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Welcome to personal finance for long-term investors, where we believe Benjamin Franklin's advice that an
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investment in knowledge pays the best interest both in finances and in your life. Every episode teaches you personal
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finance and long-term investing in simple terms. Now, here's your host, Jesse Kramer. Welcome to Personal
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Finance for Long-Term Investors, episode 134. My name is Jesse Kramer. By day, I
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work at a fiduciary wealth management firm helping clients nationwide. You can learn more at
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bestinterinterest.blog/work. The link is in the show notes. By night, I write the bestinterest blog. I host
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this podcast. I also put out a weekly email newsletter, all for free. And all of these different projects help busy
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professionals and retirees avoid mistakes and grow their wealth by hopefully simplifying their taxes, their
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investing, and their retirement planning. Today, we're going to do a deep dive episode all about Monte Carlo
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analysis. I've been working on this one for a little while. I've gotten some
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good questions over the years from listeners like you about kind of wanting to go into the into the details under
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the hood of Monte Carlo analysis. You're going to hear me say Monte Carlo analysis a lot today, just FYI. But
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before we dive into the details, we'll do our usual thing. We will do a review
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of the week. This one is from App Trail 1 who left a five-star review and said, "Gold Medal Podcast. This podcast checks
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all the boxes. Accurate, accessible, honest, and unbiased. All sprinkled with a little bit of dry wit." Well, Apptrail
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1, thank you very much for those kind words. You can shoot me an email to [email protected]
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and I'll get you hooked up with a supersoft podcast t-shirt. Now, on to the main course today, the big show.
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We're talking about Monte Carlo analysis. What it is, what it isn't, how
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to use it, all those kind of things. If you haven't heard of it before, of course, we'll introduce it. But, you
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know, this very quick preamble, I want you to think today that that Monte Carlo analysis, the thing I'm about to talk
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about, it's not a crystal ball, but it is a a stress testing tool. It is a very
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excellent conversation starter. It's not necessarily predictive though, right?
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It's not a predictive machine. It's not a crystal ball. The outputs from Monte
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Carlo analysis, things like a success percentage, for example, can be a double-edged sword. It does provide
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important data on whether you're thinking about retirement in the right way. But it can also hide different
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faulty assumptions. It can mask overconervatism. It can mislead you emotionally. So today, we're going to
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dive into all those kind of things. and and hopefully you leave today understanding why I begin with that
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preamble. Today we're going to cover specifics like why we need to do in-depth analysis like Monte Carlo
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analysis. What's going on under the hood of a Monte Carlo analysis? How does it
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actually work? Especially how does it work in financial planning? I mean, that's why we're here. Where can Monte
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Carlo analysis go wrong? How do you cautiously but accurately or or at least helpfully interpret the results of a
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Monte Carlo analysis? And how do you make sure your Monte Carlo analysis of course isn't misleading you, right? How
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do we get led down the right path with these results instead of one of the many different wrong paths? So, let's start
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with why do we need to do in-depth analysis in the first place? I would argue that most of us have big hopes,
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but also maybe some big fears. We have big questions certainly about retirement. How do we answer those
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questions? Some questions can only be answered via conversation and and looking within. Some questions are
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certainly subjective in nature. But plenty of questions, plenty of these retirement questions at least are
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objective, right? They're fact-based. They're based on numbers, plain and
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simple. But retirement has a bunch of different moving pieces and a lot of different numbers. And there are a lot
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of different ways that someone can begin to analyze their retirement plan. And when I say analyze, I mean use some form
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of numerical methods to determine if you can retire, how successful your retirement might be, how much you can
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spend, etc., etc. There are plenty of free, widely available online retirement calculators that usually provide
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something like a retirement score or a probability of success based on average static assumptions. And I think the key
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word there is static. And you'll hear me today use the word static and dynamic a
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few different times. So when I say a static assumption, these type of free online calculators, they often assume
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static unchanging earnings, static unchanging savings rates, static investment returns, 7% per year, 9% per
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year. It's just a a static one-sizefits-all constant return. And that's fine. It's fine to use that type
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of tool as a general readiness check for retirement. But it's about as loose a
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rule of thumb as as you could muster. You know, if that type of static calculator suggests that you should, and
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I'm using air quotes, that you should have four times your income saved, but
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you only have three times your income saved, well, I still wouldn't know enough to know if you're truly behind or
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not because the tool itself is too coarse to come up with that kind of conclusion. But then if that calculator
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suggests that you should have four times your income saved and here you only have
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one times your income saved, well then I might wager that you need to dig into the details a little bit deeper because
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there's a good chance you're actually behind on your retirement savings. My
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point there is that these coarse static online calculators, they're just that
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they're coarse. It's hard to walk away with really good really good outputs and
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really good direction when you're using such a course tool. Now, if we dial up
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the complexity a little bit more, we might focus on something like a, you know, quote unquote deterministic cash
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flow modeling. So, this type of tool allows for much more detail, such as specific retirement spending, maybe
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part-time work in retirement, big one-time expenses like your daughter's wedding or buying that vacation home. It
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helps you project future income, project future expenses, your net worth. It helps you project your net worth on a
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year-by-year basis. You can model in social security income, pension income, basic tax calculations in a way that is
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certainly more detailed than the previous course analysis, but also personalized to you, specific to your
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timeline. And that is all great. That is something we're looking for. We're
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moving in the right direction here. The problem though is is that exercise, this
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deterministic cash flow model. It typically assumes a static investment return. The focus is on cash flow and
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the dynamic cash flow. It's not necessarily on dynamic investment returns. We know though that investment
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returns are a really vital component of retirement planning and of course that investment returns are dynamic in
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nature. So that's where if we go one step further in terms of kind of the complexity of analysis types, that's
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usually where Monte Carlo analysis comes into play. Monte Carlo analysis is a a more sophisticated analysis type that
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tests your financial plan, your retirement plan against thousands of different possible market scenarios
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rather than testing it against only one average static market return. So Monte Carlo might simulate uh market
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volatility to determine the probability of success. You know, as in there's an
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85% chance of success that your money will last you to age 95. Success is another one of these big words that we
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will come back to multiple times today. We need to talk about exactly what success means in this context and and
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like I said, we'll come back to that. So, done well, a Monte Carlo analysis takes your unique projected cash flow
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that we just kind of talked about a minute ago. So, your unique retirement cash flow and then it layers that cash
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flow on top of a wide range of possible investment returns. And then it asks which types of investment returns, which
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series and sequence of investment returns could lead to bad retirement outcomes could lead to really good
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retirement outcomes. I mean what is the range of of possibilities in your retirement future? So for example, we
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could walk away from a from a Monte Carlo analysis and look at it and say well in this analysis we ran most kind
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of standard typical investment return series actually caused our retirement plan to fail. Oh, that doesn't really
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sound good. Or maybe we walk away and say, listen, only the worst of the worst investment series actually cause this
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retirement plan to fail. And that certainly sounds much better. Or maybe if we go even a step further, we'd see
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that our retirement cash flow actually survives even the very worst possible investment series that this Monte Carlo
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analysis could muster. And then we should ask ourselves, well, maybe we're being too conservative then, right? If
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our retirement fails, even the the worst scenario possible, are we just underspending our retirement? But before
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diving further into the specifics of financial planning, Monte Carlos, because we will get into those
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specifics, I want to take a quick step backward. Maybe it's the little engineer
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in me. So, how do Monte Carlos work in general? I think that's actually a really important place to start. And
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heck, why do we even call them Monte Carlos in the first place? So, let's talk about that. Let's talk about how
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Monte Carlo analysis works in general. When you hear Monte Carlo simulation, Monteol analysis, Monte Carlo, anything,
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you should think lots and lots of random trials. That's what I want you to think.
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Lots and lots of random trials. For example, one might ask themselves, well, how often in poker Texas Holdem does a
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player get dealt two aces. Now, a true statistician would be able to use actual statistical equations to answer that
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question, like probability equations. It's not that hard of a question to answer if you understand how many cards
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are in a deck and how statistics probability works. But you could also use Monte Carlo analysis to answer that
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question. You could teach a computer, you could program a computer to randomly deal out a twocard hand and then have it
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repeat that random dealing another billion times, which for a computer, it's pretty fast to do a billion kind of
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those simulations. And over those billion random trials, a really accurate probability of a two ace hand will
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become apparent. So cards and dice and darts and I suppose other games of chance are easy targets for Monte Carlo
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simulations, but Monte Carlo can also be used on on much more complex stuff like
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weather predictions. For example, when you see one of those um spaghetti plots that a hurricane weather forecaster
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might use to show all the various paths that a hurricane might take, that's often a product of a Monte Carlo
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simulation. So these meteorological models use fluid dynamics. And fluid dynamics, one of my former courses as an
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engineer, fluid dynamics is famously unpredictable. And that unpredictability can be somewhat tamed or at least
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understood by using Monte Carlo methods. So if you're unsure if the winds are
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going to shift faster or slower, well, you simulate both those outcomes a million times. If you're unsure whether
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the wind is going to shift east or west, you simulate both a million times. Is the hurricane going to move over warmer
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waters or colder waters? You know, a hurricane's path might depend on a million different variables. And a Monte
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Carlo simulation allows you to take those million inputs, vary them a few different ways, and then output a huge
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range of results. And one set of inputs might say that Miami is going to get crushed by this hurricane, but another
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set predicts that the hurricane won't even make landfall. And over millions of
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simulations, a probability distribution emerges. And you might say, you know, Miami is is only going to get hit by
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this hurricane in 1/100th of a percent of the results. And that's a probability
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that you can probably live with. Now, Monte Carlo, I'm learning now kind of after the fact, was the name of a famous
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casino in Monaco, which is that what French like citystate, very wealthy citystate. And uh two mathematicians,
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including the very famous John Vanoyman, developed the Monte Carlo method. And they named it after the gambling house,
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kind of thinking to themselves that they could use this method to look at darts and dice and and cards. But the original
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purpose for Monte Carlo analysis was not fun and games. It was working on nuclear
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weapons at Los Alamos. I suppose if you're unsure how a high energy neutron will penetrate into fisionable uranium,
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you might simulate that neutron's path a billion times and see what the results
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look like. We still use Monte Carlo analysis for a lot of different purposes today. We used it at my old job where we
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were designing and analyzing and building and testing satellite telescope systems. In fact, Monte Carlo analysis
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exists in lots of places in engineering. Here's an example. You know what turbulence feels like in a plane. Many
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of you have probably flown in a plane. Some flights are very smooth. Some flights have a couple bumps and some
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flights cause you to kind of yell out in terror and feel sick to your stomach. That's turbulence. Well, an orbital
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rocket flies about 30 times faster than a commercial jet. So, you better believe
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there's some serious turbulence, you know, atmospherical turbulence when your
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telescope is getting a rocket ride out in space. But due to the again the chaotic nature of fluid dynamics, just
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like in the hurricane example, how do you model out that turbulence accurately? And more importantly, how do
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you model out the turbulence as it combines with some of the other uncertainties in your engineering
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design? Well, one way of doing it is by brute force of lots and lots of random trials. Monte Carlo analysis comes to
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the rescue in that situation. But in any numerical analysis, including Monte Carlo analysis, one of the the biggest
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takeaways is that garbage in equals garbage out. And I suppose what I mean there is that, you know, the accuracy of
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your inputs determines the accuracy of your outputs. If I'm trying to simulate,
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you know, the odds of a particular poker outcome, but I accidentally model the deck of cards of having 50 cards instead
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of having 52 cards, I'm going to get the wrong answer. Everything else about my
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model could be correct and elegant and and programmed uh efficiently, but if I made a major poor assumption like the
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wrong number of cards in a deck, it's going to mess up my entire analysis. So,
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that's an important lesson as we pivot towards specifically discussing Monte
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Carlo analysis for retirement planning. So, let's start thinking about how your
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one unique retirement might play out. So, you're sitting there at 50 or 55 or
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60 or 65 years old. And you probably know some of the following. You probably know what your current asset base looks
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like. You know how much you have in the bank, how much you have in retirement accounts and a taxable brokerage. You
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know how much your your house is worth, all those kind of things. Hopefully, you
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understand your cash flow. You have good data on what you could spend. Importantly, you have a reasonable
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understanding about how your spending will change over time. You might not be totally precise there. After all, you
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know, our crystal balls are foggy, but you know, if you have, again, a big wedding to pay for, you know, if you
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have ACA premiums, uh, when you're early retired or something like that, you you
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understand roughly what your future spending looks like. Hopefully, you know, at least some of the details of
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your withdrawal strategy. You're aware of which accounts you'll be withdrawing
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assets from over time. If you're unsure, I recommend checking out episode 121 of
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this very podcast for some detail on the the generally accepted retirement withdrawal framework. What you don't
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know though, among other things, you certainly don't know how capital markets
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will behave over your 20 or 30 or 40 or more years of retirement. How will stocks and bonds and other assets
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perform? Will you be subjected to a a generous or a harsh sequence of returns? That's something you don't know. And the
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first blush reaction is to simply assume some fixed rate of return. Maybe be a little conservative with your fixed rate
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of return and do that for the full 30 or 40 years of your retirement. And if all
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you have is a simple pocket calculator, that's probably the best you can do. And
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that's not a bad place to start. And that's one of those methods we talked
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about before is try to get really accurate with your cash flow and then assume some constant rate of returns and
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see how your retirement plan performs. But thanks to better and better computing over recent decades, we can do
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a Monte Carlo analysis here. Remember, Monte Carlo analysis simply means lots and lots of random trials. So instead of
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assuming one stream of investment returns and a constant stream of investment returns at that, we can
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assume many different streams of investment returns with some randomization in there. We can assume
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thousands of different variations on how the market could perform. And each one of these trials, each one of these
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thousand trials is really a a full financial life story in and of itself. A single simulation projects year-by-year
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portfolio returns, cash flows, taxes, withdrawals from today through the end of the planning horizon. Whatever you
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know you define what the end of your planning horizon is. So here the sequence of returns is really a feature
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of the analysis. It's not some sort of bug that we're trying to avoid. We want
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to see sequence of returns risk here. We want to see how impactful it is. And by
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by shuffling the order of returns across these hundreds, if not thousands of different trials, the Monte Carlo
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analysis will explicitly capture that sequence of returns risk. And that's something that these deterministic
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projections that we talked about before, they completely ignore sequence of returns risk. We're going to spend more
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time in a few minutes talking about the specific inputs that go into a Monte Carlo analysis. But first, I actually
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want to dive into the output before we dive further into the inputs. We want to talk about the outputs. The output of a
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Monte Carlo run is a distribution of outcomes. It's not one single answer. It's a distribution of answers. So
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instead of one final portfolio value, you get a wide spread of final portfolio values. Some are terrific looking, some
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are mediocre, and some are catastrophic. Some show that you run out of money. But
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that's the entire point of of doing the analysis in the first place. One thing
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that you see as an output is the so-called success rate. It's probably the headline output, the most viewed,
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the most cited output of a Monte Carlo run. In short, the success rate says out of these thousand different trials or
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however many you run, out of these thousand different trials, how many of those trials ended as a success? And
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typically, by default, success means that you die with at least $1 of positive net worth. But it's very easy
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to tweak success if you want to to be like, I want to die and gift $100,000 to each of my five grandkids. So now you
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need to die with 500 grand and that's a success. or something like that. The
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most common result again is the percentage of trials that are successful. So by getting a 82% success
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rate, it means that 82% of the trials met your success criteria but 18% failed your success criteria. And that's useful
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to know. But that simple number, 82% success, that simple number ignores the wide range of possible successes. It
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also ignores the wide range of possible failures. It ignores the scenarios where
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you you almost ran out of money and you probably would have been really really anxious about running out of money, but
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technically it was a success. It also ignores the scenarios where you barely ran out of money and you were really
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okay except for bouncing the the final check of your life and it calls that a failure. The point is that success rate
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and those distributions of outcomes, it's a pretty blunt way of viewing the output of a Monte Carlo analysis.
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Another common output from Monte Carlo is the average net worth at death. That's another common output. If you
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look at the thousand scenarios that you run, it'll just say, let's take an
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average of what the net worth is in all thousand scenarios and report that back to the user. This too is is useful, but
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a very blunt metric. It can be helpful to see how much your average net worth has changed over time in all the
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scenarios. But this one average kind of papers over all of the important details. I think I actually use this
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either in writing or in a previous episode. Imagine we have one scenario where you die with $10 million and then
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we average that with four scenarios where you die with zero. You actually run out of money. The average of those
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five scenarios, 10 and then four zeros, the average is $2 million, which sounds great. If I told you, yeah, your average
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retirement, you will die with $2 million. A lot of you will say, well, that sounds good to me. That's kind of
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what I'm looking for. But what you might not realize is that the real important
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outcome from the data I just quoted you is that 80% of those five trials actually fail, right? Four of the five
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are just abject failures. So again, the the average net worth at death, there's
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some usefulness in that output. It masks over some important details. But okay, let me pause there cuz I'm going to come
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back in a little bit and I'm going to talk about better ways, at least in my
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opinion, to to examine Monte Carlo outputs and results. Here's a quick ad and then we'll get back to the show. I
00:19:09
love getting your questions and some of you ask me questions about the wealth management firm I work for in Rochester,
00:19:14
New York. Others ask about the Best Interest blog and this podcast, Personal Finance for Long-Term Investors, which
00:19:19
operate without advertising, without pushy sales, and with no payw walls. How can the blog and podcast stay afloat
00:19:24
without me dumping my own money into it? Well, to answer both those questions, I
00:19:28
want to point you to episode 78 of Personal Finance for Long-Term Investors. I intentionally recorded
00:19:33
episode 78 to shine light on those topics and inform you how you are actually helping and can continue
00:19:38
helping these projects carry forward. So if you've ever been curious about the
00:19:41
business of my blog and podcast or if you're curious about my day job in wealth management, please check out
00:19:46
episode 78 and let me know what you think. But let's pivot now. Let's talk a
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little bit more in depth under the hood about how Monte Carlo simulations work and and therefore once we establish that
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we can start to understand more about how Monte Carlos might go wrong, how to interpret the results that we get from
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one of these analyses, how to make sure that your Monte Carlo analysis isn't
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misleading you in some way. And and usually the the first place, the most common place where a Monte Carlo
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simulation or any kind of simulation or analysis can go wrong, it's just like in
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engineering, there's an aspect of garbage in, garbage out. And what that really means is that if the inputs into
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your mathematical model are bad, if that's the garbage in, then you just know that the outputs are going to be
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bad too. That's the garbage out. So it's really important that we try to provide
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really accurate inputs and that we try to understand how the the inputs that we choose actually affect the outputs that
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we end up getting. So anyway, that's why it's it is imperative that we understand
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the inputs here such that they play such a vital role in determining our outputs.
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It's almost like the Play-Doh machines, right? Where you you stuff some Play-Doh
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in and you turn the crank and it shapes the Play-Doh in a certain way and then you get your output. It can be really
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important to understand what's going on when you turn that crank. And in the
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Monte Carlo world, there really three unique ways that the math is is kind of done underneath the hood when you're
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turning the crank. And I just want to talk on those three different ways really quick. The first method is called
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the independent identically distributed method or independent identically distributed returns because really we're
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talking about investment returns here. The second method is is newer but very interesting. It's called the block
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bootstrap approach. And then the third and the most common one is the statistical distribution approach.
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They're all very similar but they have some pretty important nuance differences. To explain them simply, I
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want you to imagine that you're maybe creating a video game. It's kind of like
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Sim City or The Sims. you're building some sort of society in your game and for whatever reason it's important in
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this game that you you model out the heights and the weights of the individual little digital people in the
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game. If you were given that problem, there are probably a few different ways that you could do that. The first idea
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that you think of is you could look up some health study that has actual height and weight data from, you know, a
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100,000 different people. You upload all these heights and weights into your game
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and then whenever you need to assign the height and weight to someone, some little digital person in your game, you
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could pull some of the actual real data that you've already uploaded, some of
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the real data from from actual humans. The second idea is that you could kind of do the same thing except maybe you
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could upload it in blocks because what you'd say is, well, if I have a family,
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let's say I'm trying to determine the the height and weight of one of my
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digital families, like their height and weights are probably all going to be correlated in some way. At least for the
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kids, they are, right? Because if you have the same genetics, if you have the same parents, your height and your
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weight is probably correlated to your siblings in some way. That's a second method you could use. But then the third
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method is is much more statistical in nature. You could just upload the average height for all people, the
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average weight for all people, and then some information about how those data are distributed. For example, you might
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describe them as, you know, two bell curves with some standard deviation of 3 in in height and 20 lb in weight or
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something like that. And then when you need to create a person in the game with a height and a weight, you simply use
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some sort of random number generator to say, "Hey, take my averages, see where
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my random number falls in a bell curve, and assign that height and assign that weight to my digital person." So the
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first method is using only real data points. The second method is also using only real data points, but it recognizes
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that often these data points can be correlated to one another. They can be kind of grouped together. And then the
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third method is using real data originally to create a statistical model, but then technically it's it's
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really making up new data points that just so happen to fit within that that same model. And these are the three ways
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that retirement Monte Carlo simulations are are basically done. The first way it's the independent identically
00:23:42
distributed return. In this method, we are only using real data. We are not creating some sort of investment return
00:23:48
or series of returns out of thin air. We are not saying that 10% is the average.
00:23:52
So please create some sort of random number between minus30 and plus 50% and make that my return this year. That's
00:23:58
not what we're doing. Instead, we are taking historical returns, real returns.
00:24:03
Sure, we're selecting them at random and then we are kind of laying them in series with one another. So if we need
00:24:09
say 50 years worth of returns to simulate our retirement, we would select 600 cuz that's 50 years 600 months. We
00:24:16
would select 600 random monthly returns from the existing historical return set to create a random time series and we'd
00:24:24
string those 600 months together. And this detail I'm about to say is important in order to preserve the
00:24:30
correlations between assets in each month. each asset would receive its, you know, appropriate return from the same
00:24:37
original month. So, what I mean is that we aren't selecting a stock return from
00:24:41
October of 1982 and pairing that up with a bond return from January of 1960, putting them in the same month in our
00:24:48
simulation. You know, the stock returns and the bond returns for month one are both going to be from October of 1924.
00:24:56
And then the stock returns and the bond returns for month two are both going to be from January 1991. We're going to
00:25:01
rinse and repeat that process month by month until we've filled up our entire
00:25:05
simulation. And that's just one retirement run, right? So again, if we go back to the 50 years or the 600
00:25:11
months, we put together a series of 600 monthly returns. And that's simulation
00:25:16
one. But then we do it again, again, all random. We do it again for simulation two and then again for simulation three
00:25:23
over again and over again and over again another thousand times. So in that way we built our Monte Carlos simulation
00:25:29
with all these random returns. They are real data right from the real uh data set. We just kind of randomized the
00:25:35
order in which those returns occur. Now this next method is a variation of that independent identically distributed
00:25:42
return. This one's called the block bootstrap method. And the idea here is that we shouldn't take one month from
00:25:47
1928 and then follow it up with a month from 1961 and then a month from 1949 and
00:25:52
1997 etc. So that's what the independent identically distributed return method
00:25:57
does. But the reason that the block bootstrap method does something different is that we've kind of realized
00:26:03
that whatever happened in that one month in 1928 is connected in some way to the
00:26:07
month before it and the month after it and the month before that and the month after that. These economic cycles and
00:26:12
bull markets and bare markets can take a long time to play out. So it doesn't
00:26:16
make sense to pull one random month at a time. It might not even make sense to pull one random year at a time. Instead,
00:26:22
the block bootstrap method suggests we take much longer blocks of returns. The most common length being 10 years, 10
00:26:29
years of returns at a time. So, to create a a 30-year retirement simulation, we would grab three
00:26:35
different 10-year blocks of returns and use those to simulate one full 30-year retirement. Each of those 10-year blocks
00:26:42
would have some true long-term economic patterns within it. And then we do it again with three different 10-year
00:26:49
blocks. And then again with three different 10-year blocks from that. And every time we're randomly selecting the
00:26:55
start date for these different 10-year blocks. We do that over and over again hundreds if not thousands of times.
00:27:01
Again, now we have a a true Monte Carlo simulation with thousands of individual trials all randomly selected. But rather
00:27:07
than having 30 or 40 or 50 years worth of totally random monthly returns, the block bootstrap method carries some some
00:27:15
real economic oomph with it because the blocks have some sort of underlying patterns that more closely resemble
00:27:22
reality. But that brings us to the third approach which I think is the most common approach. It's certainly the one
00:27:27
that I've seen most frequently. It's the statistical distribution approach. So
00:27:31
this is the method that requires you to input an average rate of return based on
00:27:35
the performance history of your specific assets or the performance history of your specific portfolio allocation. You
00:27:42
input the average return. You also input information about the distribution of returns. For example, bell curves or
00:27:49
normal distributions or Gaussian distributions. These all mean the same thing. A bell curve is a normal
00:27:55
distribution is a Gaussian distribution. It's probably the most common type of
00:27:58
statistical distribution that people are used to. even if they might not be aware
00:28:02
that they're used to it. You know, you're used to the fact that 95% of American men are somewhere between 5
00:28:08
foot five and six foot five. And that it's much more rare, but not unheard of
00:28:12
to see someone who's 6'8. And it's much rarer still to see someone who's 7 foot
00:28:16
tall, you know, etc., etc. That's because height data matches a bell curve. It matches a normal distribution.
00:28:23
This is the the mathematical distributions that just so happens to accurately describe the way that height
00:28:28
data is is distributed amongst a large population. And whenever you're trying
00:28:33
to describe the way a bell curve looks, it's important that you define the average of that curve, it's also very
00:28:39
important that you define the standard deviation or a measure of how quickly the bell as it were kind of dissipates
00:28:46
down as you move away from the average. But there are other types of distributions too. There's lognormal
00:28:52
distributions and power law distributions and posson distributions that all actually do have a place in
00:28:57
financial modeling of some sort. But the big problem, the really big problem with
00:29:02
using a statistical distribution to model out investment returns is that no single statistical distribution seems to
00:29:09
model investment returns that accurately. So I'll say that again. There's no statistical distribution that
00:29:16
models investment returns as accurately as we would want them to be modeled. For
00:29:20
example, the normal distribution, the classic bell curve. This is the distribution that many Monte Carlo
00:29:25
simulations actually use. You know, if your Monte Carlo software of choice is using an average return plus a standard
00:29:32
deviation, I'd wager under the hood, it's using the standard bell curve, the
00:29:36
normal distribution, the Gaussian distribution. But the problem is that the stock market, investment markets,
00:29:41
they have these crazy days and crazy weeks and crazy months, maybe even crazy years that we've heard of before. We're
00:29:47
used to the idea that yeah, once in a while, like April of 2025, the market did happen to drop was it 9% in one day
00:29:55
and then the next day it was back up 9% again. The problem is that if we if we use the way a standard distribution
00:30:02
really works and if we look at how infrequent those really rare events are supposed to happen, you know, a standard
00:30:07
distribution says that something might be a 1 in 1,000-year event. And if we're
00:30:13
talking about the height of a human and we say, well, you know, if we're going
00:30:17
to get a human who's 9 ft tall, odds are one is going to be born every 500 years,
00:30:22
well, there aren't many 9 foot tall people walking around. And and the bell curve's way of describing the frequency
00:30:27
of 9 foot tall men actually plays out in reality. But when we use a a bell curve,
00:30:32
a normal distribution to describe the stock market, we have these events that should be one in every 1,00 years. But
00:30:39
we see those events occurring way too frequently in real life because the normal distribution has very thin tails.
00:30:45
The normal distribution says that, you know, the market dropping 8% in one day ought to occur once in a million trading
00:30:52
days except it's occurred eight different times in the last 10,000 trading days. It's just not a accurate a
00:30:58
way of describing market behavior. The good news though, the good news it seems, is that the normal distribution
00:31:05
does become more and more acceptable at longer time frames. Meaning, if we needed to model daily market returns,
00:31:12
the normal distribution would be a very bad choice of doing that. Totally inappropriate. There are just too many
00:31:17
crazy days up or down in the market for the normal distribution to to accurately
00:31:21
model it. But if I'm modeling one-year returns, then suddenly the bell curve
00:31:26
becomes not perfect per se, but certainly better, much better. It turns out that the most accurate way to create
00:31:32
real world market returns is just to use an actual data set of real world market
00:31:37
returns and not to let a statistical distribution kind of create those market returns for you out of thin air. But
00:31:43
many commercially available Monte Carlos software packages do use a bell curve, a
00:31:48
simple bell curve, a normal distribution to distribute and randomize their returns. Balden, for example, Balden is
00:31:55
a a very popular DIY financial planning software which includes a Monte Carlo package and a lot of again a lot of DIY
00:32:02
financial planners out there use it. It has a a Monte Carlo simulation based on a normal distribution curve, a bell
00:32:08
curve, and it uses the average rate of return and the reasonable standard deviation that you input into the
00:32:13
software to create your Monte Carlo output. Now, if you're really curious, it it seems like the uh lelass
00:32:20
distribution is at least better at capturing the the fat tail nature of the stock market. And I can throw a link
00:32:25
into the show notes that kind of has an overlay of a bell curve and a lelass curve if you want to see them on top of
00:32:30
each other. But I don't have a statistics degree and most of you don't either. And the good news is that we
00:32:35
don't really need a statistics degree to be able to run a Monte Carlo analysis,
00:32:38
right? We're all going to be okay. I think it's just worth understanding that
00:32:41
there are different ways to do any sort of numerical method underneath the hood.
00:32:46
And it's just important to know that the method that you choose has a bearing on
00:32:49
the output. It might not impact the output a ton. And that's where if you if
00:32:54
you understand what's going on under the hood, then you can make that kind of
00:32:57
value judgment for yourself. Am I willing to accept any sort of error that's going to occur because of the the
00:33:03
numerical method that I'm picking here? I think that's why they say there's a
00:33:06
very common phrase in the world of numerical modeling that all models are wrong, but some are useful. Like like if
00:33:12
you're doing some sort of statistical method like the ones we've been describing, you know that your answers
00:33:17
are going to be wrong. Like no answer that you get from any Monte Carlo is going to perfectly match the one unique
00:33:24
pathway that lies ahead of you for the next 30 years. Of course, that's not going to happen, right? We know that.
00:33:30
But we also know that these models can be really useful in explaining the range of possible outcomes that we might live
00:33:35
through. As long as you understand what kind of errors you're introducing into
00:33:37
your range, you're going to be okay. But next, whether we're using a a normal
00:33:41
distribution or some other distribution, the important inputs of a Monte Carlo actually aren't quite done yet. Because
00:33:47
with any distribution, we need to define the parameters of that distribution. We
00:33:52
need to define what average actually means and we need to define the you know how quickly the curve flattens out how
00:33:58
the data is dispersed within the distribution and whether we're talking about a normal distribution or a lelass
00:34:04
distribution or for many statistical distributions that term is called the standard deviation or the variance and
00:34:10
in investment lingo we might use the word volatility instead. We're trying to
00:34:14
define how volatile these assets are. So in Monte Carlo software, you'll often be
00:34:18
asked to define the average return and the standard deviation of that return. Maybe you'll be asked about your
00:34:23
portfolio as a whole or maybe you'll be asked to define those data points for
00:34:27
specific asset classes. So perhaps the software will already have those numbers programmed in so that you don't have to
00:34:33
put them in. If you're inputting specific asset classes, so you know, data for stocks, data for bonds, data
00:34:39
for real estate, etc., Then you'll also probably have to input the correlation
00:34:43
for those assets. Stocks and bonds and real estate and all other capital assets. They don't behave independently
00:34:49
in a vacuum. They behave in a dynamic world where they influence one another. So the return profiles of different
00:34:54
assets are correlated to one another to varying degrees and a proper Monte Carlo
00:35:00
analysis will account for that. And maybe this goes without saying, but the more inaccurate your return assumptions
00:35:05
are, the more error will end up in your results. So if you decide to be extra conservative with your return
00:35:10
assumptions, then you have to accept that your failure rate is going to be skewed higher than it otherwise would
00:35:15
be. If you underestimate the standard deviation in your portfolio, which yes, we can call volatility. If you
00:35:20
underestimate the volatility in your portfolio, then your investment ride will be far smoother inside the Monte
00:35:26
Carlo than in real life. And you'll probably end up underestimating just how
00:35:31
the sequence of returns risk might negatively affect you. So typically, this is where we go consult history. We
00:35:37
look at historical market data. We ask ourselves questions like how has this asset or this portfolio performed over
00:35:42
time? What's been the average return? How volatile has it been? And that becomes your input or at least it should
00:35:48
heavily heavily guide your inputs. And then you let the Monte Carlo run. Okay. So, what's going on uh when we let the
00:35:54
Monte Carlo run? Again, the the main inputs into a Monte Carlo simulation are typically your unique future cash flows,
00:36:00
namely how much withdrawal out of the portfolio you're expecting on an annual
00:36:04
basis. And then your portfolio return profile, both the average and the average return and the expected
00:36:10
volatility. And one thing you you might have noticed by now is that since the return and the volatility and even the
00:36:14
correlation between assets are single entries, you're usually assuming that you will have one portfolio allocation
00:36:21
throughout retirement in a Monte Carlo run. So that's the way that most Monte
00:36:25
Carlo programs operate. Even if we know that a real retirement might look different. So even in something as
00:36:31
dynamic as a Monte Carlo simulation, it requires even more dynamism to say, well, is this retiree going to be
00:36:37
changing their investment allocation over time? Are they going to retire at 6040 and then eventually go down to
00:36:42
50/50 only then to realize they have too much money and they're saving it for
00:36:46
charity and their heirs and they go back up to 7030? Like most of these software
00:36:50
packages can't handle that level of dynamic complexity. So they just assume you have one single static allocation
00:36:57
throughout your retirement. Even if we know that might not be the way that your specific retirement will go. But the
00:37:03
program takes your portfolio return, your expected volatility, and some form of a random number generator to create a
00:37:09
decadesl long sequence of returns that your portfolio will endure. And those returns combined with your future
00:37:15
outflows. That's all you need to see how your portfolio will fare in retirement.
00:37:19
And then the program does it again using the same average return and the same volatility, but now a different set of
00:37:24
random numbers to create a different decadesl long sequence of returns. Now that's you have a a second version of
00:37:30
how your portfolio fares. You can do that 10 times. You can do it a thousand times. You could do it a million times
00:37:35
if you had enough time to do it. But either way, you now have many many many hypothetical retirement paths to
00:37:41
examine. How do we interpret those results? How do we make sure your Monte Carlo isn't really misleading you in
00:37:47
some way? As I alluded to earlier in the episode, the the surface level results from Monte Carlo runs can leave out a
00:37:54
little too much detail, and I think we need to go deeper. So, first, I think we need to discuss what failure means in
00:38:00
the context of retirement planning or or Monte Carlo analysis. Typically, Monte Carlo programs define failure as not
00:38:06
meeting your end goal. And most of the time, that end goal is not running out of money at death. So to have at least
00:38:12
$1 is deemed a success and having anything less than $0 is deemed a failure. The second most common end goal
00:38:20
though usually involves leaving assets to heirs or to charity at death. So again, if you want to leave five kids
00:38:26
each $100,000, then in that case, anything less than $500,000 at death would be deemed a failure because you've
00:38:32
said that's your end goal. That's pretty black and white. But imagine this. You
00:38:36
retire at 55, things go well for a couple years, and then the market throws you some rough curve balls. You hit a
00:38:42
pretty bad sequence of returns in your first decade, and you know that some of the smart ideas we discuss here, you
00:38:47
decide to pair back your spending a little bit. You do what you can to mitigate sequence of returns risk as
00:38:52
best you can. And because you take this evasive action, your retirement gets back on track, so to speak, and you live
00:38:57
happily ever after. That is something that all of us have the power to do in our retirements. But a Monte Carlo
00:39:04
analysis, at least the commercially available ones, do not model that for us in the least. Now, why not? Because what
00:39:10
I just described there was dynamic. Based on the bad sequence of returns that we lived through in my
00:39:15
hypothetical, we dynamically decided to decrease our spending. And then when things turned around, we dynamically
00:39:21
decided to increase our spending. Again, Monte Carlo programs don't do that. Or
00:39:25
again, it's not that they can't. It's just that such many different degrees of
00:39:30
dynamic modeling. That's a challenge. It's a difficult challenge to program
00:39:33
and to be fair, it becomes a challenge for end users like us to implement that in a clear way. You know, if you're
00:39:39
looking at some sort of online software program where you have to input a hundred different numbers in order to
00:39:45
quote unquote model your retirement, it might prevent you from taking that next step and actually modeling it in the
00:39:50
first place. It's too intimidating. It's too difficult. So in order to keep Monte
00:39:54
Carlos simulations kind of on the simpler side, there are some dynamic things that we would normally do in real
00:39:59
life that just are not part of standardly available software packages. But what it really means is that the
00:40:06
failure percentage from Monte Carlo analysis is misleading. What failure percentage what it really should be
00:40:11
thought of is the likelihood that you might need to be dynamic in your retirement decisions. When your Monte
00:40:17
Carlo run gets a 80% pass rate, it means that in 80% of realistic market conditions, you could literally go on
00:40:24
autopilot and not run out of money. But in the other 20% that are deemed failures, but in those 20% of simulated
00:40:31
market conditions, you would have to make a dynamic decision in order to achieve retirement success. It doesn't
00:40:37
mean that you outright failed. It means that you would have failed if you were just on spending autopilot and
00:40:42
withdrawal autopilot. But if you decided to be dynamic, you might have succeeded.
00:40:46
Here's a quick ad, and then we'll get back to the show. I send a free weekly
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00:41:02
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thousands of people like you are already reading, a free white paper to help you
00:41:32
plan for retirement, and you can sign up for free at bestinterest.blog. And that brings us to another um kind of
00:41:39
challenge when it comes to interpreting Monte Carlo outputs. Imagine you run a Monte Carlo simulation with a thousand
00:41:44
different trials. And somewhere in those thousand trials, you're going to find
00:41:49
two of those thousand that just so happen to be on either side of the failure line. You know, one of them
00:41:54
happens to terminate with $129 in positive net worth and that's technically a success and the other one
00:42:01
terminates with negative $400 in net worth and that's technically a failure.
00:42:06
So you could have started retirement with a couple million dollars and and gone through decades of spending and
00:42:10
investing and at the end of it all a measly you know $561 ends up being the difference between these two trials
00:42:18
between success and failure. Yeah, that is how the simulation works. It's not a
00:42:22
bug. It's not a problem. But it is important that you understand these inner workings because Monte Carlo
00:42:27
analysis can be presented to you as black and white as successes and failures. But those two scenarios I just
00:42:33
described where after 30 years, $560 determine the difference between success and failure, those two scenarios are
00:42:41
almost exactly the same shade of gray. It's just that we we draw a little dotted line somewhere in that gray area
00:42:47
and that dotted line happens to divide positives from negatives. If we're not
00:42:52
careful, we'll falsely believe that the the dotted line actually separates black
00:42:55
from white. It doesn't. It separates gray from gray. Gray from slightly darker gray. I guess I reached out to uh
00:43:02
Carsten Yesi. If you don't know Carsten, he goes sometimes by the nickname Big E
00:43:06
N. Big E R N as in early retirement now. Ern, Big N. He's the author of the highly acclaimed financial independence
00:43:13
blog, Early Retirement Now. And Ern, I'll call him Ern even though his first
00:43:17
name is Carson. He's definitely one of the biggest kind of foremost thinkers
00:43:21
when it comes to safe withdrawal rate research such as the 4% rule. And I was noodling on the problem I just described
00:43:28
or or part of the problem of Monte Carlo analysis. So I wrote to him and I said,
00:43:31
you know, hey Ern, my understanding is that retirement withdrawals can be a slippery slope. If when your retirement
00:43:36
nest egg drops below a certain threshold, your risk of running out of money increases dramatically. Or perhaps
00:43:43
set another way, as your retirement accounts drop, there is an exponential, not linear, there is an exponential
00:43:49
increase in the probability that you eventually run out of money. The sequence of returns risk is a terrific
00:43:54
example of this fact. So to that end, it begs the question, why do we pass fail safe withdrawal rates based on 0 left in
00:44:02
the account? We start retirement at 100% funded. We withdraw 4% or some other percentage per year adjusted for
00:44:09
inflation. And usually markets have increased our nest egg to 200% or 300% or more by the time of death. And we
00:44:16
define 0% as a failure when really I'm not interested in the point when we hit
00:44:20
0%. By that time it's too late to change my future. I'm much more interested in
00:44:26
when should I be worried? When do I need to take evasive action? Is it when my accounts hit 80% of where they started?
00:44:33
Is it when they hit 50% of where they started? Is it something else? Is there an inflection point where you believe it
00:44:38
makes sense for a retiree to seriously consider a change in course as their probability of failure has just become
00:44:44
too high? So, that was what I wrote to Ern, and he got back to me with something that he called a conditional
00:44:49
success rate. I'll link to this article in the show notes. This conditional success rate helps retirees contemplate,
00:44:55
you know, it's been Years since I retired and my portfolio is currently down X%. How does that compare to
00:45:02
history? How worried should I be compared to other time periods? I know my retirement hasn't failed yet, but
00:45:08
surely I'm in a worse position now than other historic periods. Surely because
00:45:11
my portfolio is down, I ought to consider tightening the belt and spending less. You know, we can compare
00:45:16
that to other probabilistic games. We can compare it to poker. You know, hey, I'm playing poker now that a king has
00:45:22
come out in the flop. What's the conditional success of my pocket pair of queens in in Beijian terminology, right?
00:45:29
This is adjusting your priors. Your future probabilities change or I guess I should say your current probabilities
00:45:34
change based on this new information. So, you have to adjust your prior assumptions. And part of my question
00:45:41
essentially is how should retirees adjust their priors as they move throughout retirement? More applicable
00:45:47
to to real life, how and when should retirees adjust their spending throughout retirement? This is a
00:45:52
question of course that many people are attempting to answer in in many interesting ways, but it's always good
00:45:56
to have some analytical rigor behind whatever your your answers are going to be. So, here's a cool tidbit from Big
00:46:01
Earns Research to kind of fill you in on my thought process. Imagine our retiree
00:46:06
retires with $1 billion and they have a 30-year retirement goal ahead of them. They're invested in 75% stocks and 25%
00:46:13
bonds. They're following the 4% withdrawal rule and adjusting for inflation. And we're going to use real
00:46:18
historical market data here. It turns out that this retiree, their plan, their rigid plan, right, which isn't dynamic
00:46:24
at all. It's the 4% rule. It's going to fail in about 1.8% of all historical
00:46:29
scenarios. But let's take that tidbit. It fails 1.8% 8% of the time. Let's now
00:46:35
fast forward 10 years into this person's retirement. Can we look at that 10-year
00:46:39
mark and can we start to see if they're on the road to long-term success or the
00:46:43
road to long-term failure? In other words, for some of the trials, by year 10, their accounts will have gone down.
00:46:48
Sometimes they'll they'll have less than $750,000. Sometimes they'll have less than
00:46:53
$500,000. They started with a million, but then for other trials, their accounts will have gone up. They'll have
00:46:58
more than 1.25 million after 10 years, even with all their spending. And if we had to guess, I would wager that we'll
00:47:04
see far fewer failures from the second group, I guess maybe from the third group. We'll have far fewer failures
00:47:09
from the people who accounts where their accounts have gone up and we'll see more
00:47:13
failures from the people who 10 years in their accounts have gone down. That just
00:47:17
seems to be the way that the sequence of return risk works. And this is called conditional probability. We ask how does
00:47:24
the probability change once we add a specific condition. Perhaps the condition is you made it to year 10 and
00:47:29
you're already down to $750,000. Or the condition is you made it to year 10 and you're already up to $1.25
00:47:36
million. And what big earns analysis shows is that if at year 10, if a retirees portfolio was already down
00:47:43
below $500,000, then they have a 13% chance of outright retirement failure and they only have a 33% chance of dying
00:47:51
with more than the 500k in their portfolio at that time. So there's a twothirds chance that their portfolio is
00:47:57
going to continue going down from there. And that's a scary thought if you're
00:48:00
only 10 years into retirement. But if at year 10 your portfolio had already grown
00:48:05
to 1.25 million or more, there wasn't a single outcome where you died with less
00:48:11
than $500,000. Not one. So in the first bad scenario we just outlined, you had a
00:48:16
twothirds chance of dying with less than 500,000. But in the good scenario, you have zero chance of dying with less than
00:48:22
500,000. In fact, in the good scenario, you have a 75% chance of dying with more
00:48:27
than $2 million. The takeaway again is that the early years of retirement absolutely set your path. Your end
00:48:34
results are conditional upon the way your early years unfold. And therefore, if you can make smart, dynamic decisions
00:48:41
in your early years, you will be shifting the conditional probabilities in your favor. A Monte Carlo simulation
00:48:47
for all its power does not do this. It's on you to understand that fact. For me,
00:48:52
one of the keys of understanding a Monte Carlo output also lies in percentiles. You've probably heard percentiles
00:48:58
before. Wow, that baby is so fat. Yeah, he's in the 98th percentile for weight.
00:49:03
Okay, that means that baby is heavier than 98% of his peers. So, the 75th percentile shows us the result that is
00:49:10
better than 75% of the other simulations and worse than 25% of the simulations. The 10th percentile shows us one of the
00:49:17
worst outcomes. In fact, the 10th percentile is worse than 90% of the outcomes in the simulation. And by
00:49:24
comparing different percentiles in a Monte Carlo run, we begin to understand our range of reality. Financial planning
00:49:30
is very much about understanding your range of potential outcomes. Imagine I compare the 10th percentile to the 90th
00:49:36
percentile. In doing so, I now understand 80% of my total range of outcomes. Then I can start to ask, well,
00:49:43
just how disperate is that 80% range? If it's all over the map, say I retire at
00:49:48
age 55 with $3 million and the Monte Carlo simulations range from abject failure at the 10th percentile up to $15
00:49:56
million in terminal wealth at the 90th percentile. If it's that wide of a range, it tells me some interesting
00:50:01
things. Specifically, it might tell me that the volatility in my portfolio subjects me to some significant sequence
00:50:08
risks. Now, the good sequences are 5xing my money before I die, but the bad sequences are leading to retirement
00:50:15
failure. So, hm, I might want to play around with that result. I might want to adjust my allocation, my volatility
00:50:21
accordingly. If I'm comparing the 99th percentile though to the first percentile, well, I expect that range of
00:50:27
outcomes to be pretty big no matter what because it contains almost all of the simulations, including the ones that
00:50:32
have some really crazy sequences built into them. you might as well just be looking at the the maximum and the
00:50:37
minimum, the total range. But by cutting 5% or 10% or even 20% of the data on either end, that still leaves you with
00:50:44
this big majority chunk of data in the middle that captures most of the possible outcomes and defines your
00:50:50
likely range that you could have to deal with. If you're feeling particularly
00:50:54
statistical, I guess you might have to know that 34% on either side of the mean average of a normal distribution, that
00:51:01
34% captures one standard deviation. So by measuring between the the 16th percentile, that's 50 minus 34 in one
00:51:10
direction. Between the 16th percentile and the 84th percentile, which is 50 + 34. So between the 16th and the 84th
00:51:17
percentiles, you're capturing 68% of the results that occur within one standard
00:51:22
deviation. either direction of the average. It's just a handy way of making use of Monte Carlo outputs without
00:51:28
focusing so much on the worst of the worst outcomes or the best of the best outcomes. And then you can ask yourself,
00:51:33
you know, how many scenarios in that range that I've just described, how many
00:51:36
of those scenarios end up starting to slip down that slippery slope toward running out of money? And I mean, how
00:51:42
many of those scenarios end up on the other side where you're compounding faster than you can spend such that you
00:51:47
die with three times or five times or 10 times what you started retirement with?
00:51:51
So, I think that's one really good way to to look at Monte Carlo results. But
00:51:54
then I also really recommend you understand a middle-of the road failure scenario. So, yes, a failure scenario.
00:52:01
So, let's say for example, your initial Monte Carlo run has an 80% chance of
00:52:06
success. That's great. I recommend you grab a couple of the outputs, a couple
00:52:10
of the trials from the middle of the 20% of failure scenarios and really dig into
00:52:16
those couple of outputs. Try to understand at what point in this failure trial did things start going off the
00:52:22
track. What could you have done differently? Again, if you were allowed to be dynamic, if it's if you're living
00:52:27
your life, what could you have done differently in that trial specifically in terms of spending less? How many
00:52:32
years of lowered spending and what magnitude of lowered spending could have turned that particular failure into a
00:52:39
success? I think that will really help you understand again that that's a really good way of going beyond the
00:52:44
surface level results of a Monte Carlo and really digging into the detailed results. The very last thing I'll I'll
00:52:50
go over in this episode. What are some other common Monte Carlo mistakes? A really easy one to avoid and this can be
00:52:55
very consequential is understanding how your particular Monte Carlo software handles inflation. For example, do you
00:53:02
need to account for inflation when you input your spending numbers or does the program ask you for a particular rate of
00:53:08
inflation and then it automatically inflates your spending numbers for you? Do you need to account for inflation in
00:53:14
the portfolio return assumptions or not? You know, nominal returns or real returns. Considering long-term inflation
00:53:20
averages out to like 2 to 3% per year, you need to get inflation right. You need to make sure you model it
00:53:26
accurately. Double counting inflation, thus reducing your real return by an extra 2 to 3% per year, creates way more
00:53:34
failure than reality ought to, but then not counting your inflation at all, thus
00:53:38
increasing your real return by an extra 2 to 3% return. Well, that's going to
00:53:42
create way more success than reality otherwise would. So, you've got to get inflation right. Overall, these Monte
00:53:48
Carlo programs, they're great tools to to examine where you're at and to
00:53:52
understand the range of possibilities that your future might go through. But importantly, your unique future, it only
00:53:58
partially exists in that range of possibilities that a Monte Carlo is going to create for you. There are a lot
00:54:03
of dynamic aspects in retirement planning. And a Monte Carlo analysis captures some of that dynamism amazingly
00:54:09
well, but it does not capture all of it. Other parts of retirement the Monte Carlo has to assume are static and then
00:54:15
you, the user, you have to infer from there. So, if I didn't answer any specific questions you have about a
00:54:21
Monte Carlo analysis, by all means, feel free to email me to jessebinest.blog. blog. I know this was a wonky one. I
00:54:27
don't expect any of us to be out there writing our own Monte Carlo software or
00:54:32
uh getting statistics PhDs to understand what's really going on under the hood.
00:54:36
It's just one of those things where a Monte Carlo, like many things in this world, it's got a bit of that
00:54:40
double-edged sword to it. And as long as you understand how to use that sharpness
00:54:44
for your benefit and not to get injured by it, I think you'll be in a good place. So, thank you as always for
00:54:49
listening. >> Thanks for tuning in to this episode of Personal Finance for Long-Term
00:54:53
Investors. If you have a question for Jesse to answer on a future episode, send him an email over at his blog, The
00:54:59
Bestin Interest. His email address is [email protected]. Again, that's jessevestinterest.blog.
00:55:08
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personal podcast meant for education and entertainment. It should not be taken as
00:55:29
financial advice and it's not prescriptive of your financial situation.