Annuity

SIMPLE ANNUITY Annuity – a sequence of payments made at equal (fixed) intervals or periods of time Payment interval – the time between successive payments Annuities may be classified in different ways, as follows. Annuities According to Simple Annuity General Annuity payment An annuity An annuity interval and where the where the interest period payment payment is not interval is the the same as the same as the interest period interest period According to Ordinary Annuity Due time of Annuity A type of payment A type of annuity in annuity in which the which the payments are payments are made at made at the beginning of end of each each payment payment interval interval According to Annuity Certain Contingent duration An annuity in Annuity which An annuity in payments begin which the and end at payments definite times extend over an indefinite length of time Term of an 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 payments to be made during the entire term of the annuity Present value of an annuity, P – sum of present values of all the payments to be made during the entire term of the annuity ORDINARY ANNUITY

(1 + 𝑗)𝑛 − 1 𝑗 1 − (1 + 𝑗)−𝑛 𝑃=𝑅 𝑗 𝐹= 𝑅

Example: 1. Suppose Mrs. Remoto would like to save P3,000 at the end of each month, for six months, in a fund that gives 9% compounded monthly. How much is the amount or future value of her savings after 6 months? 2. In order to save for her high school graduation, Marie decided to save P200 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? 3. Suppose Mr. Lee would like to know the present value of her monthly deposit of P3,000 when interest is 9% compounded monthly. How much is the present value of her savings at the end of 6 months?

The cash value of cash price is equal to the down payment plus the present value of the installment payments. (CASH VALUE = Down Payment + P)

4. Mr. Ribaya paid P200,000 as down payment for a car. The remaining amount is to be settled by paying P16,200 at the end of each month for 5 years. If interest is 10.5% compounded monthly, what is the cash price of his car? 5. Paolo borrowed P100,000. He agrees to pay the principal plus interest by paying an equal amount of money each for 3 years. What should be his annual payment if interest is 8% compounded annually? ACTIVITY Find the present value P and amount F of the following ordinary annuities 1. Quarterly payments of P2,000 for 5 years with interest rate of 8% compounded quarterly 2. Semi-annually payments of P8,000 for 12 years with interest rate of 12% compounded semi-annually Answer the following problems. 3. Peter started to deposit P5,000 quarterly in a fund that pays 1% compounded quarterly. How much will be in the fund after 6 years? 4. The buyer of a lot pays P50,000 cash and P10,000 every month for 10 years. If money is 8% compounded monthly, how much is the cash value of the lot? 5. How much should be invested in a fund each year paying 2% compounded annually to accumulate P200,000 in 5 years?

GENERAL ANNUITY General Annuity – an annuity where the payment interval is not the same as the interest compounding period. General Ordinary Annuity – a general annuity in which the periodic payment is made at the end of the payment interval.

General Ordinary Annuity (1 + 𝑗)𝑛 − 1 𝐹= 𝑅 𝑗 1 − (1 + 𝑗)−𝑛 𝑃=𝑅 𝑗

Example: 1. Mel started to deposit P1,000 monthly in a fund that pays 6% compounded quarterly. How much will be in the fund after 15 years? 2. A teacher saves P5,000 every 6 months in a bank that pays 0.25% compounded monthly. How much will be her savings after 10 years? 3. Ken borrowed an amount of money from Kat. He agrees to pay the principal plus interest by paying P38,973.76 each year for 3 years. How much money did he borrow if interest is 8% compounded quarterly? 4. Mrs. Remoto would like to buy a television set payable for 6 months starting at the end of the month. How much is the cost of the TV set if her monthly payment is P3,000 and interest is 9% compounded semi-annually?

A cash flow is a term that refers to payments received (cash inflows) or payments or deposits made (cash outflows). Cash inflows can be represented by positive numbers and cash outflows can be represented by negative numbers.

The fair market value or economic value of a cash flow (payment stream) on a particular date refers to a single amount that is equivalent to the value of the payment stream at that date. This particular date is called the focal date.

Example: 1. Mr. Ribaya Received two offers on a lot that he wants to sell. Mr. Ocampo has offered P50,000 and P1,000,000 lump sum payment 5 years from now. Mr. Cruz has offered P50,000 plus P40,000 every quarter for five years. Compare the fair market values of the two offers if money can earn 5% compounded annually. Which offer has a higher market value? 2. Company A offers P150,000 at the end of 3 years plus P300,000 at the end of 5 years. Company B offers P25,000 at the end of each quarter for the next 5 years. Assume that money is worth 8% compounded annually. Which offer has a better market value? Activity Find the present value and amount F of the following. 1. Semi-annual payments of P500 at the end of each term for 10 years with interest rate of 5% compounded semi-annually 2. Annual payments of P1,000 at the end of each term for 8 years with interest rate of 6% compounded quarterly Answer the following problems. 3. On a girl’s 10th birthday, her father started to deposit P5,000 quarterly at the end of each term in a fund that pays 1% compounded monthly. How much will be in the fund on his daughters 17th birthday? 4. The buyer of a lots pays P10,000 every month for 10 years. If money is 8% compounded annually, how much is the cash value of the lot? 5. In order to save for her high school graduation, Kathrina decided to save P200 at the end of every three month, starting the end of the second month. If the bank pays 0.250% compounded monthly, how much will be her money at the end of 5 years? 6. Mr. Bajada paid P200,000 as down payment for a farm. The remaining amount is to be settled by paying P16,200 at the end of each month for 5 years. If interest is 5% compounded semi-annually, what is the cash price of his farm? 7. A television set is for sale at P13,499 in cash or on installment terms, P2,500 each month for the next 6 months at 9% compounded annually. If you were the buyer, what would you prefer, cash or installment?

DEFERRED ANNUITY Deferred annuity – an annuity that does not begin until a given time interval has passed Period of Deferral – time between the purchase of an annuity and the start of the payments for the deferred annuity Present value of a deferred annuity 𝑃=𝑅

1 − (1 + 𝑗)−(𝑘+𝑛) 1 − (1 + 𝑗)−𝑘 −𝑅 𝑗 𝑗

Where R = regular payment i = interest rate per period n = is the number of payments k = number of conversion periods in the period of deferral Example 1. On his 40th birthday, Mr. Ramos decided to buy a pension plan for himself. This plan will allow him to claim P10,000 quarterly for 5 years starting 3 months after his 60th birthday. What one-time payment should he make on his 40th birthday to pay off this pension plan, if the interest rate is 8% compounded quarterly? 2. A credit card company offers a deferred payment option for the purchase of any appliance. Rose plans to buy a smart TV set with monthly payments of P4,000 for 2 years. The payments will start at the end of 3 months. How much is the cash price of the TV set if the interest rate is 10% compounded monthly? Activity Find the period of deferral in each of the following deferred annuity problem (one way to find the period of deferral is to count the number of artificial payments) 1. Monthly payments of P2,000 for 5 years that will start 7 months from now 2. Annual payments of P8,000 for 12 years that will start 5 years from now 3. Quarterly payments of P5000 for 8 years that will start two years from now 4. Semi-annual payments of P60,000 for 3 years that will start 5 years from now 5. Payments of P3000 every 2 years for 10 years starting at the end of 6 years

Fill in the blanks in the statements that follow. 6. A loan is P30,000 is to be repaid monthly for 5 years that will start at the end of 4 years. If interest rate is 12% converted monthly, how much is the monthly payment? a. The type of annuity illustrated in the problem is a ___________________. b. The total number of payments is ___________. c. The number of conversion periods in the period of deferral is _________________. d. The interest rate per period is _____________. e. The present value of the loan is ____________. Answer the following problems completely 7. Emma availed of a cash loan that gave her option to pay P10,000 monthly for 1 year. The first payment is due after 6 months. How much is the present value of the loan if the interest rate is 12% converted monthly? 8. Adrian purchased a laptop through the credit cooperative of their company. The cooperative provides an option for a deferred payment. Adrian decided to pay after 4 months of purchase. His monthly payment is computed as P3,500 payable in 12 months. How much is the cash value of the laptop if the interest rate is 8% convertible monthly?

Source: General Mathematics TG 2016 Prepared by: SHS Mathematics Department