Lesson 7 annuity
- 1. SIMPLE ANNUITY ALAN S. ABERILLA
- 2. ANNUITY – a sequence of payments made at equal (fixed) intervals or periods of time. ◦ Annuities may be classified in different ways, as follows: ANNUITIES According to payment interval and interest period Simple Annuity – an annuity where the payment interval is the same as the interest period. General Annuity – an annuity where the payment interval is not the same as the interest. According to time of payment Ordinary Annuity or Annuity Immediate – a type of annuity in which the payments are made at the end of each payment interval. Annuity Due – a type of annuity in which the payments are made at the beginning of each payment interval. According to duration Annuity Certain – an annuity in which payments begin and end at definite times. Contingent Annuity – an annuity in which the payments extend over an indefinite (or indeterminate) length of time.
- 3. Term of annuity, t – time between the first payment interval and last payment interval Regular or Periodic payment, R – the amount of each payment Amount (Future Value) of an annuity, F – sum of future values of all the payments to be made during the entire term of the annuity Present value of an annuity, P – sum ofpresent values of all the payments to be made during the entire term of the annuity Annuities may be illustrated using a time diagram. The time diagram for an ordinary annuity (i.e., payments are made at the end of the year is given below. Time Diagram for an n-Payment Ordinary Annuity R R R R R . . . R 0 1 2 3 4 5 n
- 4. Example 1. Suppose Mrs. Remoto would like to save P 3,000.00 every month in a fund that gives 9% compounded monthly. How much is the amount or future value of her savings after 6 months? Given: Periodic payment, R = P 3,000.00 Find: amount (future value) at the end of the term, F. term, t = 6 months interest rate per annum i(12) = 0.09 number of conversions per year, m = 12 interest rate per period, j = 0.09/12 = 0.0075 Solution: (1) Illustrate the cash flow in a time diagram 3,000.00 3,0000.00 3,000.00 3,000.00 3,000.00 3,000.00 0 1 2 3 4 5 6
- 5. (2) Find the future value of all the payments at the end of term (t = 6) 3,000.00 3,000.00 3,000.00 3,000.00 3,000.00 3,000.00 0 1 2 3 4 5 6 3,000.00 3,000.00(1 + 0.0075) 3,000.00(1 + 0.0075)2 3,000.00(1 + 0.0075)3 3,000.00(1 + 0.0075)4 3,000.00(1 + 0.0075)5 ◦
- 6. (3) Add all the future values obtained from the previous steps 3,000.00 = 3,000.00 3,000.00(1 + 0.0075) = 3,022.50 3,000.00(1 + 0.0075)2 = 3,045.169 3,000.00(1 + 0.0075)3 = 3,068.008 3,000.00(1 + 0.0075)4 = 3,091.018 3,000.00(1 + 0.0075)5 = 3,114.20 P 18,340.89 Thus, the amount of this annuity is P 18,340.89
- 7. The future value F of an ordinary annuity is given by: Amount (Future Value) of Ordinary Annuity: (1 + j)n – 1 where: R is the regular payment F = R j is the interest rate per period j n is the number of payments
- 8. Example 2. In order to save for her high school graduation, Marie decided to save P 200.00 at the end of each month. If the bank pays 0.250% compounded monthly, how much will her money be at the end of 6 years? Given: R = P 200.00 j = 0.0025/12 = 0.0002083 m = 12 t = 6 years i(12) = 0.250% = 0.0025 n = mt = (12)(6) = 72 periods Solution: F = R {(1 + j)n – 1/j)} F = P 200.00 {( 1 + 0.0002083 )72 – 1)/0.0002083 )} F = P 14,507.85
- 9. Present Value of an Ordinary Annuity: 1 - (1 + j)-n where: R is the regular payment P = R j is the interest rate per period j n is the number of payments
- 10. Example 3. Mr. Ribaya paid P 200,000.00 as down payment for a car. The remaining amount is to be settled by paying P 16,200.00 at the end of each month for 5 years. If the interest is 10.5% compounded monthly, what is the cash price of his car? Given: R = P 200.00 j = 0.105/12 = 0.00875 m = 12 t = 5 years i(12) = 10.5% = 0.105 n = mt = (12)(5) = 60 periods Down payment = P 200,000.00 Find: Cash value or cash price of the car Solution: P = R {(1 – (1 + j)-n/j)} P = P 16,200.00 {(1 – ( 1 + 0.00875 )-60)/0.00875)} P = P 753,702.20 Cash Value = Down payment + Present value = P 200,000.00 + P 753,702.20 Cash Value = P 953,702.20
- 11. Periodic payment R of an Annuity: (1 + j)n - 1 (1 + j)n - 1 F = R R = F/ j j (1 – (1 + j)-n 1 - (1 + j)-n P = R R = P/ j j Where: R is the regular payment P is the present value of an annuity F is the future value of an annuity j is the interest rate per period n is the number of payments
- 12. Example 4. Paolo borrowed P 100,000.00. He agrees to pay the principal plus interest by paying an equal amount of money each year for 3 years. What should be his annual payment if interest is 8% compounded annually? Given: P = P 100,000.00 j = 0.08 m = 1 t = 3 years i(1) = 8% = 0.08 n = mt = (1)(3) = 3 periods Find: Periodic payment R Solution: P = R {(1 – (1 + j)-n/j)} then R = P / {(1 – (1 + j)-n/j)} R = P 100,000.00 / {(1 – ( 1 + 0.08)-3)/0.08)} P = P 38,803.35 Thus, the man should pay P 38,803.35 every year for 3 years
- 13. ACTIVITY 7 Solve the following: 1. Linda started to deposit P 2,000.00 quarterly in a fund that pays 5.5% compounded quarterly. How much will be in the fund after 2. The buyer of a house and lot pays P 200,000.00 cash and P 10,000.00 every month for 20 years. If money is 9% compounded monthly, how much is the cash value of the lot? 3. Rebecca borrowed P 150,000.00 payable in 2 years. To repay the loan, she must pay an amount every month with an interest rate of 6% compounded monthly. How much should he pay every month? 4. Mr. Sarsonas would like to save P 500,000.00 for his son’s college education. How much should he deposit in a savings account after 6 months for 12 years if interest is at 1% compounded semi- annually? 5. A television is for sale at P 17,999.00 in cash or on terms, P 1,600.00 each month for the next 12 months. The money is 9% compounded monthly. Which is lower, the cash price or the present value of the installment terms?
- 14. KEEP SAFE GOOD LUCK GOD BLESS Sir A