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Thread: Maximizing Profit

20080928, 08:02 #1
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Maximizing Profit
Hi! I would really appreciate some help with this math problem. I have no idea how to start...
The amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on the money. The bank can reinvest this money at 12%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)
Thanks!

20080928, 08:59 #2
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Re: Maximizing Profit
0%, of course. Unless it's in a US bank's money market. Then they can pay less. <img src=/S/grin.gif border=0 alt=grin width=15 height=15>
<IMG SRC=http://www.wopr.com/w3tuserpics/DocWatson_sig.gif>

20080928, 09:08 #3
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Re: Maximizing Profit
Doc,
I think you missed the point, if the bank pays 0% then no money gets invested. If they pay 8% then they get twice as much invested as if they pay 4% etc.
StuartR

20080928, 09:26 #4
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Re: Maximizing Profit
Are you sure that we should do your homework for you?
I don't think we have enough information to solve the problem, but here is a simplistic approach:
<span style="backgroundcolor: #FFFF00; color: #FFFF00; fontweight: bold">Let the interest rate paid by the bank be n%.
The profit of the bank is 12%n% of the amount deposited.
The amount is c·n² where c is some constant.
The total profit is (12n)%·c·n² = (c/100)·(12·n²·n³)
To find the maximum, we must calculate the derivative of this function and solve for 0.
The derivative is (c/100)·(24·n3·n²) = (3·c/100)·n·(8n)
This is zero if n=0 (but this clearly corresponds to a minimum) or n=8.
So the optimal interest rate for the bank is 8%.</span hide>

20080928, 09:41 #5
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Re: Maximizing Profit
You obviously don't live in the US these days. We just learned last week that the money we have in "safe" investments like money markets might be at risk. But we have been "encouraged" not to take it out and stuff it under our mattresses. But, I was just being glib.
My other answer would have been the same as Hans', but without the formula. <img src=/S/smile.gif border=0 alt=smile width=15 height=15><IMG SRC=http://www.wopr.com/w3tuserpics/DocWatson_sig.gif>

20080928, 10:04 #6
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Re: Maximizing Profit
Doc, this is a real worry for us as well due to the amount you are compensated if the bank you use goes belly up. In the UK, you are only compensated to a maximum of £35,000 pounds (each if a joint account). So you are stuffed if you have more than that. Some people thought they would be clever and I have heard they split their accounts if they had more than £35K but apparently this is per bank not per account and as some banks are subsidiary you still lose out.
It is a very controvertial subject so I will stop my little rant now <img src=/S/grin.gif border=0 alt=grin width=15 height=15>Jerry

20080928, 10:08 #7
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Re: Maximizing Profit
My goodness Hans, all that sophisticated maths just to get the same 8% as I had already given.
StuartR

20080928, 10:20 #8
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Re: Maximizing Profit
Isn't that what maths is all about? In the end, after all those calculations, it always amounts to X = X <img src=/S/grin.gif border=0 alt=grin width=15 height=15>

20080928, 11:38 #9
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Re: Maximizing Profit
We get $100,000 per bank, per person federally insured. But there are no regulations about 0% or negative interest (where the shares of your money market account are worth less than the $1 each you were told they would always be worth when you deposited the money in the account).
World economies are scary these days. <img src=/S/shocked.gif border=0 alt=shocked width=15 height=15><IMG SRC=http://www.wopr.com/w3tuserpics/DocWatson_sig.gif>

20080928, 13:49 #10
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Re: Maximizing Profit
<hr>If they pay 8% then they get twice as much invested as if they pay 4% etc<hr>
This is wrong. The question said that "amount of money deposited in a bank is proportional to the square of the interest rate", so if the rate doubles, the amount deposited is four times as much.Regards
John

20080928, 17:19 #11
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Re: Maximizing Profit
I can understand you having trouble with this. The wording is both confusing and confounding! Ask whoever set the problem to restate it in selfexplanatory wording.
Alan