PctLife appear to shift weights in the 0

to 6 year maturity range to the maturity range beyond 6 years. This is intuitively consistent

with the positive relation between liability maturity and portfolio level duration.

Overall, the evidence for the liability habitat is quite strong.

4.4 Evidence on Horizon Habitat

We now turn to a cross-sectional analysis of the horizon habitat. Following the second

model implication discussed in Section 2.3, we infer the horizon habitat effect via the relation

between risk aversion and portfolio durations.

We continue to follow the cross-sectional regression approach. The dependent variable

is the time-series mean of an insurer’s portfolio duration (DURL, DURS, and DURC). The

explanatory variable is each of the five risk aversion proxies (RA) defined earlier—a dummy

for mutual insurers (MUTUAL), a dummy for non-affiliated insurers (INDEP), a dummy for

firms not paying dividend (NODIV), the negative of the logarithm of firm age (YOUNG),

and the negative of the logarithm of the risk based capital (LOWCAP). If a variable is time

varying, it averaged over time for an insurer before being used in the regression.

The results are reported in Table 8. For PC insurers, when INDEP, NODIV, and YOUNG

are used to explain the three portfolio durations, we consistently obtain significantly nega-

tive coefficients (except an insignificantly negative coefficient for YOUNG when explaining

DURS). For life insurers, when INDEP, NODIV, YOUNG, and LOWCAP are explanatory

28

variables, we consistently obtain negative coefficients for explaining all three portfolio du-

rations. Thus, there appears to be a negative relation between risk aversion and portfolio

durations. Recall that the relation between portfolio duration and risk aversion depends

on the relative magnitude of the horizon habitat duration and the duration of the oppor-

tunistic part of the portfolio. The observed negative relation indicates that insurers’ horizon

habitat duration tends to be shorter than the durations of their opportunistic portfolio com-

ponents. In the context of the level duration, this can be further interpreted as insurers

having relatively short investment horizon.

However, the evidence is not clear-cut. For PC insurers, the coefficients of MUTUAL

and LOWCAP are insignificantly positive. For life insurers, the coefficients of MUTUAL

are positive and significantly so when explaining DURS and DURC. Such strong results for

MUTUAL contrast equally strong but opposite results for other risk aversion proxies.15

The discussion in Section 2.3 suggests a need to control for 1-αt when inferring from

the relation between risk aversion and portfolio durations. Thus in an additional set of

regressions we use the interaction of (1-LTV) with a risk aversion proxy as the explanatory

variable. The results are similar to those based on the corresponding risk aversion proxy itself.

The only difference is that the interaction of (1-LTV) with NODIV does not significantly

explain portfolio durations. For brevity we do not tabulate the results.

Finally, we perform cross-sectional regressions using the portfolio weights in various ma-

turity bins as the dependent variables. The results reveal that the negative relation between

portfolio level duration and several risk aversion proxies, i.e, INDEP, NODIV, YOUNG,

LOWCAP, is achieved by these firms in the form of shifting portfolio weights from long-

maturity bins to short-maturity bins, notably the first two bins (i.e., 0 to 6 years). This is

consistent with the behavior of investors with a short investment horizon.

15Recent studies, e.g., Berry-Stolzle, Nini and Wende (2012), suggest that mutual insurers could alter-natively issue surplus notes to raise capital. Thus they may not tightly financially constrained. This isconsistent with the negative correlations between MUTUAL and NODIV for both life and PC insurersreported in Panel B Table 2 – mutual insurers are more likely to pay dividends than stock companies.

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4.5 Portfolio Response to Term Structure Changes

Habitat can be viewed generally as the stability of portfolio exposure to risk factors or

stability of portfolio weights at certain maturities. In an even sharper way, habitat is the

insensitivity or inelasticity, of portfolio exposure or portfolio weights with respect to interest

rate conditions. In this part of analysis we examine insurers’ portfolio response to term

structure changes. Based on individual insurers’ portfolio reaction, we further examine the

third implication of the model, that is, liability and risk aversion reduce portfolio reaction

to term structure changes.

When measuring the response of portfolio duration to interest rate factors, it is desirable

to control for a passive effect—because portfolio durations are functions of interest rates,

portfolio durations can change with interest rates even when portfolio holdings are constant.

We control for this passive effect by computing an adjusted portfolio duration, which is the

observed portfolio duration minus a “target duration.” The “target duration,” representing

the passive impact of interest rate changes, is the duration of a portfolio with time-averaged

weights and under the prevailing term structure of a given year. To obtained the time-

averaged portfolio weights, in each year we group the expected cash flows (coupons and

principals) of an insurer’s portfolio into semi-annual maturity bins (as opposed to the coarser

3-year bins used in previous analysis).16 In each year we compute the cash flow weights

in each maturity bin, and then average the weight for each bin over years for the given

insurer. Essentially, the adjusted portfolio duration is the duration of portfolio weights’

active deviation from the average.

We measure individual insurers’ portfolio response to term structure changes by time

series regressions. For each insurer, the dependent variable is one of its three adjusted

portfolio duration measures and the explanatory variable is the corresponding factor. We

also use the portfolio weights for the 10 three-year maturity bins as the dependent variables

16The reason for using semi-annual maturity bins is that the predominant coupon frequency for govern-ment bonds is semi-annual. A small number of bonds in our sample have weekly, monthly, and quarterlycoupon frequencies. So using semi-annual bins may result in some approximation error. We have performedcalculations to gauge the magnitude of the approximation error and find it to be quite small.

30

and regress them onto each of the three factors. Recall from Section 3.4 that by construction

our portfolio weight measure is exogenous to term structure changes. So there is no need

for further adjustment of the weights. Since we have relatively short time series and the

three factors are relatively uncorrelated, we use one factor at a time in regressions. Portfolio

duration’s sensitivity to a factor is measured by the regression R-square and the regression

coefficient.

Table 9 reports the cross-sectional distributions of the regression R-squares and the co-

efficients. For PC insurers, the average regression R-squares are 0.26, 0.21, and 0.17 for the

three adjusted portfolio durations. For life insurers, the average R-squares are 0.23, 0.17,

0.14. Duo to a relatively short sample period involved, a component of the R-square may

be due to in-sample over-fitting. Also notice the large cross-sectional standard deviations

of the R-squares—they are as high as the mean values. An inspection of the R-squares for

individual portfolio weights in the 10 maturity bins reveals the same pattern.

The cross-sectional statistics on the regression coefficients further reveal something in-

teresting. By going through these coefficients we hope to understand if there are common

strategies underneath insurers’ responses to term structure changes. The average coefficient

for the level duration response is positive for both types of insurers, while those for the slope

and curvature duration responses are negative. This seems to suggest that on average insur-

ers increase the level duration in response to a rising interest rate level while reducing their

exposure to the slope and curvature risks when these two factors are on the rise (note that a

rising Nelson-Siegel slope factor should be interpreted as a flattening yield curve). However,

such a mean response pattern is overwhelmed by the large dispersion in insurers’ responses.

The cross-sectional standard deviations of the coefficients are one order of magnitude higher

than the mean. Therefore, a more important pattern to note from this table is that insurers

behave very differently from each other in their response to term structure changes, despite

the common market conditions they face.

In the presence of large cross-sectional differences in insurers’ responses, an interesting

question is to what extent such differences can be explained by difference in habitat. We

31

address this question by cross-sectional regression analysis, following the third implication

of the model described in Section 2.3. Given the large dispersion in insurers’ responses, we

do not wish to take a view on what should be the correct direction of portfolio response

when performing empirical tests. However, as pointed out in Section 2.3, liability and risk

aversion should both negatively affect the absolute value of portfolio duration’s response to

factors. Therefore, we use the absolute value of the coefficient from the first-stage time series

regression for individual insurers (hereafter ABSCOEFF) as the dependent variable in the

cross-sectional regression. The explanatory variable of the cross-sectional regression is LTV,

or one of the give risk aversion proxies.

Table 10 reports the results from this analysis. For PC insurers, the liability ratio LTV

negatively affects ABSCOEFF of portfolio duration responses for all three factors, and signif-

icantly so for the DURC response to the curvature factor. For life insurers, LTV significantly

negatively affects ABSCOEFF for the duration response to all three factors. Therefore, lia-

bility muffs the sensitivities of portfolio risk exposure to the term structure factors.

An inspection at the results for the ABSCOEFF of portfolio weight response suggests

that LTV significantly reduces portfolio weights’ sensitivities to factors mainly at the short-

term maturities, i.e., maturities below 6 years. This holds for both PC and life insurers.

At longer maturities, the relation between LTV and ABSCOEFF tends to be positive. The

relation of LTV with ABSCOEFF tends to be significantly positive for bin #10. However,

as noted earlier, due to the exogenous supply shocks at the 30-year maturity we should not

read too much into the results for this bin.

As for the five risk aversion proxies, their relations with ABSCOEFF also tend to be

negative, although less clear-cut. For each type of insurers, there are 15 pairs of relations

based on three portfolio duration measures and 5 risk aversion proxies. For PC insurers,

there are 10 counts of negative relations, among which 3 are significantly negative at the

10% confidence level. For life insurers, there are also 10 counts of negative relations, with

4 significantly negative. For both types of insurers, two risk aversion proxies, NODIV and

YOUNG, tend to exhibit positive relations with ABSCOEFF. Their relations with ABSCO-

32

EFF are always positive in the case of duration responses to the slope and curvature factors.

This is contradictory to their role as risk aversion proxies under the horizon habitat effect.

When we examine the relation of risk aversion with ABSCOEFF at portfolio weight

level, we find patterns consistent with those at the portfolio duration level. For PC insurers,

whose weights concentrate on short maturities, the relation between a risk aversion proxy and

ABSCOEFF at short maturities (i.e., below 6 years) is always the same as the relation at the

portfolio duration level discussed above. For life insurers, their weights tend to spread out

more across maturities so the relations at all maturities matter. The risk aversion proxy with

most pervasively negative impact on ABSCOEFF across all maturities is INDEP. MUTUAL

and LOWCAP tend have negative relations with ABSCOEFF at short maturities but positive

relations at maturities. For NODIV and YOUNG the opposite holds—positive relation at

short maturities and negative relation at long maturities. Interestingly, these patterns are

the same across the weight responses to all three factors, a puzzle for the horizon habitat

effect.

Finally, in addition to ABSCOEFF, we have also examined the portfolio sensitivity based

on the R-squares (R2) obtained from regressing portfolio durations and portfolio weights onto

the factors for individual insurers. We find that across insurers, LTV is pervasively negatively

related to R2. The relation between risk aversion proxies and R2 is further mixed. Since

the results are consistent with those based on ABSCOEFF, for brevity we do not tabulate

them.

To sum up, there is a pervasive pattern that a high liability ratio dampens portfolio

reaction to term structure factors, consistent with the liability habitat effect. There is also

evidence, but less-clear-cut, that risk aversion muffs portfolio reactions.

5 Concluding Remarks

The preferred habitat hypothesis of term structure has attracted considerable attention re-

cently. However, so far we have little empirical evidence that speaks to the microeconomic

foundation of this theory. Our study attempts to fill the gap. Using data for an impor-

33

tant group of bond market players–insurance firms, we detect habitat-like behavior in their

government bond portfolios. The maturity of insurers’ claim liabilities is an important deter-

minant of their bond portfolio durations. Further, insurers with stronger liability concerns

exhibit more restrained portfolio reaction to term structure changes. The analysis on insur-

ers’ horizon habitat suggests that insurers’ investment horizons tend to be relatively short.

We also find some, although less pervasive, evidence that risk aversion reduces the magni-

tude of portfolio reaction to term structure changes. Our analysis highlights institutional

investors’ liabilities as an important source of inelastic demand for government bonds and

complements the macroeconomic studies that reply on the implication of preferred habitat

to examine the effect of supply shocks to the term structure.

34

Appendix A

A.1. Horizon Habitat: “Change of Numéraire” and Log-linearization

Let the one-period log riskfree rate from time t-1 to t be rft. Let Rht and rht be the gross return

and log return of a zero-coupon bond maturing at time H, for the period from time t-1 to t. This

bond pays off a value of $1 at maturity and its time-t price is BHt .

Define Ŵt = Wt/BHt . Essentially, W

∗t is wealth expressed in the units of the H-maturity bond.

This in the same spirit of the “change of numéraire” procedure in Detemple and Rindisbacher

(2010). Since BHH = 1, WH = ŴH , and the investor’s problem in (1) is equivalent to:

MaxE0(Ŵ 1−γH1− γ

) (14)

subject to the budget constraint:

Ŵt+1 = ŴtRpt+1Rht+1

Recall that R1t+1 = BHt+1/B

Ht is the return to the H-maturity bond. The indirect utility function

Jt, combined wit