Answer with Explanations:
Given:
Simple bank pays 8.6 percent simple interest.
Compound bank pays x percent interest compounded annually.
Assume an equal amount deposited in an account of each bank.
Find x such that at the end of 8 year the future values are equal.
Solution:
Assume an initial deposit of P in either account.
We calculate the future value after 8 years.
Simple bank:
future value, Fs = P(1+8*0.086) = 1.688P (8.6% simple interest)
Compound bank:
future value, Fc = P(1+x)^8 ( x% compounded annually)
In order that both future values are the same, we equate Fc and Fs
Fc = Fs
P(1+x)^8 = 1.688P
simplify
(1+x)^8 = 1.688
Take 8th root on both sides
(1+x)^(8*(1/8)) = 1.688^(1/8)
simplify
(1+x)^(1) = 1.688^(1/8)
take 8th root on right and simplify
1+x = 1.0676319
x = 1.0676319-1 = 0.0676319
Therefore the Compound Bank would offer an interest of 6.76%, rounded to 2 decimal places.
The rate that the bank should set if it wants to match First Simple Bank over an investment horizon of 8 years is 6.76%.
First step is to calculate the total interest using this formula
Total interest=Interest rate per period× Number of periods
Let plug in the formula
Total interest=.086×8
Total interest= .688
Now let calculate the bank rate using this formula
(1 + r)^n - 1
Set the two equal
(.086)(8) = (1 + r)^8 - 1
Hence:
r=(1+.688)^(1/8) - 1
r = 1.688^(1/8) - 1
r = .0676×100
r= 6.76%
Inconclusion the rate that the bank should set if it wants to match First Simple Bank over an investment horizon of 8 years is 6.76%.
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