CHAPTER 7 FINANCIAL MATHEMATICS Page Cotets 7.1 Compoud Value 116 7.2 Compoud Value of a Auity 117 7.3 Sikig Fuds 118 7.4 Preset Value 121 7.5 Preset Value of a Auity 121 7.6 Term Loas ad Amortizatio 122 Exercise 126 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. 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 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) = 1.095 i = 9.5% Whe compoudig periods are more frequet tha oce a year, the P i = P0 1 + m m where m is the umbers of times per year compoudig occurs. Example 2 116
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 = 2 2 5 = 10 000 1+ 5%, =, (. ) = $12, 2 10 10 000 1 025 800 7.2 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 0 1 2 3 I geeral, we have : 10,000 10,000 10,000 1.08 10,800 (1.08) 2 11,664 Compoud Sum : $32,464 ====== 0 1 2 3 1 R R R R R R(1 + i) R(1 + i) 2 R(1 + i) 1 S Defie S = compoud sum 117
R = periodic receipt = legth of auity S = R+ R( 1+ i) + + R( 1+ i) + R( 1+ i) 2 1 [ 2 1 1 ( 1 ) ( 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, 0 1 2 3 10 replace the machie a a a a the a + a(1.08) + a(1.08) 2 + + a(1.08) 9 = 8,000 a (. 108 ) 10 1 8, 000 008. = a = $552.23 118
(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) 0 1 2 9 10 replace the machie a a a a the Example 5 a( 1+ i) + a( 1+ i) 2 + + a( 1+ i) 10 = 8, 000 [ + (1 + i) + + (1 + ) ] 9 = 8, 000 a (1 + i)1 i Sikig Fud Schedules i a( i) ( ) 10 1+ 1 1+ 8, 000 i = a = 511.33 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. 119
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 $300,000.00 1 $66,576.24 0 $66,576.24 $66,576.24 $233,423.76 2 $66,576.24 $5,326.10 $71,902.34 $138,478.58 $161,521.42 3 $66,576.24 $11,078.29 $77,654.53 $216,133.11 $83,866.89 4 $66,576.24 $17,290.65 $83,866.89 $300,000.00 $00000.00 TOTALS $266,304.96 $33,695.04 $300,000.00 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 12.55088%, 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 $2,472.03 $5,254.33 $514,745.71 3 $2,472.03 $659.46 $3,131.50 $8,385.83 $511,614.18 26 $406,344.99 27 $2,472.03 $50,999.87 $53,471.91 $459,816.90 $60,183.10 28 $2,472.03 $57,711.07 $60,183.10 $520,000.00 $00000.00 TOTALS $69,216.84 N/A N/A N/A N/A 120
7.4 Preset Value Example 7 Suppose you are offered the alterative of either $1,500 at the ed of three $x today. If your savig accout pays a 10% iterest p.a. Fid x. years or 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 Ed of year 121
1 R 1+ i 0 1 2 3 R R R R 1 R 1+ i 2 1 R 1+ i A A = [ + R ( 1 i) 1 + ( 1+ i) 2 + + ( 1+ i) ] 1 ( 1+ i) = R i PV factor of a auity 7.5.1 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. 122
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, 000 7. 722 = $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 $920 2 130 46 84 836 3 130 42 88 748 4 130 38 92 656 5 130 34 96 560 6 130 28 102 458 7 130 23 107 351 8 130 18 112 239 9 130 13 117 122 10 130 8 122 0 $1,300 $300 $1,000 ===== ==== ===== 123
Example 9 Settig Up a Amortizatio Schedule Jaice ad Phil Gratham have a mortgage of $61,818.44 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 $61,818.44 1 $10,666.67 $618.18 $10,048.49 $51,769.95 2 $10,666.67 $517.70 $10,148.97 $41,620.99 3 $10,666.67 $416.21 $10,250.46 $31,370.52 4 $10,666.67 $313.71 $10,352.96 $21,017.56 5 $10,666.67 $210.18 $10,456.50 $10,561.06 6 $10,666.67 $105.61 $10,561.06 $00000.00 TOTALS $64,000.02 $2,181.59 $61,818.44 N/A 124
A B C D E Paymet Regular Paymet Paymet Outperiod Paymet to to Stadig Iterest Pricipal Balace 0 $61,818.40 1 $10,657.83 $603.27 $10,054.56 $51,763.88 2 $10,657.83 $505.15 $10,152.68 $41,611.21 3 $10,657.83 $406.08 $10,251.76 $31,359.45 4 $10,657.83 $306.03 $10,351.80 $21,007.65 5 $10,657.83 $205.01 $10,452.82 $10,554.83 6 $10,657.83 $103.00 $10,554.83 $00000.00 TOTALS $63,946.98 $2,128.54 $61,818.44 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 $189,347.51 36 $1,930.04 $1,577.90 $352.14 $188,995.37 37 $1,930.04 $1,574.96 $355.08 $188,640.29 125
EXERCISE : FINANCIAL 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 2000. (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, 1998. 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? 126
9. Mr. Leug ivests $1,000 at the ed of each year for 20 years i a ivestmet fud which pays iterest at 13% p.a. compouded aually. The fud pays the iterest at the ed of each year ad at the same time Mr. Leug receives each iterest paymet, he deposits it ito his bak accout which pays iterest at 10% p.a. compouded aually. How much moey does Mr. Leug have at the ed of 20 years? 10. Mr. Cheug buys a house ad borrows $95,000 from the ABC Fiacial Compay. The loa is to be repaid with mothly paymets over 30 years at 15% p.a. compouded semiaually. The iterest rate is guarateed for 5 years. After exactly two years of makig paymets, Mr. Cheug sees that iterest rates have dropped to 10.5% p.a. compouded semiaually i the market place. He asks to be allowed to repay the loa i full so he ca refiace. ABC agrees to reegotiate but sets a pealty exactly equal to the moey the compay will lose over the ext 3 years. Fid the value of the pealty. 11. A debt of $80,000 is to be amortized with $2,500 payable every moth. The iterest rate is 11% compouded mothly. Costruct a amortizatio schedule showig the last three etries to complete the repaymet of the debt. 12. The ABC compay has a mortgage of $65,000 o a property ad has bee makig regular paymets of $3,500 every three moths. If the iterest rate is 8%, compouded quarterly, what would be the first three etries i the amortizatio schedule which shows repaymet of the loa? 13. A debt of $180,000 is to be amortized by usig a sikig fud. The paymets are to be made mothly over the ext five years ad the iterest rate is 12%, compouded mothly. Costruct a sikig fud schedule showig the first ad last two etries. Show the book value for each of the three periods i the table. 14. A city has just borrowed $7.5 millio for twety years through the sale of bods. The moey is to assist i udertakig major sewer recostructio. The city has started a sikig fud to hadle repaymet of the bod issue whe it becomes due. The fud is to accumulate moey at 8%, compouded aually. The iterest rate o the bod is 10% aually, payable every 6 moths. Roudig all calculatios to the earest dollar, fid: (a) (b) (c) the aual paymet to the sikig fud. Costruct a partial schedule for periods 2, 17 ad 18; the aual budget expediture to hadle the iterest o the debt ad the sikig fud; the book value of the debt after 15 years, 127
(d) the etries for the schedule i part (a) if the iterest rate o the sikig fud had bee 8%, compouded semi-aually 128