Confusion on Retirement Math

[Gotera]

I recently read the article The Shockingly Simple Math Behind Early Retirement ,http://www.mrmoneymustache.com/2012/01/13/the-shockingly-simple-math-behind-early-retirement/, and I was a bit confused on why the graph used to display "Saving rate" and "Years to retirement" is exponential as opposed to having a linear trend. The graph says that on an annual income with a saving rate of 64% one can retire after 10.4 years, leading to $652,376. Though when given a saving rate of 96% one can retire after just 1 year, leading to $68,880. Why would a person be able to retire after a year with just 69k as opposed to the other who retires with 652k?

[AnswerIs42]

If you have a savings rate of 96%, then you have the ability to live off 4% of your salary (probably in a tent or van or something?). In that case, 4% of your $68,880 per year would be enough to live off, because you're already doing it.

[neo von retorch]

Rules of Shockingly Simple Math
1. Given that income is constant, spending and savings are inversely correlated, diametrically opposed, a ratio going one way or the other.
2. Given that spending is constant from year to year, lower spending means you need less total invested to cover your spending needs forever.

In other words, if you have an income of $50k and spend $25k, each year, you save $25k. You also need $625k invested to cover your expenses forever. If there was no return on investment (before retiring), it would take a lofty 25 years to reach retirement ($625/$25k = 25 years.) Fortunately, investing reduces this timeline down to ~16-17 years.

If you have an income of $50k and spend $10k/year, you're saving $40k/year, and you need $250k saved to cover expenses forever. The lower spending resulted in higher spending and a lower target! Before investment returns, you can hit $250k in just over 6 years - with investing, it could be a year or two shorter!

In your example, the savings rate is so high... that the spending rate is very low, and they don't need much saved at all to retire.

[G-dog]

Why exponential? Because this assumes you have invested the money and are earning compounding interest.

Why two different times (and amounts) to retirement? Expenses! If you are saving 96% of your pay - your expenses are only 4%, which happens to be the assumed safe withdrawl rate (SWR).

If your expenses/year are:
 $40K, at 4% SWR you need $1MM
 $4K, at 4% SWR you need $100K

Some people are more comfortable with a 3% SWR, others are less conservative and are comfortable with a 5% or higher SWR.

ETA: cross-posted with NVR.

If neither explanation makes complete sense (AHA) - please do ask more questions.

[jlcnuke]

A savings rate of 96% resulting in $68,880 after one year implies an income of $71,750. Thus the total equations is "saving+spending=income", or $68,880+x(spending)=$71,750, solving for spending we get $2,880. If you spend $2,880/year, you don't need a large portfolio to support that spending level. A 64% savings rate on the same salary has a spending level of over $25k, almost 10 times higher. Thus it needs about 10x the investment portfolio to support that much spending.

[Roland of Gilead]

It really gets even better than that because the lower your income needs, the more likely you will gain benefits from government and pay a smaller tax.

But that is not really what the math question was about.

[MDM]

...why the graph used to display "Saving rate" and "Years to retirement" is exponential as opposed to having a linear trend.

"Time to FI" equation derivation: http://forum.mrmoneymustache.com/ask-a-mustachian/fire-in-8-years/

[Scortius]

It's not actually exponential, it's inverse linear. It just happens that inverse linear looks somewhat like an exponential distribution pdf.

[MDM]

n = ln((S + i*E/WR) / (S + i*A)) / ln(1 + i)

See link in reply #6.

[neo von retorch]

It's not actually exponential, it's inverse linear. It just happens that inverse linear looks somewhat like an exponential distribution pdf.

You're probably fun at parties. No, not the sarcastic version of that comment. I mean it. Especially if you play board games.

[Scortius]

It's not actually exponential, it's inverse linear. It just happens that inverse linear looks somewhat like an exponential distribution pdf.

You're probably fun at parties. No, not the sarcastic version of that comment. I mean it. Especially if you play board games.

Ha, did we just become best friends???

[G-dog]

It's not actually exponential, it's inverse linear. It just happens that inverse linear looks somewhat like an exponential distribution pdf.

To be fair, MMM called the graph exponential in his write-up.

[neo von retorch]

It's not actually exponential, it's inverse linear. It just happens that inverse linear looks somewhat like an exponential distribution pdf.

You're probably fun at parties. No, not the sarcastic version of that comment. I mean it. Especially if you play board games.

Ha, did we just become best friends???

Probably! Two of my friends are really good at math - and good at beating me at board games. I was "good at math" relative to class-mates in elementary/high school, but my skills have softened with age.

[Eric]

Unless you can actually save over 90% of your income, I wouldn't use the literal numbers of that graph.  It's simply an illustration of the power of saving more and more of your income.  Most of us can't swing a 90+% savings rate, so we will have to work for many years during which both our incomes and expenses will fluctuate, and of course the stock market is not going to return a constant return either, so your actual time to retirement will never match exactly.  It's more of a thought exercise as opposed to a future prediction. 

[DreamFIRE]

Yeah, 90% is very high.  My best is 82%, and that's using the MMM formula of adding the employer contribution to my take home pay and savings but excluding dividends/gains/interest on my stash.

But, despite having a savings rate of 70% or higher for years, I don't want to be limited in FIRE to that same amount.  With more free time to travel and enjoy entertainment, I fully expect to spend considerably more than I am now.

[2Birds1Stone]

Yeah, 90% is very high.  My best is 82%, and that's using the MMM formula of adding the employer contribution to my take home pay and savings but excluding dividends/gains/interest on my stash.

But, despite having a savings rate of 70% or higher for years, I don't want to be limited in FIRE to that same amount.  With more free time to travel and enjoy entertainment, I fully expect to spend considerably more than I am now.

82% is damn impressive!!