Introduction to retirement spending models

When most people think of retirement planning, one of their first questions is, "What will it cost me to live as a retiree?" Answering this question can seem overwhelming. This article aims to help answer this question by describing models of retirement spending.

Investors primarily use retirement spending models to estimate the Total Personal Savings they will need at retirement. The calculation needs to include price inflation. Often, nominal retirement spending is split into two independent components: real spending, and an inflation adjustment. This article deals only with the real spending component. For inflation, see: Inflation and retirement spending.

Estimating your retirement spending is easier if you break it down into two steps. First, estimate an initial spending at the start of retirement; and second, extrapolate that as retirement progresses until eventual death.

Overview

Initial spending

Generally, two types of models initial spending:

• Replacement rate (or ratio) models estimate the total income needed in retirement by multiplying total income before retirement by a constant, the replacement rate. Generic estimates for the replacement rate range from 0.7 to 0.85, but you can find more personalized replacement rates in tables published by the GSU/Aon RETIRE Project.
• Budget or Expense models have various forms. They all estimate total retirement spending by working with estimates of spending in numerous, smaller budget categories. The most common type of budgeting approach uses a Single budget to describe all retirement expenses. Using a Dual budget (with a lower essential spending and a higher preferred spending) give you a more flexible framework for extrapolating spending as retirement progresses.

Extrapolation

Models that extrapolate spending until death usually build on estimates from one of the initial spending models. These models group into five broad categories:

• The Constant (Real) Spending model assumes that the real spending at the start of retirement continues unchanged until death. Although this approach is widely used, it does not match a typical retiree's spending pattern. It usually (but not always) overestimates the savings needed at retirement.
• The Stages of Retirement model breaks up retirement into three or four age ranges, each of which has a different real spending pattern. Spending within each range can be constant, or it can vary. This model does a much better job of representing retirees' diverse spending patterns.
• Investment Returns Dependent models allow retirement spending to vary based on how total savings or investment returns change over time. These models are best combined with Dual Budget models of initial retirement spending. These models match retirees' tendency to vary spending based on their net worth.
• Flexible Spending models allow you to incorporate numerous individual categories of spending into a total spending plan. Each spending category can have its own start and stop age, as well as spending amount. This model is excellent at representing large step changes in spending during retirement. Many retirement calculators will combine elements of this model with others already described. This combination supplies a very flexible framework for accurately describing a diverse range of retiree spending patterns.
• Life Cycle models calculate how a household should smooth their spending over their entire lifetime, not just in retirement; they provide a detailed spending and savings plan that you can use before retirement. Doing this gives you a similar lifestyle both before and after retirement.

There are several retirement calculators that implement these models. For some specific examples, see: Retirement calculators and spending.

Introduction

Answering the question "What will it cost me to live as a retiree?" can seem overwhelming for many people. Coming up with an exact answer is probably impossible, because there are simply too many future unknowns.

But that does not mean that you cannot create a useful estimate of retirement spending. Some guidelines can help you do this. These guidelines are models of retirement spending.

When thinking about future retirement spending, there is always a question of how much detail to put into the models and guidelines. This question is at the heart of any Economic model: How much can the real situation be simplified and yet still be useful?. There are many models of retirement spending, ranging from the extremely simple to the detailed and complex. This article reviews the main modeling approaches, explains their key simplifying assumptions, and gives a general feel for their strengths and weaknesses.

Safe Withdrawal Rates

Many studies on retirement spending start with an estimate of future retirement savings and then estimate the spending that might be achievable in retirement. They often focus on estimating the maximum spending that will not lead to exhausting your savings too early, the so-called safe withdrawal rate. These models treat the maximum amount of savings as an independent variable. They adjust spending adjusted so that savings last as long as planned.

This article series works from the other side of the equation and treats retirement spending as the independent variable in the retirement planning process. You first develop an estimate of your desired spending in retirement. From there, the planning process makes assumptions about your longevity, inflation and investment returns, and then estimates the future savings you would need to sustain that retirement spending.

Inflation

This discussion excludes the effects of inflation on your retirement spending.

This is not meant to imply that inflation is an unimportant consideration. Just the opposite, inflation is probably the retiree’s worst enemy![1][2] But most retirement planning approaches treat inflation as an independent, adjustable variable. That way, they separate nominal retirement spending into two components: real spending,[note 1] and inflation adjustments.

Retirement calculator models

There is a strong connection between retirement spending models and retirement planning calculators. Most retirement planning software, either explicitly or implicitly, assumes some model of retirement spending. The software's adjustable options reflect the choice of spending model.

Unfortunately, some retirement calculators, although they have many great strengths, are needlessly weak in incorporating retirement expenses.[3]

Retirement planning software's usefulness is limited by its least realistic assumption. It is not useful to develop retirement software having numerous investment type choices along with a detailed Monte Carlo treatment of potential future returns, but then estimate real retirement spending with only a simple replacement rate model.

The Retirement calculators and spending article contains an extensive list of calculators, categorized by the retirement spending model(s) they use. Refer to this article for real-world examples of the spending models discussed here.

Organization

Classifying retirement spending models is useful because it emphasizes similarities and differences. This article uses the following classification to organize retirement spending models:

Initial Retirement Spending

• Replacement Rate models
• Single Budget models
• Dual Budget models

Spending as Retirement Progresses

• Constant (Real) Spending models
• Stages of Retirement models
• Investment Returns Dependent models
• Flexible Spending models
• Life Cycle models

Initial retirement spending

These models develop an estimate anticipated spending early in retirement, often focusing on spending for the very first year of retirement. There are three main approaches for developing this estimate:[note 2]

1. Replacement rate (or ratio) models
2. Single budget (or expense) models
3. Dual budget models

Replacement rate models

Replacement rate (or ratio) models are the simplest and most generic approach to estimating retirement spending. Rather than directly estimating spending, replacement rate models estimate pre-tax, gross income after retirement. This model is based on the following equation:

Gross Income (retired)   =  Gross Income (pre-retirement)  ×  Replacement Rate

Assuming you have an idea of your gross income just before retirement, you only need a replacement rate multiplier to derive a gross income just after retirement. If you need an actual retirement spending amount, you can calculate it from the gross income by subtracting federal, state and local taxes.

In its simplest form, this model assigns the same replacement rate to everyone: 0.75 being a commonly suggested value (that is, you need 75% of your pre-retirement income for retirement), but sometimes a range of 0.70 to 0.85 is suggested.[note 2][4]

This type of replacement rate model is extremely generic, and for many retirees it is the least accurate. Replacement rates change with key household characteristics such as gross income and marital status.[5][6] For tables containing these less generic replacement rates, as well as a more detailed discussion, see: Replacement rate models of retirement spending.

Although many free online retirement planning calculators use the replacement rate model,[note 3] these calculators are typically very simple. Even when using simple online calculators, replacement rates from the GSU/Aon tables are better than a generic estimate (for example, using 0.75 for everyone). If you are using a more detailed retirement calculator, adjust the GSU/Aon replacement rates to increase their accuracy.

Strengths Weaknesses
• Provides a quick and easy estimate of gross income in retirement.
• Well suited for use by people who are far from retirement.[note 2] The uncertainties in the GSU/Aon replacement rates will likely be similar to uncertainties in other aspects of the retirement plan.
• GSU/Aon replacement rates need further adjustments for best accuracy.
• Require a supplemental calculation to obtain after-tax spending.
• People far from retirement will need to estimate their gross income just before retirement in order to use replacement rates.[note 4]
• Generally less accurate than a Budget model approach for people who are close to retirement.[note 2]

Single budget models

Budget (or expense) models come in various forms. Their common factor is that they estimate total retirement spending by working with estimates of spending in numerous, smaller budget categories. The most common type of budgeting approach uses a single budget for all retirement expenses.

There are two typical approaches to obtain a single budget spending estimate: Current Spending, and Bottom-up.

Current Spending starts with your total (current) spending just before retirement.[7] You then examine each spending category to see what adjustments, either up or down, you anticipate on retirement. These spending adjustments are incorporated to arrive at a total spending just after retirement.

Bottom-up develops a personalized spending budget from scratch by estimating your retirement spending for every conceivable budget category.[note 5][note 6] You should use a detailed budget worksheet to make sure that you do not overlook any category of retirement spending. Summing these category estimates gives a total spending estimate. This approach is more time consuming than the Current Spending approach, but can give you a better estimate. It also gives a better starting point for projecting how the budget might change as your retirement progresses.

For links to detailed worksheets useful for developing a single budget spending model, see Budget models of retirement spending. Many retirement planning calculators also incorporate worksheets for developing budget spending estimates.[note 7] However it is better to develop the budget using an independent, detailed worksheet. This minimizes the possibility that you overlook important spending categories. For example, spending on big but infrequent purchases is very important,[note 6] but most calculators' built-in worksheets often omit this.

Strengths Weaknesses
• Can supply more accurate spending estimates than a replacement rate model.
• Better for handling spending estimates for big, infrequent expenses than a replacement rate model.
• A Bottom-up budget model provides a particularly good starting point for incorporating real spending changes over time.
• Requires more time and effort than a replacement rate model.
• Realistic spending estimates are difficult to determine unless you are already tracking your personal spending.

Dual budget models

Dual Budget models are a direct extension of the Bottom-up budget model that incorporate two total spending estimates.[note 8][8] The first or Essential budget is the lowest level of retirement spending that you can can accept. The second or Preferred budget is a higher level of retirement spending that you want. Your spending in any year is assumed to fall between these two budgets.

The easiest way to estimate the Dual budget models is to use worksheet designed to do this.[8][9] Each budget category on such a worksheet accepts two entries: an essential spending amount and a discretionary spending amount. Their sum gives the preferred budget spending estimate.

Dual budget spending models are particularly useful when combined with [[Withdrawal methods] that allow real spending to vary. Two examples are the Constant Percentage method and the Floor and Ceiling method.[10] The Dual budget model has an added restriction on calculating total savings at retirement: your real spending cannot rise above the preferred budget nor fall below the essential budget.

Life Cycle models of spending indirectly use Dual budgets. These optimize your discretionary spending (this is your spending above your essential budget) across your entire remaining lifespan, both before and after retirement.[note 9] In Life Cycle models the sum of the essential and the discretionary spending may be either less than or greater than an independently estimated preferred budget.

Strengths Weaknesses
• Provides a more nuanced model of retiree spending that a single budget model.
• Better suited for combining with Withdrawal methods that allow variable real spending.
• More work to develop dual budget models than a single budget model.

Spending as retirement progresses

The models previously described supply a starting point for the more important problem of modeling how real spending changes as retirement progresses. A large number of such models have been proposed and incorporated into retirement calculators. The section below summarizes these models. For more detailed descriptions of all the following models, see: Models of spending as retirement progresses.

An important criterion for evaluating retirement spending models is how well they can describe the actual spending patterns of retirees. These actual spending patterns primarily come from various Surveys of retirement spending. From these surveys, as well as from the practical experiences reported by professional financial planners, certain key facts are visible:

• The average retiree exhibits a slight drop in real spending at retirement, followed by a steady decline in real spending as they age into their late 70’s or early 80's.[11][12] This decline in real spending is voluntary and not a result of limited financial resources.[note 10]
• A substantial fraction of retirees (25%[13] to 28%[14] before the 2008 recession, rising to almost four out of ten[15]) enter retirement involuntarily. They exhibit a sharp drop in real spending at retirement.[13] If involuntary retirement was health related, these retirees may subsequently exhibit a medical expense induced, real increase in total spending near their death.
• A smaller percentage of retirees (roughly 12%[note 11]) exhibit an increase in real spending at retirement.[note 12][16] This is often driven by a jump in travel or other leisure activities. After a certain time period these special activities end, and real spending drops to a lower level, often closer to that of "average retirees."

Constant (real) spending models

This is the simplest of all approaches used to model spending as retirement progresses. It assumes that your real spending at the time of retirement will continue unchanged until death. It does not ignore inflation, but as explained in the Introduction, treats it as a separate, adjustable variable in the retirement planning process. In terms of spending, this model states that spending for any year in retirement equals spending in the first year multiplied by an inflation adjustment.

Comparison with Retiree Spending. Constant real spending models do not reflect the reality that the average retiree’s spending steadily drops during retirement. Many retirement planners have pointed this out as a major shortcoming.[note 10][17] They also point out that these models overestimate the total savings needed at retirement.

A constant real spending model could be appropriate in if your retirement is involuntary and your financial resources are limited. This situation would force you to immediately drop down to an essentials only spending budget. If you were living at your minimum acceptable level, you would expect your nominal spending to grow at around the rate of inflation (constant real spending) as you are forced to pay ever increasing market prices for essential goods.

Although Life Cycle spending models sometimes produce a constant real spending in retirement, they do this from an entirely different Economics Perspective.[18] For more, see Life Cycle models below.

Calculators. Free online retirement calculators most commonly use a Constant spending model. This is especially true for calculators that are deterministic, less so for those that use Monte Carlo or Historical Returns approaches. There are however some that let you choose different models. For more, see Retirement calculators and spending.

Strengths Weaknesses
• Easy to understand, so they are useful for illustrative (teaching) purposes.
• Easy to implement in a retirement calculator or spreadsheet.
• Does not match the reality of how average retirees spend money as they age.
• Often leads to an overestimation of the total savings needed at retirement.

Stages of retirement models

Studies have shown that a typical retirement splits into three to four Stages or Phases.[19] Within each stage, retirees tend to exhibit similar patterns of physical activity and spending.[note 6][note 12] Stages of Retirement spending models account for this.

The very simplest Stages of Retirement model assume that the real spending within each Stage is constant. All changes in real spending occur between the stages. If set up to model spending dropping by ages 75-80, it approximates an average retiree spending pattern. If it uses a budgeting approach to estimate the real spending within each stage, you can think of these as Step Change Budgets. But a changing (often dropping) replacement rate could just as easily be used to model each successive stage.[note 10]

Studies have also shown that retirees steadily reduce their spending as they age, rather than as a series of sharp drops.[11][note 10] Following on from this, several retirement planners have suggested that Stages of Retirement models can be made even more realistic by allowing real spending to vary gradually within one or more of the stages.Ty Bernicke’s Reality Retirement Planning model[17] and William Bengen’s Prosperous Retirement model[20] are well known examples of this approach.

Comparison with Retiree Spending. Using three to four Stages of Retirement, each of which can have a different real spending, allows much better modeling of retiree spending patterns. All three of the basic patterns discussed earlier can be mimicked. And if the stages allow for a gradual annual spending drop rather than just a constant real spending, then an even better match to actual retiree spending is possible.

Calculators. Calculators with a Stages of Retirement spending model are much less common than those with constant real spending model. Only a few of the calculators in the Retirement calculators and spending article explicitly include this model. Some of these include Ty Bernicke’s Reality Retirement Planning approach.

Strengths Weaknesses
• Allows the retirement calculation to incorporate diversity in retiree spending patterns with aging.
• Significantly improves model accuracy without excessively increasing complexity.
• Needs additional work to estimate budgets or replacement rates for each Stage.
• There is some uncertainty in the most realistic annual percentage spending drop to age 75.
• Somewhat more difficult to implement in a retirement calculator than constant real spending.

Investment returns dependent models

Investment Returns Dependent models let retirement spending vary based on how total savings or investment returns change over time. It unsurprising that retirees would cut back on their spending during times when they do not feel financially secure.[21] For someone who is retired and who has stock market investments, stock bear markets would motivate them to reduce spending. Conversely, if they feel financially secure (for example, during a stock bull market), they would probably increase their real spending.[22] These periods of increased and decreased spending can be modeled by Withdrawal Methods that depend on on either the previous year’s total savings or its investment return. An extensive summary of these methods is given on the Variable Withdrawals in Retirement (archived page from the Bob's Financial Website).

A Dual Budget model is a natural complement to variable withdrawal methods. The Essential budget spending level would set a lower limit on spending as retirement progressed. The Preferred budget spending level would set a corresponding upper limit on spending. These constraints combine with those of the withdrawal model, providing a better approximation of the retirement savings needed.

Comparison with Retiree Spending. By itself an Investment Returns Dependent spending model cannot adequately reflect the key observations on retiree spending discussed earlier. It cannot match the steady decline in real spending as retirees age. But it does supply an element of reality that other models lack: that is, it can reflect how retirees alter their spending in response to changes in their net worth. A combination of these two spending tendencies would be very beneficial.

Calculators. Calculators with an Investment Returns Dependent spending model are much less common than those using a constant real spending model. The Retirement calculators and spending article shows that more calculators have some variation of this model than those that have a Retirement Stages model.

Strengths Weaknesses
• Reflects the tendency of retirees to adjust spending up or down as their net worth changes.
• Nicely complements variable withdrawal methods used to ensure savings survival in retirement.
• More realistic than a Constant Spending model.
• Does not match the reality that average retirees spend progressively less money as they age.
• Less able to mimic the broad range of retiree spending patterns than a Stages of Retirement or a Flexible Spending model.

Flexible spending models

A Flexible Spending model allows a total spending plan to incorporate multiple individual spending categories. Usually there is no need to enter every conceivable budget category as a separate spending item. Rather, these models usually let you first enter a base spending amount from another model (Constant Spending, Stages of Retirement, and so on). Then you can enter any spending that still is not properly represented as a separate, individual spending item. Typically the only information required is the starting age, ending age and spending amount for each category.

This flexibility is particularly important if you anticipate one or more large expenses that would abruptly start or stop during retirement. Example include repaying a home mortgage, buying a vacation home, or paying for long-term medical care.

You can often use a Flexible Spending model to mimic the spending pattern of a Stages of Retirement spending model, by using one flexible spending category to contain all the spending within one of the retirement stages.

Comparison with Retiree Spending. Provided there are enough flexible spending categories/entries available, this model can do an excellent job of representing the diversity of retiree spending patterns discussed earlier. But if it offers only a small number of categories or entries available, it cannot model a steady decline in retiree real spending through their late 70’s or early 80’s.

Calculators. Calculators with a Flexible Spending model are much less common than those with a constant real spending model. However, the Retirement calculators and spending article offers a handful of free calculators that do include variations of this model. Most paid-for retirement calculators have this spending model.

Strengths Weaknesses
• Can provide a personalized description of retirement spending for a broad range of retirees.
• Best model for handling large changes in spending that might occur at irregular times during retirement.
• Entering the personalized spending data can be very time consuming for some calculators.
• Not well suited for mimicking investment returns dependent retiree spending.
• Additional effort required to incorporate this model into a retirement calculator.

Life Cycle models

A Life Cycle model predicts how a household should smooth their discretionary consumption over their entire lifetime, not just in retirement.[note 9][18] Utility theory predicts that people would be happiest if they neither over-spend nor under-spend each year as they age, whether retired or not. Life Cycle models try to incorporate this. They account for all types of pre-retirement expenses, generating a year by year savings plan consistent with a lifetime of smoothed discretionary consumption.

It is difficult to estimate how much you would need to save each year before retirement so that you could maintain a comparable lifestyle each year after retirement, especially if you have to account for uncertain future investment return, life span, and health care expenses.[23] Calculators offering the Life Cycle approach overcome these difficulties using Dynamic programming.

Comparison with Retiree Spending. If they use a fixed retiree lifespan and no longevity risk aversion, Life Cycle models predict a constant (real) discretionary spending target throughout retirement. This is clearly different from the average retiree spending patterns discussed earlier. But when Life Cycle models use more realistic assumptions (for example, longevity risk aversion and probabilistic lifespans), they correctly predict that real spending decreases with age.[24]

Calculators. Calculators include Life Cycle models less frequently than other spending models. This is unsurprising, given the complex programming required to implement the model. The Retirement calculators and spending article shows just a few calculators that use the Life Cycle model.

Strengths Weaknesses
• Gives specific annual spending and saving goals both before and after retirement.
• Can recommend optimum age to start Social Security.[18]
• Can fully integrate federal and state taxes into the calculation of optimal spending.[18]
• Simple implementations that give a constant real spending will not match average retiree spending patterns.
• Complicated to implement fully.

Notes

1. In economics real refers to the purchasing power net of any price changes over time. For more, see Real versus nominal value (economics), from Wikipedia.
2. For details, see: Kenn B. Tacchino, "Determining the Amount a Client Needs for Retirement," Chapter 5 in Financial Decisions for Retirement, (The American College Press, 2005), pp5.1-5.55.
3. See Retirement calculators and spending for specific examples.
4. You estimate this gross income by either extrapolating your current income using historical wage growth from the National Average Wage Index, or by replacing current income with that of someone in the same profession who is close to retirement.
5. See: Walter Updegrave, "Create a Retirement Plan That Works for You", chapter 5 in We’re Not in Kansas Anymore: Strategies for Retiring Rich in a Totally Changed World (Crown Business Publishers, 2004) 304pp.
6. For details, see: Robert C. Carlson, "How Much Will You Need", Chapter 3 in The New Rules of Retirement: Strategies for a Secure Future, (John Wiley & Sons, Inc., 2004) 288 pp.
7. Examples of retirement planning calculators which utilize a built-in budget worksheet include the T. Rowe Price Retirement Income Calculator, the Otar Retirement Calculator and the U.S. Dept. of Labor Retirement Calculator.
8. For details, see: Paul Merriman, Live it Up Without Outliving Your Money, revised and updated edition (John Wiley & Sons, Inc., 2008), 206pp.
9. Described in: N.Charupat, H. Huang and M. Milevsky, Strategic Financial Planning Over the Lifecycle, (Cambridge University Press, 2012).
10. See: Kenn B. Tacchino and Cynthia Saltzman, "Do accumulation models overstate what’s needed to retire?", Journal of Financial Planning, Vol. 12 (Feb. 1999), pp 62-74.
11. See: Michael Hurd and Susann Rohwedder, The Retirement Consumption Puzzle: Anticipated and Actual Declines in Retirement Spending, NBER Working Paper 9586 (March 2003).
12. See: Michael Stein, The Prosperous Retirement: Guide to the New Reality, (Emstco, LLC, 1998), 312 pp.

References

1. John Howe. "Inflation – A Retiree's Worst Enemy". Archived from the original on January 18, 2019.
2. Scott Pearson (2008). "Inflation: The greatest enemy of your retirement plan".
3. John Turner; Hazel Witte (2009). "Retirement Planning Software and Post-Retirement Risks" (PDF). The Society of Actuaries.
4. "Desired Replacement Rate". U.S. Office of Personnel Management. Archived from the original on November 12, 2011.
5. Bruce A. Palmer (2008). "2008 GSU / Aon RETIRE Project Report" (PDF). Center for Risk Management and Insurance Research (Georgia State Univ.). Archived from the original (PDF) on January 23, 2014.
6. "2008 Replacement Ratio Study: A Measurement Tool for Retirement Planning" (PDF). Aon Consulting. 2008. Retrieved August 29, 2023.
7. "How to Estimate Your Retirement Expenses". the balance. October 23, 2021. Retrieved August 29, 2023.
8. Helen Modly (August 31, 2005). "Helping Clients to Accurately Estimate Retirement Expenses". Morningstar Advisor.
9. "MetLife Retirement Income Worksheet" (PDF). MetLife. 2009. Archived from the original (PDF) on December 8, 2011.
10. For more on the Constant Percentage method, see: Withdrawal Methods. For more on the Floor and Ceiling method, see: Variable Withdrawals in Retirement (archived page from the Bob's Financial website).
11. J. Fisher; D. Johnson; J. Marchand; T. Smeeding; B. Torrey (2005). "The Retirement Consumption Conundrum: Evidence from a Consumption Survey" (PDF). Center for Retirement Research at Boston College Working Paper WP 2005-14.
12. Yung-Ping Chen; John C. Scott; Jie Chen (2007). "Retirement Spending and Changing Needs during Retirement: Summary of Regression Analysis" (PDF). Society of Actuaries Annual Meeting.
13. Erik Hurst. "Understanding Consumption in Retirement: Recent Developments" (PDF). Archived from the original (PDF) on May 31, 2013.
14. Marie-Eve Lachance; Jason Seligman (2009). "Involuntary Retirement: Prevalence, Causes and Impacts" (PDF).
15. Anna Rappaport. "The Golden Glitch – Expanding Longevity and Shrinking Work Lives".
16. Robert Carlson (October 17, 2017). "How to Vary Spending During Retirement". Bob Carlson’s Retirement Watch. Retrieved August 29, 2023.
17. Ty Bernicke (2005). "Reality Retirement Planning: A New Paradigm for an Old Science" (PDF). Journal of Financial Planning. Archived from the original (PDF) on May 9, 2010.
18. Laurence Kotlikoff (2008). "Economics' Approach to Financial Planning" (PDF). Journal of Financial Planning. Retrieved September 1, 2023.
19. M. Cowell; R. Helman; A. Rappaport; S. Siegel; J. Turner (2008). "The Phases of Retirement and Planning for the Unexpected" (PDF). Society of Actuaries.
20. William P. Bengen (2001). "Conserving Client Portfolios During Retirement, Part IV" (PDF). Journal; of Financial Planning. Archived from the original (PDF) on September 10, 2011.
21. Kevin J. Lansing (July 11, 2011). "Gauging the Impact of the Great Recession". FRBSF Economic Letter. Retrieved September 1, 2023.
22. Lonnie K Stevans (2004). "Aggregate consumption spending, the stock market and asymmetric error correction". Quantitative Finance, Vol. 4.
23. John Scholz; Ananth Seshadri; Surachai Khitatrakun (2006). "Are Americans Saving "Optimally" for Retirement?" (PDF). Journal of Political Economy, Vol. 114.
24. Moshe Milevsky; Huaxiong Huang (2011). "Spending Retirement on Planet Vulcan: The impact of longevity risk aversion on optimal withdrawal rates" (PDF). Financial Analysts Journal, Vol. 67, No. 2. Archived from the original (PDF) on November 15, 2012.