The second analysis in that paper identifies an optimal way to purchase annuities when the retiree can choose to purchase any amount of annuity at any time. The authors refer to this as an “Anything Anytime” annuity market and it is the scenario in which most Americans will find themselves.
While most Americans can choose to purchase life annuities in pieces rather than being limited to a single purchase in an All or Nothing market, we can't generally undo these transactions because there isn't a healthy secondary market where we can sell life annuities. We can annuitize but we can't economically un-annuitize.
Second, Anything Anytime considers public and private pension income including Social Security and similar benefits equivalent to income from life annuities from insurance companies so they all meet the optimal income-purchase requirement. A household that needs $100,000 in annuity income and expects $5,000 in Social Security or pension benefits, for example, needs only purchase another $5,000 of life annuity income. (All or Nothing largely ignored private and public pension income.)
Third, if the math for All or Nothing seems challenging, Anything Anytime quickly dives into complex utility functions and differential calculus so I suggest you do what I do in these cases: trust the math and concentrate on the conclusions.
The Anything Anytime analysis of Milevsky  explores whether there is an optimal strategy for annuitizing when a utility-maximizing  retiree can purchase more annuities at any time in any amount but cannot “un-annuitize.” It determines that there is an optimal strategy and that it is roughly (I paraphrase here) as follows:
Retirees should annuitize some amount of their wealth at the beginning of retirement (recalling that Social Security and other pensions count). Should their wealth increase as they age relative to their annuity income they should then annuitize more but if wealth remains steady or declines they should “stand pat” with existing annuity income.In other words, this research finds that the optimal annuitization path for a utility-seeking retiree is to start with a base of annuity income at the beginning of retirement and ratchet it upward if and when her wealth increases relative to her annuity income.
Anything Anytime recommends an immediate purchase of some amount of annuity income (or claim of pension or Social Security benefits) early in retirement and that is a major difference from All or Nothing. The latter sought the optimal age for a one-time purchase of annuities and found that it is typically later in retirement. The paper recommends that everyone needs some amount of annuity income, but since nearly all Americans are covered by Social Security or a public pension, nearly everyone will have some annuity income.
The optimal amount of annuity income to purchase at a given age is determined by a ratio, w/A (wealth relative to annuity income), in which w is total liquid wealth and A is total annuity income. Note that A is annuity income and not the face value of the policy. (For example, if you purchase a $100,000 annuity that pays out $4,000 per year, A refers to $4,000, not $100,000.) "Liquid wealth" means wealth exclusive of the face value of annuities. In my example, a retiree whose wealth was $500,000 and who purchased a $100,000 annuity would be left with $400,000 of liquid wealth.
Calculating this ratio (w/A) and determining a household's utility function account for the difficult math. This wealth-to-income ratio is determined by solving a differential equation of a utility function for a given time in retirement. Since utility functions are quite difficult to identify for an individual the calculation would be difficult even if the math weren't.
It is important to note that the wealth-to-income ratio is a function of time and wealth so it changes as we age. It isn't a fraction, like 50%, that we can calculate and assume that our annuity income should always equal half our wealth, but rather a fraction that needs to be calculated periodically as we age and our wealth changes. We can't know it accurately in advance because we can't predict our future wealth accurately.
Setting aside the difficulty of calculating an optimal wealth-to-income ratio for an individual household, there is much to be learned from this study. To begin with, there exists a mathematically optimal strategy for annuitizing wealth in retirement that involves establishing an initial amount of annuity income early in retirement and adding to annuity income as we age and our wealth increases. If our wealth does not increase, purchasing more annuity income is suboptimal.
This strategy fits well with a number of annuity strategies, noting that these strategies come more often from the Safety-first school and economists than from the Probabilists school and stock market devotees. Annuity-laddering strategies to avoid locking in the worst payouts during times (like these) of low interest rates work well with this strategy. The strategy to purchase multiple annuities from multiple insurers to mitigate the default of a single insurer does, as well.
Many households will be reluctant to hand over a large chunk of retirement savings to an insurance company. Establishing some annuity income early in retirement and planning to possibly buy more income later should ease this anxiety by making the purchases smaller and reducing regret.
In the paper's conclusion the authors state:
"In this case which we label anything anytime, individuals annuitize a fraction of wealth as soon as they have opportunity to do so – i.e. they do not wait – and they then purchase more annuities as they become wealthier."Not waiting somewhat contradicts the advice of many economists and planners to delay claiming Social Security benefits as long as possible. Given the benefits of delaying those claims, perhaps waiting just a few years might be better advice.
The difficult calculation of the initial wealth-to-income ratio can be approximated by applying floor-and-upside principles and buying a “comfortable” amount of flooring. That amount may not be mathematically optimal but we know that owning some amount of annuity income early in retirement is part of an optimal strategy.
Should our wealth increase and our floor no longer feel adequate, we know that purchasing more is also part of an optimal strategy. Should our wealth not increase, instead, purchasing more income is likely to be sub-optimal. Though we may not be able to calculate the precise optimal amount to purchase, we have a better understanding of when to buy more.
I do have concerns with this strategy as it might apply to households at the bottom end and top end of wealth. A household whose retirement savings become so large that they need only spend a small percentage each year probably has enough safety margin to stop buying more annuity income when wealth increases. At some point, the fortunate retiree will probably feel that his floor is adequately sized.
At the other extreme, there may come a time when the retiree's wealth declines so significantly that she wishes to divert more assets from investments to annuities, which is contrary to the Anything Anytime strategy of standing pat on annuity income when wealth declines.
Recall that the paper addresses utility-maximizing retirees, those that seek the greatest economic satisfaction given diminishing marginal returns. A real-life retiree who amasses enough wealth might change his goal from utility maximization to growing a legacy portfolio and a retiree who loses much of her wealth might begin to value bankruptcy avoidance more than optimal utility.
Tables 4a and 4b from Milevsky  are shown here for your convenience. z0 is that difficult-to-calculate ratio of optimal wealth-to-income (w/A). Recall from my previous post that γ (gamma) is the coefficient of relative risk aversion. (Higher gammas are more conservative investors.)
The first table assumes existing annuity income of $25,000 and the second assumes $50,000. Compare the optimal annuity spending for existing annuity income, current wealth, and risk aversion. For example, the retiree who already has $25,000 of annual annuity income in Table 4a should spend more money on annuities than the retiree in Table 4b who already has $50,000 of annuity income – at any level of risk aversion.
As Milevsky  notes, with other conditions remaining the same, retirees will tend to purchase more annuity income when they perceive greater market risk, are less risk-tolerant, have better health, and have greater wealth relative to annuity income. The paper also shows the value of purchasing annuities with low fees and the value of a retiree's own person health assessment (subjective hazard rate) compared to the insurance company's opinion (objective hazard rate).
A more practical approach that incorporates the findings of Milevsky  might be to purchase enough annuity income early in retirement to provide a comfortable floor when added to Social Security and pension income. As your wealth increases, assuming it does, purchase more annuity income if the floor no longer feels adequate. Purchasing multiple, smaller annuities over time from multiple insurers may help overcome reluctance to "hand over your savings to an insurance company."
The strategies of establishing a floor of secure income early in retirement with Social Security benefits, pensions and life annuities, laddering annuity purchases over time, and diversifying among multiple insurers gain an economic endorsement from this research.
When is the best time to purchase a life annuity? Annuitization and Asset Allocation suggests that the answer to this question depends on whether the retiree will make a one-time purchase or can stagger purchases as she ages. For the latter, the answer is not an age but a path that may involve multiple smaller purchases.
No matter what the research says some retirees are never going to buy an annuity and some are never going to invest their savings in the stock market. I'll share some thoughts on that in my next post.
 Annuitization and Asset Allocation, Moshe Milevsky and Virginia Young, 2007.
 Utility maximizing. Economists use the term "utility" as a measure of satisfaction, joy, or happiness. Utility is based on individual preferences and not solely on dollar value as one individual might value an additional dollar of income differently than another individual would. A single individual might also value a dollar differently in different situations. A utility-maximizing retiree seeks maximum satisfaction, which may not be the same as maximum consumption.