V$.$ Accumulating Money

Joseph reads a lot about people who are success oriented. He loves to learn about

courage, risk taking, and as he describes it$,$ "the road less traveled." His local bookstore

has a large business section where he has found biographies of entrepreneurs and

maverick corporate leaders. He also finds fascinating some of the books he has seen on

financial planning and ways to accumulate wealth. One interesting savings plan he read

about challenges the reader to put aside one full paycheck at the end of the year as a

"holiday present to yourself." Joseph had never thought about saving in that way, and

wondered if it would really accumulate much savings.

$($a$)$ He decided to test the numbers by seeing how much money he would

accumulate by a retirement age of $65$ if he put one paycheck away at the end of

each year. Right now, that would mean depositing $$1,000$ at year$-$end for the

next $35$ years. Assuming he makes one yearly deposit of $$1,000$ at $5\%$

compounded annually, how much interest would he earn?

$($b$)$ Joseph was surprised at how large the sum would be and then realized that he

would be able to put more money away in future years because most likely, his

salary would go up$.$ He also thought that he could invest the money over the

long term at a higher interest rate, so he redid the calculations with a $$1,500$

annual year$-$end deposit, at $8\%$ for $35$ years. What was his result?

$($c$)$ Joseph was amazed at how much he could save in this manner and decided to

design a detailed savings plan based on projected yearly increases. He realized

that he could not start depositing $$1,500$ now, but that he would be able to

deposit more than that in the future. If he were able to deposit $$1,000$ at the

end of each year for the next $5$ years at $8\%$ compounded annually, $$1,500$ at the

end of years $6-10$ at $8\%$ compounded annually, and $$2,000$ at the end of years

$11-35$ at $5\%$ compounded annually, how much would he accumulate at the end

of $35$ years? Assume that any balances from earlier depositing periods would

continue to earn the same rate of annual interest.

$($d$)$ By how much does the result differ from the amount calculated above for $$1,500$

deposited for $35$ years? What accounts for the difference?

$($e$)$ If Joseph decided that he wanted to have $$300,000$ accumulated in $30$ years by

making an annual payment at the end of each year that would earn $12\%$

compounded annually, what would his sinking fund payment be$?$ Use the

appropriate table to determine the answer.