### Executive Summary

Monte Carlo analysis is a superior retirement planning approach to the standard "straight-line" retirement projection, because it implicitly considers not only average returns, but a range of potentially volatile returns, allowing the prospective retiree to understand how the retirement plan might fare in various scenarios. However, with the additional capability to illustrate a range of volatile returns - potentially across multiple investments or asset classes - comes a greater burden to craft appropriate investments assumptions for the Monte Carlo analysis. Otherwise, there's a risk that the Monte Carlo analysis could mis-state the risks of various retirement portfolios.

In this week’s #OfficeHours with @MichaelKitces, my Tuesday 1PM EST broadcast via Periscope, **we discuss how Monte Carlo investment assumptions impact retirement planning projections, and particularly how important it is to include an appropriate correlation matrix when including multiple investments or asset classes in a Monte Carlo projections. As failing to include a proper correlation matrix will generally lead to a retirement projection that understates risk and overstates the probability of success!**

The key issue is that when selecting investment assumptions for a Monte Carlo analysis, there are three core points of data that are necessary for each investment: expected return, volatility, and correlation. And correlations are much more challenging assumptions, because it's necessarily to make an assumption for the relationship between each investment *and every single other investment paired to it!* And a rising number of investments necessitates dramatically more correlation pairs - as a result, 3 investments requires 3 correlations, but 5 investments requires 10 correlations, and 10 investments requires assumptions for a whopping 45 correlation relationships!

And the reality is, given the underlying math of a Monte Carlo analysis, even making no correlation assumption *is *an assumption. It's just an implicit assumption of *zero *correlation... which is actually a very *dangerous* assumption, because, in the real world most investments don't have zero correlation. Yet by assuming zero correlation when the correlation is actually higher, the projection ends out overstating the benefits of diversification, and therefore understating the risk to the retiree and overstating their probability of success in retirement!

In the end, this doesn't mean it's bad to have a diversified portfolio, but it's crucial to recognize that adding more investments to a Monte Carlo analysis doesn't necessarily make it more "accurate", and in fact will decrease the accuracy of the projection unless the entire correlation matrix is properly included (with appropriate assumptions). As a result, a better practical approach for many advisors may be to utilize simpler two-asset-class portfolios of stocks and bonds for Monte Carlo purposes... as while this may slightly understate the benefits of having a fully diversified portfolio, at least it won't overstate the benefits, and it is far easier to help a client adjust to a retirement that is going better than expected, than to adapt to one that is going worse!

*(Michael’s Note: The video below was recorded using Periscope, and announced via Twitter. If you want to participate in the next #OfficeHours live, please download the Periscope app on your mobile device, and follow @MichaelKitces on Twitter, so you get the announcement when the broadcast is starting, at/around 1PM EST every Tuesday! You can also submit your question in advance through our Contact page!)*

## #OfficeHours with @MichaelKitces Video Transcript

Welcome, everyone! Welcome to Office Hours with Michael Kitces!

This week, I want to talk about a more technically complex question than what I usually cover in Office hours. Monte Carlo analysis, and, specifically, the kinds of asset allocation investment assumptions that we have to make when we're doing Monte Carlo projections. I find Monte Carlo to still be a very misunderstood methodology for analyzing retirement plans, and especially when it comes to the impact that our investment assumptions can have on the results of a Monte Carlo projection.

A case in point example is the question that came to me recently from Jim, who asked:

“Michael, I use a robust asset allocation in my client portfolios. Some elements of risk parity and what I’d consider alternative asset classes, although it's all publicaly traded. And my challenge is no financial planning software really has robust asset allocation built in. It has to be customized manually. And I'm concerned because I think we're in an investment climate where stocks are expensive, and bonds are expensive, and Monte Carlo may not fully adequately measure this challenge. And so, this is why I use a more robust asset allocation model, except I don't want to over-engineer the software modeling the outcomes, and this is the challenge. I'm doing financial planning software demos, I'm getting hung up because if I build out custom asset classes, I have to project future returns, and then I have to measure correlations across every asset class, and there isn’t even always an input for correlation, so what should I do?”

This is a great question. This is kind of the double-edged sword of Monte Carlo analysis in the first place. It's a very capable tool for doing very robust modeling of the risks of a client's retirement plan. Frankly, far beyond what you can do just with a simple straight line projection, where you just assume steady growth rates. But the bad news is, as Jim notes, is that it's way more complex to figure out what those investment assumptions should be in the first place.

## Constructing A Correlation Matrix In Monte Carlo Analysis [Time - 2:09]

When it comes to investment assumptions for Monte Carlo analysis, the reality is, there's kind of three core points of data that you need for each investment. Number one is the expected return. An expected return is pretty straightforward. If you want to model how a retirement portfolio is going to grow with various asset classes, you need an assumption about how each asset class will grow.

The second input is volatility. This is one of the key additions you get with Monte Carlo analysis. The ability to show a range of possible outcomes, which we can quantify based on the probability that will occur using a measure of volatility. So, most commonly, we use standard deviation because it happens to make the math of Monte Carlo analysis a little easier from a programming perspective. But then there's a third one that a lot of people forget, which is correlation. Because ultimately, we’re trying to model the volatility of an overall portfolio.

And the key insight of Markowitz going all the way back to modern portfolio theory 60 years ago, is that the volatility of a portfolio may be lower than the volatility of the individual assets in the portfolio, if they're not perfectly correlated, right? Because there’s that possibility that one will zig while the other one will zag. And so, to capture not just volatility of asset classes, but volatility of the whole retirement portfolio, we need not only the volatility of the asset classes, but their correlations to each other.

Unfortunately, this is the problem, because getting correlations is actually harder than the other inputs. Getting expected return is pretty straightforward. I can look at long-term historical returns as an average, maybe I can use fundamental valuation metrics, look at earnings growth plus dividends yields. We have lots of different ways to create reasonable assumptions of expected return in the future.

But volatility is a little bit messier. We can look at how volatile the investment’s been historically (maybe in various market environments) and get some reasonable estimates of forward-looking volatility. But for expected returns in volatility, the key is that you just have to estimate it once. Each asset class in the portfolio gets one estimate of return, one estimate of volatility. Correlations though, are messier, because you have to measure the relationship with each investment to each other possible investment. So, if we have A and B, two assets, only one correlation, right? The connection between A and B. But if we have three investments A, B, and C, well, now it's a little messier because there is the relationship between A and B, A and C, B and C. Or it's like a little triangle, three possible connections. And as we add more investments from there, the numbers don’t just get larger, they get much larger.

So, if you have 5 investments, there are actually 10 different pairs of all the different things to connect to each other. If you've got 8 investments, there are 28 correlation pairs. If you've got 10 investments, there are 45 different correlation pairs that you have to measure and put into your financial planning software to model this. And so, the challenge that arises as kind of illustrated in Jim’s question is that, as you add more investments into a Monte Carlo retirement projection, it isn't just as straightforward as adding each investment in an estimate of its future return and volatility, you need a correlation. You need the correlation of that investment to every single other thing in the portfolio, which makes this a lot messier. And unfortunately, the reality is, you have to have a correlation matrix.

## The Impact Of A Monte Carlo Correlation Matrix [Time - 5:07]

It's built into the math of Monte Carlo analysis.

If you don't make any correlation assumptions, they’re still in the math. You're just implicitly assuming that the correlation is zero. And the problem is that, in the investment world, that is a really dangerous assumption. Most investments, particularly the ones we care about, that are risky and volatile, don't have a correlation of zero, right? If there’s a correlation of zero, it would as completely random about whether one goes down or the other goes down or not, or they zig or zag.

But in the real world, we know, most equity asset classes have a decently high correlation to each other. So the correlation between large cap and small cap is usually between like 0.7 and 0.9. Domestic international stocks is kind of similar in recent years, and even alternative asset classes often have surprisingly high correlations.

A lot of people owned real estate as a diversifier and an alternative asset class in the past, and then what happened in the financial crisis? Correlations were high, and they all went down together. Not only did it not have a non-zero correlation, it actually had a correlation that was pretty close to one. And so, here's why this matters, if you add lots of investments to show effective diversification in your Monte Carlo analysis, and you don't make appropriate assumptions about the correlation matrix, you will understate the portfolio risk and overstate the probability of success, by just defaulting to zero correlation.

A zero correlation would be amazing if we could get it, but if you don't recognize how high the correlations really are, you end up grossly overstating how much diversification is actually helping you, and understating the risk. Which ironically means, if the reason you don't like doing things like Monte Carlo analysis in the first place is you don't think it takes into account the risks of the marketplace when you try to add in more investments, then adding more investments without accounting for correlations, actually makes it worse.

The more investments you add without modeling the correlations, the worse the projection will even further overstate the probability of success and understates the risks. So, adding in more investments doesn't make the problem better, it makes it worse!

And it's actually even worse than that because what we are increasingly learning about correlations in markets is that they're not stable. They change over time. There are a lot of asset classes that had a fairly low correlation until the financial crisis when it mattered, and then the correlations went up. Which means, ideally, you need to not only project every correlation in the correlation matrix (every possible connection to each other), but also how they will change over time. And the reality is, unfortunately, that I don't know any financial planning software that can handle that. It's theoretically possible. I mean, you can make investment models that model volatility, and correlations that change over time, but it's really complex modeling. It's even more sensitive to assumptions, and so you have to do even more work to figure out what the assumptions should be, and then you need a more complex model to do it, which we usually don't have.

And that all assumes you can get the data in the first place, which unfortunately, for a lot of alternative asset classes, we don't have. They haven't been around long enough to give us data about how the volatilities and correlations might change in rising rate environment, or flat, or falling rates, or in a recession or a depression, or in rising inflation, if and when inflation ever finally gets underway again.

## Adding Investments Reduces The Robustness Of Monte Carlo Analysis? [Time - 8:11]

But the fundamental point here is that just adding investments to a Monte Carlo analysis doesn't necessarily give you a more robust retirement projection. In fact, if you don't accurately consider and model all the correlations between the assets, adding in more investments to the portfolio and the Monte Carlo projection, makes it worse. Because saying, "I'm not sure, let's just say there is no correlation as an assumption", actually grossly overstates the true value of diversification, and makes you understate the risks even more. As a result, even in our own advisory firm, we do Monte Carlo analysis, we actually don't show the full range of portfolio diversification that we own for clients. We actually model a relatively simple, two asset class portfolio of stocks and bonds. Because we actually know that's relatively stable.

We may slightly understate the benefits of diversification, because we do own portfolios that are more diversified than two asset classes, but that's the point. I'd rather be conservative, understate the benefits of diversification, than loading lots of investments into a Monte Carlo model without being certain that the correlations are right, overstate the benefits of diversification, tell clients they can spend more, and then have a crisis occur. At least with our approach, to the extent that diversification improves the results of the portfolio, we will beat the original projections. It's a lot easier to figure out with clients how to handle a retirement that’s going better than expected, rather than one that's going worse.

Of course, the challenge remains that we are in a low return environment. And we do want to be starting to consider the danger of sequence of return risk, especially in a low-return world. Unfortunately, the ideal way to model this would be showing something like lower returns in the next decade, because there's a high probability of below-average stock returns, given high valuations, and below average bond returns, given low yields. And then maybe assume returns normalize thereafter. Because the truth is, when the 10-year yields 2 and a half, I'm pretty sure the return on a 10-year, in 10 years, is going to be about 2 and a half. Because I can do the math of holding a bond’s maturity. I have no idea what yields are going to be in the 2030s and the 2040s. I don’t know what valuation is going to be out then. So I would probably just model that after the next 10 years. I'll assume that returns are going to average out, having no information to show otherwise.

But unfortunately, there's no Monte Carlo software that can actually show that, what I like to call regime-based retirement projections. Where we're in a low return regime for a decade and then we normalize. It's possible mathematically to do it, the software just doesn't do it now. Which means, the only alternative is to haircut long-term returns, which is what we actually do in practice. We reduce long-term returns by about 1% or 100 basis points, recognizing some of the risk of the low return environment.

Notably, it's actually not much more than that. As I published previously on the blog, even in high valuation environments, it only reduces long term (multi-decade returns), by about 1%. Now, over the next 10 years, returns are potentially reduced more from their long-term historical averages, but these things do tend to average out in the long run. I mean, you can look at the historical data, we’ve published quite a bit on it.

It’s important to realize that you can always adjust for the risk of things like today's valuation environment or return environment by just adjusting what probabilities you accept in the first place. I find this is often missed out as a way to handle Monte Carlo risks. In other words, in the past, you might have said, “We all want all clients have at least a 90% probability of success in their retirement plan before moving forward.” And now you might say, “We want at least a 95% probability of success.” So we went from a 90% threshold to a 95% threshold, as a way to handle elevated risk to the environment. To sort of acknowledge, if we're in a higher risk environment, we want a higher threshold to feel safe.

But the bottom-line though is just to recognize that this idea that adding more investments into a Monte Carlo analysis doesn't necessarily make the projection more accurate. If you don't accurately include correlations for every possible connection, and you just use the default assumption (which is usually a correlation of zero), you actually make the projection drastically less accurate and overly aggressive. By assuming diversification is going to bring you way more than it realistically will.

And it's a challenge. If you handle a lot of investments, you need a lot of correlations. Again, 8 investments need 28 correlation pairs. 10 investments need 45 different correlation pairs. And if you're not confident in your ability to determine all those different correlation combinations, it's actually better, safer, and more conservative for clients, to just not include them in the model. Do a simpler projection, with fewer asset classes. Certainly still invest in a more diversified way, and give yourself a chance to beat the projections. But including lots of projections in the Monte Carlo analysis when you don't have the correlations, doesn't make it better. It makes it worse, more tenuous, less robust, more aggressive, more prone actually to being wrong.

I hope that helps, as a little food for thought around Monte Carlo analysis, investment assumptions, including lots of investments in the Monte Carlo analysis, and the kinds of issues that you need to consider when running these projections. I find, for a lot of advisors, we're trying to make it better by adding in more and we actually make it less valuable in the process.

This is Office Hours with Michael Kitces. Normally, 1:00 p.m. East Coast time on Tuesdays, although I was a little bit late today because as you can see in the background here, I’m at the FPA NorCal Regional Conference. Thanks for joining us everyone, and have a great day!

**So what do you think? Does including more asset classes without their corresponding correlations make Monte Carlo projections less robust? How do you set Monte Carlo assumptions for your clients? ****Please share your thoughts in the comments below!**

Jason says

Great topic!

Eric Weigel says

My sense has always been that Monte Carlo simulations have been used by “smart” people to convince less technically sophisticated people of something that by simply looking at the assumptions you would have concluded anyway. My issue is not with the simulations per se but with the lack of disclosure of the assumptions used in the analysis. I also believe that for simulations to be useful there must be internal consistency in the data used – after all means, standard deviations and correlations are derivatives of the underlying return distributions. There is a whole field within asset allocation called “estimation risk” that deals with this issue. The research generally concludes that errors in means (or expected returns in this case) are much more important than errors in standard deviations and correlations (Ziemba, Chopra paper). In my own practice I use a set of 10 year forward expected returns, volatilities and correlations that are part science/history and part judgement based on our starting point for valuations, growth and yield levels. Not perfect, but fully disclosed. One can then at least have a rational discussion of how logical these assumptions are.

Josh S says

Some software uses returns on cash as high as 4%. A 20/80 stock to bond port had a 6.5% ROR! I couldn’t believe it. This wasn’t long ago either.

Yet the monte carlo basically showed EVERYONE with a 95% or better risk. Crazy.

Must be a way to increase equity correlation number as VIX rises? How far back does VIX go?

Very interesting observation. Has anyone developed a “better mouse trap”? MoneyGuidePro certainly doesn’t factor in correlation, but how would you suggest we implement a correlation matrix?

I don’t think Emoney does either.

MoneyGuidePro does factor in correlation. Both the historical and projected correlation matrixes are published on the Default Capital Market Assumptions document which can be found under the additional resources tab, under the support tab.

Modeling correlations is important. With current assumed correlation of 0 as default when no inputs are provided results are not reliable. It will be interesting to know if changing the default correlation to some other positive number say 0.5 or 0.6 would make the results better. Better in terms of not overstating benefits of diversification and not understating the risks.

Great article Michael! In the article you state, “We actually model a relatively simple, two asset class portfolio of stocks and bonds.” Do you use a specific software to do this or is this something that your company built in house?

Thanks!

Megan,

Sorry for the confusion.

We use standard financial planning software (in our case, MoneyGuidePro). I meant that the portfolio return and volatility assumptions we use for the retirement portfolios in our planning projections are based on a simple 2-asset-class portfolio of stocks and bonds.

We also have a slightly more complex 5-asset-class version that we use, tied to our benchmarks, but we analyzed portfolio volatility separately (based on aggregate results of the entire mix) and simply enter the final total portfolio parameters into the planning software, and not the underlying individual asset classes.

– Michael

I’m late to the party here considering this post was from over a month ago. However, I’m curious, Michael…since MGP does factor in correlation with the asset classes, why do you only use a simple, 2 asset class portfolio? Why not just utilize MGP’s capabilities when modeling your strategic asset allocations?

Matt,

Because the correlation history at the asset class (and especially individual security) level is very limited, which means it’s not clear what to input at the asset class level. In addition, correlations do not appear to be stable, and MGP doesn’t have an ability to model more dynamic correlations.

More generally, the problem is that without clear underlying data, adding more granular assumptions to Monte Carlo projections doesn’t necessarily improve its accuracy. In fact, you risk reducing its accuracy, as you create more assumption parameters that may be modeled incorrect. The sum of the parts can actually be LESS valuable than the whole, here.

– Michael

In the Monte Carlo Simulation, if you keep the return sampling of all assets in the same time period and chronicle order, the correlation is naturally embedded in the sampled return series. It is the random sampling without the chronicle structure that force the user to assume a zero correlation assumption.

Excellent blog post, keep the technical topics coming! Appreciate all that you do for us!

Michael says: “I find, for a lot of advisors, we’re trying to make it better by adding in more and we actually make it less valuable in the process.”

Very well said! I’m the author of a popular MC retirement planning tool and I’ve struggled with this challenge for over a decade. We want to capture every detail and nuance of each plan, but if we’re not careful, the noise can easily overpower the signal. Even worse, as Michael points out, being careless as we try to add precision can make our estimates less reliable.

IMO, Monte Carlo analysis shouldn’t be expected to provide “the answer.” Rather, it should be used to shed light on the range of possible answers and to help advisors and their clients explore the relationships between results and their underlying assumptions. At its best, MC analysis is a decision support tool that offers the ability to perform robust sensitivity analysis.

The main challenge most retirees and their advisors face is how to manage the tradeoff between risk and retirement lifestyle. The power of MC simulation is in its ability to educate and convey information about the contours of that tradeoff. When guided by skilled hands, this power can help retirees and future retirees make choices that are better aligned with their goals and values.

Another great one Michael.

Question, what if you just use a customized “all in” portfolio, say 4.5% returns with a 10% standard deviation? Essentially just one portfolio. tThat’s what I’ve done, haven’t used it with clients or anything but been thinking about this a lot. What am I missing?

Josh,

An “all-in” portfolio volatility (and return) works. The whole point of the correlation matrix of the underlying assets is to bring them back together to calculate the portfolio volatility.

Just be cautious about how you determine the portfolio’s all-in volatility. If you calculate from the underlying asset classes, be cognizant of the correlation matrix. If you use trailing return data to calculate a volatility, be certain if goes back far enough to get a reasonable range of bull and bear markets…

– Michael

Is there a way to subscribe to “office hours” video cast, Michael? First I’ve seen of this. Love it

Josh,

You can download the Periscope app and follow me there at https://www.periscope.tv/michaelkitces and you’ll get a real-time notification when Office Hours goes live every week! 🙂

– Michael