You want to be able to withdraw 35000 from your account each

You want to be able to withdraw $35,000 from your account each year for 25 years after you retire. If you expect to retire in 30 years and your account earns 7.4% interest while saving for retirement and 6.5% interest while retired:
Round your answers to the nearest cent as needed.

a) How much will you need to have when you retire?
$

b) How much will you need to deposit each month until retirement to achieve your retirement goals?
$

c) How much did you deposit into you retirement account?
$

d) How much did you receive in payments during retirement?
$

e) How much of the money you received was interest?

Solution

(a)   The formula for annuity payments is p = P*[r(1+r)ⁿ ] / [ (1 +r)ⁿ -1] where p is the annuity payment, P is the principal amount, r is the period interest rate in decimals, and n is the total number of payments. Here, p = $ 35000,n=25,r=6.5/100=0.065.We have 35000=P[0.065(1+0.065)²⁵ ]/ [ ( 1+ 0.065)²⁵ -1] or, 0.065 *P(4.827699)/ (3.827699) = 35000 or, P = 35000(3.827699)/[ 0.065(4.827699)] = 133969.465/ 0.31380 = $ 426926.27

(b)   The formula for the maturity value of an annuity is F = p [{(1 + r)ⁿ – 1} / r],where F is the future value of the annuity stream to be paid in the future, p is the amount of each annuity payment, r is the interest rate per term in decimals and n is the number of periods over which payments are made. Here, F = $ 426926.27, r is 7.4/1200= 74/12000 = 37/6000, n = 30*12 = 360. Then, 426926.27 = p[ ( 1+ 37/6000)³⁶⁰ -1 ] / ( 37/6000) or, 426926.27 = p( 9.14478 -1 )/( 37/6000) = p*(8.14478)*6000/37 =or, p = (426926.27)*37/ [(8.14478)*6000 ] = 15796271.99/ 48868.68 = $ 323.24

(c)   The amount deposited into the retirement account is $ 323.24*360= $ 116366.10

(d)   The amount received is $35000*25 = $ 875000. Therefore, the amount of interest received =             $ 875000 - $ 116366.10 = $ 758633.90