$Compound Interest question$

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Erin deposits $10,000 in an account that pays 4% interest compounded quarterly.

(a) Compute the balance of your bank account value after one year.
(b) Determine the effective annual yield.

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Let $P_0$ be the initial balance and $I$ be the yearly interest rate compounded n times during the year. Then the future value after $m$ time steps is computed by the formula
\[P(m)=P_0(1+\frac{i}{n})^m.\]
Here $I=4\%$, $n=4$, and $P_0=\$10,000$. Hence the future value after m quarters is
\[P(m)=10000(1+\frac{4\%}{4})^m \Rightarrow P(m)=10000(1+\frac{1}{100})^m=10000(1.01)^m. \]
(a) After one year $m=4$ (after 4 quarters)
\[P(4)=10000(1.01)^4=\$10,406.04.\]
(b) To compute the effective annual rate divide the interest you receive after 1 year by the total amount, i.e.
\[\frac{10,406.04-10,000}{10000}=.04060401=4.060401\%.\]

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$Compound Interest question$

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