I am trying to see if I have completed this problem I am working on correctly. I think I have, but I am not a finance person and my tutor is unavailable this week. I am trying to finish this problem for a business class. Have I done this correctly?

Thanks in advance.
Renee' Scruggs



Problem & Answer
3. Sue Sussman started a paper route on January 1, 1998. Every three months, she deposits $500 in her bank account, which earns 4 percent annually but is compounded quarterly. On December 31, 2001, she used the entire balance in her bank account to invest in a contract that pays 9 percent annually. How much will she have on December 31, 2004?

Answer
Sue will have $12, 780 on December 31, 2004.

Calculations
1998-2001
Quarterly earnings = principal x quarter payment + 4%
= $500 x 1 + 4%
= $500 + $20
= $520
Annual earnings = $520 x amount of periods/quarters
= $520 x 4
= $2080

FV (2001) = $2080 X amount of periods/years
= $2080 x 3
= $6240

2001-2004
Quarterly earnings = principal x quarter payment + 4%
= $500 x 1 + 9%
= $500 + $45
= $545

Annual earnings = $545 x amount of periods/quarters
= $545 x 4
= $2180

FV (2001) = $2180 X amount of periods/years
= $2180 x 3
= $6540

Total value = 1st period (1998-2001) + 2nd period (2002-2004)
= $6240 + 6540
= $12, 780

I don't think this is correct. First, since interest is compounded quarterly each contribution of $500 grows by a factor of 1.01^q, where q is the number of quarters the money sits from the time she deposits it until Dec 31, 2001. Thus the $500 she deposits on March 31, 1998 has 15 quarters of interest earned to Dec 31, 2001, and so it grows to 500*1.01^15. Similarly, the monet deposited on June 30 1998 grows to 50*1.01^14. And so on. Thus as of Dec 31, 201 she has:

500*(1.01^15 + 1.01^14 + 1.01^13 +... +1.01^2 + 1.01 +1).

The last 1 is because she deposits $500 on Dec 31, 2001, and so it doesn't grow at all. Think of it as 1.01^0.

She then takes this amount and gets 9% annually for 3 years, so multiply the above by 1.09^3. As I interpret the question she no longer is making $500 deposits after Dec 31 2001.

From this I get an answer of $11,174.72