Sandra Field From: US Posts: 10 Votes: 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. |

Anonymous From: - 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. |

DrProdigy From: US Posts: 13 Votes: 26 |
The formula for continuous compound interest is: Y=Y0ert where Y is the amount obtained after t years at a certain r rate,and Y0 is the amount that was present originally. Therefore, to know the amount that this guy would have when he is60 we need to find what to plug in t, for t when he is60 t=60-25=35, and for 65 t=65-25=40. the value ris the percent and it's to be divided by a 100, thereforer=5/100=0.05 in both cases and your equations would look likethis: a) Y=2500e0.05*35=14368.50669 b) Y=2500e0.05*40=18472.64025 Hope it helps, if you want to know how that formula is derivatedlet me know... |

MythJoe9 From: US Posts: 57 Votes: 114 |
At age 25 t=0 initial time m(t) = m0 ert m(0) = 2500 r= 5% if age is 60 ==> t=60-25=35 m(t) = 2500 e0.05*t m(35) =2500 e0.05*(35) = 14386.5 $ AT AGE 65 t= 65-25= 40 m(40) = 2500 e0.05*(40) =18472.6 $ |

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