Question

Suppose the interest rate is 9.4 % APR with monthly compounding. What is the present value of an annuity that pays $ 85 every three months for four ​years? ​(Note: Be careful not to round any intermediate steps less than six decimal​ places.)

Answer 1

Sol:

Interest rate (r) = 9.4% with monthly compounding

PMT = $85

Period (n) = 4 x 4 = 16

To determine present value (PV) of an annuity:

Effective annual rate = (1+9.4%/12)^12 - 1We have to convert monthly compounding rate into effective annual rate:

Effective annual rate = (1 + 0.094/12)^12 - 1

Effective annual rate = (1.007833)^12 - 1 = 0.098157 or 9.82%

Now we have to find Quarterly compounding rate:

Quarterly compounding rate = ((1 + 0.098157)^(1/4) - 1) x 4

Quarterly compounding rate = ((1.098157)^1/4 - 1) x 4 = 0.094738 or 9.47%

Present value (PV) of an annuity = PMT x (1-(1/(1 + r)^n)) / r

Present value (PV) of an annuity = 85 x (1-(1/(1+0.094738/4)^16))/ 0.094738/4

Present value (PV) of an annuity = 85 x (1-(1/(1+0.0236845)^16)) / 0.0236845

Present value (PV) of an annuity = 85 x (1-(1/(1.0236845)^16)) / 0.0236845

Present value (PV) of an annuity = 85 x 0.312390/ 0.0236845 = $1121.12

Therefore Present value (PV) of an annuity is $1121.12