### White Paper: How Much Risk Should You Take? A Better Approach.

When you hire a financial advisor, sign up for a 401(k) at work, or otherwise get started with a new investment plan, one of the first questions you’ll try to answer is “what is your risk tolerance?” Risk tolerance, an investor’s ability and willingness to tolerate losses for the prospect of greater eventual gains, is typically the measure that is used to determine an investor’s *strategic asset allocation*, the percentage mix of stocks, bonds, and cash that will serve as the foundation of the portfolio. This process can actually be thought of as the first of the two separate steps of modern investment management; first, one determines risk tolerance and strategic allocation, next; for a given risk level, one tries to optimize the portfolio to maximize return using investment strategies within the major asset classes, i.e. *tactical allocation *and *security selection*.

Basically everyone agrees that ultimately the first step is by far the more important one. Back in the 80s, some highly influential research found that about 90% of the risk and return characteristics of a portfolio were explained by its strategic asset allocation, with the remainder explained by active investment strategy. While there’s been back and forth over the years about whether this is the right number or how to interpret it, the general consensus among financial professionals is that over the long run, how much you’re invested in stocks matters much more than exactly what kinds of stocks you’re invested in. You might think then that the determination of risk tolerance would get the majority of attention from researchers and practitioners relative to optimization of risk-adjusted returns, but of course you’d be completely wrong. While risk tolerance explains perhaps 90% of the difference in investor returns, I’d say academics and advisors probably spend less than 10% of their time researching its determination or evaluating it in their clients. As a result, there are countless books and papers on the topic of enhancing investment strategies using mathematically elegant theories and methods, and countless more hedge funds, ETFs, and other financial products on the market to help investors implement them, but when it comes to the first, most basic question of personal investing, “How much risk should you take?”, most people, even most professionals, still rely mostly on fuzzy subjective evaluations and traditional rules of thumb. I think we can do better.

This is a long one. If you are looking for a more concise version of how this method compares to current options and what we do better then read HERE. If you are looking for guidance on how to get your wealth to provide for your financial freedom then schedule a free 15-minute consultation with us using THIS link.

### The Current Problem

So how do financial advisors determine the risk tolerance of their clients and recommend strategic asset allocations currently? The most common methods are age-based rules and personal attitude questionaries. Age-based rules set the proportion of stocks to bonds as some function of the client’s age, on the assumption that younger investors have a longer time horizon and so can afford to take more risk and should be more in stocks, while older and retired investors must preserve what they have and so should be more in bonds. One of the oldest, simplest age-based rules is to hold “100 minus your age in stocks”, e.g. a 30-year old should invest 70% of her portfolio in stocks, a 40-year old should invest 60% of his portfolio in stocks, and so on. This is a rule so old nobody seems to know when or where it originated. Nowadays, you might more commonly hear “120 minus your age in stocks”, perhaps due to longer life expectancies, but in any case, it should be clear that such rules are very simplistic and don’t give any weight the investor’s individual circumstances.

These days, most investors have grown accustomed to age-based rule approaches as they form the basis for the now extremely popular industry of target-date retirement funds, which have been almost universally adopted as the default options in 401(k) plans. These funds invest in a diversified portfolio of equities and fixed income and gradually shift their balance from riskier stocks to safer bonds and cash as the current date approaches the target retirement date usually specified in the fund’s name. The shape of these funds’ asset allocation over time is often called the *glide path*. Below I illustrate the glide path of the five largest target-date fund managers by plotting their current equity allocation for each of their funds for various target retirement years.

While the exact numbers at each point vary a bit for each fund family, they clearly all follow a fairly similar path, and while the allocation rule isn’t something quite as simple as “100 minus your age in stocks”, it’s pretty darn straightforward nonetheless. The industry-standard approach is to invest somewhere around 90% in stocks at a basically constant rate for the first 15-20 years of an investor’s career, gradually become more conservative over the 15-20 years leading up to retirement, and then level off at about 30-40% stocks in the post-retirement period of the investor’s lifecycle.

How did these managers come up with these numbers? If you look at their fund literature, you generally won’t find they say much about this. Some of them have published white papers that discuss how a particular glide path is optimal based on historical experience using assumptions about how an average investor saves in their career and spends in retirement. But historical tests of these strategies are fraught with challenges (as we’ll see below), and anyway they necessarily rely on assumptions about what the average investor looks like. What should be clear about target-date funds and age-based rules in general is that they are cookie-cutter solutions, and cannot take into account the personalized circumstances and objectives of an individual investor.

The other tool that is commonly used to evaluate risk tolerance is personal attitude questionnaires that seek to asses an investor’s psychological disposition to financial risk. These typically ask a series of questions that try to gauge how averse to losses versus seeking of gains an investor is. They may also ask the investor to assess the how risky their objective financial circumstances are, like how stable their income is or how many dependents they have. Sometimes these questions can be pretty silly. I’ve seen one that asked “What would you do if you inherited a million dollars?” where one of the options was “Go to Las Vegas and try to double it.” More importantly, it’s often not clear how these subjective evaluations should translate into asset allocation recommendations. Suppose you have an investor who says she has “very stable” income and “moderately high” tolerance for loss. Does this mean she should be 90% invested in stocks, or just 80%? It’s not clear.

One popular provider of risk tolerance tests for financial advisors, Nitrogen, uses a clever method to try to be rigorous about this. They ask investors a series of questions in which they have to choose between a certain gain of a specific dollar amount or a 50/50 bet with a higher average payoff but the possibility of a loss. Once enough of these choices have been made, the test can calculate the investor’s willingness to take risk for a given reward, and this can be translated into an asset allocation recommendation based on the historical risk/return profile of different stock/bond allocations[mfn]Disclosure: RHS Financial uses Nitrogen to help assess client risk tolerance in conjunction with the other methods described in this post.[/mfn].

The problem with any subjective risk tolerance questionnaire, no matter how rigorous, is that even if the investor’s attitudes are consistent one day to the next, and even if that investor doesn’t panic when risk eventually does materialize (both of which are often dubious assumptions in my professional experience), the resulting recommendation may give the investor what he wants but not necessarily what he needs. Just because an investor thinks of himself as aggressive and willing to take risks doesn’t mean that’s what’s most appropriate for him to achieve his financial objectives. In practice, probably most advisors use a combination of objective age-and-circumstance-based and subjective survey-based methods to arrive at a recommendation that seems appropriate from a holistic perspective; but while this is unlikely to lead to recommendations that are totally crazy, there’s no good reason to believe they are optimal in any precise sense. These conventional methods are ultimately just fancier ways of using rules of thumb.

### A New Approach

So how can we do better? We have to set up the problem correctly. The primary objective investors generally have in choosing a strategic asset allocation is to minimize the probability that they will fail to meet their financial needs (typically, running out of money in retirement). A secondary objective is to maximize expected total wealth (if portfolios A and B both have a 99% probability of success but portfolio B results in twice as much wealth on average, then B is preferable to A). This is a minimax problem, and there are mathematical toolkits we can use to solve it. In particular, Monte Carlo Tree Search is an algorithm that is well suited to help an investor determine how much risk she should take in order to best achieve her financial goals.

Readers who know a bit of finance theory and/or are familiar with my writings on the topic may know that the portfolio allocation decision is often framed in terms of “mean variance optimization” (or just, “portfolio optimization“). This is a set of statistical techniques that allow an investor to determine the optimal allocation to an arbitrary number of assets to maximize the expected return of a portfolio, *given an assumption of the desired level of risk*, as well as assumptions about the expected return and risk of each asset in consideration. This is part two of the two-part framework I described above, where part one is determining what is the desirable level of risk is in the first place. The nice thing about portfolio optimization is that with modern computers, given an assumption about the desired level of risk, an optimal solution can be found within a fraction of a second, even for portfolios consisting of hundreds of assets. One might reasonably wonder, can’t you just use the same framework to then determine the optimal level of risk to take?

Unfortunately, no. The problem of determining how much risk you should take is technically much, much harder than determining how you should invest assuming you already know how much risk you should take. Indeed, it is technically NP hard (standard portfolio optimization is NP-complete). For you non-computer-scientists out there, the upshot of this is that using standard methods you can solve part two of the investment decision problem, “how do I invest assuming I know how much risk I can take” within a few milliseconds, but solving part one “how much risk should I take” using the same methods could easily take trillions of years or more, which I have found is longer than most clients are willing to wait for financial advice. Clearly, different methods are required.

Enter Monte Carlo tree search. Monte Carlo methods – that is, simulating very many different paths an outcome can take by repeated random sampling from some distribution – have been commonplace in the financial planning industry for a couple decades now. But mostly they have only thus far been used to evaluate how durable a portfolio is *given its current positioning*. Monte Carlo tree search takes this one step further and simulates not just a large set of possible outcomes that can unfold, but also a large set of possible choices that could be made along the way, then solves for the set of choices that delivers the best set of outcomes by some metric. A familiar use case for this method is in AI systems that play certain kinds of games like chess or go. A chess playing AI will simulate playing many different possible moves, then many different possible responses to each of those, then many different possible responses to each of those, and so on for up to several moves out on the game board. It will then select as its *current* move the one that had the highest likelihood of ending up on the best position as many moves out as the simulation was able to run. One can think of the possible future game states as a tree that branches out more and more as additional time periods or possible moves are considered. And the number of branches increases very rapidly – not exponentially, but as a combinatorial explosion – modern game playing systems may analyze trillions or more of possible game states to make their decisions (modern game-playing AIs use many other sophisticated techniques besides brute force tree search, but this is the system that is at their core and will serve our purposes for asset allocation)[mfn]An astute early reader pointed out to us that approximate solutions to this approach can be solved in polynomial time using Dynamic Programming optimization, although doing so with reasonable precision may still be quite computationally intensive. Both dynamic programming and Monte Carlo tree search are numerical simulations that face trade-offs on efficiency versus precision but will arrive at the same conclusion given enough compute. I will continue to discuss the topic from the Monte Carlo tree search perspective throughout this article.[/mfn].

For asset allocation over a lifecycle, we know that the markets in the future will unfold in some basically random sequence of returns, which we can simulate by repeated random sampling over a distribution. We want to know: given all the possible paths the market can take over the rest of my life, what is the optimal asset allocation to adopt *now*, knowing that I am able to update my allocation at any time in the future based on the circumstances at that time?

### Detailed Example of Alice

The statistical setup for this is fairly straightforward: imagine we have a young investor, let’s call her Alice, with a 70 year time horizon who wants to know her optimal mix of stocks versus bonds. 1) We simulate say, 5,000 possible return sequences for stock and bond markets over the next 70 years; 2) we simulate each possible stock-bond portfolio mix (in say, increments of 10%) over each period and calculate the returns on that portfolio in each of the 5,000 different market states; 3) we plug in her assumptions about how much money she has now, plans to save over her career, when she’ll retire, and how much she’ll withdraw in retirement, then calculate how much her portfolio is worth for each of the set of portfolio mixes over each of the possible return sequences; 4) we calculate all the possible end-states and determine which asset allocation decision path resulted in the best final outcome, trace it back to the allocation it makes at the start of the simulation, and use that as our recommended asset allocation.

This might sound convoluted, but it ultimately just boils down to solving a bunch of arithmetic. The main practical issue is figuring out how to simplify the problem enough that it doesn’t result in so much arithmetic that even a computer can’t handle it. For static Monte Carlo simulations on a given portfolio, computers can easily run thousands or even millions of market trials in a brief amount of time. But if we consider different combinations of portfolios the numbers get out of hand quick. In our example above, if we run 5,000 simulations of 70 annual market returns that’s 350,000 market states per portfolio. If we consider each 10% increment between 100% stocks and 100% bonds that’s 11 portfolios, so 3.85 million market states. But we’re not trying to find the best once-and-for-all portfolio, but the best portfolio path to travel, so we’re considering additional combinations over time. If we’re considering just a one-step path – say, the portfolio to hold before retirement and the portfolio to hold after it – that’s 11 x 11 = 121 portfolio combinations and 42.35 million market states. If we tried to consider the optimal portfolio for each year in the simulation that would give us 2.76 x 10^78 market states, which is larger than the number of atoms in the galaxy and would take your computer until the end of time to calculate. So we have to simplify the problem by considering only a few steps in our simulation; fortunately, we quickly get diminishing marginal benefits on additional steps anyway so considering just three or four gives qualitatively similar results as including many additional steps.

We at RHS Financial have built an application that does this for our clients. It takes the estimates of a client’s current net worth, life expectancy, retirement date, annual savings, and retirement spending, simulates thousands of future market returns, then uses Monte Carlo tree search to find their personalized asset allocation glide path that maximizes their probability of success, providing a recommendation for their current strategic allocation as well as the likely future path it will take. Let’s consider a few examples.

We’ll start with our young investor Alice from above who is just getting started in her career and investing and has a 70 year time horizon. She has no investible assets now but is saving $10,000 per year and expects to gradually save more every year until she’s saving $40,000 by retirement (in today’s dollars). She hopes to be able to spend $60,000 of portfolio value annually in retirement. Using our software, the personalized glide path recommendation for her looks like this:

The far left of the graph at time = 0 represents the recommended allocation for today and as we move to the right we see how that allocation is projected to change over time. Not surprisingly, the simulation recommends that Alice to be fully invested in stocks for now and throughout the early part of her career. Then in the years leading up to retirement the allocation shifts to a much larger proportion in bonds to get more conservative. So far, this looks broadly similar to how allocations for target-date funds unfold, although the exact shape is a bit different. But then something surprising happens: after retirement at year 40, the glide path pulls a U-turn and starts shifting back into stocks, ramping up to a 100% stock allocation once again by the end of her life. This is one of the most robust and counterintuitive findings we see from this exercise, and I will discuss its significance and rationale below, but for now let’s move on.

In order to arrive at this personalized glide path, the application has to simulate the investor’s wealth over the course of their life under thousands of different scenario, and we can plot these. We customarily display the projected growth of the investor’s wealth in the median scenario as well as the 5th percentile scenario to illustrate downside. Plotting these trajectories and the area between them yields this graph:

In this case, Alice’s wealth in the median-case scenario will only continue to grow post-retirement, as the returns on her investments outweigh her withdrawals. In less rosy scenarios her portfolio value will eventually dwindle, but in the 5th percentile case she still has nearly a half million dollars left at the end of her life expectancy. Of course, there are still another 5% worth of scenarios below the boundary of the orange line here, so we also calculate the absolute probability of success, indeed this is the number the application is trying to maximize. In this case Alice’s probability of success is 98.5%, meaning there’s a 1.5% chance Alice would run out of money before the end of her 70-year time horizon under our assumptions. This number represents the best we can do with strategic asset allocation alone, so any efforts to improve it will have to come from the investor saving more and/or spending less.

This tool provides a guide to how much risk you take not only today but over the course of the rest of your life. But of course you won’t want to use it just once and then take that to guide all your investment decisions for the next several decades. Markets fluctuate and life plans change, so as an individual’s finances evolve over time they will want to refer back to this tool and use it iteratively every handful of years or so, especially in the years around major life events like retirement.

Taking our example investor Alice above once again, let’s say 30 years have passed and she is now 10 years away from her planned retirement. In our initial projection we saw that in the median case her portfolio should be worth about $1.5 million by this point. Let’s say everything has gone according to plan and that’s indeed how much money she has now. We rerun the application with her and these are the results we get now:

The recommended glidepath bears a resemblance to what we saw originally but is subtly different. While the original plan had her projected to be in a 60/40 portfolio by this point, now it is saying she should still be as much as 90% in stocks and to ramp down to 60/40 five years ahead of her retirement, and after retirement, her ramp-up back into equities is now scheduled to happen much faster. The projected 5th percentile of wealth has risen to nearly $2 million and the probability of success is now 99.6%. What has happened is that by staying on track this far Alice has avoided 30 years worth of “worst case scenarios” so her probability of success has slowly converged toward 100%. Her wealth is now well positioned to provide for her retirement and she has enough cushion that she can afford to take more risk with her investments on average without fear of adverse market returns derailing her.

Now let’s say things hadn’t gone so smoothly the first 30 years. Whether due to poor market returns or failing to save as much as anticipated, 30 years have passed and Alice now only has $1 million. If we run our tool on this alternative scenario instead the results looks like this:

Alice is still in pretty good shape here. Her probability of success is 98.8%, marginally better than things originally looked 30 years ago, but to get there the investment plan has changed substantially. Instead of remaining mostly in equities as originally planned, Alice should shift to a much more conservative portfolio of 40% stocks, 60% bonds and hold that until retirement. She will still gradually increase her allocation to stocks once again in retirement, but now at a much slower pace than we initially predicted.

Our tool provides investors with guidance on the all-important question of how much risk they should take that is personally customized to their unique circumstances. Using it iteratively, investors can explore multiple possible scenarios to consider, for example, how soon they can retire and how much they’ll be able to spend in retirement, and immediately see both the probability of success and investment allocation implications. But what is the intuition behind the results it gives? The allocation strategy sometimes resembles those of more familiar glide paths, but other times flips it on its head, especially post-retirement. And two seemingly similar investors who might normally be in the same target-date retirement fund may get totally different recommendations using this tool.

### Sequence-of-Returns Risk and Optimal Equity Glidepaths

Let’s get at the intuition behind this approach by first asking, why bother holding bonds at all? Stocks are a higher returning asset than bonds; for any given holding period an investor is more likely to build the most wealth by being fully invested in stocks versus holding some allocation to bonds. And if you’re spending money out of your portfolio, your appreciation is more likely to keep up with your withdrawals if you’re investing in a higher returning asset.

The intuition most people have is that while stocks have better returns than bonds, they are also riskier and can therefore experience longer periods of loss. Retirees don’t necessarily have a long enough time horizon to withstand a protracted equity bear market and might run out of money before an all-stock portfolio can recover. As one’s time horizon only shrinks with age, so too should one’s stock allocation. This is *almost* correct, but the math is actually more subtle, and the conclusion that risk tolerance should only decline with age is false. In fact, the main risk with stocks in lifecycle planning is not their volatility as such, but the so-called *sequence-of-returns risk*, which is concentrated around the point of retirement but actually declines as one moves forwards or backwards from that date. Basically, just as large stock market losses generally don’t matter very much to young investors, so too they actually don’t generally matter much for very old investors, but they can be devastating for investors in the very beginning years of retirement.

To illustrate, let’s consider an investor who has just retired and expects to live another 30 years. He has a portfolio worth $1 million dollars and plans to spend $40,000 from it each year (a common 4% withdrawal rate). Now, let’s imagine two simplified examples of how stock market returns could unfold over the next 30 years: in the first case, which I’ll call “Good Start”, the stock market delivers a constant 12% per year for the first 20 years, then a negative 5% per year loss for the following 10. In the second case, “Bad Start”, it’s the other way around, 5% losses every year for the first decade, then 12% annual returns after that. (Though the constant return assumption here is obviously unrealistic, these on average roughly correspond to historical worst-case-decade and best-case-two-decade episodes and in either case average out to an annualized 6% return over the full 30 years, which is a reasonable long run average.) For bonds, we’ll again simplify and assume they earn a risk-free constant 1% per year no matter what. (All these dollar and return figures are assumed to be real, inflation-adjusted.)

Now, for both of these cases, let’s consider five different allocation strategies our investor will choose from: being 100% invested in stocks, being 100% invested in bonds, being invested in a constant 60% stock/40% bond portfolio, and then two dynamic portfolios. The first will start at 60/40 and then decrease the allocation to stocks by 1% each year (a falling equity glide path as commonly practiced by e.g. target-date funds), and the other will start at 60/40 and then *increase* the equity allocation by 1% per year, a *rising* equity glide path, similar to what we’ve seen in some of the simulations above.

If we project how much wealth the investor has over the course of his remaining lifespan under these different strategy/scenario combinations it looks like this:

On the left side, in the Good Start scenario, the portfolio successfully sustains lifetime withdrawals in every case but the All Bonds strategy. In this example, bond returns fail to keep pace with the rate of withdrawals and our investor runs out of money in the final years of his lifespan. It is possible to be too conservative in retirement. In the Good Start scenario, the total growth of wealth for each of the strategies and their decline in the final decade is intuitively directly related to their average exposure to stocks. Being fully invested in stocks the whole time results in the greatest terminal wealth, but also experiences the fastest decline in the final decade. The rising equity glidepath, which has an average 75% exposure to stocks over the period performs better than 60/40, which performs better than the falling equity glidepath which has an average 46% exposure to stocks.

Now consider the right side, Bad Start scenario. The All Bond strategy, with its constant returns, follows the exact same path and goes bust near the end just as before, but the path of the other four strategies is dramatically different. The All Stocks strategy goes bust first, after being decimated in the bear market of the first decade, the portfolio is no longer large enough to sustain portfolio withdrawals even once the market starts recovering. The next worst performing strategy is the falling equity glide path. Though it suffers the least of the four strategies with equity allocations in the first decade, as it continually shifts away from stocks as time passes it forgoes the higher returns it now needs to keep up with withdrawals. 60/40 goes bust around the same time as All Bonds. The only strategy that survives is the rising equity glide path. While it take a big hit in the early period, shifting into the higher returning asset allows it to bolster returns in the later period just enough to provide for withdrawals over the entire 30 year time horizon.

I want to emphasize here that the *average* return on the stock market was the same in both scenarios, as was the length and magnitude of the bear market. A dollar invested and held in stocks would have grown to $5.78 in either case. What mattered for the ultimate outcome of our investor was not how bad the risk of the market was, but when it occurred. Exposure to equity bear markets early in retirement was devastating, exposure to equity bear markets late in retirement was inconsequential, just as we commonly intuit it to be early in an investor’s career. The strategy that best navigates this sequence-of-returns risk is one that is most conservative at the exact point of retirement and then becomes more aggressive as time moves forward (or backward) from there. A rising equity glide path in retirement generally outperforms traditional approaches whether markets do well or poorly in the early stages of retirement and are therefore generally favored by our Monte Carlo tree search approach.

This result is not widely appreciated in the financial planning world, but it is not new or original to us. In 2013 the popular financial planning blogger and researcher Michael Kitces wrote, along with Wade Pfau, Reducing Retirement Risk with a Rising Equity Glide-Path, an article whose title clearly points in the same direction as mine here. In the article, Pfau and Kitces write:

[W]e find, surprisingly, that rising equity glidepaths in retirement have the potential to actually reduce both the probability of failure and the magnitude of failure for client portfolios. In other words, just as equity exposure can be more beneficial for those who are very young, so too can greater equity exposure in the later years of retirement actually help, especially in those scenarios where returns in the early retirement years are poor and favorable returns – with a healthy amount of equity exposure – are crucial to allow the portfolio to last. In essence, the optimal equity exposure for a portfolio over an accumulation/decumulation lifetime may look less like a slow and steady downward slope, and more like the letter U, in which the stock allocation is the lowest at the point when lifestyle spending goals are most vulnerable to absolute losses in wealth (the retirement transition itself), but is greater in both the earliest years and also the latest.

Pfau and Kitces’ paper, as well as others like it in the literature, essentially use a much more restricted version of the Monte Carlo tree search approach I am discussing here, in their case considering 121 branches corresponding to the 11 x 11 set of combinations of stock/bond allocations in 10% increments at the start and end of retirement. Our advancement here is to make this approach fully generalizable to the individual investor, allowing for an arbitrary set of glide path steps, an arbitrary number of years before and after retirement, and arbitrary market assumptions, all customized to the end-investor’s personalized financial circumstances.

### Key Findings

Having developed this application and now used it for a variety of both actual and hypothetical clients, we have gleaned some valuable general insights into the strategic asset allocation puzzle and the question of how much risk should investors take, some of which run contrary to conventional advice. Here are some of our key takeaways:

**Most investors should gradually increase allocations to stocks in retirement:** This is probably the most counterintuitive insight from our approach, but it has been independently verified by many others in the academic literature, it just hasn’t yet percolated into the general investment public yet. While the exact stock/bond mix that is recommended at retirement, and how quickly the glidepath ramps down and ramps back up varies significantly depending on the assumptions made about the investor, in nearly every conceivable case what we find is that the recommended allocation to equities rises in the years following retirement. As I described above, this is due to the fact that the sustainability of a retirement spending plan does not in fact only grow more fragile as time horizon declines, but rather it is the most contingent on the value of the portfolio in the years immediately surrounding retirement. Once one is more than about a decade or so away from retirement (whether before or after), the superior growth potential offered by stocks generally outweighs the risk of their greater volatility.

**Most investors should downshift their risk later, but much faster and significantly, than conventional glide paths suggest: **As we saw above, most target-date funds keep an equity allocation around 90+% for their first two decades, then slowly ramp down over the next two decades leading up to retirement, usually arriving at a 60/40 or 50/50 stock/bond mix on the eve of retirement and continuing to get slightly more conservative from there. In most realistic scenarios we have considered, the “downshift” phase of becoming more conservative in the years leading up to retirement is usually both more swift and more severe. We may see, for example, a recommendation to move from 100% equities to just 30 or 40% in the space of just five or ten years prior to retirement. The exact timing and scaling of this is again highly dependent on investor assumptions, but the upshot is because sequence of returns risk is so highly concentrated around the date of retirement, conventional glide paths that start getting more conservative 20 years out from retirement are just leaving money on the table, but remaining as much as 60% in stocks on the eve of retirement might mean bearing more risk than is prudent.

**Having “too much” money allows for higher equity allocations; having too little demands it: **A “problem” that some of our clients have is that they have done such a good job saving over the course of their career and have been so frugal with their expenses that they have reached a point, whether they are retired or not, that there’s basically no way that they’ll ever outspend their money. We sometimes have to coach such clients that they can indulge themselves a bit more if they want to, but increasingly we are also coaching them that they should take more financial risk as well. In our experience, people who fall into this category tend to have a very conservative attitude towards risk, but if you are fortunate enough to reach this point in life, being too conservative with your investments is just leaving money on the table that could be eventually bequeathed to your family or the charities you support. Consider a retired investor who expects to live another 30 years and has $5 million in assets but only spends $50,000 from it per year. Whether she invests 100% in stocks or in a more conservative portfolio conventional for her age like 40% stocks, her probability of success is essentially 100% either way. But by choosing the more aggressive 100% stock portfolio she could expect to leave behind over $23 million to her heirs in the median case scenario. With a conventional conservative portfolio that number would be cut nearly in half to just under $13 million.

Conversely, consider an investor with a bit more “lifestyle ambition”, who has $500k, wishes to retire early in 10 years and live another 40 after that, spending $100k per year (and saving $80k/year until then). Such an investor has a reach that somewhat exceeds his grasp. His probability of success is only 64.2%, more likely than not he actually could pull it off, but these are not the sort of odds you want to bet the rest of your life on. You really don’t want to wake up one day 20 years after you’ve stopped working and realize you’re out of money. The most important thing a financial advisor can tell this investor is that he needs to either postpone his retirement, save more, or spend less in retirement (or all three). But so long as we *are* working on these assumptions, the recommended allocation to equities for this investor is 100% for the rest of his life, because that’s the only portfolio that can possibly deliver the growth necessary to finance his future expenditures. Adopting a moderate 60/40 portfolio would reduce the probability of success down to 44.6%.

**Most investors should aim for the “Goldilocks U-turn”: **Most people in the midst of their career have more or less the same financial goal: work long enough and save enough money to provide for a long and comfortable retirement with a high degree of certainty, with enough cushion left over to most likely provide some decent-sized inheritance to their heirs. What exact numbers go into this formulation will vary from person to person, but if everything is done in the right balance then the asset allocation glidepath that will result will be the U-turn shape we’ve seen above, in which investors hold mostly stocks when they’re young, become more conservative as they approach retirement, then ramp back up into stocks in their old age. This is the shape we see associated with probability of success in the 90%-99.8% range that most investors wish to see, which is not so cautious as to require heroic levels of savings, but not so aggressive as to not allow adjustments to be made as necessary. As probability of success rises to 99.9% and higher the allocation glide path flattens out to 100% equities, indicating that the investor can afford to spend more on themself if they wish to. As probability drops below 90% the curve bends towards 100% equities for the opposite reason, indicating that the investor is likely living beyond their means. The happy medium is found where sequence-of-returns risk is minimal but not entirely eliminated around the date of retirement, corresponding to a U-shaped allocation glide path.

**On average, investors should be taking more equity risk than is conventionally recommended: **We have seen that using this approach, investors on a sustainable financial trajectory should be fully or almost fully invested in equities both in their youth and in their old age, moving significantly into bonds only in the 10-15 years or so around their retirement date. Furthermore, no matter their age, investors whose financial plans are significantly overfunded should be entirely in equities (because they can easily afford to), while investors whose financial plans are significantly underfunded should be entirely in equities (because they cannot afford not to). Altogether, it’s clear that the advice here is that for most investors most of the time they should be more invested in equities than is conventionally recommended by age-based rules or offered by standard target-date funds. While the average lifetime allocation to stocks in a typical target-date retirement fund appears to be in the neighborhood of 60%, for our approach that figure is probably closer to a range of 80-90%. The notable exception, of course, is for those investors just a few years away from, or a few years into, their retirement, who in many cases may be taking too much equity risk if following traditional glide paths. Incidentally, these are also the investors who tend to be at the peak of their lifetime net worth, so when we consider dollar-weighted averages over either an investor’s lifetime or across society as a whole, it’s not clear that following our approach here would shift the relative balance in stocks versus bonds all that radically. Nonetheless, we think that this approach offers an insightful new lens through which to view strategic asset allocation and financial planning that may greatly enhance financial outcomes for investors at any stage in life.

### Historical Simulations

So far this discussion has been purely theoretical. But before we throw conventional asset allocation advice out the window, investors may want to know how the outcomes of following this approach would compare to other traditional strategies when tested on actual historical market data. Supposing an investor in the past adopted this approach and iteratively updated his asset allocation over the course of both his accumulation and retirement stages of life, how would his probability of success and terminal wealth compare with different strategies?

We can and will test this, but first we should note that this is a more challenging problem than it might seem at first due to the limitations on available data. We have very complete and accurate data on prices, returns, interest rates, and valuations for the US stock and bond markets going back to 1926 (more approximate data goes back well into the 19th century but we will not be considering that here). That’s nearly a century of data, which might sound like a lot, but when we consider the fact that the investment time horizon of a young investor is typically 50+ years, that means that we don’t even have enough historical returns data to test two completely independent, non-overlapping investor lifecycles. Basically, compared to the lifetime of an investor, market history is short, so any inferences we draw from the historical data might be influenced by luck and reduces our statistical confidence.

And we have good reason to believe that this kind of small sample bias is going to be significant in this case because over the last century the US has been an especially lucky country. Every year Credit Suisse publishes the Global Investment Returns Yearbook, which details long-term returns for various markets, including the real annualized returns on stocks, bonds, and bills (cash) for 21 countries since 1900.

The US is notable in that it has had the third-best stock market returns of any country, with an annualized real return of 6.4% compared to the global average of 5.0%, although its bond and cash returns have been much closer to the global average. If we rely on US history for financial planning or tests of asset allocation strategies we may form an overly optimistic view of what market returns are plausible and how sustainable an investment portfolio may be in retirement. Unfortunately, the raw data Credit Suisse uses for their yearbook is not publicly available and the commercial databases commonly available to financial advisors only have international market data going back a handful of decades. In practice, most financial planners do seem to assume that the expected returns on stocks are basically equivalent to the historical returns that have been realized in the US[mfn]A note about the merits of international diversification: some investors observe the superior long-term returns of US stocks and make an argument for American investment exceptionalism. In this view, the US market has outperformed, and will likely continue to outperform the rest of the world, due to e.g. its relatively more investor-friendly legal system, culture of entrepreneurialism, more diversified and sophisticated economy, etc. But examining the other countries in the top 5 undermines this narrative and points to other factors that matter more. South Africa, Australia, New Zealand, and Sweden might seem like a fairly random collection of countries that don’t have a lot in common with the US or each other. One thing they all have in common, however, is that they were relatively unharmed by the two World Wars, the events surrounding which seriously dragged down the returns of the global average. Since the end of WWII, US returns have been much closer to average, and if we think that in any future global conflict the US, being the global hegemon, might be at risk of much more serious damage, then we should seriously doubt that American markets are destined to indefinitely outperform.[/mfn].

This raises an important point, what numbers should we use for expected returns? The Monte Carlo tree search method requires us to make an assumption about the distribution of returns for stock and bond markets; how should we define that? This is a matter I glossed over in the previous sections, but let’s investigate it now. As I mentioned, for long term projections long term historical returns are often assumed. But we know that valuations or yields are often a better predictor of market returns over intermediate periods than long run averages, so we could assume that the expected returns are equivalent to whatever current real yields are for a more timely estimate. It’s questionable how relevant current yields are going to be 40+ years out in the later parts of the tree search, but the point of this approach is to use it iteratively over the course of one’s life, so one can regularly ask the more specific question, rather than “how much risk should I take?” instead, “how much risk should I take given current expected returns?” This introduces an element of tactical asset allocation on top of the strategic allocation function that may potentially add further value over the investment time horizon.

With all these caveats and conditions in mind, let’s run some horseraces between different strategies. We’ll first consider US market history from 1926-2022, then consider an expanded global sample. We’re going to test two different kinds of investors, Youngsters, who have a 60-year investment horizon, the first 30 years of which they are working and savings, followed by 30 years of retirement and spending, and Retirees, who have a 30-year horizon over which they are spending down their portfolio. Each of these investor types is going to follow one of six different asset allocation strategies over the course of their lifetime:

- 100% Bonds: hold their wealth in bonds over their lifetime
- 100% Stocks: hold their wealth in stocks over their lifetime
- 60/40 Portfolio: hold their wealth in a 60% stock/40% bond portfolio, rebalanced annually, over their lifetime
- Traditional Glidepath: follow an age-based stock allocation rule, rebalancing each year to reduce stock allocation by 1%, for a maximum stock allocation of 90% at the start, 30% at the end, and 60% at retirement
- Historical Returns Based Monte Carlo Search: every year allocate to stocks and bonds based on the results of the Monte Carlo tree search algorithm using the latest client input data and assuming expected returns are equal to historical averages.
- Yield Based Monte Carlo Search: every year allocate to stocks and bonds based on the results of the Monte Carlo tree search algorithm using the latest client input data and assuming expected returns are equal to current real yields[mfn]I calculate real yields for stocks and bonds as follows: for stocks, I use a forward-adjusted estimate of the earnings yield based on the inverse CAPE ratio. For bonds, I use the current nominal yield minus an inflation forecast. Prior to 1970 the forecast is based on long-term average inflation; after 1970 I use expected inflation from the Survey of Professional Forecasters.[/mfn].

For each of our two types of investors trying six different strategies we will run tests on as many cohorts of them as we have available years of data. The first cohort of Youngsters will start in 1926, save and invest in the market each year until they retire in 1956, then spend down on their portfolio until they die in 1986. We observe what their terminal wealth was for each of the six possible strategies, then we run the next cohort: 1927-1987, and so on until we run out of test years in 2022, then we will compare the results of each strategy across the various cohorts overall. Note that for the Youngsters with a 60-year horizon, our 96-year sample window only gives us 37 cohorts, all of which have some degree of overlap with the others, so this is definitely a small sample like I warned about above.

We finally have to make an assumption about the investors’ wealth, savings during working years, and spending during retirement years. We’ll start by assuming the Youngsters start with no wealth, save $20,000 (real) while working, then we’re going to test different spending levels to see at what point do they start to break. We’ll start with $70k of spending.

We run the tests for our Youngsters for each strategy and observe what their terminal wealth is in each cohort. To compare outcomes we’ll look at what the overall probability of success was (what percent of cohorts still had positive wealth at the end of their simulation), as well as the dollar value of wealth in both the 5th percentile case and the median case.

For this level of retirement spending relative to savings, all of these asset allocation strategies were mostly successful for our 37 cohorts with the exception of investing only in bonds. This underscores the point that in order to build a well-funded retirement nest egg you need to take a certain amount of risk to earn better returns earlier in life. Similarly, the other strategy that doesn’t make it 100% of the time is the constant 60/40 portfolio, which failed in 2 of the cohorts, or about a 5% rate. For the dynamic strategies, the traditional age-based glidepath and both versions of the Monte Carlo search strategies were 100% successful. The Monte Carlo search strategies both outperformed the traditional glidepath in terms of the median outcome, but the 5th percentile outcome was a bit more mixed.

But the most surprising thing here is that not only was an all-stock strategy successful 100% of the time, it resulted in greater terminal wealth than any of the diversified strategies not only in the median case but also in the less fortunate cases, as illustrated by the 5th percentile results. This seems to suggest that investors over history would improve their upside and their downside results by ignoring asset allocation altogether and putting all their money in stocks forever!

Before we jump to that conclusion let’s push on the plan a little harder and see what’s next to give. Next we raise the retirement spending level up to $80k and observe the outcomes.

The traditional glidepath is no longer 100% successful, and the 60/40 portfolio fails half the time. The Monte Carlo search strategies remain 100% successful and more robust than conventional approaches, but 100% Stocks still dominates.

Let’s push it even further now and consider a $100k spending rate.

No strategy is 100% successful in this case, but the Monte Carlo search approaches outperform all other strategies including 100% stocks. Using Monte Carlo search now seems to be much more robust than traditional allocation strategies, but it’s surprising how robust investing in all stocks has been in all but the most aggressive of financial plans. How much more value does using this approach really add compared to just buy-and-hold stocks?

This is where we remember our caveats from earlier. Not only has the US stock market performed unusually well over the last century, these cohorts of investors that started working after 1925 but died before 2023 got a disproportionately good share of these good returns. Over the last century the worst two rough patches for the US stock market have been the Great Depression, from 1929 to the early 40s, and the decade of the aughts, 2000-2009. But in our test period none of our early cohorts retires before or during the Great Depression, while our last cohort retires in 1992 and so the dot-com bust and financial crisis only happen late in retirement after the portfolio has swollen in size during the early retirement period coinciding with the the dot-com boom.

The inferences we can make about these test cases is limited. The number of investor lifetimes we can simulate is small, and they all share an unusually good stock market history. We can increase the number of observations available to us by just shrinking the investment time horizon, looking only at the latter half: retirement. So now let’s look at our Retirees group of investors. Each cohort of these investors will start with $1 million and plans to spend $40k each year for 30 years (a 4% rule). Shrinking the horizon down by 30 years increases the number of cohorts up to 67, some of which are now non-overlapping, and a few of which will have the misfortune of retiring on the eve of the Great Depression, so while the sample size is still relatively small, we should see greater variation in investor luck over different lifecycles.

The dominance of an all-stock portfolio has disappeared, as 6% of the 100% Stock strategy cohorts fail in retirement – those that retired around the Great Depression. The traditional approaches do better in terms of probability of success, although the declining-equity-glidepath doesn’t achieve 100% like the constant 60/40 does. But the dynamic Monte Carlo approaches are the most robust, not only achieving 100% success in both cases, but outperforming in the 5th percentile case compared to any other strategy, whether using historical or yield-based expected returns. And using the yield-based approach increased the median outcome to be roughly as good as stocks, in other words, better downside protection than traditional asset allocation strategy, with upside about as good as all stocks.

How does the MC Yield approach manage this feat? Looking at the example of someone who retires in 1929 on the very eye of the Great Depression is illustrative. This is pretty much the worst case scenario for the equity investor, and as the chart of the simulated growth of wealth below shows, the 100% Stocks strategy eventually goes bust. But while the other strategies manage to stay afloat, the MC Yield strategy runs away from the pack and finishes with a better-than-average terminal wealth of about $10 million!

It should be clear what has happened to the 100% Stocks strategy: in the early days of retirement, this investor’s portfolio is utterly devasted by the market crash. Though the US market started to recover by 1934 and eventually returns soared during the post-WWII boom, they weren’t enough to make up for lost ground in the face of regular withdrawals for spending and eventually the portfolio goes bust in the 1950s (later than I would have guessed). The previous sentences should give a clue to what explains the stellar performance of the MC Yield strategy in this cohort. The allocation of this strategy over the period in question looked like this:

The Monte Carlo tree search approach is usually maximally conservative on the eve of retirement. Add to this, in the late 20s as stocks soared their earnings yields came down lower and lower relative to bonds, making bonds look even more prudent and attractive to a conservative investor at the time. This investor therefore parks nearly all his wealth in bonds and basically sits out the great crash that kicked off the Depression. By 1932 though, US stocks were about as cheap as they’d ever gotten, before or since, and using up-to-date yield estimates would have told you stocks now offered much higher returns than bonds. So in year 4 of retirement this investor shifts to 100% stocks and just… stays there for the rest of his life. He therefore participates in the enormous recovery of 1933-36 as well as the later post-WWII boom years. The former of which elevate his portfolio to the point where sequence-of-returns risk is no longer a concern, and the latter of which just provide icing on the cake of a very successful retirement.

### Bootstrapping Alternate Histories

We’ve now looked at US history and found that investors who had followed our Monte Carlo search strategy would have on average fared better than by following more traditional allocation advice across a variety of situations, and furthermore that by using up-to-date yields as our expected return inputs rather than historical average returns that the Monte Carlo search strategy seems to lead to greater average outcomes. We’ve also noted that the 96 years of history we have makes for a small sample size and the unusually good stock returns of the US in the last century inflates the probability of success for all these strategies – especially the 100% Stocks strategy – compared to what we might reasonably expect going forward. We would ideally like to test this on a much longer sample of historical data that is more representative of long-run global distribution of returns.

The statistical technique of bootstrapping allows us to do just that. Bootstrapping involves repeatedly resampling slices from one or more datasets and using them to form some new sample of arbitrary length but maintains the same statistical structure as the data that was originally used to generate it. It is often used to study what could possibly happen with some statistical series given a limited amount of data.

For our purposes, I use bootstrapping to simulate 500 years of realistic market data by resampling over US market data from 1926-2022 (as above), international developed market data from 1979-2022, and emerging market data from 1994-2022. To do this, I randomly select a twelve-month long slice of returns data from one of the samples, then stich it together with another randomly selected twelve-month slice, and so on. So for example, the first year might correspond to returns from March 1964-February 1965 in the US, then the second year to returns from July 2002-June 2003 in emerging markets, then the third year to returns from April 1988-May 1989 in developed markets, and so on, randomly resampling through market history until we’ve stitched together 500 years of simulated market returns. I follow the same procedure for bonds but due to data limitations I restrict myself to the US market 1926-2022 (as mentioned, US bond market returns have been very similar to global averages over the long term)[mfn]Because we are interested in testing the predictive value of current yields and yields, unlike returns, are relatively persistent over time, I technically use a modified random resampling technique by which the next slice in the sequence is found by observing the yield at the end of a given period and randomly matching it with a different slice whose starting yield is close to it within the range of observed statistical persistence. This allows us to preserve the empirical stochastic relation between yields and returns with each other and over time while still generating substantially different alternate histories.[/mfn].

Using bootstrapped returns in this way, we can generate a time series of market returns that closely resembles the long-run global distribution and allows us to observe again and again alternative universe versions of all the great market events of the last century: the Great Depression, the post-WWII boom, the inflationary 70s, the Japanese bubble and bust of the 80s, the dot-com bubble and bust, the meteoric rise of China in the 00s, the ’08 financial crisis, and so on, all played on repeat in a smash-cut of snippets. This way we can test how different market events and different sequences of them would have effected investors at every stage of their lifecycle, giving us a way to test these strategies not just on the contingent history that actually happened, but on a much larger possibility space of what plausibly *could* happen[mfn]The technically inclined reader may be wondering if simulating bond returns independently of stock returns is a reasonable assumption. Empirically, the correlation of historical stock and bond market returns at an annual horizon has not been significantly different than zero, so we believe it is a reasonable assumption. This mathematically convenient fact also allows us to simplify the tree-search process by assuming independent distributions.[/mfn].

When we test against the global bootstrapped series, we quickly see that the US offers a rose-tinted history and it’s been more difficult to sustain a given level of retirement spending with more average returns. Here, I show the results for a Youngster as before who saves $20k/year the first 30 years, but then spends only $50k in their 30-year retirement phase.

The results on this much larger test on 500 years of simulated data are broadly consistent with what we’ve seen so far: over the course of an entire investment lifetime, following a traditional asset allocation strategy like a 60/40 portfolio or an age-based glidepath is about as good as just leaving everything in stocks, but updating your allocation dynamically over the course of your life using the Monte Carlo search approach results in the highest probability of success, especially when using up to date expected returns based on yields.

We can also compare the results for Retirees using a 4% spending rule over a 30-year horizon.

Here we see, as with the US simulation, holding a more diversified portfolio is marginally more likely to be successful than all stocks, but the conventional advice of a declining equity glidepath again fares worse than simply holding a constant 60/40. In contrast, the Monte Carlo search approaches, which typically start retirement with a conservative portfolio then rotate back into equities later in life, achieve higher likelihoods of success, especially the yield-based model, which is not only significantly more likely to succeed but also has the highest expected terminal wealth.

These simulations give us confidence that this approach to strategic asset allocation is more robust and will serve our clients better than the traditional advice on the topic. But we should not forget the importance of advice itself. These simulations offer a fair test of the performance of different allocation strategies but they are oversimplified in the sense that they don’t allow for any adaptation over a lifespan. Financial planning is a lifelong process, and investors may need to course-correct over time. If your financial plan is not on a sustainable trajectory, you may need to save more or retire later, for example. This application is therefore helpful to us not just in choosing the client’s allocation but in guiding their financial resources over time, helping them achieve the most with the least.

In conclusion, our tool offers an innovative perspective on strategic asset allocation and investment risk management. By tailoring recommendations to individual circumstances and adapting to changes over time, it challenges traditional approaches and potentially enhances financial outcomes for investors at any life stage.

If you are looking for guidance on how to get your wealth to provide for your financial freedom then schedule a free 15-minute consultation with us using THIS link.

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