How much money is enough? We're obsessed with this question, aren't we? Research has found that there is a connection between wealth and happiness, but only up to a point. The beneficial effects of wealth taper off almost entirely once a comfortable living standard is reached.

Image from Sven Hoppe

This living standard would obviously be different across individuals, cultures, and economies. What almost everyone agrees with, however, is that it would be an inefficient use of our time to accumulate wealth that we never use. Wealth, among other things, is only a means to an end. We should earn as much as we need and no more.

It is common to see this theory, and advice based on it, in popular media. What I haven't yet seen is a mathematical analysis, even a simplistic one, to come up with a formula for this amount -- wealth that we need to comfortable retire and never having to worry about money at all -- commonly known as **FU money**.

FU money is defined as: *any amount of money allowing infinite perpetuation of wealth necessary to maintain a desired lifestyle without needing employment or assistance from anyone*. With only interest (return on investments) as an income, this should last all of the remaining years of my life, while accounting for my lifestyle and inflation. I was curious to see what the formula would look like and this is what I came up with. I made certain assumptions to simplify the exercise but the point is to come up with an analytical formula and understand its nature to see the impact of each variable.

First, let me define a few terms:

**Sn: Savings at the start of the nth year (counting from 0).**

En: Expenses for the *nth* year

*i*: % Rate of return (like interest) on my savings

*f*: Inflation rate (% increase in my expenses every year, assuming that my lifestyle remains the same)

*N*: Numbers of years remaining in my life (estimated)

Every year, we earn some income as interest on our savings, spend some money as that year's expenses, and transfer the remaining back to the savings. When interest income is less than expense for the year, we will need to take funds out of savings. With time, savings would dwindle away each year and ultimately reach zero. Our hope is that this point comes just after our dealth so that our entire life is taken care of.

So our goal is to find S0 (initial savings) such that Sn becomes 0 when *n*=*N*. To get this, first we need to write Sn in terms of S0, E0, *i*, *f* and *n* and then solve it for *n*=*N* such that Sn=0.

Let's use *I*=(1+*i*) and *F*=(1+*f*) to simplify our equations:

En=E0(1+*f*)n=E0Fn

Savings at the start of *nth* year can be written as: savings at the start of the (*n*−1)th year, plus interest earned, minus expenses for the year. However, to avoid a cashflow problem, money has to be kept aside at the start of the year for expenses and therefore interest will be earned only on the remaining amount.

By using *k*=*n*, we can get Sn in terms of S0:

There it is. The relationship is actually quite straight forward but we get to learn what the important variables are. What really matters is r : the ratio of inflation to interest and not their absolute values. Also, your FU money turns out to be a direct multiple of the cost of your desired lifestyle. As an example, taking *f*=7%, *i*=10%, and *N*=40 years, I need savings equal to 25 times of this year's expenses.

As a special case, when *r*=1: S0=*NE*0

Here is a simulation in Excel which verifies that the formula is correct.

Typically, *r* is always less than 1 (Think about it: if inflation were higher than interest rate, borrowing money would be impossible). But *r* can be higher than 1 if you are really bad at investing. I plotted *P* as a function of *r* for *N*=30 below. You can clearly see how much of a difference *r* makes to the FU money. For *r*=0.5, *P*=2 meaning that you just need savings worth two times your current expenses to not have to work for next 30 years. When *r*=0.7, this factor becomes 3.3. It grows to 9.5 for *r*=0.9, 30 for *r*=1 and 164 for *r*=1.1. **Being good at investing is absolutely critical to retiring early.**

You'll have to be really good at investing to bring *r* below 0.9 (although this is much easier if you live in low-inflation countries).

Needless to say, this analysis is quite simplistic. Big life events like marriage, chlidren and so on will affect your lifestyle immensely and have to be considered. Also, if you want to leave some money when you die (say for family or charity), that has to be accounted for too.

How Much is Enough? A Formula for FU Money [Nilesh Trivedi]

*Nilesh Trivedi is an MBA-turned Ruby hacker who loves making good, honest software for fun and profit. When he's not thinking up and implementing big ideas, he loves to play guitar, read books, and explore the amazing wonderland called India.*

## Comments

"any amount of money allowing infinite perpetuation of wealth necessary to maintain a desired lifestyle without needing employment or assistance from anyone."

Yeah that's exactly what I've always thought. As long as you're not eating into your metaphorical money making machine it's all good. The mining and IT booms now mean there are a lot of people under 30 with around half a million saved (well in my circles anyway), so if played correctly you could already begin semi-retirement now, as long as you don't go nuts and try to live a lavish lifestyle.

I'm guessing you aren't yet married with Kids? That'll soon sort out all that money you've got saved up :)

This assumes you want to get married and/or you want to have kids.

I agree with this article, but the one thing it doesn't really take into account is the fact that some people would like to have a job whereby the income they receive from that job is a compliment to the satisfaction they receive from it: the main reason they want to have a job. I know a few people that would do their job even if they didn't get paid for it (admittedly these people could financially afford to hypothetically not get paid for their job as they are already wealthy but still).

...but then again if you enjoy working and you get paid enough then there is no real reason to workout how much money you would need in the long-run as your savings aren't really decreasing...

I need savings equal to 25 times of this year’s expenses.I'd argue that any number under 8 figures isn't FU money. It's comfortable life money or partial early retirement money. Which isn't at all the same as being able to say FU to everyone and maintain the lifestyle that you want for the rest of your life. Or you have really boring lifestyle plans and fewer than average relatives with an emotional hold over your finances.

I suspect that there is an outside possibility (very outside, but still plausible) that I might see FU money sometime in the next 5 years. By plausible I mean I don't expect it to happen - but it's more likely than playing the lottery and likely enough that I actually felt the need to figure out what FU money would be for me. First key figure is what you'd spend in the first 2 months - which I assume includes paying off my debts and my parents debts, and setting up my parents retirement and things like buying the house I hope to use as my home for the rest of my life and establishing the base furnishings and etc. I then have an estimate for how much it would cost to maintain the lifestyle to which I'd like to become accustomed. I estimate inflation at 3.5% and interest at 5.5% as worst case figures and project that with my health and my expectations about healthcare advances within my planning horizon - I almost certainly won't live past 110.

For me, that puts minimum FU money at 17 million. Anything less than that, I'd want to actively work on increasing my wealth through either more active pursuit of a better interest return, or through continuing to try and increase my wealth somehow through working. $18 million would be my comfort point for that starting ground and $20 million would be the point where I'd upscale the lifestyle to which I'd like to become accustomed.

You think that reaching $17 million is plausible for you in the next 5 years? Where do you work?

FYI if you go to his site, the article is much easier to read and you can download the XLS.

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