# Assistance on whether numbers are correct? Financial mathematics annuities problem

You would like to have $750,000 when you retire in 25 years. How much should you invest each quarter if you can earn a rate of 4.2% compounded quarterly? a) How much should you deposit each quarter? b) How much total money will you put into the account? c) How much total interest will you earn? You would like to have \$750,000 when you retire in 25 years. How much should you invest each quarter if you can earn a rate of 4.2% compounded quarterly?

future value = (payment)[((1+i)^n-1)/i]

750000 = P[(1.042^(4*25)-1)/0.042]

750000 = P[1433.4]

payment = \$523.23 a) How much should you deposit each quarter?:: 523.2307 b) How much total money will you put into the account?::$52,323.07

c) How much total interest will you earn?:: 750,000-52323 = 697,676.93

• Is it 4.2% interest per year or per quarter? We need to know that to give you the correct answer. – cocoahomology Dec 19 '16 at 7:12
• compounded quarterly – onlyZoe Dec 19 '16 at 18:11
• These textbook questions are almost always written this ambiguously. So you assume the given rate is an annual rate. – JPKowal Jan 13 '17 at 9:24

Before you commit to a final answer, you should use some reasoning. Is it reasonable that you only deposit \$52 323 in total but have \$750 000 at the end of 25 years with the low interest rate of 4.2% per year? If we earned 4.2% interest every 3 months, that is more than 17% interest per year (due to compounding), which does make the answer reasonable. But I don't think this is the intent of the question, since it is supposed to model a real-world retirement investment plan.
a) The formula you are using is $FV = \frac {P\left[\left (1+i\right)^n - 1\right]}{i}$, but here $i$ should be the interest rate per compounding period. For 4.2% interest per year compounded quarterly, there are 4 compounding periods, so $i=\frac{0.042}{4}=0.0105$. For a future value of \$750 000, this is:$\$750000 = \frac {P\left[\left (1.0105\right)^{100} - 1\right]}{0.0105}$, so $P = \frac{$750000}{\frac {\left (1.0105\right)^{100} - 1}{0.0105}}=\$4275.13$.
b) You make 100 deposits, and so deposit $100\times\$4275.13=\$427 513$ in total.
c) The total interest earned is $\$ 750 000 - \$427 513=\$ 322 487\$.