Regarding mortality table risk, by definition it is impossible for all pensioners to live more than the average life expectancy. However, mortality tables may be outdated, underestimating life expectancy. This means that the estimated (reported) annuity interest rates, calculated using these outdated mortality tables, will be biased downward.<sup>6</sup>

# **2.1 Modigliani-Miller and the Expected Elasticity**

From the perspective of a pensioner, whether a home-made annuity is feasible is an important question. If the age at death is known with certainty, it is feasible using a ladder of risk-free bonds. If the pensioner cannot build the ladder, the LICO cannot provide one either without assuming risk. Recognizing the longevity risk, the pensioner should be willing to give up a fraction of his/her pension in exchange of longevity insurance. However, if longevity risk is largely diversifiable at no cost, in a competitive annuity industry the pensioners should not be charged for it.<sup>7</sup> An annuity can be interpreted as a long-term bond guaranteed by a LICO. If annuities are priced (as corporate bonds are) in highly competitive and informationally efficient markets, the main determinants of their interest rates should be the long-term, default-free rates and LICO credit risk. General credit-risk results should also be directly applicable to this case. Applying Acharya and Carpenter’s analysis of corporate defaultable bonds (2002), and assuming that long-term matching risk-free assets do exist, the price of an annuity _At_ should be _At_ = _Bt_ 1( − _ct_ ) (1) where _Bt_ is the price of a matching default-free bond and _ct_ is the value of the default option held by the LICO equity holders, expressed as a fraction of the value of the default-free bond. It corresponds to the option of buying the risk-free bond with the LICO’s assets, so the underlying asset is the riskless bond and the exercise price is the assets’ value. The default option is valuable if the asset-liability ratio is “high”, if there is significant mismatching and/or if assets are risky. However, it is important to note that, given an asset-liability ratio, investing in riskier assets (with higher interest rates or spreads) not necessarily implies that higher interest rates should be paid on annuities. Modigliani and Miller (1958) prove that equity absorbs the risks in the first place and that only when risk becomes “large” is it shared with debt holders.

In any case, determining whether the option to default is valuable is an empirical matter. Considering that debt-equity ratios have been much lower than their legal maximum (11 versus 15),<sup>8</sup> despite mismatching, that LICO portfolios tend to be conservative, and that risk-ratings have been in general above (local) AA, we conjecture that the default option is not significantly valuable.

> 6 For example, assume constant annual payments for 25 years and an interest rate of 4 percent. If we wrongly assume that the payments will take place only during 22 years, the reported interest rate will be 3.2 percent.

> 7 Here we think of diversification at the LICO shareholders’ level.  We do not mean to say that longevity risk is diversifiable within each LICO.

> 8 However, we need to keep in mind that these leverage ratios are calculated with outdated lifeexpectancy tables such that the true economic leverage may be larger.

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