AAuctioning Annuities ∗ Gaurab Aryal † , Eduardo Fajnzylber ‡ , Maria F. Gabrielli § , and Manuel Willington ¶ February 2, 2021
Abstract
The performance of a market for annuity contracts depends on the interaction be-tween retirees and strategic life insurance companies with private information abouttheir costs. We model this interaction using multi-stage and multi-attribute auctionswhere a retiree–auctioneer–maximizes her expected present discounted utility . We es-timate the model parameters using rich administrative data from Chile and find thatretirees with low savings value firms’ risk-ratings the most. The estimates also suggestthat almost half the retirees who choose an annuity do not value bequest, and firms aremore likely to have low annuitization cost for retirees in the top two savings deciles.Counterfactuals show that under the current mechanism, private information aboutcosts harms only these high savers. Implementing English auctions and prohibitingthe use of risk-ratings lead to higher pensions, but only for these high savers.
Keywords : Annuity Contract, Annuitization Costs, Auctions, Mortality.
JEL : D14, D44, D91, C57, J26, L13.
Most countries have social security programs to help provide retirees with financial security.However, these programs are experiencing enormous pressure to remain solvent and viable. ∗ We are thankful to Lee M. Lockwood, Leora Friedberg, Gaston Illanes and Fernando Luco for theirhelpful suggestions. We also thank the seminar/conference participants at SEA 2019, APIOC 2019, 2020ES NAWM, NYU Stern and ES World Congress 2020. Aryal acknowledges financial support from theBankard Fund for Political Economy at U-Va., TIAA Institute and Wharton School’s Pension ResearchCouncil/Boettner Center. Fajnzylber and Willington acknowledge financial support from CONICYT/ANIDChile, Fondecyt Project Number 1181960. The content is solely the responsibility of the authors and doesnot necessarily represent official views of the TIAA Institute or Wharton School’s Pension Research Council. † University of Virginia, e-mail: [email protected] ‡ Universidad Adolfo Ib´a˜nez, e-mail: [email protected] § Universidad del Desarrollo and CONICET, e-mail: [email protected] ¶ Universidad Adolfo Ib´a˜nez, e-mail: [email protected] a r X i v : . [ ec on . GN ] J a n or example, the (OECD, 2019) notes that “...pressure persists to maintain adequate andfinancially sustainable levels of pensions as population aging is accelerating in most OECDcountries.” At the same time, too many people do not have enough retirement savings.Policymakers have proposed several fiscal measures to improve these programs, but there isa growing belief that it is fruitful to also use a competitive, market-based system to providebetter retirement products, e.g., annuities, (Feldstein, 2005; Mitchell and Shea, 2016).Despite this, there is little empirical research about how a market for annuities works,or how the demand and strategic supply interact to determine equilibrium pensions andretirees’ welfare. In this paper, we answer these questions in the context of the annuitiesmarket in Chile where firms have private information about their annuitization costs andthey compete in multi-stage multi-attribute auctions for savings of risk-averse retirees withdifferent mortality risks and preferences. An annuity is an ideal retirement product becauseit insures against longevity risk (Yaari, 1965; Brown et al., 2001; Davidoff, Brown, andDiamond, 2005), so a better understanding of an annuity market can help many retirees.For instance, retirement income in Chile is considered low, and according to the an-titrust authority (Quiroz et al., 2018) pensions are low because Chile uses first-price-auction-followed-by-bargaining as a pricing mechanism, and retirees have poor understanding of therole of firms’ risk-ratings (AA+ vs. AA), which soften competition. There is a proposal inChile to use English auctions and prohibit the use of risk-ratings. Using our model estimateswe evaluate the effect of this proposal on pensions and retirees’ welfare. Our counterfactualssuggest that these changes lead to an insignificant increase in pensions except for those withhigh savings. The estimates of the cost distributions suggest that the current system iscompetitive for those with low savings, and for those with high savings, firms are more likelyto have lower costs. So, using English auctions improves pensions only for the high savers.These and our other estimates of bequest preferences and welfare can be useful for coun-tries that have adopted the “Chilean model” or are considering private a market for annuities.For example, the SECURE Act of 2019 incentivizes businesses and communities in the U.S.to band together to offer annuities but is silent about “designing” such a market.Chile provides an ideal setting to study and evaluate a market for annuity contracts. Itis one of the first countries in the world to adopt a market-based system for annuities. In1981, Chile replaced its public pay-as-you-go pension system with a new system of privatelymanaged individual accounts. Moreover, since 2004, all retirees use a centralized exchange(known as SCOMP) to choose between an annuity, from among those offered by insurancecompanies, or a programmed withdrawal option, which is a default “self-insurance” product.SCOMP provides access to high-quality administrative data that span more than adecade. We observe everything about retirees that firms observe before they make their2articipation decisions. In particular, for each retiree, we observe her demographic informa-tion, savings, names of the participating firms and their offers for different types of annuities(e.g., immediate annuity, an annuity with ten years of guaranteed payments), her final choice,and her date of death, whenever applicable. All annuities are fixed and standardized.We propose a flexible but tractable model of demand and imperfectly competitive supplyfor annuities to capture the key market-features. There are at least four main componentsof a retiree’s demand for annuity: her savings, mortality risk, preferences for bequest andfirms’ risk-ratings, and the monthly pensions. So, we model each retiree as an “auctioneer”who chooses a firm and an annuity that gives her the highest expected present discountedutility . To determine each pension’s expected present discounted utility, we follow the extantliterature and assume that the preferences are homothetic with constant relative risk aversionutility and retirees’ mortality follow Gompertz distribution.In Chile, there is uncertainty about the role of firms’ risk-ratings in retirees’ decisions.To capture this uncertainty, we assume that retirees are rationally inattentive decision-makers, and do not know their preferences for risk-ratings. However, they learn about theirpreferences in the first stage by processing some costly information. We use the discretechoice framework in Mat¨ejka and McKay (2015) to model the first stage’s decision process.In the second stage, we assume that retirees know their preferences.On the supply side, we assume that life insurance companies observe everything aboutthe retirees. They know their annuitization costs before participating in a retiree-auction. The per-dollar annuitization cost is also known as the
Unitary Necessary Capital (UNC),and it captures the cost of making a survival-contingent stream of payments. In particular,UNC is the expected amount of dollars required to finance a stream of payments of onedollar until the retiree’s death and any proportional obligations to her surviving relativesif any. For example, if the UNC is 200, it means that the firm’s expected cost to providea pension of $
100 is $ Berstein (2010); Alcalde and Vial (2016, 2017); Morales and Larra´ın (2017) and Illanes and Padi (2019)also use data from Chile. However, they focus on demand (retirees) and we consider both sides of the market. Bequest preference affects demand for annuities (Kopczuk and Lupton, 2007; Lockwood, 2018; Illanesand Padi, 2019; Einav, Finkelstein, and Schrimpf, 2010), but we let its distribution to have a “mass-at-zero.” Fajnzylber and Willington (2019) find that in Chile, adverse selection is present in the dichotomouschoice between an annuity or programmed withdrawal, but conditional on choosing an annuity, there is noadverse selection across different types of annuities. Toidentify preferences for firms’ risk-ratings and the conditional distributions of annuitizationcosts, we use only the second-stage data. We can express the chosen pension as a sum of thedifference in utility from the two-most competitive firms’ risk-ratings and the losing firm’sannuitization costs. These two competitive firms’ identities vary with retirees and with themthe differences in risk-ratings and the cost for the losing firm, which allow us to apply theidentification strategies from random coefficient models and English auctions to our setting. Our estimates suggest that those who have higher savings have lower information process-ing costs. This result is consistent because those with more considerable savings tend to bemore educated and possibly have better financial literacy. Interestingly, we find that thosewho use sales-agents or directly contact insurance companies behave as if they care a lotmore about risk-rating than others. One interpretation of this result is that while everyonestarts with a prior that puts much weight on the risk-ratings, those with lower informationprocessing cost revise their weights downwards.Approximately half of all retirees who choose annuity show no preference for a bequest.There is, however, considerable heterogeneity among those who value bequests: those in thelowest and highest savings quintiles, on average, care 1.92 and 2.82 times more about theirspouse than themselves, respectively.Using our demographic information, we also estimate the survival probability for eachretiree. Comparing the expected mortality with the model-implied annuitization costs, wefind that retirees who may live longer have higher annuitization costs. We find significantheterogeneity in these costs across retirees’ and across retirees’ savings. However, the averageannuitization costs do not increase with savings, which is different than what one expectsfrom other studies, e.g., Attanasio and Emmerson (2003), who document a negative correla-tion between wealth and mortality. The average costs do not increase with savings quintilesbecause for the top two quintiles with 0.14 probability, firms’ annuitization costs are lessthan actuarially fair self-annuitization costs. This probability drops to 0.06 for the rest.To quantify the effect of asymmetric information on pensions and retirees’ ex-post ex-pected utilities, we simulate the equilibrium pension under the assumption that the firmsobserve each other’s annuitization costs while shutting down the risk-ratings. We find that Existing literature (e.g., Lockwood, 2018) identifies the bequest preference only indirectly from savings. Our identification strategy does not rely on optimal bidding in the first stage, which involves submittingbids for several types of annuities. Without the first-stage model, however, we cannot determine the ex-anteexpected profit, so we cannot identify the entry costs. However, we find that these changes do not translate into large gains in ex-postexpected present discounted utilities because either the pensions do not increase (for thosewith lower savings) or they increase (for those with high savings). These retirees have higherpensions than other retirees, and because of diminishing marginal utilities, the utility gainsfrom English auctions are minimal.In the remainder of the paper we proceed as follows. In Section 2 we introduce theinstitutional detail, and in Section 3 we describe our data. Section 4 presents our model andSection 5 discusses its identification. Sections 6 and 7 present estimation and counterfactualresults, respectively and Section 8 concludes. Appendix includes additional details.
The Chilean pension system went through a major reform in the early 1980s, when it tran-sitioned from a pay-as-you-go system to a system of fully funded capitalization in individualaccounts run by private pension funds (henceforth, AFPs). Under this system, workers mustcontribute 10% of their monthly earnings, up to a pre-determined maximum (which in 2018was U.S. $ Upon reaching the minimum retirement age –60 years for female and 65 years for men–individuals can request an old-age pension, transforming their savings into a stream of pen-sion payments. In this paper, we focus only on those retirees who have savings in theirretirement accounts, that are above a certain threshold, who can, and must, participate in Adding a reserve price has an insignificant effect on the outcomes because the benefit of reserve pricedecreases with the number of bidders, and in our sample, there are at least 13 to 15 potential bidders. This maximum, and annuities in general, are expressed in
Unidades de Fomento (UF), which is a unit ofaccount used in Chile. UF follows the evolution of the Consumer Price Index and is widely used in long-termcontracts. In 2018 the UF was approximately equivalent to U.S. $ Regulation
The Chilean government regulates and supervises AFPs, who manage retirement savingsduring the accumulation phase, and life insurance companies, who provide annuities duringthe decumulation phase. In addition, at the time of retirement, the government providessubsidies to workers who fail to save enough during their work-lives (Fajnzylber, 2018).Moreover, the life insurance industry is heavily regulated. The current regulatory frame-work for life insurance companies providing annuities recognizes that the main risks associ-ated with annuities are the risk of longevity and reinvestment. Longevity risk is taken care ofthrough the creation of technical reserves by insurers that sell annuities, which consider self-adjusting mortality tables. The government also regularly assesses the risk of reinvestmentvia the Asset Sufficiency Test established in 2007. Under this regulation, every insurancecompany is required to establish additional technical reserves, if and when there are “insuffi-cient” asset flows, following the international norm of good regulatory practices in insuranceindustries. Bankruptcy among life insurance companies is rare in Chile, but the governmentguarantees every retiree pensions up to 100% of the basic solidarity pension, and 75% of theexcess pension over this amount, up to a ceiling of 45 UFs (see footnote 7). Thus, there isenough safety nets for retirees to feel protected in case of a bankruptcy.
Retirees participating in the electronic market have three main choices: Programmed With-drawal (PW), immediate annuity (IA), and deferred annuity (DA). Under PW, savingsremain under AFP management and is paid back to the retiree following an actuarially fairbenefit schedule. In the event of death, remaining funds are used to finance survivorshippensions or, in absence of eligible beneficiaries, become part of the retiree’s inheritance. PWbenefits are exposed to financial volatility and provide no longevity insurance so that, barringextraordinarily high returns, the pension steadily decreases over time.Under both IA and DA, the retiree’s savings are transferred to an insurance company ofher choice that will provide an inflation-indexed monthly pension to her and her survivingbeneficiaries. In the deferred annuities, pensions are contracted for a future date (usually The threshold is currently established as the amount required to finance a
Basic Solidarity Pension ,which is the minimum pension guaranteed by the State. Retirees with insufficient funds will receive themfrom the AFP based on a programmed withdrawal schedule. There is a fourth, rarely chosen, pension product which is a combination between a PW and an IA. If an annuity includes, for instance, a 10 year guaranteed period, the full pension will bepaid during this period to the retiree, eligible beneficiaries or other individuals. Once theguaranteed period is reached, the contracts reverts to the standard conditions (implying acertain percentage of the original pension and only for eligible beneficiaries). For illustration of how benefits change with the annuity products and marital status,consider a male retiree who is 65 years old, has a savings of U.S. $ (cid:5) ’ in Figure 1-(a)), but after that his beneficiaries getsnothing (blue ‘ (cid:5) ’ in Figure 1-(b)). But if he chooses an annuity with GP=20, then whilealive he gets lower pension (compare red ‘+’ and blue ‘ (cid:5) ’ in Figure 1-(a)), but if he dieswithin 20 years of retirement, his beneficiaries get a strictly positive amount (purple ‘x’ inFigure 1-(b)) for 20 years, and after that it they get nothing. If he was married, then evenwith GP=0 and DP=0 (blue ‘ (cid:5) ’ in Figure 1-(c)), the beneficiaries will get a positive amount(blue ‘ (cid:5) ’ in Figure 1-(d)) after the retiree dies. The process of buying an annuity begins when a worker communicates her decision of con-sidering retirement to her designated AFP. We assume that she is then exogenously matchedwith one of four intermediaries or “channels” who can help her choose a product and firm.Out of these four channels, two (AFP and insurance company) are free and the other two(sales-agent and independent advisor) charge fees. Retirees must also disclose informationon all eligible beneficiaries. The AFP then generates a
Balance Certificate that containsinformation about the total saving account balance (henceforth, just savings), and her de-mographic characteristics. Then the decision process can be described in the following steps: Another rarely chosen clause is the spouse’s percentage increase, which maintains the full payment tothe surviving spouse, instead of the mandated 50% or 60% for regular contracts. In our sample, 99 .
9% of the chosen annuities correspond to contracts with 0, 10, 15, or 20 years of GP. The main beneficiaries are the retiree’s spouse and their children under age 24.
Benefit Schedules, by Annuity Type
65 70 75 80 85 90 95 100 105 110
Age P en s i on ( $ ) (a) Pensions for Unmarried Retiree GP=0, DP=0GP=20, DP=0GP=0, DP=3GP=20, DP=3
65 70 75 80 85 90 95 100 105 110
Age P en s i on ( $ ) (b) Bequests for Beneficiaries of an Unmarried Retiree GP=0, DP=0GP=20, DP=0GP=0, DP=3GP=20, DP=3
65 70 75 80 85 90 95 100 105 110
Age P en s i on ( $ ) (c) Pensions for Married Retiree GP=0, DP=0GP=20, DP=0GP=0, DP=3GP=20, DP=3
65 70 75 80 85 90 95 100 105 110
Age P en s i on ( $ ) (d) Bequests for Beneficiaries of a Married Retiree GP=0, DP=0GP=20, DP=0GP=0, DP=3GP=20, DP=3
Note.
The figure shows the survival-contingent benefit schedules for retirees and their beneficiaries for arepresentative retiree in our data, who is a 65 years old male and with savings of U.S. $ mortality table . GP standsfor guaranteed period (in years) and DP stands for deferred period (in years).
1. The retiree requests offers for different types of pension products (described above). Upon request, insurance companies in the system have 8 business days to make anoffer (for every requested annuity products).2. These offers (i.e., bids) are collected and collated by the SCOMP system and presentedto the retiree as a
Certificate of Quotes.
The certificate is in the form of a table, onefor each type of annuity, sorted from the highest to the lowest pensions along with thecompany’s name and risk-rating.
3. The retiree can choose from the following 5 options: (i) postpone retirement; (ii) fill anew request for quotes (presumably for different types of annuities); (iii) choose PW;(iv) accept one of the first-round offers for a particular type of annuity; or (v) negotiatewith companies by requesting second-round offers for one type of annuity. In the lattercase, firms cannot offer lower than their initial round offers, and the individual can Retirees can request quotes up to 13 different variations, including PW and annuities with differentcombinations of contractual arrangements. In the case of guaranteed periods, the certificate also includes a discount rate that would be applicablein the event of death within the GP. In absence of legal beneficiaries, other relatives can receive the unpaidbenefits in a lump sum, calculated with the offered discount rate. For an example see Figure 1. We have data on the annuity market in Chile from January 2007 to December 2017. Weobserve everyone who used SCOMP to buy an annuity or choose PW during this period.As mentioned before, we observe everything about a retiree that participating life insurancecompanies observe about them before they make their entry decisions and their first-roundoffers. We observe all the offers they received, their final choice, and whether they choseit in the first round or the second round. Our working assumption is that retirees use thefirst-round offers to decide between different types of annuities in most cases. Furthermore,conditional on choosing the annuity type, they bargain with companies for better pensionsin the second-round.
Table 1:
Share of Pension Products
Product Obs. %PW 78,161 32.7Immediate annuity 87,115 36.4Deferred annuity 73,272 30.6Annuity with PW 343 0.9Full Sample 238,891 100
Note.
The table shows the distribution of retirees across different annuity products. We restrict ourselvesto annuities with either 0, 10, 15 or 20 years of guaranteed periods or at most 3 years of deferment.
We focus on individuals without eligible children considering retirement within ten yearsof the “normal retirement age” (NRA), which is 60 years for a woman and 65 years for aman. The result is a data set with 238,891 retirees, with an almost even split between PW,immediate annuities, and deferred annuities, see Table 1. Less than 1% of retirees chooseannuity with
PW and so we exclude them, leaving a total of 238,548 retirees. In Table 2, we present the sample distribution by retirees’ marital status, gender, andage at the time of their retirement. Around 56% retire at their NRA, and 79% retiree ator at most within three years after NRA (rows 2 and 3), and married men are half of allretirees. Retirees also vary in terms of their savings; see Table 3. The mean savings in A firm that does not offer in the first-round cannot participate in the second-round. The fourth option allows retirees to split their savings into PW and annuity.
Age Distribution, by Gender and Marital Status
Retiring Age S-F M-F S-M M-M Total
Before NRA 1,871 1,771 4,714 22,142 30,498At NRA 20,789 22,475 17,114 72,572 132,950Within 3 years after NRA 14,470 16,797 4,447 19,086 54,800At least 4 years after NRA 6,900 6,715 1,251 5,434 20,300Full Sample 44,030 47,758 27,526 119,234 238,548
Note.
The table displays the distribution of retirees, by their marital status, gender and their retirementages. Thus the first two columns ‘S-F’ and ‘M-F’ refer, respectively, to single female and married female,and so on. NRA is the ‘normal retirement age,’ which is 60 years for a female and 65 years for a male. our sample is $ $ $ Savings, by Retirement Age and Gender
Mean Median P25 P75 NRetiring Age
Before NRA 185,660 129,637 73,104 245,857 30,498At NRA 89,907 60,023 41,521 103,680 132,950Within 3 years after NRA 115,666 87,126 54,353 135,562 54,800At least 4 years after NRA 141,673 101,594 58,815 168,202 20,300Full Sample 112,471 74,515 46,449 132,356 238,548
Gender
Female 97,308 81,180 51,817 121,633 91,788Male 121,955 69,372 43,818 147,184 146,760Full Sample 112,471 74,515 46,449 132,356 238,548
Note.
Summary statistics of savings, in U.S. dollars, by retiree’s age at retirement, and by retiree’s gender.
A retiree receives approximately 10.6 offers for several types of annuity, and the number ofoffers increases with savings. For example, retirees with savings at the 75 th percentile ofour sample get an average of 12 . th percentile get an average of7.8 offers. It is reasonable to assume that retirees with higher savings are more lucrativefor the firms, and therefore more companies are willing to annuitize their savings. If thosewith higher savings, however, also live longer than those with lower savings, then it meansthat annuitizing higher savings are costlier for the firms. To determine which of these twoopposing forces dominate, we estimate the annuitization costs and mortality by savings.Moreover, there is also substantial variation in the pensions offered across life insurancecompanies and retirees; see Table 4. For an immediate annuity, retirees get an average offer10f $ $ $
479 for immediate annuities and $
412 for deferred annuities, while for men, theyare $
631 and $
473 respectively. These features are consistent with men having larger savings and shorter life expectancy than women (see Table 7).Table 4:
Monthly Pension Offers, by Annuity type and Gender
Savings Savings Savings Savings SavingsAnnuity Type Gender Mean Median Q1 Q2 Q3 Q4 Q5
Immediate Female 479 414 202 288 385 510 857Male 631 435 200 269 372 585 1329Full Sample 570 423 201 278 378 556 1152Deferred Female 412 374 190 258 349 463 714Male 473 356 187 241 331 529 1019Full Sample 446 365 189 248 339 500 882
Note.
Summary of average monthly pensions (in U.S. dollars) offers received in the first-round.
In our empirical model, we rationalize this variation in pension offers by allowing firmsto have heterogeneous costs (UNCs) of annuitization. We assume that only the firm knowsits annuitization cost, which can depend on retirees’ savings. An important exogenous factoraffecting UNCs is the market interest rate, which affects the opportunity cost of offering apension at retirement. Our sample spans a decade, so we observe substantial variation ininterest rates, which causes exogenous variation in annuitization costs.
Once the participating companies make first-round offers, one for each type of annuity theretiree requests quotes for, she can either choose from one of those offers or buy a PW, orinitiate the second-round bargaining phase. Table 5 displays the distribution across thesestages. Most retirees who choose PW choose that in the first round (98.1%), and mostretirees (86.9%) who choose annuity choose in the second round. As we can see, 2,979retirees opt for the second round but choose an annuity quote from the first round.In Table 6 we present information about the chosen annuities: (i) the total number ofaccepted offers by the type of annuity; (ii) the average number of first-round and second-round offers received for the chosen annuity; (iii) the number of accepted second-round offers;(iv) the average percentage increase in pension offers from first-round to second-round (onlyfor the accepted choice); (v) the percentage of retirees who requested at least one second-round offer; (vi) the percentage of retirees who chose the highest paying alternative; and(vii) the percentage of retirees who chose a dominated option, in terms of either pension11able 5:
Number of Retirees who choose in First- or Second-Round
Round/Choice PW st round nd round Total st round 76,690 18,001 0 94,6912 nd round 1,471 2,979 139,407 143,857Total 78,161 20,980 139,407 238,548 Note.
Round refers to whether retirees chose in the first- or in the second-round. (with the same risk-rating) or risk-ratings (with the same pension) or both.Table 6:
Summary of Accepted Annuities
GP st Round Offers nd Round Increase Requested nd Round Best DominatedImmediate
Total
Deferred
Total
Note.
The table shows the number of chosen annuities by type of product, the average number of first-roundoffers received for the chosen annuity, the number of accepted offers that resulted from second-round offers,the average percentage increase between the first-round and second-round offers (for the accepted choice),the percentage of individuals who requested at least one second-round offer, the percentage of retirees whochose the highest paying alternative option and the percentage of individuals who chose an offer that wasdominated by another alternative with same (or better) credit rating.
From Table 6, we see that some retirees do not choose the annuity with the highestpension. One way to rationalize this behavior is to recognize that besides pensions, retireesalso care about firms’ risk-ratings. After all, risk-rating is a proxy of financial health, and itis also widely advertised as such. A retiree can prefer lower pensions from healthier firms toa higher pension from a less healthy firm.This rationalization, however, begs the following follow-up questions: Is there an objec-tive (i.e., correct) trade-off between pension and risk-rating, and should it be homogeneousor vary across retirees? If it is heterogeneous, should it increase or decrease with savings?On the one hand, because of the regulation, those with lower savings are less exposed to therisk of firms defaulting than those with higher savings; those with higher savings should caremore about the risk-ratings than those with lower savings. On the other hand, savings posi-tively correlated with education, so those with higher savings will process publicly availableinformation about past defaults. Thus in the case of Chile suggests that retirees should not12are much about the risk-rating. Finally, how does this trade-off vary with preferences forbequests? To determine which of these countervailing forces dominate and how pensions andutilities would change under alternative market rules, later we estimate a structural model.
A determinant of annuity demand and supply is the retiree’s expected mortality. We observeevery retiree when they entered our sample, i.e., their retirement age and their age at deathif they die by the end of our sample period. Using this information, we estimate a mixedproportional hazard model (defined shortly below) and use the estimated survival functionto predict the expected life conditional on being alive at retirement.Let the hazard rate for retiree i with socio-economic characteristics X i at time t ∈ R + ,that includes includes i ’s age, gender, marital status, savings and the year of birth, be h it = lim dt → d Pr( m i ∈ [ t,dt ) | X i ,m i ≥ t ) dt = h ( X i ) × ψ ( t ), where m i is i ’s realized mortality date, ψ ( t ) is the baseline hazard rate. Furthermore, let the hazard function ψ ( t ) be given byGompertz distribution, such that the probability of i ’s death by time t is F m ( t ; λ i , g ) =1 − exp( − λ i g (exp( g t ) − λ i = exp( X (cid:62) i τ ).The identification of such a model is well established in the literature (Van Den Berg,2001). The maximum likelihood estimated coefficients of the hazard functions suggest asmaller hazard-risk is associated with younger cohorts, individuals who retire later, withfemales, those who are married, and those with higher savings. Using these estimates, wereport the median expected lives, by gender and savings quintile, and their standard errorsin Table 7. Overall, 50% of males expect to live until 86 years, and 50% of females expectto live until they are 94.9 years old. As we can see, those who have larger savings also tendto live longer than those with lower savings.
We observe retirees with one of the four intermediary channels (AFP, Insurance Company,Sales Agent, or Independent Advisor) to assist them with their annuitization process. If andwhen such an intermediary’s incentives do not align with those of a retiree, then retireesdo not always choose the “best” option for them. The misalignment of incentives may beparticularly relevant for sales-agents, who receive their intermediation fee only if the retireechooses the sales agent’s firm. In other words, it is possible and very likely that those withsales-agent would appear to value the non-pecuniary benefits of a company more than the For robustness, we estimated the Gompertz model using data from before the introduction of SCOMP.The estimates are qualitatively the same. For instance, the predicted median expected life at death is 85and 96 for males and females, respectively. Both of these results are available upon request.
Median Expected Life, by Savings Quintile
Savings Male Female Overall
Q1 85.15 93.80 86.89 (5.79) (6.03) (5.82)
Q2 85.86 94.24 87.64 (5.81) (6.06) (5.84)
Q3 86.45 94.83 88.23 (5.83) (6.09) (5.88)
Q4 87.62 95.48 89.40 (5.88) (6.12) (5.95)
Q5 90.87 97.25 93.52 (6.01) (6.21) (6.11)
Total (5.82) (6.09) (5.94)
Note.
The table shows the predicted median expected life at the time of retirement implied by our estimatesof the Gompertz mortality distribution. Standard errors are reported in the parentheses. pecuniary benefits. We allow preferences for risk-ratings and information processing coststo depend on the channel to capture this effect.To account for observed differences among retirees, we consider the money’s worth ra-tio (henceforth, mwr ), which is the expected present value of pension per annuitized dollar.If mwr = 1 then it means the retiree expects to get $ mwr offered in thefirst-round (left panel) and mwr accepted by the retirees (right panel). The mean and themedian mwr of the offers, by channels (AFP, Insurance Company, Sales Agent, Advisor), are(0 . , . , . , . . , . , . , . . , . , . , . . , . , . , . mwr .We use a multinomial Logit model to consider if observed differences among retirees canexplain the differences in their channels, see Table 8. In particular, we estimated the log-odds ratio of having one of the three intermediary channels relative to the AFP and foundthat some characteristics and the channel are correlated. For instance, those who have lowersavings, retire early, are male or unmarried are more likely to use sales-agents than AFP.We treat the channel as exogenous for model tractability. There are two reasons whywe believe this is not a strong assumption in our context as it might appear. First, severalanecdotal evidence from Chile suggests that most people rely on word-of-mouth when itcomes to a channel. Second, and as mentioned previously, we observe everything the firmobserves about a retiree when making the first-round offers. When we estimate the preferenceparameters, we estimate them separately for several groups that we define based on age,14igure 2: CDFs of Offered and Accepted MWR, by Channel
Note.
Distributions of the offered and chosen mwr (left panel vs. right panel), by channel.
Table 8:
Intermediary Channel - Estimates from Multinomial Logit
Regressors \ Channels Insurance Company Sales-Agent Advisor
Savings ( $ million) 0.629*** -0.857*** -0.130***(0.128) (0.0436) (0.0447)Age 0.0131 -0.0408*** -0.0816***(0.00857) (0.00189) (0.00218)Female 0.437*** -0.0588*** -0.124***(0.0546) (0.0120) (0.0140)Married 0.0245 0.0620*** 0.0874***(0.0491) (0.0107) (0.0127)Constant -5.029*** 2.333*** 4.326***(0.560) (0.123) (0.142)N 238,548 238,548 238,548 Note . Estimates of multinomial logit regression for channels, where the baseline choice is AFP. Standarderrors are in parentheses, and ∗∗∗ , ∗∗ , ∗ denote p-values less than 0 . , .
05 and 0 .
1, respectively. gender, savings, and channels. Estimating preference parameters separately for each groupallows us to control the effects of the potential selection on unobservable characteristics.For instance, from Table 9, we see that channels affect the outcomes. Out of 109,786retirees who choose AFP, only 25.1% choose the second-round, whereas the shares are 85.2%,92.0%, and 87.8% for Insurance Company, Sales Agents, or Advisors, respectively. Most ofthose who choose PW have AFP, and those with sales-agents are least likely to choose PW.Our empirical framework can capture the effect of channels on outcomes. In particular,we posit that channels affect the cost of acquiring information about the importance of risk-rating. For instance, we allow those retirees who use sales-agents to act “as if” they havea higher cost of acquiring information about the trade-off between risk-rating and pensions.We assume that in the first stage, retirees are rationally inattentive with respect to their15able 9:
Retiree choices, by Intermediary Channel
N Requests nd Round Chooses PW Chooses in nd Round
AFP 109,786 0.251 0.661 0.235Company 2,169 0.852 0.066 0.817Sales-agent 79,120 0.920 0.030 0.907Advisor 47,473 0.878 0.066 0.846Full Sample 238,548 0.603 0.328 0.584
Note.
Proportion of retirees separated by their choices and their channel. preference for risk-ratings, but they know their preferences in the second-stage.
In our sample, we observe 20 unique life insurance companies, and they differ in terms oftheir annuitization costs, which are unobserved, and in terms of their risk-ratings. Table10 shows the distribution of risk-ratings. The ratings mostly remain the same over time,and most companies have high (at least AA) risk-ratings. For our empirical analysis, wetreat these ratings as exogenous, and group them into three categories: 3 for the highest riskrating of AA+, 2 for all the risk-ratings from AA to A, and 1 for the rest.Table 10:
Risk-Ratings
Rating Frequency % Cumulative %
AA+ 155 24.64 24.64AA 245 38.95 63.59AA- 171 27.19 90.78A+ 2 0.32 91.1A 15 2.38 93.48BBB+ 1 0.16 93.64BBB 6 0.95 94.59BBB- 15 2.38 96.98BB+ 19 3.02 100Total 629 100
Note.
The table shows the distribution of quarterly credit-ratings from 2007-2018.
Although there are 20 unique firms, not all of them are active at all times, and notall participate in every auction. On average, 11 companies participate in a retiree-auction,which suggests that the market is competitive. We define potential entrants (for each retiree-auction) as the set of active firms that participated in at least one other retiree-auction inthe same month. In our sample, retirees have either 13, 14, or 15 potential entrants.16he participation rate, which is the ratio of the number of actual bidders to the numberof potential bidders, varies across our sample from as low as 0.08 to as high as 1, with meanand median rates of 0.73 and 0.78, respectively, and a standard deviation of 0.18. Thus,it is likely that a firm’s decision to participate depends on its financial position when aretiree requests quotes and this opportunity cost of participating can vary across retirees. To capture this selection, in our empirical application, we follow Samuelson (1985) to modelfirms’ entry decisions, which posits that firms observe their retiree-specific annuitization costbefore entry. This entry method is a reasonable assumption in our setting because firms havesophisticated models to predict retirees’ mortality and the expected returns from the savings.We treat firms as symmetric bidders with annuitization costs independently and iden-tically distributed with some (unknown) distribution function. We do not observe firms’annuitization costs, and so, we cannot directly test this assumption. However, we can per-form a diagnostic test and check if the firm-specific pension (bid) distributions are different.If they are not different from one another, then our symmetry assumption is a reasonablefirst step.However, to perform this test, we have to “control” for all relevant factors that can affectthe pension. For instance, retirees with high savings can be lucrative because the total gainfrom annuitizing their savings will be large. However, as we have seen above, these retireesare expected to live longer. To compare the bids across firms, we have to estimate the expected discounted life for each retiree, which we refer to as
U N C i where the subscript i refers to retiree i . This U N C i is different from U N C j , where the latter refers to a firm j ’scost. We formally define U N C i when we present our model’s supply side, and in AppendixA.1 we detail how we use the estimates from the mortality distribution to calculate U N C i .But for now, it sufficient to know that U N C i depends on i ’s estimated mortality and thediscount factor. A retiree who expects to live longer will have a larger U N C i and will becostlier for firms to annuitize, but these costs are unobserved.For each of the 20 firms, in Figure 3 we present the histograms and scatter plots ofmonthly pension per annuitized dollar (which is known as the monthly pension rate) andthe U N C i s of all the retirees that the firms make offers to in the first stage. Using pensionrates instead of pensions allows us to compare across different retirees. As we see, indeed U N C i and pension rates are negatively correlated, and there are no differences across firms. Using a Poisson regression of the number of participating firms on the retiree characteristics, we findthat one standard deviation increase in savings, which is approximately $ We tested this selection by estimating a Heckman selection model with the number of potential biddersas the excluded variable and found strong evidence of negative-selection among firms.
U N C i ’s, we can compare pensions across firms. We normalize the offeredFigure 3: Pension Rates and
U N C i for each Firm Note.
These are histograms and scatterplots of monthly pension rate, i.e., the ratio of monthly pension toannuitized savings, and the
U N C i of the retirees the firms make an offer. There are twenty firms, so thereare twenty sets of four subfigures each. Clockwise, the first sub-figure is the histogram of U N C i , and thesecond sub-figure is the scatterplot of the pension rates (on the x-axis) and U N C i (on the y-axis). The thirdsub-figure is the histogram of the pension rates, and the last sub-figure is the scatterplot of U N C i and thepension rates. pension rates (ratio of monthly pension to annuitized savings) across firms and compare thedistributions across firms. We say that firms are asymmetric if the distributions are different18nd symmetric otherwise. For each firm we estimate Pension-Rate i,j = constant + β × U N C i + β × Age i + β × Gender i + β × Marital Status i + β × Spouse’s Age i + β × Guaranteed Months i + β × Potential Bidders i + ε i,j , (1)using ordinary least squares method, and predict the residual ˆ ε i,j for retiree i and firm j .In Figure 4 we show the Kernel density estimate of the firm-specific distribution of ˆ ε i,j . Wecan see that these 20 distributions are very similar, so it is reasonable to say that firms havesymmetric cost distribution.Figure 4: Distributions of Homogenized Pension Rates, by Firms -5 -4 -3 -2 -1 0 1 2 3 4Homogeized Monthly Pension per Annuatized Dollar 10 -3 CD F Firm 1Firm 2Firm 3Firm 4Firm 5Firm 6Firm 7Firm 8Firm 9Firm 10Firm 11Firm 12Firm 13Firm 14Firm 15Firm 16Firm 17Firm 18Firm 19Firm 20
Note.
Kernel estimates of the distribution residuals ˆ ε ij from Equation (1), one for each firm. In this section, we introduce our model. To model the demand, we consider the decisionproblem facing a retiree who uses SCOMP to choose a company to annuitize her savings.To model the utility from an annuity, we closely follow the extant literature on annuities,particularly, Einav, Finkelstein, and Schrimpf (2010), with a modification that accounts forheterogeneous preferences for firm characteristics.As we have shown before, retirees do not always choose the best offer. To rationalizethis, we posit that besides the pecuniary aspect of an annuity, retirees also care about acompany’s risk-ratings, which is a proxy for the likelihood of default. That said, we assume19hat all retirees have a prior that puts much emphasis on risk-rating, and only those whospend some resources learning about the likelihood of default will update their prior andchoose accordingly. To capture the trade-off between pension, risk-ratings, and informationgathering, we follow Mat¨ejka and McKay (2015) and model the retiree as a rationally inat-tentive decision-maker. If a retiree chooses to go to the second-round bargaining, we assumethat she knows her risk-ratings preferences.On the supply side, we model the imperfect competition using an extensive form gamewhere the first stage is a first-price auction with independent private value and endogenousentry (Samuelson, 1985). If there is a second stage, then it is multilateral bargaining withone-sided asymmetric information. The winner of the game is not always the firm thatoffers the highest pension because the probability of winning depends on the bids and thepreferences for risk-rating and bequest, which can vary across retirees.
Here, we consider the problem faced by an annuitant i who has already decided which annuityproduct to choose (e.g., an immediate annuity with 0 guaranteed period) and is consideringbetween J i firms who have decided to participate in the auction for i ’s savings S i . The retireewill choose the firm that provides her the highest indirect utility.We assume that the utility from an annuity consists of three parts: the expected presentdiscounted utility from the monthly pension that the retiree enjoys while alive, utility shegets from leaving bequest (if any) to her kin, and her preference for firm’s risk rating.Retirees may value the risk-ratings because they may dislike firms with lower risk-ratings.However, they may not know the “correct” weight to put on these risk-ratings. To capturethis uncertainty, we model retirees as rationally inattentive decision-makers. We explain thisaspect shortly below, but for ease of exposition, we begin without rational inattention.Let ( θ i , β i ) denote i ’s preferences for bequest and risk-rating, respectively, and givensavings S are distributed independently and identically across retirees as ˜ F θ ( ·| S ) × F β ( ·| S )on [0 , θ ] × [ β, β ]. To capture the fact that retirees might not be able to afford bequest, andtherefore will act as someone who does not care about bequest we allow ˜ F θ to have a masspoint at θ = 0. Letting ζ ∈ (0 ,
1) be the probability that the retiree has θ i = 0, and let F θ ( · ) = ζ × H (0)+(1 − ζ ) × ˜ F θ ( · ) where, H (0) is a Heaviside function and ˜ F θ is the continuousdistribution on (0 , θ ] , θ < ∞ .Let P ij denote the pension offered by firm j to retiree i . Given the type of annuity andthe pension P ij , i ’s expected mortality and the mortality of her beneficiaries determine thebequest, which we denote by B ij ( P ij ). Whenever it is clear from the context, we suppress20he dependence of B ij on P ij . Let i ’s indirect utility at retirement from choosing an annuitywith pension and bequest ( P ij , B ij ) from firm j with risk rating Z i,j ∈ { , , } be U ij = U ( P ij , B ij ; θ i ) (cid:124) (cid:123)(cid:122) (cid:125) i ’s discounted utility + β i × Z j , (cid:124) (cid:123)(cid:122) (cid:125) i ’s preference for j ’s risk-rating − U i ( S i ) , (cid:124) (cid:123)(cid:122) (cid:125) outside utility (2)where the utility U i ( S i ) is the utility associated with the outside option.Next, we explain the expected present discounted utility, U ( P ij , B ij ; θ i ). For simplicity,consider only the first month after retirement, and let q i be the probability of being aliveone month after retirement. Then, the expected present discounted utility will be U ( P ij , B ij ( P ij ); θ i ) = u ( P ij ) × q i + θ i × v ( B ij ( P ij )) × (1 − q i ) , where u ( P ij ) is the utility from P ij , and v ( B ij ) is the utility from leaving a bequest B ij .Thus, the marginal utility from leaving a bequest B ij upon death is θ i × (1 − q it ) × v (cid:48) ( B ij ).Now, let us consider two periods after retirement. We have to adjust the probability thatthe retiree survives two periods given that she is alive at retirement and take into accountthat the bequest left upon death will also change, which in turn depends on whether theannuity product under consideration includes a guaranteed period.In practice, we do not know for how long i expects to live. So, to determine expectedlongevity at retirement, we estimate a continuous-time Gompertz survival function for i and her spouse (if she is married) as a function of her demographic and socio-economiccharacteristics. Once we have the survival probabilities, the expected discounted utilitiesbecome the product of u ( P ij ) and the discounted number of months i expects to live, wherethe discount factor is the market interest rate.Even with a bequest, U ( P ij , B ij ( P ij ); θ i ) has an intuitive structure. It is a sum of twoterms, one of which is the product of u ( P ij ) and the discounted number of months i expectsto live, and the other term is the product of v ( B ij ) times the discounted number of months i ’s beneficiaries expect to receive B ij . Legally, i ’s spouse is entitled to 60% of i ’s pension,and 100% during the guaranteed periods, the amount B ij may change over time.Thus, we can write U ( P ij , B ij ( P ij ); θ i ) as U ( P ij , B ij ( P ij ); θ i ) := u ( P ij ) × D Ri + θ i (cid:16) v (0 . × P ij ) × D Si + v ( P ij ) × D S,GPi (cid:17) ≡ ρ i ( P ij ) + θ i × b i ( P ij ) , (3)where D Ri is the discounted expected longevity of the retiree (in months, from the momentthe annuity payments start) and D S,GPi is the discounted number of months that the spouse21or other beneficiaries) will receive the full pension because of the guaranteed period. Fur-thermore, D Si is the discounted number of months that the spouse will receive 60% of theretiree’s pension. If the annuity has a deferred period, then the retiree gets twice her pen-sion until the annuity payment begins. So ρ i ( P ij ) = u ( P ij ) × D Ri + u (2 P ij ) × D R,DPi where D R,DPi is the expected life during the deferred period. However, a retiree can have additional wealth, besides S i , that she can use for con-sumption or bequest, especially those who are wealthy. However, we do not observe herconsumption (after retirement) or her wealth, so following the literature (Mitchell et al.,1999; Einav, Finkelstein, and Schrimpf, 2010; Illanes and Padi, 2019), we assume that re-tirees have homothetic preferences. In particular, we assume that all retirees have CRRAutility u ( c ) = v ( c ) = c (1 − γ ) − γ with γ = 3. Homothetic preferences imply that the retiree’sannuity choice does not depend on the unobserved wealth. In Appendix A.1 we detail thesteps to estimate ρ i ( P ij ) and b i ( P ij ).Substituting (3) in (2) we can express i ’s indirect utility from annuity P ij from firm j as U ij = ρ i ( P ij ) + θ i × b i ( P ij ) + β i × Z ij − U i ( S i ) . (4)Thus (4) shows that there is a trade-off between higher pensions and lower risk-ratings, butas mentioned above, we assume that i does not know her β i , but only its distribution.We follow Mat¨ejka and McKay (2015) and assume that before the retirement processbegins, i has a belief that β i i.i.d ∼ F β ( · ) with support [ β, β ], and if i wants to learn herpreference, she has to incur information processing cost, valued at α > i has first to decide how much to spend learning about β i , and after thatmake the decision. Let σ : [ β, β ] × P → Γ := ∆([0 , J +1 ) denote the strategy of a re-tiree with preference parameter β , with offered pensions P i := ( P i , . . . , P iJ ) ∈ P . Thestrategy is a vector σ ( β, P i ) ≡ ( σ ( β, P i ) , . . . , σ J ( β, P i ) , σ J +1 ( β, P i )) of probabilities, where σ j ( β, P i ) = Pr( i chooses j | β, P i ) ∈ [0 , P i ). Then by adapting Mat¨ejka and McKay These “discounted life expectancies” also have interpretation in terms of the annuitization costs. As-suming firms use the same mortality process as us and invest retirees’ savings at an interest rate equal tothe discount rate, then D Ri is the necessary capital to provide a one-dollar pension to the retiree until shedies. Similarly, D S,GPi is the necessary capital to finance a dollar of pension for the beneficiaries once theretiree is dead and until the guaranteed period expires. Finally, D Si is the necessary capital to finance adollar of pension for the beneficiaries between the retiree’s death or the guaranteed period is over (whicheveroccurs later) and until the spouse dies. The gains from trade between retirees and insurance companies comefrom the differences in risk-attitude between retirees and life insurance companies and potential differencesbetween the discount rate of retirees and firms’ investment opportunities. For simplicity, we are disregarding survival benefits during the deferment period. Deferred periods inour sample are at most three years. Thus, death probability is quite low. i chooses j is given by σ ij ( P i ) = σ j ( β i , P i ) = exp (cid:16) log σ j + Uijα (cid:17)(cid:80) Jk =1 exp (cid:16) log σ k + Uikα (cid:17) +exp ( E Uiα ) , j = 1 , . . . , J exp ( E Uiα ) (cid:80) Jk =1 exp (cid:16) log σ k + Uikα (cid:17) +exp ( E Uiα ) , j = J + 1 . (5) Next, we present the supply side, where J insurance companies participate in an auctionrun by “auctioneer” i with characteristics X i ≡ ( S i , ˜ X i ). For simplicity, we suppress thedependence on X i and treat J as fixed, but account for selection in our empirical application.Companies differ in terms of their U N C s. Thus, if j can annuitize i cheaper than j (cid:48) , then j has an advantage over j (cid:48) because all else equal, j can offer a higher pension. Let U N C Rj be j ’s unitary necessary capital to finance a dollar pension for the retiree. Similarly, we mustconsider the costs related to the bequest, which may come from two sources: a guaranteedperiod, during which after the death of the retiree the beneficiaries receive the full amountof the pension, and the compulsory survival benefit, according to which the spouse of theretiree receives after the retiree died and after the guaranteed period is over, 60% of thepension until death, see Equation (A.1). We denote by U N C
S,GPj and
U N C Sj the presentvalue of the cost of providing these two benefits. Then, j ’s expected cost of offering P ij is C ( P ij ) := P ij × ( U N C Rj + 2 × U N C
R,DPj + 0 . × U N C Sj + U N C
S,GPj ) ≡ P ij × U N C j . (6)Here, the 2 in (6) follows from our assumption that the life insurance company made thepension payments during the deferred period. Let U N C i be the unitary cost of a pensioncalculated with the retirees’ discount rate and the mortality process we estimate. For thesame retiree i , firms’ U N C s may differ from
U N C i due to the differences in their (i) mortalityestimates, (ii) investment opportunities, and (iii) expectations about future interest rates. For these reasons, it is more likely that only firm j knows its U N C j . Moreover, the ratio of U N C j to U N C i captures j ’s margin from selling an annuity to i . Henceforth we call thisratio r ij ≡ UNC j UNC i , j ’s relative cost of annuitizing a dollar.We assume the cost r ij is private and is distributed independently and identically acrosscompanies as W r ( ·| S ), with density w r ( ·| S ) that is strictly positive everywhere in its support[ r, r ]. Thus, we assume that firms are symmetric, and this is consistent with what we observein the data; Figure 4. Allowing the cost distribution to depend on S captures the fact that Firms may also have different expectations about the future’s interest rates than the retirees. j ’snet present expected profit from offering P ij , to a retiree i with S i is E Π Iij ( P ij ) = ( S i − P ij × U N C j )) × Pr( j is chosen by offering P ij | P i − j )= S i × (1 − r ij × ρ ∗ i ( P ij )) × σ ij ( P i ) , (7)where ρ ∗ i ( P ij ) ≡ P ij × U N C i /S i is the money worth ratio ( mwr ) computed using the retirees’discount rate, and σ ij ( P i ) is the probability that i chooses j given the vector of offers P i .Considering the second round, and denoting by ˜ P ij the second-round offer of firm jS i × (1 − r ij × ρ ∗ i ( P ij )) × σ ij ( P i ) + σ iJ +1 ( P i ) × E Π IIj ( ρ ∗ i ( ˜ P ij ) | r ij , P i ) , (8)is its ex-ante expected profit, where σ iJ +1 ( P i ) from (5) is the probability that i takes thebargaining option in the second round with expected profit given by E Π IIj .The two rounds are connected. First, more generous offers on the first round may lowerthe retiree’s probability of going to the second round. Second, and more importantly, eachfirm’s first-round offer is binding for the second round: a firm cannot make any second-round offer below its first-round one. Our focus in the empirical analysis will be on thesecond round. For the first period, it suffices for our purposes to argue that firms will nevermake first-round offers that, if accepted by the retiree, would render expected non-positiveprofits.Now, when we include the fact that i might request offers from A i types of annuities,insurance companies have to solve a multi-product bidding problem. As mentioned in thetiming assumptions, once i receives all the offers { P aij : a ∈ A i , j ∈ J } , she chooses a ∗ ∈ A i and then chooses the firm. Thus, with a slight abuse of notations, we can express the expectedprofit of a firm j ∈ J from an auction where i requests offers for A i types of annuities as E Π ij := (cid:80) a ∈ A i E Π ij ( a ) × Pr( i chooses a |{ P bi } b ∈ A i ; θ i ) . Thus, in the first round, when choosing P aij , firm j has to consider the competitionfrom other firms for product a and all other types of annuities in A i \{ a } . It also has toconsider competition from its offers P bij , b ∈ A i , b (cid:54) = a , which is the self-cannibalizationconsideration facing multi-product firms. Determining the equilibrium bidding strategiesfor the first-round auction, although conceptually straightforward, will require us first todetermine the equilibrium in the bargaining phase. However, irrespective of the first-roundoffers, to estimate F β and W r it is sufficient to only consider the equilibrium outcome in thesecond round. Under the assumption that by the second round, the retiree would alreadyknow her β i and has already decided which a ∈ A i to choose, the choice problem facing24he retiree is relatively straightforward: choose the offer that maximizes the utility (3).Henceforth, we focus only on the second-round bargaining, which is relatively simpler tomodel and to use for estimation.This multi-product feature means to characterize the equilibrium that first-round offersrequire solving a multi-dimensional bidding problem. The problem becomes more compli-cated when we consider that when making the first-round offers, it is unlikely that firmsknow ( β i , θ i ).In our empirical application, we only use the chosen offers from the second-round to inferthe annuitization costs’ distributions. Moreover, in the second round, it is more reasonable tothink that firms can learn retirees ( β, θ ) from the retirees. First, there are many interactionsbetween firms and retirees, so it is reasonable to assume that the firms will be able to (atleast) update their priors belief about β i . Second, given our assumption that retirees choosethe type of annuity in the first round, it is reasonably likely that in the second round, firmswill be able to know more about θ i than they did in the first round.We recognize that this is a strong assumption, but it allows us to keep the second-roundbargaining game tractable. If retirees’ preferences were their private information, it wouldlead to a bargaining game with two-sided asymmetric information. Even then, we wouldhave to make assumptions about firms’ updated beliefs about ( β i , θ i ), and if and how theupdating varies across retirees. So from here, we assume that firms know ( β i , θ i ) for thosewho opt for the second round.We model the second-round as an alternating offer bargaining process. The game’stiming is as follows: In an arbitrary order, firms sequentially choose whether to improvetheir previous offer by a fixed amount ε or to “stay.” The process ends after the round withall firms consecutively choosing to stay. Finally, the retiree then chooses any of the offers.In Lemma 1 we formalize the analysis, with the proof in the Appendix A.3.Before we proceed, we introduce some new notation. Let P max ij be the maximum firm j can offer to i without losing money, i.e., P max ij solves C ( P max ij ) = P max ij × U N C j = S i , orequivalently 1 = r ij × ρ ∗ i ( P max ij ) and let j ∗ i denote the firm that can offer the highest utilitywithout losing money, i.e., j ∗ i := arg max j ∈ J ρ i ( P max ij ) + θ i × b i ( P max ij ) + β i × Z ij . Lemma 1.
In the bargaining game, firm j ∗ i wins the annuity contract and, as ε goes to zero,ends up paying a pension ˜ P ij ∗ i such that β i × Z ij ∗ i + θ i b i ( ˜ P ij ∗ i ) + ρ i ( ˜ P ij ∗ i ) = max k (cid:54) = j ∗ i (cid:110) β i × Z ik + θ i b i ( P max ik ) + ρ i ( P max ik ) (cid:111) . (9)25he symmetric behavioral strategies that sustain this perfect Bayesian equilibrium are:1. For the retiree, choose whichever firm made the best offer (including non-pecuniaryattributes), i.e., retiree i chooses firm j ∗ i if j ∗ i = arg max j ∈ J ρ i ( ˜ P ij ) + θ i × b i ( ˜ P ij ) + β i × Z ij , where ˜ P ij refers to the last offer of firm j (or to its first-stage offer if it did not raise itduring the bargaining game).2. For a firm j , play I iff ˜˜ P ij + ε < P max ij and β i × Z ij ∗ i + θ i b i ( ˜˜ P ij ∗ i ) + ρ i ( ˜˜ P ij ∗ i ) < max k (cid:54) = j ∗ i (cid:110) β i × Z ik + θ i b i ( ˜˜ P ik ) + ρ i ( ˜˜ P ik ) (cid:111) , where ˜˜ P ik refers to the standing offer of firm k (or to its first-stage offer when we arein the initial round of the bargaining game). In this section, we study the identification of the model parameters, which include theconditional distribution of bequest preferences F θ ( ·| S ), the distribution of preferences forrisk-ratings F β ( · ), the distribution of costs W r ( ·| S ), and the channel- and savings-specificinformation processing cost α . We observe outcomes of the annuity process described abovefor N retirees who choose one of the several annuity products, where N is large.For each retiree i ∈ N we observe her socio-economic characteristics X i = ( ˜ X i , S i ), herconsideration set A i , which is the list of annuity products that she solicits offers for, theset of firms ˜ J i who could participate, the set of participating firms J i ≥
2, their risk-ratings { Z j ∈ R : j = 1 , . . . , J i } and their pension offers each product and the implied discountedexpected utilities ρ ia := ( ρ a , . . . , ρ J i a ) for all a ∈ A i . For each offer we can determine thecorresponding bequest, if any. So, for each a ∈ A i we also observe the implied discountedexpected utilities from bequest b ia := ( b a , . . . , b J i a ).Let D i ∈ { , . . . , J + 1 } denote i ’s choice in the first-stage, such that D i = j means i chose firm j , and D i = ( J + 1) means i chose to go to the second-round. Conditional on D i = ( J + 1), we also observe j ’s final choice and the chosen company’s identity.26 .1 Distribution of Bequest Preference Here we study the identification of the distribution of the preference for bequest F θ ( ·| S )with support [0 , θ ]. To this end, we rely on the fact that we observe her final choice for eachretiree, which means we know her chosen bequest. Comparing the chosen bequest and theforegone bequests, we can identify her bequest preference. In this exercise, we use only thewinning firms’ offers to “control” for the effect of risk-rating on choices.For intuition, let’s consider the case where the consideration sets have only two annuityproducts, where product 1 offers a smaller bequest–and larger pensions–than product 2. Let a ∈ { , } denote the two products. Using (4) we can write the utility from product a as U ij ∗ i a = β (cid:62) i Z ij ∗ i + ρ ij ∗ i a + θ i × b ij ∗ i a − U i ( S i ) , where j ∗ i ∈ J i is the firm chosen by retiree i . Let χ i ∈ { , } denote i ’s choice a = 1 or a = 2. Suppressing the index for retiree and winning firm, χ = 1 if and only if U ≥ U , orequivalently θ ≤ ρ − ρ b − b . Then the probability that a retiree with characteristics X choosesthe annuity with the smallest bequest isPr( χ = 1 | X ) = F θ | S (cid:18) ρ − ρ b − b (cid:12)(cid:12)(cid:12) S (cid:19) = F θ | S (cid:18) − ∆ ρ ∆ b (cid:12)(cid:12)(cid:12) S (cid:19) . The left hand side probability Pr( χ = 1 | X ) can be estimated, and we also observe the “in-difference ratio” (cid:110) ∆ ρ ∆ b (cid:111) . So if there is sufficient variation in ˜ X across retires, and sufficientvariation in the indifference ratios across retirees and firms, we can “trace” F θ ( ·| S ) every-where over [0 , θ ]. Formally, if for t ∈ [0 , θ ] there is a pair { ∆ ρ , ∆ b } in the data such that t = − ∆ ρ / ∆ b then the distribution is nonparametrically identified.If there are more than 2 products in the consideration set, i.e., A ≥