Question

Rose has saved $100,000. If she decides to withdraw $2500 at the beginning of each month when interest is 6.25% compounded semi-annually, for how long can she make withdrawals

Answer 1

Answer:

n = 45.064 months or

n = 45 months (approximately)

Explanation:

Calculations:

Step 1:

Effective annual rate = (1+r/n)ⁿ - 1

current rate is semi-annual n=2

Effective annual rate = (1+6.25%/2)² - 1

Effective annual rate = 6.35%

Step 2:

Using annuity formula to find time period

P = A(1 - (1+ r)^-n)/r

Here,

P = $100,000

A = $2,500

r = 6.35% per annum

r = 6.35%/12 = 0.529167% per month

n = ?

So, using the formula

100,000 = 2,500 (1 - (1+0.529167%)^-n)/0.529167%

100,000/2,500 X 0.529167% = 1 - (1+0.529167%)^-n

0.2116668 = 1 - 1/1.00529167ⁿ

1/1.00529167ⁿ = 1 - 0.2116668

1/1.00529167ⁿ = 0.7883332

1.00529167ⁿ = 1/0.7883332

1.00529167ⁿ = 1.26849916

Apply 'log' on both sides

n log 1.00529167 = log 1.26849916

n X 0.0229208394 = 0.103290183

n = 0.103290183/0.0229208394

n = 45.064 months

n = 45 months