Nelson Collins decided to retire to Canada in 10 years. What amount should Nelson deposit so that he will be able to withdraw $80,000 at the end of each year for 25 years after he retires? Assume Nelson can invest money at 7% interest compounded annually. (p. 322)
To find the amount Nelson should deposit to be able to withdraw $80,000 at the end of each year for 25 years, we need to use the concept of present value of an annuity.
First, let's calculate the present value of the annuity. The present value of an annuity formula is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
PMT = Amount of annuity payments per year ($80,000 in this case)
r = Interest rate per compounding period (7% or 0.07 as a decimal)
n = Number of compounding periods (25 years in this case)
Substituting the provided values into the formula:
PV = $80,000 * (1 - (1 + 0.07)^(-25)) / 0.07
Simplifying:
PV = $80,000 * (1 - (1.07)^(-25)) / 0.07
Next, calculate the future value (FV) of the amount found above at the end of 10 years using the compound interest formula:
FV = PV * (1 + r)^n
Where:
FV = Future Value
PV = Present Value (which we just calculated)
r = Interest rate per compounding period (7% or 0.07 as a decimal)
n = Number of compounding periods (10 years in this case)
Substituting the values:
FV = PV * (1 + 0.07)^10
Finally, we need to solve for the present value (PV) by rearranging the formula:
PV = FV / (1 + r)^n
Substituting the values:
PV = FV / (1 + 0.07)^10
Now we can calculate the final answer.