Question

Annuity problem help :(

How much money would have to be deposited every quarter, beginning 3 months from now, if a company wanted to have $75,000 at the end of 3 years? Assume the interest rate is 11% per year, compounded continuously.

Answer 1

At the rate of r=11% compounded continuously, the deposited amount, P would be increased by a factor of \(e^{rt}\) for every period of t.
For t=1/4 year = 3 months, and r=11%, the factor becomes
\(R=e^{0.11/4}=e^{0.0275}=1.0278816\) approximately.
If an amount P is deposited 3 months from now, the amount at the end of three years will be \(PR^{11}\).
Similarly, if P is deposited 6 months from now, the amount at the end of three years will be \(PR^{10}\).
Considering all 12 deposits (including the last payment at the end of 3 years), the total amount is
\(75000=P(R^{11}+R^{10}+R^{9}+...+R^{2}R^{1}+R^{0})\)
Factoring the right hand side,
\(75000=P\frac{R^{12}-1}{R-1}\)
Solve the equation to get the quarterly payment P.

Answer 2

thank you and sorry I forgot to respond properly :)

Answer 3

No problem! You're welcome! :)