From Alexander Weis and Gerd Kommer
This post was updated in April 2025.
Anyone who is thinking about retiring today, in ten or even in 20 years, but whose name is not Bill Gates, Jeff Bezos or Elon Musk, will ask themselves whether their assets will be sufficient to maintain the desired standard of living until the end of their life.
The usual way to answer this question is to assume a portfolio return and "portfolio withdrawal rate" for a given asset, assuming that both variables will occur unchanged each year until the end of the observation period (which typically coincides with the estimated remaining life expectancy).
Speaking of remaining life expectancy: There are numerous calculators available on the Internet to calculate remaining life expectancy, e.g. b. this. They provide the “mean” remaining life expectancy, also known as the median remaining life expectancy. According to her, there is a 50% probability that the person in question will live longer than statistically comparable cases and a 50% probability that they will live less long. However, in order to be on the safe side for the purposes of a Monte Carlo simulation, one should expect a remaining life expectancy that the person in question statistically only exceeds with a probability of 10% or 20%. You can calculate this, for example this one internet-based computer.
Back to calculating the portfolio withdrawal rate: The usual calculations can be made quite easily on various financial portals or with a spreadsheet program. These are usually linear, “deterministic” calculations; so-called “point estimates”. Although such estimates are better than no calculation at all, they still have limited cognitive value. Limited because they ignore the very fundamental aspect of the uncertainty of future returns: the uncertainty of the portfolio's returns in the future, i.e. their fluctuations and especially the specific sequence of these fluctuations. Such point estimates unrealistically assume that the path into the future is a simple, linear, fluctuation-free path.
Linear calculation methods ignore the fact that portfolio returns fluctuate around an average in the future and the specific upward or downward fluctuation cannot be predicted for a specific year. This birth defect of the usual linear method can be cured to a small extent - but not really - by, for example, making differently optimistic and pessimistic point estimates ("worst case", "best case", "medium case" or "base case").
Do you want to ask questions like “do I have enough assets?” or “How much withdrawal per month can I afford?” or “When is the earliest I can stop working?” To answer the question more meaningfully over a long period of time than with point estimates, one has to move on to a more sophisticated forecasting technique Monte Carlo simulation.
In a Monte Carlo simulation (“MCS”), a simple mathematical algorithm is used to find a solution to a stochastic (probability mathematical) problem.
MCS as a simulation and prediction technique was developed during World War II by famous mathematicians Stanislaw Ulam and John von Neumann as part of a nuclear weapons project at the Los Alamos Scientific Laboratory in the USA. Due to the project's confidentiality, a codename was required for the new procedure. The two chose “Monte Carlo Simulation” because Ulam's rich uncle occasionally indulged in gambling at the Monte Carlo Casino in Monaco.
MCS is now used in practically all scientific disciplines such as biology, chemistry, mathematics and physics; So it is not a new method and not one that is only used in financial economics for forecasts (actually simulations).
In the investment context relevant to our purposes, an MCS involves a computer generating a few hundred or thousands of possible different cases, so-called “iterations,” based on assumptions about the expected return and volatility (uncertainty of return) of a portfolio, the remaining life expectancy of the investor, and his periodic portfolio additions and withdrawals (an iteration or “case” is a single, individual forecast). The computer then sorts the resulting cases from best to worst case and prepares this “bouquet” of iterations with simple statistical key figures in tabular and graphical form for the user. The user can then make important decisions based on these results.
The minimum number of iterations needed to reach a reliable result depends on what you want to simulate. For the purposes of simulating an investment portfolio, which are simple compared to solving specific problems in biology or physics, 1,000 to 10,000 iterations per simulation are usually considered sufficient.
In technical terms, MCS helps to better understand and manage the so-called sequence of returns risk. Return sequence risk means that the specific order of fluctuating monthly or annual returns during the observation period then has a high impact on the overall return and thus also on the final value of the portfolio, provided that funds are added to or withdrawn from the portfolio over time. Only in a portfolio that does not experience any additions or withdrawals during the period under review does the sequence of returns risk play no role for returns and final assets, but the complete absence of additions or withdrawals never occurs in the typical context for MCS anyway, because it is precisely about determining a sustainable withdrawal rate.
MCS can be carried out methodologically in different ways. The most common method assumes an average return (annual return or monthly return) and volatility of these returns. [1] With these two input values, the computer then calculates thousands of iterations (individual individual forecasts) based on an assumed statistical normal distribution of future returns. This standard procedure is referred to as “classic MCS”.
Another method is to use historical portfolio returns (e.g. monthly returns from the last 50 years) to randomly draw individual period returns like from an urn and string them together (after drawing, the return is returned to the urn). This is then repeated thousands of times by the computer. This process is called “bootstrapping with replacement”. It is less common, but just as relevant and meaningful as the classic MCS.
In the two previous methods, a random sequence of periodic portfolio returns is assumed in the simulation.
A third method, however, uses complete historical return sequences (e.g. the monthly returns from 1970 to the present in the same order as they occurred historically). The simulation element in this method consists of randomly selecting the starting point within the historical data series, which is then repeated by the computer several thousand times. For the iterations whose starting point is so late within the historical data series that the end point would be behind the last historical monthly return in the data series, the simulator starts again "at the beginning" at the first data point in order to be able to generate a sufficiently long forecast - comparable to a Pater Noster elevator that always "runs through". This method is called the “Historical Sequences Method” in technical jargo. In particular, it does not eliminate the weak “regression to the mean” present in the historical stock market data (Wikipedia entry on regression to the mean here). The same applies to bond data, where the historical data contains a slight “anti-regression to the mean”.
If all three methods are used in a given case, then the results differ in the sense of: B. the “failure rate”/“bankruptcy rate” (the proportion of all individual forecasts for which the portfolio is not sufficient until the end of the observation period) is something like this:
- Classic MCS – typically produces the highest failure rate among the three methods, i.e. the relatively least pleasant results
- Bootstrapping with Replacement MCS – usually produces the average failure rate among the three methods
- Historical Sequences MCS – usually delivers the lowest failure rate among the three methods, i.e. the relatively most optimistic results
In advice books and on financial portals you can often read that a “4% withdrawal rate” is “feasible”, “sustainable” or “a good rule of thumb”. [2] The 4% figure comes from an interpretation of the historically first academic study on the topic of “sustainable withdrawal rate”, which has now been outdated by scientific progress. In 1994, a good 30 years ago, the US economist William Bengen published the first systematic study on this (Bengen, 1994). In the years that followed, the so-called “4% rule” became widespread in the financial industry and in advice literature. According to this rule, one can withdraw 4% of the initial value in monetary units plus inflation from a 50/50 stock-bond portfolio each year without running the risk of ever completely depleting the portfolio. This corresponds to a 100% portfolio survival probability and a 0% failure rate. From today's perspective, however, the 4% rule is clearly over-optimistic (McQuarrie 2025). One can only warn against their uncritical transference to one's own circumstances. However, due to lack of space, we cannot go into the various reasons for the lack of realism of the 4% figure here.
It is occasionally heard that the assumption of a normal distribution of security returns within the classic MCS is an overly optimistic representation of the reality on the capital markets, since in reality there are “fat tails”, i.e. extreme upward and downward return values that lie outside a normal distribution. However, moving away from the normal distribution assumption in MCS, as described here, tends to lead to better results than with the classic method. As a result, the “normal distribution criticism” probably misses the point here (Tharp, 2017).
There is no general answer to which of the three methods is “the right” or “the best”. A final clarification of this question would require much longer historical data series than are available; probably more than 1,000 years of capital market returns instead of just 30 to 120 years (depending on the country), as is actually the case.
Regardless of the method chosen, the following applies: Looking into the future, the development of investments is uncertain and the longer the forecast period, the greater the spread of possible end assets. This uncertainty has a particularly strong impact on the return and thus on the final asset value for portfolios that experience additions or withdrawals over time (which is the rule). An MCS attempts to mathematically model this uncertainty for the investor's insight purposes. The consideration of the uncertainty of future returns, especially their timing, fundamentally distinguishes MCS from the linear portfolio development or withdrawal calculations mentioned at the beginning, which do not take uncertainty into account and are therefore only suitable to a very limited extent for answering the investor questions relevant here.
So much for theory. But now to practice, that is, to concrete constellations in which MCS can be used sensibly.
MCS can be used to assess how likely it is that a household's assets will be sufficient for a given standard of living for a given period of time. However, MCS can also be used much earlier, namely to find out how long and how much a household still needs to save before it can retire. For this purpose, two phases of the underlying portfolio are basically depicted in the simulation: a savings phase (asset accumulation) and a withdrawal phase (asset use). In the savings phase, the investor household accumulates assets, i.e. the additions to the portfolio exceed the withdrawals; In the withdrawal phase, the household consumes all or part of the saved assets, i.e. the withdrawals from the portfolio exceed the additions.
If you carry out an MCS long before you want to retire, you can derive particularly useful insights and conclusions for the present. Particularly useful because there is still time to make fundamental decisions based on the MCS results in the present, e.g. B. saving more, spending less or working longer. Ten or twenty years later, such adjustments would probably come too late.
In general terms, MCS offers an additional plausibility check for the fundamental economic question that every household should ask itself: “Is my long-term financial plan reasonably realistic and feasible?” If an MCS shows that the answer to this question is not a "yes" combined with a reasonably good feeling, you can sit down again and modify your plan - many years before things actually burn out and cannot be repaired.
How do you go about carrying out an MCS? First, you determine the input variables: (a) the amount of existing assets, (b) the monthly savings or withdrawal rate, (c) the observation horizon (usually this is the estimated remaining life expectancy) and (d) the expected portfolio return and volatility. The MCS computer program then calculates e.g. B. 1,000 or several thousand possible cases.
In the following table we have presented the results of an exemplary test simulation for the Meierhofer household for illustration purposes. We use the classic MCS method. The couple Paul and Anna Meierhofer are both 50 and plan to retire relatively early in ten years at the same time as 60; The savings phase will therefore last another decade. After that, the assets saved should last for 30 years. It is a 60/40 (risky/low risk) portfolio. The detailed description of the input variables can be found below the table.
Table: The results of a Monte Carlo simulation based on a normal distribution (“classic MCS”) for the example case of the Meierhofer household
► Source: Own calculations using the MCS tool from Gerd Kommer Invest GmbH. ► All figures real (adjusted for inflation) and in EUR; Since real returns were calculated, neither an estimate of inflation nor an adjustment of the savings and withdrawal rates to inflation are necessary. ► Underlying assumptions: The household invests EUR 50,000 annually in its portfolio for the next ten years and then withdraws EUR 50,000 annually for 30 years; the initial assets of the household are EUR 1,000,000; the arithmetic (real/inflation-adjusted) portfolio return is 2.9% p.a. (after costs and taxes) and the standard deviation of the annual returns is 12.2%. ► Column 1: Age of household; ► Column 2: Household cumulative deposits into the portfolio; ► Column 3/4/6: The 10th/50th/90th “percentiles” of household wealth; ► Column 5: Arithmetic average of household wealth (final wealth); ► Column 7: Standard deviation of final wealth; ► Column 8: “Success Rate” of the portfolio = the percentage of cases in which the household assets last until the end of the specified observation period (“portfolio survival probability”).
To interpret the table: In our example of the Meierhofer household, a total of EUR 1,500,000 was paid into the portfolio (column 2). In the worst 10% of cases, the portfolio would not have lasted until the age of 85 (column 3). In 50% of cases, the household's portfolio would still have been worth over EUR 1,240,000 at the end of the period (column 4). In the best 10% of cases, the portfolio would have grown to more than EUR 5,260,000 at the end of the period (column 6). In 18% (100% - 82%) of cases, the assets were not sufficient until the household reached the age of 90 (column 8).
Overall, the result in this example can be described as unsatisfactory because in 18% of cases the portfolio does not last until the age of 90. This failure rate is likely to be intolerably high for some households. With such an outcome, the household must consider how it wants to change its plan. It would be conceivable to increase the savings rate (i.e. lower the standard of living now), postpone the retirement date to a later date, lower the withdrawal amount in retirement, a combination of these measures, or the “belt-tightening” approach, which we will discuss below.
Would an increase in the equity quota in the portfolio (i.e. an increase in the expected return) be a solution for the Meierhofers? Not necessarily. A frequently observed phenomenon in MCS in relation to the variation of the asset allocation is as follows: the more “aggressive”, i.e. more equity-heavy/risky, the asset allocation chosen, the better the medium and good cases become and the worse the around 10% to 30% of the worst cases become. In other words: the good cases become even better and the bad cases become even worse to the extent that the equity quota is at the expense of the bond quota. Conversely, there is a movement towards a more conservative, i.e. more bond-heavy, asset allocation. An increase in the equity quota in the portfolio beyond a certain value will not lead to an improvement, but rather a worsening of the failure rate. The famous basic law of economics is expressed here once again: “There is no free lunch”.
However, an improvement (= reduction) in the failure rate can be achieved by introducing a “dynamization” of the withdrawal rate. This means that the annual (or monthly) withdrawal is temporarily reduced based on a rule after a year of poor capital market development. One could loosely describe this as “temporarily taking the belt-tightening approach in bad times”. However, most MCS systems cannot display such dynamizations. The MCS tool from Gerd Kommer Invest GmbH can do it.
Let's come to a summary assessment of the MCS process and what it can do and what it cannot do:
- MCS exerts a positive pressure on the investor to deal with the uncertainty that always exists in reality about the future value development of a portfolio, especially if withdrawals are made from this portfolio in the long term. This gives the investor a better feeling for the probabilities of different outcomes than is the case with simple deterministic return and final net worth calculations.
- By varying the inputs and assumptions on which an MCS is based, the investor can see - better than with the deterministic method - what the most important levers are for long-term wealth creation and how “non-linear” these levers often work. This means that MCS can help you make decisions (e.g. reducing monthly portfolio withdrawals) when it is not too late.
- Standard market MCS applications cannot reflect annual withdrawals that are temporarily reduced for precautionary reasons in response to a stock market decline. To the extent that investors are willing to temporarily reduce their withdrawals in bad market years (a realistic assumption in our opinion), the results of an MCS without dynamizing the withdrawal rate are too pessimistic. Gerd Kommer Invest uses an MCS with a dynamization function.
- The confrontation with the often surprisingly high level of uncertainty (e.g. the spread of final assets) over periods of 20 to 50 years causes some private investors to feel uneasy because they are not used to probabilistic to make considerations, e.g. B. to interpret spreads of final assets or portfolio survival probability.
In summary, we can say that Monte Carlo simulation is a very useful tool when it comes to retirement-related questions such as “how much do I need to save?”, “do I have enough assets?”, “how soon can I stop working?” or “How much withdrawal per month can I afford?” to answer. In this respect, MCS is superior to many other forecasting methods, especially the standard method of simple point estimation that is common in the financial industry.
Endnotes
[1] The MCS must be fed the arithmetic average of returns, not the geometric average typically reported in how-to books and on the Internet.
[2] The withdrawal rate in percent is defined as the absolute withdrawal amount in the first year relative to the initial portfolio value. This amount of money is then increased annually by the inflation rate, so that the amount of money, adjusted for inflation, remains the same over time. The “4%” is numerically correct only in the first year.
literature
Bengen, William (1994). “Determining Withdrawal Rates Using Historical Data”; Journal of Financial Planning: Oct. 1994; Internet reference: here
Schreiber, Schreiber/Weber, Martin (2023): “Saving in old age – withdrawal strategies for the retirement phase”; University of Mannheim; Internet reference: here
Tharp, Derek (2017): “Does Monte Carlo Analysis Actually Overstate Tail Risk in Retirement Projections?”; www.kitces.com; July 5, 2017; Internet reference: here
McQuarrie, Edward (2025): “How the 4% Rule Would Have Failed in the 1960s: Reflections on the Folly of Fixed Rate Withdrawals”; Social Sciences Research Network; 06 Feb 2025; Internet reference: here