Question

8. -12 points HarMathAp 12 6.3.043.ML A small business owner contributes $4.000 at the end of each quarter to a retirement account that earns 6 compounded garterly (a) How long will it be until the account is worth at least $150,0007(Round your answer Up to the nearest quarter.) Ask Your Teacher quarters (b) Suppose when the account reaches $150,000, the business owner increases the contributions to $6,000 at the end of each quarter. w more years? (Round your answer to the nearest dollar) Need Help? 9. 1/1 points Previous Answers HarMathAp1264.003.MI. Find the present value of an annulty of $8000 paid at the end of each month period for years the interest My Notes Ask Your Teacher compounded manually (Round your answer to the nearest Gent.) od 2M ofte 9 DOLL

Answer 1

(a) Amount deposited each quarter = P = $4000

Quarterly Interest rate = r = 6%/4 = 1.5%

Future Value required = FV = $150000

Let number of quarters be n

Hence, FV = P(1+r)^n-1 +....+ P(1+r)² + P(1+r) + P = P[(1+r)ⁿ -1]/r

=> 150000 = 4000[(1+0.015)ⁿ -1]/0.015

=> (1+0.015)ⁿ = 150000*0.015/4000 + 1

=> (1.015)ⁿ = 1.5625

=> n = ln(1.5625)/ln(1.015) = 30 quarters

(b) Value in account after 30 quarters = X = $150000

We need to find the value in account after n = 15*4 = 60 quarters

Quarterly Interest rate = r = 6%/4 = 1.5%

Amount deposited = P = $6000

Hence, Value after 60 more quarters = X(1+r)ⁿ + P(1+r)^n-1 +....+ P(1+r)² + P(1+r) + P = X(1+r)ⁿ + P[(1+r)ⁿ -1]/r

= 150000(1+0.015)⁶⁰ + 6000[(1+0.015)⁶⁰ -1]/0.015

= $943,770.88