Annuity with Simple Interest for Fractions of a Year

Problem: Deposits of 500 each are made into an account on the first day of every January and July beginning on January 1, 1999. Suppose that the effective annual interest rate is i=0.04, and interest is credited only on December 31 each year, with simple interest credited for fractions of a year. On what date should the account be closed in order that the closing balance be nearest 10000?

My approach was probably pretty naive, and a bit off since I didn't seem to get exactly the right answer. For 1999, I took 500(1.04)+500(1.02)=1030 to get the balance at the end of that year. To get the balance at the end of 2000, I took 1030(1.04)+1030=2101.20, where 1030(1.04) is the accumulated value of the 1999 deposits and 1030 is the value of the new deposits in 2000. Proceeding similarly, I got a balance of 9490.65 at the end of 2006, hence 9990.65 on January 1, 2007.

I found that $\displaystyle 9990.65(1+0.04*\frac{9}{365})=10000.50$, and this is the closest one gets to 10000 using a whole number of days. So I answered January 10, 2007. But the back of the book says January 11, 2007. Can anyone see where I may have gone wrong?

Re: Annuity with Simple Interest for Fractions of a Year

Yours seems correct.

Can't answer your question: we don't know what method was used to get book's answer.

Re: Annuity with Simple Interest for Fractions of a Year

Seems like a good enough answer to my question to me. :) Thanks for looking at the problem!