What are your chances of losing money on the MSCI world index? (2024)

Conventional investing wisdom dictates that highly diversified low cost index funds offer the best returns in the long-term for unsophisticated investors. Capitalizing on market inefficiencies is out of reach for everyone except big Wall Street firms that employ massive computational power to engage in high frequency trading. Those that favor stock picking based on fundamental or technical analysis might as well lay out tarot cards and bet on horse races, since almost no “professional” investor can consistently outperform the broader market.

Based on the market’s historical performance, most investing advice suggests you can expect an average annual return of about 10%, which is reduced to an effective 7–8% if one accounts for inflation. Based on this magical number, one can perform a simple calculation using a dollar cost averaging (DCA) investing strategy to show impressive exponential wealth growth.

DCA is a simple strategy whereby a fixed amount of money, perhaps adjusted for inflation, is invested at fixed intervals. For example, one buys into an exchange traded fund (ETF) for 1000 EUR each month. The idea of DCA is that you make no attempt to predict the highs and lows of the market. By buying at fixed time intervals, you will buy both low and high which average out, allowing you to benefit from the generally upward trend in the market.

To illustrate our example, if you invest 1000 EUR every month and get an annual return of 7%, the progression of our wealth over 30 years is visualized in the following plot:

The total amount you invested over 30 years is 360 k€, but thanks to compounding returns your portfolio is expected to be worth a bit over 1.2 M€. You have multiplied your wealth by a factor of 3.4!

This type of basic analysis can be read anywhere on the internet, but it misses a lot of nuance. The market does not monotonically increase, it wiggles up and down considerably as a result of all kinds of unpredictable events in the world. To illustrate, here is the evolution of a portfolio that invested 10 k€ into the MSCI world index in 1979 [source]:

Sure, looking at this entire picture, your wealth increased by a factor of 75 over 40 years. But had you invested your 10k in the year 2000 at the height of the dot-com bubble, you would have had to wait 15 years to break even.

No one knows with any certainty what the market will do in the future, but we can simulate thousands of possible futures to estimate probabilities of certain outcomes. For example, what are the chances of losing money by investing in the index using a DCA strategy? That is what I will try to answer in this article.

If the market is entirely unpredictable, there is no way to make any kind of forecast. So we have to make a number of assumptions about how the market evolves. For that we base ourselves on historical data.

Let’s assume that the market goes up or down each month by some factor (1+r). If r>0 the market goes up, if r<0 it goes down. r is drawn randomly from some unknown distribution. We assume that our historical data samples this distribution. We can derive r from the historical data by calculating the quotient between consecutive months and subtracting 1. The following figure shows a histogram of the result:

The data shows two things:

• The mode (~0.01), mean (0.007) and median (0.01) of the distribution are larger than 0. That means that there are more months in which the value increases than months where it decreases, i.e. we expect a positive trend over long timescales.
• The distribution shows a much wider tail on the negative side than on the positive side, which reflects the old adage: “Stocks take the stairs up and the elevator down”

To make forecasts using Monte Carlo simulations, I will make two assumptions:

1. r is selected from the distribution randomly and uncorrelated with previous values of r or with time. That means that the selection of r each month is an independent event that does not depend on any other variable.
2. the distribution from which r is drawn is static: it does not change over time.

I will return to these assumptions later.

Approach

I ran DCA simulations where I invest 1000 EUR each month over 30 years. Each month, the total value of the portfolio is multiplied by the return factor (1+r) where r is selected randomly from the list of historical r values.

To account for the fact that the historical data is simply a sample of some unknown distribution, I attempt to reduce the sampling bias using bootstrapping. So in the simulation procedure, I bootstrap the historical data a few 1000 times, and for each bootstrapped returns distribution I simulate a few 1000 DCA runs. Overall I performed 10 million simulations, which took about 1 minute in total to calculate.

Results

The value of the portfolio after 30 years for the millions of simulations with bootstrapping is shown in the following logarithmic histogram:

Note the logarithmic x-axis. On a linear axis, it was impossible to get a good sense of the distribution, since the right tail spans multiple orders of magnitude. With a log scaled x-axis, the distribution looks very similar to a normal distribution, meaning that the actual distribution approaches a lognormal distribution. This makes sense because the process is a multiplicative random walk instead of an additive one.

The total invested amount was also plotted as a red line (360 k€). There are some simulations where the total portfolio value after 30 years is lower than the invested amount: this happens in about 3.1% of the simulations. So you have a 97% chance to break even or make a profit over 30 years, not accounting for inflation.

This calculation can be performed over the entire investing period. This is shown in the figure below:

The longer you continue investing, the lower the chance of ending up at a loss. If you invest for only a few years, the chance of making a loss is substantial, as you are very susceptible to sudden drops in the market. As you invest longer, the years in which you expect to make a profit will counterbalance the bad years. Patience is the name of the game.

Besides calculating the chance of making a loss, other scenarios can be explored. The mean and median portfolio values over all simulations are 1.9 M€ and 1.3 M€ respectively. So you have a 50% chance to to multiply the amount you invested by a factor of 3.6 or more. Additional percentiles can be easily calculated from this distribution, for example you have 90% chance to do better than 530 k€, a gain of about 50%. You can also do the opposite calculation, e.g. what is the chance of at least doubling your money? Turns out, about 80% chance.

The average annual rate of return that corresponds to a particular result can also be calculated. The value V of a portfolio in year n is approximately given by the following expression:

where C is the amount invested per year and r is the average annual return. This is not exact if investments happen at a higher frequency than once per year, but the error is minor. Since V, C and n are known, r can be calculated.

For the median r=8% and for the mean r=10% approximately. So we end up with the same numbers that are typically quoted for annual return by investing in the index, which is a nice confirmation that this approach is sensible. The advantage is that with this approach we can start to estimate likelihood of outcomes instead of focusing on single numbers.

It’s important to remember that because I make predictions 30 years into the future I get an extremely wide distribution. Predictions on shorter timescales yield a much more narrow distribution. As new information becomes available, some paths get excluded, thereby updating and narrowing the probability distribution.

In performing our simulations, I made two important assumptions:

• I assume the return in one month is an independent event
• I assume the probability distribution from which the return is drawn does not change over time.

Let’s have a look whether these assumptions are actually valid.

Assuming returns are random

To validate the first assumption, let’s look at r of the historical data over time:

This plot pretty much looks like random noise. To verify whether it is, we can make an auto-correlation plot, which shows the correlation between the r time series and itself lagged by different amounts:

This shows that there is no strong evidence to suggest past and future r are correlated, suggesting our first assumption is reasonable.

This analysis does not exclude the possibility that combinations of past r do correlate with future r, or are related via hidden variables. For example, we could imagine that a cataclysmic event like a pandemic is followed by mostly negative return months. Such ideas could be explored using recurrent neural networks or hidden Markov models. However, adding complexity increases the chances of over-fitting our data, i.e. we discover patterns in the data that do not generalize. Hence, I did not explore this option further.

Assuming the distribution of r is static

There are a number of ways in which time itself could influence r. For example, there could be a seasonal periodic effect (e.g. returns are better in January than August), or there could be a continuous evolution over time (e.g. returns were better in the 80’s than in the 00's).

We can take subsamples of our time series, and check whether the r distributions are significantly different using the Kolmogorov–Smirnov (KS) test.

To check whether the month has any effect, the data can be split by month and compared against all other months using the 2-sample KS test. The resulting p-values are shown by color in the matrix below:

The colormap was adjusted so that only squares with a p-value smaller than 5%, which we take as our cut-off for significance, are highlighted. Six month-pairs show a significantly different distribution. The most dissimilar distributions are those in September and November, with a p-value of around 1%. They are shown for comparison below:

It would seem that September has relatively worse returns than November, and at least according the the KS test there is a low probability r is drawn from the same distribution for this month. This would suggest there is some periodic influence of time on r.

On the other hand, we have made 66 comparisons, which means that purely by chance we expect between 3 and 4 of those comparisons (5%) to give a p-value smaller than 0.05.

By using a binomial distribution, the chance to get 6 or more significantly different comparisons of 66 comparisons is only about 5%. Therefore, it is unlikely that differences are purely down to chance, and there is likely a small periodic effect of time, i.e. the month has an influence on the distribution of returns.

We can do a similar analysis for returns over contiguous blocks of time. Suppose we take blocks of decades: 1980s, 1990s, 2000s, 2010s and 2020s. The analysis shows that there are multiple decades with significantly different returns distributions, as indicated by the p-value plot:

The most dissimilar decades are the 1990s and the 2010s, as shown by the histogram:

The 90s saw a much bigger spread, with much bigger drops, but also larger gains. The 2010s were more tempered. Interestingly, both mean and median are very comparable. The 2000s showed a distribution quite similar to the 90s, except with negative values even more strongly represented.

We can run all the simulations again using only a subset of the data to check the effect on the results.

For example, we could assume that our future distribution of r will be similar to the past decade. If we run the simulations again with only the r data since 2010 we get nearly the same results as before with all the data, with a 3% chance to lose money after 30 years.

However, if we run the simulations with data from 1990–2010, a period which saw both the bursting of the dot-com bubble and the housing market crash, we see a completely different picture emerge:

In this situation, the chance of making a loss after 30 years is 35%. The mean and median portfolio value after 30 years are a measly 490 k€ and 760 k€ respectively.

In our simulations and analysis we have not taken inflation into consideration. Hence, the probabilities of making a loss that are reported here are underestimations, since you need to end up with a higher portfolio value than just the sum of what you invested to beat inflation.

Additionally, we have not taken into consideration costs. Buying ETFs, typically the vehicle used for investing into the index, may require you to pay transaction fees to your broker. Holding ETFs also incurs a relatively small management fee that is factored into the price. These factors will slightly decrease the expected returns.

If we assume the returns on the MSCI world index in the past 40 years are a good predictor for future performance, then it is a good investment. On average, non-inflation adjusted returns are around 10% per year, as is typically quoted in most sources. Using all the historical data, MC simulations show we expect to make a profit with 97% probability over 30 years (not accounting for inflation). Shorter investing periods show higher probabilities for making a loss.

However, the biggest assumption we had to make was that past returns over the last 40 years are representative for the future. If we cherry-pick for our historical reference the 20-year time period between 1990 and 2010, during which there was serious upheaval in the market, then the results of the simulations look much less rosy: 35% chance to make a loss over 30 years.

What will actually happen depends on what you believe. Are we embarking on a new era of exponential growth due to recent breakthroughs in AI? Or is this a dot-com style bubble waiting to burst, and is civilization headed for collapse due to climate change? As a former scientist who believes in the finite nature of the planet, I lean towards team doom, but only time will tell.

What I hope is that this article has at least given you an appreciation of the immense uncertainty that is inherent to investing, even if it is investing in an index.

The code containing the analysis for this post can be found in the following Github repository:

If I make errors of reasoning in this post, please let me know in the comments so I can try to correct them. If there are mistakes in the code, please make an issue or pull request on Github.