Saturday, August 11, 2018

The Critical Factors of Portfolio Ruin Aren't Predictable

Probability of ruin and sequence of returns risk are probably the most widely-discussed topics in all of retirement finance and perhaps the least understood.

Probability of ruin is not sequence of returns (SOR) risk. The sequence of portfolio returns we experience after retiring is one determinant of premature portfolio depletion (ruin) but so are life expectancy, the market returns, themselves, the volatility of those returns, the amount we choose to periodically spend and the value of our portfolio.

For a given sequence of returns, the probability of prematurely depleting our savings increases if we expect to live longer or spend more, start out with a smaller portfolio, receive better average market returns or experience less volatility of those returns. As I will explain below, some of these factors have a significantly larger impact on expected terminal (end-of-retirement) wealth than others.

The fact that some of those key variables, our life expectancy and the size of our portfolio, invariably change as we age tells us that probability of ruin also changes as a result of aging. The amount we need to spend annually might also change over time, as might our expectations of future portfolio returns and these will also alter our updated estimate of probability of ruin.

But, the size of our portfolio and our life expectancy are certain to change as we age. They are critical factors of portfolio survival and I suspect nearly everyone would agree that he or she can't know how much money will be left in the retirement-funding portfolio in 10 or 20 years or whether he or she will live that long.

This should dispel the notion some have that a 95% probability of success at the beginning of retirement remains 95% throughout retirement. It probably changes the next year, perhaps meaningfully. That also means that spending 4% of initial portfolio value could become far riskier or far less risky as we age.

“Sequence risk” is introduced when we periodically spend from or invest in a volatile portfolio of stocks and bonds. If we plan to sell stocks every year for the next 30 years, we have no idea today what the selling price will be when those 30 times arrive. That uncertainty of future selling prices creates sequence risk.

Notice I said, “or invest in a volatile portfolio.” When we are accumulating a retirement portfolio with periodic stock purchases before retiring, we don’t know future purchase prices today, either, and that uncertainty also creates sequence risk.

The best way to see the cause of sequence risk is to look at what happens when it isn’t present. Any given thirty years of market returns, for example, will result in the same terminal portfolio value for a buy-and-hold strategy regardless of the order of those returns.

Imagine three years of portfolio returns of 10%, -7% and 12%. These equate to growth rates of 1.1, 0.93 and 1.12, respectively. Multiply those in any order and you get a three-year growth factor or 1.146.  One dollar invested returns $1.15 after three years. The sequence of the returns doesn’t matter.

When you add (save) or subtract (spend) numbers from each of those years, however, no matter where those numbers come from (constant-dollar spending, constant-percentage spending or whatever) the order of the sequence does matter. This is sequence risk. We see sequence risk when we periodically spend from or invest in a volatile portfolio. We see no sequence risk with a buy-and-hold portfolio, so the sequence risk comes from either periodic savings or periodic withdrawals.

The critical factors of portfolio ruin aren't predictable.
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This periodic spending, if too large, can result in depleting our portfolio after retirement, so we are exposed to both sequence risk and a “risk of ruin.” Losing 100% of a savings portfolio, however, is extremely unlikely and while we save for retirement we have sequence risk but almost zero probability of ruin.

So, probability of ruin and sequence risk aren’t the same thing. A poor sequence of returns combined with unsustainable spending can lead to ruin after retirement but a good sequence of returns decreases probability of ruin given the same average return.

The cost of sequence risk is lost compounding of returns. When we have a losing year with a buy-and-hold portfolio, we lose money. When we spend from a volatile portfolio we also lose money during that same losing-market year but our portfolio balance further loses the money we spend plus all potential future compounded gains on the amounts we sold.

Losses hurt more when we spend from a volatile investment portfolio than when we buy and hold. This is why it takes longer for a spending portfolio to recover from a bear market than it takes a buy-and-hold or accumulation portfolio.

(It is often noted that the market recovered fairly quickly after the Great Depression when dividends are considered. A buy-and-hold portfolio would have, too. An accumulation portfolio would have recovered even faster as cheap stocks were subsequently purchased. But, a retiree's spending portfolio would have recovered much more slowly, assuming the portfolio had survived, of course.)

Losses early in retirement hurt more than later losses because those earlier losses leave less capital to compound over time. As Michael Kitces has explained, good returns late in retirement aren't helpful if your portfolio doesn't survive long enough to see them.

The best possible sequence of your annual portfolio returns would result if those returns happened to materialize ordered from best annual return in the first year to worst return in the last. The opposite order would be the worst. That’s why we’re warned that significant portfolio losses early in retirement are the most severe.

Of course, we have no control over the sequence of returns we receive nor can we predict the sequence.

Sequence risk never completely goes away. It is present in a 30-year retirement (and greater in the early years) and it is present in a 5-year retirement (and greater in the early years). Note that a 30-year retirement will eventually become a 5-year retirement if we live long enough.

The challenge of savings decumulation is to optimally spread one's portfolio over one's remaining lifetime but a healthy individual's lifetime is unpredictable. Will sequence risk be reduced when a 60-year old reaches 85? That depends on how much longer the 85-year old will live, how much of her wealth remains and how much she will spend. It requires a new calculation of safe spending based on these new variable values.

A reduced range of life expectancy reduces that component of risk compared to 25 years earlier. However, the amount of wealth we will have 25 years into the future is wildly uncertain. If the retiree's portfolio performs well, she may reach age 85 with reduced probability of ruin compared to age 65 because she has greater wealth and fewer years to spread it over. If her portfolio performs poorly, however, she may reach age 85 with fewer years to fund but far less wealth to fund them and, therefore, increased probability of ruin.

Many SWR analyses suggest that risk decreases because the safe withdrawal percentage increases as we age. Those analyses estimate a safe withdrawal rate when a retiree experiences a 30-year retirement beginning with initial savings of say, a million dollars, and an SWR for a 10-year retirement beginning with the same million dollars.

Risk then appears to decrease with age because the analysis assumes the retiree will have the same million dollars with 10 years remaining as he had with 30 years remaining.  But, in real life there is no guarantee that the retiree will still have a million dollars after 20 years.

An SWR model of historical market returns since 1928 with 4% spending produced a maximum TPV after 20 years of $10.8M and a minimum non-zero TPV of $106K. With continued 4% spending, the former scenario would clearly have a far lower probability of ruin than the latter after 20 years. Add the risk of future portfolio value back into the mix and sequence risk doesn't diminish.

Said differently, the percentage of your remaining portfolio that can be safely spent increases as you age because your life expectancy decreases. The problem is knowing "the percentage of what?" Spending 7% of $106K isn't better than spending 7% of $10.8M even though 7% is larger than 4%.

Probability of ruin doesn't always decline with time but it does change as our savings balance and our remaining life expectancy change. We need to recalculate periodically.

We can estimate a terminal portfolio value (TPV), say after 30 years, for a given sequence of returns and we can estimate how often that will deplete the portfolio in less than 30 years (probability of ruin). These are two different measures. TPV says, "you might have this much money left at the end of retirement", while probability of ruin tells us the likelihood that amount will be more than zero.

The EarlyRetirementNow blog[1] estimates the impact of sequence of returns on the sustainable withdrawal rate* and summarizes its findings: "Precisely what I mean by SRR matters more than average returns: 31% of the fit is explained by the average return, an additional 64% is explained by the sequence of returns!" 

However, the sequence of returns explains 100% of portfolio ruin. To illustrate, we can take a series of portfolio returns that result in premature portfolio depletion (ruin) and rearrange those exact same returns in a better way that avoids premature depletion. We simply swap some of the poor early returns with better late returns. As I explained above, doing so doesn't change the average portfolio return we would receive but it does increase the resulting terminal portfolio value. The difference between success and failure is the sequence, not the returns, themselves.

Focussing on portfolio ruin, however, can be misleading. Sequence risk can dramatically decrease consumption (standard of living) in retirement without resulting in portfolio depletion. (This happens when you end retirement with a small portfolio value that is greater than zero.)

As Jason Scott told me years ago, probability of ruin treats a scenario that successfully funds 29 years as a failure and a scenario that successfully funds 50 years of retirement as no better an outcome than one that funds 30 years. I would add that for a retiree who lives less than 30 years, all three scenarios are winners. It's important to also model life expectancy.

Readers often comment that variable-spending strategies eliminate sequence risk. They don't but they can lower the probability of portfolio depletion by not foolishly spending the same fixed amount annually when savings dwindle. Reducing the chances of depleting the portfolio, however, comes at the expense of lower spending.

Think of it this way: a poor sequence of returns reduces our wealth. We can ignore that reduced wealth and keep spending the same constant amount, risking portfolio depletion, or we can spend less (variably) when our savings are stressed. Either way, we have less wealth so variable spending didn't eliminate the consequences of a poor sequence of returns. It simply changed the impact of sequence risk from portfolio depletion to a lower standard of living.

There is a problem with variable spending strategies, though I still consider them vastly superior to mindless constant-dollar strategies. There is no guarantee that the varying amount you can safely spend every year will maintain your standard of living.

If I am stranded on a desert island with a limited water supply, I can choose to drink decreasing amounts as the supply dwindles but at some point, I can't drink less and survive. Likewise, when variable "safe" spending drops below non-discretionary spending for a sustained period I still have to buy food and pay the mortgage even if that entails an "unsafe" level of portfolio spending. Variable spending isn't a flawless strategy but it seems more sound than the alternative.

I mentioned that the sequence of your future portfolio returns can’t be predicted but the risk can be mitigated. We can do this by spending less from the portfolio, for example, or by changing bond-equity allocations. Sequence risk is moderated by safety-first advocates by ensuring an acceptable income from assets not exposed to market risk in the event of portfolio failure.

To summarize some key characteristics of sequence risk:
  • The sequence of future returns is critical for the survivability of a spending portfolio — but unknowable.
  • Sequence risk and the "safe" amount we can spend vary throughout retirement. They can become much safer or much riskier. We need to modify the amount of portfolio withdrawals to compensate — if we can. 
  • Sequence risk can be helpful or harmful and it has different impacts (generally better) during the accumulation phase than after retirement.
  • Sequence risk can result in portfolio depletion (ruin) or lowered standard of living after retirement but probably not before.
  • The sequence of returns matters more than average returns. To avoid premature portfolio depletion you need a fortunate sequence of portfolio returns about twice as badly as you need really good returns.
  • Althought we can't predict or control our sequence of future portfolio returns, the risk it introduces can be mitigated in various ways.
  • Sequence of returns explains most of sustainable withdrawal rate and all of portfolio ruin.
  • The portfolio return of the first five and ten years of a 30-year retirement are much better predictors of a sustainable withdrawal rate than the mean return for 30 years.[1] You can experience good average returns for thirty years and see your portfolio fall to a poor sequence of those returns or experience mediocre average returns and be saved by a good sequence.
  • A terrible bear market isn't required to sink a retirement portfolio. To quote Michael Kitces, "a “merely mediocre” decade of returns can actually be worse than a short-term market crash..."[2] Retiring in the 1960's was a perfect example. Retiring around the beginning of the Great Depression offers a similar example of how a shorter period of dramatic losses can also result in portfolio failure.
  • Sequence risk never goes away but it can become quite small if your wealth is (or becomes) very large relative to your spending needs and remaining life expectancy — in other words, when your portfolio performs well throughout retirement. Sequence risk can become quite high under the opposite circumstances.
The key takeaways are that the sequence of the returns your retirement portfolio experiences is a major determinant of portfolio survival and is about twice as important as your mean portfolio return. The most important factor is how long you will be retired. And, neither of these is predictable for an individual household.

EarlyRetirementNow's analysis calculates the safe withdrawal rate that would deplete the portfolio in exactly 30 years.


[1] The Ultimate Guide to Safe Withdrawal Rates – Part 15, Early Retirement Now blog.

[2] Understanding Sequence Of Return Risk – Safe Withdrawal Rates, Bear Market Crashes, And Bad Decades, Michael Kitces, Nerd's Eye View blog.

Thursday, July 12, 2018

Monte Carlo and Tales of Fat Tails

I recently read a white paper[1] claiming to show that Monte Carlo (MC) simulation "creates fat tails" and suggesting that constant-dollar withdrawals (the "4% Rule") are historically 100% safe.

Before you log onto E*TRADE for that stock-buying binge, let me explain how I come to a totally different conclusion.

The paper asserts that the reason Monte Carlo models produce different results than the historical data model is the absence of mean reversion in the paper's MC model or perhaps a general flaw in the Monte Carlo technique. The paper presents no statistical evidence, however, of either fat tails or mean reversion and I can't find any in the paper or in my own MC models.

Let's start with a definition of "fat tails."  The term has multiple meanings[2] but in this context, it describes a sample that is more likely to include extreme draws than a normal distribution would predict. A few extreme draws from a normal distribution isn't evidence of fat tails; it is simply evidence of tails.

For example, it is possible (though improbable) to draw an annual market return of 80% from a normal distribution with a mean of 5% and a standard deviation of 12% because a normal distribution has tails that are infinite. A single draw, however, tells us nothing about the probability of extreme draws, which is the definition of fat tails. If our model were to produce many extreme draws – more than a normal distribution would predict – then we would have evidence of fats tails. There are also statistical measures that indicate fat tails, though the paper doesn't report any.[2]

The major flaw in the analysis appears to be the use of a naive Monte Carlo model based solely on normally-distributed market returns. (I say "appears" because the paper reveals little about how the model was constructed but the results are telling). Portfolio survivability is too complex to be modeled by such a simple strategy and it is wrong to blame "Monte Carlo" for the results of a poorly constructed model that happens to use Monte Carlo.

David Blanchett and Wade Pfau wrote on this topic in 2014[3]:
"But this argument is like saying all cars are slow. There are no constraints to Monte Carlo simulation, only constraints users create in a model (or constraints that users are forced to deal with when using someone else's model). Non-normal asset-class returns and autocorrelations can be incorporated into Monte Carlo simulations, albeit with proper care. Like any model, you need quality inputs to get quality outputs."
There are no normal distributions in the real world, only samples that seem likely to have been drawn from a normal distribution. Historical annual market returns, as you can see in the following histogram, appear to be such draws.

The historical data model doesn't use this distribution to create sequences of returns, though. It uses rolling 30-year sequences of these returns, changing only the first and last of 30 years for each new sequence, which distorts the distribution significantly, as shown below. That red distribution doesn't look very normal, does it? Rolling sequences also reduce sequence risk, so we won't find as much as we might otherwise. MC-generated sequences of market returns will be independent and that is a primary reason that MC provides different results than the historical data model, not fat tails or mean reversion.

While our only available sample of historical annual returns data seems likely to have been drawn from a normal distribution, not all draws from that normal distribution create a realistic market return sample. A draw from a normal distribution of annual market returns might legitimately represent a theoretical 120% annual market loss or gain but the former would be impossible for a real portfolio and the latter extremely unlikely.

These are not draws that should be used by an MC model of retirement portfolio returns, at least not when the goal is to measure tail risk. As Blanchett and Pfau note above, "There are no constraints to Monte Carlo simulation, only constraints users create in a model. . ." There is no constraint that says an MC model must use unrealistic scenarios simply because they are drawn from a normal distribution. This MC model is meant to model real-life capital markets, not a distribution that exists only in theory.

The sequence of market returns is critical to portfolio survivability. The historical data shows no strings of more than four market losses or more than 15 consecutive annual gains. This isn't predicted by a normal distribution in which the sequence of returns is purely random but it can be modeled with Monte Carlo. There appear to be market forces that constrain normally-distributed market return sequences and a model based solely on a normal distribution of market returns will not account for these market forces.

Blanchett and Pfau note that autoregression can be incorporated into MC models. This is important for interest rates and inflation rates, which tend to be persistent. Mean reversion, or "long-term" memory of market returns, can also be modeled if one has a strong opinion regarding the existence of mean reversion in the stock market and a strong opinion of the lag time. The authors further note that a proper MC retirement model also incorporates random life expectancy rather than assuming fixed 30-year retirements.

In short, the things the paper complains about "Monte Carlo" not doing are all things an MC model can do but the researcher's model simply doesn't.

An MC model that limits market returns and sequences of returns to appropriately reflect empirical market performance will eliminate most of the anomalies cited in the white paper but it raises another concern: the paper's analysis appears to be a comparison of the historical data model results to a single MC simulation.

I refer to the reference to the (single) maximum "$26M" terminal portfolio value generated by the MC model and to a single probability of failure. MC models should provide a distribution of possible maximum TPVs and probabilities of ruin, not a single result, and that requires running the model many times.

Running the MC model once might produce a maximum TPV of $26M but a second run with different random market returns might produce a maximum TPV of $6M. We run the MC model many times to estimate how likely various TPVs and probabilities of ruin are. There is no single answer.

(To explain more simply, I have a basic MC probability of ruin model much like the one in the paper. I set it to run 1,000 thirty-year scenarios. The first time I ran this model it calculated a maximum terminal portfolio value of $6.8M. I ran the same model again with nothing changed except that it calculated a new set of random market returns for another 1,000 scenarios. The maximum TPV was $10.4M. The third time it produced $9.5 M. The maximum TPV changes each time the random market returns are updated.

I automated the process and ran the MC model 1,000 times with 1,000 different random market returns each.  Maximum TPVs ranged from $4.7M to $41M but the most common maximum TPV was around $10M. This is why we don't stop after running the MC model once and estimating a maximum TPV (in this case) of $6.8M, or a single probability of ruin, for that matter.)

This extremely large, improbable terminal portfolio value is not a fault of Monte Carlo analysis but the result of a naive model of market returns and sequences of those returns that poorly approximates capital markets as we currently understand them. It is also a point estimate.

(As an aside, I'm not sure why we should be concerned about overly-optimistic TPVs in this context.  This is an analysis of portfolio survivability, which is a function of poorly-performing scenarios.)

Is a $26M terminal portfolio evidence of fats tails? Many portfolios that large over many MC simulations might be but a single result tells us nothing about whether it is more or less likely than a normal distribution would predict. Then there's the other issue – terminal portfolio values aren't normally distributed.

Following is a histogram of TPVs created by the historical data model and a log-normal distribution of those results in red.

The white paper notes that some MC-generated terminal portfolio values are larger than a normal distribution would predict. However, TPVs, as you can see in the chart above, are log-normally distributed, not normally-distributed, and should be expected to be larger than a normal distribution predicts. A log-normal distribution is the expected result of the product of n (30) annual normal distributions and a fat right tail is the expected probability density of a log-normal function. If TPVs were normally distributed, some would be less than zero.

Is accepting unrealistic scenarios always a bad thing? This depends on the model's purpose. William Sharpe's RISMAT model[5], for instance, doesn't bother excluding them nor does the research I'm currently co-authoring. The same unrealistic scenarios are included in every strategy tested and filtering them out wouldn't change the comparisons. A small number of unrealistic scenarios is easy to deal with.

The paper in question, however, uses Monte Carlo analysis specifically to measure probability of ruin and this purpose is overly sensitive to unrealistic scenarios because they're the ones that generate results counted as portfolio failures (and large TPV). There will probably be only a relative handful of failed scenarios and adding in a few more failures from unrealistic scenarios can have a dramatic impact on the percent of failures (probability of ruin).  If you insist on trying to estimate tail risk this way, then you should use only realistic scenarios.

To my earlier point, the questionable validity of using MC models specifically to estimate tail risk doesn't disqualify all MC models of retirement finance. As Blanchett and Pfau say, not all cars are slow.

Back to the white paper's claims, no statistical evidence of fat tails or mean reversion is provided and I can find neither of these in these results. I certainly see no evidence of 100% success in the results. I mostly see evidence that a naive MC model provides strange results but I would have guessed that.

Joe Tomlinson wrote a follow-on post[4] to that Blanchett-Pfau piece in which he raised several important points. One is that the selection of metrics is critical when analyzing MC results. In fact, I would argue that estimating a probability of ruin metric is a poor use of MC models since low-probability events are unpredictable.

Tomlinson also makes the point that "The measures being applied by researchers may be more useful than those provided in financial-planning software packages, which provides an opportunity for software developers to introduce new measures to improve the usefulness of their products." So, perhaps an important finding of this paper can be gleaned from the phrase "Monte Carlo analysis (as typically implemented in financial planning software). . ."

If most MC models available to planners are indeed as naive as this white paper suggests and we are using those models to calculate probability of ruin (not my preferred use), then we really do have an MC problem. But it isn't fat tails or the lack of mean-reversion modeling.

So, do Monte Carlo models of retirement finance generate fat tails? I don't see evidence of that. Do they create unrealistic scenarios? Maybe, but that depends on the specific software you're using and its purpose, not on the Monte Carlo statistical tool.

Monte Carlo can be a powerful tool for retirement planning but only if used correctly and for the right application. Estimating tail risk is probably not a good application.


[1] Fat Tails In Monte Carlo Analysis vs Safe Withdrawal Rates. Nerd's Eye View blog.

 Fat Tail Distribution: Definition, Examples.

[3] [The Power and Limitations of Monte Carlo Simulations, David Blanchett and Wade Pfau, Advisor Perspectives.

[4] The Key Problem with Monte Carlo Software - The Need for Better Performance Metrics, Joe Tomlinson.

[5] Retirement Income Scenario Matrices (RISMAT), William F. Sharpe.