Question

You agree to deposit $500 at the beginning of each month into a bank account for the next 24 months. At the end of the 24th month, you will have $13,000 in your account. If the bank compounds interest monthly, what annual interest rate will you have earned?

Note: Please post the formula used to solve the question and list the steps taken to reach the answer, please don't use excel. I provided a list of formulas, please state the one you used.Thank you.

01POPO TABLE 5-7 Summary of Time Value of Money Equations Calculation Equation Future value of a single payment FV, = PV(1 + ) Present value of a single payment "L(1 + 0"] Future value of an annuity [(1 + A" – 1] FV of an annuity = PMT PV=Fyl 1 Present value of an annuity 1 – (1 + nan PV of an annuity = Php 1 - 4, + PV of an annuity 1] Future value of an annuity due Present value of an annuity due Effective annual rate (EAR) FV,(annuity due) = future value of an annuity X (1 + n) Pannuity due) = present value of an annuity x (1 + n) APR or quoted annual rate EAR = 1 + compounding periods per year (m) APRn Future value of a single payment with nonannual compounding PVI 1 + Present value of a perpetuity PV = PP Notations: FV, = the future value of the investment at the end of n years n= the number of years until payment will be received or during which compounding occurs p= the annual interest or discount rate PV = the present value of the future sum of money m= the number of times compounding occurs during the year PMT = the annuity payment deposited or received at the end of each year PP = the constant dollar amount provided by the perpetuity

Answer 1

since deposits are made at the beginning of every moth we use the formula

future value of annuity due = futurevalue of annuity x (1 + r)

future value of annuity = PMT[ (1 + r)^n - 1 / r ]

where PMT = monthly payments

r = rate of interest (per period)

n = number of periods

future value = 13000

so , 13,000 = 500[(1 + r)^24 - 1 / r ] x (1 + r)

[(1 + r)^24 - 1 / r ] x (1 + r) = 13000 / 500

  [(1 + r)^24 - 1 / r ] x (1 + r) = 26

  using future value of annuity due table( or using scientific calculator) we can see that value 26 occurs at r =0.635%

nominal  Yeraly rate = 0.635 x 12 = 7.62%

(in calculator inputs will be ( n = 24 , PV=0 ,PMT=500,FV=13000 , i/n = ? )