A Formula for FU Money

FU money is defined as: any amount of money allowing 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 for this (while considering inflation and life expectancy) 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. I have ignored things like taxes, big life events etc. to keep this simple.

I keep hearing this from people: if you can beat inflation by 3%, your FU money is `100/3 = 33` times your current annual spending. I suspected this to be incorrect because (i) this doesn't take care of my finite little life. Who wants perpetual income long after I'm dead? and (ii) (interest - inflation) did not seem to be the critical variable. I did the analysis and it turned out that the variable that really matters is (1 + interest)/(1 + inflation). This is close to (interest - inflation) but may differ significantly in low-inflation countries. Also found that planning for infinite income may overestimate the FU money by as much as `50%`!

First, let me define a few terms:

`S_n` : Savings at the start of the `n^(th)` year (counting from `0`).
`E_n` : Expenses for the `n^(th)` 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 `S_0` (initial savings) such that `S_n` becomes 0 when `n = N`. To get this, first we need to write `S_n` in terms of `S_0`, `E_0`, `i`, `f` and `n` and then solve it for `n = N` such that `S_n = 0`.

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

`E_n = E_0(1+f)^n = E_0F^n`

Savings at the start of `n^(th)` 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.

`S_n = S_(n-1)+(S_(n-1)-E_(n-1))i-E_(n-1)`
` = S_(n-1)I-E_(n-1)I`
` = S_(n-2)I^2-E_(n-1)I-E_(n-2)I^2`
` = S_(n-k)I^k-E_0(F^(n-1)I+F^(n-2)I^2+ ... + F^(n-k)I^k)`
` = S_(n-k)I^k-E_0F^(n-1)I((I/F)^k-1)/(I/F-1)`

By using `k=n`, we can get `S_n` in terms of `S_0`:

`S_n = S_0I^n-E_0I((I^n-F^n)/(I-F))`

Let's define `P=S_0/E_0` and `r=F/I`. Now we can find `S_0` in terms of `N` such that `S_N = 0`:

FU Money = `S_0 = E_0(1-r^N)/(1-r)`

Note that `r = F/I = (1+f)/(1+i)`. For example, 3% inflation and 10% interest means `r = 1.03 / 1.10 = 0.936`. Don't calculate `r` to be `0.03 / 0.10 = 0.3`.

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`: `S_0 = NE_0`

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

Download the excel sheet to try it out and verify for yourself.

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).

We can also find the year when a given amount of savings will last - by solving for `N` in terms of `S_0`:

`N = log(1 - (S_0/E_0)(I-F)/I)/log(F/I)`

`  = log(1-P(1-r))/log(r)`

Needless to say, this analysis is quite simplistic. Big life events like marriage, chlidren etc. will affect the 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.

Discuss this on Hacker News or LifeHacker and let me know what you think.