How to calculate time until FI

[Fabian]

Hi, I asked rePete about this but it's not clear to me yet and I don't want to mess up his journal so I repeat the question here.
His answer was:
The algorithm is time_FI = [N(expense_month) - networth_EREaccounts] / savings_rate.
N is the desired withdrawal factor. Right now, I use N = 1/.04 = 25.

expense_month is how much I spent that month.

savings_rate is monthly income less expense_month
If I use this formula with N=30 to be more conservative I get:
Expenses: 1500

Savings: 500

Net Worth: 0
timeFI= (30*1500-0)/500 = 90 months = 7.5 years
That can't be correct...
Who can help?

[Fabian]

Jacob, please move this to the correct forum - I can't do it -- sorry!

[BeyondtheWrap]

OK, well most of your inputs are monthly, so I guess that the withdrawal rate should be too. So for a withdrawal rate of 4% per year, N should be equal to 12*25 = 300. So your N=30 should rather be 12*30 = 360.
Then we have time_FI = (360*1500-0)/500 = 1080 months = 90 years. Much more realistic.

[Fabian]

Thanks! That's what I thought. But it was too discouraging to believe...

[aquadump]

90 years is depressing! As Jacob's book points out, you will gain on compound interest during those years.
This formula, or model, is better used for large savings rates, such as above 50%. It's fun to play around with the model to note how reducing expenses, as opposed to increasing income, makes FI much more achievable as well as decreases the time to it.
Here's just one way to explore the above.

If you haven't checked it out and are interested in this graph, make sure to read Jacob's Chapter 7.

[jacob]

It's equation 7.15 and 7.16 in the book. The equation above is a linearized version and does not work when compounding or drawdown is taken into account. In other words it does not give the same results as traditional retirement calculations do. The full equations will.

[gibberade]

Ahh, the book has equations for FI in it too? I really can't wait till it arrives!

[Fabian]

rePete, your image doesn't show up!
jacob, I bought your book (Kindle for Mac version). I don't know what you mean by equation 7.15 or 7.16. And I don't know which one is chapter 7 because my book has no chapter numbers and no paging at all. I really dislike that as it's hard to refer to your book.
I also don't understand all the mathematical formulas in the book. I would have loved to get an easy to understand formula after all these equations. Can you provide it here please?

[jacob]

@Fab - Ah, I see. This is the upside or downside of the different formats: classic vs hypertext. Paging is impossible---it would make no sense since you can resize the font at will which would completely kill the page count. Numbering hyperlinks is editorially anathema. Thus, I was a bit in a bind, when I laid out the kindle version. That's part of the reason why it took a week to convert it. If I had to do it again, I could do it in a few hours.
It's the second last equation in the section called Financial Independence and Investing.
You're looking for M=log(1+i*P0/p*(1-r)/r) / log(1+i)
In your example

P0/p = 25 (or 30)

i = 0.04

r=0.25

M=number of years of work.
Math guys will probably recognize that they can Taylor expand this they'll get rePete's formula in the limit.
Just plug those numbers into a calculator. Unfortunately, this equation does not reduce to anything simple. It's already as simple as it gets [without simplifying into something that gives wrong results outside the limit conditions.]
The easy way is to use the graph in the figure "The time M it takes to grow ..."
Follow the 4% curve. Read it off where the x-axis is 25%. The y-axis looks by my poor eyesight to be around 35-40 years of work.

[jacob]

The graph you really need to look at is the last graph. Pick a return rate you believe in. I'd pick 3 or 4%. Then pick your savings rate. Now you can see how many years you need to work.
For instance, 4% is a traditional inflation-adjusted rate. You can see if you save the usual 15%, you'll be working for about 40 years which is about right.

[jacob]

But please don't see this as a predictive science. Read the "Investing and reasonable return rates" section right after. Markets are not predictable. What the equations do is to give you a rough indication. If you save 75%+ I can say "less than a decade", but I can't say 5.91 or 7.26 years. Similarly, those middle class 401k owners were told to save $1,000,000 for their retirement and look what happened to them last year. Suddenly many were down to half that.

[Fabian]

jacob, thanks so far. It's almost clear. I've put the formula in Excel.
What does this mean?:

P0/p = 25 (or 30)?
What's P0 and p? Why 25 or 30 in my case?

[Fabian]

According to the book P0 is the "fund" - is that net worth??? And p are expenses?

[jacob]

Yes, the fund is liquid net worth, the amount you have invested, say P0=$200,000. p is the amount spent per year, say $8000/year.
P0/p = 200000/8000 = 25 years
I used 25 and 30 because that's what you guys used above.

[aquadump]

Agh, sorry. Life was easier with chmod . . .

Link

[Fabian]

I don't get it... (((
For me P0 = 0 and - for now - p = 18000
That comes out to 0 years to work...
Here's my spreadsheet:

http://bit.ly/btR3jx
It's editable.
At the moment I have to work longer the more net worth I get...?!?

[AlexOliver]

Fab, I think your P0 in that case is 450000 to 540000, depending on if you're using 25 or 30. If you input that into your spreadsheet, you get 35-38 years left to work.

[Fabian]

Hm. Are you sure, Alex? Is P0 the amount I want to have one day? Or the amount I have now? If it's the amount of money I want to have what's the use of this formula?

[jacob]

Yes, P0 is the amount you want to have at day 0, when you begin your retirement and draw down the money. All equations assume you start with 0 savings.
I edited your spreadsheet to fit your numbers.
Now, what your spreadsheet currently does it to calculate the fund size you'd need to live forever. It stands to reason that those who spend 60 years accumulating it won't need it to last forever. They can therefore accumulate less. Hence they don't need to work as long. There's no analytical (equation) expression for this equation. It's N+M=however many years on has left to live. I used 80, because I'm an optimist assuming everybody will live to 100. Numerically the solutions are given in the last graph.
So rather than get a headache crunching equations for now, just use the last graph. Plug in your 25% savings rate at the x-axis and read off the working years on the 4% curve. (Or the 3% curve, if you're conservative or the 6% curve if you're very optimistic).