The Shockingly Simple Maths – Simulations

Last week I said that the what started my journey to financial freedom was reading the post the shockingly simple maths behind retiring early from Mr Money Mustache. When I read that somehow everything seemed to click for me. It all just boils down to your savings rate.

However, lately, I have been reading some post stating that in reality, it’s not as simple as what Mr MMM lays out. And it’s true. There are many simplifications made with this model. But isn’t that the same with any model?

The Reasons

There are two reasons that I often read as to why this retirement model does not work. The first is that the market doesn’t give you smooth steady returns and instead gives you different returns every year, some good years and some bad years.

The second reason is that what you are withdrawing each year will vary depending on the amount you have invested. Some years it will be more than you need, and other years it will be less. The wild swings in withdraws, which is your income, make it highly unrealistic for anyone to be able to support their lives day to day.

So to try and see if this reasoning was true I decided to do some simple simulations using historical data from the NZX50 and S&P500. There are two things that I wanted to try and figure out.

  1. Does the saving rate rule hold true
  2. Does the 4% withdrawal rate hold true

It’s said that because real world returns are not constant and fluctuate that the 4% rule doesn’t work. So let’s have a look.

Saving Rate

Let’s say that you are ambitious and want to save 50% of your income. You look at the savings rate and retirement table and work out that you should have enough to retire on in 17 years. Does this work with real-life data.

Savings Rate (%)Working Years Until Retirement (yr)
0Never
1051
2037
3028
4022
5017
6012.5
708.5
805.5

Here’s the scenario, you have an income of $50K and invest $26k each year into a passive index fund that tracks just the NZX50 in scenario 1 and the S&P500 in scenario 2. For the savings rate rule to hold true after 17 years, you should have $650k to start drawing down on in retirement.

I choose an investment rate of $26,000 because it is 50% it is halfway between 50% of the median personal income in New Zealand, at $49,000, and how much we spend each year if we exclude the mortgage, at $27,000. Assuming the mortgage is paid off by retirement.

Hopefully, this makes the numbers more achievable and realistic, although I can appreciate that you would have to make sacrifices to save 50% of your income.

Simulating the savings rule

So how does that stack up? Below is the graph for all 17 year investment periods between 1949 and 2017 for the NZX50

And below is the graph for all 17 year periods between 1949 and 2017 for the S&P500

Investment portfolio simulation S&P500over 17 years

Results NZX50: If you had invested $26,000 each year in the NZX50 for 17 years you could expect an average portfolio size of $954,000. The largest portfolio was $3,324,000, and the minimum was $458,000

So you wouldn’t be able to retire with an income of $26,000 using the 4% rule on the minimum simulated portfolio. But 34 out of the 51 portfolios did reach the goal of $650,000. That’s a success rate of 67%

Results S&P500: If you had invested $26,000 each year in the S&P500 for 17 years you could expect an average portfolio size of $954,000. Yes, the average is exactly the same as the NZX50. The largest portfolio was $2,052,000 and the minimum was $474,000

So you wouldn’t be able to retire with an income of $26,000 using the 4% rule on the minimum simulated portfolio. But 38 out of the 51 portfolios did reach the goal of $650,000. That’s a success rate of 75%

Overall, 71% of the simulations reached the target investment portfolio size of $650,000 in 17 years. Stations would probably call this a likely chance, not certain, but not impossible either.

4% Withdrawal Rate

Using the savings rate calculations there is a likely chance that you will achieve your goals in the time specified. Now let us assume that you have achieved the savings goal of investing $650,000 and see if the 4% withdraw rate works.

Here’s how the simulation works- Start with $650,000. Withdraw 4%. And multiply the rest by the annualized market return.

It’s harder to summarize these results. The portfolio size varies massively, and hence so to does the amount withdrawn each year. I mean, just look at the following two graphs and see what I mean.

Value of Portfolio after starting 4% withdrawal rate- the 4% rule of investing
Value of Portfolio after starting 4% withdrawal rate- the 4% rule of investing

What you can see from the simulations is that when using the 4% rule your portfolio tends to continue to grow. That’s because the average return of these indices is higher than 4%.

The goal of these simulations was to see whether or not the initial investment of $650,000 could generate a consistent return of $26,000 or more each year. In that regard, the 4% withdrawal rate failed, with only 15% of simulations returning $26,000 or more each year. The remaining portfolios generally returned less than $26,000 for the first few. After which they achieved more than $26,000 withdrawal.

12% of the simulations failed to average more than $26,000 per year. These tended to be the simulations with the shortest number of years to compound, so their averages would probably improve given more time.

Strictly speaking, the 4% rule only worked for 18% of the simulations. These are the simulations that produced a withdrawal of $26,000 per year, every year.

Below is a chart of average withdrawal amount for all simulations. There is a healthy number of simulations that produce an average withdrawal of between $40k to $60k. Which should easily be enough to retire on

average withdrawal rate using the 4% rule from simulations using historical data

Final Words

Now to answer the following questions;

  1. Does the saving rate rule hold true
  2. Does the 4% withdrawal rate hold true

Using historical data from the NZX50 and S&P500 the savings rate table from Mr Money Mustaches original post is likely to work. It worked for 71% of simulations.

The results for the 4% withdrawal rate is also likely to work, with 88% of simulations averaging more than $26,000 per year. But early on it is highly likely that 4% withdrawal will be less than $26,000. Only 18% of simulations returned more than $26,000 per year, every year.

From these simulations, I think the savings rate and 4% rule is likely to work, but I would likely add a couple of years of extra savings to improve the odds.


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