_9 Periodic Payment of an Ordinary Annuity

Periodic Payment of an Ordinary Annuity To find periodic payment: when amount is given: RS 

Si 1  i n  1  

when present value is given: RA 

Ai 1  1  i  n   

Examples: 1. How much monthly deposit must be made for 4 years in order to accumulate P12, 500 at 6% compounded monthly?

2. Mr. Tan wants to buy a car worth P650, 000. He can pay P250, 000 down payment and the balance payable every end of the month for 5 years. How much must he pay monthly at 3% compounded monthly?

3. Mr. Reyes wants to accumulate P797, 500 at the end of 10 years to put his son to college. Determine the amount he would have to invest at the end of each year at 4 compounded annually. 7 % 5

4. On April 30, 2000, Alberto invested P106, 250 at 15 3 % compounded bi – monthly. 5 The investment is to be paid out in 25 equal payments at the end of each two months starting on June 30, 2000. What is the size of the payment and on what date is the payment due?

Finding the Interest Rate of an Ordinary Annuity The formula for finding the rate of an ordinary annuity is derived from a quadratic equation using the given values of n, R, S, and A. Given the present value: nR    n  1 i  6  n  1 i  12 1  A   0 2

2

Given the amount: nR    n  1 i  6  n 1 i  12 1  S   0 2

2

Then use the Quadratic Formula to solve for i:

 b  b  4ac i 2a 2

NOTE: Take the positive root if A is given in the problem and take the negative root if S is given. r Since i  m , then to solve for the nominal rate we have r  i  m  .

Examples: 1. At what rate compounded monthly is P3, 200 the present value of an annuity of P200 paid at the end of each month for 1 1 years? 2

2. Payments of P500 each are made every year. At what rate compounded annually will these payments amount to P10, 000 in 13 years?

• Example 3: An item can be purchased for P45,000 cash or for P10,000 down payment and a payment of P4,500 every six months for 6 years. Find the interest rate compounded semi-annually.

• Example 4: At the end of every three months, Mr. Magaling puts P10,500 in an investment house. If the account amounted to P259,940.37 at the end of 5 years, at what rate compounded quarterly was the interest earned?

• Additional Examples: 1. A loan of P8 838.51 is to be discharged by making 15 semiannually payments of P940 each. At what rate compounded semiannually is the interest charged on the loan?

• 2. Payments of P300 each made every year. At what rate compounded annually will these payments amount to P8 336.42 in 22 years?