Question

Sanaa has been saving money in the same account for 20 years. She started with $3000. For the first 6 years, the account paid j(12) = 7% . For the next 7 years, it paid j(12) = 9%. For the last 7 it paid j(12) = 6% compounded.

a) How much money is in her account today?

b) Suppose the account had paid the same effective annual rate r for all 20 years, and that her starting balance and ending balance were as above. Solve for r (to 2 decimal places). (If your answer is 3.06% write 3.06 not 0.0306.

c) Write the formula you used to compute your answer to b)

Answer 1

a.) How much money is in her account today?

Ans. $3000 * (1.07)^6*(1.09)^7*(1.06)^7 = $12,375.14976

b. Computation of effective annual rate

let effective annual rate be r

FV = PV*(1+r)^20

=> 12,375.1496 = $3000 * (1+r)^ 20

=> (1+r) = (12,375.1496/ 3000)^(1/20)

=> r = 1.073424394 - 1

=> r = 0.07342 or 7.342%

c. The formula used above is

FV = PV *(1+r)^n

=> r = (FV/PV)^(1/n) - 1

where FV = Future value

PV = Present value

n = time period

r = effective interest rate