Ecoomcs Notes The follow otes are used for the ecoomcs porto of Seor Des. The materal ad examples are extracted from Eeer Ecoomc alyss 6 th Edto by Doald. Newa, Eeer ress. Notato Iterest rate per perod. Number of perods reset sum of moey uture sum of moey Ed-of-perod cash recept or dsbursemet, (ex. lke mothly paymet Uform perod-by-perod creases or decreases of (ex. crease your mothly paymet by $0 each moth Uform rate creases or decreases of (ex. crease your mothly paymet by 0% each moth r Nomal terest rate per perod Iterest: Compoud: Moetary fee for ret or loa moey. Compoud s whe the terest s chared or credted to the oral balace at the ed of each perod. Sce the ew balace s ow dfferet the ext terest wll also be dfferet. Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Smple Iterest Smple terest s the terest computed o the oral sum wthout compoud. If you borrowed Dollars for years at a terest rate of per year wth o compoud the you wll owe erest I terest ad add the oral sum wll be Or ( Example: You borrow $000 for 5 year from a fred who chares you % aual smple terest. How much wll you owe at the ed of the 5 years. $ 000( (0.(5 $800 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Compouded Iterest Compouded terest s the terest chared o the curret sum. The curret sum chaes as the terest fee s added perodcally. Suppose you wll borrow Dollars for years compouded yearly the: fter the st year you balace s ( fter the d year your balace s ( ( ( ( fter the rd year your balace s ( ( ( ( ( d fally after the th year your balace wll be ( ( ( ( ( ( So eeral ( We refer to ths formula as the sle paymet compoud amout factor ad s wrtte as: ( /,, d the sle paymet compoud amout formula s the as: ( /,, Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Example: You borrow $000 for 5 years from a Bak who chares you % terest per year compouded aually. How much wll you owe at the ed of the 5 years. 000 $000( (0. 5 $587.0 Ths sum s more tha the $800 because your fred does ot chare you terest o the terest you owe hm whle the bak does. Example: The same problem as before but ow the bak compouds mothly stead of yearly. How much wll you owe at the ed of the year. 0. 5( 60 $000( $000(.0 $550.09 Ths s more because compouded yearly meas that wth the year t s smple terest where as ow you oly et smple terest wth each moth stead of each year. We could also have used a tle lookup. rst fd the value of ( /,, for 0.0 ad 60. The multply ths amout by $000. Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Uform Seres Compoud mout Uform paymet seres s where you have perodcal paymets. Lke you pay back the loa us mothly paymets. Suppose you wll depost Dollars each year for years to a savs accout that pays terest per year compouded yearly. How much wll you have at the ed of the years? If we oly cosder the st year s depost the after the years we wll have: ( We rse to the power of stead of because we wll make our st depost at the ed of the st year ad by the we wll oly have years to left. The coveto wth uform paymets,, s to make the st paymet or depost after the st perod. Now f we oly cosder the d year s depost the after the years we wll have: ( Cosder oly the rd year s depost the after the years we wll have: ( d fally cosder oly the th year s depost the after the years we wll have: 0 ( 5 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Now f we cosder all deposts we smply add the s. We et ( ( ( actor out the we et [( ( ( ] Now we create a ew equato multply by ( [( ( ( ( ] ( Now we subtract the two equatos so that we ca elmate all the terms but the frst ad last. ( To et [( ( ( ( ] [( ( ( ] [( ] d smplfy: ( I the eeral case we have perods stead of so we have: ( We refer to ths factor as the uform seres compoud amout factor ad s wrtte as: ( /,, So the uform seres compoud amout formula s: ( /,, 6 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Example: You depost $500 aually for 7 year to a savs accout that pays 6% terest per year compouded yearly. How much wll you have ths accout at the ed of the 7 years? Note the frst paymet s year. ( 0.06 $500 0.06 7 $,96.9 Or you could use the lookup tle for ( /,, us 0.06 for ad 7 for. The multply ths factor by $500 to et. 7 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Uform Seres Sk ud If we take the verse of the uform seres compoud amout factor to et ve the we have: ( We refer to ths factor as the uform seres sk fud factor ad s wrtte as: ( /,, So the uform seres sk fud formula s: ( /,, Example: You wat to save $000 for a trp you wll take years by mak yearly deposts to a savs accout that pays 6% aually compouded aually. How much wll I have to depost each year? Note the frst paymet s year. 0.06 $ 000 $56. ( 0.06 Or you could use the lookup tle for ( /,, us 0.06 for ad for. The multply ths factor by $000 to et. 8 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Uform Seres Captal Recovery If we substtute for us the sle paymet compoud amout formula ( the we have ve : ( ( ( ( So the uform seres captal recovery formula s ad s wrtte as ( ( ( /,, Example: You wat to take a trp ow that wll cost $000 so you chare t to your credt card ad make equal mothly paymets be wth ext moth. The credt cards compouds mothly ad has a terest rate of % aually. How much should I pay each moth to payoff ths amout year? Note we dvde the terest by to et the terest per perod. 0.0( 0.0 000 ( 0.0 $ $88.85 Or you could use the lookup tle for ( /,, us 0.0 for ad for. The multply ths factor by $000 to et. 9 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Uform Seres reset Worth If we take the verse of the uform seres captal recovery we et a formula to compute ve : ( ( ( ( So the uform seres preset worth formula s ad s wrtte as ( ( ( /,, Example: You calculated that you ca pay as much as $50 per moth for year towards the paymets for a ew computer ad the store offers 8% terest o there mothly compouded paymet pla. What s the maxmum amout of moey that I ca sped o the computer? (.05 50.05(.05 $ $55.8 Or you could use the lookup tle for ( /,, us 0.05 for ad for. The multply ths factor by $50 to et. 0 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
rthmetc radet reset Worth rthmetc radet s where you have perodcal paymets that crease or decrease by a fxed amout every perod. We ca cosder ths to be a fxed perodc paymet plus a separate amout that cremet perodcally. 0 To derve a equato for the cremetal part we cosder each separately as a ad add them toether the covert to a : 0 0 0 0 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
wth ( ( 0 ( Now f we cosder all terms ad add them we et: ( ( ( 0 actor out the we et 0 [( ( ( ] Now we create a ew equato multply by ( [( ( ( ] ( Now we subtract the two equatos so that we ca elmate all the terms but the frst ad last. ( To et [( ( ( ] 0 [( ( ( ] [( ( ( ] Next we remove the the bracket to et [( ( ( ] Now let [( ( ( ] w So we have w Now we multply w by ( to et Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda [ ] ( ( ( ( w ad subtract [ ] ] ( ( [( ( ( ( ( w w we et [ ] ( ( w ad [ ] ( ( ( w plu back we et ( ( ad ( ( I the eeral case wth perods stead of we have ( Ths formula does ot have a ame but t ves ve. Multply by the sle paymet preset worth factor to express terms of we et ( ( (
Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda Smplfy we et ( ( ( ( ( ( ( ( ( Now the eeral case wth perods stead of we have ( ( We refer to ths factor as the arthmetc radet preset worth factor ad s wrtte as:,, / ( So the arthmetc radet preset worth formula s:,, / ( Example: You wsh to set asde suffcet moey to a accout so that he may take perodc paymets at the ed of each year follow the schedule: Year $0 Year $50 Year $80 Year $0 Year 5 $0 How much should he have to depost ow f the bak pays 5% terest per year? Note the frst paymet s year.
0 0 80 50 0 We ca break the problem to 0 0 0 50 80 0 0 0 0 0 60 90 0 0 0 5 5 ( 0.05 ( 0.05 5(0.05 0 $0 5 0.05( 0.05 0.05 ( 0.05 $ 5 $766 Or you could use the lookup tle for (/,, ad (/,, us 0.05 for ad 5 for. The multply these factors to et: $ 0( /, 5%, 5 $0( /, 5%, 5 $0(.9 $0(8.7 $766 5 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda 6 rthmetc radet Uform Seres If we multply the ve factor by the sk fud factor ( ve we et ( ( ( ( ( ( So we have ( We refer to ths factor as the arthmetc radet uform seres factor ad s wrtte as:,, / ( So the arthmetc radet uform seres formula s:,, / ( Example: You have a mache that requres a aual mateace cost accord to the follow schedule: Year $6,000 Year $,000 Year $8,000 Year $,000 What s equvalet costat aual mateace cost f the terest s 0%?
.000 6,000,000 8,000 6,000 6,000 6,000 6,000 0 6,000,000 8,000 $6,000 $6,000( /,0%, $6,000 $6,000 0. ( 0. $6,000 $6,000(.8 $,87.0 Example: You have a mache that requres a aual mateace cost accord to the follow schedule: Year $,000 Year $8,000 Year $,000 Year $6,000 What s equvalet costat aual mateace cost f the terest s 0%? 7 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
,000 8,000,000 6,000,000 6,000,000 8,000-0 $,000 $6,000 5,7 0. ( 0. 8 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
eometrc radet reset Worth eometrc radet s where you have perodcal paymets that crease or decrease by a fxed percetae every perod. Cosder that the perodc paymet amout s to start at $00 ad crease by 0% each perod. The the paymet schedule wll be: erod $00.00 erod $00.00 0% (00.00 $00( 0.0 $0.00 erod $0.00 0% (0.00 $00( 0.0 $.00 erod $.00 0% (.00 $00( 0.0 $.0 erod 5 $.0 0% (.0 $00( 0.0 $6. erod $00( 0.0 - I the eeral case we have ( Where s the uform rate of cash flow crease or decrease, s the frst paymet ad s the paymet at year Now to fd the preset worth of the sum of the paymets we fd the preset worth of each dvdual paymet ad sum them toether. 9 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda 0 The preset worth of the th paymet s ( ( ( ( ( ( ( Now we sum them all N terms up to et N x x ( To solve ths we let ( a ad b Now we have ( N x x a b a L
Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda Multply both sdes by b we et b L We subtract the two equatos to et b a L L to et a b Solve for to et b b a b a a b a b ( ( Now f we replace the oral values for a ad b we et ( ( ( We smplfy to et
Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda ( ( ( ( ( ( ( So we have ( ( We refer to ths factor as the eometrc seres preset worth formula ad s wrtte as:,,, / ( Example: The mateace cost of a mache s $00 ad creases by 0% each year. What s the preset worth of the cost for the frst 5 years us 8% terest? $80. 0.0 0.08 0.08 ( 0.0 ( $00 5 5 If the we use
Nomal ad Effectve Iterest So far we have dealt wth terest terms of terest per perod. There has ot bee ay compoud wth a perod. We use the effectve terest rate per perod for the terest for a perod. If we cosder that there s compoud wth a perod the we must dstush betwee cosder the effects of compoud or ot. Ths s commo because we typcally specfy terest terms of a year perod or yearly terest whle baks typcally compoud several tmes per year. The terest for the compoud perod s cosdered effectve terest ad s deoted by. The terest for a larer perod that cludes compoud wth the perod the f we cosder the effect of compoud we refer to ths terest as effectve terest, a ad f we do ot cosder the effects of compoud we refer to ths terest as omal terest, r. Effectve terest rate per compoud perod. r - Nomal terest rate per perod. Ths s the terest rate ot cosder the effects of compoud dur the perod. a - Effectve terest rate per year. Ths s the aual terest rate per year cosder the effect of ay compoud dur the year. m - The umber of compoud perods per terest perod. I the prevous examples we specfed the terest per perod so we dd ot have to worry out ths. The omal ad effectve terest was the same. We have the follow formulas a r m m r m m ( Cosder a bak loas $00 at.5% per 6 moth perod compouded every 6 moths. The effectve terest rate per 6 moth perod s.5%, the omal yearly terest s 5% ad the effectve yearly terest rate s 5.06% m r 5% a 5% 0.05 0.05065 5.06% Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Example: You depost $5000 o the be of the year a bak that pays 8% omal terest per year, compouded quarterly. You wsh to wthdraw all of the moey 5 equal yearly paymets made at the ed of each year. How much should wthdraw at the ed of each year? We have 5 perods where we wll wthdraw however we have compoud perods per year for a total of 0 compoud perods. To use the formulas we must adjust the problem so that the umber of perods s the same for both wthdraws ad compoud. We ca fd the effectve yearly terest ad have 5 perods or fd 0 equal quarterly wthdraws ad the take each roup of wthdraws ad fs a equvalet wthdraw at the ed of the year. The frst method appears much smpler. The effectve yearly terest rate s a 8% % 0.0 ( 0.0 0.08 8.% Now we have 5 perods each wth a effectve terest rate of 8.% ad the yearly wthdraw amout ca be computer as: W $ 5000( /,8.%,5 $,60 lteratvely, f we do ot use the effectve terest formulas we wll solve the problem as follows. rst assume you have 0 wthdraws ad compute the wthdraw amout the compute ad amout for the yearly wthdraws. $5000( /,%,0 $06 W $06( /,%, $,60 Both solutos work ad ether are computatoally complex however the soluto wth the effectve terest s more straht forward. Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
reset Worth Up to ow we have see how to covert cost from oe form to aother. or example we covert a preset worth,, to a future worth, by us the sle paymet compoud amout formula (the / formula. I ths secto you wll see how to make decsos based o ecoomc effcecy of several alteratves. Sce we eed to compare the cost or beefts we must put all cost a sle form. Typcally the preset worth,, s chose. The ma dea ths secto s to covert all cost ad beefts to preset worth the compare the varous preset worth for hher beeft or lower costs. Ths to cosder are the tal cost, perodc cost or mateace, salvae value, lfe durato ad perod of aalyss. I the frst example we oly cosder the perodc beeft. Example: compay wats to buy a mache that wll reduce the cost of operato. Each mache cost the same, has the same lfe durato ad has o salvae value. Mache saves $00 / year for 5 years whle mache B saves $00 the frst year ad each addtoal year t saves $50 less. Wth terest at 7% aually whch should the compay buy? 00 00 5000 50 00 5 years 5 years W of beeft W of beeft W of 00(/, 7%, 5 $,0 W of B 00(/, 7%, 5 50(/, 7%, 5 $,57.65 Mache B ves a better savs overall. Notce B the total amout of savs s the same as but we et more moey up frot whch we may vest. I the ext example we oly cosder the tal cost. Example: cty wshes to stall sewer ppes the cty. The cty may sped $00 mllo ow ad ths wll last 5 years at whch pot they wll eed to spet aother $50 mllo to crease ts capacty. lteratvely the cty ca sped $00 mllo ow ad put the larer ppes ad ot eed to uprade 5 years. The cty s terest rate s 6%. Should the cty sped the extra $00 mllo ow to save sped $50 mllo 5 years? Two stae system W of cost $00 $50(/, 6%, 5 $8.6 mllo 5 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
Larer system W of cost $00 mllo The cty wll save $8. mllo f they stall the smaller ppes ow ad redue t 5 years. Bascally what ths meas s that f we take $8.6 mllo ad vest t at a terest rate of 6% after 5 years we wll have $50 mllo. So stead of pay $00 mllo ow we pay oly $00 mllo ad put a addtoal $8.6 mllo the bak for the uprade 5 years. We save $00 mllo $8.6 mllo $8. mllo. Sometmes the there s a salvae value. Ths s moey that you et by sell the used equpmet at the ed of ts lfe. Ths moey arrves after the products lfe ad therefore eed to be coverted to preset worth. Example: compay wats to buy a scale. It uses a terest rate of 8%. Scale cost $,000 ves a aual beeft of $50 ad has a $00 salvae value after ts 6 year lfe. Scale B cost $,000 ves a aual beeft of $600 ad has a $700 salvae value after ts 6 year lfe Whch scale should the compay purchase? W of Beefts -$000 $50(/, 8%, 6 $00(/, 8%, 6 $ W of beefts B -$000 $600(/, 8%, 6 $700(/, 8%, 6 $5 Scale B has more beefts ad therefore should be purchased. Sometmes the aalyss perod s ot equal for the dfferet alteratves. We eed to make the aalyss perod the same for all alteratves order to compare them. If the alteratve ves perodc beefts the the system wth the shorter lfe wll smply ve $0 beeft for the perod after ts lfe. Ths s ot always possble sce sometmes equpmet s smply eeded ad does ot have a measurle perodc beeft. Cosder the ret of a buld. buld s ot someth that you ca measure the beeft of hav oe versus ot hav oe. You smply eed oe to operate ad both buld may ve the same beefts, f they are the same sze for example. buld that last a extra 0 years s deftely beefcal but the beeft s ot represeted the ecoomc equatos. I ths case the aalyss perod must be the same. To make the aalyss perod the same oe ca fd a commo multple umber of perods however ths s ot always feasble. Cosder oe lfe to be 7 years ad the other. Example: compay wats to buy a mache t eeds for ts operatos. Its terest rate s 7%. Mache cost $500 has a useful lfe of 5 years ad a salvae value of $00. Mache B cost $600 has a useful lfe of 0 years ad a salvae value of $5. 6 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda
00 00 5 500 500 600 W of cost $500 ($500 $00(/, 8%,5 - $00(/, 7%, 0 $,5 W of cost B $600 - $5(/, 7%, 0 $,5 Mache B has a lower cost ad therefore ths mache s chose. Whe equalz the aalyss perod t s commo to use the same alteratve several tmes to crease ts perod. However oe also may cosder the fact that wth certa tems the replacemet cost may be dfferet. or example f the mache s a computer oe ca assume that after 5 years the ext oe wll be much cheaper. Sometmes the aalyss perod fte for example cosder a cty pla to make roads. Cosder the stuato where we wat to set asde a sum moey so that the terest ves a fxed amout each perod for a defte perod. To do ths our tal balace must ot chae over tme. The perodcal yeld s oly from the terest. Sce for a sle perod we have. We ca see that the perodc terest s. So the. Example: We wat to set asde a sum of moey that wll ve $50 per year forever for the mateace of a raveste. How much should we set asde f terest s assumed to be % ad ot cosder flato? 50 f. 50 50 We kow that $ 50 0.0 We should set asde $,50 so that the terest wll ve $50 per year deftely. 7 Dr. erado ozalez, Uversty of Cetral lorda, Orlado lorda