ARGH! Need maths help!

[blamebrampton]

I was once good at maths.

I know I should know how to do this, but I cannot remember enough to see if my method is effective or not. I found a website that will let me punch in numbers and give me an answer, but I want to check it! So I am hoping that shocolate  or someone similarly gifted is up and about.

I start with $50. Every week, I add $50. I have a compounding interest rate of 9.96%. I compound it annually, or monthly (two results). What do I have at the end of 21 years?

More than happy to do all the actual working if someone can remind me of what the formulae are.

Comments

[franalan.livejournal.com]

This doesn't seem to be too horribly wrong...
http://en.wikipedia.org/wiki/Compound_interest

Sorry I can't be more help...I've never been any good at this type of maths. Give me triple differential equations and I'm fine. Interest rates and anything using real nulbers instead of symbols...eeeep! No can do!

[blamebrampton.livejournal.com]

Alas, doesn't work for the continual additions.

It shouldn't be that hard ... but I can't do it!

Really it's:
50*52*21 (the amount per week, times weeks, times years), plus
1.0083*50*251 (monthly interest rate, times first month's money, times all months bar the first, plus
(1.0083 squared)*50*250, plus
(1.0083 cubed)*50*249, plus ...

And I KNOW I can work out how to do this, if only I could remember how!

Weeks in 21 years

[ewen]

FWIW, there are more than (52 * 21) = 1092 weeks in 21 years. There's more like (((365*21)+5) / 7) = 1095 weeks (actually over 1095.5 weeks; and depending on how the leap years fall there could be one more day in those 21 years). Just one of the annoying things about trying to do financial calculations on a weekly basis with rates that are yearly. (Payroll calculations for, eg, monthly allowances paid weekly are even more annoying; such things should be prohibited by the Geneva Convention.)

The formulas here (which was suggested by sherryillk) look plausible, and pretty simple to calculate (no series calculations), but I've not done enough finance to be sure of the derivation. (Hence using brute force to solve such problems.)

Ewen

Re: Weeks in 21 years

[blamebrampton.livejournal.com]

Quite right! And, as I realised after the first three minutes of trying to check independently, the money goes in weekly and is compounded monthly, so that won't fit easily unless I take an average monthly amount, which I may as well do, actually ...

Back to sherryillk's formula! It wouldn't work for me for m>c, but maybe it will if m=c!

I knew I would regret not practicing my maths as much as my French and Italian one day!

Interest

[ewen]

I've never studied finance in detail, so I may be overlooking some summary formula, but I've always solved such savings problems with brute force, viz by calculating a series of weekly totals taking the new deposits into account and the new interest (eg, one week's fraction of the annual interest) into account as appropriate. (FWIW, the wikipedia compounding interest page assumes unchanged principal, just interest being added, as does this textbook chapter.) Any programing language, or even a big spreadsheet, allows that sort of iteration (it's only about 1000 rows providing you don't want daily compounding).

Note that IME most savings accounts are actual daily compounding, with interest credited 1-4 times a year, rather than only compounding once or twice a year. But given your hypothetical interest rate (9.96%! does anyone pay even half that these days?!) perhaps you have a hypothetical account too that really does only compound once or twice a year.

My brute force perl script suggests that with weekly compounding (irrespective of when it's credited), you'd have over $185,000 at the end (with exactly how much depending a bit on the details of leap years you happen to cross, and hence number of weeks involved). This calculator suggests you'd have a bit over $174,000 if the interest were only compounded annually (alas it doesn't have a biannual compounding option, only daily/monthly/quarterly, but the figure ought to be somewhere in between those two values -- or more precisely between the quarterly and yearly figures).

I hope that helps,

Ewen

PS: Of those amounts about $55,000 is weekly cash contributions and the rest is interest. (1.0996)^21 is nearly 7.5 (so approx 7500%), but obviously not all of the cash is there at the start to get interest, so figures in the $170,000-$185,000 range are quite believable.

Re: Interest

[ewen]

PPS: The above assumes that you are getting 9.96% after tax, an even less believable situation. If your tax rate is, eg, 33% then the effective annual interest rate is closer to 6.6% and the totals will correspondingly be much smaller. Although typically the tax is only deducted when the interest is credited, not when it is compounded, so daily compounding with annual crediting can work out slightly better from this point of view. (If you're getting 9.96% interest after tax, at present, I'd be very cautious about the risk of the investment -- the chances of ending up with $0 would seem noticeably high.)

Ewen

Re: Interest

[blamebrampton.livejournal.com]

HEE! Yes, totally unreal world ;-) It's basically looking at what a real-world investment with regular inputs would have done if invested in the ASX 21 years ago.

The last time I did anything like this, I proved that you would have been about $10 billion richer if you followed one particular financial adviser for 30 years. Of course, he was the little columnist who turned up for work on the train and wore a nice hat but old jacket, because he could only be happy investing in things he didn't know anyone on the board of ...

[sherryillk.livejournal.com]

Try this: http://www.mathhelpforum.com/math-help/business-math/60410-i-have-question-annuity-calculation.html

It seems to be along the lines you are looking for... But it seems to yield a pretty bad-ass looking equation that makes me want to run for the hills... >

[blamebrampton.livejournal.com]

Cheers dear! Alas, it does not work for me when m>c :-(

[sherryillk.livejournal.com]

But the equation states that it will work when m > c, m < c and m = c, just not when it's compounded continuously...

Anyway, I went and tried both scenarios and I got $196,083.16 when it's compounded once annually and $184,053.28 when it's compounded monthly. I tried to be as accurate as I can but it's hard when the only calculator you have is your computer one... >< It really made me wish I had my graphing calculator but alas, it's been lent out to my brother...

[blamebrampton.livejournal.com]

I've lost track among all the calculations tonight, but that sounds familiar, except monthly must be > than annually, so it can't be right :-(

It's too late, I am off to write fic!

[sherryillk.livejournal.com]

Whoops, went and rechecked everything. That was actually my mistake. I accidentally mixed the 9 and 6 when I copied it from my handwritten math... But it did make me check all my math again, and new answer are $173,563.05 for annually. The monthly number remains unchanged though.

[shocolate.livejournal.com]

*buckles under the pressure*

having a maths degree doesn't seem to help in the Real World...

[blamebrampton.livejournal.com]

HEE! Well, to be honest, the only thing degrees in English and Philosophy are good for is coming up with convincing explanations for why you're late for work ...

[oceaxe.livejournal.com]

Agreed. Though in my case, it's theater and philosophy.

[blamebrampton.livejournal.com]

You would have the skills to be able to avoid giggling during said explanations ;-)

[oceaxe.livejournal.com]

Indeed. I have used "My basement flooded," "I had a terrible nosebleed," "I had a 'female' medical emergency," "I ran over a cat," and many others. The 'female' medical emergency is the best though, because they will literally NEVER inquire about that.

[phoenixacid.livejournal.com]

If I remember it correctly,

By year:

Firstly, we calculate the amt of savings in a year = $50 X 4 weeks X 12 months = $2400

Total savings at the end of 1st yr
= 2400 + (9.66/100)(2400)
= 2400(1.0966)

Total savings at the end of 2nd yr
= (2400 + 2400(1.0966))(1.0966)
= 2400 (1.0966 + 1.0966^2)

Total savings at the end of 3nd yr
= (2400 + 2400(1.0966) + 2400(1.0966)^2)(1.0966)
= 2400 (1.0966 + 1.0966^2 + 1.0966^3)
.
.
.

Total savings at the end of 21st yr
= 2400 (1.0966 + 1.0966^2 + 1.0966^3 + ... + 1.0966^21)
= 2400 (1.0966 [1.0966^21 -1]/[1.0966 -1]
= 161682.8889

(using the summation formula of geometric progression, Sn = a(1-r^n)/(1-r))

Similar for month...

I have no idea how accurate this is. It's what I learn in secondary school (and I'm now doing applied math and not finance lol)

[phoenixacid.livejournal.com]

Oh, it's Sn = a(r^n-1)/(r-1)) if |r| >= 1, that's why we use it the other way around....

[blamebrampton.livejournal.com]

For the annual, yes. For the monthly, sadly no, because it's compounded monthly on a principle increased weekly. AND I CAN'T DO EEEET!

*Weeps at yet another area of stupidity ...*

[phoenixacid.livejournal.com]

It should be the same?

amt of savings in a month = $50 X 4 weeks = $200

Total savings at the end of 1st month
= 200 + (9.66/100)(200)
= 200(1.0966)

... and so on for 21X12 months?

Or did I get your question completely wrong?

Compounded monthly on a principle increased weekly... erm, sorry does it mean you +50 weekly and the interest per month is 9.96%?

(this is what happens when you learn high school maths in Malay)

[blamebrampton.livejournal.com]

+50 weekly, interest will be 0.83% per month 9.96/12, but then there aren't always 4 weeks in a month and Argh! It's nearly midnight, I'll have coffee and handle it in the morning. When I am sure it will be magically easy!

[phoenixacid.livejournal.com]

Lol, good luck, hon. I'm sorry I can't be more help; it's nearly bedtime for me too. :) Have a good rest alright!

[phoenixacid.livejournal.com]

BTW is the answer 163840.8696 using your online calculator? (I used the formula I've given above)

[blamebrampton.livejournal.com]

No, but I think that if we factor in the extra weeks, it would be quite close ;-)

[http://users.livejournal.com/_inbetween_/]

Since this sounds exactly like the deal the bank pestered me about, I'd like to add that if then withdrawn at once and not paid monthly till your death, it won't be 10 percent but much less plus more taxes. I'd be happy if I'm mistaken about the deal you're considering, and hope you do live to see 100 and make them pay till then, hah.

[blamebrampton.livejournal.com]

Heh! Well, people in my family often do live over 100 ... Nah, it's for a magazine article, purely hypothetical :-)

[http://users.livejournal.com/_inbetween_/]

I'm glad to hear that, or I'd have regretted my lost ties with OZ even more! :)

[annes-stuff.livejournal.com]

When do you need it?

I might be able to work it out using Excel.

[blamebrampton.livejournal.com]

Ta luv ;-) Mr B has offered to do it if he stops killing aliens before 1. *Gives him A Look*

[annes-stuff.livejournal.com]

Ah yes, given what he does for a living he would be the logical choice.

All the same if he can't pull himself away from the spaceships let me know.

[arcanetrivia]

What I wanna know is where the heck you are getting an interest rate that fantastic, since my best account right now is slightly above 2%.

[blamebrampton.livejournal.com]

All hypothetical. Though if you want something good at the moment, see if you can invest in the Australian Stock Exchange, it's trending nicely. Buy in after a US-led dip ;-)

[blamebrampton.livejournal.com]

NB I am an editor. Speak to a trusted financial adviser. Then ask to see their records for the last 10 years and if they Goldman Sachsed it, run like the wind ;-)

[alethiaxx.livejournal.com]

I'm fairly certain you just combine the formulas for capital accumulation and future value of a series:
Balance = P(1+i)^n + c/i[(1+i)^n-1]
P=principal, i=interest/period, n=# periods, c=contribution/period

You've probably figured it all out by now, though :)

[mabonwitch.livejournal.com]

Er...my sole contribution is that my work calculates weeks per month as 4.3. Good luck!

[uminohikari.livejournal.com]

The only problem is that months and weeks don't match up exactly D: So unless you just use brute force, you'd end up with pretty crazy equations..