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?

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

To calculate the amount in the retirement account under different scenarios, we need to use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = Amount in the account
P = Initial deposit
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years

Given:
P = $3,000
r = 8% = 0.08
n = 1 (Assuming interest is compounded annually)

a) Calculation when the savings program starts at age 40 and retirement is at age 65 (25 years):
t = 25 years

A = 3000(1 + 0.08/1)^(1*25)
A = 3000(1.08)^25
A ≈ $12,014.41

So, when the individual retires at age 65, the account will have approximately $12,014.41.

b) Calculation when the saver defers retirement until age 70 and continues contributions (30 years):
t = 30 years

A = (3000(1 + 0.08/1)^(1*25))(1 + 0.08/1)^(1*5)
A = 3000(1.08)^25(1.08)^5
A ≈ $38,629.31

The additional money in the account if the saver defers retirement until age 70 and continues contributions is approximately $38,629.31.

c) Calculation when contributions stop at age 65 but retirement is at age 70 (5 years):
t = 5 years

A = 3000(1 + 0.08/1)^(1*25)
A ≈ $12,014.41

The additional money in the account if the saver discontinues contributions at age 65 but does not retire until age 70 is also approximately $12,014.41.

To calculate the account balance at various scenarios, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = Final account balance
P = Initial deposit
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years

Now let's calculate the answers to the given questions:

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?

To calculate the account balance at age 65, we need to determine the number of years the savings program has been running. Since the person starts at age 40 and retires at age 65, the number of years is:

t = 65 - 40 = 25

Using the formula mentioned above, with the given values:

P = $3,000
r = 8% = 0.08
n = 1 (interest is compounded annually)
t = 25

A = $3,000(1 + 0.08/1)^(1 * 25)

Calculating this, we get:

A = $3,000(1.08)^25 = $16,633.17

So the account balance when the individual retires at the age of 65 will be approximately $16,633.17.

b) How much additional money will be in the account if the saver defers retirement until age 70 and continues the contributions?

To calculate the additional money in the account if retirement is deferred until age 70, we need to calculate the account balance at that age. Considering the person has been contributing for 30 years (= 70 - 40), the number of years is:

t = 70 - 40 = 30

Using the same formula, with the given values:

P = $3,000
r = 8% = 0.08
n = 1 (interest is compounded annually)
t = 30

A = $3,000(1 + 0.08/1)^(1 * 30)

Calculating this, we get:

A = $3,000(1.08)^30 = $39,800.30

So the account balance when the individual retires at age 70 with continued contributions will be approximately $39,800.30.

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?

To calculate the additional money in the account if contributions are discontinued at age 65, we need to calculate the account balance at age 70 but consider only the contributions until age 65. The number of years contributions were made is:

t = 65 - 40 = 25

Using the same formula, with the given values:

P = $3,000
r = 8% = 0.08
n = 1 (interest is compounded annually)
t = 25

A = $3,000(1 + 0.08/1)^(1 * 25)

Calculating this, we get:

A = $3,000(1.08)^25 = $16,633.17

So the account balance when the individual does not contribute beyond age 65 but retires at age 70 will still be approximately $16,633.17.