Gregorio Impavido, Craig Thorburn, Mike Wadsworth world.<sup>17</sup> The AEW is a utility based measure the amount of numeraire that a fully annuitized individual would need to achieve the same indifference curve in a nonannuitized world.<sup>18</sup> The first measure is easiest to implement as it does not involve any assumption on the shape of individuals’ indifference curves. All three measures require assumptions on: 1) the individual discount rate; 2) survival probabilities; and 3) annuity payouts. The complete terms structure of interest rate should be used to derive individuals’ discount rates as these vary with time and age. Obviously, unsatisfactory assumptions need to be made in less developed financial markets where bonds with sufficiently long duration are usually not issued. Cohort mortality tables for annuitants should be used to derive survival probabilities in order to endogenize longevity improvements<sup>19</sup> and eliminate the effect of adverse selection.<sup>20</sup> Less developed countries hardly have population mortality tables and level less frequently have cohort or easily available annuitant tables. As in the case of individual discount rates, other unsatisfactory assumptions need to be made when using these measures in less developed financial markets.

Mitchell _et al_ . (1999) use the MWR and the WE frameworks to evaluate nonparticipating, single-premium, immediate, nominal,<sup>21</sup> individual life annuities in the US.<sup>22</sup> In their calculations, the authors: 1) use the term structure of Treasury and corporate<sup>23</sup> bonds interest rates to discount future annuity payouts; 2) consider the effect of tax treatment of annuities to distinguish between before-tax and after-tax interest rates; and 3) use both cohort mortality tables for the population as a whole and for the annuitant population. They find that: 1) there exists a substantial variation in the annuity benefits paid by different insurers; 2) the expected present value of annuity payouts to a man aged 65 in 1995 was valued between 81% and 93% of the annuity premium,<sup>24</sup> implying that consumers buy a considerably less than fair insurance product; 3) the internal rates of return of annuities are between 1% and 2% below that yielded by Treasury or corporate bonds, mirroring the result in 2); and 4) that the negative margin on annuity premiums 17 For a risk averse consumer and a given level of utility, the insurance value of an annuity would cause, _ceteris paribus_ , WE to be lower than the non annuitized wealth.

18 Essentially, WE and AEW are similar to the money metric indirect utility functions that are constructed by means of the expenditure function in welfare analysis. Notice, that the EW and AEW would be one the reciprocal of the other in a world where individuals have a linear utility function and where, therefore, the ratio of the two would yield the marginal rate of substitution between annuitized and non-annuitized assets. 19 Period mortality tables report the mortality experience of a person at a given point in time. Cohort mortality tables report the mortality experience of a person at a given point in time, contingent to his/her year of birth (also called dynamic mortality tables). Cohort mortality tables incorporate longevity improvements of specific cohorts so that the survival probability at time _t_ of an individual born at time _t – n_ is generally lower than the survival probability at time _t + n_ of an individual born at time _t_ .

20 See later on the discussion on adverse selection.

21 Or in British tradition “level”.

22 The argument made for focusing on this specific subset of annuities is that these are the only type of annuities that are unambiguously linked to the insurance of longevity risk during retirement. 23 Although constructed by adding a constant to the Treasury bonds yield curve.

24 The difference is due to the use of population or annuitant mortality tables, and of the Treasury or corporate yield curve. _Ceteris paribus_ , expected present values of annuities are higher when annuitant mortality tables are used as annuitants have a higher life expectancy than the population as a whole. They are also higher when the Treasury yield curve is used, as this implies assuming a lower subjective discount rate.

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