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Where, R = regular payment j = equivalent interest rate per payment interval n = number of payments Examples: 1. Semi-annual payments of P8000 at the end of each term for 12 years with interest rate of 12% compounded quarterly, find its present value. Given: R = 8000 c = 4 (c = number of compounding; for quarterly, it is 4) n = 2(12) = 24 (we will multiply 12 years to 2 since payment is done semi-annually) i* = 0.12 (it is it because the interest rate of 12% is compounded quarterly) Since Semi-annual payments z compounded quarterly, let us convert the 12% compounded quarterly to semi-annually. Solve for j: F1 = F2 1 21 P(1 + = ) = P(1 + !) Substitute the given to the formula: (1 + ;) = (1+ 7) P = PJ-(1+j)" (1 + =) = (1 + 0.12, #2 P = 8000_-(1+0.0609)-24 0.0609 (1 + =) = (1.03) P = P99,573.23 = (1.03) - 1 NI-NI-1 = 0.0609 7 j = 0.0609 2. To pay for his debt at 12% compounded semi-annually, Ruben committed for 8 quarterly payments of P24,491.28 each. How much did he borrow? Given: R = 24,491.28 c = 2 i = 0.12 n = 8 Since Quarterly payments = compounded semi-annually, let us convert the 12% compounded semi-annually to quarterly Solve for j: F1 = F2 P(1 + = ) = P(1 + 5) (1 + 4 ) = (1 + 0.12, (1 + 7 ) = (1+0.06)274 7 = (1 + 0.06)24-1 j = 0.029563014 Substitute the given to the formula. P = RJ-(1+j)" P = 24,491.28_(1+0.0296)- 0.0296 P172,211.71 What's More Directions. Give what is asked in each number. Show your solution. 1. Sam and Alfie are brothers. After college graduation and being finally able to get a good job, they plan for retirement as follows. Starting at age 24, Sam deposits P15,000 at the end of each year for 30 years. Starting at age 42, Alfie deposits P30,000 at the end of each year for 20 years. Who will have the greater amount at retirement if both annuities earn 12% per year compounded annually? 2. A television (TV) set is for sale at P13,499 in cash or on installment terms, P2,550 each month for the next 6 months at 9% compounded annually. If you were the buyer, what would you prefer, cash or installment