### Author Topic: The not-so-simple math behind early retirement  (Read 7186 times)

#### barbaz

• Posts: 75
##### The not-so-simple math behind early retirement
« on: June 25, 2015, 01:28:40 AM »
When you save 50% of your income with 5% interest, you need 17 years to have saved 25 times your expenses, according to the article and networthify.com. There's even an excel formula (NPER) to calculate how many years it takes to reach a savings goal.

But in the case of retirement, the goal is moving because of inflation. When I need 20 years to save enough money, I will need 25 times of my expenses-as-in-twenty-years, which can be eighty (!) percent higher than today's (assuming 3% inflation).

So, I'm looking for a way to calculate the inflation adjusted time to retirement (assuming my income grows with inflation, so that the savings rate stays constant). Is anyone able to do the math?

#### FXF

• Bristles
• Posts: 496
• Age: 31
• Location: Los Angeles
##### Re: The not-so-simple math behind early retirement
« Reply #1 on: June 25, 2015, 02:18:49 AM »
I have an excel sheet that does that for me.

For each month starting from now I know my expected needs and can easily take inflation into account. (I have inflation set to 1.5% btw. not 3%).
Then for every month I can say How much I would need at that time to live off considering the 4% rule.
It's not difficult but I cannot just post the equations. I might be able to clear up the table and post that.

#### jfer_rose

• Bristles
• Posts: 422
• Age: 39
• Location: Urban Dweller
##### Re: The not-so-simple math behind early retirement
« Reply #2 on: June 25, 2015, 04:40:28 AM »
I adjust my expected interest rate to account for inflation when I do calculations like this. So for example, if you expect the market to return 7% over time and think inflation would be 3%, that means you do the calculations with a 4% interest rate. The idea is that you would be doing your calculations with the assumption that 3% of your returns are eaten by inflation. I expect that the 5% number you quote was already accounting for inflation since many people say 7% returns are typical and inflation has been pretty low (so they would have assumed 2% inflation).

#### deborah

• Walrus Stache
• Posts: 5698
##### Re: The not-so-simple math behind early retirement
« Reply #3 on: June 25, 2015, 04:55:09 AM »
Everything is moving - income tends to increase with inflation, as do interest rates. So you have two options

Pretend everything stays in todays \$, ignore the fact that your income will increase, just assume there is no inflation, and use interest rates after inflation. Each year, you then adjust to the inflation that happened that year.

OR

Pretend there is a predictable inflation rate, income increases with inflation, use current interest rates, and each year adjust all your assumptions to the inflation that happened that year.

I always did the first.

#### ender

• Magnum Stache
• Posts: 3808
##### Re: The not-so-simple math behind early retirement
« Reply #4 on: June 25, 2015, 05:51:43 AM »
So, I'm looking for a way to calculate the inflation adjusted time to retirement (assuming my income grows with inflation, so that the savings rate stays constant). Is anyone able to do the math?

I just use before-inflation returns and ignore inflation. So instead of 8% or 7% estimates, I would use 5% or 4%, assuming inflation is 3%.

Makes everything easier - my income can be in 2015 dollars, 401k/IRA contributions have 2015 limits, I can rationalize things (ie \$30k@4% withdrawal in 2035 makes sense, etc).

It's not 100% accurate but close enough for my purposes.
« Last Edit: June 25, 2015, 09:49:50 AM by ender »

#### Zaga

• Handlebar Stache
• Posts: 1411
• Age: 37
• Location: North of Pittsburgh, PA
##### Re: The not-so-simple math behind early retirement
« Reply #5 on: June 25, 2015, 06:38:46 AM »
I adjust my predictions and target each year based on personal spending.  Because my personal inflation rate won't be the same as the national inflation rate.

#### barbaz

• Posts: 75
##### Re: The not-so-simple math behind early retirement
« Reply #6 on: June 25, 2015, 09:28:03 AM »
Basically, I had two problems with after-inflation returns:

1) I don't know my actual savings target. Right now, I would need 500k, but by the time I will reach that it likely won't be enough.
2) When I combine interest and inflation, I get confused when I also want to include tax.

#### MDM

• Walrus Stache
• Posts: 6713
##### Re: The not-so-simple math behind early retirement
« Reply #7 on: June 25, 2015, 10:44:33 AM »
So, I'm looking for a way to calculate the inflation adjusted time to retirement (assuming my income grows with inflation, so that the savings rate stays constant). Is anyone able to do the math?

Yes, if I understand the question correctly.  See http://forum.mrmoneymustache.com/ask-a-mustachian/fire-in-8-years/msg620478/#msg620478.

As it says there:
After a bunch of algebra similar to that in the previous post we get:
n = ln((S + (r - i)/(1 + i)*E/WR) / (S + (r - i*(1 + i))*A)) / ln((1+r)/(1+i))

If A = 0, the results of the equations for "n" are identical because (1 + nominal rate) / (1 + inflation rate) = (1 + real interest rate).  When A is not zero, there is little practical difference between assuming 0% inflation and real returns vs. non-zero inflation and nominal returns.

So, after all that...just use current values for retirement-type expenses and savings, and use real returns - simpler and accurate.

#### Dicey

• Walrus Stache
• Posts: 5635
• Age: 59
• Location: NorCal
##### Re: The not-so-simple math behind early retirement
« Reply #8 on: June 26, 2015, 10:56:45 AM »
This discussion always reminds me of one of my favorite quotes:

"The man who says it can't be done is generally interrupted by someone doing it."

The answer is akin to horseshoes. Close enough gets the job done. You don't have to drill down to the last penny. A key point is the earlier you start, the easier and faster it is, and the fewer actual dollars you have to save, to reach your ER and lifetime goals. Forget about the exact answer, just get a move on!
I did it! I have a journal!
A Lot Like This
And hell yes, I am still moving confidently in the direction of my dreams...

#### barbaz

• Posts: 75
##### Re: The not-so-simple math behind early retirement
« Reply #9 on: June 26, 2015, 01:57:39 PM »
The answer is akin to horseshoes. Close enough gets the job done. You don't have to drill down to the last penny. A key point is the earlier you start, the easier and faster it is, and the fewer actual dollars you have to save, to reach your ER and lifetime goals. Forget about the exact answer, just get a move on!
True. Of course, I'm not going to wait with saving until I have figured to the minute when I will be able to retire. Still, forgetting to adjust the savings rate with inflation (which can easily happen if your income stagnates) will drag out the retirement for years. This important fact is cleverly hidden when you make all calculations in today's dollars.

#### sol

• Walrus Stache
• Posts: 5290
• Age: 40
• Location: Pacific Northwest
##### Re: The not-so-simple math behind early retirement
« Reply #10 on: June 26, 2015, 02:21:26 PM »
Everything is moving - income tends to increase with inflation, as do interest rates. So you have two options

Pretend everything stays in todays \$, or...
Pretend there is a predictable inflation rate, income increases with inflation, use current interest rates, and each year adjust all your assumptions to the inflation that happened that year.

I always did the first.

I've always done the latter, because it's the only way your calculation makes sense along the way.  Otherwise in 2015 you're making projections based on 2008 dollars from when you made the spreadsheet, and you have to calculate your current net worth in 2008 dollars.  How confusing is that?

The other counter argument to the OP's question, discussed here often, is that your expenses do not have to inflate over time.  Most of the retirees here retired by some combination of growing their stash and reducing their expenses.  For these people, inflation was negative.

But even if it's not, it's not hard to calculate.  Make a spreadsheet with a column for expenses each year, and inflate that number by your chosen inflation rate.  Make another column for your actual net worth in that year, either observed or calculated using actual (aka "nominal" aka "what is reported on your investment statements") investment returns in current dollars.  When column B is 25x column A, you're there.  Easy.

From there you can complicate things if you want.  For example, if you're keeping a mortgage in retirement then it is unlikely to increase with inflation (other than property taxes) so you can subtract off that fixed expense and only inflate the rest of your living expenses.

I've never understood how anyone can make sense of a future calculation using "real" returns with inflation subtracted out.  What a headache.

#### MDM

• Walrus Stache
• Posts: 6713
##### Re: The not-so-simple math behind early retirement
« Reply #11 on: June 26, 2015, 02:28:39 PM »
Still, forgetting to adjust the savings rate with inflation (which can easily happen if your income stagnates) will drag out the retirement for years.

And a quick estimate of "by how many years?" can be made from the table below, for the inputs given.  The various "dt/dX" items are the estimated change in time to retirement for a given change in the particular "X".  E.g., if annual savings dropped from \$20000/yr to \$19000/yr, the time to FI would change by ~dt/dS * (change in S) = -0.6 yr/\$K * -\$1K = 0.6 more years.

For an interactive version see the 'Misc. calcs' tab in the case study spreadsheet.

And yes, this may well be TMI, but that's what one gets at times in the MMM forum.... ;)

 WR 4% S 20000 E 35000 A 20000 r 6% t, time to FI 21.1 yr E/WR 875000 dt/dr -1.5 yr/% dt/dS -0.6 yr/\$K dt/d(E/WR) 1.4 yr/\$100K dt/dA -4.9 yr/\$100K dt/dE 3.6 yr/\$10K dt/dWR -3.1 yr/%

#### SwordGuy

• Magnum Stache
• Posts: 3071
• Location: Fayetteville, NC
##### Re: The not-so-simple math behind early retirement
« Reply #12 on: April 22, 2016, 10:34:29 AM »
Everything is moving - income tends to increase with inflation, as do interest rates. So you have two options

Pretend everything stays in todays \$, or...
Pretend there is a predictable inflation rate, income increases with inflation, use current interest rates, and each year adjust all your assumptions to the inflation that happened that year.

I always did the first.

I've always done the latter, because it's the only way your calculation makes sense along the way.  Otherwise in 2015 you're making projections based on 2008 dollars from when you made the spreadsheet, and you have to calculate your current net worth in 2008 dollars.  How confusing is that?

I use current dollars and take the inflation out of the return %.

That is because I project from where I am NOW, so each time I make a new projection, it's based on current dollars.

I don't make projections NOW based on stuff I projected in 2008.   That would be silly.  I'm not interested in figuring out where I would be  in 2025 if my 2008 projections were correct for 2016.   I'm interested in making my projections for 2025 based on where I am now.

The only thing old projections are good for is to compare where I am now with where I thought I would be in prior projections.  By analyzing the differences I can attempt to make better projections going forward from where I am now.