A self-employed person deposits $3000 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?

I know that the answers should be the following but I do not know how to write out the problems
a. 219,318
b. 120,531
c. 17,671

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?

Clown Bot: Ah, retirement planning, a classic source of financial fun! Let's crunch some numbers to find out how much our self-employed friend will have in their Keogh account.

To calculate this, we can use the formula for compound interest:

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

Where:
A = the future amount in the account
P = the annual deposit
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years

In this case, the annual deposit is $3000, the interest rate is 8%, or 0.08 as a decimal, and the account has 25 years to grow until the age of 65.

Using the formula, we have:
A = 3000(1 + 0.08/1)^(1*25)

Now let's break out the calculator, clown style, and do the math. Drum roll, please...

a. The amount in the account when our person retires at 65 is approximately $219,318! Ta-da!

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

Ah, our friend wants to extend the retirement age and keep contributing to their Keogh account. Let's see how those extra years and contributions make a fabulous difference!

We need to calculate the amount from the previous question (a) at age 65, and then add the additional annual deposits of $3000 from age 65 to age 70.

Using our previous amount of $219,318, we now have 5 more years and 5 additional contributions. The interest and contributions compound magically!

A = 3000(1 + 0.08/1)^(1*5)

Shazam! The additional money in the account if they defer retirement until age 70 is approximately $120,531! Double the fun!

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?

Our friend is ready to call it a day at age 70, but decides to stop contributing to the Keogh account at age 65. Let's calculate the additional amount that will have accumulated during these last 5 contribution-free years.

We've already determined that the amount at age 65 is $219,318. Now we need to calculate the additional interest without any contributions for the final 5 years.

A = 219318(1 + 0.08/1)^(1*5)

And voila! The additional amount in the account if they stop contributing at 65 but don't retire until 70 is approximately $17,671. That's what I call a retirement magic trick!

To solve these problems, we can use the formula for the future value of an ordinary annuity:

FV = P * ((1 + r)^n - 1) / r

Where:
FV = Future value of the account
P = Annual deposit
r = Interest rate per compounding period
n = Number of compounding periods

Let's solve each problem step-by-step:

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?

Step 1: Calculate the number of compounding periods.
Since the person starts saving at age 40 and retires at age 65, there are 25 years of saving.
Number of compounding periods, n = 25

Step 2: Calculate the future value using the formula.
FV = 3000 * ((1 + 0.08)^25 - 1) / 0.08
FV ≈ $219,318

Therefore, the amount in the account when the individual retires at age 65 is approximately $219,318.

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

Step 1: Calculate the number of compounding periods.
Since the person saves until age 70, there are 30 years of saving.
Number of compounding periods, n = 30

Step 2: Calculate the future value using the formula.
FV = 3000 * ((1 + 0.08)^30 - 1) / 0.08
FV ≈ $340,849

To find the additional money, we deduct the amount calculated in part a from the new value:
Additional money = $340,849 - $219,318
Additional money ≈ $121,531

Therefore, there will be approximately $121,531 additional money 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?

Step 1: Calculate the number of compounding periods.
Since the person stops saving at age 65 but continues to earn interest until age 70, there are 5 additional years of interest.
Number of compounding periods, n = 5

Step 2: Calculate the future value using the formula.
FV = 3000 * ((1 + 0.08)^5 - 1) / 0.08
FV ≈ $19,676

To find the additional money, we deduct the amount calculated in part a from the new value:
Additional money = $19,676 - $219,318
Additional money ≈ $-199,642

Therefore, there will be approximately -$199,642 (a decrease) in the account if the saver discontinues the contributions at age 65 but does not retire until age 70.

To calculate the answers to these questions, you can use the formula for calculating the future value of an annuity. The formula is:

FV = P * [(1 + r)^n - 1] / r

Where:
FV = Future Value
P = Payment amount per period
r = Interest rate per period
n = Number of periods

Now, let's break down each part of the questions and calculate the answers:

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?

In this scenario, the individual starts saving at age 40 and retires at age 65. The annual deposit is $3000, and the interest rate is 8%. We need to calculate the future value of the annuity over 25 years.

Using the formula, we have:
P = $3000
r = 8% or 0.08
n = 25

FV = 3000 * [(1 + 0.08)^(65-40) - 1] / 0.08
FV = 3000 * [(1.08^25 - 1) / 0.08]
FV ≈ $219,318

Therefore, when the individual retires at age 65, there will be approximately $219,318 in the account.

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

In this scenario, the individual delays retirement until age 70 and continues to make the annual deposits of $3000. So now, we need to calculate the future value of the annuity for an additional 5 years (from 65 to 70).

Using the formula again:
P = $3000
r = 8% or 0.08
n = 5

FV = 3000 * [(1 + 0.08)^5 - 1] / 0.08
FV = 3000 * [(1.08^5 - 1) / 0.08]
FV ≈ $60,265

Therefore, if the saver defers retirement until age 70 and continues the contributions, there will be approximately an additional $60,265 in the account.

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?

In this scenario, the individual stops making annual deposits of $3000 at age 65 but continues to earn interest until age 70. We need to calculate the future value of the current balance over the 5-year period.

Assuming the balance at age 65 is the same as the value calculated in part a -- approximately $219,318:

P = 0 (there are no additional deposits)
r = 8% or 0.08
n = 5

FV = 219,318 * (1 + 0.08)^5
FV ≈ $267,989

Therefore, if the saver discontinues the contributions at age 65 but does not retire until age 70, there will be approximately an additional $267,989 in the account.