CHAPTER 1: MATHEMATICS OF INVESTMENT

Why do you need to know nvestments, bonds, stocks, nterests? Why s there a need to nvest your hard earned money? Whether you just want to save for that phone or tablet that you wanted to buy; or you re savng for a gft that you wanted to gve to your parent s brthday; or you as a parent plannng on usng your credt card n payng your chld s tuton fee payable n 6 months wth 0% nterest; all these reasons why you plan nvest or save your money, entals knowng some thngs about nterest rates, and deas n nvestment or fnance. Studyng the mathematcs behnd fnance and nvestment Havng the knowledge n basc concepts n busness mathematcs or the mathematcs of nvestment may help you decde whether to use that credt card for a 5% nterest compounded monthly or a smple nterest for a perod of 6 months. Some topcs mght shed lght on whch banks would gve a hgher nterest rate for your savngs. As a young couple startng a famly, one mght plan for ther chldren s future by understandng stocks and bonds or fund accumulatons. These are but a few reasons for nvestng your hard earned money. In ths chapter, we wll dscuss the concepts of smple and compound nterest as well as smple and general annutes. The notons of stocks and bonds wll be vewed as a smple fnancal nstrument. Prepared by: Francs Joseph H. Campeña 1

1.1 Smple Interest In fnancal transactons an nterest s the amount pad by a borrower to a lender for the use of money over a perod. Interest that s pad as a percent of amount borrowed or nvested s called smple nterest. The formula for smple nterest s gven by the followng: Smple Interest Where, I = Prt I s = nterest earned (owed) P = prncpal amount nvested (or borrowed) r = annual rate of nterest t = term or tme frame n years Example 1. Suppose Kko wanted to nvest an amount Php 50,000.00 for 2 years at a fnancal nsttuton that gves a smple nterest of 3% per year. The nterest rate was gven to Kko by the fnancal nsttuton on the assumpton that he cannot wthdraw the nvestment wthn the 2-year perod. How much s Kko s earnng on the nvestment after the 2-year perod? Soluton: The followng can be obtaned from the problem: P 50, 000, r 0. 03, t 2. I Prt 50,000 0.03 2 3, 000. From ths we conclude that, the nvestment earned Php3, 000. 00. Prepared by: Francs Joseph H. Campeña 2

Interest can be vewed as a lender or a borrower. Sometmes f we are the nvestor, we consder the value of our nvestment after a gven perod. In ths case we ntroduce the concept of future values or accumulated values or maturty value. Future Value F = P + I s F = P(1 + rt) Where, I s = nterest earned (owed) P = prncpal amount nvested (or borrowed) r = annual rate of nterest t = term or tme frame n years Example 2. Aprl wants to borrow Php40, 000.00 from a bank that gves an annual nterest rate of 4.5%. However, she only wants to borrow the fund for a 9-month perod and wll be able to pay the bank mmedately after 9 months. How much nterest s she gong to pay from borrowng the amount of money? What s the accumulated value of the amount borrowed after the 9-month perod? Soluton: The followng can be obtaned from the problem: P 40, 000, r 0. 045, t 0. 75 snce she only borrowed the fund for 9-months whch s ¾ of a year. I s Prt Php1, 350. 00. 40,000 0.045 0.75 1, 350. From ths we conclude that, the nterest due s Prepared by: Francs Joseph H. Campeña 3

In ths example the accumulated value of the amount borrowed s F = P + I s that s; the sum of the prncpal amount or the amount borrowed and the nterest. Thus, after nne months, Aprl wll pay the bank Php 41, 350. 00 Example 3. What s the smple nterest rate appled f an nvestment of Php37,500 accumulates to Php41,812.5.00 n the perod of 5 years? Soluton: We note that the nterest earned by the nvestment s Php4312.5 that s, I 4312. 5. From the formula I Pr t, we have r I Pt 4,312.5 37,500 5 0.023 2.3% Example 4. The repayment on a loan was Php16,275. If the loan was for 15 months or 1.25 years at 6.8% nterest a year, how much was the prncpal? Soluton: Based from the gven we have the followng: F 16, 275, r 0. 068, and t 1. 25 Snce F P rt 1, we have F 16,275 P 15,000. 1 rt 1 0.068 1.25 Prepared by: Francs Joseph H. Campeña 4

Dfferent ways of expressng tme/term of a loan or nvestment. Sometmes the term of nvestment s not gven n years. The term or tme frame gven n certan problems maybe stated n days or months. In cases where the tme s expressed n months t s easy to express t n years. But when the term/tme s gven n days we use a tme factor such as the followng: Actual tme 360 Actual tme 365 Ordnary Smple Interest or Bankers Rule Exact Smple Interest Approxmate tme 360 Approxmate tme 365 Actual tme Number of days untl the repayment date except the orgn date. Approxmate tme Assumes that every month contans 30 days. Remark The Bankers Rule or Ordnary Smple Interest s appled whenever a gven problem does not specfy the tme factor to be used. Example 5. Fnd the actual and approxmate tme from May 1, 1983 to September 15, 1983. Actual Tme May 30 June 30 July 31 Aug 31 Sept 15 137 31-1=30 Approxmate Tme May 29 30-1=29 June 30 July 30 Aug 30 Sept 15 134 Prepared by: Francs Joseph H. Campeña 5

Example 6. Fnd the actual and approxmate tme from Aprl 15, 2008 to December 21, 2008. Actual Tme Aprl 15 May 30 June 30 July 31 Aug 31 Sept 30 Oct 31 Nov 30 Dec 21 250 30-15=15 Approxmate Tme Aprl 15 30-15=15 May 30 June 30 July 30 Aug 30 Sept 30 Oct 30 Nov 30 Dec 21 246 Example 7. Fnd the actual and approxmate tme from June 25, 2008 to Nov 18, 2008. Actual Tme June 5 July 31 Aug 31 Sept 30 Oct 31 Nov 18 146 30-25=5 Approxmate Tme June 5 30-25=5 July 30 Aug 30 Sept 30 Oct 30 Nov 18 143 Prepared by: Francs Joseph H. Campeña 6

EXERCISES 1. Determne the Actual and Approxmate number of days n the gven orgn and repayment dates. Orgn Date Repayment Date Actual Tme Approxmate Tme A May 22, 1995 July 09, 1995 B January 06, 1997 November 06, 1997 C March 03, 2007 October 11, 2007 D February 04, 1990 November 05, 1992 E March 02, 2005 November 05, 2006 2. Joseph borrowed Php 5,000 on November 2, 2015 from Arthemus, whch s to be repad on May 21, 2016 at 6.2% smple nterest per year. Fnd the amount to be repad. How much wll the nterest be at the repayment date f the followng tme factors are used? a. Bankers Rule b. Exact Smple Interest c. Approxmate 360 d. Approxmate 365 3. How much should Mark pay to Mchele f he borrowed Php 10,000 on June 25, 2015 and f the prncpal and nterest are to be pad on November 18, 2015 at 15% smple nterest per year? Use the followng tme factors. a. Bankers Rule b. Exact Smple Interest c. Approxmate 360 d. Approxmate 365 Prepared by: Francs Joseph H. Campeña 7

4. At what smple nterest rate wll a sum of money double tself n 5 years? 5. If Wendy wants to nvest her Php25,000, how many years wll t take for her savngs to accumulate to Php 40,000 f she nvested her savngs to a fnancal nsttuton that provdes a smple nterest rate s 4.5% per year? 6. An amount of P12, 500 s nvested at 3.25% smple nterest for 3 years. Complete the table below. Tme (n years) Prncpal Rate Interest Future Value 0 P12, 500 0.0325 P0 P0 7. W 1 hat 2 wll 3 be the maturty value of P15, 500 f t s borrowed at 10.5% rate for 10 months? 8. How much should Mrs. Dolores nvest today n a tme depost wth 5.5% nterest f she expects to have P175, 000 for hs son s educaton at the end of 5 years? 9. Mr. Pascual, an arlne owner, decded to nvest P2.5 mllon to fund hs department handlng spare parts replacement. How long wll t take hs nvestment to accumulate to P3.325 mllon f the bank s nterest rate s 5.5%? 10. FJC Prntng would lke to nvest a certan amount n a bank that wll accumulate to P157, 800 n 550 days for the replacement of a prnter. If the bank offers 8% nterest, how much must be nvested at the start of the term under 11. Use Banker s Rule to compute the smple nterest of P10, 000 nvestment at 10% smple nterest rate from Aprl 14, 2004 to November 18 of the same year. 12. The unversty treasurer puts half a mllon pesos to a tme depost offerng 7% for 2 years. How much s n the fund at the end of the term? 13. What rate was appled to a 4-year loan of P278, 000 n whch the maturty value s P372, 520? Prepared by: Francs Joseph H. Campeña 8

1.2 Compound Interest Consder an nvestment whose tme frame s dvded nto equal ntervals. If an nterest s computed after an nterval and s beng added to the prncpal and thereafter earns an nterest, then the dfference between the orgnal prncpal and the total amount after the whole tme frame s called compound nterest. The compound amount or the accumulated value of the prncpal s the sum of the prncpal and the compound nterest. In ths stuaton, we see that the nterest s beng converted nto a prncpal and thus we use the phrase nterest s compounded or nterest s converted. Consder an nvestment amount P place on a fnancal nsttuton that gves a compound nterest where the nterest rate per converson perod s. After one converson, the total amount due to the nvestor s P + P. The new prncpal at the end of the frst converson s now P + P = P(1 + ). At the end of the second perod, the accumulated value now becomes, P + P + (P + P) whch s equvalent to P + P + P + P 2 = P(1 + ) 2. Thus, the new prncpal after the second converson s P(1 + ) 2. In a smlar manner, at the end of the thrd perod or thrd converson, the accumulated value becomes P(1 + ) 2 + P(1 + ) 2 = P(1 + ) 2 (1 + ) = P(1 + ) 3. Now, followng the patterns that we see, after the n th converson, the accumulated value becomes P(1 + ) n. Thus we have the followng formula: Formula F = P(1 + ) n Where, F = compound amount or accumulated value of P at the end of the term P = present value or orgnal prncpal j =nterest rate per year = nterest rate per perod n= total number of conversons perods m = number of conversons per year t =term = j n = tm Prepared mby: Francs Joseph H. Campeña 9

In the context of compound nterest, the nterest rate per annum or per year s called the nomnal rate of nterest. Thus when a gven nomnal rate s sad to be compounded quarterly, that means n a gven year there wll be 4 conversons. Smlarly, when we say compounded monthly, the conversons are made every month therefore n a gven year, there wll be 12 conversons. Example Fnd the compound amount and nterest: a. If Php 2,500 s nvested at 13% compounded quarterly for 12 years Soluton Gven: P = 2,500; j = 13% ; t = 12; m = 4 n = tm = 4 12 = 48 = j m = 0.13 3 = 0.0325 F = P(1 + ) n = 2,500(1.0325) 48 = Php 11, 605. 47 b. If Php 3,700 s nvested at 12% compounded sem-annually for 5 years Soluton Gven: P = 3,700; j = 12% ; t = 5; m = 2 n = tm = 2 5 = 10 = j m = 0.12 2 = 0.06 F = P(1 + ) n = 3,700(1.06) 10 = Php 6, 626. 14 Example Fnd the present value of Php 2,850 due n 5 years f money s worth 10% compounded quarterly. Soluton Gven: F = 2,850; j = 10% ; t = 5; m = 4 Prepared by: Francs Joseph H. Campeña 10

n = tm = 5 4 = 20 = j m = 0.1 4 = 0.025 P = F (1 + ) n = 2,850(1.025) 20 = Php 1, 739. 27 Example How much must be nvested today n a savngs account to realze Php 9,000 n 4 years, f money earns at the rate of 4% compounded quarterly? Soluton Gven: F = 9,000; j = 4% ; t = 4; m = 4 n = tm = 4 4 = 16 = j m = 0.04 4 = 0.01 P = F (1 + ) n = 9,000(1.01) 16 = Php 7, 675. 39 Example What rate compounded annually wll double any amount prncpal f t s nvested n 6 years? Soluton Let x be the amount to be nvested. m = 1; t = 6 n = 6. We want to fnd j. F = P(1 + ) n 2x = x (1 + j 6 1 ) 2 = (1 + j) 6 6 j = 2 1 j = 12. 25% Example What nomnal rate converted sem-annually wll make Php 20,000 amount to Php 50,000 n 8.5 years? Soluton Prepared by: Francs Joseph H. Campeña 11

We want to fnd j. Gven: m = 2; t = 8.5 n = 17. F = P(1 + ) n 50,000 = 20,000 (1 + j 1 ) 17 j = 11. 08% 2.5 = (1 + j 17 2 ) 17 j = 2( 2.5 1) Example When wll Php 30,000 earn nterest of Php 15,000 f t s nvested at the rate of 7.5% converted annually? Soluton We want to fnd t. Gven: F = 45,000; P = 30,000 ; j = 7.5%; m = 1 F = P(1 + ) n 45,000 = 30,000 (1 + 0.075 1 ) n 1.5 = (1 + 0.075) n log1.5 n = = nlog(1.075) log1.5 n = 5.6065 t = 5. 6065 log(1.075) Example. When wll a prncpal double tself f the nterest rate s 14% compounded quarterly? Soluton We want to fnd j. Gven: F = 2P ; j = 14%; m = 4 F = P(1 + ) n 2P = P (1 + 0.14 4 ) n 2 = (1.035) n log2 = nlog(1.035) n = log2 n = 20.1488 t = 5. 0372 log(1.035) Sometmes we may want to compare whch nterest rate would provde a hgher nterest when ther nterest payments are not the same. We then have to resort to convertng these nterest rates to a common nterest payment. Ths s the noton of effectve rates of nterest. For nstance, whch nterest rate gves a hgher nterest for an nvestment of 1 Peso (Php1), an 8% Prepared by: Francs Joseph H. Campeña 12

compounded sem-annually or a 7.9% compounded monthly. If 1 peso s nvested at a rate of 8% compounded sem-annually, then at the end of the year t accumulated to S = 1 (1 + 0.08 2 )2 = 1.0816. Whle an nvestment of 1 peso at a rate of 7.9% compounded monthly accumulates to S = 1 (1 + 0.08 12 )12 = 1.082999507. The effectve rate of nterest of compounded m tmes a year can be computed as r = (1 + ) m 1 Example Determne the effectve rate of nterest for each of the followng nomnal nterest rate j compounded m tmes a year. j m = j r = (1 + ) m 1 m 2% 4 0.5% (1 + 0.5%) 4 1 = 4.0625% 3% 3 1% (1 + 1%) 3 1 = 7% 1.5% 12 0.125% (1 + 0.125%) 12 1 = 3.10989% 5% 4 1.25% (1 + 1.25%) 4 1 = 24.6289% EXERCISES 1. Mary Joy deposted P14,500 n a bank that pays nterest at 2% compounded annually. If no wthdrawal s made, how much does she have n her account after 5 years? 2. If the prncpal nvested by Anka s P50,000 and the nterest rate gven by Peso Fnancal Inc. s 2.5% compounded quarterly, how much dd she earned at the end of 5 years? 3. Fnd the compound nterest earned at the end of 2 year and 5 months of an nvestment fund amountng to P24,500 f t s nvested at 3.5% compounded monthly. Prepared by: Francs Joseph H. Campeña 13

4. If money can be nvested at 7% compounded monthly, fnd the present value of 55,300 whch s due after 2 years and 11 months from today. 5. Jame wants to have P45,000 n 2 years to buy a new computer. How much money should he nvest today n a fund that earns 5% compounded quarterly to get ths amount after 2 years? 6. A 35,000 prncpal earned an nterest of P8,500 at the end of 7 years. At what nomnal rate, compounded annually, was t nvested? 7. At what rate compounded monthly should P25,000 be deposted n a bank to gan an nterest of P4,500 n 3 years? 8. If P48,000 s nvested at the rate of 2.5% compounded quarterly, when wll the compound amount be P70,000? 9. When wll P80,000 grow to P95,000 f t s nvested at 4.5% compounded quarterly? 10. If P135,650 s the maturty value of a sum nvested at 3.2% compounded sem-annually for 9 years and 6 months, fnd the present value and the compound nterest earned. Prepared by: Francs Joseph H. Campeña 14

1.3 Annutes An Annuty s a sequence of equal payments made at equal perods or tme ntervals. These payments may be made annually sem-annually, quarterly or at other perods. Some examples are: Monthly payments of rent Monthly wages Annual premums on a lfe nsurance polcy Quarterly payments for a car loan Annual payments on a bond In our study of annutes, we need to be famlar wth the followng terms: payment nterval, term, perodc payment, smple annuty. DEFINITION The payment nterval s the tme between successve payments of an annuty. The term of an annuty s the tme between the frst payment nterval and the last payment nterval. The perodc payment, denoted by R, s the amount of each payment. A Smple annuty s an annuty n whch the payment nterval s the same as the nterest perod. If the payment ntervals are not the same as the nterest perod, then the annuty s called a general annuty. Example 16. A four-year lease agreement between Alfred and Thrfty Mall Inc. (TMI) ndcates that, Alfred pays TMI Php 100,000 at the end of every year f the agreed nterest rate s 5% compounded quarterly. In ths example, the payment perod s a whole year. However, the nterest perod s quarterly or every 3 months. Hence, the annuty s a general annuty. Prepared by: Francs Joseph H. Campeña 15

Example 16. Remy needs to repay her debt to Marvn by payng Php 150 at the end of every 6 months for 4 years n a bank account that gves an nterest rate of 4.35% compounded semannually. Ths s an example of a smple annuty snce the payment ntervals are the same as the nterest perod. Perodc Payment 150 150 150 150 150 150 150 150 0 1 2 3 4 Term : 4 years There are three types of smple annutes and they are Ordnary Annutes, Annuty Dues and Deferred Annutes. These annutes dffer only by the date of the frst regular payment and hence, t would be natural to see a relatonshp between the formulas for the accumulated value or present value of an annuty compared to the other DEFINITION An Ordnary Annuty s an annuty n whch the payments are made at the end of each payment nterval. Perodc Payment R R R R R R 0 1 2 3 4 n-1 n 2 nd nterval 1 st nterval Term: n payments Prepared by: Francs Joseph H. Campeña 16

The sum or accumulated value of the annuty or amount of an ordnary annuty, denoted by S, s gven by S = R + R(1 + ) + R(1 + ) 2 + + R(1 + ) n 2 + R(1 + ) n 1 = R (1 + )n 1 The present value of an ordnary annuty, denoted by A s just the sum of the present value of each of the payments and s llustrated below: Perodc Payment R R R R R R 0 1 2 3 4 n-1 n A 1 = R(1 + ) 1 A 2 = R(1 + 2 ) A n 1 = R(1 + ) n+1 A n = R(1 + ) n A = A 1 + A 2 + A n = R(1 + ) 1 + R(1 + ) 2 + + R(1 + ) (n 1) + R(1 + ) n = R 1 (1 + ) n Example 17. Mchele wants to venture nto the food busness stands. She plans to create a fund by makng deposts of Php 30,000 at the end of every 6 months n a bank that gves 4% nterest compounded sem-annually, how much money wll be n the fund after 4 years? Perodc Payment 30k 30k 30k 30k 30k 30k 30k 30k 0 1 2 3 4 Term : 4 years Prepared by: Francs Joseph H. Campeña 17

Soluton: In ths example we see that the regular payments are R = Php 30,000 and the nterest rate per perod s = 0.04 = 0.02. Snce payments are made twce a year for 4 years, the total number of 2 payments s n = 8. Thus to get the accumulated value of the annuty we have S = R (1 + )n 1 = 30,000 (1 + 0.02)8 1 0.02 = Php 257, 489. 07 Example A father wants to nvest Php 1.5M n a fnancal nsttuton that gves 8% compounded monthly so that he could gve a monthly allowance for hs daughter for the next 15 years. How much wll the monthly allowance be? Soluton: In ths example, the monthly allowance for 15 years s unknown and that the present value of these allowances should amount to Php 1.5M where the rate of nterest = 0.08 = 1 and 12 150 n = (12)(15) = 180. A = R 1 (1 + ) n R = 1 500 000 = R [ 1 (1 + 1 150 ) 180 1 500 000 [ 1 (1 + 1 180 150 ) 1 ] 150 1 150 ] = Php 14 334.78 Thus, the monthly allowance of the daughter from based from the nvestment s Php 14,334.78. Example Maro wants to accumulate Php 100,000 by makng deposts of Php 500 at the end of every month n a cooperatve bank that gves 5% nterest rate compounded monthly. At least how many payments should he make to the bank to attan hs nvestment goal? Soluton: The followng are gven n the problem: S = 100,000, R = 500, = 0.05 and we wsh to fnd n. 12 Prepared by: Francs Joseph H. Campeña 18

In the formula, S = R [ (1+)n 1 ], solvng for n gves us n = log[s, thus pluggng n the known log[1+] values we obtan n 145.78 146. Maro should at least make 146 deposts amountng to Php 500 each to accumulate Php 100,000 n the sad bank. R +1] However, sometmes we want to accumulate a certan amount of money by savng or depostng a regular payment wthn a specfed tme frame. We then look for banks that can provde such an nvestment scheme and thus we need to check whch banks offers an nterest rate that wll accommodate the gven stuatons. If A s gven, then solvng for n the equaton (n 2 1) 2 + 6(n + 1) + 12 (1 nr ) = 0, provdes us that answer. However, f S s gven, then solvng for A n the equaton (n 2 1) 2 6(n 1) + 12 (1 nr ) = 0 gves us the answer. S These equatons are of quadratc n form and the use of a quadratc formula to solve for leads us to Lastly to solve for the nomnal rate, we have Where m s the number of conversons per year. = b ± b2 4ac 2a j = m Example John bought a car worth P984,000. He pad P90,000 cash and agreed to pay P15,500 at the end of every month for 10 years. At what nomnal rate compounded monthly was he charged? Soluton: The followng are gven n the problem: A = 894,000, R = 15,000, t = 10, m = 12 and we wsh to fnd j. (120 2 1) 2 + 6(120 + 1) + 12 (1 120(15000) 894,000 ) = 0 14399 2 + 726 12.1611 = 0 Prepared by: Francs Joseph H. Campeña 19

We now use quadratc formula to solve for. = 726 ± 7262 4(14399)( 12.1611) 2(14399) Snce the other value of s -0.064 whch s not possble, we have = 0.0133. The nomnal rate s j = (0.0133)(12) = 0.1596 = 15.96%. Example Sheena wanted to save P500,000 n 15 years. She was proposed by a fund manager to nvest P7,500 every quarter at Sgma Investment Group. Fnd the nterest rate gven by Sgma Investment Group. Soluton: The followng are gven n the problem: S = 500,000, R = 7,500, t = 15, m = 4 and we wsh to fnd j. (60 2 1) 2 6(60 1) + 12 (1 (60)(7,500) 500,000 ) = 0 3599b 2 354 + 1.2 = 0 We now use quadratc formula to solve for. = 114 ± ( 114)2 4(399)( 84) 2(399) We obtan two values of, whch are 0.0948 and 0.0035. To determne whch s the rght nterest rate, we have to solve of S usng both values. When = 0.0948, S = 7500 ( (1+0.0948)60 1 ) = 18,048,483.6 0.0948 When = 0.0035, S = 7500 ( (1+0.0035)60 1 ) = 499,769.62 0.0035 Prepared by: Francs Joseph H. Campeña 20

Note: Snce we rounded off the values of, we wll not get the exact value of P500,000. From the computed values of S, the nomnal rate gven by Sgma Investment Group s 1.4%. Example Earl s nterested n buyng a lot worth P2,500,000. He agrees to gve a down payment of P500,000 and to pay P23,500 every month for as long as necessary. He wll be charged an nterest of 8.5% monthly. a. How many monthly payments of P23,500 must he make? b. How much s the fnal payment f t s to be made on the same day as the last P23,500 payment? c. How much s hs fnal payment f t s to be made one month after the last P23,500 payment to completely pay off the lot? Soluton: The followng are gven n the problem: = 2,000,000 R = 23,500, = 0.085 12. a. In the formula, A = R [ 1 (1+) n We have n = payments of P23,500. log[1 (2,000,000)(0.085 12 ) ] 23,500 ], solvng for n gves us n = log[1 A R ]. log[1+] log[1+ 0.085 12 ] = 130.82. Ths means that Earl wll be makng 130 full b. Gven that n=130.82, t tells us that we don t need 131 full payments of P23,500. We must compute for the addtonal amount to be pad on the 130 th month. We wll be usng the 130 th month as the comparson date. We now solve the value of 2,000,000 and total payment at the 130 th month. 2,000,000 (1 + 0.085 12 ) 130 = 5,006,487.38 Prepared by: Francs Joseph H. Campeña 21

130 0.085 (1 + S 130 = 23,500 ( 12 ) 0.085 12 1 ) = 4,987,232.01 The addtonal amount wll be 5,006,487.38 4,987,232.01 = 19,255.37. The fnal payment wll be P42,755.37. c. Usng the value computed from (b), the fnal payment to be made one month after the last P23,500 payment s 19,255.37 (1 + 0.085 12 ) = 19,391.76 DEFINITION An Annuty Due s an annuty n whch the payments are made at the begnnng of each payment nterval. Perodc Payment R R R R R R 0 1 2 3 4 n-1 n 2 nd nterval 1 st nterval Term: n payments The sum or accumulated value of the annuty due or amount of an annuty due, denoted by S, s gven by S = R(1 + ) n+1 + R(1 + ) n + R(1 + ) n 1 + + R(1 + ) 2 + R(1 + ) = R (1 + )n+1 1 R Prepared by: Francs Joseph H. Campeña 22

The above formula for the accumulated value of an annuty due s equvalent to S = R ( (1 + )n+1 1 ) 1 = R ( (1 + )n+1 1 ) = R ( (1 + )n+1 (1 + ) ) = R ( (1 + )n 1 ) (1 + ). The above formula shows the relatonshp of the accumulated value of an ordnary annuty S and the accumulated value of an annuty due, S ; that s S = S(1 + ). The present value of an annuty due, denoted by A s just the sum of the present value of each of the payments and s llustrated below: Perodc Payment R R R R R R 0 1 2 3 4 n-1 n A 1 = R(1 + ) 1 A 2 = R(1 + 2 ) A n 1 = R(1 + ) n+1 A = R + A 1 + A 2 + A n 1 = R + R(1 + ) 1 + R(1 + ) 2 + + R(1 + ) (n 1) = R + R 1 (1 + ) (n 1) Equvalently, we have A = R 1 + [ 1 (1+) (n 1) ] = R [ +1 (1+) (n 1) ] = R(1 + ) ( 1 (1+) n ) Example A father wants to set up an educatonal fund at Thrft Bank of the Phlppnes for hs daughter by makng a depost of Php 10, 000 at the begnnng of every quarter of the year for 10 years. If money s worth 2.5% n Thrfty Bank of the Phlppnes, how much money s n the fund at the end of 10 years? Prepared by: Francs Joseph H. Campeña 23

10k 10k 10k 10k 10k 10k 10k 10k 10k 10k 10k 10k CHAPTER 1: MATHEMATICS OF INVESTMENT Soluton: Perodc Payment 0 1 2 3 4 5 6 7 8 9 10 Term : 10 years or 40 regular deposts The annuty formed by the deposts s an annuty due wth regular payments R = 10,000, n = 40 and = 0.025 = 0.00625. We then have the followng: 4 S = R ( (1 + )n 1 ) (1 + ) = 10,000 ( (1 + 0.00625)40 1 ) (1.00625) 0.00625 = Php459,325.14 Thus, at the end of 10 years, the fund wll be Php459,325.14. Example A smart phone can be bought by makng a cash payment of P5,000 and 6 quarterly payments of P4000 each, the frst of whch s due on the date of purchase. Fnd the cash value of the smart phone f money s worth 6% compounded quarterly. Soluton: The followng are gven n the problem: R = 4,000, = 0.06 = 0.015, n = 6. 1 (1 + ) n A = R(1 + ) ( ) = 4,000(1.015) [( 1 (1.015) 6 )] 0.015 = 23,130 CV = A + DP 4 = 23,130 + 4,000 = Php 27,130. Prepared by: Francs Joseph H. Campeña 24

Example Joshua wants to vst Japan 3 years from now. He wll be needng a sum of P100,000 for hs trp. How much must he put asde n hs travel funds every year startng now f money s worth 6% compounded annually? Soluton: The followng are gven n the problem: S = 100,000, = 0.06 t = 3. S = R ( (1 + )n 1 ) (1 + ) 100,000 = R [( (1+0.06)3 1 )] (1.06) R = 0.06 100,000 [( (1+0.06)3 1 )](1.06) 0.06 R = 29,633 DEFINITION A Deferred Annuty s an ordnary annuty n whch the frst payment s made at some later date. Perodc Payments R R R R R 0 1 2 d d+1 d+2 d+3 d+4 n+d-1 n+d 2 nd nterval n th nterval 1 st nterval No. of deferred perods: d perods Term: n payments Prepared by: Francs Joseph H. Campeña 25

The sum or accumulated value of a defer annuty denoted by S d s equvalent to that of the accumulated value of an ordnary annuty. As for the present value of a deferred annuty, we use the notaton A d and s computed as A d = A(1 + ) d where A s the present value of an ordnary annuty, s the nterest rate appled for every nterval and d s the number of deferred payment ntervals or perod. We can see that the accumulated value and present value of the three types of smple annutes are related as shown below. Present Value Ordnary Annuty Annuty Due Deferred Annuty S = R (1 + )n 1 S = R ( (1 + )n 1 ) (1 + ) S d = S Accumulated Value A = R 1 (1 + ) n 1 (1 + ) n A = R(1 + ) ( ) = A(1 + ) A d = A(1 + ) d EXERCISES 1. Fnd the amount of a Php 100 ordnary annuty payable sem-annually for 3 years f money s worth 5% converted sem-annually. 2. A farmer bought a farmng tractor. If t was purchased under the followng terms: Php 50,000 down payment and Php 4,500 payment every month for 5 years. If money s worth 6% compounded monthly, fnd the cash prce of the car. 3. Francs wants to accumulate a fund for hs daughter s graduaton gft. He deposts Php 1,500 every 3 months for 9 years n a savngs account that pays 4% compounded quarterly. How much would he save n the account at the end of 9 years? (assume that no wthdrawals were made and the startng balance of the account s Php0.00) 4. A student nvests Php 500 every 6 months at 4% compounded sem-annually. Fnd hs savngs n 12 years. Prepared by: Francs Joseph H. Campeña 26

5. Anka purchased a property and pays Php 250,000 cash and the balance s to be repad n 20 annual payments of Php 10,000 each. If money s worth 4.5%, what s the cash value of the property? 6. Fnd the amount of an ordnary annuty whch pays Php 850 at the end of each 3 months for 5 years f money s worth 8% compounded quarterly. 7. Every 6 months for 5 years, a father deposts Php 3000 n a trust company for hs daughter s educaton. If the money earns at 16% compounded sem-annually, how much wll be n the fund after the 7 th depost? After the last depost? 8. Mrs. Bautsta s a chef fnancal offcer of TCB Caterng Servces. He s proposed to the company that they offer a retrement plan for a company employee who s now 55 years of age. The plan wll provde an annuty due of Php7,000 every year for 15 years upon retrement at the age of 65. The company s fundng the plan wth an annuty due of 10 years. If the rate of nterest per year s 5%, what s the amount of nstallment that the company should pay to fund ths retrement. 1.4 Stocks and Bonds Stocks are also known are equty securtes, whle bonds are smlar to promssory notes. These two fnancal nstruments are commonly used by bankers, traders, and busnessmen n ther fnancal portfolos. Stocks represent shares of ownershp of a company. By a share, we mean a unt of ownershp of a corporaton s proft and asset. For example, ABC Tradng Co. has released 100 shares for ts proft and assets. If Francs bought 50 shares from ABC Tradng Co. then he has 50% ownershp of the profts and assets of the company. People who buys stocks usually receves a certfcate ndcatng the pertnent detals of the stock lke the company name, name of stock holder, certfcate number, number of shares owned and par value of the share. There are two types of stocks, the common stocks and the preferred stocks. Prepared by: Francs Joseph H. Campeña 27

Common stock: represents a share of company's asset and proft. Holders of common stock can vote n electon of the board of drectors (normally one vote per share). The board of drectors oversee the management of the company, but do not drectly run the company. Common stock s hgh rsk and hgh return. Although common stocks yeld hgher return than other stocks, common shareholders stand to lose most when a company goes bankrupt. Preferred stock: Holders of preferred stock, n most cases, cannot vote. On the other hand, they are guaranteed a fxed dvdend before any dvdends are dstrbuted to other shareholders. In the event of bankruptcy and lqudaton, shareholders of preferred stocks are pad off after credtors and before common shareholders. When a certan company sells stocks for the frst tme n publc t s generally called an IPO, short term for ntal publc offerng. In the lterature, some refer to ths as Gong Publc. The paper works and detals of ths publc offerng s beng handled by fnancal nsttutons or nvestment banks called Underwrters. If for nstance, an ndvdual would lke to buy all the shares of stocks that a company s sellng, the ndvdual should pay a market cap equvalent to the prce per share multpled to the number of shares beng sold. There are two ways of earnng from stocks these happens when earnngs are pad out to you n the form of dvdends or there s an ncrease n share prce. Two types of stock market namely: Prmary market and Secondary market. Prmary market s where a company ssues ts shares for the frst tme va an IPO. Secondary market s commonly known as stock market where prevously-ssued stocks are traded wthout the nvolvement of the companes whch ssued them. Some well- known stock market nclude the New York Stock Exchange, NASDAQ, London Stock Exchange and Hon Kong Stock Exchange. Here are some mportant tps when plannng to nvest n stocks. Invest n companes you understand and whose busness make sense to you. Prepared by: Francs Joseph H. Campeña 28

If you already bought some stocks or shares, stay for a whle and do not expect to gan a lot n your frst week or frst month. Usually tradng n stocks gves a reasonable return n the long term. Try not to panc when you see fluctuatons n your stock values, but rather make nformed decson. Once you have conceptualzed an nvestment plan for yourself, you need to determne the amount that you want to nvest. Check out the stocks you plan to acqure and set the tmeframe you ntend to keep your stock. And the mportant part s to follow your plan. However, askng for fnancal advce from credble analysts may also help you make a wse nvestment decson. Corporaton ssue bonds to whoever wants to buy them. When you buy a bond, you are actually lendng money to the corporaton that ssues the bond. The corporaton n return, promses to pay nterest on specfed ntervals for the length of the loan. In earler tmes, bond certfcates actually have prnted coupon whch can be detached to collect the nterest. Physcal possesson of the certfcate was proof of ownershp. Prepared by: Francs Joseph H. Campeña 29

http://s28.photobucket.com/user/randellt/meda/phlppneusbond.jpg.html http://meda.lveauctongroup.net//9372/10324011_1.jpg?v=8cda5b95d2756e0 There are several types of bonds. These are Government Bonds, Corporate Bonds and Zero- Coupon Bonds. Government bonds are bonds ssued by the government to fund programs, meet payrolls, pay ther blls or some economc strategy of the government. Whle Corporate bonds are bonds ssued by busness or companes to help them pay expenses or fund some company expanson program. These bonds are usually hgher rsk than government bonds but can earn more consderably. Another type of bonds s called Zero-coupon bonds. These bonds make no coupon payments but nstead s ssued at a consderable dscount to par value. Comparng stocks and bonds, we see that stocks are equtes whle bonds are thought of as debts or loans. When t comes to the growth of your money, stocks may have varyng nterest whle Prepared by: Francs Joseph H. Campeña 30

bonds are usually fxed as stated n the bond certfcates or contracts. Whle stocks may gve a hgher yeld rate compared to bonds, t s usually rsky and unstable. Bonds markets, unlke stock or share markets, often do not have a centralzed exchange or tradng system, whle stock or share markets, have a centralzed exchange or tradng system. In the Phlppnes, the Phlppne Stock Exchange located n Makat s the natonal stock exchange n the country. Stocks pay dvdends to the owners, but only f the corporaton declares a dvdend. Dvdends are a dstrbuton of a corporaton's profts. Bonds pay nterest to the bondholders. Generally, the bond contract requres that a fxed nterest payment be made every sx months. Prepared by: Francs Joseph H. Campeña 31