Chapter 4. More Interest Formulas

Transcription

1 Chapter 4 More Iterest ormulas

2 Uform Seres Compoud Iterest ormulas Why? May paymets are based o a uform paymet seres. e.g. automoble loas, house paymets, ad may other loas. 2

3 The Uform aymet Seres s The seres s: ed-of-perod cash recept or dsbursemet a uform seres, cotug for perods, the etre seres equvalet to or at terest rate. 3

4 paymets terms of mout s vested at the ed of each year for four years = (+) 3 + (+) 2 + (+) + = (+) - + (+) (+) 2 + (+) + () Multply by (+) (+) = (+) +(+) (+) 3 +(+) 2 +(+) (2) Subtract eq () from eq (2) 4

5 (+) = (+) = [( + ) ] %,, 5 Where (/, %, ) s called uform seres compoud amout factor

6 lso, we ca solve for terms of %,, 6 Where (/, %, ) s called uform seres skg fud factor

7 Example 4- ma deposts $500 a credt uo at the ed of each year for 5 years. The credt uo pays 5% terest, compouded aually. t the ed of 5 years, mmedately after the ffth depost, how much does the ma have hs accout? Ma s pot of vew = $500, = 5%, = 5 Credt Uo pot of vew 7

8 8

9 Example 4-2 Jm wat to save a uform amout of moey at the ed of each moth order to save a $000 at the ed of each year Bak pays 6% terest, compouded mothly How much would he have to depost each moth? Gve: = $000, = 6/2 = ½%, = 2 moths =? 9

10 ormula relatg ad %,, ) %,, / ( 0 Where (/, %, ) s called uform seres captal recovery factor

11 lso, we ca solve for terms of %,, Where (/, %, ) s called uform seres preset worth factor

12 Example 4-3 You borrowed $5000 ad wat to repay fve equal ed-of-year paymets. The frst paymet s due oe year after you receve the loa. Iterest o the loa s 8% terest. What s the sze of each of the fve paymets?.e. wth terest at 8%, a preset sum of $5000 s equvalet to fve equal ed-of-perod dsbursemets of $252 2

13 Example 4-4 vestor has a purchase cotract o some mache tools. He wll be pad $40 at the ed of each moth for a 5-year perod The vestor offers to sell you the cotract for a $6800 cash today. If you otherwse ca make % per moth o your moey, would you accept or reject the vestor s offer..e. f you take the cotract, you wll be pad $40 per moth for a perod of 60 moths. That meas a total of $8400 over the 5-year perod = $40 = % = 60 moths We eed to determe f the cotract s worth $6800 = (/, %, 60) = 40(44.955) = $

14 Example 4-4 It s clear that f we pay $6800 for the cotract ad receve $40 per moth, we wll receve less tha the % per moth terest (whch s the terest that we ca otherwse make). Therefore the offer should be rejected. OR, we ca say that the $40 per moth paymet s equvalet to $ Therefore f we pay $6800 for a beeft of $40 per moth, we wll lose moey OR, we ca say that the $6800 wll gve me more tha $40 per moth f vested at the gve terest rate of %, whch meas that vestg the $6800 the purchase cotract wll be a loss whe compared to vestg t the other vestmet opportuty (whch gves a % terest rate) Therefore, Reject the offer. 4

15 Example 4-5 Suppose that we decded to pay $6800 for the cotract example 4-4. What mothly rate of retur would we obta o our vestmet? (See the text book for the soluto) al aswer: = 0.722%, whch meas that the mothly rate of retur o our vestmet s 0.722% per moth. 5

16 Example 4-6 Usg 5% terest rate, compute

17 Example 4-7 Usg 5% terest rate, compute

18 Relatoshps Betwee Compoud Iterest actors = (+) = (/,,) = /(+) = (/,,) Compoud mout factor (/,,) = / (/,,) reset Worth factor =[(+) ]/[( + ) ] =(/,,) =[( + ) ]/[(+) ]=(/,,) Uform Seres Captal Recovery actor (/,,) = /(,,,) Uform Seres reset Worth actor ={[(+) ]/}=(/,,) ={/[(+) ]}=(/,,) Uform Seres Compoud mout actor (/,,) = / (/,,) Uform Seres Skg ud actor 8

19 Relatoshps Betwee Compoud Iterest actors ( /,, ) How? = (+ ) - + (+ ) ( + ) - = [(+ ) - + (+ ) (+ ) - ] = [(/,,)+(/,,2)+...+(/,,)] sce = (/,, ) We coclude that (/,, ) = (/,,)+(/,,2)+...+(/,,) ( / J ( /,, ),, J) J ( /,, J) or Example: (/, 5%, 4) = (/, 5%, ) + (/, 5%, 2) + (/, 5%, 3) + (/, 5%, 4) 9

20 Relatoshps Betwee Compoud Iterest actors ( /,, ) ( /,, ) J J ( /,, J) ( /,, J) = + (+ ) + (+ ) (+ ) - = [ + (+ ) + (+ ) (+ ) - ] = [ + (/,,) + (/,,2) (/,,-)] sce = (/,,) We coclude that (/,,) = + (/,,) + (/,,2) (/,,-) 20

21 Relatoshps Betwee Compoud Iterest actors (/,,) = [( + ) ]/[(+) ] (/,,) = {/[(+) ]} We start wth a detty: (+) = + (+) = + [(+ ) ] Now dvde by (+ ) to get [ (+) ]/[(+) ] = /[ (+) ] +. Ths gves: (/,,) = (/,,) + 2

22 rthmetc Gradet Suppose you buy a car. You wsh to set up eough moey a bak accout to pay for stadard mateace o the car for the frst fve years. You estmate the mateace cost creases by G = $30 each year. The mateace cost for year s estmated as $20. Thus, estmated costs by year are $20, $50, $80, $20, $240. $20 $50 $80 $20 $

23 rthmetc Gradet We break up the cash flows to two compoets: = ad G = = (/,5%,5) = 20 (/,5%,5) = 20 (4.329) = 59 2 = G (/G,5%,5) = 30 (/G,5%,5) = 30 (8.237) = 247 = + 2 = $766. Note: 5 ad ot 4. Usg 4 s a commo mstake. 2 Stadard orm Dagram for rthmetc Gradet: perods ad - ozero flows creasg order 23

24 rthmetc Gradet reset Worth rthmetc Gradet Uform Seres rthmetc Gradet (/G,,) = { [(+) ] / [ 2 (+) ] } (/G,,) = { (/ ) / [(+) ] } (/G,,) = G [(+)--]/2 (/G,5%,5) = {[(+) ]/[ 2 (+) ]} = {[(.05) ]/[ (.05) 5 ]} =

25 rthmetc Gradet Example 4-6. Mateace costs of a mache start at $00 ad go up by $00 each year for 4 years. What s the equvalet uform aual mateace cost for the machery f = 6%. Ths s ot the stadard form for usg the gradet equato, because the year-oe cash flow s ot zero We reformulate the problem as follows. 25

26 rthmetc Gradet = =00 + G = The secod dagram s the form of a $00 uform seres. The last dagram s ow the stadard form for the gradet equato wth = 4, G = 00. = + G (/G,6%,4) = (.427) = $

27 rthmetc Gradet Example Wth = 0%, = 4, fd a equvalet uform paymet for Ths s a problem wth decreasg costs stead of creasg costs. The cash flow ca be rewrtte as the DIERENCE of the followg two dagrams, the secod of whch s the stadard form we eed, the frst of whch s a seres of uform paymets. 27

28 rthmetc Gradet =24000 G= = - G 2G 3G = G(/G,0%,4) = (/G,0%,4) = (.38) = 5,74. 28

29 rthmetc Gradet Example d for the followg dagram wth = 0% J Ths s ot the stadard form for the arthmetc gradet. However, f we sert a preset value J at the ed of year 2, the dagram from that pot o s stadard form. Thus: J = 50 (/G,0%,4) = 50 (4.378) = = J (/,0%,2) = (0.8264) = $80.9 OR oe le: = 50 (/G,0%,4) (/,0%,2) = 50(4.378)(0.8264) 29

30 Geometrc Gradet I arthmetc gradet, the perod-by-perod chage s a uform amout, G. I geometrc gradat, the perod-by-perod chage s a uform rate, g. Hece we ca defe the geometrc gradet (g) as a uform rate of cash flow crease/decrease from perod to perod. Example Suppose you have a vehcle. The frst year mateace cost s estmated to be $00. The rate of crease each subsequet year s 0% (g). You wat to kow the preset worth of the cost of the frst fve years of mateace, gve = 8% rthmetc gradet le

31 Dervato of Geometrc Gradet ormula The cost ay year s - = (+g) -, ort example: the cost the thrd year s 3 = (+g) 2 Where: g = geometrc gradet uform rate of cash flow crease/decrease = Value of cash flow at year = Value of cash flow at ay year The preset worth of ay cash flow s = (+) = (+g) - (+) = (+g) - (+).[(+)(+) - ] = (+g) - (+) - (+) - 3

32 we obta..., ad for values the orgal Replacg ) ( yelds from eq () Subtractg eq (2) (2) to get he eq by Multply t (), ) ( ) ( ad ) ( Let ) (... ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( b a b b a ab a b ab ab ab ab b b ab ab ab a g b a g g g g g g x x Dervato of Geometrc Gradet ormula 32

33 use, If the s The expresso brackets where, ) ( g g g g g g g g g g geometrcserespreset worth factor Dervato of Geometrc Gradet ormula 33

34 Example 4-2 The frst-year mateace cost for a ew automoble s estmated to be $00, ad t creases at a uform rate of 0% per year. Usg a 8% terest rate, calculate the preset worth of cost of the frst 5 years of mateace. Soluto g g = $00, g = 0%, = 8% $

35 ) Cosder the stuato of a perso depostg a$000 to a bak that pays 2% terest, compouded mothly. 2% terest, compouded mothly, meas that the bak pays % every moth. = %, = 2, = $000 = 000(/, %, 2) = 000(.0) 2 = $26.8 Nomal & Effectve Iterest B) Cosder the stuato of a perso depostg a$000 to a bak that pays 2% terest,. How much would be the savgs accout at the ed of oe year for case () ad case (B) 2% terest, meas that the bak pays 2% every year. = 2%, =, = $000 = 000(/, 2%, ) = 000(.2) = $

36 Nomal & Effectve Iterest Nomal terest rate per year, r, s the aual terest rate wthout cosderg the effect of ay compoudg. (t s the 2% the prevous example) Effectve terest rate per year, a, s the aual terest rate takg to accout the effect of ay compoudg durg the year. (I the prevous example, a = 26.8/000 = 2.68%) Effectve terest rate per terest perod,. (t s the % used the prevous example) m = Number of compoudg subperods per tme perod (It was the 2 compoudg perods used the prevous example) 36

37 Dervato of the effectve terest rate equato If a $ depost was made to a accout that compouded terest m tmes per year ad pad a omal terest, r : Iterest rate per subperod = r/m Total the accout at the ed of year = $(+r/m) m We ca determe the effectve terest rate by deductg the $ prcpal amout Therefore, effectve terest rate per year s r a m OR just substtute a ( ) m m r / m toget 37

38 Nomal & Effectve Iterest Commo omeclature egeerg Ecoomcs Whe we say the terest rate s 6% compouded mothly, we mea: - r = 6% per year (omal terest rate) 2- = r/m = 6/2 = 0.5% per moth (terest per perod) 3- a = (+r/m) m (Effectve terest rate per year) = (+0.005) 2 = 6.678% Whe we say terest rate s 6% per moth, we mea: - = 6% per moth (terest per perod) 2- r = 72% per year (omal terest rate) 3- a = (+r/m) m (Effectve terest rate per year) = (+0.06) 2 = 0.22% 38

39 Example 4-4 If a savgs bak pays ½ % terest every 3 moths, what are the omal ad effectve terest rates per year? = ½% (effectve terest rate per terest perod) m = 4 (umber of compoudg subperods per tme perod Nomal terest rate per year r = 4 ½% = 6% Effectve terest rate per year a a 6.% m r m OR Effectve terest rate per year m a = (+0.05)4 = 0.06 = 6.% 39

40 Example 4-5 loa shark leds moey o the followg terms: If I gve you $50 o Moday, you owe me $60 o the followg Moday: (a) What omal terest rate per year (r ) s the loa shark chargg? (b)what effectve terest rate per year ( a ) s he chargg? (c) If the loa shark started wth $50, how much moey he would have at the ed of the year? a) Nomal terest rate per year (r)? rst we eed to calculate the terest rate per perod (week) = (/,, ) 60 = 50(/,, ), therefore, (/,, ) =.2 = 20% per week Nomal terest rate per year = 52 weeks 0.20 = 0.4 = 040% 40

41 Example 4-5 (b) Effectve terest rate per year ( a )? a r m m , ,04,30,400% OR m ,04,30,400% a (c) How much at the ed of the year? $655, 200 4

42 Example 4-6 O Jauary, a woma deposts $5000 a credt uo that pays 8% omal aual terest, compouded quarterly. She wshes to wthdraw all the moey fve equal yearly sums, begg December 3of the frst year. How much should she wthdraw each year? Nomal terest rate r = 8% compouded quarterly. Therefore, the effectve terest rate per terest perod = 2% W W W W W $5000 = 2% per quarter = 20 quarters I example 4-3, we used = (/,, ). Ca we do the same thg here? We ca t apply t drectly sce the compoudg perod does ot match the aual wthdrawals. 42

43 Example 4-6 Soluto Compute the effectve terest a per year a = ( + r/m) 4 = ( /4) 4 = = 8.24% W W W W W 0 $ = 8.24% per year = 5 years W = 5000(/, 8.24%, 5) Use / formula sce 8.24% does ot exst tables 43

44 Example 4-6 Soluto 2 Compute the equvalet uform cash flows,, at the ed of each quarter = 5000(/, 2%, 20) = 5000(0.062) = $ $5000 = 2% per quarter = 20 quarters Now, we ca calculate W from for each oe year perod W = (/, 2%, 4) = 306(4.22) = $ W W W W W 44

45 Cotuous Compoudg I all prevous examples, we used perodc compoudg, where the durato of the terest perod was a fte umber (e.g. a year, sx moths, oe moth, oe weak, etc. Table Nomal & Effectve Iterest expressed percet Effectve rates, a = ( + r/m) m - Nomal rate Yearly Sema. Mothly Daly Cotuously r m = m = 2 m = 2 m = By cotuous compoudg, we mea to crease the umber of compoudg perods (m) to fty. 45

46 If Cotuous Compoudg ( ) we have m compoudg subperods the year, r m m ( m wll be the umber of the subperods years) To obta a formula correspodg to cotuous compoudg, we eed to crease m ad make t very large;.e. m m r or cotuous compoudg :. lm m m mportat lmt calculus s lm x Therefore, set x r/m, ad m becomes (/x)(r) x0. lm x. x r becomes Therefore, for cotuous compoudg, e e x0 e Effectve terest rate per year for cotuous compoudg r r r x e a e r 46

47 47 Cotuous Compoudg The followg formulas apply for cotuous compoudg. They are obtaed by substtutg = e r to the prevously studed formulas. r e e e r e e r e e e r e e r e r e r r r r r r r r r r r r,, / reset Worth Uform Seres,, / Compoud mout Seres,, / Captal Recovery,, / ud Skg,, / reset Worth,, / Compoud mout

48 Example 4-7 If you were to depost $2000 a bak that pays 5% omal terest, compouded cotuously, how much would be the accout at the ed of 2 years? e r $2000, 2000e r omal terest rate 0.05 umber of years $

49 Example 4-8 bak offers to sell savgs certfcates that wll pay the purchaser $5000 at the ed of 0 years but wll pay othg to the purchaser the meatme. If terest s computed at 6%, compouded cotuously, at what prce s the bak sellg the certfcates? e r $5000, r omal terest rate e umber of years0 $2744 Therefore, the bak s sellg the $5000 certfcates for $

50 Example 4-20 If a savgs bak pays 6% terest, compouded cotuously, what are the omal ad the effectve terest rates? Nomal terest rate 6% per year Effectve terest rate e r e % 50