8. Assume you just turned 28. You want to accumulate $3.5 million by your 65th birthday. Beginning next year, you will invest equal amounts each year until your 60th birthday. Your twin brother has the same goal, but he will wait 4 years before making his first payment. Assume he also will make equal payments into the same mutual fund until his 60th birthday. If both accounts earn a 12.5% annual rate of return, how much more will your brother have to invest each year (relative to you) in order to have the same $3.5 Million at age 65?

Question

8. Assume you just turned 28. You want to accumulate $3.5 million by your 65th birthday. Beginning next year, you will invest equal amounts each year until your 60th birthday. Your twin brother has the same goal, but he will wait 4 years before making his first payment. Assume he also will make equal payments into the same mutual fund until his 60th birthday. If both accounts earn a 12.5% annual rate of return, how much more will your brother have to invest each year (relative to you) in order to have the same $3.5 Million at age 65?

Answer 1

To solve this problem, we need to calculate the equal annual investments required for both you and your twin brother in order to accumulate $3.5 million by age 65.

First, let's find out the number of years you will be making investments. Since you just turned 28 and plan to invest until your 60th birthday, you will be investing for a total of 60 - 28 = 32 years.

Now, let's calculate the future value of your annual investments. We will use the future value of an annuity formula, given by:

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

Where:
FV = future value (desired amount)
P = annual investment
r = annual interest rate
n = number of years of investment

Plugging in the values for your case:
FV = 3,500,000
r = 12.5% = 0.125
n = 32

We need to solve for P, so rearranging the formula, we have:

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

Now, we can calculate the annual investment required for you.

P = 3,500,000 * 0.125 / ((1 + 0.125)^32 - 1)
P ≈ $9,268.76

Therefore, you would need to invest approximately $9,268.76 each year until your 60th birthday.

Next, let's calculate the number of years your twin brother will be investing. Since he will wait 4 years before making his first payment and will invest until his 60th birthday, he will be investing for a total of 60 - 4 = 56 years.

Using the same formula and calculations as above, we can find the annual investment required for your twin brother:

P = 3,500,000 * 0.125 / ((1 + 0.125)^56 - 1)
P ≈ $4,045.16

Therefore, your twin brother would need to invest approximately $4,045.16 each year until his 60th birthday.

To find the difference in annual investments between you and your twin brother, we subtract your annual investment from his:

Difference = Twin Brother's Annual Investment - Your Annual Investment
Difference = $4,045.16 - $9,268.76
Difference ≈ -$5,223.60

Since the difference is negative, it means your twin brother would need to invest approximately $5,223.60 less than you each year to have the same $3.5 million at age 65.