WHY DO MARKETS FAIL TO FULLY INSURE
AGAINST RECLASSIFICATION RISK? Contract
Incompleteness vs. Limited Commitment in the Life
Insurance Industry
Denrick Bayot
University of Chicago
Abstract
In addition to insuring mortality risk, life insurance contracts protect policyholders against the
risk of rising premia associated with a policyholder’s change in risk status by providing long-term
contracts. Contracts purchased in the life insurance market, fail to fully protect policyholders
against such risk. We examine the role of limited commitment and contract incompleteness in
explaining why the market fails to provide full insurance against reclassification. Limited commitment arises since contracts are nonbinding, and policyholders with favorable ex-post mortality risk
terminate a contract if its price cross subsidizes higher-risk agents. Insurance companies front load
premia to subsidize future premia and mitigate a policyholder’s incentive to replace a contract.
However, credit constraints prevent the level of front-loading necessary to prevent ex-post low-risk
policyholders from replacing a contract, leading to a break-down in full reclassification risk insurance. On the other hand, life insurance contracts do not account for changes to a policyholder’s
valuation for life insurance coverage, which is why contracts are incomplete. Ex-post inefficient
insurance provision arises with such contract incompleteness since reclassification risk insurance
provides subsidy to high-risk individuals that may no longer value life insurance coverage relative
to its actuarially fair insurance cost in the future. Thus, a tradeoff between ex-post inefficient
insurance provisions and protection against reclassification naturally arises. In equilibrium, firms
balance such trade off leading to contracts that partially protect policyholders against reclassification risk. We calibrate a life cycle model of insurance demand and lapsation behavior to match
the patterns of life insurance purchases observed in the Survey of Consumer Finance as well as
lapsation rates and mortality claims from an industry-wide experience study. After leveraging
this model to investigate counterfactual contract environments, we find that limited commitment
alone cannot explain the failure of full reclassification risk insurance in the term-life market, and
we largely attribute such failure to contract incompleteness.
1
Introduction
Consider a scenario where there are gains to a long-term agreement between a firm and a
consumer, but the consumer can default on such agreement. Firms employ lock-in mechanisms
1
to ensure consumer participation, and such mechanisms allow both parties to capture the gains
achieved through commitment. This paper discusses a cost to binding these consumers that
naturally arises when there are non-contractible, idiosyncratic uncertainties about the value
consumers place on their relationship with firms. These uncertainties, paired with the lock-in
mechanisms used to mitigate default, lead to ex-post inefficient contract participation.
We illustrate this scenario in the market for term-life insurance contracts. In addition to
providing insurance against mortality risks, insuring against reclassification risk (an increase in
premia caused by an unfavorable evolution of mortality risk) is a significant component of life
insurance contracts. However, a consumer/policyholder can walk out of a contract simply by
ceasing to pay the premia, and policyholders with favorable ex-post mortality lapse a contract
if its price cross-subsidizes agents with higher mortality risk. Such strategic lapse behavior
plays a key role in shaping contract profiles in the life insurance industry (Hendel and Lizzeri
(2003)). Policy payment schedules are front-loaded to subsidize subsequent premia, thereby
reducing the policyholder’s incentive to lapse.
We find that policyholders seek insurance against reclassification risks and prefer contracts
with longer level-term premium duration. A lion’s share of contracts purchased specify a
20-year or more level-term duration (LT20+contracts). Despite the heavy-front loading of
premia in these longer-term contracts, almost 40% of policy holders lapse halfway through
the level-term duration. This persistent lapsation points to the existence of shocks affecting
the policyholder’s demand for mortality risk insurance, such as changes to bequest motives
and liquidity-constraint shocks. Lapse patterns in the Health and Retirement Survey indicate
that newly divorced agents are more likely to lapse than married individuals. Indeed, divorce
increases lapsation probability by about 300%. Patterns of life-insurance holdings across marital
status support this view: divorced-male and never-married male individuals share the same life
insurance holding patterns. This suggests that at all stages of the lifecycle, changes to bequest
motives, especially owing to a recent divorce, affect the policyholder’s valuation of mortality
risk insurance.
This paper proposes that front-loading premia in these contracts leads to excessive ex-post
mortality-risk insurance provisions. The subsidy of future premia locks in future policy holders
whose ex-post valuation for mortality risk insurance is less than the cost of providing such
insurance. We emphasize that this inefficiency stems from an inability to contract uncertainties
that affect contract valuation. Were firms able to contract these uncertainties, contingent
rebates could provide appropriate incentives for policy holders to efficiently lapse. Thus this
paper highlights the inefficiency that arises from an incomplete contract environment when
contract participation is not enforceable.
Although the theoretical literature on incomplete contracting has matured, empirical analyses of the effect of incomplete contract environments continues to lag, as does work that
quantifies the extent of the inefficiency arising from these environments. Our paper fills this
gap by quantifying the ex-post inefficient contract participation that arises from the incomplete
market setting in term-life contracts, as discussed above. To our knowledge, we are the first to
move beyond empirical analyses of the economic arrangements caused by incomplete contracts
and expand the process to include welfare analysis.
2
We use this calibrated model to impute the welfare cost associated with contract incompleteness and limited commitment. To do this, we consider fixed-payment contracting environment
where firms charge an entry payment at the time of contracting and a fixed annual premia.
Firms in this environment can specify the contract length for up to age 70, and we consider the
equilibrium contracts offered for a 40-year old male individuals. We calculate equilibrium contracts in an environment where consumers can commit to not recontracting with another firm
(full commitment) and firms can specify participation contingencies that ensure efficient ex-post
life insurance holdings (quasi-completeness). We then compare the equilibrium contracts in this
baseline environment to the ones that arise without full commitment and quasi-completeness.
We find that moving from a quasi-complete and full commitment environment to one with
limited commitment and contract incompleteness leads to a substantial welfare loss (more than
$1B dollars). Moreover, the problem arising from limited commitment accounts for only 17%
of this loss.
Policyholders in the life insurance industry continue to face substantial reclassification risks.
Indeed, most contracts purchased typically shield the agents against reclassification risk for at
most 20 years. Hendel and Lizzeri (2003) argue that such failure arises from the policyholder’s
limited-commitment and credit constraints. In particular, they surmise that a policyholder’s
credit constraint prevents the level of front-loading necessary to prevent strategic lapsation. The
anticipated default of healthy policyholders then leads to a break-down in reclassification risk
insurance. We investigate this premise and find that limited commitment alone fails to explain
the break-down in full reclassification risk insurance observed in the life-insurance industry.
In particular, we find that the average contract length in quasi-complete equilibrium without
full commitment is almost equal to the contract environment’s upper bound length (27.5 years
versus 30 years). Instead, we find that failure of full long-term insurance or reclassification risk
insurance in the life insurance market can be mostly attributed to contract incompleteness. For
example, we find that the average contract length in an incomplete contract environment with
full commitment is 18.2 years. Adding limited commitment to this environment decreases the
average contract length by less than two years (16.7 years).
Our result suggests that the non-contractible uncertainty in the policyholders’ valuation for
life insurance coverage led to a failure of full reclassification risk insurance in the life-insurance
market. The intuition on why such contract incompleteness led to this failure is as follows. By
its very nature, a reclassification risk insurance provides subsidy to unhealthy individuals. Thus,
individuals in an unhealthy state in the future will receive a discount even without contract
front loading. The same unhealthy individual may no longer value life insurance coverage
relative its contracting cost in the future. Simply put, the individual’s willingness-to-pay for
life insurance coverage may no longer exceed the actuarially fair insurance cost (spot price).
But, if the individual’s willingness-to-pay exceeds that of the discounted price, then she will
choose to maintain life insurance coverage, leading to ex-post inefficient insurance provisions. In
a perfectly competitive market, policyholders ultimately bear this inefficiency and is reflected
in the ex-ante contract pricing. Thus, in an environment with contract incompleteness in
the policyholders’ valuation for life insurance coverage, a tradeoff between ex-post inefficient
insurance provisions and protection against reclassification naturally arises. In equilibrium,
3
firms balance such trade off leading to contracts that partially protect policyholders against
reclassification risk.
Although this paper focuses on the term life insurance industry, examples of contract features that bind agents are present in many markets with limited commitments. Prepayment
penalties in mortgages, infidelity clauses in prenuptial agreements and early-termination fees
in utility markets are but a few examples of these mechanisms. Policy makers have questioned
whether these lock-in mechanisms lead to inefficient agreements, akin to the ex-post inefficient
life insurance holdings observed in this paper. Indeed, such concerns resulted in a series of judicial decisions and regulations that prevent firms in many industries from drawing agreements
that bind consumers.1
Our analysis does not favor these laws. While our paper acknowledges the ex-post inefficiencies that could arise from these lock-in mechanisms, we do not claim that such mechanisms
are ex-ante inefficient and should therefore be eliminated. In fact, we acknowledge the necessity
of these mechanisms when contracts are not enforceable but there are gains to commitment.
Rather than prevent agents from drawing contracts that help policyholders commit, regulators
should understand the nature of the ex-post loss and seek alternative solutions to mitigating
these losses. One such solution, which this paper advocates, encourages firms to compete in a
more complete contracting environment.
2
Life Insurance
We focus our analysis on term life insurance policies. Unlike cash-value policies, term lifecontracts simply provide coverage to a policy holder’s beneficiary for a specified time frame
and the simplicity of their terms makes them fairly homogenous. These contracts can be
compared in terms of the coverage period, schedule of payments and the face value (amount
paid to the beneficiary in the event of the policy-holder’s death within the coverage period)
specified. If the policyholder survives after the coverage period, the contract ceases and no
additional benefits are given to the policyholder.2
Almost all term life contracts guarantee fixed coverage until the late stages of the lifecycle
(typically 85), provided policyholders continue to pay premia,3 but these contracts vary in their
payment schedule. Annual renewable term (ART) contracts have premium levels that increase
over time, while level-term (LT) contracts fix the premia for a specified number of years with
1
See, for example, the recent cap on prepayment penalties imposed by the Dodd-Frank Act Section 1414.
For the cellular phone industry, a summary of recent cases and state laws against early termination fees can be
found here: http://cell-phone-termination-fee.whocanisue.com/.
2
While cash-value insurance (particularly, whole-life insurance) account for a large share of the number of
policies issued (51.8% versus 23.9% for term insurance according to the 2009 LIMRA persistency study), term
insurance account for a large share of the face amount in force (52.3% by 2004). This is the case since cashvalue policy owners tend to purchase contracts with lower face value. The average face amount exposed for all
whole-life contracts in 2004 was $ 39,000, while buyers of term policies purchased contracts with an average face
value amount of $309,000. This suggests that agents primarily used cash-value policies as a savings instrument
rather than to insure their beneficiaries.
3
A small percentage (approximately 1%) provide decreasing death benefit (Jr. Kenneth Black (2013))
4
increases thereafter according to a pre-specified (current) schedule of payments akin to an ART
contract. 4 The degree of premium front loading varies by contract type, with longer level term
contracts exhibiting higher up-front payments.
Policyholders face non-trivial reclassification risk
The degree of risk reclassification policyholders face is not trivial. Life insurance risk categories tend to be coarse. Agents who qualify for contracts in the standard LI market are
typically placed in 2-4 risk categories (e.g., standard, preferred, preferred plus, etc.), but not
all consumers qualify for life insurance contracts.
Health conditions, such as heart failure, diabetes, or cancer, can prevent an agent from participating in the standard insurance market (Hendren (2013)). While there are some significant
risks to changes in risk category, the biggest reclassification risk stems from not being able to
requalify for a standard-underwritten insurance contract. Indeed, we find that agents have a
high likelihood of falling into a substandard category.
Table 1.1 tabulates the one-year hazard rate of a 40-year -old male individual falling into a
substandard category based on the National Health and Nutrition Examination (NHANES) III
Survey. We use the age of first exposure to a disease that would prevent a prospective insuree
from participating in the standard life insurance market to create this “life table”. We follow
Hendren (2013) in using a list of diseases most-cited by life insurance underwriting guidelines
that would preclude a prospective policyholder from participating in standard life insurance.
An agent is in a substandard class if he/she has had one of the following conditions: cancer
(except for melanoma), heart failure, diabetes or obesity. Male individuals face substantial
reclassification risk; the risk of falling into a substandard class for a 40-year-old male amounts
to almost 8% within the next 10 years.5
As pointed out by Hendel and Lizzeri (2003), font-loading contracts enables insurance companies to insure against reclassification risks despite the policy holders’ incentive to strategically
lapse. In particular, front-loading premia in these contracts allows for a premium subsidy in
subsequent periods and reduces the agent’s incentive to lapse and repurchase a new contract.
In their analysis, Hendel and Lizzeri (2003) find that all term contracts offered in 1997 tended
to be front-loaded in that the ratio of premium payments over mortality risk declines over time.
Since their analysis, the characteristics of ART contracts have drastically changed. premia for
ART contracts during the first few years of a contract dropped substantially, perhaps due to
reduced search costs (Brown and Goolsbee, 2002). However, such a decrease was paired with
a steep rise in premia over the contract duration. Thus, ART contracts offered in the market
today no longer exhibit their front-loaded characteristic and closely resemble spot-market insurance contracts. Given this change and the high risks that policyholders face of falling into
a substandard risk category, it doesn’t come as a surprise that life-insurance buyers shifted
4
The contract also includes a “guaranteed” schedule. In practice, insurance companies do not deviate from
the “current” scheduled specified in the contract.
5
Note that bunching every five years is operative. We attribute this to agents rounding ages when reporting
age of first diagnosis.
5
Interval
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
Exposure Count
5172
4979
4809
4627
4467
4347
4191
4066
3933
3814
3723
3571
3478
3343
3235
3138
3011
2913
2807
2709
2618
Hazard (One-YearI)
0.0064
0.0034
0.0044
0.0054
0.0036
0.0110
0.0064
0.0057
0.0074
0.0052
0.0212
0.0064
0.0104
0.0102
0.0093
0.0137
0.0086
0.0093
0.0114
0.0137
0.0302
Survival Rate
1.0000
0.9966
0.9921
0.9866
0.9831
0.9721
0.9658
0.9603
0.9531
0.9480
0.9277
0.9217
0.9120
0.9026
0.8942
0.8817
0.8740
0.8658
0.8558
0.8441
0.8180
Table 1.1: Substandard Classification Risk
This table tabulates the risks of falling into an uninsured class by age. We used the age of first exposure to the disease that renders the individual
uninsurable (exposure to non-melanoma cancer, heart failure, diabetes and obesity) as the failure event. Individuals that did not report a disease at
the time of the survey are counted as a censored observation. We use individuals ages 40 and above from the NHANES III that did not report exposure
to one of the diseases before age 40.
their purchases away from these ART contracts towards LT contracts. In fact, LT contracts
with level term durations lasting 20 years or longer are the most commonly purchased term
contracts today. Figure 1.2 highlights this fact, where we observed that LT contracts account
for almost the entire share of term contracts purchased across all age groups.
Nevertheless, policyholders continue to face nontrivial reclassification risk post level term.
For example, almost 15% of 40-year old policyholders own LT 10 contract, and, as mentioned
earlier these policyholders face a nontrivial risk of not being able to qualify for a new insurance
contract. Why do agents policyholders opt to expose themselves to nontrivial uninsurability
risk? Hendel and Lizzeri (2003) argue that credit-constrained individuals cannot afford the
level of front-loading necessary to commit a life-insurance policy, causing them to purchase
shorter term contract. Given that the cost of life insurance coverage is small relative to average
annual consumption (see table 2), we find such reason suspect.
6
Figure 1.2: Distribution of Contract Types by Age
This figure displays the lapsation pattern of level term contracts. Panel A provides the mean lapse rate, while panel B calculates the cumulative
proportion of individuals that lapse the contract by year. This figure is baed on a large-scale industry-wide experience study from a top actuarial
consulting firm. Cumulative lapse rate is calculated based on the mean lapse rates. We exclude policies that were issued for individuals under age 20
and over age 65.
Uncertainty in the valuation of life insurance coverage
In this paper, we argue that any uncertainty the policyholders’ valuation for life insurance
coverage leads to a failure of reclassification risk insurance. How? By its very nature, a
reclassification risk insurance provides subsidy to unhealthy individuals. So those that end
up in an unhealthy state in the future will always receive a discount even without contract
front loading. This same unhealthy individuals may no longer value life insurance coverage
relative its contracting cost in the future. Simply put, that individual’s willingness-to-pay
life insurance coverage may no longer exceed the actuarially fair insurance cost. However, if
the individual’s willingness-to-pay exceeds that of the discounted price, then she will choose
to maintain life insurance coverage leading to an ex-post inefficient insurance provision. In a
perfectly competitive market, policyholders ultimately bear this inefficiency and is reflected in
the ex-ante contract pricing. If such cost exceed the value to insuring reclassification risk, then
the equilibrium contracts limit the level of protections agents receive against such risk. In this
section, we provide evidences that suggest that policyholders do face tremendous uncertainty
in how they value life insurance coverage.
7
Figure 1.3 illustrates the extent of lapsation patterns for level-term contract holders and
depicts the average lapse rate for both LT10 and LT20 contracts. LT20 policyholders are less
likely to lapse than LT10. This is expected given that premia for LT20 are more likely to
be front-loaded relative to LT10. Despite the heavy front-loading of both contract types, we
find substantial lapsation. More than half of LT10 contract-holders lapse before the end of the
level term, while almost 40% of LT20 contract-holders drop out halfway through the level-term
duration. Not surprisingly, LT10 contracts exhibit a jump in lapsation (“shock lapse”) in the
post-level term period with almost all existing contract holders dropping two years after the
level-term period.
Figure 1.3: Mean Lapse Rate by Contract Type
This figure displays the distribution of term contract by type. ART includes all contract with non-level premia. LT10 are level term contracts with a
10-year level premium duration. Similarly LT20+ include contracts with level term duration lasting more than 20 years. The figure is based on policies
issued in between the years 2004 and 2011 and comes from a large-scale actuary experience study.
In the HL framework, the front-loading level necessary to achieve full insurance against
reclassification may not be achieved if agents find the large outlay of premia too costly. Distortions relating to optimal consumption smoothing are likely to be limited given that premia
for these contracts tend to be small relative to average annual household consumption and
8
income. For example, the level premium for a 40-year-old male who purchases 20-year term
insurance with a half-a-million coverage amounted to an annual premium of $592 in 2004. This
expenditure accounted for about 1.3% of the median household income.67 .
It thus begs the question why firms fail to increase the level of front-loading to curtail the
sizable lapsation observed in these contracts. Our theory provides an alternate explanation to
this question. Changes to the valuation of life insurance contracts, owing perhaps to changes
in bequest motives or income shocks, makes front-loading contracts costly even without credit
constraints. The front-loading of contracts leads to inefficient insurance take up when there
are ex-post uncertainties to the value of the component of LI contracts that insures against
mortality risks. Given this cost, firms shy away from excessive premium front-loading.
We do not dismiss the notion that some of these lapses are strategic and involve a replacement of LI contracts. In fact, our theory does not preclude the existence of strategic lapsation.
We find, however, that the contracts offered in the market leave little incentive for individuals to
strategically lapse due to a favorable ex-post mortality shock. Table 1.2 tabulates the average
premia (based on the lowest 15 contracts observed in Compulife) of standard-class level term
contracts for a non-smoking male individual. We use a face value amount of 1 million dollars as
an illustration. Notice that it does not behoove the policyholder to lapse an LT contract and
repurchase the same type of LT contract, unless the agent wishes to extend the level-premium
duration. Of course, an agent in an LT20 contract can lapse and repurchase an LT10 contract.
The savings for doing this, however, are quite minimal. For example, the savings for a 30-yearold male during the first few years is less than $100 annually and is non-existent after the 8th
year. Moreover, dropping a contract during those early years lessens the level-term period and
exposes the agent to a higher reclassification risk. Admittedly, the savings from this type of
lapsation increases for older individuals, although these individuals would have to drop their
LT 20 contract by year 4-5 to realize some form of savings.
Actuarial studies dating back to the 1980s emphasize the role of lapsation in deteriorating the insured pool over time, 8 and understanding lapse behavior continues to be of keen
interest in the life insurance industry (Black and Skipper, 2012). Unsurprisingly, various explanations abound as to why individuals drop their existing life insurance contracts beyond
the “replacement hypothesis” (i.e., the conjecture that individuals strategically lapse). Most of
these studies examine lapsation behavior for cash-value life insurance contracts and focus on
two main hypotheses: the interest-rate and the emergency-fund hypothesis. The interest-rate
hypothesis strictly applies to cash-value insurance policies since they are related to lapsation
behavior due to households taking advantage of higher market interest rates. The emergency
fund hypothesis, which suggests that households drop their insurance when faced with wealth
or income shocks, applies to both term life and cash-value policies, and various empirical works
find evidence consistent with this hypothesis (see, for example, Fier and Liebenberg (2013) and
surrender and lapse rates with economic variables (2005)).
6
Based on the SCF 2004 survey.
During the sample periods Hendel and Lizzeri (2003) analyzed, insurance loads were substantially higher.
Credit constraints may have played an important role during these periods
8
See for example, Dukes and MacDonald (1980), and Shapiro and Snyder (1981))
7
9
Age
30
32
34
36
38
40
42
44
46
48
LT10
929.5
933
940
991
1112.5
1250
1424
1634.5
1889.5
2197
LT20
1184
1243.5
1309
1418
1590.5
1808.5
2120.5
2476
2896.5
3408
Age
50
52
54
56
58
60
62
64
66
68
70
LT10
LT20
2555.5
4002
2971.5
4745
3475
5581.5
4092.5
6764.5
4883
8169.5
5869.5
9875
7167
13180.5
8902.5 16699.5
11022.5 26423.75
13738
39416
17220 46546.67
Table 1.2: Level-term Prices
This table tabulates the prices for lever-term contracts in 2004. premia are calculated for a standard-class 40-year old individual with a coverage
amount of $1M. We take the 15-lowest price in the Compulife Historical Data and use average the premia.
In this subsection we explore the role of changes to a policyholder’s bequest motives in
determining lapsation. Bequest motives, particularly protection against income loss or expenses
associated with death, are cited as the primary reasons why families purchase life insurance
(LIMRA Barometer 2014). Indeed, term-life insurance holding patterns from the SCF data
bolster this view (see Table 1.3). In particular, we find that adult men who remained single
at the time of the survey (i.e., those who have never married) are less likely to own termlife insurance than currently married individuals (22% vs. 49%). Moreover, the face-value
of contracts held by married individuals tends to be larger ($100k) relative to the contracts
purchased by men who never married. This difference in ownership rate and amount purchased
holds even when one controls for age, assets, income and number of children. We also find that
take-up rates by divorced males and the face-value of these LI purchases parallels that of the
individuals who never married.
It’s not obvious why divorced males are less likely to hold term-life insurance contracts than
their married counterparts. It’s tempting to conclude, based on these patterns, that lapsation or
termination of contracts ensues after a divorce or moments leading to a divorce. This pattern may
simply reflect ex-ante marital attributes (present at the beginning of marriage) that affect the likelihood
of a divorce. For example, in efficient household models that account for limited commitment (Voena,
2012), households with a bad match quality at the time of marriage are less likely to purchase insurance
during marriage and are more likely to divorce. We do not observe the take-up rates for these divorced
individuals during their marriage years so we cannot test the hypothesis that lapsation leads to this
stark difference in ownership rates.9
Following Fang and Kung (2012) and Fier and Liebenberg (2013), we used the Household and
Retirement Survey (HRS) to examine the impact of divorces on lapsation. The HRS panel contains
information on life insurance holdings and purchase behavior for older individuals (above 50). Besides
9
We explored using the SCF’s short 2007-2009 panel. We find no changes in the marital status, however,
over these two-sample periods.
10
VARIABLES
Ownership
(2)
(3)
(4)
0.00697
(0.0152)
-0.00754
(0.0168)
-0.0933***
(0.0210)
-0.192***
(0.0157)
-0.131***
(0.0148)
0.0278*
(0.0157)
0.101***
(0.0167)
0.0397**
(0.0180)
0.134***
(0.0207)
38.94***
(11.36)
2.986
(11.32)
-33.41**
(14.90)
-105.0***
(13.36)
-91.14***
(10.62)
0.487***
(0.0127)
0.00604
(0.0155)
-0.0213
(0.0170)
-0.166***
(0.0204)
-0.241***
(0.0159)
-0.191***
(0.0148)
0.0383**
(0.0160)
0.113***
(0.0172)
0.0341*
(0.0185)
0.430***
(0.0172)
NO
9,212
0.067
NO
9,212
0.073
YES
9,212
0.123
(1)
Age: 40-60
Age: 50-60
Age: 60-65
Never Married
Divorced
0.00234
(0.0154)
-0.0455***
(0.0160)
-0.209***
(0.0184)
-0.270***
(0.0151)
-0.198***
(0.0147)
1 Child
2 Children
3 or More children
Constant
Controls:
Income + Networth
Observations
R-squared
Face Amount
(5)
(6)
298.0***
(8.606)
41.58***
(11.43)
16.74
(12.61)
-7.510
(16.58)
-86.36***
(13.82)
-87.77***
(10.54)
27.59**
(11.85)
46.01***
(12.10)
34.21**
(14.60)
264.5***
(12.37)
32.90***
(11.29)
-0.790
(12.55)
-22.90
(16.53)
-76.10***
(13.65)
-72.53***
(10.60)
30.27***
(11.58)
45.57***
(11.87)
35.81**
(14.41)
195.1***
(19.74)
NO
3,008
0.042
NO
3,008
0.048
YES
3,008
0.093
Table 1.3: Life Insurance Demand Patterns from the SCF Households
Source: Survey of Consumer Finance 2001, 2004 and 2007 waves. Notes: We restrict our households to households with male head of the household in
between the ages 30 and 65. We exclude households with networth above the 95% in our analysis. Networth is calculated using the standard SCF’s
program for calculating networth. We use the CPI index to adjust all monetary variables (networth, family income and face-value amount) so that
they are in 2004 $. Sample weights are used in the regression. Lastly, we use the first implicate in our analysis.
11
allowing us to track changes in life insurance holdings, the HRS explicitly asked individuals whether
they “voluntarily” lapsed a life insurance policy between the two waves.10
We mimic the analysis in Fang and Kung (2012) to identify lapse determinants, but we focus on
lapsation behavior without contract replacements (i.e., optimal lapsation). We only use HRS waves
after 2001: HRS 2002, 2004, 2006, 2008, 2010. This restriction allows us to focus on optimal lapsation
as opposed to strategic lapsation. Life insurance premia started to decline during the early 1990s and
continued to do so in a monotonic fashion until 2001. Thus, individuals were likely to strategically
lapse and waited on repurchasing a policy until premia stabilized to their lower, competitive level.11
Furthermore, we restricted our analysis to male individuals less than 85-years-old who were married
during the first sample wave (2002). Individuals without a life insurance policy throughout the sample
or those that only owned a life insurance policy during the last wave (2010) are excluded in the analysis.
After taking into account all of our restrictions, our data set contains 5,408 males. The bulk of these
individuals were only observed for two years, with about 42% surveyed for at least 3 years.
Lapsation for the individuals in our HRS sample tends to be lower (averaging around 4.8% in
between the sample waves) than the reported average annual lapse rate observed in Figure 1.2. We
expect this discrepancy given that the HRS sample is comprised of older individuals who face relatively
minimal income shocks. Moreover, life insurance holdings in the HRS data include cash-value holdings,
and these policies tend to exhibit lower lapse rates (LIMRA persistency study, 2008). The majority
of these reported lapses were voluntary (77%) and almost all voluntary lapses during our sample
period are optimal lapsations. In particular, roughly 94% of these lapses were not associated with an
insurance replacement.
We considered two sets of shocks that possibly affect lapsation. The first set is comprised of
characteristics that affect life insurance demand but are unrelated to health or mortality risk. These
include wealth (measured as the log of income), percent change in income, the logarithm of the ratio
of medical expense to income and changes to marital status. The second set of shocks captures the
policy holders’ risk class and includes health specific variables. These include various health conditions,
BMI, self-reported health status and a dummy variable (whether the individual had been admitted to
a hospital in between waves).
Table 1.4 reports a reduced-form logit model of voluntary lapsation behavior without insurance
replacement on these two groups of shocks and determinants. 12 All models control for age, education
and year fixed effects. We find that recently divorced individuals as well as those with high medical
expenditures relative to their wealth are likely to lapse. This holds true even when one controls for
health shocks. Moreover, these health shocks do not appear to be correlated with the first set of lapse
determinants, given that the coefficients on the first set of shocks do not change when one controls for
the policyholder’s health characteristics.
We also find evidence of anti-selective behavior (i.e., healthier individuals are more likely to lapse),
which is consistent with our conjecture that unhealthy individuals are less likely to completely opt out
10
In particular, the survey asked whether the individual lapsed an LI policy. Furthermore, they ask if the
“lapse or cancellation [was] something [the agent] chose to do, or was it done by the provider, [an] employer, or
someone else [besides the agent]?”.
11
See, for example, the slides from the 2013 SOA Life & Annuity Symposium which provide evidence of this
behavior: https://www.soa.org/Files/Pd/Las/2013/2013-las-session-28.pdf
12
We omit reporting the self-reported health category and the dummy variable capturing hospital visits; these
variables were insignificant and followed signs consistent with the idea that individuals with higher mortality
risk are less likely to lapse.
12
VARIABLES
(1)
Shocks
(2)
Health State
(3)
Shocks + Health State
-11.93
(7.314)
-0.300*
(0.169)
-0.171
(0.142)
-0.423
(0.272)
-0.00824
(0.0905)
0.000293
(0.00143)
-14.99**
(6.298)
-0.00996
(0.0206)
1.477**
(0.656)
-1.241
(1.016)
1.180
(1.136)
-0.000531
(0.000775)
0.0455*
(0.0254)
0.00790
(0.0690)
-0.329*
(0.188)
-0.207
(0.154)
-0.358
(0.278)
0.00513
(0.0901)
4.11e-05
(0.00138)
-11.09
(7.377)
9,095
Yes
HH
13,315
Yes
HH
9,018
Yes
HH
%4 Income
-0.0108
(0.0224)
Recently Divorced
1.474**
(0.653)
Recently Married
-1.202
(1.008)
Recently Widowed
1.178
(1.117)
Mortgage-to-Wealth
-0.000437
(0.000619)
Log Medical Expense
0.0405*
(0.0208)
Log Income
0.0392
(0.0707)
Cancer
Heart Attack
Stroke
BMI
BMI2
Constant
Observations
Demographic FE
Cluster
Table 1.4: Lapse Determinants
Note: Logit estimates are based on a sample from the 2002-2010 HRS Wave. We restrict the sample to individuals that owned life insurance for since
2002 or purchased a life insurance contract before 2010. Only male individuals under the age of 85 are considered in the sample. Lapsation is defined
as a voluntary lapsation without contract replacement. Models in columns (2) and (3) also include other health shocks, such as self-reported health
and hospital visitation in between waves. These variables were not statistically significant.
13
of the life insurance market. Simply put, risk class affects the valuation of the life insurance contract,
where individuals with a high mortality risk are more likely to maintain some form of life insurance
coverage. This result casts doubt on an assumption typically made in risk-reclassification models,
which assume that agents may simply have no need for an insurance contract in subsequent periods
(e.g., a change in bequest motive) independent of their risk class. In these models, optimal lapsation
(lapsation without replacement) does not depend on the risk class. 13
To put these numbers into perspective, we used estimates from the logit model with both sets of
shocks and imputed the likelihood of a voluntary lapsation without replacement (optimal lapsation)
conditional on various divorce status and health conditions (e.g., cancer and heart attack). Values for
the covariates are calculated at the mean for an individual with a “poor” reported health status. We
find that divorce increases the likelihood that agents optimally lapse by more than three-fold (from
8% to 29% for an individual with cancer or a history of heart attack). Such a substantial percentage
increase in lapsation is true across different types of self-reported cancer/heart attack conditions. Antiselective behavior is present for individuals with cancer or a history of heart attack, though divorce
has a larger effect on lapsation behavior. Specifically, while divorce increases the probability of lapsing
by about 300%, the existence of cancer or heart condition merely decreases the probability of lapsing
by about a third.
Using the HRS has some disadvantage in that it does not distinguish between term policy and
cash-value policy. Moreover, life-insurance demand wanes during retirement years (see table 1.3, and
one must be cautious when extrapolating the effect of divorces on the lapsation of policies to younger
individuals. Nevertheless, we believe that the results from the HRS data, paired with the evidence
on life insurance holdings patterns of younger individuals, suggest that divorce leads to substantial
lapsation during all adulthood-lifecycle stages.
3
Lifecycle model of life insurance demand
Results from the previous section suggest that one cannot solely attribute the policyholder’s lapsation
behavior to a strategic lapsation. Voluntarily lapsed contracts in the HRS sample were likely to not
be replaced by another contract. Moreover, we find that these type of lapsation are largely drive by
factors unrelated to mortality risks, such as changes to a policyholder’s bequest motives when couples
divorce. Within the context of our theoretical model, these results aline to the existence of a nondegenerate ex-post contract valuation φ that is not contracted upon. Our theoretical model points
out to an ex-post inefficiency inherent in this environment; that is, excessive and inefficient insurance
against mortality risks are likely to ensue over the contract duration.
Quantifying the extent of this ex-post inefficiency, not only requires observing factors affecting life
insurance demand unrelated to risk, but also requires knowledge of how front loading prevents efficient
lapsation. For example, in the HRS data one would need to know the level of front-loading in premia
the policy holder faces when choosing to lapse. Furthermore, one must tease out the variations in
premium front-loading that cannot be attributed to selection. Consumers with different ex-ante belief
of how likely they are to value the contract ex-post 14 may sort into contracts that vary in their degree
of front-loading; in this case, the relation between front-loading and lapsation may simply reflect the
unobserved component of ex-post valuation.
13
14
As in the Daily, Hendel, and Lizzeri (2008) and Fang and Kung (2010) model.
Within the context of our model, this amounts to differences in the distribution of θ
14
We do not know of a dataset that contains household-level information on lapses and also exogenous
variation in premium front-loading,15 but we would like to be able to understand the magnitude of
this inefficiency in the life insurance industry. To do this, we create a lifecycle model of insurance
demand and lapsation behavior that takes into account the relevant shocks that household face when
determining optimal life insurance holdings. We calibrate this model to match patters on LI holdings,
lapsation and mortality claims found in the data. We then use this model to estimate the welfare
losses associated with contract incompleteness and limited commitment.
3.1
The lifecycle model
We employ a unitary household lifecycle model with bequest motives for married individual. Male
policy holders account for most of the life insurance policies purchased (Hong and Rı́os-Rull (2012)).
So, our model focuses on life insurance demand on contracts with the male spouse as the primary
insurance holder (i.e., the beneficiaries include everyone in the household but the husband). Following
the literature on annuity demand (De Nardi, French, and Jones (2009), and Lockwood (2010)), the
household head in our model receives utility from bequeathing individuals at the time of the husband’s
death. This bequest motive can be fulfilled through savings or the purchase of a life insurance contract.
Households in our model face various risks that affect their demand for life insurance and the need
to insure against mortality risks. Thus, agents in our model decide on the optimal savings, insurance
purchase and life insurance policy lapsation.
We are not the first to apply lifecycle household model in life insurance demand. Examples can be
found in Hong and Rı́os-Rull (2012), Hosseini (2007), Hosseini (2008) and Inkmann and Michaelides
(2012). But, none of these models account for reclassification risk insurance and assumes that agents
face a static, spot-market contract choice set. In these models agents purchase a spot-market contract
that is equal to its actuarially fair value plus some insurance load. To our knowledge, we are the first
to construct a model that allows for a richer contracting set, one that insures against reclassification
risk. In particular, we mimic the contract set currently displayed in the market, which comprises
primarily of level term contracts.
This section proceeds as follows. We first provide details to each of the model’s components:
the shocks agents face, the contracting set, the budget constraints and the agent’s problem. We
then discuss parametric assumptions used in our calibration. This subsection is then followed by a
discussion on the first-stage calibrated parameters (parameters that can be estimated independent of
the lifecycle model) and the information used to impute these values. We then discuss the moments
used to calibrate the rest of the parameters that rely on the lifecycle model for identification. Finally,
we discuss the fit of our calibrated parameters.
Health state and Morality risk We consider two health states ht in our model: insurability
(ht = 1) and uninsurability (ht = 0). Agents in our model start the lifecycle with an insurable
health state, but health state evolves stochastically according to the law πh (ht+1 |ht ). Mortality risk
depends on the health state in each period. We denote the probability of not surviving in period t + 1
conditional on being alive in period t by δtht .
15
The HRS data provides virtually no information on premia paid. In particular, the survey stopped collecting
information on premia for term-life insurance in the year 2000.
15
As mentioned earlier, standard LI underwriting typically precludes individuals from purchasing
when they fall into a health category deemed uninsurable. We use adverse health conditions that
most life insurers use in their criteria in defining uninsurability. In particular, a person is uninsurable
if she has the following health condition: cancer (except for melanoma), heart failure, heart attack,
diabetes and extreme obesity. In most cases, LI underwriting prevent individuals from purchasing
standard LI contracts if they’ve had a history of these adverse health conditions. We thus assume
that πh (ht+1 = 1|ht = 0) = 0.
Life Insurance Contracts
Previous life-cycle models of demand typically assume that households can only purchase spot-market contracts. As mentioned earlier, this assumption does not align
with the observed contracts in the market. In our model, households can purchase long-term contracts,
and we focus on the two-types of contracts typically purchased in the market: 10-year and 20-year
lever term.
We characterize these contracts by their per-dollar premia, Pt , face value, Ft and tenureship, dt .
These contracts are non-binding so that households at any point in time can choose to lapse by ceasing
payment. Households in our model pay the premia at the beginning of the period, and a payment
insures the agent against mortality risk in that period. At the end of the level-term, households are
left uninsured and can either continue being insured against mortality risk in the spot-market market
or purchase a new level-term contract.
The linearity assumption of premia in face value is not an innocuous assumption and reflects the
limited role of adverse selection in the life insurance industry. In practice, most contracts are linear
with some bulk discounts for contracts with large face values (Cawley and Philipson (1999)).
Income shocks and the household budget constraint We allow for income shocks to
capture the effect of liquidity constraints on lapsation. In particular, household income follows the
following process:
ln yit = µi + it
it = ρi,t−1 + ηit
In this equation yit is the per-period income and exp(µi ) captures the income at the beginning of the
household’s lifecycle, and the innovation ηit is assumed to be independent across time and individuals.
These permanent income shock end at retirement, and during such phase households earn a pension
equivalent to their end-of-retirement income multiplied by a replacement ratio. A household in possession of a term contract with premia Pt and face value Ft faces the following capital accumulation
constraint if she chooses to keep the contract and pay the premia:
at+1 = [at + yit − (ct + Pt F )] (1 + r)
In this equation at denotes the asset, ct is the per-period household expenditure beyond lifeinsurance purchases and r is the market interest rate. We assume that individuals cannot leave debt
so that paired with a positive mortality risk we have at ≥ 0. Households may opt out of keeping or
purchasing a long-term contract, in which case the asset accumulation follows the usual form:
at+1 = [at + yit − ct ] (1 + r)
16
.
Preferences and Household Problem Our model of household preference closely follows the
unitary household model with bequest motive in Lockwood (2013) in that the agent receives utility
from leaving a bequest at the time of his death. Bequest can either come from assets accumulated
and the face value of a life insurance contract held, if any. In particular, if the agent dies in between
period t and t + 1, has accumulated assets at+1 for use in the next period, and purchased a contract
with face value F , then the agent receives the following bequest utility:
v(at+1 + F |Mt , ξt )
Bequest motives depend on the marital state Mt and an unobservable preference shock ξt . These
unobservable preference shock are persistent and follows a random walk: ξt = ξt−1 + ηt . Let Ct =
(Pt , F, dt ) denote the agents current life insurance holding with Ct = ∅ with no insurance holdings.
Agents in each period can choose to maintain their contract, lapse and purchase a new contract among
the set Ct0 of contracts available in the market, assuming the agent is insurable (ht = 1), or opt out
of the insurance market. Thus the agents choice set Ct (ht , Ct ) depends on the health status and
current-periods insurance ownership:
Ct (ht , Ct ) = Ct0 ∪ C̃t (Ct ) ∪ ∅ if ht = 1
Ct (ht , Ct ) = C̃t (Ct ) ∪ ∅ o.w.
Here C̃t ((Pt , Ft , dt )) = (Pt , Ft , dt + 1) if the duration is within the level-term period; otherwise, we
let it be the empty set. The empty set represents the option to opt out of the life insurance market,
and agents in this case can only bequeath through savings. Let ωt = (at , Ct , yit , ht , Mt , ξt ) denote the
agents current-period state. The agent in each period chooses his optimal consumption, savings and
life-insurance purchase decision. The following bellman equation describes this intertemporal decision:
h
i
ht
ht
0
0
Vt (ωt ) =
max
u(c
)
+
β
δ
v(a
+
F
|M
,
ξ
)
+
(1
−
δ
)E[V
(ω
|ω
,
c
,
C
)]
t
t+1
t
t
t+1
t+1
t
t
t
t
t
0
0
0
ct ,Ct =(Pt ,F 0 ,dt )
s.t. budget constraint and
Ct0 ∈ Ct (ht , Ct )
3.2
3.2.1
Calibration and Parameterization
Parametric form
The households felicity function takes on the standard constant relative risk aversion (CRRA) form :
1−σ
u(c) = c1−σ . Our parametric form of bequest motives v(·|Mt , ξt ) closely follow Lockwood’s threshold
crossing model but allow for bequest motives to vary across age, marital status and an unobservable
preference shock. In particular, we assume the following form;
!
σ
φt (Mt , ξt )
(κ(Mt ) + b)1−σ
vt (b|Mt , ξt ) =
1 − φt (Mt , ξt )
1−σ
17
φ(·) captures the bequest intensity and admits a logit specification:
φt (Mt , ξt ) =
exp(αo + αm Mt + αξ ξt )
1 + exp(αo + αm Mt + αξ ξt )
κ(Mt ) = κo (1 − Mt ) + κm Mt reflects the threshold consumption level below which households prefer
not to bequeath an actuarially-fair mortality insurance contract as discussed by Lockwood (2013).
We assume a parametric survival function when estimating the mortality risk. As in Finklestein and
Porterba (2004), we model the hazard function using the Gompertz distribution and estimate these
hazard functions by health type. To be exact, for each health type h, we let the the survival function
with respect to mortality take on the form Sh (t) = exp −λh γh−1 (exp (γh t) − 1) . These functions are
h (t)
then used to impute each period’s mortality risk δtht = Sh (t−1)−S
.
Sh (t−1)
3.2.2
Assumptions on the stochastic components of the model
Agents in our model face income uncertainty, divorce shocks (i.e., Mt is stochastic), uninsurable
health shock, mortality risks and an unobservable bequest shock. With the exception of mortality
risk and health shocks, we assume that shocks are drawn independently of one another. We make
such assumption since the processes are estimated from various data sources (see next subsection).
Admittedly, medical studies linking divorce to poor health conditions flood the medical literature
(Sher and Noth (2013)). Thus, our estimates on the gains to divorce contingent rebate–conversely,
the lock-in inefficiency–is likely to be conservative figure (bias downwards). This is the case since the
cost of locking consumers in is higher for unhealthy individuals with high mortality risk. While we
do not impose any parametric assumption when estimating the income process, we assume that ηit
is normally distributed in our simulations. We also assume that the unobservable bequest shock’s
innovation follows a normal distribution; that is, ηξ ∼N (0, 1) so the preference innovation has variance
αξ2 .
3.2.3
First-stage parameters
We divide our estimation procedure into two stages. Our first-stage estimates involve parameters that
are identified without solving the lifecycle model. These include the income process, reclassification
and divorce risk. We then use these estimates as calibrated parameters in the lifecycle model. We
calibrate the remaining parameters by matching simulated patterns based on our lifecycle model. In
particular, we seek out values of the parameter that result in simulated lifecycle profiles that closely
matches the patterns of life-insurance demand, lapsation and mortality experience in the data.
Income We use information from the biannual 1997-2011 waves of Panel Study of Income Dynamics
(PSID) to estimate the household income process. We use the PSID’s measurement of pretax total
family income including transfers. Details of the sample selected in our analysis can be found in
Appendix B.I.
Our estimation consists of two parts. First, we project this measurement on annual dummies, headof-the-household’s age, education and race. We then match the moments of the estimated residuals to
estimate the parameters of the income-shock process (ρ and ση2 ) using a standard method of moment
18
ρ
0.5895
(0.0228)
ση
0.0053
(0.0005)
Table 1.5: Income-process estimates
Source: PSID 1997-2011. Refer to the main text for the estimation strategy.
approach. 16 In our analysis, the time window in between t and t + 1 consists of a two year period.
Table 1.5 reports these estimated parameters, where we find considerable dispersion in the innovation
and a persistent shock. We use these estimates and assume that η follows a N (0, σ 2 ) in our lifecycle
simulations.
Reclassification risk and the uninsurable mortality risk
We employ the NHANES III to
estimate reclassification risk. The NHANES III provides information on the age at which an exhibited
an uninsurable health condition. We previously discussed this risk in the life table tabulated in table
1.1. As mentioned earlier, there appears to be some bunching around certain years, most likely due to
respondents rounding their age in the survey. We attempt to smooth this by fitting a nonparametric
survival curve that captures reclassification risk. In particular, we first take the restricted sample
and fit a cox proportional model with sex and smoker proportionally affecting the baseline hazard.
We then non parametrically fit the baseline survival function using the a cubic specification on log
survival function: ln S R (t) = βoR + β1R t + β2R t2 . Details of our sample restriction and the variables
from NHANES used to define uninsurability can be found in Appendix B.II.
Table 1.6 lists the estimated parameters for this survival function as well as the effect of sex
and smoker-status on the hazard function. Consistent with the lifetable found in Table 1.1, we find
considerable reclassification risk. For example, the probability of falling into uninsurable category
within 10 years for a 45-year old male individual , conditional possessing an insurable health status
at this age, is approximately 14.34%. Not surprisingly, smoking status substantially increases the
reclassification risk (the hazard rate increases by 23 .61%)
We restrict our sample to uninsurable individuals and estimate a cox-proportional hazard model
on mortality risk. As in the reclassification risk model, we allow sex and smoker status to affect
the baseline hazard model. We use a Gompertz distribution to fit the baseline hazard. While our
data allows us to non-parametrically estimate this baseline function, we find that the Gompertz
distribution provides a good fit relative to the nonparametric fit. This assumption is also consistent
with our parametric assumption on the mortality risk of policyholders (insurable individuals), where
we make such assumption for computational tractability (see second-stage estimation section). Table
1.7 tabulates the parameters of the Gompertz distribution.
Divorce risk Our model takes divorce risk as exogenous, and we let divorce risk vary by age.
Admittedly, divorce risk varies by marital duration but this effect can be captured by the age-specific
divorce rate. Considering both age and duration specific divorce probabilities significantly increases
the dimension the state space in our computation. We use the NSFG 2006-2010 data on married
16
In particular, we match the following moment conditions E[2it ] = ση2 and E[it it−j ] = ρj ση2 for j >= 1. A
diagonal weighting matrix is used, and we calculate the standard errors using a bootstrap method.
19
(1)
First-stage cox (exp(β 0 x))
t
(2)
ln Ŝt
0.0174***
(0.000251)
-0.000303***
(2.88e-06)
t2
Female
-0.274***
(0.0357)
0.212***
(0.0563)
Smoker
Constant
-0.252***
(0.00476)
Observations
R-squared
19,999
20,005
0.990
Table 1.6: Cox Regression
This table reports estimates of the uninsurable hazard function. Column 1 reports the first-stage cox-proportion hazard function where uninsurability
is the failure event. We then nonparameterically fit the estimated base-line survival curve as a quadratic function of time. The estimates for such fit
is displayed in column (2). Refer to the main text for the estimation strategy. Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1
Parameters
Female
Smoker
Constant
Observations
(1)
ln λ
(2)
γ
-0.377***
(0.0426)
0.820***
(0.0685)
-9.220***
(0.274)
0.128***
(0.00505)
3,285
3,285
Table 1.7: Baseline Death Hazard for Uninsurable Adults
This table reports estimates of the death hazard function for uninsurable class using the NHANES III data and the Mortality File Supplement.
Calculations are based on the MLE, where we use the parametric form h(t) = λ exp γt. Estimates are based on a sample of adult individuals age 30
and over. Time is reset so that t = 0 corresponds to the 30th year. Standard errors are calculated using the replicate weight method suggested in the
guidebook.*** p<0.01, ** p<0.05, * p<0.1
20
male individuals ages 30 and above to calculate the two-year divorce rates by age. We first estimate a
cox proportional hazard model of divorce without parametrically specifying the baseline hazard and
allow it to depend on marriage at the time of marriage. We then non parametrically fit the estimated
survival function and use this to calculate the marital survival function for a 30-year old male. Figure
1.4 plots the divorce risk across the lifecycle considered and shows a nontrivial rate of divorce up until
age 55. In fact, conditional on being married at age 30, a male individual has a 20% chance of getting
divorced within 10 years.
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
30
35
40
45
50
55
60
65
Age
Figure 1.4: Marital Survival Function
We use the NSFG 2006-2010 data on married male individuals ages 30 and above to calculate the two-year divorce rates by age. We first estimate a
cox proportional hazard model of divorce without parametrically specifying the baseline hazard wit age of marriage as a factor affecting the hazard
rate. We then non parametrically fit the estimated survival function and use this to calculate the marital survival function for a 30-year old male.
Life insurance Pricing We use data provided by Compulife to impute the term-life insurance
prices. For each age, we take the average of the 15 lowest premia in the dataset for a nonsmoker,
standard class individual issued in 2004. Table 2.1 lists these premia.
Other Calibrated Parameters We do not estimate the CRRA parameter σ but use previous
studies to calibrate its value. In particular, as in Lockwood (2013) we set this value to 3. The annual
discount rate is set at 3%.
3.3
Second-stage calibration and simulation procedure
Our second-stage procedure calibrates the remaining preference parameters characterizing bequest
motives (αo , α1 , α2 , αm , αξ ) and the parameters governing the insured’s mortality risk profile (βI , λI ).
21
To do this, we use our lifecycle model and simulate insurance purchase behavior, lapse decision and
the insured’s mortality experience and chose parameters that match the patterns in data. We target
the following moments in our calibration:
1. Ownership Rate: [m1 ] and [m2 ] - We take a sample of male-household heads from the SCF
(waves 2001, 2004 and 2007) that were married at some point in their life. These include
divorced individuals at the time of the survey. We then calculate the average ownership rate
conditional on current marital status: An individual owns a term-life insurance contract if she
holds a contract with a face value amounting to at least $75,000. We make such restriction
to better approximate individual-life ownership as oppose to group-life or employer-sponsored
policies that tend to have low face amounts. We also restrict the sample to household heads
in between the age of 30 and 50. The average ownership rate for divorced individuals (m̂1 ) is
equal to 28.14%, while 53.17% of married individuals owned a term-life insurance in our sample
(m̂2 ).
2. Face value: [m3 ] and [m4 ] We take a subsample of households discussed in the preceding
paragraph that owned a life insurance at the time of survey and calculate their average face
amount. The face value of insurance purchased by divorced individuals in our sample amounts
to about $219k, while married individuals purchase contracts with substantially higher face
values ($310k).
3. Lapse rate: [m5 ]. We use information for a large-scale actuary persistency and experience study
to impute the 10-year lapse rate of LT20 and consider only contracts issued for male policy
holders with ages in between 35 and 45. We restrict the sample to policies issued for a nonsmoker, fully underwritten contract with a face value exceeding $50k. Approximately 37.62%
of these policy holders lapse by year 10.
4. Mortality rate: [m6 ], [m7 ], [m8 ], [m9 ] We use the same large-scale actuary to calculate mortality
rates during the first 10 years for polices issued at various age groups: 30-34, 35-39, 40-44, 45-49,
and 50-54. We take the total number of claim counts within the first 10-years of the contract
and divide it by exposure counts (measured as a year and policy). Thus, one can view this
statistic as the average mortality rate within the first 10-year of the contract duration. These
numbers are .00035, .0004704, .000814, .001272, and .002024 , respectively.
We start the lifecycle for a married individual at age 30. We initialize the unobservable bequest
parameter by setting ξ = 0 across all simulation, but we allow for heterogeneous starting values for
income and assets. The distribution of initial income and assets is drawn from a sample of married
male individuals ages 25 to 40 in the SCF. We draw (with replacement and randomly up to the SCF
household weights) from a sample of 500 individuals, and for each individual we simulate 1000 lifecycle
decisions.
To match the first two patterns in the data, we take a cross-section from our simulated life profile.
To be exact, for each simulated life, we choose a particular point in the simulated lifecycle profile
based on the age distribute of the subsample used to estimate m1 and m2 . We then use this simulated
cross-sectional data to calculate the simulated moment moments on take up rates and face-value by
marital status. For the moments relating to lapse and mortality rates, take our simulated lifecycle
and emulate the experience and lapse data. We then use this data to calculate simulated lapse and
22
mortality rates by age group. Simulated method moments are formed for each parameter guess β, say
mSim (β).
Our calibration method uses the same technique as a simulated method of moment approach.
We minimize the distance between the the simulated and the empirical moments, where we use the
diagonal variance-covariance matrix of the empirical moments variance Σ. In particular, our calibrated
parameters is based on minimizing the following objective function:
min (mSim (β) − m̂)0 Σ (mSim (β) − m̂)
(1)
β
Although we use the same empirical approach as SMM, we do not feel comfortable calling our calibration as estimates and imputing standard errors at this stage. First, we make no formal identification
claims, but we suspect that the moments m1 through m5 identify the parameters relating to bequest
motive. Similarly, the survival function for insured individuals are likely to be identified by attempting to fit the mortality rates m6 through m9 . Also, given that our use of multiple datasets, properly
accounting properly for standard errors is a nontrivial task, and we feel that such exercise, while
important in itself, is secondary to the paper’s intent. We are currently working on addressing these
two issues, but at the moment we use our calibrated parameters for our counterfactual analysis.
3.4
Model fit
Table 1.8 displays the calibrated values found by minimizing the expression (1) using a simulated annealing algorithm, β̂. This table also provides a comparison between the simulated moments mSim (β̂)
evaluated at β̂ and our observed empirical moment. With the exception of the average face value
of term-life holding ([m4 ] and [m5 ]) our model reasonably fits the targeted patterns in the data.
Threshold-crossing level of consumption in our model ($6, 613.1 for divorced individuals and $5, 488.5)
for married individuals) is substantially lower than the ones estimated in Lockwood (2013) for older
individuals. This result doesn’t come as a surprise given the large take-up rate of term life insurance
in the data for ages 30-50 individuals.
Parameter
αξ
κo
αo
αm
κm
o
λo
γo
Estimated Parameter
0.0990
6, 613.1
1.2898
1.1047
5, 488.5
exp(-11.4947)
0.0825
Moment
Divorced Ownership Rate
Marriage Ownership Rate
Avg Face
Avg Face (Married)
First-10 year lapse
30-35 Mort
35-40 Mort
40-45 Mort
45-50 Mort
50-55 Mort
Empirical
28.14%
53.17%
219k
310k
37.62
.0004
.0005
.0008
.0013
.002
Table 1.8: Second-stage parameter estimates
23
Matched
27.16%
50.78%
123k
183k
37.14%
.0004
.0005
.0007
.0008
.001
4
Welfare Loss: Fixed-Payment Contracting Environments
To impute the welfare loss that stem from the policyholder’s lack of commitment and incomplete
contracting, we consider a fixed-payment contracting environment. In particular, we define the set of
all feasible fixed-payment contracts as follows:
Definition 1. A feasible fixed-payment contract signed in date t is a tuple Ct = {pto , pt , T̄ t , lt } that
specifies:
1. an entry payment pto : Ωt → R,
2. a fixed annual premia pt : Ωt → R,
3. a contract length: T t : Ωt → N+ ,
t )T −t , where for each j = 1, · · · , T − j lt
4. and a participation rule lt = (lt+j
t+j : Ωt × Ωt+j → {0, 1}
j=1
dictates wether a person receives coverage in period t + j. Hence, for each ωt+j and t with
t (ω , ·) = 0.
t + j > T t (ωt ), we have that lt+j
t
The contract above mimics the standard term-life insurance market that specifies a level premia
for a specified number of years. Unlike the life insurance contracts we observe in the market, we allow
for a richer contracting set, one that allows for contracts to depend on the agent’s current state ωt ,
including the unobservable bequest shocks. We also allow firms to front load or back load payments
via an entry payment, pto . Lastly, the contracts specify a contingent participation rule that depend on
the realized states in the future. Such rule allow firms to eliminate any ex-post inefficient life insurance
holdings if possible. We thus look at environments where this rule is left unrestricted (we refer to this
environment as a quasi-complete contracting space) and compare it with an environment where no
such rule exists (an incomplete contracting space).
−t
Given a set of contracts specified for each age t, say (Ct )Tt=1 , let Vt (ωt , Ct (ωt )|(Ct+j )Tj=1
) denote
each policyholder’s value function for contract Ct , conditional on the set of contracts the policy holder
−j
can possibly purchase in a future date (Ct+j )Tj=1
. In particular, policy holders at any date t0 > t
can possibly drop a contract and repurchase the new contract Ct0 . Similarly, let Rt (ωt , Ct (ωt )|(Cj )j6=t )
and Mt (ωt , Ct (ωt )|(Cj )j6=t ) be the expected revenue and mortality/claims cost associated, respectively,
with the contract Ct . We are now in a position to characterize a set of equilibrium contracts in the
fixed-payment contract environment.
Definition 2. A set of contracts (CtE )Tt=1 is an ¯l-equilibrium contract in the fixed-payment contract
environment if
1. Each policy has a load of ¯l:
E )T −t )
Mt (ωt ,CtE (ωt ),(Ct+j
j=1
E )T −t
Rt (ωt ,CtE (ωt ),(Ct+j
j=1
= 1 − ¯l, for any ωt ∈ Ωt and t = 1, · · · , T .
2. For any other fixed-payment contract with the same load factor ¯l, we have that
−t
E T −t
Vt (ωt , Ct (ωt )|(Ct+j )Tj=1
) ≤ Vt (ωt , CtE (ωt )|(Ct+j
)j=1 )
for any ωt ∈ Ωt and t = 1, · · · , T .
24
EI
EII
EIII
EIV
(1)
Participation
(2)
Annual Premia
(3)
Length
(4)
Face
(5)
Take-up
(6)
Total 4 Welfare
-3,548
-5,242
2,432
592
1,775
2,378
1,240
1,682
30
18.2
27.5
16.7
353k
181k
257k
172k
63%
49%
59%
43%
$1,081.43 (3.51%)
$160.21 (0.52%)
$1,312.51 (4.26%)
Table 1.9: Average Equilibrium Outcome
We set the load factor to 42.1%, which is the average load observed for the LT10 and LT20 contracts found in the market. Columns 1-4 report the
average participation fee, annual premia, contract length, and face value across all the equilibrium contracts for a 40-year old individual. Columns 5
and 6 calculates the take-up rate and welfare loss (moving from EI to an alternate environment) for all contracts found in equilibrium. A monte-carlo
simulation is used to calculate the averages. Averages are taken over the distribution of assets and income found in the SCF data for ages 35-45 male
household head. Marital states and bequest shocks are drawn independently when forming these average profiles. We set the maximum length to age 70.
EI reports the results for a contracting environment with full commitment and quasicompleteness. EII reports the results for a contracting environment
with full commitment only. EIII reports the results for a contracting environment with limited commitment and quasicompleteness. Lastly, EIV reports
the results for a contracting environment with limited commitment and contract incompleteness.
In this definition, ¯l is the load associated with each contract. An equilibrium dictates that no
other firm can offer a set of contract that makes some consumers better off. In this section, we
consider four types of contracting environments and consider the equilibrium contracts found in these
environments. We first consider the baseline contracting environment where we assume that firms can
force individuals to drop out of the contract in a given period–that is, the rule lt is unrestricted. We
refer to such environment as a “quasi-complete” contracting environment since it allows for efficient
ex-post participation. We also consider two forms of policyholder commitment.
Table 1.9 details the average contract profile for an insurable 40-yr old male individuals in various
contracting environments. A monte carlo simulation is used to calculate the averages. Averages are
taken over the distribution of assets and income found in the SCF data for ages 35-45 male household
head. Marital states and bequest shocks are drawn independently when forming these average profiles.
We set the maximum length to age 70. We set the load factor to ¯l = 42.1%, which is the average
load observed for the LT10 and LT20 contracts found in the market. 17 . While this load may appear
high, its value is within the range found in other front-loaded contracts, such as the long-term care
market (Brown and Finkelstein, 2007). Lastly, we restrict contracting to insurable lives and continue
to assume that uninsurable lives must self insure through savings. 18
17
We use the compulife average profile that were used in the lifecycle calibration. We then take simulated
claims to calculate the expected cost
18
For the contracting environment with limited commitment, we recursively calculate each equilibrium contracts Ct (ωt ). We do not calculate each equilibrium contract but sample the state space Ωt and interpolate the
values. We use a product uniform distribution to sample the points for each period t. We then use a linear
interpolation method to calculate the equilibrium contracts for the unsampled points. Calculation of equilibrium contracts in an environment with full commitment can independently calculated since lapsation behavior
do not depend on contracts specified in future dates.
25
4.1
Full Commitment Environment
In this subsection, we consider the case where policyholders can fully commit to paying the premiums whenever the firm provides life-insurance coverage in that period. We refer to this contracting
environment as a full-commitment environment.
t )T −t is left
The first row reports the average contract profile in an environment where lt = (lt+j
j=1
unrestricted (EI), while the second row reports the average contract profiles where the participation
rule is set to 1 whenever the contract duration is within the contracting length (EII). Thus, in this
environment firms provide insurance coverage regardless of the realized states within the specified
contracting length. We refer to this environment as an incomplete contracting environment.
In the contract environment EI, the full contract length is observed. This is unsurprising given
that there are no cost to extending the contract length. If it’s optimal for firms to restrict life insurance
coverage in later periods, the participation rule allows the firm to do so without shortening the contract
length. In both full-commitment contracting, we observe that entry payments are negative so that
contract payments are back loaded. To see why this is the case, notice that our model restricts
borrowing so that the insurance companies act as a lender our contracting environments. Indeed, the
backloading of contracts decreases with assets. Notice, however, that the average backloading payment
increases in the environment EII. The further back-loading of contracts allow for higher annual premia,
which curtails inefficient life insurance holdings in the future. Indeed, we find that contracts in this
environment are structured to mitigate inefficient lapse.
To see this, consider the contract environment CI, where we find that in year 10 only 5.1% of the risk
t )T −t are uninsurable. This proportion is lower
pool covered according to the participation rule lt = (lt+j
j=1
than the expected uninsurable class in year 10, contingent on being insurable at age 40 (approximately
9%). Thus in a complete contracting environment, it is inefficient for life insurance companies to insure
some high-risk class (uninsurable) individuals. In contrast, we find that the proportion of insured lives
covered in year 10 that are uninsurable increases to 7.7% in the environment EII, which suggest that a
high degree of inefficient life insurance coverage for the uninsurable in year 10. While this proportion
is higher than what we observe in EI, it pales in comparison to the case when we restrict the entry
payment to zero. When we calculate the equilibrium in a setting where firms are not allowed to
backload contracts and operate in an incomplete contracting environment, we observe that 10.2% of
the insured individuals in year 10 are uninsurable.
Unsurprisingly, premia increases in the environment EII, and the contract length shortens almost
12 years. Extending the contract length comes at a cost since it increases the excessive life insurance
holdings.
We calculate the welfare loss associated with moving from a contract environment CI. To do this,
we calculate for each state ωt and equilibrium contract in environment Ej, say CtEj (ωt ), the agents
willingness to pay to exchange such a contract with the equilibrium contract observed in EI, say
CtEI (ωt ). That is, how much assets is the agent willing to forego to trade the status-quo contract
for a contract found in EI. We find that movig from a “quasi-complete” contracting environment to
a “complete contracting” environment amounts to an industry-wide loss19 that is greater than one
billion dollars. This figure amounts to about $100 loss per policy issued.
19
This figure is taken over contracts observed and purchased across all ages. We then multiply this aggregated
amount by a scaling factor to reflect the industry size
26
4.2
Limited Commitment Environment
We relax the full-commitment assumption by allowing policyholders to strategically lapse in this
subsection. In particular, consumers can drop their coverage and repurchase a new contract if she is
insurable. As in the previous subsection, we consider an environment with an unrestricted participation
rule (EIII) and environment where firms must provide coverage within the specified contract length
(EIV).
The environment EIII is similar to the environment analyzed in the Hendel and Lizzeri (2003)
framework. Unsurprisingly, we observe contract front-loading and find that contracts require policyholders to pay a participation fee ($2,432 on average). Such front-loading of contracts mitigate
the consumer’s incentive to strategically lapse. When comparing the contracts to the contracts observed in environment EI, we find that the front-loading of contracts minimally distorts intertemporal
consumption. In particular, the loss associated with contract front-loading amounts to a mere $160
million.
Average contracts in environment EIV mimics the contracts observed in the industry during our
sample period. Front-loading is minimal ($592 on average) and the annual premia lies roughly in
between the annual premia observed for LT10 and LT20 contracts. Unsurprisingly, contract length
is substantially reduced in this incomplete contracting environment. Lastly, we find that the loss
associated with limited commitment merely accounts for 17% of the welfare loss in environments with
contract incompleteness and limited commitment.
4.3
Welfare Gains of a Divorce-Contingent Rebate
We consider the welfare effect of a divorce contingent rebate. We first consider a counterfactual were
a regulator requires firms to offer a specified rebate if the contracts lapsed within the first ten years
and the policy holder changed marital status prior to terminating the contract. The rebate amounts
to a specified percentage of the total premia paid in the contract. Using the calibrated parameters,
we calculate the per-period value function contingent on the rebate VtR ((at , ωt−a ), where ωt−a is the
set of state variables apart from assets and R is the rebate percentage (i.e, if the consumer has paid
m amount into the policy before lapsing she receives m × R upon lapsing if she happens to be divorce
at the contract termination date). For each ωt−a and at , we defined the policy holders willingness to
pay, W R (at , ωt−a ), as the solution to the following equation:20
VtR ((at − W R (at , ωt−a ), ωt−a ) = Vt ((at , ωt−a )
(2)
In particular, W R (at , ωt−a ), is the amount of wealth the policy holder is willing to forego for the
addition of a divorce-contingent rebate in her existing contract. Let ECtR (at , ωt−a ) denote the expected
claims cost for contract issued in date t; that is, the (discounted) face value times the probability that a
claim will be initiated during the contract duration. Similarly, let Et PtR (at , ωt−a ) denote the expected
(present-value discounted) stream of premia firms expect from the contract contingent on the rebate
R issued at time t and ERtR (at , ωt−a ) be the present-value discounted expected rebate cost; these
values accounts for the consumers lapsation behavior contingent on a rebate environmnt. Given this
20
For each value ωt in the discretized state space, we calculate W R (·) using a gradient/newton-based optimization method by minimizing the squared distance between the two value functions.
27
notation, welfare estimates contingent on ωt can be defined as follows:
GtR (ωt ) = ECt0 (at , ωt−a ) − ECtR (at , ωt−a ) − EPt0 (at , ωt−a ) − EPtR (at , ωt−a )
− ERtR (at , ωt−a ) − W R (at , ωt−a )
(3)
Based on our theoretical discussion, a rebate induces efficient lapsation and reduces claims for
individuals whose valuation for insuring mortality risks is lower than the mortality insurance cost;
the first term in square bracket should reflects the effect of rebates in reducing claims cost. Rebates
naturally lower the firm’s expected costs since agents are more likely to lapse; the second square bracket
in equation (3) captures this loss. Moreover, the consumer’s valuation of a rebate (her willingness to
pay for a rebate) is less than the actual (present-value discounted) rebate amount, which is the firm’s
expected present-value cost of providing such divorce-contingent rebate. Thus, the last term in square
brackets should be negative. The intuition behind this result comes from viewing the rebate as a lottery
and the willingness to pay for such lottery as the certainty equivalent valuation of such gamble. Also,
consumers forego mortality risk insurance.
We draw samples from the SCF data of married male individuals between the ages 30-50 to
estimate the expected welfare gains in the population.21 In our counterfactual analysis, we restrict
rebates to LT20 contracts, and, in particular, require firms to rebate a percentage of the total premia
paid within the first 10 year of the contract. We fix premia for both LT10 and LT20 contracts at their
observed levels. Our draws allow for heterogeneity in income, asset level, and age: (ωi , agei ). For each
20 (ω ), and only include the
draw, we use the agents optimal purchase insurance purchase decision,Page
i
i
individuals that optimally purchase an LT20, conditional on a rebate environment, in our calculation.
Thus, our total welfare calculation can be summarized as follows:
"
#
X
R
R
20
Total Welfare Contingent on R ≡ G =
Gage
(ωi )Page
(ωi ) × Scaling Factor
i
i
i
The scaling factor scales up our estimates to reflect the industry size.22 We take the same sample
of 500 individuals used to estimate term-life ownership in the model calibration, and, for each, draw
we simulate 1000 policy experiences (lapsation behavior and mortality experience). We thus have total
of 500,000 simulated policy behavior used to calculate the expected revenues, rebates and claims cost.
We find that total welfare is maximized by offering 19% rebate. The welfare gains amount to
about $32.48 million if firms allowed offered this divorce-contingent rebate. This amounts to about
10.5 basis points of the annual premia collected in 201023 . Industry wide profits decreases (accounting
for the loss in revenue from higher lapsation, rebate cost and reduced expected claims) cost by roughly
$72.57 million , but this amount is less than the agents aggregate willingness to pay for the rebate
contracting feature (about $105.05 million).
21
We use the same weighting as in the preceding subsection.
We use the ACLI estimates of 3.8 million term policies issued in 2010. Thus our scaling factor is equal to
the 3.8 million divided by the number of policies simulated that purchased a contract.
23
Based on the ACLI 2010 Life Factbook, the LI iindustrry collected about $79,621 million in individual life
premia in 2010. Approximately 39% of these policies were term-life contracts.
22
28
5
Conclusion
This paper discussed an inherent inefficiency that naturally arises in incomplete contract markets
with limited commitment. We show that efforts to lock in agents in this setting lead to inefficient
ex-postcontract participation. Such loss can be corrected if firms compete in a richer, more complete
contracting environment.
We illustrated this inefficiency in the market for term life insurance contracts. The front-loading of
premia in these contracts and reduction of future premia allows firms to insure against reclassification
risk by reducing the policyholder’s incentive to drop a contract and shop for a possibly less expensive
policy. We argue that such reduction in future premia, paired with changes to a policyholder’s bequest
motives due to a recent divorce, may result in inefficient mortality insurance holdings.
We calibrate a lifecycle model of life insurance demand that matches patterns of life insurance
holdings observed in the Survey of Consumer Finance data, where we account for health, income and
marital shocks. Then we use this lifecycle model to analyze the welfare gains of a divorce-contingent
rebate, and we find modest welfare gains when firms operate in this counterfactual setting.
Our welfare analysis leverages the relation between non-contractible unobservable bequest shocks
and divorce rates. Since bequest motives are driven by policy holders’ marital status and policy holders
face nontrivial divorce risk, we examine welfare gains to introducing divorce-contingent rebates. Of
course, other unobservable factors or bequest motives beyond marital status may affect the valuation of
a life insurance contract. Thus our welfare calculation provides a lower bound figure for the inefficiency
caused by incomplete contracting. Viewed in another manner, our exercise estimates the amount of
“money left on the table” when divorce-contingent contracts are not drawn up.
Given the nontrivial gains to a rebate-contingent contract, one may wonder why no such contracting
feature exists in term-life contracts. Our contract feature is simple to implement. Thus, it is unlikely
that the standard transaction-cost argument rationalizes its absences. The regulations in place, which
discourage pricing insurance contracts based on anticipated lapsation, may explain the industry’s
hesitation in offering a rebate-contingent contract. Justifying these contracts to regulators immediately
requires admission of lapse-based pricing, which is a “taboo topic in the life insurance industry”
(Gottlieb and Smetters (2013)). Thus our results encourage embracing lapse-based pricing in termlife insurance contracts.
Although this paper focuses on life insurance contracts, there are other markets where one suspects
excessive contract holdings. In the subprime mortgage market, for example, concerns over prepayment penalties locking homeowners to their existing homes has led regulators to recently cap such
penalties.24 However, this law eliminates any benefits associated with borrower commitment, which
exist especially for borrowers with subprime-credit ratings. Subprime borrowers face great uncertainty
in their future credit ratings, and those who exhibit favorable shocks that lead to better credit (e.g.,
positive income shocks) are more likely to refinance their loans. If borrowers were able to commit to
a loan, then firms could anticipate better ex-post risk exposure and offer lower rates, thereby insuring
borrowers against the possibility of not being able to refinance their existing loans.
We admit that borrowers face much uncertainty in valuing their existing homes, and prepayment
penalties may prevent borrowers from moving into another, more desirable home. However, our
analysis suggests that preventing lenders from adding prepayment-penalty clauses is not the solution
to correcting this distortion. Rather, regulators should allow the addition of prepayment-penalty
24
See the recent implementation of the Dodd-Frank Act Section 1414
29
waivers provided homeowners sell their existing homes.
The subprime mortgage market is not the only market where a firms’ ability to employ lock-in
mechanisms has been limited. Scrutiny from policy makers and consumer advocates has led to a series
of judicial proceedings against the use of early-termination fees (ETF) in the cellular-phone industry.
This pressured firms in this industry to dispose of or cap ETF in their agreements.
In general, our paper advocates competition in a more complete contract setting as opposed to
simply eliminating these lock-in mechanisms. Of course, our results beg the question why some markets
do not currently operate in a more complete contracting environment. This paper does not attempt
to answer this question, but we believe that regulators may play an important role in allowing or
nudging markets to do so.
30
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31
A
A.1
Appendices
Appendix A: The Income Process
We use the 1997-2011 waves of PSID data to estimate the household income process. We use the
imputed gross household income (“total family income”) but use the CPI index to convert the nominal
income measurement into real income with the year 2005 as the base. We restrict the sample to
households with the a male household head ages 30 to 66. We exclude observations with missing
information on the household heads education, race and age. Individuals that report a negative
income are excluded in this sample, as well as individuals with income below the 1 percentile and
above the 99th percentile.
We estimate the income process in two stages. First, we estimate the following wage equation:
0
ln(yit ) = βo + βe0 educationi + βr0 ∗ racei + δt + βa0 ageit + β2,a
age2it + it
We then use the fitted residual ˆit and use a standard method of moment approach to estimate the
residual process. In particular we match the following moment conditions E[2it ] = ση2 and E[it it−j ] =
ρj ση2 . We use a diagonal weighting matrix and use a bootstrap method to calculate the standard errors.
(1)
ln yit
VARIABLES
High School Graduate (0,12)
Some College or College Graduate (12,16]
Graduate Degree (more than 16 years)
Black
Other
Age
Age2
Constant
0.409***
(0.0258)
0.712***
(0.0249)
0.942***
(0.0309)
-0.540***
(0.0183)
-0.136***
(0.0339)
0.0776***
(0.00581)
-0.000807***
(6.26e-05)
8.617***
(0.130)
Observations
R-squared
37,211
0.234
Appendix Table I: Wage Equation
This regression controls for year fixed effect. Standard errors are clustered at the household level.
32
A.2
Appendix B: NHANES III Data
We use the NHANES III interview file and link this with the publicly available NHANES III Linked
Mortality File, which provides mortality follow-up data from the date of NHANES III survey participation (1988-1994) through December 31, 2006. The NHANES file contains data for 33,994 individuals
ages 2 months and older, but we strictly use the “Adult” data file. We also restrict our analyses to
agents above the age of 29. The adult data file contains questions pertaining to the “age when 1st
told” the person had a certain disease. We use this information to impute the uninsurability risk. An
agent is uninsurable if she’s had a history of cancer (except for melanoma), diabetes, stroke, heart
attack or obese.
33