Mary said there is no difference between an 8% annual interest rate compounded quaterly and annual interest rate. Explain to mary why she is wrong. with examples
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If an 8% annual interest rate is compounded quarterly, the rate per quarter is 8/4 = 2%.
So an amount $100, in a year would become $100*( 1 + .02)^4
=> $100* 1.02^4
=> $100* 1.0824
=> $108.24
If the compounding is done annually the amount would increase by 8% or become $100*(1 + .08)
=> $108.
So, there is a difference of 24 cents in the two cases.
Even if we take the notional annual interest rate to be the same, as the period of compounding decreases (3 months instead of 12 months), the amount compounded increases after a year.
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To find the difference between (i) 8% annual interest compounded quarterly and (ii)8 % interest annually.
Let Mary invests $10000 with 8% interest quarterly.
So at at each quarter she gets a quarterly compound interest of (8/4)% = 2% or 0.02 per dollar.
The amount A after the 4 quarters is given by A = Principle * (1+r/4)^4, where r = 8% by the formula for compound interest with interest.
So at the end of the year $10000 becomes with interest $10000(1+0.2)^4 = $10824.32.
I Mary invests the principle P = $10000 in annual interest 8%, then the amount A with interest at the end of the year = A = P(1+r)^1 = $10000(1+8%)^1 = $10000(1+ 8/100) = $100001.08 = $10800.
The difference Mary should know is that she earns a higher amount ($10824.32-10800) = $24.32 if she invests in a quarterly compounding system.
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