# At Age 25, Annual Deposits Of \$2500 Into An IRA,...

 Sandra FieldFrom: USPosts: 10Votes: 10 At age 25, annual deposits of \$2500 into an IRA, that pays an annual 5% interest rate compounding conyinuously, begins. Assuming paymentsre made as a continuous income flow, how much will be in the account at age 60? At age 65? Please show all work.

 AnonymousFrom: -Posts: -Votes: - The first two answers given above are NOT correct because theyassume that a single deposit of \$2,500 is made at age 25. You arebeing asked to find out how much you will have at age 60 or 65 if\$2,500 is deposited every year continuously over the years untilretirement. First we need to determine how much we would have after oneyear if we deposited \$1 today and made no more payments. Ifinterest were 5% compounded annually this amount would be just\$1.05. However, interest is being compounded continuously sothe corresponding accumulated value after a year would be: lim[(1+δ/n)n, n-> ∞] = eδ,where δ=0.05 in this problem. Actuaries call δ the "forceof interest". Note that e0.05= 1.05127, so we have a little morebecause interest is compounded continuously rather than just once ayear. Now payments are being made into the IRA continuously so weneed to find their accumulated value after n years. This isgiven by the following integral: Integral[eδ*tdt] taken over the limits of integration 0 < t < n where n isthe number of years until he reaches age 60 or 65. Do the integration which is very easy and you get(en*δ-1)/δ. Now plug in δ=0.05, letn be 35 or 40 years, and multiply the answer by 2500 because youare depositing \$2500 annually rather than just \$1. You obtain about\$237,730 to age 60 and \$319,453 at age 65. If no interest were paid(δ=0) you would have only \$87,500 to age 60 and \$100,000 toage 65. You can readily see the power of compound interest.