Friday, May 13, 2011

Early retirement calculations - part I

The most common retirement equations you need
In order to achieve (extreme) early retirement, there are a couple of things you want to find out:
  • When can I retire?
  • How big does my nest egg need to be, so I can keep living on my investments?
  • How can I retire sooner?
Behind all these questions, an answer is hiding. One that mathematical formulas can show us.

When can I retire?
I already covered this question in a previous post.

n = (((income-saved)*12/swr)-P)/saved/12
n = number of years you need to save/compound your money
income = your current monthly income
swr = safe withdrawal rate, this is typically 4% or for conservative people 3%. I pick 4%
P = principal (the money you start saving with) 
saved = amount you can save from your income each month

Lets take the following assumptions:
income = 1400 EURO
saved = 600 EURO
A 27 year old guy named Joske starts with 25000 EURO.

According to the above formula, he needs to do 30 years of saving to become financially independent.

I'll use the same numbers in the next eqations.

How much money will I have?

If you want to retire (extremely) early, you need to save money. The pile of money you end up with, should be big enough so you can live from it. To make this happen while you are still alive, there's a magical thing called compound interest. You save, invest and reinvest your interest and magic will happen.
Actually, it's not magic, it's a nice mathematical formula, that goes like this:

K = P*(1+APR)^n
K = required nest egg (final Kapital with a capital K)
P = principal (the money you start saving with)
APR = the interest you earn on your money
n = number of years you need to save/compound your money

For more information on how to get to this formula, you can watch this excellent movie that I found on youtube:

Deriving the compound interest formula

 That guy explains everything very clearly.

So how much money do I need? Well, lets use an example.
Suppose we start with 25000 EURO as I already mentionned above. This would be our principal.
 So we have:
P = 25000
APR = 0.05 (I use 5% interest, that's more interest than the banks give you, but you can do better if you invest wisely)
n = 30 (see above)

K = 25000*(1+0.05)^30 = 108047.50 EURO

That's less money than I expected... but it takes 30 years to get!

How can I retire sooner?
The biggest factor in that equation is the amount of money you can save from your paycheck, as was also explained in my previous post regarding early retirement calculations.
So save, save, save...

You'd like to do more than just save? Okay, than lets see what else we can do...

We could make sure we make more money, so the interest gets bigger and that too will have an effect on compounding.

Suppose we make twice as much, meaning an APR of 10% (=0.1).
K, P and the APR are known, but we need to know n. Thats a problem, we'll have to refactor the formula...

K = P*(1+APR)^n
K/P = (1+APR)^n
ln(K/P) = ln((1+APR)^n)
--> if you do the same thing with both sides of the equation, it stays the same
and now for a cool mathematical trick:
ln(K/P) = n*ln(1+APR)
--> Yes people, that is a legal move!

n = ln(K/P)/ln(1+APR)

So for our example, this formula says:
n = ln(108047.50/25000)/ln(1+0.1) = 15.3572 years

Thats approximately 15 years less than our previous 30 years if we can double our interest rate!

The mathematical formulas show us that there are 2 things we can do, to reach our early retirement goals:
  1. Save as much as we can from our pay check
  2. Get a high interest on that money, by investing wisely
Option 1 is by far the most important one, because it's one we can have complete control over. So start saving... ;)