Question about "Shockingly Simple Math"

[leif20]

Hi Guys,

Wondering if someone can shed some light on this topic. 
Post in question: http://www.mrmoneymustache.com/2012/01/13/the-shockingly-simple-math-behind-early-retirement

In the spreadsheet the formula for "Investment Gains This Year (% of one year's take-home Income)" is given as:
http://i.imgur.com/56VbHJw


I am not sure why the savings rate is divided by 2 in the equation.  It's probably something simple that I'm overlooking, but why is that factor there?

Thanks!

[mskyle]

Hi Guys,

Wondering if someone can shed some light on this topic. 
Post in question: http://www.mrmoneymustache.com/2012/01/13/the-shockingly-simple-math-behind-early-retirement

In the spreadsheet the formula for "Investment Gains This Year (% of one year's take-home Income)" is given as:
http://i.imgur.com/56VbHJw


I am not sure why the savings rate is divided by 2 in the equation.  It's probably something simple that I'm overlooking, but why is that factor there?

Thanks!

OK, there are two parts to this equation: first (everything in parentheses before the plus sign) is the growth of your starting stash/beginning of year stash - it's pretty straightforward.

After the plus sign is the growth of the stash you will accumulate during the year. Since presumably you are saving the money evenly throughout the year, you divide it by 2, because at the beginning of the year you will have none of that money, at the end of that year you will have all that money - on average you will have half that money. The money can't grow until you save it. It's an approximation, but not a terrible one.

Here's the second part of the equation written out:
(Total Savings for Year / 2) * Annual Investment Return

Since everything is reckoned in "years of take-home pay", savings rate is equal to savings for the year; cutting this in half gets you your average balance for the year.

[leif20]

Since presumably you are saving the money evenly throughout the year, you divide it by 2, because at the beginning of the year you will have none of that money, at the end of that year you will have all that money - on average you will have half that money. The money can't grow until you save it. It's an approximation, but not a terrible one.

Here's the second part of the equation written out:
(Total Savings for Year / 2) * Annual Investment Return

Since everything is reckoned in "years of take-home pay", savings rate is equal to savings for the year; cutting this in half gets you your average balance for the year.

Ah, that's the missing link! Thank you very much. 

[gerardc]

BTW, totally negligible but if you're interested in how accurate the /2 approximation is, the exact formula for perfectly even contributions (i.e. you contribute fractions of cents every instant) would be

Total savings * Return * (1/ln(1 + Return) - 1/Return)

instead of

Total savings * Return * 0.5

which is very close to a factor of 0.5 for low interest rates:

Annual return (Front load)	Annual return (Even contributions)
1%	0.4991708%
2%	0.9966996%
3%	1.4926104%
4%	1.9869268%
5%	2.4796716%
6%	2.9708672%
7%	3.4605355%
8%	3.9486977%
9%	4.4353747%
10%	4.9205869%
11%	5.4043540%
12%	5.8866956%
15%	7.3253544%
20%	9.6962990%
25%	12.0355029%
30%	14.3448406%
40%	18.8805365%
50%	23.3151731%
75%	34.0205220%
100%	44.2695041%

You need to do an integral to get to this result. Getting the value for investing evenly every trading day or after each paycheck would be a little trickier, but doable with some code.