Optimization of interest rates

Question

I'm not sure how to set up this equation. It's an extra credit assignment. Or even how to interpret it. At first I thought the interest rate needed to be maximized to attract the most deposits, but it seems that the given equation x=100r makes that a flat rate anyway. So I guess I need to maximize the interest rate vs the 12% interest the bank gains from reinvestment? 

 

The amount of money 𝑋 deposited in a bank is related to the interest rate 𝑟
as 𝑋 = 100𝑟
The bank can reinvest the money at 12%.
What is the interest rate 𝑟 that the bank should pay to maximize profit? [Hint: the
profit is the amount of money the bank makes by reinvesting the money minus the
interest the bank has to pay for the deposited money]

Answer 1

Let P = the bank's Profit:

 

     P = Revenue - Expenses

     P = (0.12)X - X·r

     P = (0.12)(100r)- (100r)r

     P = 12r - 100r²

 

Take the derivative of P wrt r, set it to zero, and solve for r.

Answer 2

The profit the bank makes is .12X minus rX (the amount of interest that they pay). If you substitute in 100r for X, you get a formula for the profit (P)that can be graphed.

 

P = .12(100r)-r(100r)

 

P = 12r - 100r²    or    P = -100r² + 12r

 

The bank makes no profit if the rate is 0.12, (= 12%), as they pay out as much in interest as they make.

To find the maxima of this graph, you can either graph this or you can find the maxima by finding when the derivative of the function is zero.  The derivative of the function defines a line tangent to the curve. At the maxima of the graph, the tangent line will be horizontal, having a slope of zero.

 

The derivative of the graph is -200r + 12

 

Setting this equal to zero       - 200r + 12 = 0

                                                       12 = 200r

                                                       .06 = r

 

The graph has a maxima at r = 0.06, or 6% interest.