Tuesday, March 21, 2017

Annuities: Anything Anytime

In my previous post, Annuities: All or Nothing, I discussed a paper entitled, "Annuitization and Asset Allocation" [1], written by Dr. Moshe Milevsky and Dr. Virginia Young in 2007. The authors developed models for two annuity markets. The first, referred to as “All or Nothing”, calculates an optimal age for purchasing a life annuity one time in retirement, but the authors also found that there is a better way.

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 [2007] explores whether there is an optimal strategy for annuitizing when a utility-maximizing [2] 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 [2007] 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 [2007] 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 [2007] 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.



REFERENCES

[1] Annuitization and Asset Allocation, Moshe Milevsky and Virginia Young, 2007.

[2] 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.





Monday, March 6, 2017

Annuities: All or Nothing

I recently reviewed recommendations for the optimal age to buy a life annuity and found that a number of sources recommend that men purchase in their mid-70s and women about six years later. Some of the recommendations could be traced back to a paper entitled, Annuitization and Asset Allocation [1], written by Moshe Milevsky and Virginia Young in 2007. You can find a link to the paper below in the References section, but be forewarned that it isn't for the mathematically faint of heart.

The authors develop models for two annuity markets. The first, referred to as “All or Nothing”, calculates an optimal age for purchasing a life annuity once in retirement. This single-purchase limitation might be the result of a retiree preferring to make a single annuity purchase, a pension plan that limits the participant to a single purchase, or a country's annuity market.

The second analysis 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 strategy as “Anything Anytime” and it is the scenario in which most Americans will find themselves. In this post, I'll review All or Nothing and save the more complex Anything Anytime analysis for next time.

For those of you who are not interested in the math, I provide a link to an Excel spreadsheet[2] at Github in the References section below that does the heavy lifting for you. If you're even less interested in the math, skip down to the last five paragraphs beginning with "How can an individual use this information?" No one will know.

Milevsky [2007] shows that the optimal age to purchase a life annuity is when the retiree's “force of mortality”[4] – more on that in a minute – is greater than the following constant:
where γ (gamma) is the coefficient of relative risk aversion, μ (mu) is the expected market return, σ (sigma) is the standard deviation of those returns and r is the risk-free rate[6].

γ, the coefficient of relative risk aversion, is difficult to know for an individual but research shows that it is often between 1 and 2 for the typical investor. An investor with a high risk tolerance will have a low coefficient of risk aversion like γ=1. A more conservative but still typical investor might have risk aversion γ=2. An individual with a γ=5 has very low risk tolerance.[7] A γ=1 retiree is far more likely to invest in the stock market than a γ=5 retiree.

The equity risk premium is the excess return demanded by stock investors above the risk-free rate. When the market returns 9% and a risk-free bond returns 5%, the equity risk premium is 4%. It is the expected market return less the risk-free rate (μ - r in the equation above). This would simply mean that an investor could get a 5% return with no risk and that he might earn 4% more than that by taking on the risk of investing in stocks. Historically, the equity risk premium has ranged from about 3.5% to 5.5%.

Think of M in the equation above as the return you expect from your investments (the term in square brackets) adjusted by your risk aversion (the term 1/(2γ) ). M considers not only your expectations of future market returns but also how much you value those returns based on your level of risk aversion. A risk-taker would value a 12% return, for example, more highly than a conservative investor would value that same 12% return.

If your risk aversion is γ =1, meaning you're a typical risk-taker, then the hurdle you need to exceed to convince yourself to annuitize is one-half (1/2γ) your expected risk-adjusted market return, but if you're less of a risk-taker (γ =2), the hurdle is a much lower one-fourth of that return, meaning you will probably annuitize sooner than an investor with γ =1.


When to purchase annuities with the All-or-Nothing strategy.


People who expect higher returns on their investments and value those returns more (because risk bothers them less) will annuitize later in retirement than those who don't because they think they can do better investing than buying an annuity. They expect to do so well in the market that they won't need to annuitize as early. Some are so confident in their investment prowess that they won't buy an annuity ever. Retirees with lower market expectations and a higher aversion to risk will annuitize sooner. Some won't ever invest in the stock market.

Milevsky [2007] identifies the optimal age to annuitize in the All-or-Nothing scenario as the age at which retirees will expect to do as well in the annuity market with mortality credits[5] as in the investment market. This introduces the second half of the optimization equation as well as the paper's big idea regarding All-or-Nothing purchases:
"One can then think of the hazard rate [force of mortality] as a form of excess return on the annuity due to the embedded mortality credits and the fact that liquid wealth reverts to the insurance company when the buyer of the annuity dies."
While M quantifies how much the retiree values expected excess returns from her investment portfolio after adjusting for her risk aversion, the force of mortality quantifies how much he or she can expect as a similar “excess return” from an annuity. When the retiree values his expectation of investment returns equally with an annuity's excess return he has reached the optimal age to annuitize.

Let me try to say all of this more simply.

Milevsky [2007] shows that the optimal age to buy an All-or-Nothing annuity is when the retiree values the return on an annuity the same as she values the return that she expects from her investments. In order to get a fair comparison, the paper uses the excess return on investments (expected return less the risk-free rate) to compare to the excess return on an annuity. Since some investors don't mind a lot of risk and some do, the excess return on investments is adjusted to reflect the risk tolerance of individual investors. (The annuity has no market risk.) And lastly, the excess return on an annuity is considered to equal the individual retiree's force of mortality, which is determined primarily by the retiree's gender and age.

The second half of the optimization equation is the retiree's force of mortality that can be calculated with the Gompertz function[3]:
Force of mortality is the instantaneous probability of a person's death at a given age conditional upon reaching that age. It is the probability that a person who reaches the age of 75, for example, will die before his or her next birthday. You can play with force of mortality parameters a bit at the web page in References[4].

The following graph shows the optimal age for the retiree used in the All-or-Nothing example in Milevsky [2007], a female with risk-aversion coefficient λ=1. The tables that follow and include this example are taken from that paper. The optimal age to annuitize in the All-or-Nothing scenario occurs when the blue curve of the force of mortality for this individual crosses the “M” red line of the value of expected market returns (the hurdle). This intersection is the age at which the retiree expects to do as well purchasing an annuity as investing the same amount of money.


When the retiree values expected investment returns more the red line (M) moves up and the optimum annuitization age (the intersection of the two curves) increases. When the retiree is less optimistic about doing well in the market, is more risk averse, or both the red line moves lower and the optimal annuitization age moves left. The green "M" line in the graph below shows the result for a retiree with the same market expectations but greater risk aversion (λ=2). (This can also be seen by altering market return and risk-aversion inputs in the aforementioned spreadsheet[2].)


Using these same inputs, Milevsky[2007] provides a range of examples in Tables 1a and 1b provided below for your convenience.



Notice the column labeled “Value of Delay.” These entries show that the longer you wait before the optimal age the more value you receive from purchasing an annuity. Once you pass the optimal purchase age, delaying longer means an annuity would provide less value than an earlier purchase would have provided, but still more than the adjusted portfolio return (M). You can see this on the graph but the tables show only that the value of delaying becomes negative.

To calculate the optimal annuitization age directly in the All-or-Nothing model I'll simplify the math just a bit. The optimal age (x) for purchasing an All-or-Nothing annuity is:
How can an individual use this information for their own annuity purchase decision in an All-or-Nothing scenario?

Economic studies are much better at teaching us how things work than at predicting outcomes for an individual household. The coefficient of risk aversion is difficult to tie down for an individual household and future market returns are at best a guess, so calculating an optimal annuitization age isn't nearly as exacting a process as these equations might seem to imply.

So, what can we learn from playing around with these equations?
  • The optimal age for annuitizing in the All-or-Nothing strategy is rarely much less than age 70 or much greater than 80 unless you are highly risk-averse, in which case you should simply ignore the math and buy an annuity early in retirement.
  • Males will purchase sooner than females because males of the same age have a shorter life expectancy than females
  • Retirees less tolerant of market risk will purchase annuities sooner
  • Retirees who are less optimistic about future market returns will annuitize sooner.
Play around with the spreadsheet[2] to see how the parameters affect the optimal age to annuitize.

Before you invest a lot of time in the All-or-Nothing analysis, though, Milevsky [2007] follows it with an analysis of an “Anything Anytime” strategy in which the retiree buys some initial amount of annuity upon retiring and then buys more if and when wealth increases. This strategy better fits the U.S. annuities market and my instincts about retirement finance.  See that analysis at Annuities: Anything Anytime.

The math gets a lot harder, though.





If Obamacare Exits, Some May Need to Rethink Early Retirement, New York Times, February 27, 2017. 



REFERENCES

[1] Annuitization and Asset Allocation, Moshe Milevsky and Virginia Young, 2007.


[2] Download Excel spreadsheet to calculate the optimal age for annuitization in the All-or-Nothing model.


[3] The Gompertz function can estimate continuous life expectancy by fitting a curve to a discrete life expectancy table. The fit is achieved by manipulating the m and b terms. Milevsky [2007] fits the curve to the Individual 2000 basic Annual Mortality table with projection scale G. For females m= 92.63 and b=8.78 and for males, m=88.18 and b=8.78 using this table.


[4] 
Force of mortality explained. It is analogous to hazard rate in reliability engineering. It is the probability, for example, that you will die at age 70 conditional upon your having lived to age 70.


[5] Understanding The Role Of Mortality Credits – Why Immediate Annuities Beat Bond Ladders For Retirement Income, Michael Kitces.


[6] Examples in this post and Milevsky [2007] assume γ=1 or 2, μ=0.12, σ=0.20 and r= 0.06.


[7] Estimating the Coefficient of Relative Risk Aversion for Consumption from Gordon Irlam's AACalc website.