Friday, January 22, 2016

Death and Ruin

An MBA and a med student walk into a bar.

(Stop me if you’ve heard this one.)

My son and I had just shot a round of sporting clays (“catch-and-release hunting”) on a hot summer day and we stopped into City Tap in Pittsboro on the drive home for a couple of ice cold, locally brewed adult beverages and a lunch of disgusting chili dogs, and by “disgusting” I mean “outstanding”.

Here in the South we often have a couple of beers after shooting because several generations of experience have taught most of us that having the beers before shooting is sub-optimal in so many ways.

I soon began complaining about the quality of much of the retirement finance research dealing with portfolio survival, as clay shooters often do (OK, not really). My son is on a “Physician-Scientist” track and spends much of his time researching patient survival. He immediately noticed that portfolio survival research isn’t terribly different than patient survival research and that medical research has better tools to study this than we have in financial research.

Portfolio survival is a lot like patient survival but medical researchers have better tools to study it.
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We had a “you-got-peanut-butter-in-my-chocolate” moment and decided to co-author a paper studying portfolio survival by using two medical research survival study tools, Kaplan-Meier analysis and Competing Risks analysis. Our paper hasn’t been published, yet, but here’s a sneak preview.

In clinical trials, information on some patients will be incomplete. Some patients will drop out of the trial, move away, or the trial will end without discovering the patient’s eventual outcome. Kaplan-Meier analysis is a statistical technique that makes as much use as possible from the incomplete data available for these patients. Kaplan-Meier analysis also removes patients who are no longer at risk from the calculated probabilities.

Kaplan-Meier analysis divides time into intervals and for each interval survival probability is calculated as the number of patients surviving (or portfolios, or whatever is being studied) divided by the number of patients still at risk. In portfolio survival studies, simulated retirees who die or go broke are no longer at risk of going broke before they die. They should be removed from the denominator of the probabilities. (This is referred to as “right-censoring.”)

The probability of a patient or portfolio surviving to any point in time (or age, in our study) is estimated from the cumulative probability of surviving each of the preceding time intervals. This all no doubt sounds very complicated, and it is, but it is easier to see if we look at a Kaplan-Meier curve.

The following graph is the result of a portfolio survival study assuming annual withdrawal rates of 3%, 4% and 5%. The analysis used actuarial tables to generate random lifetimes for a retired couple of the same age retiring at age 65. (Double-clicking any Retirement Cafe´ graphic will enlarge it in a separate window.)

To read the curves, for example, a retiree who withdraws 5% annually from her retirement savings portfolio (blue curve) has about an 80% probability of portfolio survival if she survives to age 90. With 4% withdrawals (green curve), she has about a 93% probability of portfolio survival if she lives to age 90.

Reading the curves more generally, retirees don't outlive their savings before about age 80 to 85 assuming any reasonable (say, 5% or less) annual withdrawal rate. Then, the percentage of outlived portfolios heads south, the higher the withdrawal rate, the steeper the fall.

Portfolio survival studies typically calculate a single lifetime probability of survival, commonly quantified around 95%, that doesn’t show how that probability decreases with age, or the effects of random lifetimes. Kaplan-Meier curves do.

Most portfolio survival studies calculate absolute "lifetime" probabilities, or the percentage of all retirees in the study whose portfolios can be expected to fail at some point in their lifetimes. Kaplan-Meier analysis calculates conditional probabilities, meaning the percentage of failed portfolios expected among retirees who are still alive and haven't already depleted their savings.

There is a big difference between losing most or all of your retirement savings, thereby losing some standard of living, and experiencing financial setbacks so dire as to lead to bankruptcy. In my previous post, Why Retirees Go Broke, I suggested that very few retirees will go broke due to sequence of returns risk though there is a possibility of losing standard of living for that reason. Our portfolio survival research, combined with bankruptcy rates by age, is part of that argument.

As the following graph shows, few simulated portfolios are outlived before age 80 to age 85 (blue curve) but most bankruptcies (red curve) are filed before that age – portfolio ruin accelerates about the same time that new bankruptcy filings become negligible. By about age 83, nearly everyone who is going to go bankrupt has, while portfolio ruin has just begun. Thus, the probability that a retiree filed bankruptcy due to sequence of returns risk is quite small – retirees go broke for several reasons, but sequence of returns risk doesn't appear to be a major contributor.

(Note the vastly different scales of the y-axes. Portfolio survival probabilities range from about 80% to 100% in this scenario while bankruptcy probabilities are always less than half a percent. I present the graph this way to emphasize the timing of the two events.)

We applied Kaplan-Meier analysis to a portfolio survival model that included random lifetimes to provide greater insight into the portfolio ruin process. This should help you understand how your risk of outliving your savings – as a function of market volatility – will change as you age. (There are other "non-market" reasons that you might deplete your savings, like devastating medical expenses. Portfolio survival studies typically only look at ruin due to poor market returns.)

We also considered another statistical method from the medical research field, competing risks analysis.

I’ll discuss that in the near future.

Special thanks to my son, Cary, of whom I am obviously ridiculously proud, for collaborating on this research and for helping me explain it in a blog post.


  1. Dirk, ah the benefits of having a smart son that allows for different disciplines to collaborate. We are all the beneficiaries. I really like the curve display of portfolio survival. More meaningful (with age dimension added) to me than just typical 95%. Can't wait to see the competing risk analysis. Brad

    1. That's the key, Brad – you may have a lifetime probability of ruin of 5% but early in retirement it's 0%. It changes over time.

      My other son tells me that Cary has a "scary brain", but let it be known that I out-shoot him more often than not. Well, for now, anyway.


  2. The "Death and Ruin" series of blog posts is very interesting. I think one of the things that it does is quietly highlights the critical importance of Social Security benefits over the past half-century. the concept of "portfolio ruin" is largely a problem for the middle-class and wealthy since about half of the population has little in the way of a portfolio and is effectively entirely dependent on Social Security and any pensions.

    Similarly, bankruptcy does not necessarily mean a large change in the standard of living for many, especially the poor since they don't have assets to lose unless their SS benefits can be garnisheed.

    The progressive nature of Social Security benefits calculations means that the working poor can largely replace their previous income with SS if they worked for much of their life and paid FICA taxes. Similarly, SS can largely replace their income for the lower middle-class. Any pensions (still common in the public sector) are also important.

    Another key area that makes portfolio ruin more of an issue for the more well-off is that there does appear to be a statistical linkage between income and life expectancy. So people with smaller portfolios to begin with are more likely to "withdraw" from the study earlier.

    So bankruptcy for the elderly is more likely to occur to people with assets who get hit with something like a massive medical bill. Ultimately, those people are likely to retain rights to their SS benefits, pensions, their house (assuming that wasn't a major cause of bankruptcy), and at least some of their retirement assets (especially 401ks and annuities) so that even the bankruptcy is likely to be a hit to standard of living, not a fundamental ability to put dinner on the table.

    The population with a large enough portfolio to make it to an age where they can have portfolio ruin are also usually going to have decent SS benefits and possibly pensions/annuities as well. So they may still have enough income to match the median household income even if their portfolio is fully depleted. Their pensions/annuities may have lost substantial ground to inflation but the SS benefits shouldn't have.

  3. Very interesting. Thanks!

    What is the composition of the portfolio?

    Also you generated random lifetimes, but I didn't see any mention of where the portfolio performance came from. Was it backtested or simulated?

    I look forward to seeing the paper when it is available.

  4. Actually, I did mention "simulated" in the post.

    We didn't use a portfolio allocation directly. We simulated a log-normal distribution with a real mean return of 5% annually and a standard deviation of 11%, which roughly equates to a 50% equity portfolio since 1928, assuming 3% inflation. What it might be going forward is speculation.

    The purpose was to model the risk. Changing these parameters wouldn't have a large impact on those findings. You should observe the general characteristics of the curves and not apply the absolute findings to your own situation. You should also use your own expected returns for whatever allocation you select.

    Lastly, I'm not a fan of backtesting. Backtesting suggests that you believe future returns will look like past returns and I see little support for that assumption. (Past returns are just one of countless scenarios that might have happened.) We don't have enough actual data on market returns to be confident that we understand the underlying processes, which is why we run simulations.

    Simulation can make the same assumptions, which is why we shouldn't believe that simulations are predictive. They're great for studying underlying mechanisms, but not at predicting your future.

    I'll let you know when and where the paper is published. I'm expecting summer.


  5. Interesting post, but I'm curious: what's the advantage of using KM vs just weighting the simulation results by the probability of surviving to the age at which the portfolio is exhausted (calculated from the actuarial tables you are using)? For instance, say the first simulation result sampled from your log-normal distribution fails after 20 years, and the probability of surviving to 85 is 0.5. The second simulated result fails after 5 years, and the probability of surviving to 70 years is 0.8. Then the calculated probability of exhausting the portfolio before dying after two simulations would be (0.5+0.8)/2.

    1. Fred, I'm really not sure what the probability you suggest calculating would actually mean. Part of what you are suggesting (weighting) calculates the probability that a retiree would be both alive AND have savings remaining. KM calculates the cumulative probability of portfolio survival conditional upon reaching a specific age.

      Conjunction is the probability that both A and B will occur. Conditional probability is the probability that A will occur given the assumption that B has already occurred.

      Conditional probability and conjunction are two very different measures. I'm not sure what your proposed statistic would measure, but KM tells us the percent of portfolios that will be expected to have survived when a retiree reaches each age throughout retirement, in other words, the cumulative probability of portfolio survival given that the retiree has survived to age X.

      Lastly, and importantly, KM removes portfolios from the probability's denominator when they are no longer at risk of outliving their savings and the calculation you suggest does not.

      If you're still curious, I suggest walking through the steps to calculate KM probabilities here.

      Thanks for the question!

  6. No, I'm calculating the probability that the retiree is alive and that she has exhausted her savings. Perhaps it's a different question than the one you are attempting to answer with the KM estimator, but isn't it really what a retiree wants to know? Graphs of portfolio failure rates vs age are only useful if you are fortunate (or perhaps unfortunate!) enough to know how long you are going to live.

    1. The probability that a retiree is still alive at a given age and has exhausted her savings is simply the product of the probability that she is still alive at that age and the cumulative probability of portfolio failure to that age. The shortcoming of that "absolute probability" calculation, which isn't the calculation you are suggesting, is that it doesn't show the timing of ruin. An 82-year old woman with a depleted portfolio might have depleted it anytime between the ages of 65 and 82. K-M graphs show that timing and enable us to say, "If you live to age X, your probability of failure will increase this much."

      Absolute probabilities have other uses, as I will show in a future post about competing risks analysis.

      My bigger concern is your statement that "Graphs of portfolio failure rates vs age are only useful if you are fortunate (or perhaps unfortunate!) enough to know how long you are going to live."

      The usefulness of a K-M analysis is the understanding it provides of how the risk of ruin changes with age and spending rates. This type of research will never be useful to predict the future of a single retiree. Model's explain the underlying financial processes. They're not very good at predicting the future. (Nothing is.)

  7. Interesting discussion.

    The calculation I'm proposing (and actually do, in my own calculator) is similar to what you suggest, as I mentioned in my first post. Randomly sample m sequences of returns from a portfolio distribution for N years (say 50), so there will be m experiments of length N years. For each experiment, determine if the portfolio is exhausted at t < N. If it isn't exhausted, then assign 0 to the result. If it is exhausted, then assign 1*p(S(t)) where p(S(t)) is the probability of living to age t given an initial starting age. Sum up all of the results and divide my m and that gives you the probabiliy that the retiree will be ruined before dying.

    Why is it important to know the probability of ruin at a given age? It's pretty obvious that someone who lives to be 115 has a much higher chance of running out of money than someone who lives to be 70, but why would knowing that probability be helpful? I suppose you could argue that it would not be good if the chance of failing at a relatively young age is high for given withdrawal rate and a retiree faces the prospect of living solely on SS; on the other hand, that same information is reflected in a higher failure rate in the age independent calculation. And, finally, what about error bars? They are going to be pretty big for later ages and that might limit how much you can actually say about survival probability at, say, 90. So, ultimately, I don't think this type of analysis is particularly helpful.

    1. I don't believe it is important to know the probability of ruin at a given age for an individual retiree. I do believe it is important to know how that probability changes with age.

      You're still focused on predicting results for an individual retiree, and that isn't the point of the research. (And it isn't achievable.)

      Thanks for writing.

  8. This could be useful for pricing longevity insurance.

  9. Love your blog. Might I suggest a summary paragraph at the end of each blog where you summarize the key take-aways. It would help me and I'd bet others. Thanks.

    1. You certainly may. I always try to do that, but I don't always remember. Sometimes, as in my upcoming post, I write a summary of key points of the past few posts.

      I'll try to be more vigilant.

      Thanks for the suggestion!