Annuity and future equivalent values

Question

I need some clarification on the formulas to use for these questions.

Q1. If $\$30,000$ is deposited now into a savings account that earns $7\%$ per year, what uniform annual amount could be withdrawn at the end of each year for 10 years so that nothing would be left in the account after the 10th withdrawal?

$A = P\,i\frac{(1+i)^N}{(1+i)^N-1}$ is the formula I assume I need to use here right? since we're given the present value, interest and years and need to find the annuity.

Q2. A future amount of $\$100,000$ is to be accumulated through annual payment, $A$, over 20 years. The last payment of $A$ occurs simultaneously with the future amount at the end of year 20. If the interest rate is $10\%$ per year, what is the value of $A$?

Same for this but I need to use the future value equation for annuity right?.

Answer 1

1. Without withdrawals, your initial capital $P=10,000$ at $i=10\%$ will produce in $n=10$ years the sum $S=P(1+i)^{10}= 19,671.51$. The stream of withdrawals $A$ have to produce the same amount in $10$ years. Or alternatively, the stream of withdrawals $A$ have to produce the same present value $P$. So you have, using present value, $$ P=Aa_{\overline n |i}=A \frac{1-(1+i)^{-n}}{i}\quad\Longrightarrow A=\frac{P}{a_{\overline n |i}}=1,423.78 $$ or, using future value, $$ P(1+i)^{n}=As_{\overline n |i}=A \frac{(1+i)^{n}-1}{i}\quad\Longrightarrow A=\frac{P(1+i)^{n}}{s_{\overline n |i}}=1,423.78 $$
2. The future value $S=100,000$ is obtained by a stream of payments $A$ in $n=20$ years at $i=10\%$. So we have $$ S=As_{\overline n |i}=A \frac{(1+i)^{n}-1}{i}\quad\Longrightarrow A=\frac{S}{s_{\overline n |i}}=1,745.96 $$