1. The Utility Value of Longevity Risk Pooling: Analytic Insights, Milevsky & Huang 2018
- One of the main theoretical contributions in this paper is to argue that when pre-existing (fixed) pensions are included in a lifecycle model one has to be (very) careful about how to define annuity equivalent wealth and the value of longevity risk pooling. This is due to the wealth depletion time which complicates the discounted utility analysis. Although the concept of a wealth depletion time is explained in Leung (2002, 2007) or in Lachance (2012), within the context of the Yaari (1965) lifecycle model, it doesn't appear to be well known.
- It might seem odd to split up the objective function in this manner, but in fact when p [pensionized income] > 0, there is a qualitative change in optimal consumption at some point during the horizon t ∈ [0, ∞). That is, liquid wealth is actually depleted and the optimal consumption rate ct* = p from that point onward. That is the ͳ value we select. Until that wealth depletion time consumption is sourced from from both pension income and wealth. But after t ≥ ͳ consumption is exactly equal to the pension, the individual has run out of liquid (non-annuitized) funds and Wt = 0, for t ≥ ͳ.
- To be crystal clear we are not imposing this on the problem. It actually is the optimal policy, as elaborated on by Leung (2002, 2007) and carefully explained in the (textbook) by Charupat, et al. (2012), chapter #13. We can't emphasize enough how critical this (seemingly minor) point is to the calibration of lifecycle models in general and the computation of AEW [annuity equivalent wealth] values in particular. At the risk of flogging a dead horse, if one assumes all pension income is capitalized and discounted to time zero, or if the pension income is added to optimal consumption as a scaling afterthought, the wealth depletion time will be lost in the backward induction algorithm.
2. The existence, uniqueness, and optimality of the terminal wealth depletion time in life-cycle models of saving under uncertain lifetime and borrowing constraint, Siu Fai Leung 2006[7]
Mostly, this paper was too challenging for me… My comments
to myself in the margins were things like "WTF!?" [as in: I feel like a clueless moron] and "no extractable usefulness for us [retirees]"
and "I am stopping here!" Though I didn't really stop nor is it useless. I did what
most people do that don't have the math, I skipped forward to the conclusion.
- …it means that there is no optimal way to consume the endowed wealth if it is too large.A minute increase in S0 can change the model from one with an optimal solution to one without. [p 476. not a WDT quote but an interesting comment]
- Terminal wealth depletion is an intrinsic and important property of Yaari’s [42] life-cycle model of saving under uncertain lifetime and borrowing constraint.
- …the optimal terminal wealth depletion time, if such exists, must be unique. There are no multiple-solution indeterminacies in the model.
- …if the equation that determines the optimal terminal wealth depletion time has multiple isolated solutions, the largest solution will be the only possible optimum.
- …if the equation that determines the optimal terminal wealth depletion time has a continuum of solutions, the smallest solution will be the only possible optimum.
- …a general formula can be utilized to locate the optimal solution.
- …it is necessary to use the optimality test developed in the paper to verify whether a candidate solution for the terminal wealth depletion time is indeed optimal.
- …it is possible that none of the solutions for the terminal wealth depletion time is optimal, in which case the control problem does not have an optimal solution.
- These new and subtle results not only are of theoretical interest but also have practical significance especially for simulation studies on life-cycle models of saving under uncertain lifetime and borrowing constraint. [back to simulation. Why didn't he tell me that in the first place]
- While simulation analysis is certainly valuable, it would be even better if analytical results are available to guide the analysis and interpret the findings. [I thought this was interesting. It's either true or a PhD full-employment comment or maybe both]
- As
pointed out in Leung (1994, 2000), previous studies utilizing Yaari’s
model of uncertain lifetime and borrowing constraint have failed to
recognize that terminal wealth depletion is an intrinsic and important
property of the model. This paper highlights the important role of
terminal wealth depletion in the theoretical analysis and demonstrates
that erroneous and misleading results will be obtained if it is ignored in
the investigation. Although the mathematical derivations are complicated
by the presence of terminal wealth depletion, tractable analytical results
can still be obtained. These results, which
- have never before appeared in the literature, are very useful in understanding the life-cycle effects of social security.
- Therefore, t* [WDT] is solely determined by (26). The equation reveals that t* depends on [initial wealth, interest rates, subjective discount rates, risk aversion, maximum possible lifetime, income, and conditional survival probabilities] in a complex way. [emphasis added]
- The existence of terminal wealth depletion essentially truncates the time of the life-cycle decision from [max lifetime] to t*.
- Proposition 2 shows that, if terminal wealth depletion occurs after retirement, then an increase in the social security benefit will generally result in earlier terminal wealth depletion, higher consumption and lower wealth throughout the interval [0, t*). [an exercise like this has been done by this amateur by modeling a DIA at age 85 and seeing wealth lower and consumption higher with a known and fixed target for t*]
- The findings are also intuitively reasonable. An increase in [soc sec] benefits the individual without any cost to him. The extra income enables him to raise consumption, reduce saving and deplete his wealth earlier. In this case, the uniform social security system displaces private savings.
- By ignoring terminal wealth depletion…The conclusions thus obtained will likely be erroneous and misleading…
4. Optimal onset and exhaustion of retirement savings in a life-cycle model. Marie-Eve Lachance 2012
- Retirement can be interpreted as a limit case of precautionary savings where income declines with certainty.
- While
the previous literature has often focused on wealth as a determinant of
consumption and saving behavior, our solution indicates that the evolution
of this behavior over the life-cycle is explained by the shape of the
function lambda(t). This function captures the combined effect of income profiles,
risk aversion, time preferences, investment return, and mortality.
6. Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin, Habib, Huang, Milevsky 2017
- In other words, from a lifecycle perspective, ruin is not a scenario or outcome that should be avoided at all costs. Rather, the rational objective should be to slowly and smoothly deplete financial resources accounting for the declining probabilities of living to very old ages. And, if a by-product of this behavior is that financial wealth is expected to hit zero at some distant point, so be it -- provided there is some pension income to fall back on. In other words, ruin should not be feared if annuities are part of the retiree's portfolio. We provide the stochastic model to justify this claim. [emphasis added]
- When
one has access to some sort of pension annuity income (i.e. social
security income, DB pension, or income annuities), running out of money
during retirement shouldn't necessarily be feared or avoided. It might in
fact be optimal.
- …in this paper we solve the full Merton (1971) model in which investment returns as well as lifetimes are random, but one in which pension annuities are also available. Moreover, we show that in such a world -- with parameters properly calibrated to real world values -- it is optimal to exhaust ones financial resources before becoming a centenarian. Now, of course, this does not imply that one starves to death. Rather, if indeed the retiree reaches that age they should plan to live off their pension annuity income (if it is available). Stated bluntly, if there is only a 5% chance of reaching the age of 100, it is quite rational to (i.) assume that you won't and (ii.) reduce your consumption to the minimal pension level, if you do. [emphasis added]
- While the 4% withdrawal rate is consistent with life-cycle consumption smoothing, it is so only under a very limited set of implausible preference parameters.
- The main insights in a deterministic framework of Milevsky and Huang (2011) are as follows: 1. The initial spending rate critically depends upon a retiree's risk aversion and pre-existing pensions. 2. The optimal consumption (i.e. sum of all pensions and withdrawals from the account) is a declining function of age. Retirees should consume more today than what they consume in the future. … 4. Wealth trajectory declines with age and retirees with sufficient pensions spend down their wealth well ahead of reaching an advanced age. 5. The rational reaction to portfolio shock is non-linear and depends upon pre-existing pensions. 6. Converting some of the initial investible wealth into a stream of lifetime income increases consumption at all ages even when interest rates are low. [emphasis added]
- When [returns are deterministic, (some rate assumptions I didn't follow), and pension income] is held constant, one expects consumption to decline as risk aversion [gamma] rises. However, when returns are stochastic, this phenomenon isn't quite true. For example, in the case of a 65-year old male, Table 2, we observe consumption rising and later falling with an increase in [gamma].
- Initial consumption of a retiree is not only a function of the growth rate, discount rate, and risk aversion but it is also a function of the mortality rate.
- We also observe that while pension levels have an impact on the optimal spending strategy, optimal spending appears to be more sensitive to changes in asset allocation.
- WDT is solved both numerically and using Monte Carlo Simulations. Observe that as pension levels increase WDT tends to decrease. On the other hand and as expected, WDT rises with increasing risk aversion.
- It is
also important to note, that wealth depletion time is shortened only in
the context of higher pension levels. In other words, consumption is not
zero but it is equal to pension when wealth gets depleted. Clearly if a
retiree does not have any pensions we see that wealth is depleted at the
terminal age; on the other hand, for those retirees with high levels of
pension income relative to their investible wealth, wealth is depleted much
earlier than the terminal age.
7. Evaluating Retirement Strategies: A Utility-Based Approach - Estrada & Kritzman 2018,
This is not directly a WDT reference paper. On the other hand, since they create an analytical tool that is a "coverage ratio" of x/y where x is the number of years spending is covered by a portfolio and y is the planning (lifetime) horizon, it felt vaguely similar. So I took a look at it here:
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