Question

2. (4pts) Deposits of $500 are made at the beginning of every quarterly year for 20 years in order to accumulate an annuity paying $X at the end of each year for 10 years, with the first payment starting at the end of year 21. Find X if the nominal interest rate is 6% convertible monthly. (Answer: $10,651.72) 3. (3pts) You want to accumulate $10,000 at the end of five years by equal quarterly deposits at

Math Interest Theory/ Financial Math

Please Use Formulas

Answer 1

here first we need to find interest rate per quarter

Effective annual rate = (1 + r/n)^n - 1

where r = nominal rate ; n = number of compounding periods

it is given interest rate of 6% is monthly compounded so effective annual rate is = (1 + 6/12)^12 - 1 = 0.061678

but the payments are made quartelry so we need to find out quartely rate

quartely rate is (1 + r)^4 - 1 = 0.061678

solving above for r it is = 0.015075

now we need to find out futurevalue of annuity due

future value of annuiyt due = p[(1+r)^n - 1 / r](1+r)

where p = quartely payments = $500

r = rate of interest = 0.015075

n = number of periods = 20 x 4 = 80

future value of annuity = 500[(1+0.015075)^80 - 1 / 0.015075](1+ 0.015075)

= 77,778.16

now we can find out the value of 'X' using present value of annuity formula

present value of annuity = P[1 - (1 + r)^-n / r]

present value of annuity = 77,778.16

where P = annual payments ( which we need to find out)

r = interest rate = 0.061678 (yearly compounded value of 6% we already calculated above)

n = number of periods = 10

77,778.16 = P [ 1 - (1 + 0.061678)^-10 / 0.061678 ]

P = 77,778.16 / 7.301933273

P (X) = $10,651.72