# FI A CIAL MATHEMATICS

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1 CHAPTER 7 FI A CIAL MATHEMATICS Page Cotets 7.1 Compoud Value Compoud Value of a Auity Sikig Fuds Preset Value Preset Value of a Auity Term Loas ad Amortizatio 123 Exercise 127 Objectives: After workig through this chapter, you should be able to: (i) (ii) (iii) (iv) (v) explai the term compoud iterest; explai the meaig of a auity; calculate the compoud amout ad the preset value of a auity; set up a amortizatio schedule for the amortizatio of debt; set up a sikig fud schedule ad explai some of the applicatios of sikig fuds.

2 7.1 Compoud Value Defie the followig terms : P 0 = pricipal, or begiig amout at time 0. i I = iterest rate = total amout of iterest eared. P = pricipal value at the ed of periods The P may be calculated as follows : P 1 = P 0 + I = P 0 + P 0 i = P 0 (1 + i) P 2 = P 1 + P 1 i = P 1 (1 + i) = P 0 (1 + i) 2 P 3 = P 2 + P 2 i = P 2 (1 + i) = P 0 (1 + i) 3 M M P = P 0 (1 + i) where (1 + i) is called the Compoud Value Iterest Factor. Example 1 A loa of \$35,000 made today is to be repaid by a sigle paymet of \$42,000 two years from ow. Fid the aual iterest rate. 42,000 = 35,000 (1 + i) 2 (1 + i) 2 = 1.2 (1 + i) = i = 9.5% Whe compoudig periods are more frequet tha oce a year, the i P = P m where m is the umbers of times per year compoudig occurs. Example 2 m 117

3 Fid the amout to which \$10,000 will grow after five years if semiaual compoudig is applied to a stated 5% iterest rate. P 5 i = 5% m = = %, =, (. ) = \$12, Compoud Value of a Auity Def : A auity is defied as a series of paymets of a fixed amout for a specified umber of years. Each paymet occurs at the ed of the year. Example 3 Suppose you are to receive a three-year auity of \$10,000 ad deposit each aual paymet i a savig accout payig 8% iterest. How much will you have at the ed of the third year? Ed of Year I geeral, we have : 10,000 10,000 10, ,800 (1.08) 2 11,664 Compoud Sum : \$32,464 ====== R R R R R R(1 + i) R(1 + i) 2 R(1 + i) 1 S Defie S = compoud sum 118

4 R = periodic receipt = legth of auity S = R+ R( 1+ i) + KK + R( 1+ i) + R( 1+ i) 2 1 [ ( 1 ) KK ( 1 ) ( 1 ) ] = R + + i i + + i ( 1+ i) 1 = R i Compoud Value Factor of a auity. 7.3 Sikig Fuds Example 4 - Whe a sum of moey will be eeded at some future date, a good practice is to accumulate systematically a fud that will equal the sum desired at the time it is eeded. Moey accumulated i this way is called a sikig fud. A machie acquired at the begiig of this year is expected to last 10 years ad its replacemet price is estimated to be \$8,000. What aual provisio must be made to esure sufficiet fud is available if moey ca be ivested at 8% per aum? (i) If the sikig fud a is to be set aside at the ed of each year, replace the machie a a a a the a + a(1.08) + a(1.08) a(1.08) 9 = 8,000 a ( 108. ) , = 119

5 a = \$ (ii) Suppose the firm wish to start the fud ow (i.e., set aside a sum of moey a at the begiig of each year) replace the machie a a a a the Example 5 a( 1+ i) + a( 1+ i) 2 + K + a( 1+ i) 10 = 8, [ + (1+ i) + + (1+ ) ] = 8, 000 a(1+ i) 1 K i Sikig Fud Schedules i a( i) ( ) , 000 i = a = The View Royal Fire District eeds a ew fire truck which will cost \$300,000. The district is able to arrage the ecessary fiacig provided a sikig fud is established to provide for repaymet of the debt. The loa must be repaid i four years. Moies for repaymet will come from a tax icrease o the lad owers who are part of the fire district. The iterest cost o the loa must be paid every six moths as per the loa agreemet with the provice. If the district ears 8% compouded aually o the sikig fud, if the iterest rate o the loa is 12% compouded semi-aually ad if the paymet must be made aually to the sikig fud, aswer the followig questios. a) Set up a sikig fud schedule to show the accumulatio of the fud ad the book value of the debt for each year. 120

6 b) Show the total aual expese associated with the iterest paymet ad the sikig fud paymet. A B C D E F Regular Paymet Icrease Accumulated Accumulated Year Paymet Of I Balace Book Value Iterest Fud I the Fud Of the Debt 0 \$300, \$66, \$66, \$66, \$233, \$66, \$5, \$71, \$138, \$161, \$66, \$11, \$77, \$216, \$83, \$66, \$17, \$83, \$300, \$ TOTALS \$266, \$33, \$300, N/A N/A Example 6 Partial Sikig Fud Schedules A small muicipality is settig up a sikig fud with aual paymets to repay a debt of \$520,000. The sikig fud will ear %, compouded aually. If the fud is to accumulate the desired sum of \$520,000 over 28 years, costruct a partial sikig fud schedule which shows the fud at periods 3, 27 ad 28. A B C D E F Regular Paymet Icrease Accumulated Accumulated Year Paymet Of I Balace Book Value Iterest Fud I the Fud Of the Debt 2 \$2, \$5, \$514, \$2, \$ \$3, \$8, \$511, \$406, \$2, \$50, \$53, \$459, \$60, \$2, \$57, \$60, \$520, \$ TOTALS \$69, N/A N/A N/A N/A 121

7 7.4 Preset Value Example 7 Suppose you are offered the alterative of either \$1,500 at the ed of three years or \$x today. If your savig accout pays a 10% iterest p.a. Fid x. x( 110. ) 3 = 1, 500 x= 1, 500 ( 11. ) 3 = 1127, \$x is defied as the preset value (PV) of \$1,500 due i 3 years whe the applicable iterest rate is 10%. Fidig preset values (discoutig) is simply the reverse of compoudig. Recall that P = P ( + i) 0 1 Therefore P 0 = P ( 1+ i) = 1 P ( 1+ i) PV factor Whe discoutig periods are more frequet tha oce a year, the 1 P0 = P i ( 1+ ) m m 7.5 Preset Value of a Auity Defie : A = preset value of a auity of years R = periodic receipt = legth of auity i = iterest rate 122

8 Ed of year R R R R 1 R 1+ i 1 R 1+ i 2 1 R 1+ i A A = [ R ( 1 i) 1 ( 1 i) 2 K ( 1 i) ] 1 ( 1+ i) = R i PV factor of a auity Preset Value i Perpetuity A = R i 7.6 Term Loas ad Amortizatio A term loa is a busiess loa with a maturity of more tha oe year. Ordiarily, term loas are retired by systematic repaymets (ofte called amortizatio repaymets) over the life of the loa. 123

9 7.6.1 Determiatio of Repaymet Schedule Example 8 Assume that a firm borrows \$1,000,000 o a te-year loa, that iterest is computed at 5% o the decliig balace, ad that the pricipal ad iterest are to be paid i te istallmets. Fid the amout of each of the te aual repaymets ad produce a repaymet schedule. Let R be the amout of repaymet R= A 1 ( 1+ i) i from A 1 (1+ i) = R i = 1, = \$130 i.e. Te istallmets of \$130,000 will have retired the oe millio loa ad provided the leder a 5% retur o his ivestmet. Total Amortizatio Remaiig Year Paymet Iterest Repaymet Balace 1 \$130 \$50 \$ 80 \$ \$1,300 \$300 \$1,000 ===== ==== ===== 124

10 Example 9 Settig Up a Amortizatio Schedule Jaice ad Phil Gratham have a mortgage of \$61, which is to be amortized over six moths with a aual iterest rate of 12%. Set up a amortizatio schedule uder the followig coditios : a) Assume the iterest rate is compouded mothly. b) Assume the iterest rate is compouded semi-aually. A B C D E Paymet Regular Paymet Paymet Outperiod Paymet to to Stadig Iterest Pricipal Balace 0 \$61, \$10, \$ \$10, \$51, \$10, \$ \$10, \$41, \$10, \$ \$10, \$31, \$10, \$ \$10, \$21, \$10, \$ \$10, \$10, \$10, \$ \$10, \$ TOTALS \$64, \$2, \$61, N/A 125

11 A B C D E Paymet Regular Paymet Paymet Outperiod Paymet to to Stadig Iterest Pricipal Balace 0 \$61, \$10, \$ \$10, \$51, \$10, \$ \$10, \$41, \$10, \$ \$10, \$31, \$10, \$ \$10, \$21, \$10, \$ \$10, \$10, \$10, \$ \$10, \$ TOTALS \$63, \$2, \$61, N/A Example 10 Partial Amortizatio Tables Omega Holdigs has arraged a \$200,000 mortgage o a piece of property. The arragemets are that the mortgage will be amortized over 20 years with mothly paymets based o a iterest rate of 10%, compouded mothly. Show the amortizatio schedule etries for the 36 th ad 37 th paymets. A B C D E Paymet Regular Paymet Paymet Outperiod Paymet to to Stadig Iterest Pricipal Balace 35 \$189, \$1, \$1, \$ \$188, \$1, \$1, \$ \$188,

12 EXERCISE : FI A CIAL MATHEMATICS 1. A sum of moey is deposited ow at 10% per aum. How log will it take for the sum ivested to double? 2. We require \$10,000 i 15 years time ad we ca deposit moey at 10% per aum. How much must be ivested ow to achieve this sum? 3. Suppose \$9,500 is ivested o the 1 st of Ja. of a certai year at 12% compoud ad \$800 is withdraw at the ed of each year. How much would remai after 12 years. 4. A Co. decide to ivest \$10,000 at the begiig of 1999 i a fud earig 12% per aum. A Co. will add a further \$3,000 to the fud at the begiig of each year, commecig i (a) What will be the value of total ivestmet i the fud at the ed of 2007? (b) Suppose A Co. ow decide to make equal aual istalmets startig at the begiig of 1999 at the same iterest rate of 12%, calculate the aual istalmets ecessary for the fud to have the same value at the ed of 2007 as i (a). 5. Suppose we deposit \$10,000 ow, ad we withdraw X at the ed of each year for 5 years so that othig is left o deposit. What is X if moey ca be ivested at 10% p.a. compoud? 6. At what rate of iterest will moey double its value i three years? Assume that the rate is compouded semi-aually. 7. Mr. Wog bought a flat o Jauary 1, 1998 at three millio dollars with dow paymets of 30% of the purchase price. The remaiig 70% is to be repaid by 180 mothly istalmets at 6% p.a. compouded mothly. The first istalmet due o February 1, Calculate the amout (rouded to the earest dollar) of each istalmet. 8. ABC Holdigs has just borrowed \$300,000 to fiace a ew lad developmet project. The repaymet requires 20 quarterly paymets with iterest 18% p.a. compouded mothly, the first paymet due 3 years from ow. What is the size of each quarterly paymet? 127

14 (d) the etries for the schedule i part (a) if the iterest rate o the sikig fud had bee 8%, compouded semi-aually. 129

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