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        riddles >> easy >> Interesting interest
        
(Message started by: Noke Lieu on Mar 10^th, 2009, 9:18pm)

Title: Interesting interest
Post by Noke Lieu on Mar 10^th, 2009, 9:18pm

A new bank has opened in Cuckooland, where interest is calculated and added daily.

It lets you name your compound interest rate (under 20%) for deposits. The only catch is that it'll offer you double that rate to take it as simple interest instead.

As expected, Simple interest starts off winning.
After how many days does it pay better to have chosen Compound interest?

Title: Re: Interesting
Post by towr on Mar 11^th, 2009, 2:23am

The interest rate is per day? Then [hide]after ~7.7 days compound is better than simple. Unless you withdraw your deposit each day and then (have a friend) deposit it again :P[/hide]

Title: Re: Interesting
Post by Immanuel_Bonfils on Mar 11^th, 2009, 5:43pm

Seems the rate is less than 20% and not 20%... (I don't know why)

Title: Re: Interesting
Post by Noke Lieu on Mar 11^th, 2009, 6:14pm

Towr... I have a different answer.
Either I've completely lost the plot (given the events of this morning, entirely plausible), or what I was expecting to ask wasn't quite what you've answered.

Even if I plug in to Excel (oh dear)
100 . 100
=a1*1.1 . 120
=a2*1.1 . 140

at day 15 it's 379.75 vs 380....

Title: Re: Interesting
Post by towr on Mar 12^th, 2009, 2:04am

on 03/11/09 at 18:14:41, Noke Lieu wrote:

Even if I plug in to Excel (oh dear)
100 . 100
=a1*1.1 . 120
=a2*1.1 . 140

I have no idea what that means. *is total excel newb*

I did 1.20ⁿ vs 1+0.40*n, since in the limit 20 is the highest interest rate you can get, and a higher interest rate is always better to get than a lower one.
n=7 : 3.5832 vs 3.8
n=8 : 4.2998 vs 4.2

Title: Re: Interesting
Post by Noke Lieu on Mar 12^th, 2009, 3:59pm

well, yeah. That's what that does, except it's

(1.1)ⁿ vs (1.2)n

Yes, for taking 20%, 7.7 days is what I get too.
But that's only for 20%.
at 10%, it's much longer.

Why not takethe 20%? who knows. It's cuckooland, after all.
perhaps somewhere in the fine print, it says we take as many fingers as you claim in percent interest...

Title: Re: Interesting
Post by towr on Mar 12^th, 2009, 4:14pm

on 03/12/09 at 15:59:51, Noke Lieu wrote:

Yes, for taking 20%, 7.7 days is what I get too.
But that's only for 20%.
at 10%, it's much longer.

At 1% it is much longer still. And at 0.1% even longer. There isn't really a limit to it. :-/

Title: Re: Interesting
Post by Noke Lieu on Mar 12^th, 2009, 5:26pm

hence, what I was after was the solution to
(1+I/100)ⁿ = (1 + 2I/100)n, giving n in terms of I, or somesuch.

as I approches 0, compound interest wins out as Iⁿ approaches...?

I was trying to dust off how to use logs, having not done them formally in years and years. and failing.

Title: Re: Interesting
Post by towr on Mar 13^th, 2009, 1:06am

on 03/12/09 at 17:26:11, Noke Lieu wrote:

I was trying to dust off how to use logs, having not done them formally in years and years. and failing.

On this problem, that's hardly a surprise. You need the Lambert W function (http://en.wikipedia.org/wiki/Lambert_W_function), rather than logs.
You can get decent approximations without it, but even that goes beyond mere logarithms. Such as perhaps using a Taylor expansion to approximate (1+I/100)ⁿ for small I.