The authors like annuities and dislike "under-annuitisation". Much of the paper is concerned with exploring the reasons for the latter, even though this only applies for the voluntary market and "Table 4.6 shows that the vast majority of annuities sold are in the compulsory market." (page 81).

The authors say that "under-annuitisation" may be "a consequence of imperfect annuity markets":

"If it is the case that under-annuitisation is a consequence of imperfect annuity markets interacting with other factors, then the policy consequence is that it may be possible to encourage greater degrees of annuitisation without changes to annuity markets if the other factors are either directly or indirectly under the control of the government." (page 89)

Most people do not share the authors' enthusiasm for annuities. They call this "the annuity puzzle", which is discussed in Section 5. The reasons for the 'annuity puzzle' starting

"Yaari's theoretical work suggests that an annuity is the best way for an individual with an uncertain lifetime to obtain a secure income... The failure of Yaari's theory to match the evidence constitutes the 'annuity puzzle'." (page 87)

They say in the Conclusions:

"The seminal paper by Yaari (1965) that demonstrates the welfare benefits of annuitising one's wealth to insure against longevity risk, sits uncomfortably with the fact that voluntary annuity markets are small. We considered a range of factors which could explain this puzzle. These included: bequests, habit formation, existence of state pension benefits, means testing, selection effects, deferred annuitisation and behavioural aspects." (page 125)

"Individuals systematically over-estimate their own individual ability." (page 106)

"It appears likely that psychological explanations, many of which might be characterised as irrational behaviour may underline much of the unwillingness to annuitise." (page 107)

Professor Yaari seems to have considered at least some of the same factors. The International Foundation for Retirement Education reports:

"(Yaari 1965) recognised that if retirees want to leave bequests to their children full annuitization is no longer optimal...Sufficiently strong bequest motives combined with the existence of Social Security and the fact that annuities are not actuarially fair could explain the limited amount of annuitization." ^x

Further reasons, not including inflation, are given in the paper why some people do not buy annuities such as:

"The demand for voluntary annuities is low, and this appears due to a combination of rational reasons due to the inflexible nature of annuity products, and a misunderstanding of the nature of mortality drag. Better explanations of the annuity products might reduce this second type of aversion." (page 10)

But "imperfect annuity markets", and for example "the inflexible nature of annuity products", does not seem to apply to compulsory annuities which are doing fine:

"Not many people do voluntarily buy annuities... But the UK market for compulsory annuities (pension annuities) is projected to grow in the future." (page 79)

With-profits annuities are explained:

"With-profits mean that the pension fund is invested in a with-profits fund of an insurance company, so that annual bonuses are generated, which allow the annuity payments to grow." (page 45)

In the case of Equitable Life payments have declined - leading to "under-annuitisation".

2. “Annuity pricing”

I have a problem with the concept of the “price” of an annuity, and other people such as Stargazer ^x also question whether this concept is valid. The problem I have is that there is a jump in the reasoning. The “price” of annuities is defined in the paper as the “money's worth”, which is calculated using a) mortality tables, b) an interest rate. In my opinion we should in the first instance just take a) mortality tables - then stop and look around.

This approach avoids half the arguments: 1. “You are using the wrong mortality tables”. 2. “You are using the wrong interest rate." - because there are no interest rate assumptions. The mortality tables I am using are PMA92 until someone suggest a better alternative. Stargazer said “No one can predict the future.”, which mortality tables such as PMA92 do to some extent. But it is actually not necessary to “predict the future”. We could instead look at the past.

The paper calculates the "money's worth" by discounting future annuity payments and comparing the result value with purchase price of the annuity. The problem with this procedure is that the resulting value is highly dependent on the rate of interest chosen for discounting. So it is possible to work backwards. Choose a value for the money's worth and then calculate the rate of interest and it is possible to argue that this is the correct value for long-term interest rates, as there is some doubt about which rate is the most appropriate, which depends for example on the maturity dates of bonds.

3. Two “yields”

Using just mortality tables we can calculate two “yields”: 1) mortality yield, 2) investment yield. The mortality yield is the annuity rate which would be obtained if annuity funds are invested at zero rate of interest (with no charges). This is 100/(expectation of life) % per annum. The investment yield is the investment return allowing for mortality at different ages. It is the rate of interest used for discounting annuity payments which produces the purchase price of the annuity.

It has been claimed that PMA92 is out-of-date and the expectations of life too low. Therefore the above table is recalculated adding 5 years to the ages when people die, that is with the mortality the same as PMA92 when five years younger.

age	years	50	55	60	65	70	74
expectation of life:	years	37.98	33.06	28.22	23.49	18.98	14.87
monthly income:	£	415	448	506	584	678	790
a) annuity rate:	% pa	4.98	5.38	6.07	7.01	8.14	9.48
b) mortality yield:	% pa	2.63	3.02	3.54	4.26	5.27	6.72
a) - b)	% pa	2.35	2.36	2.53	2.75	2.87	2.76
investment yield:	% pa	3.55	3.55	3.76	3.98	3.97	4.04

4. Conclusion

The concept of a "price" for an annuity is in my opinion unsatisfactory. It relates to insurance company charges and profits. But in this case we need to look at insurance company accounts to study the profitability of annuity business. People who buy annuities should be interested in the "yield". Rather than the "price" in the first instance we need to study the "yield". Studying the "price" is a jump into a quagmire as can be seen from this Survey of annuity pricing.

December 2006