# A self –employed person deposits \$3,000 annually in a retirement account(called a Keogh account)that earns 8 percent.

a) How much will be in the account when the individual retires at the age of 65 if the savings program starts when the person is age 40?
b) How much additional money will be in the account if the saver defers retirement until age 70 and continues the contributions?
c) How much additional money will be in the account if the saver discontinues the contributions at age 65 but does not retire until age 70?

Question ID
432420

Created
April 3, 2011 3:33pm UTC

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1. a) The person saves \$3,000 each year for 25 years (from age 40 to age 65). This is an annuity problem that can be solved using the future value of annuity formula. The future value of annuity formula is:
FV = P * [(1 + r)^t - 1] / r
where FV is the future value, P is the payment per period, r is the interest rate per period, and t is the number of periods. Plugging in the values:
P = 3,000
r = 0.08 (8% annual interest rate)
t = 25 years

FV = 3,000 * [(1 + 0.08)^25 - 1] / 0.08
FV = 3,000 * [7.03999] = \$211,199.72

So at age 65, the person will have about \$211,199.72 in the account.

b) If the person defers retirement until age 70 and continues the contributions, they will be saving for five additional years. Let's add these five additional years to the previous calculation:

t = 30 years (from age 40 to 70)

FV = 3,000 * [(1 + 0.08)^30 - 1] / 0.08
FV = 3,000 * [10.06266] = \$301,879.97

So at age 70, the person will have about \$301,879.97 in the account if they continue contributing.

The additional money due to deferred retirement and contributions is:
\$301,879.97 - \$211,199.72 = \$90,680.25

c) If the person discontinues the contributions at age 65 and does not withdraw the money, the account will continue to grow at 8% for the next five years. To calculate the future value of the account at age 70, we will use the compound interest formula:

FV = PV * (1+r)^t

where PV is the present value (\$211,199.72 from age 65), r is the interest rate, and t is the number of years. Plugging in the values:

PV = 211,199.72
r = 0.08
t = 5 years

FV = 211,199.72 * (1 + 0.08)^5
FV = 211,199.72 * 1.46933 = \$310,077.69

So at age 70, the person will have about \$310,077.69 in the account if they discontinue contributions at age 65.

The additional money due to deferred retirement without contributions is:
\$310,077.69 - \$211,199.72 = \$98,877.97