Question

Jane and John Smith have accumulated $650,000 for their retirement, which begins today. They plan to receive monthly payments from their investments, which will be paid at the beginning of each month over the next 35 years based on their estimated life expectancy. If investments are accumulating at an after-tax annual rate of 5.75%, compounded monthly, what will be the payment amount that the Smiths will receive each month to the nearest dollar?

Answer 1

here since it is compounded monthly, therefore we have to calculate the effective rate of interest.

effective rate of interest = [1 + rate_{/no. of times compounded in a year} ] ^{no. of times compounded in a year} - 1

= [1 + 0.0575 _{/ 12}]¹² - 1

= [1.00479]¹² - 1

= 1.05901 - 1

= 0.5901

=5.90%

now the formula,

present value of annuity due = [ cash flow * 1 -(1+r)^-n _{/ r}] * [1 + r]

here, r= effective rate of interest

n= no. of payments

therefore on substituting the values,

present value of annuity due= [cash flow * 1-(1+ 5.90 _{/ 100} )^-35 / 0.059] * [1 + 0.059]

650000 / 1.059 = cash flow * 1 - (1.059)^-35_{/ 0.059}

613786.59 = cash flow * 1 - 0.1345 _/0.059

   613786.59 = cash flow * 14.669

therefore cash flow monthly = 41842.43 ANSWER