Alexander deposited money into his retirement account that is compounded annually at an interest rate of 7% . Alexander thought the equivalent quarterly interest rate would be 2%. Is Alexander correct? If he is, explain why. If he is not correct, state what the equivalent quarterly interest rate is and show how you got your answer.
Let the quarterly interest rate be a%
This means that each quarter his money would increase by a%.
Let P0 be his money at the beginning of the 1st quarter.
At the end of the 1st quarter, his money will increase to P1 = P0*(1 +a).
At the end of the 2nd quarter, his money will increase to P2 = P1*(1 +a) = P0*(1+a)^2.
At the end of the 3rd quarter, his money will increase to P3 = P2*(1 +a) = P0*(1+a)^3.
At the end of the 4th quarter, his money will increase to P4 = P3*(1 +a) = P0*(1+a)^4.
We are told that his money will be increased by 7% over one year. i.e. it will be P0*(1 + 7%)
So, if 2% is the equivalent quarterly rate, then P0*(1 + 0.07) = P0*(1 + 0.02)^4
i.e. 1.07 = 1.02^4
but 1.02^4 = 1.0824
So Alexander is wrong. 2% is too large to be an equivalent quarterly rate.
The true quarterly rate is given by solving the equation,
1.7 = (1 + a/100)^4
(1 + a/100)^2 = √(1.07) = 1.0344
1 + a/100= √(1.0344) = 1.01706
a = 1.7%