Sunday, March 07, 2010

A heuristic for retirement savings

Most persons should save at least a quarter, perhaps as much as half, of their post-tax (i.e., "take-home") income.

Saving for retirement feels complicated, because there are so many decisions to make. The purpose of this essay is to enumerate a few rules of thumb I shall use for simplifying these decisions. I think there may be some social benefit in promulgating these heuristics if they produce more-sustainable retirement outcomes.

Thaler and Sunstein (Nudge, 2008) make several useful points in this area:


  • Economists disagree on exactly how much people should save for retirement.
  • Humans are lazy.
  • Default choices or settings are extremely powerful.
  • Many people, when surveyed, say that they would like to save more for retirement; and then fail to do so.
  • Tversky and Kahneman demonstrated how associations and references affect our judgement.

References and Reality

What is retirement? Television commercials portray bliss with vacations and Viagara. I suspect the average reality is quite different. A few years ago, at a hotel in Clearwater, FL, an old woman said, "Retirement isn't all roses." Indeed, retirement is just permanent unemployment. And when you are unemployed, the key drivers that determine whether you will remain solvent are:


  1. How much money you have, and
  2. How quickly, and for how long, you'll deplete that money.

The first rule-of-thumbs I shall offer will be for part (2). Part (1) is left as an exercise to the reader.


Companies selling financial products offer (but do not guarantee) to help you realize your dreams. Many provide some sort of calculator to help you estimate how much to save. While these calculators range from simplistic to sophisticated (some even with Monte Carlo simulations to obtain a distribution on investment returns), the biggest problem is that they offer no transparency or guidance on assumptions --- and the assumptions are critical. The default settings on several calculators imply that income needed in retirement can be less than current income, and your investment growth rate exceeds inflation by several percentage points. Both can be dangerous assumptions.


On the income point, if you can hardly afford to take vacations now, there is no way that all the senior-citizen discounts will accrue to allow you to take any vacations on less income. If you maintain a car to commute to work, are you going to wait for the bus to visit the public library and supermarket? If you expect to finish paying a mortgage on your home, you should not forget to plan for increasing house-repair expenses, such as a new roof or water heater; if you rent, your housing costs are not going to decrease. If you are not one of the few C-suite executives, tenured professors, or government- or union-pensioners eligible for subsidized health insurance for life, you should start studying Medicare-related issues now. Otherwise, consider that private health-insurance quotes for a 60-year-old couple today range from $5,000 to $28,000 per year; and, in the past ten years, health-insurance premiums have increased at double the rate of inflation. There goes the prescription for Viagara. In summary, I think you should plan to spend about the same amount of money each year in retirement as now. (Moshe Milevsky: Your Money Milestones (2010) cites Modigliani for this idea of smoothing consumption.)


  • Source: http://www.ehealthinsurance.com/, quotes for family health insurance for male born 1/1/1950 and female born 1/1/1952 in zip code 10022, retrieved 2/15/2010, ranged $400-2,300 per month
  • Source: http://money.cnn.com/2009/09/15/news/economy/health_insurance_costs/index.htm

On the investment growth assumption, most persons should not expect to realize inflation-adjusted returns over 30 years in excess of the +6% (net of inflation but not taxes) annualized historical S&P 500 rate. And what of taxes? The historical inflation-adjusted rate may be appropriate for estimating pre-tax-with-tax-deferral (401(k) and traditional IRA) dollars, or post-tax-with-tax-free-growth (Roth 401(k) and Roth IRA) dollars, but may be as low as +3% or +4% in a taxable account. I shall provide an example in a postscript. I am not going to discuss the suitability or the moral implications of owning stock.


Savings Pipeline

Now, how many years should you plan for? For Generation-Xers and Generation-Yers, life expectancy at birth was about 70-75 years; and there is a significant chance that you'll outlive the typical lifetime. I am roughly 35 now, and I think it's safe to estimate that I'll work 30 years and then live 30 years more.


In this case, my savings in year 1 of my career will grow (or lay dormant) for 30 years, to be spent in year 1 of retirement. My savings during year 2 of my career will grow (or lay dormant) for 30 years, to be spent in year 2 of retirement. And so on. With this pipeline, I calculated the below table to estimate the amount I need to save, given various inflation-and-tax-adjusted investment returns.












If you expect your savings to grow, compounded, by this percent each year after subtracting inflation and taxes,then, to have the same annual budget in retirement as now, assuming a schedule (years of work/years of retirement) of
30/3040/2045/1548/1250/10
your savings rate should be:
-4%77%72-53%68-38%64-29%61-23%
-2%65%52-43%45-31%40-24%35-20%
0%50%33%25%20%17%
+2%36%18-25%12-20%9-16%7-14%
+4%24%9-19%5-16%4-14%3-12%
+6%15%5-13%2-12%2-11%1-10%


To read this table, let's imagine you can invest in a 30-year certificate of deposit (CD) that returns 0% above inflation and taxes. (This product doesn't exist, but you may be able to simulate it with I bonds or TIPS. I am not going to discuss the suitability or the moral implications of owning government debt.) Then, for every $1.00 you put into the CD, you can withdraw $1.00 (today's dollars) 30 years from now. If your current post-tax income is X, you can simulate a future budget now by saving 50% of X (leaving 50% of X). If you are comfortable with the 50% you can spend today, then your lifestyle will not need to suffer much adjustment when limited to the saved 50% available in 30 years. If you save less of your income now, you are leaving yourself less to spend in retirement. You can vary your assumptions by considering:


  • Perhaps you will plan to work for 40 years and retire for 20 years. Then you can use two years of savings for each year of retirement, and (based on the same 0% tax-free, inflation-adjusted growth assumption) you could start by saving 33%. Why not 25% --- half of 50%? If you save 25%, then you are giving yourself a budget of 75% during your work years, but only 2 * 25% = 50% for each year of retirement, which violates the rule-of-thumb I suggested above. The other columns of the table show the needed savings rates for two, three, four, and five years of savings per year of retirement. Where there are percentage ranges, they indicate the saving rates needed in the first and last years of the non-retirement period. If you are saving 15% now, you are on a frontier hoping for a +6% annualized return on the 30/30 (years of work/years of retirement) schedule, or planning for the 50/10 schedule and with a 0% annualized return.
  • Perhaps you will have access to investments that return more than 0% after inflation and taxes. Good luck with that.
  • Perhaps you will receive Social Security payments or a pension. Good luck with that.

Changes in these assumptions can modify the savings fraction by perhaps a factor of two or three, but the conclusion is still robust: you should save a substantial fraction of your income.



To summarize, I am using two heuristics to estimate retirement savings:


  1. Retirement income requirements are the same as now.
  2. Using the savings pipeline and a conservative investment returns assumption, I must save a quarter to a half of post-tax income.

Pre-tax or post-tax?

In the above discussion, I've tried to use all post-tax, inflation-adjusted numbers. This is partly because it would be difficult for many persons to save 50% of their pre-tax income.


When it comes to retirement investing under United States laws, one often has three choices: (1) pre-tax funds with tax-deferral (401(k) or traditional IRAs), (2) post-tax funds with tax-free returns (Roth 401(k) or Roth IRAs), and (3) post-tax funds with taxable returns (depending on choice of investment vehicle). The benefits of (1) and (2) are that taxes are a one-time cut, either (1) at the time of withdrawal or (2) before investment; while in case (3), taxes not only take the the same one-time cut, but also reduce the realized return each year. For example, if over 30 years, your investment returns 4% each year with 3% inflation, the inflation-adjusted return will be 1% = 4% - 3%; if taxable each year with a 25% tax rate (on the 4% earned), your real return would be zero, but if taxed only at the end with the same 25% tax rate, your real return would total 26%, or about 0.77% per year. The disadvantages of (1) and (2) are that they come with restrictions on the amounts you can save and the conditions under which you can take a penalty-free withdrawal. The only difference between (1) and (2) is whether you expect your future tax rates to be higher or lower than now, and if you aren't sure, you can hedge by dividing your saving between them.

Thursday, May 28, 2009

Competition for mind-share

At the encouragement of a friend, I have decided to give blogging another try. I recall that daily writing as a high-school senior helped sharpen my thinking; and I hope it will again now.

I have changed the subtitle of this blog from "ideas scrapbook" (where I was using it as a bookmarking tool) to "competition for mind-share". Resolved: we must publicize nascent ideas clearly so that they can be criticized. This may be one personal blog out of thousands, but I hope to bring my own brand of logic, judgement, compassion to improve the signal-to-noise ratio in the blogging universe. I welcome comments.

If writing in this blog sharpens my thinking and makes me a better person, then it will have been successful. If this blog collects a significant following, then it will have been wildly successful.

Saturday, July 19, 2008

Friday, June 13, 2008

A Remarkable Tornado Photo

"The Lede: A Remarkable Tornado Photo"

An Iowa woman snaps a stunning close-up of a looming funnel cloud.

Kalman filters

Consider these for noise rejection, especially when moving to continuous-time data.

Wednesday, May 07, 2008

FFT Window references

http://www.lds-group.com/docs/site_documents/AN014%20Understanding%20FFT%20Windows.pdf
http://www.bores.com/courses/advanced/windows/files/windows.pdf

The first link has a table with window characteristics:

Window / Best for these Signal Types / Frequency Resolution / Spectral Leakage / Amplitude Accuracy
Barlett: Random / Good / Fair / Fair
Blackman: Random or mixed / Poor / Best / Good
Flat top: Sinusoids / Poor / Good / Best
Hanning: Random / Good / Good / Fair
Hamming: Random / Good / Fair / Fair
Kaiser-Bessel: Random / Fair / Good / Good
None (boxcar): Transient & Synchronous Sampling / Best / Poor / Poor
Tukey: Random / Good / Poor / Poor
Welch: Random / Good / Good / Fair

A. O. Scott: Here Comes Everyboy, Again

"Mr. Sandler did not invent the archetype of the overgrown man-child, which has been around at least since the silent era. ... Nor has Mr. Sandler been alone, over the past 15 years or so, in turning male infantile aggression into the basis of a lucrative and long-running movie career. His rivals and confreres have included Jim Carrey, Jack Black and Will Ferrell — all of them different physical and temperamental types, but all of them committed to a brazen and unyielding refusal of maturity."

Add to this list the television comedies Seinfeld, Friends, Everyone Loves Raymond, and most recently, King of Queens. I had wondered about the provenance of this archetype, and the exact nature of the thing that so repulsed me about those shows, until this essay put a name to the phenomenon.

Thursday, April 03, 2008

DoG is used in wavelets

Just noticed that derivative of gaussian is also called "DoG" when used as a wavelet basis function. But I think Daubechies are more popular.
Are you using those in your price models?
Maybe in the future. One of my coworkers is using Daubechies wavelets in his model; compared with ordinary moving-average smoothing, doing a wavelet transform, cutting off the spectrum, and then transforming back seems to offer better noise reduction with less information. You still need a power-of-two data length, like for fast Fourier transforms.

Recently, I have been using techniques from "robust statistics": medians, trimmed data, etc. This reminded me of our early experiences with median filtering. Now I think that, while the technique may be nonlinear, the point of experimental science is to make a reliable measurement. If the number itself is what we want to measure (or even ordinary arithmetic transformations thereof), then I say filter. I think the only case where I wouldn't automatically reach for a trimmed mean (where the top and bottom x% of observations are excluded before taking the mean) is if I were doing transformation to a different space, such as FFT. But for dynamic pulling experiments, I suspect FFT is a rarely-used technique.

You probably know that the reporting of a standard deviation is only meaningful if the data are close to normal. Especially in the kind of data I have, I am plagued by fat-tailed distributions, and find that ordinary means and variances are too sensitive to outliers. The robust alternative is the MAD (median absolute deviation).

Wednesday, April 02, 2008

Database Monte Carlo (DBMC): A New Strategyfor Variance Reduction in Monte Carlo Simulation

Monday, March 31st,2008
10:00 am - 11:00 am

CNLS Conference Room (TA-3, Bldg 1690)

DataBase Monte Carlo (DBMC): A New Strategy for Variance Reduction in Monte Carlo Simulation

Professor Pirooz Vakili
Boston University

A well-known weakness of (ensemble) Monte Carlo is its slow rate of convergence. In general the rate of convergence cannot be improved upon, hence, since the inception of the MC method, a number of variance reduction (VR) techniques have been devised to reduce the variance of the MC estimator. All VR techniques bring some additional/external information to bear on the estimation problem and rely on the existence of specific problem features and the ability of the user of the method to discover and effectively exploit such features. This lack of generality has significantly limited the applicability of VR techniques.

We present a new strategy, called DataBase Monte Carlo (DBMC), which aims to address this shortcoming by divising generic VR techniques that can be generically applied. The core idea of the approach is to extract information at one or more nominal model parameters and use this information to gain estimation efficiency at neighboring parameters. We describe how this strategy can be implemented using two variance reduction techniques: Control Variates (CV) and Stratification (DBMC approach can be used more broadly and is not limited to these two techniques). We show that, once an initial setup cost of generating a database is incurred, this approach can lead to dramatic gains in computational efficiency. DBMC is quite general and easy to implement -- it can wrap existing ensemble MC codes. As such it has potential applications, among others, in ensemble weather prediction, hydrological source location, climate and ocean, optimal control, and stochastic simulations of biological systems.

We discuss connections of the DBMC approach with the resampling technique of Bootstrap and the analysis approach of Information Based Complexity.

LANL Host: Frank Alexander, ADTSC