The concept behind the simulator

The starting point of many considerations about financial freedom is the so-called Trinity Study (English) from 1998. This study states that someone who wants to retire with a broadly diversified stock portfolio can withdraw between 3% and 4% of this portfolio annually despite price fluctuations, without substantially running the risk of going broke. Extremely abbreviated, this has become the well-known “4% rule” that most will already know from the FI community.

The Trinity study in the simulator

To better understand the concept behind the simulator, we will recreate the Trinity study to get started. To follow the example, I therefore recommend opening the simulator in parallel in a new browser tab. Conveniently, the simulator starts by default almost exactly with the input data of the study: We assume that the current value of our stock portfolio is $480.000 and we would like to retire with this portfolio now. According to the 4% rule we should be able to withdraw $19,200 per year, i.e. $1,600 per month from this portfolio by selling corresponding shares. At this point, let us ignore the fact that we might have to pay taxes when we sell and note that the $1,600 is already entered accordingly in the field “Expenses from FI”. Let us leave the Simulation Period at its example value of “30 years”. If we were 60 years old and want to plan a retirement horizon until our 90th birthday that would be a good value. Let us also leave the “Simulation Start” and the “FI Date” at its default value of the current month because we want to simulate a retirement at this point in time.

The worst thing that can happen to the prospective retiree is that a crash of the stock market occurs exactly at the beginning of the regular withdrawals from the portfolio. The monthly withdrawals of $1,600 then suddenly require the sale of larger amount of shares due to their price loss, so that the portfolio melts down very quickly and bankruptcy threatens before the end of the desired 30 years. In the best case, a rally of the stock market would start today instead and the portfolio would grow faster than the monthly withdrawals.

Unfortunately, of course, we don’t know how the stock market will perform over the next 30 years, so we have to make assumptions. In principle, there would be the following possibilities to get to the bottom of this problem mathematically:

The time series of the portfolio can be “viewed” for all start histories.

We can now take a closer look at all these possible histories in our simulator. The right diagram in the simulator shows in light-blue the “boundary” created by the portfolio development of all “histories”, i.e. there was no historic precendent where our portfolio would end up below the lower boundary or abocve the upper boundary. The slightly darker blue area shows the histories between the first and third quantil, i.w. in this area you can find 50% of the more average historic developments. In the middle you see the median as a grey line representing the average of all histories.

In dark-blue you see as a default always the portfolio development in the worst possible case of all calculated histories, i.e. the scenario with with the lowest final value. Let us now take a closer look at this worst-case scenario: All the graphs used are interactive, i.e. we can select the beginning of the curve and zoom into this area to see what happens there. Already in the first two months of our retirement we see the stock market collapse. Our portfolio takes a 11% hit in the first month alone and goes down another 27% in the month after. We can see here live the nightmare of an early retiree. To cover our monthly $1,600 expenses we now have to sell a lot more shares and our portfolio therefore shrinks so fast that we would be bankrupt already in February 2039. In purely arithmetical terms, at the end of our observation period of 30 years, i.e. in July 2052, we would actually still have debts to our bank of about 1.3 million dollars, since we would still continue to withdraw $1,600 per month for our living expenses. If we look at the slider at the very bottom of the page or at the description of this graph we see which of the possible histories is represented here as “worst case “. The worst possible history for our simple example case starts in September 1929 and the catastrophic price collapse that hits here is exactly the peak of the Great Depression.

If we now press the green button “Jump to best case” the picture unsurprisingly looks completely different. The slider at the bottom does not jump far at all, namely to June 1932 when the economy starts its big recovery after the crisis. The dark-blue curve now moves near the upper limit of the boundary and ends per definition on the right at the top of the boundary. If our 30 year period would start in this best of all worlds, we would have accumulated a portfolio of 17.5 million dollars at the end and our heirs would be rubbing their hands.

These two extreme cases limit the spectrum of possibilities how our portfolio should evolve over the next 30 years. Of course nobody can guarantee that we see an even more extreme scenario in the future but such a scenario at least never happened in the last 151 years. The slider below also allows us to virtually travel through all other possible “start histories “ and to look at the associated portfolio development in the right graph. If we want to go back to the worst case, just push the red button.

Here both cases are shown again side by side as an example. On the left the worst case, on the right the best case:

If you look more closely at the lower right portion of this diagram, you can also see that the minimum of the developments still lies below the zero line. This means that naively following the 4% rule is not at all a safe withdrawal stragegy. Nevertheless, the vast majority of all possibilities end up in the positive range after 30 years and the application of the 4% rule can therefore make sense, if one is aware of the residual risk. After all, that’s exactly what this simulator is designed to do, when it shows the wide spectrum of all possibilities.

Now let’s see what happens if we withdraw only 3% annually instead of the 4%. This corresponds to a monthly withdrawal of $1,200, which we therefore enter in the “Expenses from FI” field. After entering, the simulator immediately updates its data for all histories and we can see directly that the whole light blue corridor has moved significantly upwards. Zooming in near the minimum at the end of the simulation period you can see the following, on the left with 4% withdrawal rate and on the right with 3%:

The Minumum of the graph now seems to be very close to zero, i.e. there are almost no historical starting points that end in bankruptcy. A withdrawal of 3% thus appears to be much safer than applying the 4% rule, at least if one also wants the portfolio to survive extreme events such as the great depression. At this point, we can already see that we can confirm the results of the Trinity study (for the 100% equity share assumed here) nicely with the simulator. However, our simulator can do much more.

Exact calculation of withdrawal rates

To do this, we now switch to the second tab of the simulator, labeled “Calculation of Exact Withdrawal Rates”, which does just that. We do not make any changes to the settings at all and get the following overview:

The table (which, by the way, can also be exported as an Excel file at the click of a button) now shows precisely calculated withdrawal rates and associated bankruptcy probabilities. If we start at the bottom we see that a monthly withdrawal rate of exactly $1199.82 leads to a bankruptcy probability of 0%. This means

If you want, you can enter the $1199,82 (or even better the 1199,82209244152 directly from the Excel sheet if you want to get rid of the rounding effects) as “Expenses from FI” again and have a look at the development, especially for the worst case. The portfolio will end up exactly at 0 for the worst case.

The graph shown here on the right now shows the exactly calculated withdrawal rates for each virtual starting point of the price history. Exactly calculated here again means that the withdrawal rate formed for each virtual starting point would have set the portfolio exactly to zero at the end of the simulation. We see that the minimum of the withdrawal rates corresponds to the already known virtual start date in September 1929, as we have already seen on the first tab. The final box plot on the right side shows the distribution of the withdrawal rates over all histories and is therefore a compact representation of the table.

From the table, we can then also see a somewhat better interpretation of the 4% rule: if we look at the line for the bankruptcy probability of 2.5%, we get an annual withdrawal rate of 3.96%. Hence, the 4% rule accepts a bankruptcy probability of about 2.5% for a 30-year time horizon. This can nevertheless be a reasonable starting point for own considerations, especially if one has other income besides the planned depot withdrawals. But this is a topic for another article. There I will also go into detail about the income/expenses graph of the first tab, which I have completely ignored for now in this conceptual introduction.