Question

2.) David works during the summer to help with expenses at school the following year. He is able to save $250 each week for 12 weeks, and he invests it at an annual rate of 7% that is compounded monthly. When school starts, David will begin to withdraw equal amounts from this account each week. What is the most David can withdraw each week for 34 weeks?

im struggling with this problem, can I get help?

Answer 1

We can use following formula for Future value (FV) of ordinary annuity calculation of periodic savings

FV = PMT * [(1+i) ^n – 1] /i

Where FV = future value of savings =?

PMT or saving per week = $250

For monthly compounding, effective annual rate (EAR) i = (1 + r/m) ^m – 1

Where, annual rate r = 7% and m = 12 months

= (1 + 7%/12) ^12 - 1

= (1 + 0.58%) ^12 - 1

= 0.0723 or 7.23%

And i= I/Y = 7.23% is the interest rate per annum; therefore weekly interest rates = 7.23%/52 =0.139% per week

The time period n = 12 weeks

Therefore,

FV = $250 * [(1+0.139%) ^12 -1]/ 0.139%

FV = $3,023.04

Therefore the value of his savings will be $3,023.04 when school start.

David will begin to withdraw equal amounts from this account each week. The above future value will be present value in this calculation

We can use resent value (PV) of an Annuity formula to calculate the equal monthly withdrawals

PV = PMT * [1-(1+i) ^-n)]/i

Where PV of loan = $3,023.04

PMT = weekly withdrawals =?

n = N = number of payments = 34 weeks

i = I/Y = weekly interest rate = 7.23%/52 = 0.139% per week

Therefore,

$3,023.04 = PMT* [1- (1+0.139%) ^-34]/0.139%

PMT = $91.09

Therefore David can withdraw $91.09 each week for 34 weeks from his savings