help calculating savings and investing amount

[Stache In Training]

Hello All.  Trying to explain to a friend the savings coupled with investing miracle, but running into a math brain fart.

So I've convinced him to stop eating out at lunch everyday (to just eating at home once a week), and that just cutting back one meal at the average of lunch cost of $7 (low-balled it), and substitute for homemade sandwiches.  The sandwiches I'm estimating at costing $0.70 total for 2 sandwiches. 

So multiply it all out and you save $315 a year (7 times 50 work weeks, equals 350.  0.7 times 50 work weeks equals 35.  350 - 35 = 315.)  I know he's going to say that's not worth it, so I want to say, but when invested at 7%, it will become BLANK, in 10 years.

So when I put into a compound interest calculator, like this one: http://www.thecalculatorsite.com/finance/calculators/compoundinterestcalculator.php, I get a total of 5,135.15.  (The numbers: initial amount, 315. interest rate, 7%.  calculation period, 10 years.  monthly deposit, 26.25 (315/12 months).  Compounded yearly)

However, when I take MMM's rule of multiplying a monthly expense by 173 (shown here: http://www.mrmoneymustache.com/2011/04/15/getting-started-3-eliminate-short-termitis-the-bankruptcy-disease/, I get 4541.25 (26.25 X 173).  Which is obviously not what is above.  And when I take the weekly amount (315/50 work weeks =) 6.3 a week, and multiple that by 752, I get a different number: 4737.60.

So what am I doing wrong? (is it really thrown off that much by the fact that I'm saving for 50 work weeks, but investing for 52 weeks in a year?) Or what are the assumptions made in that 173?  Daily compounding? No compounding, just growth?

Am I trying to show this savings in the correct way?  Or is there a better and simpler way to explain continued savings (getting the 315 each year) coupled with investing it? I could just say, multiply by 173 for monthly, or 752 for yearly, but I think showing the math is a bit better than just saying "uses these 'magical' numbers."

Any help is greatly appreciated, thanks!

[sherr]

Maybe this thread will be useful for you: https://forum.mrmoneymustache.com/ask-a-mustachian/752-and-173/

The assumptions behind 752 and 173 are:
1) The same amount will be saved every week / month for 10 years
2) 7% annual interest
3) interest compounds monthly (monthly interest rate 7% / 12)

Your problem in the first case(the $5,135.15 number) is that you set an initial amount of 315 when it should be 0 and that you're compounding yearly instead of monthly. Change that and the calculator gives us $4,569.98. 173 * 25.26 = $4541.25. They agree (it is just an estimate after all, you'll get a little rounding error when you're multiplying by the rule-of-thumb numbers).

The problem with your second number ($4737.60) is that yes, it really does make a couple hundred dollars difference if it's 50 times a year or 52. You carefully built into the original yearly estimated savings the fact that this is out of 50 work weeks, which is fine, now you can divide by 12 to get an average monthly savings to play with. But then when you get the weekly total instead of dividing by 52 to get the *average* weekly savings that you can use with calculators / estimates, you instead divided by 50, which will give you back the *actual* amount saved per work week. The 752 rule-of-thumb assumes that's how much you're saving all 52 weeks, so you'd want the weekly average for the year, not however much it actually happens to be per week for however many weeks you actually happen to work.

It makes sense that it would make a couple hundred dollars difference. Ignoring interest completely, if we assume $6.30 per week times 2 weeks per year that we're missing times 10 years, that's $126 difference. Add in interest and it would be more. If instead we do the *average* weekly savings we come out with: 315 / 52 = 6.058.    6.058 * 752 = $4555.616, which is in agreement with the other two but again with a little rounding error.

I hope that clears things up for you.

[Stache In Training]

AHH! The fog is lifted!  Thank you! 

I set the starting amount as 315 so that they'd invest after saving for a year.  But I guess it makes more sense to just start at 0, in order to show the way it adds up.

So I think I'll just change to showing it with 52 weeks, instead of 50 work weeks.  Because for those other 2 weeks or vacation, it's not like he'd be eating at home the entire week.  The same mind-set is still applied. So that will make everything much easier.

Thank you for clearing it up!

[charm]

Hi Mr. Moustache,

Would you know the formula for coming up with the figures 752 (for weekly expense) and 173 (for monthly expense)?

[MDM]

Would you know the formula for coming up with the figures 752 (for weekly expense) and 173 (for monthly expense)?

It's the Future_value_of_an_annuity.

E.g., to get 752: ((1 + .07/52)^520 - 1) / (.07/52)

See also Excel's FV function, and the 'Misc. calcs' tab of the case study spreadsheet.