

How Much Is Enough? A Formula for FU Money - skbohra123
http://www.nileshtrivedi.com/2012/07/10/how-much-is-enough.html

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gibybo
It's not the ratio of interest to inflation that matters, it's the difference.

The easiest way to conceptualize it is to consider an inflation rate of
near-0, or even negative. The ratio would imply a months salary would be
sufficient FU money if you could invest it at even 1%.

A slightly more mathematical example:

Consider $100 in today's money. Invested at 10% annually, it will become 100 *
1.1^25 = $1,083 in 25 years. If inflation is 1%, it will require $100 *
1.01^25 = $128 for every $100 in today's money, thus the $1,083 is worth $846
in today's money. If inflation is 2%, it will require $164, or $660.

Despite the F/I ratio doubling, the difference is only ~22% after 25 years.

If you use ((1+I) - F), you can achieve the same $846 and $660 numbers with a
very straightforward calculation: S_0 * ((1+I)-F)^N

e.g. $100 * (1.10 - 0.01) = $846, $100 * (1.10 - 0.02) = $660

EDIT: Also, you mention the it is very hard to get r below 0.9, but it's not
really. Or at least, it hasn't been for the last 100 years (I am not able to
predict the next 100 years as accurately as the past). The average inflation
rate of the US Dollar for the past 100 years is roughly 3% and the average
annual return of the S&P 500 is 10%. That would produce an r = 0.3, well below
0.9.

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nileshtrivedi
I also left out explanation for the edge cases. For low enough values of r,
savings will never decrease and never reach zero. For practical values of r,
the equation is correct.

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gibybo
Ok I think I've got a counter example. First of all, I'm only going to
consider values where N approaches infinity && r < 1 && r > 0 to make the math
easier.

lim [n-> inf, r < 1, r > 0 ] of S_0 = E_0 (1-r^n)/(1-r) ->

S_0 = E_0 * 1 / (1-r)

I propose that the actual relationship is:

S_0 = E_0 / (i-f)

E_0=1,f=.07,i=.10 - Original: S_0 = 36.666..., My proposal: S_0 = 33.333...

E_0=1,f=.80,i=1.00 - Original: S_0 = 10, My proposal: S_0 = 5

So it seems your formula approximates mine at low values of i and f, but
diverges strongly at higher values.

I believe I can show mine is the correct one with a small amount of intuition
when we let f=0:

If I make 10% annually in interest, then $1mil will provide $100k/year
forever. The same thing more concisely: i * S_0 = E_0

r = F/I = (1+0/1+i)

After substitution and division, your formula is E_0 = S_0 / ((i+1)/i)

In words: If I make 10% annually in interest, then $1mil will provide
$90.9k/year forever, which is intuitively false.

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nileshtrivedi
It is not. As I mentioned in the post, you need to set aside the money for the
year's expenses at the beginning of the year. $1mil provides $90.9k/year
because (1m - 90.9k)*0.1 = 90.9k.

Interest is received at the end of the period but we need cash in hand
throughout the year to pay for the expenses.

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gibybo
Alright, so I spent another hour playing with these numbers and I think I
understand it correctly now and agree with your original math.

A combination of mistakes (F/I vs f/i, taking money out at the start of the
year) and my imperfect intuition of inflation led me astray. Thanks for
teaching me something new, and for your patience :)

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nileshtrivedi
Thank you for helping me find weak points in the post and clarify it. :)

------
sopooneo
"Research has found that there is a connection between wealth and happiness,
but only upto a limit. The beneficial effects of wealth taper off almost
entirely once a comfortable living standard is reached". But which study?

I've heard that thrown around a lot, and probably with citation a few times.
But I don't see a linked study here, and I recently read (or heard, maybe on
Radiolab?) that further studies have contradicted that. And in fact various
happiness metrics increase pretty much linearly with wealth. And they came at
it from another angle, saying that the biggest jump after basic security comes
when you have _somewhat more than others around you_. That is, are you richer
than your neighbor? Does he covet your ox?

~~~
nileshtrivedi
Absolutely. I think there's a lot more we're yet to understand about the
connection between wealth and happiness. Here is one research which was cited
in the NYTimes article:
[http://www.people.hbs.edu/mnorton/aknin%20norton%20dunn%2020...](http://www.people.hbs.edu/mnorton/aknin%20norton%20dunn%202009.pdf)

However, this point is not central to my post. My analysis is only concerned
with how much savings we need to live a particular lifestyle for a given
number of years - including the effect of inflation.

I had seen many people using heuristics like this: If you can beat inflation
by 3%, 100/3 = 33x times your annual expenditure is your FU money. That turned
out to be incorrect because of not considering the life expectancy, and cash
flow. Also the critical variable turned out to be (1 + interest rate) / (1 +
inflation rate) instead of (interest rate - inflation rate) which can be
significant. As I commented elsewhere, 21x may be sufficient instead of 33x.

That's what the post is about.

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HockeyPlayer
Conservatively, a portfolio can produce 3% over inflation forever. So you need
to start with 33x your desired annual spending. Aggressively is 5%, which
means you need 20x your spending.

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nileshtrivedi
This is an overestimate because you don't need perpetual income. You only need
income till your death. So for the case when portfolio return is 3% higher
than inflation, 21x of your spending is fine (assuming 30 years of life after
retirement).

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pmiller2
I'd just like to point out that the word "tax" does not appear in the article.
You need to be sure to include taxes inside the E_n term. This is a bit
problematic, since tax rates can and do change, but you can estimate using
historical values if you like. Other than that, it looks good to me. :)

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compsciphd
your FU money formula doesn't work. don't have time to figure out where you
might have made a transcription error.

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nileshtrivedi
I have verified the formula by simulations in excel. Perhaps you're doing
something wrong.

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gibybo
He's right. I'm not sure what your 'simulation' involves, but it's possible
you just happened to pick numbers where the ratio produces similar results to
the difference. Try inflation = 0.0001% and interest = 10%, for example.

~~~
nileshtrivedi
r stands for F/I which is (1+f)/(1+i). This terminology seems to be creating a
lot of confusion. I will add a clarification. :)

