Fair Market Value is computed by adding the Down payment and Present Value or \"Fair Market Value (FMV) = Down payment + Present Value\". In computing for the present value of the given problem, the formula to be used are the following: A. Simple Annuity the payment interval is the same as the interest period P = F (1 + j)'\" where F is the future value j - interest rate per period and is given by the formula, . T' . . . j = E where ms the annual 1nterest rate and m 1s the number of compounds. Note: annually (m =1) quarterly (m =4) semi-annually m =2 n number of actual payments and is given by the formula, n = mt where m is the number of compounds and t is time express in years. B. General Annuity the payment interval is not the same as the interest period where R is the regular payment j equivalent rate (see example 1) n number of actual payments and is given by the formula, n 2 mt where mis the number of compounds and tis time express in years. Example 1: ABCD Company offers 1" 250,000 at the end of 3 years plus 1" 400,000 at the end of 5 years. XYZ Company offers ? 35,000 at the end of each quarter for the next 5 years. Assume that money is worth 6% compounded semiannually. Which company offers a better market value? Given: ABCD Company XYZ Company ? 250,000 at the end of3 years 1? 35,000 at the end of each quarter for ?400,000 at the end of 5 years the next 5 years Find: Fair Market Value of each offer Solution: a. Illustrate the cash ows of the two offers using time diagrams. ABCD's Company 250,000 400,000 0 l 2 3 4 5 5 LU_Genera| Mathematics_Module12 XYZ's Company 35,000 35,000 35,000 35,000 0 l 2 3 20 Suppose that selected focal date is the start of the term. Compute for the present value of each offer. ABCD Company's Offer: 0 In computing for the present value of ? 250,000 three years from now, the formula to be used is P1 = F (1 +j)_". Given: Solution: F = F 250,000 p1 = F (1 +j)_" . 0.06 _ ,6 j = T = 0.03 300 v P1 250,000 (1 + 0.03) 1e n = 2 (3) = 6 P1 = P 209, 371.06 Given: Solution: F = ? 400,000 P2 = F (1 +j)_" j = 02$ = 0.03 P2 = 400,000 (1 + 0.03)-10 n = 2 (5) = 10 P2 = e297, 637.57 0 In computing for the Fair Market Value (FMV), use the formula: Fair Market Value (FMV) = P1 + P2 = 209,371.06 + 297,637.57 = 13 507,008.63 XYZ Company's Offer: Compute for present value of a general annuity With quarterly payments but with semiannual compounding interest at 6%. o Solve the equivalent rate, compounded quarterly, of 6% compounded semi annually. i4 4(5) 1'2 2(5) P 1 =P 1 (+ 4) (+ 2) 1+ i4 2 _ (1 + 0.06)10 4 _ 2 [(4) 1 1 + T = (1.03)? i(4) _ T _ 1(4) T = 0.014889156 1 (1.03); _ 1 6 LU_Genera| Mathematics_Module12 o The present value of an annuity is computed using the formula: P = R 1\"1TDn. 1 Given: R = 19 35,000 1' = 0.014889156 n = 4 (5) = 20 Solution: P _ 35 000 1(1+0.014889156)_2 0.014889156 P = 1" 601, 559.47 Therefore, XYZ Company's offer is preferable since its market value is larger than the other one. Example 2: Martha inherited a lot property from her grandfather. She wants to sell the lot because she will be moving to other country with her parents. She got two buyers with different offers on the lot that she wants to sell. Jhazzy's offer: Down payment of? 50,000 and a 19 1 million lump sum payment, 5 years from now. Nika's offer: Down payment of 1; 50,000 plus 13 40,000 every quarter for ve years. If money can earn 5% compounded annually, compare the fair market values of the two offers. Which offer has a higher market value? Given: I Jhazzy's offer I Nika's offer I Down payment of P 50,000 Down payment of P 50,000 ? 1,000,000 after 5 ears P 40,000 eve uarter for ve ears Find: Fair Market Value of each offer a. Illustrate the cash ows of the two offers using time diagrams. J hazzy's offer 50,000 1,000,000 0 l 2 3 20 Nika's offer 50,000 40,000 40,000 40,000 40,000 0 l 2 3 20 7 LU_Genera| Mathematics_Module12 Solution: Choose a focal date and determine the values of the two offers at that focal date. For example, the focal date can be the date at the start of the term. Since the focal date is at t = 0, compute for the present value of each offer. Jhazzy's offer: Since 19 50,000 is offered today, then its present value is still P 50,000. The present value of F 1,000,000 offered ve years from now is computed using the formula: P =F(1+j)'". Given: Solution: F = F 1,000,000 p = F (1 +j)n . _ 0.05 _ ,5 1 T 0.05 P = 1,000,000 (1 + 0.05) n = 1 (5) = 5 P = P78352617 Fair Market Value (FMV) = Down payment + Present Value Fair Market Value (FMV) = 50, 000 + 783,526.17 = ?833,526.17 Nika's offer Compute for the present value of general annuity with quarterly payments but with annual compounding at 5 %. Solve the equivalent rate of 5% compounded annually. F1: F2 i4 405) i1 1(t) P 1 2P 1 ( + 4) ( +1) 1+ i4 4_ (1+ 0.05)1 4 _ 1 L'(4') 1 1 =1.05_ +4 ( )4 1(4) 1 = (1.05)4 1 4 1(4) T = 0.012272234- o The present value of an annuity is computed using the formula: P = R 1'(1}T'j)_n. Given: R = F 40,000 j = 0012272234 n = 4 (5) = 20 Solution: P _ 40 000 1 (1+ 0.012272234)'20 0.012272234 P = ? 705, 572.68 8 LU_Genera| Mathematics_Module12 Fair Market Value (FMV) = Down payment + Present Value Fair Market Value (FMVI = 50, 000 + 705, 572.68 = ? 755,572. 68 Hence, Jhazzy's offer has a higher market value. The difference between the market values of the two offers at the start of the term is 833, 526.17 755, 572.68 = ? 77,953.49 Present Value and Period of Deferral of a Deferred Annuity Deferred annuity is a type of annuity that does not begin until a given time interval has passed or payments are done at some later date of a period of time. In this type of annuity, we also consider a very important concept which is the period of deferral, this refers to time between the purchase of an annuity and the start of the payments for the deferred annuity. Time Diagram for a. Deferred Annuity R* R\" R* R R 0 1 2 k k+1 k+2 k+n In this time diagram, k \"articial payments\" of R*are placed in the period of deferral. Example 1: Calculating the Period of Deferral of a Deferred Annuity A. Quarterly payments of 19 300 for 9 years that will start 1 year from now. Answer: The rst payment is at time 4 because there are 4 quarters in 1 year. The period of deferral is from time 0 to 3, which is equivalent to 3 periods or 3 quarters. B. Semiannual payments of 19 10,000 for 13 years that will start 4 years from now. Answer: The rst payment is at time 8. The period of deferral is from time 0 to 7, which is equivalent to 7 periods or 7 semiannual intervals. In solving for the Present Value of a Deferred Annuity, we are going to use the formula below. 1 (1+j)"'\"\") R 1 (1+j)\"' j 1' P=R where R = regular payment j = interest rate per period and is given by the formula, j = i where ris the annual interest rate and m is the number of compounds. Note: annually (m =1) quarterly (m =4) semiannually (m =2) monthly (m. =12) n = number of actual payments and is given by the formula, n=mt where m is the number of compounds and t is time express in years. k: = number of conversion periods in the period of deferral (or number of articial payments) and is given by the formula, n = mt where m is the number of compounds and t (period of deferral) is time express in years. 9 LU_Genera| Mathematics_Module12 Example 1: Calculating the Present Value of a Deferred Annuity Mr. Song is celebrating his 40th birthday. He decided to buy a pension plan for himself that will allow him to claim P 10,000 quarterly for 5 years starting 3 months after his 50th birthday. How much should he pay on his 40th birthday to pay off this pension plan if the interest rate is 8% compounded quarterly? Given: R = P 10,000 i (4) = 0.08 t = 5 m= 4 Find: P or the present value Solution: The annuity is deferred for 20 years and it will go on for 5 years. The first payment is due three months (one quarter) after his 60th birthday, or at the end of the 81-conversion period. Thus, there are 80 artificial payments. Number of artificial payments: k = mt = 4(20) = 80 Number of actual payments: n = mt = 4 (5) = 20 Interest rate per period: j =- _0.08 = 0.02 4 If you assume that there are payments in the period of deferral, there would be a total of k + n =80+ 20 = 100 Time Diagram: P 10,000 10,000 . .. 10,000 O 2 . . . 80 81 82 . . . 100 The present value can be solved using the formula: P = R- 1 - (1+ j)-(k+n) R . 1 - (1 + j) - 1 - (1 + 0.02)-(80+20) P = 10,000 1 - (1 + 0.02)-80 10,000 0.02 0.02 (1.02) -100 1 - (1.02)-80 P = 10,000 - 10,000 0.02 0.02 P = 430,983.5164 - 397,445.1359 P = 33, 538.38 Therefore, the present value is P 33, 538. 38. 10 LU_General Mathematics_Module 12Activity 2: Annuities in Real-Life! In the previous lesson, you have learned about deferred annuity and its applications in real life. Your task now is to nd for reallife situations showing an application of Deferred Annuity in your community. Follow the given format below. Example: 1. Search the page of an appliance store and check how much a certain appliance costs if it is (a) paid in full, or (b) paid by installment or you can also interview someone who is working in an appliance store in your barangay. Compare the payment in cash and in installment basis. Which is better to pay in cash or in installment basis? Why? Annuities in My Community Problem/Situation: Regular Payment Period of Deferral Annual Interest Rate Number of Actual Payments Interest Rate per Period Number of conversion periods in the deferral Solution