Joseph has just retired and is trying to decide between two retirement income option as to where he would place his life savings. Fund A will pay him quarterly payments for 25years starting at $1000 at the end of the first quarter. Fund A will increase his payments each quarter thereafter by $200 and the last payment will be $20800.
Fund B will pay $1000 at the end of the first quarter and the payments will increase at a constant rate of 18% p.a, compounding quarterly thereafter.
How much will be available in each fund at the end of 25 years if Joseph’s required interest rate is 12% p.a compounded quarterly?
none
FV = $3,989,205.95
constant ratio
FV=$4,157,992.40
To calculate the amount available in each fund at the end of 25 years, we will first determine the future value of Fund A and then the future value of Fund B.
1. Future Value of Fund A:
Fund A is giving quarterly payments that increase by $200 every quarter. The last payment after 25 years will be $20,800.
We can calculate the future value of Fund A using the formula for the future value of an annuity:
FV = P * [(1 + r)^n - 1] / r
In this formula:
- FV is the future value of the annuity.
- P is the amount of each payment, which is $1,000 initially.
- r is the interest rate per period, which is 12% divided by 4 (since it's compounded quarterly), giving us 0.03.
- n is the number of payment periods, which is 25 years multiplied by 4 (since it's compounded quarterly), giving us 100.
Plugging these values into the formula, we get:
FV = 1000 * [(1 + 0.03)^100 - 1] / 0.03 ≈ $2,006,961.02
So, at the end of 25 years, Fund A will have approximately $2,006,961.02.
2. Future Value of Fund B:
Fund B is increasing payments at a constant rate of 18% per year, compounded quarterly.
To calculate the future value of Fund B, we can use the formula for the future value of a growing annuity:
FV = P * [(1 + r)^n - (1 + g)^n] / (r - g)
In this formula:
- FV is the future value of the annuity.
- P is the amount of the initial payment, which is $1,000.
- r is the interest rate per period, which is 12% divided by 4, giving us 0.03.
- g is the growth rate per period, which is 18% divided by 4, giving us 0.045.
- n is the number of payment periods, which is 25 years multiplied by 4, giving us 100.
Plugging these values into the formula, we get:
FV = 1000 * [(1 + 0.03)^100 - (1 + 0.045)^100] / (0.03 - 0.045) ≈ $3,613,080.43
So, at the end of 25 years, Fund B will have approximately $3,613,080.43.
Therefore, Fund A will have approximately $2,006,961.02, and Fund B will have approximately $3,613,080.43 at the end of 25 years.