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. To determine how much will be in the account when the individual retires at age 65 if the savings program starts when the person is age 40, we can use the future value of an annuity formula:
FV = P * [(1 + r)^n - 1] / r
where FV is the future value of the account, P is the annual deposit amount, r is the interest rate, and n is the number of years.
In this case, P = $3,000, r = 0.08, and n = 65 - 40 = 25 years.
FV = $3,000 * [(1 + 0.08)^25 - 1] / 0.08
FV ≈ $3,000 * [8.1452] ≈ $244,356
So there will be approximately $244,356 in the account when the individual retires at the age of 65.
b. To find out how much additional money will be in the account if the saver defers retirement until age 70 and continues the contributions, we can recalculate the future value of the account with n = 70 - 40 = 30 years.
FV = $3,000 * [(1 + 0.08)^30 - 1] / 0.08
FV ≈ $3,000 * [10.819] ≈ $324,573
The additional money in the account is the difference between the future values when retiring at age 70 and age 65:
Additional money = $324,573 - $244,356 ≈ $80,217
So, there will be approximately $80,217 additional money in the account if the saver defers retirement until age 70 and continues the contributions.
c. To find out 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, we need to calculate the future value of the account at age 65 and then calculate the future value of that amount after five more years without contributions.
First, we already calculated the future value at age 65 in part a, which is $244,356.
Now, we can calculate the future value of that amount after 5 more years without contributions using the compound interest formula:
FV = P * (1 + r)^n
In this case, P = $244,356, r = 0.08, and n = 5 years.
FV ≈ $244,356 * (1 + 0.08)^5 ≈ $358,811
The additional money in the account is the difference between the future values when retiring at age 70 without contributions and age 65:
Additional money = $358,811 - $244,356 ≈ $114,455
So, there will be approximately $114,455 additional money in the account if the saver discontinues the contributions at age 65 but does not retire until age 70.
a. To calculate how much will be in the account when the individual retires at the age of 65, we need to determine the future value of an annuity.
The formula to calculate the future value of an annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value of the annuity
P = Annual deposit
r = Interest rate per period
n = Number of periods
In this case, P = $3,000, r = 8% or 0.08, and n = (65 - 40) = 25 years.
Plugging in the values, we get:
FV = $3,000 * [(1 + 0.08)^25 - 1] / 0.08 = $201,796.34
So, the amount in the account when the individual retires at 65 will be approximately $201,796.34.
b. If the saver defers retirement until age 70 and continues the contributions, we need to calculate the future value of the annuity for an additional 5 years, from age 66 to 70.
Using the same formula as above, with n = 5 years, we get:
FV = $3,000 * [(1 + 0.08)^5 - 1] / 0.08 = $16,977.18
Therefore, there will be an additional $16,977.18 in the account if the saver defers retirement until age 70 and continues the contributions.
c. If the saver discontinues the contributions at age 65 but does not retire until age 70, we need to calculate the future value of the existing account balance for the additional 5 years, from age 65 to 70.
Using the formula for future value of a single sum:
FV = PV * (1 + r)^n
Where:
FV = Future value
PV = Present value
r = Interest rate per period
n = Number of periods
In this case, PV = $201,796.34 (calculated in part a), r = 8% or 0.08, and n = 5 years.
Plugging in the values, we get:
FV = $201,796.34 * (1 + 0.08)^5 = $289,685.68
Therefore, there will be an additional $289,685.68 in the account if the saver discontinues the contributions at age 65 but does not retire until age 70.
To calculate the future value of the retirement account, we can use the formula for compound interest:
FV = P * (1 + r)^n
where:
FV = Future Value
P = Principal amount (annual deposit)
r = Annual interest rate
n = Number of compounding periods (years)
a. To calculate the future value when the savings program starts at age 40 and the person retires at age 65:
We have:
P = $3,000
r = 8% = 0.08
n = 65 - 40 = 25 years
Substituting these values into the formula, we get:
FV = $3,000 * (1 + 0.08)^25
Calculating this on a calculator or spreadsheet gives us:
FV = $3,000 * (1.08)^25
FV ≈ $14,060.43
So, the account will have approximately $14,060.43 when the individual retires at age 65.
b. To calculate the additional money when deferring retirement until age 70 and continuing contributions:
In this case, the number of compounding periods becomes 25 + 5 = 30 years (from age 40 to 70).
Using the same formula as before, we have:
FV = $3,000 * (1 + 0.08)^30
Calculating this, we get:
FV ≈ $3,000 * (1.08)^30
FV ≈ $24,692.03
So, if the saver defers retirement until age 70 and continues the contributions, there will be approximately $24,692.03 in the account.
c. To calculate the additional money if the saver discontinues contributions at age 65 but does not retire until age 70:
In this case, there will be no additional annual contributions from age 65 to 70.
The number of compounding periods becomes 65 - 40 = 25 years (from age 40 to 65).
Using the same formula:
FV = $3,000 * (1 + 0.08)^25
Calculating this, we get:
FV ≈ $3,000 * (1.08)^25
FV ≈ $14,060.43
So, if the saver discontinues contributions at age 65 but does not retire until age 70, the account will have approximately $14,060.43.