Question

Chloe deposits $X into her bank account today. The money is sufficient to support 40 quarterly withdrawals of $700 starting in 3 months. The interest rate is 5% p.a. compounded quarterly for the first year and 6% p.a. compounded quarterly thereafter. Which of the following equations can be used to find $X (Only one correct answer)?

X=700/0.05*(1-1.05^-4)+700/0.06*(1-1.06^-36)

X=700/0.0125*(1-1.0125^-4)+700/0.015*(1-1.015^-36)

X=700/0.05*(1-1.05^-4)+700/0.06*(1-1.06^-36)*(1.05)^-4

X=700/0.0125*(1-1.0125^-4)+700/0.015*(1-1.015^-36)*(1.0125)^-4

None of the equations gives the correct X.

- Calculate X and input your answer below. (Round your answer to 2 decimal places. Do not put unit. Do not use comma separators. E.g. 1234.56)

Answer 1

Present value of annuity = Annuity * [ 1- ( 1 + r)^-n ] / r

Present value of withdrawals for the 1st year =X *{ [ 1 - 1.0125^-4 ] / 0.0125 }

Present value of withdrawals after i year = X * { [ 1 - 1.015^-36 ] / 0.015 } / 1.0125^4

700 =  X *{ [ 1 - 1.0125^-4 ] / 0.0125 } + X * { [ 1 - 1.015^-36 ] / 0.015 } / 1.0125^4  

X = 700 * 0.0125 / [ 1 - 1.0125^-4 ] + 700 * 0.015 * 1.0125^4 / [ 1 - 1.015^-36 ]

= 700*0.0125 / 0.048476 + 700 * 0.015*1.0125^4 / 0.41491

= 207.10