The REAL Mathematics behind Early Retirement

My brain was feeling a little itchy for some Math yesterday. Math was my best subject in school, and my favourite subject when I tutor. So, I decided to try out to calculate and derive the formula for Early Retirement.

After using this website (link), and MMM post (link) on how Early Retirement is only determined by the Savings Rate, I decided to try to derive the formula that calculates the number of years for retirement. This will be expressed in terms of savings rate, real rate of investment return, and withdrawal rate.

Warning

If you absolutely hate Mathematics or have a phobia of algebra or whatsoever, simply scroll all the way down to the last section for the formula, and how to use it!

Let the Mathemagic begin

Let:
Savings rate = s
Withdrawal rate = w
Monthly income = y
Real rate of investment return = r
Number of years to retirement = n

The logic

For retirement, passive income must be able to cover expenses. Passive income will be the withdrawal rate multiplied by the total net worth in investments. In other words:


Sn is the total investment sum. We have to express Sn in terms of y, r and n.

Deriving Sn

Let's go through the logic.

In the first year, you would have saved an amount of money. So, at the end of year 1, you would have x amount.

In the second year, that x amount would have grown by r% and then you add in another x amount. Thus, at the end of year 2, you will have:



Similarly, in the third year, you will have:


This is a geometric progression, and is learnt in JC mathematics. If you are not sure, you can read this website here: link. The sum of geometric progression in this case becomes:


We also know that the monthly amount saved, x, will be s.y. Savings rate multiplied by income will be monthly amount saved. Thus,


Putting Sn into the formula


We can see the y cancels out,


I take log on both sides,

The final equation and some applications

This is the section you can jump right to for the final formula.


Let's play with the numbers.

Assume we save 60% of income, real rate of return of 5%, withdrawal rate of 3%, this actually gives us 12.4 years before we can retire.


All these is just for fun. I had fun doing the algebra, hope you have fun too! (Maybe I shall give this question to my tutees to try out!)
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2 comments

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1 May 2016 at 10:26 delete

Retirement planning is so difficult!

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2 May 2016 at 18:18 delete

This is just for fun!

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