# 9.4 Annuities. Objectives. 1. Calculate the future value of an ordinary annuity. 2. Perform calculations regarding sinking funds.

Save this PDF as:

Size: px
Start display at page:

Download "9.4 Annuities. Objectives. 1. Calculate the future value of an ordinary annuity. 2. Perform calculations regarding sinking funds."

## Transcription

1 9.4 Annuities Objectives 1. Calculate the futue value of an odinay annuity. 2. Pefo calculations egading sinking funds. Soewhee ove the ainbow... skies ae blue,... and the deas that you dae to dea...eally do coe tue. Lyan Fank Bau (Autho of The Wizad of Oz) What ae you financial deas? Do you dea about owning a beautiful house? Visiting soe exotic fa-away place? Poviding a college education fo you childen? Retiing happily in cofotable suoundings? If you have any long-te plans such as these, to achieve you deas you will need to have a lage su of oney in the futue. In this section, we will discuss how you can do just that by aking a seies of egula payents ove any yeas to accuulate the oney that you will need. This type of investent is called an annuity. Copyight 2010 Peason Education, Inc.

2 9.4 y Annuities 423 KEY POINT We ake egula payents into an annuity. Annuities An annuity is an inteest-beaing account into which we ake a seies of payents of the sae size. If one payent is ade at the end of evey copounding peiod, the annuity is called an odinay annuity. The futue value of an annuity is the aount in the account, including inteest, afte aking all payents. To illustate the futue value of an annuity, suppose that in Januay you begin aking payents of at the end of each onth into an account paying 12% yealy inteest copounded onthly. How uch oney will be in this account fo a sue vacation beginning on July 1? This poble is diffeent fo those in Section 9.2. In the ealie pobles, we deposited a lup su that eaned a stated inteest ate fo an entie peiod. In this poble, we ae depositing a seies of payents, and each payent eans inteest fo a diffeent nube of peiods. The Januay deposit eans inteest fo Febuay, Mach, Apil, May, and June. Using the foula fo copound inteest, this deposit will gow to 100( ) 5 = \$ Howeve, the May deposit eans inteest fo only 1 onth and theefoe gows to only 100( ) 1 = \$101. The June deposit eans no inteest at all. We illustate this patten with the tieline in Figue 9.3. Januay Febuay Mach Apil May June July Eans inteest. Eans no inteest. FIGURE 9.3 Tieline fo odinay annuity with deposits at end of Januay,...,June. If we copute how uch each deposit contibutes to the account and su these aounts, we will have the value of the annuity on July 1. Januay 100(1.01) 5 = \$ Febuay 100(1.01) 4 = \$ Mach 100(1.01) 3 = \$ Apil 100(1.01) 2 = \$ May 100(1.01) 1 = \$ June 100(1.01) 0 =.00 Total = \$ We can expess the value of this annuity as 100(1.01) (1.01) (1.01) (1.01) (1.01) Copyight 2010 Peason Education, Inc.

3 424 CHAPTER 9 y Consue Matheatics By factoing out the 100, we can wite the value of the annuity in the fo 100[(1.01) 5 + (1.01) 4 + (1.01) 3 + (1.01) 2 + (1.01) 1 + 1]. (1) Notice that although you deposited at the end of each onth fo six onths, the fist deposit eaned inteest fo five onths, the second deposit eaned inteest fo fou onths, and so on. Following this patten, it is clea that if you had deposited the in a vacation savings account at the end of each onth fo 10 onths, the value of the annuity would have been 100[(1.01) 9 + (1.01) (1.01) 2 + (1.01) 1 + 1]. (2) Notice that in equation (1) we have an expession of the fo x 5 + x 4 + x 3 + x 2 + x 1 + 1, and in equation (2) we have an expession of the fo x 9 + x x 2 + x Fotunately, thee is a way to wite these lengthy expessions in a siple fo. Quiz Youself KEY POINT 14 Wite x 3 + x 2 + x as a quotient of two polynoials. (Do coputations that ae siila to those we did in Exaple 1.) The futue value of an annuity depends on the size of the payent, the inteest ate, and the nube of payents. EXAMPLE 1 Siplifying Annuity Coputations Show that x 5 + x 4 + x 3 + x 2 + x = x 6-1. x - 1 SOLUTION: To show this elationship, we ultiply the polynoials x 5 + x 4 + x 3 + x 2 + x and x - 1 in the usual way. x 5 + x 4 + x 3 + x 2 + x x 6 + x 5 + x 4 + x 3 + x 2 + x 1 * x x 5 - x 4 - x 3 - x 2 - x 1-1 x 6-1 This esult shows that (x 5 + x 4 + x 3 + x 2 + x 1 + 1)(x - 1) = x 6-1. Dividing this equation by x - 1, we obtain ou desied elationship. Now ty Execises 3 to 4. ] Following the patten of Exaple 1, we can pove that Retuning to ou vacation savings account exaple, we can think of 1.01 as x in equation (1). Then we can use equation (3) to siplify ou calculations. Because (1.01) 5 + (1.01) 4 + (1.01) 3 + (1.01) 2 + (1.01) we can wite x n + x n -1 + x n-2 + Á + x 2 + x = x n+1-1 x [(1.01) 5 + (1.01) 4 + (1.01) 3 + (1.01) 2 + (1.01) 1 + 1] L 100(6.1520) = \$ This is the sae aount that we found ealie. Doing siila coputations, we find that the 10-onth vacation savings account annuity has a value of 100[(1.01) 9 + (1.01) 8 + Á + (1.01) 2 + (1.01) 1 + 1] 14 = (1.01) = = L , We can genealize the pattens that we have just seen in a foula fo finding the futue value of an annuity. (3) = 100c (1.01)10-1 d L 100( ) = \$1, Copyight 2010 Peason Education, Inc.

4 9.4 y Annuities 425 FORMULA FOR FINDING THE FUTURE VALUE OF AN ORDINARY ANNUITY Assue that we ae aking n egula payents, R, into an odinay annuity. The inteest is being copounded ties a yea and deposits ae ade at the end of each copounding peiod. The futue value (o aount), A, of this annuity at the end of the n peiods is given by the equation A = R a1 + b n - 1. To calculate this expession, you should do the following steps: 1st: Find and add 1. 2nd: Raise 1 + to the n powe and then subtact 1. 3d: Divide the aount that you found in step 2 by. 4th: Multiply the quantity that you found in step 3 by R. EXAMPLE 2 Finding the Futue Value of an Odinay Annuity Assue that we ake a payent of \$50 at the end of each onth into an account paying a 6% annual inteest ate, copounded onthly. How uch will be in that account afte 3 yeas? SOLUTION: This account is an odinay annuity. The payent R is 50, the onthly ate 6% is 12 = = 0.005, and the nube of payents n is 3 * 12 = 36. Using the foula fo finding the futue value of an odinay annuity, we get TI-83 sceen veifies calculations in Exaple 2. Quiz Youself 15 Redo Exaple 2, except now assue that you ae depositing \$75 pe onth fo 2 yeas. 1 + n (1.005) A R ( ) \$1, * Now ty Execises 7 to 16. ] If you ake egula payents into an annuity fo any yeas, the value of the annuity can becoe enoous due to the copounding of inteest. Figue 9.4 shows that if the annual inteest ate is 6.6%, then in oughly 19 yeas, the aount of inteest that the annuity has eaned exceeds the aount of the deposits. The fact that the inteest cuve is ising so apidly indicates that the futue value of you account is also gowing apidly. KEY POINT With a sinking fund, we ake payents to save a specified aount. Sinking Funds You ay want to save egulaly to have a fixed aount available in the futue. Fo exaple, you ay want to save \$1,800 to tavel to Jaaica in 2 yeas. The question is, how uch should you save each onth to accoplish this? The account that you establish fo *When using you calculato, you should hold off on ounding you answes as long as possible. If you ound off too soon, you answes will diffe slightly fo the answes in this text. Copyight 2010 Peason Education, Inc.

5 426 CHAPTER 9 y Consue Matheatics HIGHLIGHT Between the Nubes Who Do You Tust? The tust that we ae efeing to is the Social Secuity tust fund. When you fist stated woking, you ay have been disayed to see a deduction fo you fist paycheck labeled FICA, which is an acony fo the Fedeal Insuance Contibutions Act. This law andates that wokes ust contibute a cetain aount of thei wages to the Social Secuity tust fund. When you contibute to Social Secuity, you ae not actually saving you oney fo you etieent, but you taxes ae paying fo soeone else s etieent. The idea is that when you etie, younge wokes will then pay fo you etieent. Howeve, soe see a huge poble with this. Right now, oughly 50 illion Aeicans eceive Social Secuity benefits with contibutions suppoted by appoxiately 200 illion U.S. wokes. If we think of this as a atio, evey peson on Social Secuity is suppoted by 4 wokes. In 1950, the atio was 16 wokes fo evey Social Secuity beneficiay. It is pojected that by 2030, the atio will be 2 wokes fo evey beneficiay and the fund will be in touble. So what can you do to potect you etieent? The govenent has been encouaging people to ake plans fo thei etieent by establishing tax-defeed annuities to guaantee that when they etie they will have oney to suppleent Social Secuity benefits. Tax defeed eans that the oney you set aside in the annuity is not taxed now but at a late date when you stat withdawing fo the annuity. As you will see in the execises, thee can be a huge financial benefit to saving this way. 250 Thousands of dollas Inteest exceeds deposits Inteest Deposits Months FIGURE 9.4 At an annual ate of 6.6%, the aount of inteest eaned in an odinay annuity exceeds the aount of deposits in about 230 onths. you deposits is called a sinking fund. You could estiate the aount to save each onth 1,800 by siply dividing 1,800 by 24 onths to get 24 = \$75 pe onth. Because you estiate ignoes the inteest that you deposits will geneate, the actual aount you would need to put aside each onth is soewhat less. Knowing exactly how uch you need to save each onth could be ipotant if you wee on a tight budget. Because a sinking fund is a special type of annuity, it is not necessay to find a new foula to answe this question. We can use the foula fo calculating the futue value of an odinay annuity that we have stated ealie. In this case, we know the value of A and we want to find R.* EXAMPLE 3 Calculating Payents fo a Sinking Fund Assue that you wish to save \$1,800 in a sinking fund in 2 yeas. The account pays 6% copounded quately and you will also ake payents quately. What should be you onthly payent? *In aking payents into a sinking fund, we will always ound the payent up to the next cent. Copyight 2010 Peason Education, Inc.

6 9.4 y Annuities 427 SOLUTION: Recall the foula fo finding the futue value of an odinay annuity: TI-83 sceen veifies calculations in Exaple 3. Quiz Youself 16 What payents ust you ake to a sinking fund that pays 9% yealy inteest copounded onthly if you want to save \$2,500 in 2 yeas? KEY POINT We use the log function to find how long it takes fo an annuity to accuulate a specified value. a1 + n b - 1 A = R. (4) We want the value A of the annuity to be \$1,800, the onthly ate is 4 = 0.015, and the nube of payents n is 2 * 4 = 8. Substituting these values in equation (4), we get Dividing both sides of the equation by , we get the onthly payent Now ty Execises 17 to 20. ] Soe Good Advice You ay be tepted to eoize a new foula to solve sinking fund pobles. This is not necessay, because once you have leaned to solve annuity pobles, you can use the sae foula (and a little bit of algeba) to solve sinking fund pobles. Soeties in woking with annuities, we want to know how long it will take to save a cetain aount. That is, in the annuity foula we want to find n. This poble is a little oe coplicated than those we have solved so fa. To solve such pobles, we will use the exponent popety fo the log function, which we intoduced in Section 9.3. We will show you how to use this popety in Exaple 4. EXAMPLE 4 1,800 = R R = 1, L \$ = R( ). Finding the Tie Requied to Accuulate \$1,000,000 6% 4 = 0.06 Suppose you have decided to etie as soon as you have saved \$1,000,000. You plan is to put \$200 each onth into an odinay annuity that pays an annual inteest ate of 8%. In how any yeas will you be able to etie? SOLUTION: We can use the futue value foula fo an odinay annuity to solve this poble: a1 + n b - 1 A = R Hee, A is the futue value of \$1,000,000, is the onthly inteest ate of 12 L , and R is 200. We ust find n, the nube of onths fo which you will be aking deposits. Theefoe, we ust solve fo n in the following equation: ( ) 1,000, A R n We begin by ultiplying both sides of the equation by : 6,670 = 200[( ) n - 1] Copyight 2010 Peason Education, Inc.

7 428 CHAPTER 9 y Consue Matheatics Quiz Youself 17 If you put \$150 pe onth into an odinay annuity that pays an annual inteest ate of 9%, how long will it take fo the annuity to have a value of,000? Next, we divide both sides of the equation by 200 and then add 1 to both sides of the equation: = ( ) n. Then we take the log of both sides: log = log( ) n. Now we use the exponent popety of the log function to siplify the equation: log = n log We divide by log and use a calculato to find n: log n = = L532. log This tells how any onths it will take you to save \$1,000,000. Dividing 532 by 12 gives 532 us 12 = yeas until you etieent. Now ty Execises 29 to 34. ] 17 Execises 9.4 Looking Back* These execises follow the geneal outline of the topics pesented in this section and will give you a good oveview of the ateial that you have just studied. 1. To calculate the futue value of the annuity in Exaple 2, we used the expession a1 + n b - 1 R. What is the eaning of? What is the eaning of? What is R? 2. What iplications do you see of the situation pesented in the Highlight egading the Social Secuity tust fund? Shapening You Skills In Execises 3 and 4, siplify each algebaic expession, as in Exaple x 7 + x x 2 + x x 8 + x x 2 + x Execises 5 and 6 ae based on the vacation account exaple at the beginning of this section. Assue that, beginning in Januay, you ake payents at the end of each onth into an account paying the specified yealy inteest. Inteest is copounded onthly. How uch will you have available fo you vacation by the specified date? 5. Monthly payent, ; yealy inteest ate, 6%; August 1 6. Monthly payent, \$200; yealy inteest ate, 3%; May 1 In Execises 7 and 16, find the value of each odinay annuity at the end of the indicated tie peiod. The payent R, fequency of deposits (which is the sae as the fequency of copounding), annual inteest ate, and the tie t ae given. 7. Aount, \$200; onthly; 3%; 8 yeas 8. Aount, \$450; onthly; 2.4%; 10 yeas 9. Aount, \$400; onthly; 9%; 4 yeas 10. Aount, \$350; onthly; 10%; 10 yeas 11. Aount, \$600; onthly; 9.5%; 8 yeas 12. Aount, \$500; onthly; 7.5%; 12 yeas 13. Aount, \$500; quately; 8%; 5 yeas 14. Aount, \$750; quately; 9%; 3 yeas 15. Aount, \$280; quately; 3.6%; 6 yeas 16. Aount, \$250; quately; 4.8%; 18 yeas In Execises 17 20, find the onthly payent R needed to have a sinking fund accuulate the futue value A. The yealy inteest ate and the tie t is given. Inteest is copounded onthly. Round you answe up to the next cent. 17. A = \$2,000; = 6%; t = A = \$10,000; = 12%; t = A = \$5,000; = 7.5%; t = A = \$8,000; = 4.5%; t = 3 Solve each equation fo x x = x = x = x = x x = = x x = = In Execises 29 34, use the foula fo finding the futue value of an odinay annuity, a1 + n b - 1 A = R, *Befoe doing these execises, you ay find it useful to eview the note How to Succeed at Matheatics on page xix. Copyight 2010 Peason Education, Inc.

8 9.4 y Execises 429 to solve fo n. You ae given A, R, and. Assue that payents ae ade onthly and that the inteest ate is an annual ate. 29. A = \$10,000; R = 200; = 9% 30. A = \$12,000; R = 400; = 8% 31. A = \$5,000; R = 150; = 6% 32. A = \$8,000; R = 400; = 5% 33. A = \$6,000; R = 250; = 7.5% 34. A = \$7,500; R = 100; = 8.5% Applying What You ve Leaned In Execises 35 40, assue that the copounding is being done onthly. 35. Saving fo a scoote. Matt is saving to buy a new Vespa. If he deposits \$75 at the end of each onth in an account that pays an annual inteest ate of 6.5%, how uch will he have saved in 30 onths? 36. Saving fo a tip. Angelina wants to save fo an Afican safai. She is putting \$200 each onth in an odinay annuity that pays an annual inteest ate of 9%. If she akes payents fo 2 yeas, how uch will she have saved fo he tip? 42. Saving to stat a business. Victoy is aking onthly payents into an annuity that pays 0.8% onthly inteest to save enough fo a down payent to stat he own business. If she wants to save \$10,000 in 5 yeas, what should he onthly payents be? 43. Saving fo consue goods. Sanda Lee is aking onthly payents into an annuity. She wants to have \$600 in the fund to buy a new convection ange in 6 onths, and the account pays 8.2% annual inteest. What ae he onthly payents to the account? 44. Saving fo consue goods. Lennox is aking onthly payents into an annuity. He wishes to have \$1,150 in 10 onths to buy execise equipent. His account pays 9% annual inteest. What ae his onthly payents to the account? Tax-defeed annuities wok like this: If, fo exaple, you plan to set aside \$400 pe onth fo you etieent in 30 yeas in a taxdefeed plan, the \$400 is not taxed now, so all of the \$400 is invested each onth. In a nondefeed plan, the \$400 is fist taxed and then the eainde is invested. So, if you tax backet is 25%, afte you pay taxes, you would have only 75% of the \$400 to invest each onth. Howeve, in the tax-defeed plan, all of you oney is taxed when you withdaw the oney. In the nondefeed plan, only the inteest that you have eaned is taxed. In Execises 45 50, we give the aount you ae setting aside each onth, you cuent tax ate, the nube of yeas you will contibute to the annuity, and you tax ate when you begin withdawing fo the annuity. Answe the following questions fo each situation: a) Find the value of the tax-defeed and the nondefeed accounts. b) Calculate the inteest that was eaned in both accounts. This will be the value of the account inus the payents you ade. c) If you withdaw all oney fo each account and pay the elevant taxes, which account is bette and by how uch? 37. Saving fo a ca. Kisty Joe deposits \$150 each onth in an odinay annuity to save fo a new ca. If the annuity pays a onthly inteest ate of 0.85%, how uch will she be able to save in 3 yeas? 38. Saving fo etieent. Cohutta is saving fo his etieent in 10 yeas by putting \$500 each onth into an odinay annuity. If the annuity has an annual inteest ate of 9.35%, how uch will he have when he eties? 39. Saving fo a vacation hoe. Wendy has set up an odinay annuity to save fo a etieent hoe in Floida in 15 yeas. If he onthly payents ae \$400 and the annuity has an annual inteest ate of 6.5%, what will be the value of the annuity when she eties? 40. Saving fo etieent. Thiep has set up an odinay annuity to save fo his etieent in 20 yeas. If his onthly payents ae \$350 and the annuity has an annual inteest ate of 7.5%, what will be the value of the annuity when he eties? 41. Saving fo a condoiniu. Kanye wants to save \$14,000 in 8 yeas by aking onthly payents into an odinay annuity fo a down payent on a condoiniu at the shoe. If the annuity pays 0.7% onthly inteest, what will his onthly payent be? Monthly Payent Nube of Yeas Annual Inteest Rate Cuent Tax Rate Futue Tax Rate 45. \$ % 25% 18% 46. \$ % 25% 15% 47. \$ % 30% 30% 48. \$ % 25% 25% 49. \$ % 25% 30% 50. \$ % 18% 25% In Execises 51 54, assue that onthly deposits ae being placed in an odinay annuity and inteest is copounded onthly. 51. Saving fo a fie tuck. The Reliance Voluntee Fie Copany wants to take advantage of a state poga to save oney to puchase a new fie tuck. The tuck will cost \$400,000, and ebes of the finance coittee estiate that with counity and state contibutions, they can save \$5,000 pe onth in an account paying 10.8% annual inteest. How long will it take to save fo the tuck? 52. Saving fo new equipent. BioCon, a bioengineeing copany, ust eplace its wate teatent equipent within 2 yeas. The Copyight 2010 Peason Education, Inc.

9 430 CHAPTER 9 y Consue Matheatics new equipent will cost \$80,000, and the copany will tansfe \$3,800 pe onth into a special account that pays 9.2% inteest. How long will it take to save fo this new equipent? 53. Saving fo a condoiniu. Kisten wants to save \$30,000 fo a down payent on a condoiniu at a ski esot. She feels that she can save \$550 pe onth in an account that has a 7.8% annual inteest ate. How long will it take to acquie he down payent? 54. Saving fo a business. Leo needs to save \$25,000 fo a down payent to stat a photo estoation business. He intends to save \$300 pe onth in an account that pays an annual inteest ate of 6%. How long will it take fo hi to save fo the down payent? Counicating Matheatics 55. What popety of the log function would you use to solve an equation of the fo y = a x fo x? 56. What is siila about the poble of finding the futue value of an annuity and finding the payents to ake into a sinking fund? How ae the two pobles elated? Using Technology to Investigate Matheatics 57. See you instucto fo tutoials fo the TI-83 to do financial calculations. Use you calculato to epoduce soe of the exaples in this section.* 58. You can find any inteactive annuity calculatos on the Intenet. Soe ipleent the calculations that you have leaned in this section and othes ipleent othe types of annuity pobles. Find soe inteesting annuity calculatos, expeient with the, and epot on you findings. Fo Exta Cedit 59. Saving fo etieent. Reconside Execise 38. Suppose that Cohutta had stated his annuity 10 yeas ealie. What would his onthly payents have been to accuulate the sae futue value in his annuity as you found in Execise 38? 60. Saving fo etieent. Reconside Execise 40. Suppose that Thiep had stated his annuity 10 yeas ealie. What would his onthly payents have been to accuulate the sae futue value in his annuity as you found in Execise 40? 61. Saving fo etieent. Calos began to save fo his etieent at age 25, and fo 10 yeas he put \$200 pe onth into an odinay annuity at an annual inteest ate of 6%. Afte the 10 yeas, he could no longe ake payents, so he placed the value of the annuity into anothe account that paid 6% annual inteest copounded onthly. He left the oney in this account fo 30 yeas until he was eady to etie. How uch did he have fo etieent? 62. Saving fo etieent. If Calos had waited until age 45 to think about etieent and then decided to put oney into an odinay annuity fo 20 yeas, what would his onthly payents have to be to accuulate the sae aount fo etieent as you found in Execise 61? (We assue that the inteest ate fo the annuity is the sae.) 63. Saving fo etieent. Exaine you solutions to Execises 61 and 62. What do you notice? Explain why this is so. 64. Copaing annuities. The diffeence between an annuity due and an odinay annuity is that with an annuity due, the payent is ade at the beginning of the onth athe than at the end of the onth. This eans that each payent geneates one oe onth of inteest than with an odinay annuity. a. How does this change the foula fo finding the futue value of an annuity? b. Use this foula to find the value of the annuity in Exaple 2, assuing that the annuity is an annuity due. Copyight 2010 Peason Education, Inc.

### 9.5 Amortization. Objectives

9.5 Aotization Objectives 1. Calculate the payent to pay off an aotized loan. 2. Constuct an aotization schedule. 3. Find the pesent value of an annuity. 4. Calculate the unpaid balance on a loan. Congatulations!

### est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

### I = Prt. = P(1+i) n. A = Pe rt

11 Chapte 6 Matheatcs of Fnance We wll look at the atheatcs of fnance. 6.1 Sple and Copound Inteest We wll look at two ways nteest calculated on oney. If pncpal pesent value) aount P nvested at nteest

### Products of the Second Pillar Pension

Óbuda Univesity e-bulletin Vol. 4, No. 1, 2014 Poducts of the Second Pilla Pension Jana Špiková Depatent of Quantitative Methods and Infoation Systes, Faculty of Econoics, Matej Bel Univesity Tajovského

### CONCEPT OF TIME AND VALUE OFMONEY. Simple and Compound interest

CONCEPT OF TIME AND VALUE OFMONEY Simple and Compound inteest What is the futue value of shs 10,000 invested today to ean an inteest of 12% pe annum inteest payable fo 10 yeas and is compounded; a. Annually

### AMB111F Financial Maths Notes

AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed

### FI3300 Corporate Finance

Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multi-peiod multi-cf time-value-of-money poblems: Geneal case Pepetuity

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.

Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

### Example: Suppose that we deposit \$1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?

Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,

### Valuation of Floating Rate Bonds 1

Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

### Firstmark Credit Union Commercial Loan Department

Fistmak Cedit Union Commecial Loan Depatment Thank you fo consideing Fistmak Cedit Union as a tusted souce to meet the needs of you business. Fistmak Cedit Union offes a wide aay of business loans and

### How Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes

How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk - Risk to shaeholdes esulting fom the use of debt. Financial Leveage - Incease in the vaiability of shaeholde etuns that

### Basic Financial Mathematics

Financial Engineeing and Computations Basic Financial Mathematics Dai, Tian-Shy Outline Time Value of Money Annuities Amotization Yields Bonds Time Value of Money PV + n = FV (1 + FV: futue value = PV

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### Converting knowledge Into Practice

Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

### The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

### Who Files for Bankruptcy? State Laws and the Characteristics of Bankrupt Households

Who iles fo Bankuptcy? State Laws and the Chaacteistics of Bankupt Households By MICHLL M. MILLR While pio papes have exained the ipact of state exeption and ganishent laws on the aveage household, this

### The Time Value of Money

he ime Value of Money Inteest Rates and Futue Value Inteest ates ae a facto in the valuation of vitually all financial instuments. While all money maket ates () ae quoted on an annual basis (PR nnual Pecentage

### YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE

YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1 - P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face

### GESTÃO FINANCEIRA II PROBLEM SET 1 - SOLUTIONS

GESTÃO FINANCEIRA II PROBLEM SET 1 - SOLUTIONS (FROM BERK AND DEMARZO S CORPORATE FINANCE ) LICENCIATURA UNDERGRADUATE COURSE 1 ST SEMESTER 2010-2011 Chapte 1 The Copoation 1-13. What is the diffeence

### Indiana Department of Revenue. Individual Income Tax Booklet IT-40. Cover Page. (Individual Forms are on the last pages.)

Indiana Depatment of Revenue Individual Income Tax Booklet IT-40 Cove Page (Individual Foms ae on the last pages.) What fom do I file? Indiana has fou diffeent individual income tax etuns. See which one

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

### Definitions and terminology

I love the Case & Fai textbook but it is out of date with how monetay policy woks today. Please use this handout to supplement the chapte on monetay policy. The textbook assumes that the Fedeal Reseve

### Define What Type of Trader Are you?

Define What Type of Tade Ae you? Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 1 Disclaime and Risk Wanings Tading any financial maket involves isk. The content of this

### Personal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit (-) Total Private Saving Rate (S Private /Y) 12/18/2009

1 Pesonal Saving Rate (S Households /Y) 2 SAVING AND INVESTMENT 16.0 14.0 12.0 10.0 80 8.0 6.0 4.0 2.0 0.0-2.0-4.0 1959 1961 1967 1969 1975 1977 1983 1985 1991 1993 1999 2001 2007 2009 Pivate Saving Rate

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### Faithful Comptroller s Handbook

Faithful Comptolle s Handbook Faithful Comptolle s Handbook Selection of Faithful Comptolle The Laws govening the Fouth Degee povide that the faithful comptolle be elected, along with the othe offices

### WELCOME TO OUR NEW NEWSLETTER

ISSUE ONE > JUNE 2013 WELCOME TO OUR NEW NEWSLETTER PARTNERS OUTLOOK Welcome to the band new Patnes Outlook newslette. It s had to believe it s June and anothe end of financial yea is aleady ceeping up.

### Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

xam #1 Review Answes 1. Given the following pobability distibution, calculate the expected etun, vaiance and standad deviation fo Secuity J. State Pob (R) 1 0.2 10% 2 0.6 15 3 0.2 20 xpected etun = 0.2*10%

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### GRADE 5 TEXAS. Multiplication and Division WORKSHEETS

GRADE 5 TEXAS Multiplication and Division WORKSHEETS Multi-digit multiplication Multiplying lage numbes is a pocess of multiple steps. Fist, you multiply: 542 6 =,252 2 You have now used up all you ones.

### BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR John R. Gaham Adapted fom S. Viswanathan FUQUA SCHOOL OF BUSINESS DUKE UNIVERSITY 1 In this lectue we conside the effect of govenment

### Who Files for Bankruptcy? State Laws and the Characteristics of Bankrupt Households

Who iles fo Bankuptcy? State Laws and the Chaacteistics of Bankupt Households Michelle M. Mille ssistant Pofesso Rutges Business School epatent of inance and Econoics Washington Pak, Roo 54 Newak, New

### In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

### Optimal Pricing Decision and Assessing Factors in. Closed-Loop Supply Chain

Applied Matheatical Sciences, Vol. 5, 2011, no. 80, 4015-4031 Optial Picing Decision and Assessing Factos in Closed-Loop Supply Chain Yang Tan Picing Science and Engineeing Depatent, FedEx Expess Wold

### TIME VALUE OF MONEY PROBLEMS CHAPTERS THREE TO TEN

TIME VLUE OF MONEY PROBLEMS CHPTERS THREE TO TEN Probles In how any years \$ will becoe \$265 if = %? 265 ln n 933844 9 34 years ln( 2 In how any years will an aount double if = 76%? ln 2 n 9 46 years ln76

### Ignorance is not bliss when it comes to knowing credit score

NET GAIN Scoing points fo you financial futue AS SEEN IN USA TODAY SEPTEMBER 28, 2004 Ignoance is not bliss when it comes to knowing cedit scoe By Sanda Block USA TODAY Fom Alabama comes eassuing news

### Controlling the Money Supply: Bond Purchases in the Open Market

Money Supply By the Bank of Canada and Inteest Rate Detemination Open Opeations and Monetay Tansmission Mechanism The Cental Bank conducts monetay policy Bank of Canada is Canada's cental bank supevises

### Trigonometric Functions of Any Angle

Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

### Review Graph based Online Store Review Spammer Detection

Review Gaph based Online Stoe Review Spamme Detection Guan Wang, Sihong Xie, Bing Liu, Philip S. Yu Univesity of Illinois at Chicago Chicago, USA gwang26@uic.edu sxie6@uic.edu liub@uic.edu psyu@uic.edu

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

### Effect of Contention Window on the Performance of IEEE 802.11 WLANs

Effect of Contention Window on the Pefomance of IEEE 82.11 WLANs Yunli Chen and Dhama P. Agawal Cente fo Distibuted and Mobile Computing, Depatment of ECECS Univesity of Cincinnati, OH 45221-3 {ychen,

Potfolio Pefoance Attibution Alia Biglova, Svetloa Rachev 2 Abstact In this pape, we povide futhe insight into the pefoance attibution by developent of statistical odels based on iniiing ETL pefoance isk

### KNIGHTS OF COLUMBUS LEADERSHIP RESOURCES. Practical Information for Grand Knights, District Deputies and Financial Secretaries

KNIGHTS OF COLUMBUS LEADERSHIP RESOURCES Pactical Infomation fo Gand Knights, Distict Deputies and Financial Secetaies Thank you fo taking on the esponsibility of being a leade in the Knights of Columbus.

### YOUR REPRODUCTIVE LIFE PLANNING GUIDE

YOUR REPRODUCTIVE LIFE PLANNING GUIDE Notheast Floida Healthy Stat Coalition 644 Cesey Blvd., Suite 210, Jacksonville, FL 32211 p. 904.723.5422 / f. 904.723.5433 www.nefhealthystat.og KEY QUESTIONS TO

### Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

### Power and Sample Size Calculations for the 2-Sample Z-Statistic

Powe and Sample Size Calculations fo the -Sample Z-Statistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the -Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.

### Chapter 11: Aggregate Demand II, Applying the IS-LM Model Th LM t

Equilibium in the - model The cuve epesents equilibium in the goods maket. Chapte :, Applying the - Model Th t C ( T) I( ) G The cuve epesents money maket equilibium. M L(, ) The intesection detemines

### Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 10 Solutions

Peason Physics Level 30 Unit VI Foces and Fields: hapte 10 Solutions Student Book page 518 oncept heck 1. It is easie fo ebonite to eove electons fo fu than fo silk.. Ebonite acquies a negative chage when

### Financing Terms in the EOQ Model

Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

### Cost/Benefit Analysis of Aquaponic Systems

Cost/Benefit Analysis of Aquaponic Systems Richad Chiang 1 PURPOSE pupose of this pape is to analyse the costs and benefits of aquaponic systems designed fo home use. Howeve, one can easily extapolate

### Simple way of teaching transistor amplifiers

ession 793 iple way of teaching tansisto aplifies ogdan M. Wilaowski Uniesity of Wyoing bstact Fo sall signal analysis a siple change fo coonly used tanconductance g to tansesistance /g leads to a significant

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Financial Planning and Risk-return profiles

Financial Planning and Risk-etun pofiles Stefan Gaf, Alexande Kling und Jochen Russ Pepint Seies: 2010-16 Fakultät fü Mathematik und Witschaftswissenschaften UNIERSITÄT ULM Financial Planning and Risk-etun

### Analysis of the purchase option of computers

Analysis of the of coputers N. Ahituv and I. Borovits Faculty of Manageent, The Leon Recanati Graduate School of Business Adinistration, Tel-Aviv University, University Capus, Raat-Aviv, Tel-Aviv, Israel

### Millennium Theory of Inertia and Gravity. Copyright 2004 Joseph A. Rybczyk

Millenniu Theoy of Inetia and Gavity Copyight 004 Joseph A. Rybczyk Abstact Investigative analysis of Keple s laws of planetay otion lead to the evelation that Newton s laws of echanics ae incoplete and

### Left- and Right-Brain Preferences Profile

Left- and Right-Bain Pefeences Pofile God gave man a total bain, and He expects us to pesent both sides of ou bains back to Him so that He can use them unde the diection of His Holy Spiit as He so desies

### 2006-2008 Energy Efficiency Rebate and Incentive Programs for Business.

The Company } A ~ Sempa Enegy utilit/ 2006-2008 Enegy Efficiency Rebate and Incentive Pogams fo Business. Lean how you business may qualify fo up to \$525,000* pe yea in ebates and incentives unde the Expess

### Trading Volume and Serial Correlation in Stock Returns in Pakistan. Abstract

Tading Volume and Seial Coelation in Stock Retuns in Pakistan Khalid Mustafa Assistant Pofesso Depatment of Economics, Univesity of Kaachi e-mail: khalidku@yahoo.com and Mohammed Nishat Pofesso and Chaiman,

### TRADING DAYS IN 2010 - LONDON DERIVATIVES MARKET INTRODUCTION

EURONEXT DERIVATIVES MARKETS LONDON MARKET TRADING DAYS IN 2010 - LONDON DERIVATIVES MARKET INTRODUCTION An oveview of the Tading Days in 2010 was issued unde London Notice No. 3203 on 13 Octobe 2009.

### Introduction to Stock Valuation. Background

Intoduction to Stock Valuation (Text efeence: Chapte 5 (Sections 5.4-5.9)) Topics backgound dividend discount models paamete estimation gowth oppotunities pice-eanings atios some final points AFM 271 -

### How to create a default user profile in Windows 7

AnswesThatWok TM How to ceate a default use pofile in Windows 7 (Win 7) How to ceate a default use pofile in Windows 7 When to use this document Use this document wheneve you want to ceate a default use

### The Predictive Power of Dividend Yields for Stock Returns: Risk Pricing or Mispricing?

The Pedictive Powe of Dividend Yields fo Stock Retuns: Risk Picing o Mispicing? Glenn Boyle Depatment of Economics and Finance Univesity of Cantebuy Yanhui Li Depatment of Economics and Finance Univesity

### Some text, some maths and going loopy. This is a fun chapter as we get to start real programming!

Chapte Two Some text, some maths and going loopy In this Chapte you ae going to: Lean how to do some moe with text. Get Python to do some maths fo you. Lean about how loops wok. Lean lots of useful opeatos.

### Finance Practice Problems

Iteest Fiace Pactice Poblems Iteest is the cost of boowig moey. A iteest ate is the cost stated as a pecet of the amout boowed pe peiod of time, usually oe yea. The pevailig maket ate is composed of: 1.

### Database Management Systems

Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012

### SLOT EXCHANGE AND PURCHASE PLANNING OF SHORT SEA SERVICES FOR LINER CARRIERS

Jounal of Maine Science and Technology, Vol. 18, No. 5, pp. 709-718 (2010) 709 SLOT EXCHNGE ND PURCHSE PLNNING OF SHORT SE SERVICES FOR LINER CRRIERS Hua-n Lu*, Shu-Ling Chen**, and Pete Lai*** Key wods

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Section 5-3 Angles and Their Measure

5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

### Confirmation of Booking

The Pesentes Rebecca Mogan Rebecca is a Taxation Consultant with the NTAA and has ove 15 yeas tax expeience. Rebecca holds a Bachelo of Ats and Law and a Mastes of Taxation. Rebecca has pesented a numbe

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exa FM/CAS Exa 2. Chapter 1. Basic Interest Theory. c 2008. Miguel A. Arcones. All rights reserved. Extract fro: Arcones Manual for the SOA Exa FM/CAS Exa 2, Financial Matheatics. Spring

### Capital Asset Pricing Model (CAPM) at Work

Capital Asset Picing Model (CAPM) at Wok Some o the Intuition behind CAPM Applications o CAPM Estimation and Testing o CAPM Intuition behind Equilibium Suppose you obseved the ollowing chaacteistics o

### Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions

Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isk-etu tadeoff ad time value of

### The impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011

The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### Loyalty Rewards and Gift Card Programs: Basic Actuarial Estimation Techniques

Loyalty Rewads and Gift Cad Pogams: Basic Actuaial Estimation Techniques Tim A. Gault, ACAS, MAAA, Len Llaguno, FCAS, MAAA and Matin Ménad, FCAS, MAAA Abstact In this pape we establish an actuaial famewok

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

### CHAPTER 10 Aggregate Demand I

CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

### BOARD OF EDUCATION Ossining Union Free School District Ossining, NY. Work Session/Special Meeting March 4, 2015 MINUTES

BOARD OF EDUCATION Ossining Union Fee School Distict Ossining, NY Wok Session/Special Meeting Mach 4, 2015 Ossining High School Libay Special Meeting: 7:00 p.m. MINUTES Pesent: Bill Kess, Boad Pesident,

### Farming: It s a Fact! Career & Technical Education, Introduction

Faming: It s a Fact! Caee & Technical Education, Intoduction Whee Does You Food Dolla Go? Mateials Compute Lab o Compute & Pojecto fo Pesentation Compute Speakes o Headphones Compute Intenet Access o Agicultual

### Solution Derivations for Capa #8

Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### Office of Family Assistance. Evaluation Resource Guide for Responsible Fatherhood Programs

Office of Family Assistance Evaluation Resouce Guide fo Responsible Fathehood Pogams Contents Intoduction........................................................ 4 Backgound..........................................................

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### Money Math for Teens. Introduction to Earning Interest: 11th and 12th Grades Version

Moey Math fo Tees Itoductio to Eaig Iteest: 11th ad 12th Gades Vesio This Moey Math fo Tees lesso is pat of a seies ceated by Geeatio Moey, a multimedia fiacial liteacy iitiative of the FINRA Ivesto Educatio

### NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING?

NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING? Daia Bunes David Neumak Michelle J. White Woking Pape 16932 http://www.nbe.og/papes/w16932

### An Epidemic Model of Mobile Phone Virus

An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### CANCER, HEART ATTACK OR STROKE CLAIM FORM

CANCER, HEART ATTACK OR STROKE CLAIM FORM Please ead the impotant infomation below: We suggest you make photocopies of any infomation sent fo you own ecods. Please be sue you policy numbe(s) is/ae witten

### Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

### DOCTORATE DEGREE PROGRAMS

DOCTORATE DEGREE PROGRAMS Application Fo Admission 2015-2016 5700 College Road, Lisle, Illinois 60532 Enollment Cente Phone: (630) 829-6300 Outside Illinois: (888) 829-6363 FAX: (630) 829-6301 Email: admissions@ben.edu

### Simple Harmonic Motion

Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

### Thank you for participating in Teach It First!

Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident