Davis, Michaels, and Company

        Now that we saw option 1 of a CD, let’s look at a second alternative with five annual payments of $2,000 assuming payments are made at the end of the each year.   This would be considered an ordinary annuity.  If payments are invested in the account paying 10 percent interest annually, the future value of this annuity would be $12,210.20.  If the payments were invested with First National Bank so it would have semiannual compounding interest, the future value of this annuity would be $25,155.79.  Notice that the value at least doubled when it is compounded semiannually.  Next, we want to break it down into payments and figure out what size payment we need to make to accumulate $20,000 with 10 percent interest annually.  We found out to accumulate $20,000 annually at a 10 % nominal rate we need to make 5 equal payments of $3,275.95.  If we look back and recall that the future value of an annuity that makes payments of $2,000 at 10 % interest annually is $12,210.20, now, we want to find what lump sum, if deposited today, would produce the same stream of payments.  Through our calculations we found out that the lump sum needed today would be $7,581.57.  We are now going to suppose payments are only $1,000 each, but made every six months.  In this case, with annual compounding, future value annuity from $1,000 every six months will be equal to the future value annuity from $2,000 every year.  This happens because the periodic payment needs to be doubled so it would be $2,000 per compounding period.  However, with semiannual compounding under ordinary annuity the future value annuity every six months is greater than the future value annuity from $2,000 every year.  We calculated this amount to be $12,577.89, showing an increase of $367. 69.   We saw what an ordinary annuity has to offer so let’s move onto option 3, an annuity due.  With an annuity of $2,000 and after five payments, we found out that the total amount would be $13,431. 22.  This amount is a higher future value annuity than that of the ordinary annuity.  If we go back and try an annuity due through First National Bank and semiannual compounding, the future value annuity comes to $26,413.57.  Note, this also doubled like an ordinary annuity but it again is higher than an ordinary annuity.  When working with this annuity due and we wanted to see what the payment breakdown would be if we wanted to accumulate $20,000, we found out it would be $2,978.14.  Again, this would be better than an ordinary annuity since you would get to pay a lower payment each time and still end with the same amount of money.  We have to look back and see that the future value of an annuity due that makes payments of $2,000 at 10 % interest annually is $13,431.22, now, we want to find what lump sum, if deposited today, would produce the same stream of payments.  Through our calculations we found out the lump sum needed today would be $8,339.73.  Unlike all the other values for annuities due compared to ordinary annuities, this present value is higher than that of an ordinary annuity, which means it is not as good.  If we suppose payments are only $1,000 each, but made every six months, it would be the same as ordinary annuity.  Future value annuity from $1,000 every six months is equal to future value annuity from $2,000 every year.  This happens because the periodic payment needs to be doubled so it would be $2,000 per compounding period.  However, with semiannual compounding under annuity due the future value annuity every six months is less than the future value annuity from $2,000 every year.   We calculated this amount to be $13, 206.79, showing a decrease of $224.43.