A. after 25 yeares, how much more will you have from the lump-sum investment than from the annuity?

B. after 25years, how much more interest will be earned from the lump-sum investment than from the annuity?





LUMP-SUM DEPOSIT: $40,000, RATE: 6.5% compounded anually, TIME: 25 years





PERIODIC DEPOSIT: $1600 at the end of each year, RATE, 6.5% componded annually, TIME: 25 years |||For the lump sum deposit, the value after 25 years at 6.5% compounded annually should simply be:





Value = ($40,000)*(1+6.5/100)^25 = $193,107.96





The interest earned would simply be the value minus the original deposit:





Interest earned = $153,107.96








For a periodic deposit, I always forget the formula so I try to derive it from first principles:





Interest for each deposit will start accruing only after it is deposited. The interest for each deposit would use the same formula as above except the number of years is different:





Interest for single deposit = ($1600)*(1+6.5/100)^n





After 25 years the interest for the 1st deposit would be:





Interest for 1st deposit = ($1600)*(1.065)^(25-2), since you wait 1 year before depositing and you have to wait another year to earn the interest.





The interest for the 2nd deposit would be:





Interest for 2nd deposit = ($1600)*(1.065)^(25-3), and so on for the remaining deposits





The generic term is:





Interest for nth deposit = ($1600)*(1.065)*^(24-n)





Note that since you wait 1 year before starting to make deposits you only make 24 deposits, and the last deposit (at the end of the 24th year, doesn't make any interest)





This forms a series of 24 items that must be summed to get the final answer;





Value = sum (from 0 to 24 of) ($1600)*(1.065)^(24-n)





You can evaluate this more easily if you do a trick math manuever by multiplying both the top and bottom of the right side by the term (1- 1.065). If you do this and carry out the math for the top part of the equation it simplifies due to cancellation of terms to:





Value = ($1600)*(1 - 1.065^(24))/(1 - 1.065)





= $86,967.40





with the interest earned = $86,967.40 - (24)*($1600)





= $48,567.40





So to answer your questions:





You will have $193,107.96 - $86,967.40 = $106,140.56 more from the lump sum arrangement,





and will have earned:





$153,107.96 - $48,567.40 = $104,540.56 more in interest from the lump sum arrangement.





Hope this helps