Adverse Selection in Annuity Markets: Evidence from the British Life Annuity Act of 1808

Transcription

1 Averse Selection in Annuity Markets: Evience from the British Life Annuity Act of 1808 Casey G. Rothschil December 21, 2007 Abstract We look for evience of averse selection using ata from an 1808 Act of British Parliament which effectively opene up a market for life annuities. We evelop statistical techniques for analyzing the awkwarly conense ata, an our analysis inicates significant levels of self-selection. The evience for averse selection is particularly strong among a sub-sample of annuitants whose annuities were purchase by profit-seeking speculators. We fin evience of aitional selection among self-nominate annuitants when the annuities were re-price to reflect the empirical longevity of the early annuitants. This pattern of selection effects is reminiscent of the early stages of an Akerlovian eath spiral, suggesting that averse selection may be an important factor unerlying the paucity of annuity sales in contemporary annuity markets. But the magnitues of the selection effects suggest that it is unlikely to be a complete explanation. Milebury College Economics Department. crothsch@milebury.eu. The author thanks James Poterba, Clauia Golin, Jerry Hausman, Dora Costa, Peter Temin, Georges Dionne, an Robert Prasch for helpful comments. Any errors are my own.

2 1 Introuction In 1808, the British Parliament passe the Life Annuity Act, effectively opening a market for government-provie life annuities. The unique features of the Act incluing surviving ata on the mortality histories of the annuitants provie an unusual opportunity to explore the empirical importance an consequences of informational asymmetries. Informational asymmetries have playe an important role in economic theory since the seminal works of Akerlof (1970), Spence (1973), an others. Stuying the empirical importance of averse selection has prove challenging in several respects. First, as emphasize by Chiappori an Salanié (2000), it is ifficult to empirically istinguish between moral hazar an averse selection. Secon, informational asymmetries between market participants are likely to apply to the econometrician as well; information which is not known by an insurance provier, for example, is unlikely to be observe by the econometrician. This has motivate the evelopment of tests relying only on ex-ante observable information an ex-post outcomes. As escribe in Dionne et al. (2001), these test for a correlation between insurance choices an ex-post risk, conitioning on information observable to the insurance provier. Informational asymmetries can pose empirical problems for these tests as well, since econometricians may not even have access to all information symmetrically known by the market participants. Thir, existing empirical evience for even the most robust consequences of informational asymmetries is less than efinitive. For example, Cawley an Phillipson (1999) fin no evience of selection in life insurance markets, Caron an Henel (2001) fin no evience in health insurance markets, an Chiappori an Salanie (2000) an Dionne et al. (2001) fin no evience of averse selection in auto insurance markets (overturning Pueltz an Snow s, 1994, suggestive evience). Some economists have recently argue that this absence of evience may be ue to countervailing informational avantages of the insurance provier (Villeneuve, 2003) or else to a fourth empirical challenge: the confouning effects of avantageous selection (DeMeza an Webb, 2001, Cohen an Einav, 2007, an Finkelstein an McGarry, 2006). Annuity markets provie a particularly interesting setting to stuy informational asymmetries, not least because many of these empirical challenges are less acute than in other insurance settings: moral hazar is plausibly negligible for annuities, an annuity proviers o significantly less risk- 1

3 classification as compare with other markets. Perhaps in part for these reasons, annuity markets are one of the few settings in which irect evience of averse selection has been ocumente (Finkelstein an Poterba, 2002, 2004, 2006). Unfortunately, the applicability an generality of this evience is limite for at least two reasons. First, the evience comes from a compulsory market, wherein iniviuals are legally obligate to annuitize their assets. Secon, a substantial literature ocuments the so-calle annuity puzzle, the observation that voluntary annuity markets are extremely unerevelope in spite of theoretical finings inicating that annuities shoul be extremely esirable for rational, forwar looking, risk averse retirees. Unerstaning this puzzle is crucial for interpreting any evience of informational asymmetries in annuity markets. This paper uses the 1808 Act to provie a novel empirical look at averse selection in the purely voluntary annuity market effectively opene thereby. In the spirit of Finkelstein an Poterba (2006), we provie evience of averse selection by comparing the population-average longevity with the longevities of several classes of annuitants. Our evience is inicative of averse selection within the risk-classification scheme employe by the annuity provier (the U.K. government); insofar as we o not have ata on the entire set of information available to the U.K. government, however, we cannot provie a pure test for informational asymmetries in the sense of Dionne et al. (2001). The paper procees as follows. Section 2 introuces an escribes the 1808 Life Annuity Act an its subsequent evolution. The Act s history then guies a series of empirical tests presente in Section 3. The awkwarly conense nature of the surviving ata from the annuities sol uner the act necessitates eveloping novel empirical techniques. Aitional etails on these techniques appear in an appenix. The Act s history also provies a unique opportunity to stuy the first stages of the type of eath spiral ynamics foun by Cutler an Reber (1998) (but not by Buchmuller an DiNaro, 2002) in health insurance markets. The evience from Section 3 is suggestive of a eath spiral. Since evience of a eath spiral woul potentially resolve the annuity puzzle, Section 4 escribes this puzzle in greater etail. It then presents an illustrative theoretical moel of such a market, an it takes a closer look at how our empirical results bear on the puzzle. Section 5 offers some brief concluing remarks. 2

4 2 The 1808 Act as a Testing Groun Prior to the Life Annuity Act of 1808, the British government s ebt consiste almost exclusively of Consols coupon bons with an infinite maturity. The explicit goal of the Act was to convert these perpetual obligations to finite-live ones by allowing the exchange of Consols for life annuities. 1 Since Consols were traable assets, this effectively opene up a market for government provie life-annuities. The Act was originally esigne to provie annuities at actuarially fair rates. The (semi-annual) payment provie by the annuity receive in exchange for a Consol was compute using the market price of Consols i.e., using long-term interest rates an a life table specific to the age of the iniviual upon whose life the annuity payments were contingent (henceforth the nominee). The life table use, known as the Northampton table, was thought to capture population-average mortality. 2 We can thus test for selection effects by looking at whether the longevity of the nominees exceee the population-average longevity preicte by this table. As we will see below, there is inee striking evience of this sort of averse selection: the mortality rates of the nominees were significantly lower than those preicte by the Northampton tables. Members of the British Parliament suspecte the inapplicability of the Northampton tables from an early stage, but it was not until 1823 that the government took active steps to aress it. At this point, Parliament commissione John Finlaison to stuy the mortality experience of the nominees of this an several earlier (an less significant) life-contingent ebt issues. 3 1 There may well have been more subtle motivations leaing to its passage. Spencer Perseval, aressing Parliament in 1808 (viz Henricks, 1856), argue that the Act woul allow the government to retire ebt at favorable interest rates without causing interest rates to rise an argument Murphy riicules as inicating Persival s esire to have his cake an eat it too (Murphy, 1939, page 6) but which is plausible if the government believe it coul extract surplus by filling a missing market. Alternatively, there coul have been a political esire to align the interests of the retire monie classes with the government by proviing for them a value service as argue by Weir (1989) for the French government-issue Tontines of the 18th century. 2 This table, publishe in Baily (1813) was base on the mortality experience of the resients of the town of Northampton. The original pricing of the annuities also inclue a 2% loa relative to the actuarially fair prices. 3 For a more thorough iscussion of the history of these life-contingent ebt issues, see Henriks (1856), Murphy (1939) an The Insurance Institute of Lonon (1969). 3

5 His report evelope a new set of life tables, known as the Finlaison tables, base on the observe mortality experience of the nominees. After some ebate an a brief suspension of the life annuity program, Parliament etermine to resume it with pricing base on these new tables. These ynamics allow us to test for the first stages of a eath spiral: if, subsequent to the re-pricing, the new mortality tables still significantly uner-preicte the longevity of the subsequent nominees, then the government woul have continue to lose money at the new prices, an aitional rouns of re-pricing woul have been require. This coul potentially have le to an unraveling of much or all of the market. We will show that the post-repricing nominees were inee significantly longer live than even the Finlaison tables preicte. An aitional feature useful for our stuy is the fact that the Northampton life tables use for pricing annuities uner the original act were poolegener tables. In contrast, the Finlaison tables were gener-specific. Since women are an were longer live than men (especially at annuity purchasing ages), we are able to test for selection effects by comparing the gener composition of purchases prior to an subsequent to re-pricing. Finally, the Act ha a particularly o feature, absent from moern annuity markets: the buyer of an annuity i not have to be the nominee. Rather, an iniviual coul buy an annuity whose payments were contingent on anyboy else s life. This provie an opportunity for speculation on lives: investors coul fin an nominate particularly healthy iniviuals, buy annuities contingent on their lives, an profit hansomely from the higherthan-average longevity of these nominees particularly if they bunle a collection of nominees an resol shares of the resulting income streams. 4 Anecotally supporting the presence of this type of averse selection is the following quote, taken from an informal history of the life annuities sol uner the 1808 Act written (J. Francis, 1853, p. 200): From 1809 this system continue. The speculators soon foun out that the Government charge for a life annuity affore a very remunerative investment, an the insurance offices mae consierable profits by purchasing an re-selling them. The Commis- 4 There was historical preceent for this sort of speculation. For example, Vele an Weir (1992) escribe a scheme known as the trente emoiselles e Genéve evise by Genevan bankers to nominate classes of healthy young girls an take avantage of a French life-contingent ebt issue in the 1770s. 4

6 sioners of Greenwich Hospital also selecte many of the most healthy of their pensioners an bought large annuities on them. We can exploit this feature to stuy ifferences between passive selection effects among self-nominees an speculative selection effects. As we will see shortly, speculative selection appears to have been much stronger: speculators were quite successful at selecting particularly long-live iniviuals, an they appear to have profite hansomely at the government s expense by oing so. 3 Data an Analysis This section escribes the sources an form of the ata available for analysis. It then evelops empirical techniques for using that ata to test for averse selection. Finally, it escribes the results of that test. An analysis of the implications of these results for eath-spiral ynamics an the annuity puzzle appears in the subsequent section. 3.1 The Data Data is available from two reports commissione by Parliament. Both reports were commissione to examine the profitability of the annuities sol uner the 1808 Act. The first is John Finlaison s 1829 report. His report contains ata on annuities sol between 1808 an The secon is an 1860 report by John Finlaison s son, Alexaner Glen Finlaison. It examines annuities sol between 1808 an Both reports focus on eveloping mortality tables for annuity buyers; consequently, they report ata in a highly aggregate form. John Finlaison s 1829 report contains a ata set for each gener. Each set consists of three columns of ata. The first column is a list of the number of annuities sol between 1808 an 1826 at each nominee age. 5 The secon column gives the number of nominee eaths (between 1808 an January, 1826) at each age. The final column reports the istribution of ages, in January 1826, of all nominees still living at that time. 5 As iscusse above, annuity buyers were not constraine to nominate their own lives. The ata we analyze is base on the ages of the nominee of a given contract, not the contract owner. 5

7 The 1860 report is in a similar format, but it contains five istinct threecolumn classes of ata. The first three ata sets escribe three istinct classes of the nominees of those parties who speculate in life annuities (A. G. Finlaison, 1860, page 14), henceforth speculative nominees. The annuities escribe in these sets were sol to an investor or investment group seeking to profit by selecting particularly healthy oler men an purchasing annuities contingent on their lives. Each of these three ata sets refers to annuities purchase after 1828; hence, these nominees o not appear in John Finlaison s ata set. Each of the three sets appears to represent one or more istinct investment portfolio put together by a group of speculators. The first contains 353 nominees age 59 to 64. The secon contains 288 nominees age 73-84, an the thir contains 34 nominees age The final two ata sets in A. G. Finlaison s 1860 report (one for each gener) contain all nominees nominate between 1808 an the en of 1850, excluing the speculative nominees escribe in the preceing paragraph. We henceforth refer to these nominee classes as non-speculative. It is important to note, however, that that they inclue any speculative nominees nominate between 1808 an 1828 as well as any speculator nominate annuitants from after 1828 that the government faile to ientify as such. The first column for each of the classes of ata in the 1860 report contains the number of nominees between 1808 an 1850 at each age. For the nonspeculative classes an the youngest speculative class, the secon column recors the number of nominee eaths between 1808 an May 8, 1854 at each age of eath. For the two oler speculative classes, the secon column reports eath ages between 1808 an June 10th, The final column reports the istribution of ages, on December 31, 1850, of all nominees still alive on May 8, 1854 (respectively, on June 10, 1856 for the two oler classes of speculative nominees). We illustrate the ata in Tables 1 an 2 an Figure 1, escribe below. Table 1 summarizes the ata from the two reports. From the 1829 report, we see that, of the 6892 annuities purchase between 1808 an Jan 1, 1826, 2077, or 30%, were male, an 5344, or 77.5% were still alive on Jan 1, From the 1860 report, we see that a total of annuities were purchase between 1808 an Dec 31, 1850 (excluing the speculative nominee classes). Of these nominees, 34% were male, an 41% were still living on May 8, Among the speculative nominees from the 1860 ata set, we see that all but 6

8 Table 1: Data Summary 1829 Report Purchase, Living, January Male Female Total Report, Non-Speculative Nominees Purchase, Living, May Male Female Total Report, Speculative Nominees (Males) Purchase, Living, May (June 10, 1856 ) Age Age Age Table 1: Summary of the available ata. The first ata column reports the number of iniviuals on whom annuities were purchase. The secon reports the number still living at the time of observation. The ata are broken own into three classes: ata on annuities purchase from ; ata on annuities purchase from excluing annuities purchase between 1828 an 1850 by speculators ; an annuities purchase by speculators between 1828 an (sources: J. Finlaison, 1829; A. G. Finlaison, 1860) 7

9 one of the nominees from the two oler nominee classes ha ie by June 10, , while nearly 85% of the nominees in the younger class were still living on May 8, Table 1 provies the first suggestive evience of selection effects: the composition of male an female nominees change in response to the movement from gener neutral pre-1828 pricing to gener-specific post-1828 pricing. To wit: males constitute approximately 30% of all early (pre-1826) nominees (i.e., ( ) ( ) ( ) ). In contrast, they accounte for about 41% of all later nominees (i.e., ). In other wors, relatively more favorable pricing for men unsurprisingly shifte the purchase patterns away from women an towars men. Table 2 presents the entire ata set for the youngest class of speculative nominees; the ata for the other classes of nominees is in similar form, but we omit them to save space. To reiterate: the ata is only available in highly aggregate form. All we observe is the total number of initial nominees, ea nominees, an living nominees at each age. Notably, we lack information both on when they were nominate an on whether an when any particular nominee ie. Figure 1 plots the three columns of ata for the non-selecte males nominate uring the perio. The tall, ark curve graphs the istribution of ages of the nominees at the time of nomination. It shows a moal nomination age of about 60 an a range of nominee ages from the late teens to the early 90s. The rightmost curve shows the age, at eath, of the males who ha ie by May 8, The lowest curve epicts the istribution of ages on Dec 31, 1850 of those nominees who ha not ie by May 8, Shortcomings of the Data Each of the 5542 non-speculative male nominees in the ata set implicitly appears twice in this ata: his age at nomination is recore, an either his age at eath or his age in 1850 is recore. But we lack information proviing any connection between these two appearances of this same iniviual for example, whether an when any particular age 65 nominee ie. This form of the ata is well suite for estimating a mortality table 6 The one remaining nominee ie at the age of 102 in March, 1857 (A.G. Finlaison, 1860, page 88). 8

10 Table 2: Age Speculative Nominee Data Set Age Number Nominate Number Die Number Living in 1854 ( ) (by 1854) (by age in 1850) Table 2: Complete ata from the youngest speculative nominee class. Column 2 reports the number of nominees of each age (nominate between 1828 an 1850). Column 3 reports the number of these nominees (by age of eath) who ha ie by May 8, Column 4 reports the number of the nominees (by age on Dec 31, 1850) who ha not ie by May 8, Source: A.G. Finlaison (1860). 9

11 Figure 1: Non-Speculative Male Nominees, Source: A.G. Finlaison (1860). 10

12 as was the purpose of the two reports but it has significant shortcomings as a tool for testing for averse selection. In principle, to test for selection effects, we simply want to compare the mortality experience of the nominees with their expecte mortality experience base on a reference unselecte population. If we ha isaggregate ata reporting the purchase age, the purchase year, an the eath age, if applicable, of each nominee, testing for selection woul thus be completely straightforwar. The aggregate nature of the ata makes eveloping such a test more challenging. 3.3 A Consistent Test for Averse Selection We wish to test the question: Di nominees live longer than we woul have expecte? To that en, we first evelop a test for the simpler case that applies to the two oler nominee classes: when we know the eath ages of all nominees (not just those who ha ie by 1826 or 1854). We then consier how to exten this case to the remaining classes, where a subset of eath ages is known an the age istribution of survivors is known. Take as given a population mortality table q = (q 0, q 1, q 2,, q 109, q 110 ), where q t enotes the age-t mortality hazar (i.e., the probability of ying before turning t+1, conitional on having reache age t) an where we assume that noboy survives beyon age 110. View the first column of ata from each nominee class as a known vector e = (e 0, e 1,, e 109, e 110 ), where e t is the number of age t nominees. Let the ranom vector = ( 0, 1,, 109, 109 ) enote the istribution of the actual eath ages of all of these nominees, an let enote the average eath age of the nominees, i.e. = n=110 n=0 n n n=110. n=0 n When the realize value of, an hence of, is known, we can simply test whether is significantly greater than we woul have expecte, given the ages at nomination an the population mortality table. Specifically, the test statistic Z = E [ e, q ] V [ e, q ], (1) has a stanar normal istribution (asymptotically) where E [ e, q ] an V [ e, q ] are, respectively, the expecte value an variance of given the mortality table q an the nomination ages e. We can explicitly compute these by simulation or with analytic methos. High Z scores are taken as evience of an aversely selecte nominee pool. 11

13 With the exception of the two oler classes of speculative nominees, we o not know. Instea, we know a vector early of eath ages of iniviuals who ie prior to 1826 (or 1854) an a vector l of ages of the iniviuals who were still alive. 7 This requires that we moify the simple test base on Equation 1. Our moifie test compares the expecte mean eath age conitional on early an l with the the unconitional expecte mean eath age. Specifically, the appenix establishes that the test statistic Z = E[ l, early, q ] E[ e, q ] V [ e, q ] V [ l, early, q ] has an asymptotic stanar normal istribution. Each term appearing on the right han sie of Equation 2 can be irectly compute to arbitrary precision by simulation. We compute E[ e, q ] an V [ e, q ] for a given mortality table q by simulating a large number (in practice: 4000) of final eath profiles for the nominee population e an computing the sample mean an variance of the average final eath age for this sample. Similarly, we compute E[ l, early, q ] an V [ l, early, q ] by simulating a large number of final eath profiles late for the still-living population l. We then compute the sample mean an variance of the final average eath age for the combination of this population an the early-ying population early. Our results are thus base on a series of Z-tests base on Equation (1) or (2) for whether the observe annuitant longevity significantly excees the longevity preicte by an exogenously given life table. 3.4 Results We first test for selection effects by testing whether early nominees live longer than preicte by the Northampton table. The results appear in Table 3. We see that the early nominees of both geners are conitionally 7 For the 1860 ata sets, it is important to note the iscrepancy between the thir column of ata an l. This occurs because the ata report the age as of December 31, 1850, but we know that all these iniviuals in fact live until May 8, Hence, we obtain the correct l by aging the population by slightly over 3 years. (2) 12

14 Table 3: Testing for Averse Selection Northampton Tables E( ) ˆσ Z-Score P-Value Unconitional Conitional Males O(10 7 ) Females O(10 75 ) Table 3: Testing for averse selection among early nominees. This table reports the results of a test for whether the iniviuals nominate for annuities live longer than preicte by the Northampton life tables escribe in the text. E() refers to the expecte average nominee eath age, once all nominees have ie. The columns labele Unconitional an Conitional report the unconitional expecte average eath age an the expecte average eath age conitional on the observe mortality history through The column labele ˆσ is the estimate stanar eviation of the ifference between these two columns. P-value refers to a one taile test. preicte to live significantly longer than preicte by the population average mortality tables, with males an females living an estimate average of.63 (i.e., ) an 1.53 years longer, respectively. This is strongly suggestive of selection amongst the early nominees. 8 Our secon test amounts to a test for aitional selection amongst annuitants nominate subsequent to repricing in This involves testing whether various populations of nominees live longer than preicte by the Finlaison life tables use for pricing after Table 4 reports the results. The final two rows of Table 4 help verify the valiity of the Finlaison life tables for escribing the mortality of the nominee population. They inicate that males are conitionally preicte to live an estimate average of.002 years fewer than the Finlaison tables woul suggest, while females are conitionally preicte to live an estimate average of.009 fewer years. Since both are statistically (an economically) insignificant, there is inee no evience that these early nominees live longer (or shorter) than Finlaison s tables preict. The first two rows of Table 4 report the results of testing whether the non-speculative nominees from live longer than the Finlaison tables preict. The first row of inicates that non-speculative male nominees are conitionally preicte to live an average of about 1 year longer than woul be unconitionally preicte, while Females nominate between 1808 an 1850 are conitionally preicte to live an average of about.33 8 Note that this conitional preiction likely unerstates the egree of averse selection, since the conitional preiction is mae uner the assumption that the survivors will age accoring to the Northampton tables. 13

15 Table 4: Testing for Averse Selection Finlaison Tables E( ) ˆσ Z P Unconitional Conitional N-S Males, O(10 20 N-S Females, O(10 4 S Males, O(10 19 ) S Males, O(10 7 ) S Males, O(10 8 ) Males, Females, Table 4: This table reports the results of a test for whether various classes of iniviuals nominate for annuities live longer than preicte by the Finlaison life tables escribe in the text. N-S an S refer to non-speculative an speculative classes, respectively. E() refers to the expecte average nominee eath age, once all nominees have ie. The columns labele Unconitional an Conitional report the unconitional expecte average eath age an the expecte average eath age conitional on the observe mortality history through 1854, 1856, or 1826 (for the first four rows, the fifth row, an the final two rows, respectively). The column labele ˆσ is the estimate stanar eviation of the ifference between these two columns. Z an P values refer to a 1-taile test of whether the conitional number of years live is significantly longer than the unconitional number of years live? years longer. Both are statistically significant at stanar levels. 9 The mile 3 rows of Table 4 inicate that all three classes of speculative nominees on whom we have ata live significantly longer than the Finlaison tables woul suggest. 10 There is thus strong evience of selection effects 9 The pre-1828 nominees are a subset of these nominees. Hence, this is a pure test for aitional selection amongst post-1828 nominees only insofar as the Finlaison tables accurately escribe the mortality of early nominees. For similar reasons, these results unerstate the egree of averse selection among the post re-pricing nominees insofar as they involve pooling the un-selecte early nominees with the selecte later nominees. 10 Speculators appear to have selecte oler nominees for a simple reason, iscusse by Francis (1853): The Finlaison tables were evelope base on actual mortality experiences of early nominees. Payouts from early annuities were cappe at age 75, which meant that there were extremely few nominees above this age prior to Finlaison s mortality estimates for age iniviuals are thus erive from the mortality experience of iniviuals who were nominate at a young age an subsequently grew ol likely significantly unerestimating the longevity of selecte oler lives. Francis relays anecotes suggesting that speculators unerstoo this well. Apparently, they even combe the countrysie looking for hale oler men with vali birth certificates an even resorte to paying local surgeons an pastors to keep them healthy a rare case 14

16 Table 5: Estimate Longevity Enhancements Panel A: Oler Nominee Classes LEF Selecte Males, % Selecte Males, % Panel B: Other Nominee Classes Estimate LEF Bouns Lower Boun Upper Boun Selecte Males, % % Non-Selecte 1850 Males 6.58% 10.74% Non-Selecte 1850 Females 1.74% 2.66% Table 5: This table reports estimates of the longevity enhancement factor (LEF) escribe in the text. This measures the percentage by which the longevity of a given group of nominees exceee the longevity preicte by Finlaison s mortality tables. Panel A reports the actual longevity enhancement factors for the olest two groups of selecte nominees from A.G. Finlaison s ata set. Panel B reports estimate upper an lower bouns for the longevity enhancement factors of the non-selecte groups an the youngest selecte group of nominees. among speculator-selecte nominees. It is worth emphasizing that this evience falls short of irectly inicating averse selection in the sense of Dionne et al. (2001), since there coul have been other symmetrically known information (e.g., township of birth). So we o not an cannot know whether the positive correlation between longevity an the ecision to purchase an annuity woul be present if we conitione on this information. Table 4 inicates that both the selecte an non-selecte nominees from A.G. Finlaison s ata set isplay higher longevity than the Finlaison mortality tables woul inicate. Table 5 attempts to measure how much higher the longevity was by computing a longevity enhancement factor (LEF) for each of the five groups in the ata set. This measures the percentage by which actual longevity exceee the anticipate longevity. Formally: LEF = Actual average eath age Expecte average eath age Expecte average eath age Average nomination age. (3) The top panel of Table 5 inicates that nominees from the mile an olest selecte nominee classes live an average of 25.37% an 94.02% longer from their nomination ate than the Finlaison tables inicate they woul. of moral hazar in annuity markets. 15

17 For the other groups, the structure of the ata prevents a irect computation of a LEF. Instea, the bottom panel of Table 5 reports an upper an lower boun on the LEF for the non-selecte males, females, an the youngest group of selecte nominees. The lower boun is compute uner the assumption that the nominees still living at the time of observation will age accoring to the Finlaison tables in effect assuming that all the longevity enhancement has alreay taken place. The upper boun is compute uner the assumption that the nominees still living at the time of observation will have the same LEF applie to the remainer of their lifetimes. 11 Table 5 inicates that the longevity of the speculative classes were substantially more enhance relative to expectations than the longevity of the non-speculative classes. The non-selecte males are estimate to have live between 6.58% an 10.74% longer than the Finlaison tables preicte. In contrast, the selecte groups are estimate to live more than 20% an as much as % longer than the Finlaison tables preicte. The longevity of the non-selecte female nominee class appears to be even less selecte than the males, with a LEF estimate to be between 1.73% an 2.66%. 12 The relative enhancement of the non-selecte males an the non-selecte females likely stems from the combination of gener-blin early pricing an the fact that the pre-1828 ata on which the Finlaison tables were base i not istinguish between speculative an non-speculative nominees. Since females are (an were) longer live than males at typical annuity purchasing ages, most speculative nominees from the gener-neutral pricing era are likely to have been female. Table 4 clearly inicates that speculators were quite goo at selecting long-live nominees. Hence, Finlaison s female life tables probably reflect the longevity of a substantially more selecte pool of nominees than the Finlaison s male life tables, an the aitional selection 11 Formally, the upper boun LEF solves: N T (A N +(E[A F ] A N )(1+LEF )) = N D A D +N L (A L +(E[A LD l] A L )(1+LEF )), (4) where: N T, N D, an N L are the number of nominees, the number of nominees who ha ie by the time of observation, an the number of living nominees, respectively; A N, A D, an A L are the average age of the nominees, the ea nominees, an the living nominees, respectively; an E[A F ] an E[A LD l] are the expecte average eath age the nominees at the time of nomination an the expecte average eath age of of the still living nominees at the time of observation. 12 These qualitative observations are robust to accounting for the fact that all of the longevity enhancement is attributable to the post-1826 nominees (in light of Table 4). 16

18 effects after repricing are thus relative smaller for females. 4 Moern Annuity Markets an the Annuity Puzzle There is a substantial an growing literature on the so-calle annuity puzzle, the observation very few iniviuals voluntarily choose to annuitize their retirement assets in spite of the benefits economic theory suggests they shoul provie. On the theoretical sie, Yaari (1965) showe that full annuitization shoul be optimal uner specialize circumstances, an Davioff et al. (2005) have recently shown that the theoretical preiction of significant annuitization by optimizing iniviuals is quite general. On the empirical sie, in contrast, Poterba et al. (2003), Johnson et al. (2004) an others have ocumente a marke paucity of life annuity purchases by househols in the Health an Retirement Survey. This paucity is corroborate by inustry sales ata which inicate a paltry $5.9 billion in immeiate annuity sales in in the U.S. in 2006 (LIMRA, 2007) an by international evience (James an Vittas, 2004). A number of papers have attempte to explain the annuity puzzle. Potential explanations inclue pre-existing (public or company pension) annuities, unfavorable pricing resulting from averse selection or aministrative loaing (Mitchell et al., 1999), within-househol risk pooling for marrie couples (Brown an Poterba, 2000), bequest motives (Frieman an Warshawsky, 1990), higher returns from alternative assets (Milevsky, 1998), the nee for liquiity to cover health shock expenitures (Sinclair an Smetters, 2004), an the option value of elaying annuitization (Dushi an Webb, 2004). Whether some combination of these explanations can fully resolve the puzzle remains an open question. The typical approach in this literature is to posit that househols are life-cycle utility maximizers typically with constant relative risk aversion, aitively separable utility functions an to compute the value such iniviuals woul place on having access to private annuity markets. There is some empirical support for this approach: Brown (2000) shows that iniviuals state intentions to annuitize are inee correlate with the theoretical value they woul place on annuitization accoring to such life-cycle moels. Nevertheless, a istinct weakness of this approach for aressing the annu- 17

19 ity puzzle is that quantitative conclusions regaring the value an levels of annuitization rely heavily on functional form utility assumptions. Our empirical methoology is free from functional form assumptions an therefore helps speak to the importance of averse selection of a possible resolution to the puzzle. We now present an abstract moel of an annuity market. This illustrative moel suggests three istinct possibilities: First, the market may exhibit no averse selection whatsoever. This happens when proviers can break even while selling annuities to the entire population of potential annuitants. Secon, the market may suffer from averse selection but still remain thick, with firms earning normal profits selling annuities to all but a reasonably small number of the least healthy potential annuitants. Finally, the market may be nearly or completely estroye by an Akerlovian eath spiral, wherein annuity proviers set progressively higher prices on their proucts, only to fin that aitional averse selection reners each new price unprofitable, an the entire market eventually unravels. 4.1 A Stylize Moel of Averse Selection in Annuity Markets Consier a sequence of perios t = 0, 1, 2,... In perio 0, a continuum of iniviuals retires. Each iniviual has unit wealth an has the option of exchanging her wealth for a life annuity with a constant per-perio payout of a. In perios 1, 2,..., t,..., iniviuals consume, if they are still alive. An iniviual who purchases an annuity with her wealth will thus have a perperio consumption of a for as long as she lives. Iniviuals who o not purchase an annuity consume in later perios by saving at an interest rate r. We take r = 0, purely for simplicity. Also for simplicity, we assume that, conitional on reaching perio t, an iniviual i has a privately known an perio-invariant probability S i of surviving to perio t + 1. Iniviuals iffer only in their survival probabilities, an we take the istribution of S i to be uniform on [0, S], where S < 1. Preferences over consumption streams c 1, c 2,, c t, are given by: V (c 1, c 2,, c t, ) = Si t ln(c t ). (5) t=1 18

20 Iniviuals who choose to purchase an annuity thus receive utility V i (a) = t=1 S t i ln(a) = S i ln(a) 1 S i. (6) Iniviuals who o not choose to purchase an annuity solve V A i Max Si t ln(c t ) t=1 (7) s.t. t=1 c t 1. ( ) Solving Equation 7 yiels c t = S t 1 i (1 S i ), an, Vi A = S i 1 S i ln (1 S i ) + S i 1 S i ln (S i ). Iniviuals offere annuities paying a per perio will thus choose to annuitize whenever ln(a) > ln (1 S i ) + S i ln (S i ). (8) 1 S i The right-han sie of Expression 8 is ecreasing in S i. Thus, for any given annuity payment a < 1, iniviuals with survival probabilities above the cutoff value S (a) solving ln(a) = ln (1 S (a)) + S (a) 1 S (a) ln (S (a)), (9) will purchase an annuity, while those with survival probabilities below S (a) will choose not to purchase annuities. (If a > 1, then the everyone wishes to buy an annuity an we take S (a) = 0.) We assume that annuities are sol in a competitive market, so that firms must break even on annuity sales. 13 Selling an annuity with payment a to an iniviual i yiels expecte profits 1 t=1 St ia = 1 S i 1 S i a. Equilibrium is then characterize by a value a such that E [ S i 1 S i a S i > S (a) ] = 1. To solve for equilibrium, we first etermine the annuity payment a ( S) that woul allow a firm to break even for a given cutoff S: a ( S) = [ S S S i 1 S i S i S S ] 1 = [ 1 S 13 Aing a loaing factor oes not change the qualitative results. ( 1 ln S ) 1 1]. (10) S 1 S 19