An annuity is developed and used based on the concept of time value of money. Please solve for the principle required when one retires so that a payment of $1500.00 is received each month for 30 years (360 months). Assume that the interest rate for the payout is 5% and the principle fund will be depleted (zero) at the end of 30 years. How much would one need to save monthly for 40 years (480 payments) to have the principle in the future needed to pay out the $1500.00 monthly for 30 years (Assume one can earn 8% on one’s savings)?
To solve for the principle required when one retires, we can use the formula for the present value of an annuity:
PV = PMT * ((1 - (1 + r)^(-n)) / r)
Where:
PV = Present value (principle required)
PMT = Payment received each month ($1500.00)
r = Interest rate for the payout (5% or 0.05)
n = Number of periods (30 years or 360 months)
Plugging in the values:
PV = 1500 * ((1 - (1 + 0.05)^(-360)) / 0.05)
Now, let's calculate the principle required:
PV = 1500 * ((1 - 1.05^(-360)) / 0.05)
PV ≈ 1500 * (1 - 0.11443) / 0.05
PV ≈ 1500 * 18.8573624
PV ≈ $28,285.04
Therefore, the principle required when one retires is approximately $28,285.04.
Now let's calculate the monthly savings required for 40 years to achieve the principle needed.
We can use the formula for the future value of an annuity:
FV = PV * ((1 + r)^n - 1) / r
Where:
FV = Future value (principle needed to pay out $1500.00 monthly for 30 years)
PV = Present value (unknown, what we are trying to calculate)
r = Interest rate (8% or 0.08)
n = Number of periods (40 years or 480 payments)
Plugging in the values:
FV = 1500 * ((1 + 0.08)^480 - 1) / 0.08
Now, let's calculate the principle needed:
FV = 1500 * ((1 + 0.08)^480 - 1) / 0.08
FV ≈ 1500 * (2.20873281995 - 1) / 0.08
FV ≈ 1500 * 1.26041 / 0.08
FV ≈ $23,756.25
Therefore, to have the principle in the future needed to pay out $1500.00 monthly for 30 years, one would need to save approximately $23,756.25 monthly for 40 years.