Variable annuities and aggregate mortality risk.

Weale, Martin ; van de Ven, Justin

This paper explores the extent to which annuitants might be prepared to pay for protection against cohort-specific mortality risk, by comparing traditional indexed annuities with annuities whose payout rates are revised in response to differences between expected and actual mortality rates of the cohort in question. It finds that a man aged 65 with a coefficient of relative risk aversion of two would be prepared to pay 75p per 100 [pounds sterling] annuitised for protection against aggregate mortality risk while a man with risk aversion of twenty would be prepared to pay 5.75 [pounds sterling] per 100 [pounds sterling]; studies put the actual cost at 2.70 [pounds sterling]-7 [pounds sterling] per 100 [pounds sterling], suggesting that unless annuitants are very risk averse it is likely that existing products tend to over-insure against cohort mortality risk.

Keywords: variable annuity; aggregate mortality risk; risk aversion

JEL Classifications: D14, D91, J11, J14

I. Introduction

The removal in the UK of the requirement that people should buy annuities with savings in pension schemes has had major implications for the UK pensions industry. The Association of British Insurers has found that 2.5bn [pounds sterling] were withdrawn from pension funds by people aged 55 and over in the first three months after the requirement to buy annuities was lifted, equal to about 1 per cent of the total value of pension funds held by people aged 55 and over. Most of the withdrawals were by people liquidating their entire pension fund, and 80 per cent were made by people under the age of 65. As the ABI acknowledges, the scale of these short-run withdrawals may reflect pent-up demand for liquidity, but it is difficult to avoid the impression that there was extensive dissatisfaction with the pension products available before the reform.

There are a number of possible causes of this dissatisfaction. One is that savers do not take a rational view of the future, but rather are myopic; that is to say that they discount the future more excessively in the short run than they do in the longer term. Discussion of behavioural economics has popularised this view, although it is facile to infer that myopia necessarily leads to under-saving. Rational people who know that they are myopic may well want to lock their money away to protect themselves from the consequences of their myopia and a thorough study of the issue suggests that myopia in fact is unlikely on its own to have much impact on traditional pensions saving (van de Ven and Weale, 2010).

Another possibility is that people do not like the income profile offered by traditional products. Without being myopic, they may rationally want front-end loading so that their income and spending decline as they age. This may reflect a view that recent retirees are in a better state to enjoy discretionary expenditure than older people, possibly due to limitations consequent on declining health. Means testing of care homes and other support for old people might enhance this effect. To the extent that pension products are commonly fixed in nominal terms rather than real terms, there is, of course, an element of front-end loading, but it is not necessarily the profile that people would choose for themselves.

Furthermore, it may be the case that people find it difficult to see long life as a financial risk against which they should insure because, unlike other things protected by insurance such as the costs of meeting storm damage or road accidents, it is seen inherently as a good thing. Thus people may be put off the insurance offered by annuities because they offer 'bad value' to people who die young, even though they might not think that home insurance offers a similarly bad deal if their house fails to burn down.

Alternatively, there is the possibility that the products made available provide an excessive degree of insurance. Annuities offer fixed money or real incomes over people's lifetimes and, as a result, they are fully protected from the financial implications of both an uncertain lifespan and uncertainty about rates of return. Until the recent reforms there was a widespread belief amongst policymakers that old people needed to be protected, as far as was possible, from all forms of financial risk, and also an assumption that people with relatively small pension pots needed more protection than those with large pension pots.

Yet another possibility is the recent decline in yields on both nominal and indexed government stock, which have pushed down annuity rates on nominal and indexed debt respectively. (1) This may have created a sense that annuities are poor value especially since returns on shares and also on housing appear to have held up rather better. This increase in the risk premium on risky assets may either reflect increased uncertainty or an increase in the cost of insurance against risk. To the extent that it is perceived as the latter, it is to be expected that people would be more reluctant to purchase annuities.

Protection against risk has costs and there are good reasons for believing that the cost of protection has risen over time. As this increase in the cost of protection has risen, it seems natural that people are likely to want to buy less protection and to carry more of the normal risks of life for themselves. In this article we first discuss the evidence and reasons why costs of protection may have risen. We then focus specifically on the costs of insuring against an uncertain lifespan. We suggest a way in which a pension product might provide full protection for uncertainty about an individual's lifespan relative to the cohort to which they belong, but much less protection against the uncertainty surrounding the mortality pattern of the cohort itself. In essence this would protect people from the major element of individual risk without obliging insurers to carry the risks associated with uncertain life expectancy. We suggest that, while protection against individual survival risk is valuable, people probably do not want to pay very much to be protected from this uncertainty surrounding aggregate mortality rates.

2. Why annuities?

In the absence of any opportunity to purchase an annuity, a rational individual facing an uncertain lifespan will typically choose a path along which consumption declines over time. A simple model of consumption, for example, supposes that people decide how much to consume at age t, [c.sub.t], to maximise expected lifetime utility [U.sub.t], given their available wealth [w.sub.t], as described by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [alpha] is the coefficient of relative risk aversion, [[phi].sub.t+1,t] is the probability of surviving to age t+1 given survival to age t, [delta] represents the annual discount factor, and r is real return to wealth. The utility maximising time profile of consumption will then follow this relationship:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

If [delta]r = 1, then it is clear that the utility maximising consumption profile will be falling over time if [[phi].sub.t] < 1 while it would be flat if the consumer were immortal ([[phi].sub.t] = l, [for all]t). If r > 1/[delta] (the return to wealth exceeds the impatience of utility), then this will tend to put upward pressure on the time profile of consumption, and vice versa if r < 1/[delta].

One of the implications of a formal behavioural model like that described by equation (1) is that the optimal profile of consumption will not see wealth w exhausted until an age beyond which survival is thought to be impossible. In practice, while the risks of surviving to 100 are perhaps now material, those of surviving to 110 are extremely small. Medical advance may increase this, but it is reasonable to suppose that the chance of that increasing materially for people currently planning their retirement is remote. Unless r > 1/[delta] by a very considerable margin (suggesting a fairly extreme form of patience), the consumption path implied by equation (2) will decline with age. This profile is tempered in context of a welfare safety net that insures a minimum income stream into the future; a subject that we return to below. For the moment, note that regardless of the assumed tax and benefits framework, the basic intuition underlying standard economic theory will continue to hold: people are considered to strike a balance between the urge to enjoy the benefits of immediate consumption, against the desire to have something left over in case they survive into the future.

The desire to retain funds in case of uncertain survival implies that people will be prone to leave accidental legacies. People who die relatively early will leave relatively large bequests while those who live into extreme old age will leave little. These arguments are not greatly affected by an explicit bequest motive. People who decide that they want to leave some particular sum to any beneficiary can put the money aside and manage their remaining resources in the manner described above.

3. Annuity pricing?

In a world where the return to wealth, r, is constant, and the survival probabilities known with certainty, then the calculation of the cost, C, of an annuity which pays 1 [pounds sterling] a year to someone from age t onwards is straightforward

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where N is the maximum lifespan.

If [V.sub.j] is the value of an annuity fund at the end of period j, then the value of the fund financing an annuity of 1 [pounds sterling] to each of n people, all of whom buy the annuity at the same age, t, will follow the following recursive relationship

[nV.sub.j+1] = [nrV.sub.j] - n[[phi].sub.j,t] (4)

In expression (4) the number of initial annuitants cancels out. Nevertheless, the whole point of the fund is that the aggregate payments decrease as the annuitised population dies out. The smaller aggregate payments made late in the annuity's life permit larger aggregate payments to be made in the initial periods of the annuity while maintaining the overall sustainability of the fund; essentially, the accidental bequests of people in the fund who die early are used to finance consumption of fund survivors. This makes it possible to sustain a higher level of consumption than would otherwise be the case.

Figure 1 illustrates the issue, with reference to four alternative consumption profiles. The first profile (real annuity) describes the constant rate of income payable by an actuarially fair inflation-adjusted annuity purchased by a 65 year-old whose expected mortality is equal to that of their respective age cohort in context of a zero real interest rate. The second profile (nominal annuity) shows the consumption path followed by someone who invests in a nominal annuity and spends the dividend in each year. It is assumed that the inflation rate and the nominal interest rate are both equal to the inflation target of 2 per cent per annum.

[FIGURE 1 OMITTED]

The third profile reports the consumption profile described by condition (2), on the assumption that [alpha] = 2 and [delta] = r = 1. This consumption profile lies well below the annuitant due to the likely payment of accidental bequests; the analysis presented here suggests that someone who self-finances consumption in retirement requires 1.71 times the starting wealth of an annuitant to achieve the same expected lifetime utility at age 65.

The fourth reported consumption profile applies the same assumptions as the third profile, with the sole exception that the real return to wealth is increased from 0 per cent per annum to 1.7 per cent per annum. This adjustment is just sufficient for the consumer to expect the same lifetime welfare as an annuitant who invests the same sum at age 65 in context of a zero real rate of return. Notably, the 1.7 per cent rate differential is probably below the prevailing long-run difference between the rates of return to government debt and equities.

The self-financed profile based on a 1.7 per cent rate of return premium consumes more than the level life annuity up to the age of 85, but at the cost of more sharply declining consumption thereafter. Indeed a notable feature of both self-financed consumption profiles reported in the figure is that they show consumption falling to very low levels at high ages. This feature can be reconciled with reality by interpreting the displayed measures of consumption as the excess over subsistence expenditure financed by the state pension and other benefits available to old people.

The substantive differences displayed in figure 1 between the self-financing consumption profiles and the inflation indexed level life annuity are also likely to be exaggerated by the indexation commonly applied to such annuities. Specifically, inflation-adjusted annuities are usually indexed to the Retail Price Index which, as a result of the way in which it is calculated, overstates the consumer price inflation rate perhaps by as much as 1 per cent per annum (even if historically the error was smaller than this). Thus a 'level' indexed annuity probably, on average, allows real consumption to rise at approximately 1 per cent per annum, in contrast to the sharp falls associated with self-financed consumption profiles; it also implies a lower initial payout. In contrast, fixed nominal annuities deliver a real consumption path that will fall by the rate of inflation, and so is better aligned to the front-loaded profile identified under self-financing.

Three key conclusions concerning the appeal of annuities are consequently highlighted by the analysis reported in relation to figure 1. First, the influence of accidental bequests permits annuities to pay a higher level of consumption than self-financed savings/consumption profiles in context of the same rates of return. Secondly, the advantage to annuities posed by accidental bequests can be offset by differences in rates of return that lie within prevailing market dispersion. Thirdly, level indexed life annuities that have commonly been sold in the UK generate a profile of income that fails to reflect the front-loading of consumption evident in (optimal) self-financed profiles. Possibly potential annuitants are aware of this without appreciating the insurance which is offered.

This analysis suggests that the weak demand for annuities might be attributable to low effective rates of return on underlying annuities, and/or to substantive differences between the desired consumption profiles, and the profile described by common annuity products. On the former of these possibilities, it is important to bear in mind that rates of return and risk are closely related, and that risk is also likely to bear upon consumption preferences. Standard economic theory suggests that risk exposure provides an added incentive to save (we return to discuss the precautionary savings motive later in the paper). If self-financed individuals also chose to take on additional risk in the pursuit of added returns--as suggested by Maurer, Mitchell, Rogalla and Kartashov (2013)--then this would also tend to flatten out their preferred consumption profile. This effect would presumably unwind some of the disincentive to annuitise associated with the dis-connect between the income profile generated by standard annuities, and that desired under self-financing.

An alternative possibility for weak demand is that the effective returns underlying annuities are commonly perceived to perform poorly when measured against market alternatives. One reason that this may be the case is if administration charges associated with provision of annuities are very large. Cannon and Tonks (2011) summarise the results of a number of studies which explore the money's worth of annuities, that is the expected value of the payout as a proportion of costs. For the United Kingdom they quote results of seven studies of nominal annuities for men and five for women aged 65, averaging across these yields a money's worth for men of 95.3 per cent and 93.1 per cent for women. The money's worth for indexed annuities (two studies for men and one for women), at 85 per cent for men and 86.7 per cent for women, is materially lower. Thus the charges associated with indexed annuities are more likely to deter purchasers than are those for nominal annuities.

It may of course be that the risks underlying annuity provision may be systematically under-appreciated. One important aspect of this risk concerns perceptions of the uncertainty underlying equities investments. Barro (2006), for example, argues that Mehra and Prescott's (1985) finding that only extreme intolerance of risk would explain the high average returns on shares is attributable to a systematic under-representation of the risks associated with stock market collapses. In a similar vein, increasing life-expectancy--which poses a substantial fiscal burden on annuity providers--has been systematically under-represented by most industry experts, (2) and it is reasonable to suppose that the same is true of the public in general.

4. Annuities with uncertain mortality rates

Piggott, Valdez and Detzel (2005) discuss the organisation of annuities in context of uncertain mortality rates. Starting with a given fund, it is possible to revise the payments to annuitants in the light of i) the amount remaining in the fund and ii) revised estimates of survival rates. If [C.sub.t] is the amount in the fund when the annuitants have reached age t, and the new estimates of survival probabilities are denoted by [[phi].sup.*.sub.j,t], then the fund can pay out a revised annuity [d.sup.*] where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

If prospective survival rates have increased, then the payment will be reduced. If, in addition, previous payments have been higher than actual survival rates imply, then the dividend will be reduced further. Structuring an annuity that adapted to evolving mortality expectations in the way described here would leave annuitants entirely protected from their individual mortality risk, while at the same time carrying the risks arising from the uncertain survival prospects for their cohort. Maurer, Mitchell, Rogalla and Kartashov (2013) use the approach of Campbell and Coco (2003) to explore how people should optimally allocate their portfolios when such annuities are one of a range of assets with uncertain returns.

However, it would seem to us better if annuities were designed to take account of the attitudes of annuitants towards risk. It is generally considered that people are risk averse; that is, they prefer a certain level of consumption to a gamble that would return the same level of consumption in expectation. The preference relation described by equation (1) can capture this view. Unless [alpha] [less than or equal to] 0, the expected welfare resulting from an uncertain amount of consumption will be lower than the welfare of consuming the expected sum; the individual is risk-averse.

The certainty-equivalence of any given uncertain financial shock is calculated by solving for the sum that they would need to consume with certainty to be as well-off as they were in context of the shock. For an individual with initial wealth w in context of an uncertain shock to their wealth with expectation zero, E([epsilon]) = 0, this problem involves solving for [w.sup.*]:

U[w - [w.sup.*]) = E{U(w + [epsilon])} (6)

If a single period is considered in isolation, then it is possible to solve for the sum that individuals would be willing to pay to be protected from uncertainty (value [w.sup.*] in equation 6). A recent retiree, however, faces a sequence of periods over which uncertainty will apply when making decisions concerning annuities, and in this context no analytical solution to the certainty equivalent problem exists. This complicates any attempt to design an annuity to respond to preferences concerning risk.

Nevertheless, the certainty equivalence problem can be solved in a dynamic context, if numerical methods are employed. Suppose that [w.sub.t], is wealth at the start of period t. Then the optimal consumption profile in context of uncertainty can be found by solving numerically the recursive problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Here [V.sub.t]([w.sub.t]) represents the maximum possible expected lifetime welfare obtainable, if an individual starts with w, at time t. [V.sub.t]([w.sub.t]) = 0 in the event of death.

Deaton (1992) shows that, on the solution path for consumption defined by equation (7), the following relationship will hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The key point is that when consumption is uncertain, a consumer will choose a path for consumption which rises more steeply than if there is no future uncertainty; this is the precautionary savings motive mentioned earlier in the paper. But, since the uncertainty of consumption is a consequence of the consumption path, and not something exogenous, it is not possible to use this equation to compute the path for consumption; instead dynamic programming methods have to be used. Nevertheless, it is clear that the greater is the value of [alpha] the steeper will be the growth path of consumption. (3)

Figures 2 and 3 show fan-charts for the payouts which would be generated by annuities that protected individuals from uncertainty concerning the timing of their own death, but not from the uncertainty concerning the future mortality rates of their respective birth cohorts (as suggested above). Figure 2 reports the case of moderate risk aversion ([alpha] = 2), and figure 3 reports the case of severe risk aversion ([alpha] = 20). (4) Both charts indicate the probability of the age-specific payout rate being in particular regions shown on the fan, together with the median value of the payout.

With ([alpha] = 2) the upward slope of the median path is scarcely perceptible. On the other hand, if ([alpha] = 20) the rising path is very clear. In both cases, however, annuitants would choose an annuity which pays out less at the start so that, in the event of mortality rates being lower than expected, it is possible to maintain payment rates to some extent. This precautionary saving means that increased payouts are more likely than reduced payouts.

[FIGURE 2 OMITTED]

It is possible to use the idea of certainty equivalents to evaluate how much individuals would be willing to pay to exchange the type of annuities described in figures 2 and 3 with otherwise similar annuities that shielded them from uncertainty concerning the future evolution of the mortality rates of their respective birth cohorts. Applying this approach suggests that annuitants with [alpha] = 2 would be prepared to pay 75p for each 100 [pounds sterling] of annuitised capital, while those with [alpha] = 20 would be prepared to pay 5.75 [pounds sterling].

Data from the pensions buy-out market in the United Kingdom (Lane, Clark and Peacock, 2008), for example, value this risk at about 2.7 per cent of the capital of an average pension currently in payment, and at 5 per cent for a 65 year-old man. More recent observations (Aegon Global Pensions, 2011) point to a higher premium of 3 per cent to 7 per cent for a typical pension portfolio. A certainty equivalent charge of even 2.7 per cent is generated by a value of a close to 10. That said, these calculations relate to an annuity bought at the age of 65. Without more information on actual transactions, it is not possible to be more specific than is outlined above. Nevertheless, these results show how annuities can be designed without exposing providers to systematic mortality risk, and at the same time they provide estimates of the maximum that, given the choice, people would be prepared to pay for complete certainty.

[FIGURE 3 OMITTED]

5. Conclusions

Annuities have proven to be an unpopular investment; nominal annuities have historically been better value than index-linked annuities, but only the latter can offer a stable consumption path for the remaining lifetime of an annuitant. One possible reason for poor value is that the risks associated with uncertain mortality rates are particularly pronounced with index-linked annuities.

Here it is shown that, unless people are extremely risk averse, the amount that they are prepared to pay to protect themselves from aggregate mortality risk is small. It follows that, rather than levy charges to cover themselves from the effects of aggregate mortality risk, annuity providers should develop products which allow annuitants to carry that risk. Nevertheless, some care is needed in their design. It is not enough simply to provide annuities whose payouts are updated on the basis of past mortality and the best estimates of future mortality since these do not provide the element of precautionary saving which risk-averse annuitants would typically require.

Of course it is also true that annuitants must be prepared to trust the calculations of the annuity providers. The annuities described here are actuarially fair on a cohort basis, but the nature of the exercise is that if mortality rates are lower than expected those who die late will be disadvantaged relative to those who die early, while if mortality rates are higher than expected the reverse will be true. With relatively modest risk aversion people should rationally be prepared to carry that risk, but ex ante people may find it difficult to distinguish the precautionary saving required from low money's worth and that difficulty may be compounded if annuity providers are not trusted.

NOTES

(1) The Monetary Policy Committee's policy of quantitative easing is often thought to have contributed to this decline. However the evidence on the matter is ambiguous (Weale and Wieladek, 2016).

(2) See, e.g. Pensions Commission (2004), First Report, Chapter I.

(3) If individuals undertake precautionary saving to protect themselves from the uncertain payments generated by a mortality-adjusted annuity, then they face the same problem as non-annuitants: they may die before their precautionary savings are put to good use. Thus the precautionary saving has to be undertaken by the annuity fund itself; only in this way can the precautionary savings themselves be annuitised.

(4) Most studies suggest a value between I and 5.

REFERENCES

Aegon Global Pensions (201 I), Paying the Price for Living Longer. What is the Right Price for Removing Longevity Risk?

Barro, R. J. (2006), 'Rare disasters and asset markets in the twentieth century'. Quarterly Journal of Economics, 121, pp. 823-66.

Campbell, J., and Coco, J.F. (2003), 'Household risk management and optimal mortgage choice', Quarterly Journal of Economics, 118, pp. 1449-94.

Cannon, E. and Tonks, I. (2011), 'Annuity markets: welfare, money's worth and policy implications', Netspar Panel Paper No 24. http://www.bath.ac.uk/management/research/pdf/tonks-cannonannuity-markets.pdf.

Deaton, A. (1992), Understanding Consumption, Oxford: Clarendon Press.

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Mehra, R. and Prescott, E.C. (1985), 'The equity premium: a puzzle', Journal of Monetary Economics, 15, pp. 145-61.

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Piggott, J., Valdez, E.A. and Detzel, D. (2005), The simple analytics of a pooled annuity fund', Journal of Risk and Insurance, 72, pp. 497-520.

Van de Ven, J. and Weale, M.R. (2010), 'An empirical investigation of quasi-hyperbolic discounting', National Institute Discussion Paper No. 355, http://www.niesr.ac.uk/sites/default/files/ publications/dp355_0.pdf.

Weale, M.R. and Wieladek, T. (2016), 'What are the macroeconomic effects of asset purchases', Journal of Monetary Economics, 79, pp. 81-93.

Martin Weale * and Justin van de Ven **

* Monetary Policy Committee and Queen Mary, University of London. E-mail: martin.weale@outlook.com. ** University of Melbourne and National Institute of Economic and Social Research. E-mail: j.vandeven@niesr.ac.uk.