Friday, April 29, 2016

A Random Walk, A Sequential Game, Part 3

In A Model of Retirement Planning, Part 1, I wrote that the challenge of retirement income planning is to best position ourselves to maintain our desired standard of living throughout an unpredictable length of retirement with somewhat-predictable future income but largely unpredictable future expenses. A mighty challenge.

In Adding Risk to the Model, Part 2, I added to the model tolerance toward the risk of losing standard of living, because within a fairly small range of expected income and expenses, some households will choose to spend more early in retirement at the risk of having less to spend late in retirement, and some households will choose the opposite. Some can live with more risk than others.

We need to add one last important characteristic to the top-level model of retirement finance, its “chained state” nature. Retirement finance is not a “set-and-forget” decision that we implement and never revisit. It's a series of moves in a sequential game.

Retirement finance is not a “set-and-forget” decision that we implement and never revisit. It's a series of moves in a sequential game.
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I often use the sailing metaphor. At the end of a day of sailing – or a year of retirement – we will find that we have drifted off course and we need to correct our heading. We can't just continue using the heading we set at the start.

Game theorists refer to this as a sequential game against nature, meaning that the game is a series of alternating moves in which Player 1 (your household) makes a move and nature (defined in game theory as "a fictitious player having no known objective and no known strategy") responds.

Although personal finances are practically time-continuous, it is easier to think of them as a series of years, or “discrete-time states”, so that’s how we plan. The age of death for a healthy person is unpredictable, but we often think of people retiring around age 65 and living until age 100, or so. In that case, retirement would consist of one to 36 discrete time states representing ages 65 through 100.

 Following is a simple state diagram for a retiree who retires at age 65 and turns out to live to age 76. Of course, life span is unpredictable for a healthy retiree, so we don’t know beforehand if our own chain will contain one state or dozens.

An individual state can be identified by the age of the retiree, so we can use the terms “state” and “age” synonymously in this example. Each state has associated with it information about income, expenses, net worth, remaining lifetime, portfolio balance, desired standard of living, risk tolerance and other critical financial information.

This information is known with the most certainty in the state that is current, in other words, at our present age. For example, we can know our current portfolio balance, interest rates, current desired standard of living, and current risk tolerance fairly well. We can't know with as much confidence what these values will be for next year, and the uncertainty increases every future year.

For example, if state zero represented 2007, the market crash in October of that year might significantly change all future expectations for portfolio balance, portfolio spending, and net worth and it might even affect our decision to delay Social Security benefits. For some households, it postponed the planned retirement date.

The following table illustrates some of the plan's forecasted financial data for each year in the diagram above as of the starting state (age 65). Age 66 data is less certain when predicted at age 65, age 67 data predicted at age 65 is even less certain, etc. (Click to enlarge.)

Large changes in expectations might also result from out-sized market gains, unexpected medical expenses or the loss of a spouse. Our view of the future can change significantly in a short time. In 2006, our forecast for 2008 would not have included a 55% market crash and a housing crash.

Also, note that the state data we are forecasting are moving targets. Income changes when we claim Social Security benefits. Life expectancy decreases at each new state. Spending, our desired standard of living, tends to decline with age. The sustainable withdrawal percentage from our savings portfolio increases with age. The purchasing power of a dollar changes. Our forecasts constantly change, but so do our targets. We can't simply say we're going to spend $50,000 a year in retirement or receive $50,000 of income annually in retirement because those numbers change over time.

What about the past?

This series or “chain” of discrete states (ages) has the characteristic that the values of next year's state are dependent only upon the information provided in the current state and what happens this year. Anything that happened before reaching the current state is no longer relevant. (Mathematicians refer to this as a discrete-time Markov chain.)

A Monopoly board provides a simpler example of a Markov chain. If your race car or thimble is currently parked on Illinois Avenue, where you will end up next depends solely on where your thimble or race car currently sits and the next roll of the dice. It doesn't matter if you got to Illinois Avenue by sitting on New York Avenue and rolling a five or States Avenue and rolling eleven. That won't affect where you will move next.

This is an important concept that points out, for example, the absurdity of a fixed sustainable withdrawal strategy basing how much you can spend in year 12 of retirement on how much savings you had at the beginning of retirement. If you reach year 12 of retirement with a half million dollars in your savings portfolio, it doesn't matter if you got there by starting retirement with $1M and depleting half of it, or by starting retirement with $250,000 and doubling it. All that matters is where you are now and what happens next.

This is also an important concept in retirement planning because the states you “land on” will be a random walk through retirement-wealth “state space” resulting from those unpredictable incomes, expenses, market returns, and lifetimes, etc.

(State-space is simply the set of all possible future states of a dynamic system – or all possible states of retiree wealth in this explanation. In the simple game of tic-tac-toe, for instance, there are 765 essentially different states that can be reached. The state space for a coin-toss consists of only a head and a tail. There are only two possible future states. In reality, there are an infinite number of possible wealth states for a retiree and time is continuous. It simplifies the explanation, however, if we imagine time in discrete years (snapshots) and a finite number of essentially-different wealth states.)

The state diagram above shows the path moving forward in a straight line, but your path will actually wander through wealth “state space” depending on the draws from those random variables, incomes, expenses, market returns, etc., as illustrated in the following diagram.

When you reach the darker-blue state at age 67 in the above diagram, for example, it won't matter how much or how little wealth you had at ages 65 or 66. Those gray states and the information they contained will no longer be relevant. At age 67, we can only guess the future positions of the light blue states and when we reach age 68 and gray-out age 67, our predictions of the position of future light blue states may change then, perhaps dramatically.

This Markov-chain, or "Markovian", nature of retirement finance has a number of implications for the retirement model. First, since year three's finances depend solely on year two's financial state plus some unpredictable events, and year two's finances are also unpredictable, predicting our future finances with any accuracy quickly becomes untenable. We are trying to predict where we will be in the future by moving an unpredictable distance and direction from an unknown starting point.

Our ability to predict future states decays quickly. We can perhaps predict a year in advance with a little accuracy, but this foresight decreases with each year beyond that and quickly becomes unpredictable. No one predicted the 2008 financial disaster in 2006.

Our ability to predict our financial future decays quickly. No one predicted the 2008 financial disaster in 2006.
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Second, thinking of retirement as a Markov chain that renders past information irrelevant means each new year of retirement becomes a new puzzle to solve, possibly quite different than the one we faced the previous year, so dynamically updating our plans becomes an obvious necessity. It also rids us of the notion that our financial situation years ago remains relevant.

The top-level model for retirement finance, then, should look something like this.
"Retirement finance is a random walk of unpredictable length ranging from one year to several decades. At a given age, only the present financial data are known with any certainty. Data from previous years can be known but are irrelevant. The reliability of forecasts of data for future states decays rapidly with time and forecasts beyond five years are probably sheer conjecture. The key determinants of retirement wealth are random variables: income, expenses, life span, and risk tolerance. Retirees can choose to spend more or less, within a reasonable range, depending on their risk tolerance. Retirees with high risk tolerance can increase spending early in retirement and consequently increase the risk of a lower standard of living in late retirement while more risk-averse retirees can decrease spending early in retirement and consequently decrease the risk of a lower standard of living in late retirement.”
Retirement finance is a random walk along a Markov chain, or to a game theorist, a sequential game against nature. Each year we make forecasts based on what we know (our current financial status and the financial environment), what we expect to happen in the future, and what unexpected outcomes we believe we might experience in the future (risks). We make our move based on this analysis and our risk tolerance. Then nature takes its turn and we repeat.

Once we have a high-level model of retirement finance, we can start to think about how to plan for it. Surprisingly, I have been able to find very little literature that addresses the best way to develop a plan. A good place to start, I think, would be to answer this question: How can you know a good plan when you see one?


  1. Dirk, really enjoy the way you explain complex topics in an understandable way. This article is meaty with information, and to me explains much of things to be considered with a retirement plan. Thanks for your blog always very thoughtful and enlightening. Brad

    1. Thanks, Brad. This is leading up to how we should develop a retirement plan. Following my thought process as it develops. More fun to write a blog when you don't know where it's headed!

  2. I'm beginning to suspect that the best strategy for us is to defend ourselves against foreseeable calamities. Our biggest two would be (i) collapse of our DB pensions, and (ii) one of us going into "care" when the other of us still needs the house to live in.

    (There's a potential calamity that's bigger yet - the collapse of Western Civilisation, or any least our corner of it. I don't think I can do much about defending us from that, though.)

    Number (ii) seems the easier - invest in Care Homes.

    Number (i) is a little trickier - I suppose we could go out of our way to invest in assets that those particular pension funds don't hold. That would lead to a pretty odd portfolio I suspect, but maybe that's as it should be.

    1. There's always LTC insurance for (ii). Joe Tomlinson thinks it's a good idea despite acknowledging its shortcomings. Is your DB plan not insured by PBGC?

      I don't think investing in LTC facilities will adequately mitigate your individual risk of needing one.

      Statistically, your biggest risk is medical expenses, and not necessarily the ones covered by LTC.

      Good luck!

  3. We're in the UK, Dirk, so medical expenses shouldn't be a worry. Amazingly, one can't buy LTC insurance here; several firms tried it out but couldn't find enough customers. (I take it that people just assume that the welfare state will provide.)

    We do have an equivalent of your PBGC, but they wouldn't provide the inflation-protection that my principal pension has, and would almost halve the widow's pension that it pays. Since my wife is in good health and comes from a long-lived family, I do our sums assuming that she'll live to 99. A heavily reduced pension and the lack of inflation-protection would be a calamity for her.

    1. Hmm. Well, here's hoping for the best with your pensions. And actually, LTC insurance isn't working well here, either. Insurance works best when you have a small probability of a very bad outcome and I'm afraid LTC has a relatively high probability of a bad outcome.

      Thanks for sharing.

  4. Brilliant part 3. Time is that thing which keeps everything from happening at once. In the retirement game it helps stop us rom spending all of our savings at the onset.

  5. Amen Dirk!

    This part is uncanily relevant to the paper I shared with you last week.

    Now if the planning profession would only catch on to the difference too.

    One of your best posts yet Dirk!

  6. Another great post, Dirk. Your random walk along a Markov chain and sequential game theory analogies are especially compelling to an actuary. I once again pointed my readers to your post and look forward to your post on how to develop a retirement plan.

  7. Thanks, Ken! Since you are following my thought process in real time, I'm looking forward to that post, too! :-)

    To my other readers, you can follow Ken at his How much can I afford to spend in retirement? blog. Ken is an actuary who sees things from a different viewpoint than I do, though we are finding we frequently come to the same conclusions. I find looking at problems from different angles extremely valuable.

  8. This is possibly your best post ever, Dirk. I especially like the diagrams, which drive home visually the dependence of each state upon events in time that absolutely cannot be predicted with any certainty.

    And yet, plan we must! Can't wait for the next installment of this suspense novel (aka life).

    1. Thank you! That is precisely what I hoped to communicate.

  9. Two thumbs up for this post. Periodic analysis is key to a happy, reduced stress life funded on savings. Two general thoughts:

    1) To extend your thought, the problem persists back to day one when you start saving. A year after day one, you could be expecting triplets and that could change your savings strategy!

    2) This problem could be simulated via a Monte Carlo simulation where you vary the earnings and withdrawal rate around some appropriate distribution curves every year (or any sub period). Sites like T Rowe Price and Fidelity alread do this for the earnings side and uses actual market data. To vary the withdrawal rate, you can rerun the analysis using different withdrawal rates (usually expressed as spend rates but that is not necessarily accurate).

    Excel could be set up to run the Monte Carlo simulation varying withdrawal rate and performance. But is suspect there are Monte Carlo specific tools that are better.

    What I encourage is that it is important to look at the distribution of the results, not solely the probability of success. For example, with a 99% success rate, if the results all collapse to zero within a year or so after the planned date, you have a lot more risk than if the accounts collapse to zero spread out over the 30 years following the planned date centered at 43 years.

    Looking at plots of the account balance is a good way to visualize the simulated future. But that doesn't yield a "nice number like 4%" which we all crave.

    In the end, we are trying to determine a level of confidence on our guess :-)

  10. I should have noted in my previous post that using the existing tools at T Rowe Price, Fidelity and Firecalc only allow modifying the withdrawal rate once at start and do not allow changing the rate each and every year in the run. To change the annual withdrawal each year, Excel or a Monte Carlo specific tool would be needed.

    1. Or, you could use those same tools at the beginning of each new year to recalculate spending based on your then-current account balance, which is what this post recommends. As I previously pointed out, due to the uncertainty of market returns, you can't know a SWR or an optimal asset allocation with any certainty, or your future portfolio balances, so SWR is a very blunt instrument.

      Thanks for writing!

  11. What a great series of eye-opening articles! Thank you for writing these. Looking forward to the next article(s).