5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

Save this PDF as:

Size: px
Start display at page:

Download "5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?"

Transcription

1 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso will ever ecouter is a istallmet loa such as a car or home loa. For these types of loas, a amout of moey is borrowed. This amout plus iterest is paid back over with fixed paymets. I geeral, paymets are made o a mothly basis. The legth of time over which the paymets are made (the term) may be as short as 36 to 72 moths for a auto loa. Or the paymets may be made over a 15 to 30 year term for a home loa. Sice these loas behave like a decreasig auity, we ca use the formulas we have developed i earlier sectios to compute the paymet o the loa. I this sectio we will compute the paymet for several differet loas ad track those paymets i a special type of table called a amortizatio table. 1

2 Questio 1: How do you fid the preset value of a auity? I Sectio 5.3, we were able to calculate the future value or paymets of auities that were icreasig i value or decreasig i value. For a ordiary auity whose preset value is PV, the future value is 1 i 1 FV PV 1i PMT i if the paymets PMT are made ito the auity which ears iterest per period i over periods. Sice the paymets are made ito the auity, the secod term is added. The future value of the auity icreases. If the paymets are made from the auity, the secod term is subtracted to give 1 i 1 FV PV 1i PMT i I this case, the future value of the auity decreases sice moey is removed from the auity. I some applicatios, we wish to fid the preset value (what must be i the accout today) so that the accout eds up with some amout i the future. The ext two examples illustrate how to fid the preset value i cases like this. Example 1 Fid the Amout Needed to Establish a Trust Fud A wealthy idividual wishes to create a trust fud for his gradso so that he may withdraw $5000 at the ed of every quarter for te years. At the ed of te years, the gradso will receive the rest of the trust which cotais $50,000. If the trust ears 8% iterest compouded quarterly, how much should be put ito the trust iitially? Solutio I this problem, the amout i the auity is decreasig sice withdrawals are beig made. However, we wish the future value of the auity to be $50,000 i te years. This meas that a larger amout 2

3 must be placed i the trust ow so that paymet may be made from it Substitute FV 50000, PMT 5000, i , ad ito to give 1 i 1 FV PV 1i PMT i PV Now solve this equatio for the preset value PV PV PV This is calculated i a TI Graphig Calculator as show below. 40 The trust must be established with a iitial deposit of $159,

4 Example 2 Reachig a Retiremet Goal A fifty-five year old ivestor wishes to retire at age 67. The ivestor has budgeted $1000 a moth that she may deposit i a ordiary auity that ears 5% iterest compouded mothly. If she wishes to accumulate $2,000,000 for retiremet, what must be i the accout today to reach that goal? Solutio His is a icreasig auity sice regular paymets are beig made a accout. Substitute PMT 1000, i, 144 ad FV ito 1 i 1 FV PV 1i PMT i ad solve for the preset value PV. This yields PV PV PV Subtract the secod term o the right from both sides 1 Divide both sides by , PV This is calculated i a TI Graphig Calculator as show below. To simplify the calculatio, the umerator is calculated first. The that aswer is divided by the deomiator. 4

5 There must be $990, i the auity today, for the value to grow to $2,000,000 i twelve years. 5

6 Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued iterest. This process is called amortizatio. I the laguage of fiace, a loa is said to be amortized if the amout of the loa ad iterest are paid usig fixed regular paymets. From the perspective of the leder, this type of loa is a decreasig auity. The amout of the loa is the preset value of the auity. The paymets from the auity (to the leder) reduce the value of the auity util the future value is zero. This iterpretatio allows us to determie the paymet PMT o a loa of PV dollars. Start with the decreasig auity formula ad set the future value FV equal to zero, This equatio is simplified to give 1 i 1 0PV1i PMT i 0i PV 1i PMT 1i 1 i 1i 1 0i PVPMT 1 0i PVPMT1 1 i Clear the fractio by multiplyig each term by i Divide each term by 1 i Simplify the fractio by dividig 1 i ito each term i the umerator Now solve this equatio for the paymet PMT: PMT 1 1i i PV Add 11 PMT i to both sides i PV PMT= 1 1 i Divide both sides by 11 i 6

7 o a Amortized Loa Suppose a loa of PV dollars is amortized by periodic paymets of PMT at the ed of each period. If the loa has a iterest rate of i per period over periods, the paymet is PMT= i PV 1 1 i We ca use this formula to calculate the paymet o ay loa that is amortized. Pay special attetio to the loa amout. Ofte the loa amout is ot the same as the purchase price because of a dow paymet. A dow paymet is a amout paid up frot that reduces the amout that must be borrowed. This amout must be subtracted from the purchase price to give the loa amout. Whe a loa is amortized for the purchase of a home, the loa is called a mortgage. A typical mortgage is paid back over a 15 or 30 year period with mothly paymets. Example 3 o a Amortized Loa A youg professor purchases a home for $149,000. He plas to take out a 30 year mortgage at a aual iterest rate of 5.75%. The mortgage requires a dow paymet of 20% of the purchase price. a. Fid the mothly paymet o this mortgage. Solutio To qualify for this loa, the professor must put 20% dow, Dow The loa amout is PV 149, , , 200. For a 30 year mortgage, there are 30 or 360 periods. The iterest rate per 75 period is i. Usig these values, the mothly paymet is 7

8 75 PMT This calculatio may be carried out o a TI graphig calculator as show below. This paymet is usually rouded up to the earest pey to isure the loa is paid off. I practice, this meas the fial paymet will be slightly less tha all other paymets. b. How much iterest is paid o this mortgage? Solutio Accordig to part a, the professor will pay a total of 360 $ or $250, over the term of the loa. Sice the loa amout is $119,200, the additioal amout paid must be iterest, Iterest $250, $119, 200 $131, The professor pays $131, i iterest o this 30 year mortgage. c. The professor has also discovered that he qualifies for a 15 year loa at a aual iterest rate of 4.85%. This mortgage also requires a 20% dow paymet. Fid the mothly paymet o this mortgage. Solutio For this mortgage, the umber of periods is 15 or The itest rate per moth is i. This leads to a paymet of 8

9 PMT The calculatio is show below o a TI Graphig Calculator. Although the iterest rate is lower for this mortgage, the shorter term leads to a higher mothly paymet of $ d. How much iterest is paid o the 15 year mortgage? Solutio The professor will pay a total of 180 $ or $168, i paymets. The iterest is Iterest $168, $119, 200 $48, The professor pays $48, i iterest o this 15 year mortgage. Eve though the 15 year mortgage has a lower iterest rate, the shorter term leads to higher paymets tha the 30 year mortgage. However, because of the lower iterest rate ad shorter term, the amout of iterest paid to the leder for the 15 year loa is almost a third of the iterest paid o the 30 year loa. I geeral, loas with shorter terms have lower iterest rates. This leads to less iterest paid for shorter term loas. The paymets calculated above are the portio of a mortgage paymet that applies to the loa. A typical mortgage paymet also icludes other amouts to cover property 9

10 taxes, homeowers isurace, ad mortgage isurace. These amouts ca icrease the overall paymet by a large amout. 10

11 Questio 3: How do you make a amortizatio table? A amortizatio table (also called a amortizatio schedule) records the portio of the paymet that applies to the pricipal ad the portio that applies to iterest. Usig this iformatio, we ca determie exactly how much is owed o the loa at the ed of ay period. Amortizatio tables are useful whe a loa is to be paid off. Recall that whe we calculated the paymet, we rouded the amout of the paymet up to the pey. Over the term of the loa, we might pay a additioal amout each moth leadig to the pricipal beig reduced more quickly tha aticipated. Whe the fial loa paymet is made, it eeds to be adjusted to isure the balace is paid off properly. Differet leders roud paymets ad iterest differetly. This may lead to slightly differet umbers i the amortizatio table. Suppose you wat to borrow $10,000 for a automobile. Navy Federal Credit Uio offers a loa at a aual rate of 1.79% amortized over moths. The paymet would be PMT Sice paymets are made to the pey, a paymet of $ would lead to a overpaymet of almost a half of a pey. While this may ot seem like much, over the term of the loa it ca add up. Fiacial istitutios eed to accurately accout for these small amouts to isure their books are balaced. A amortizatio table helps them to do this. Amortizatio tables geerally have five colums. These colums track the paymet umber, the amout of the paymet, the iterest paid i the paymet, the portio of the paymet applied to the balace, ad the outstadig balace o the loa after the 11

12 paymet is made. Let s look at how the amouts i the table are calculated. We ll do this by lookig at the differet rows of the table, oe at a time. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 The first row of the table helps us to establish the iitial balace o the loa. We call it paymet 0 sice it does ot correspod to a actual paymet. Usig this balace, we ca determie the portio of the paymet, $841.44, that is applied to the balace ad the portio that is iterest. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $ $14.92 $ $ The iterest i the paymet is calculate by multiplyig the iterest rate per period times the balace at the ed of the previous period, Iterest i 1 $10,000 $14.92 I this amortizatio table, we will roud iterest amouts to the earest pey. I practice, you should check with the leder to see how they roud iterest i the table. Sice the amout applied to balace is the differece betwee the paymet ad the iterest, Amout i 1 Applied to Balace $ $14.92 $826.52

13 This amout reduces the balace at the ed of the period, Balace at the Ed of the First Period $10,000 $ $ This strategy is also used to fill i the amouts for the secod paymet. However, i this case, the iterest is calculated usig the balace after the previous period. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $ $14.92 $ $ $ $13.68 $ $ As the balace decreases, the iterest also decreases. This meas that a larger ad larger portio of the paymet goes to payig off the balace. s 3 through 11 are carried out i a similar fashio to give the ext few rows. Remember, i this table we are roudig iterest amouts to the earest pey. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $ $14.92 $ $ $ $13.68 $ $ $ $.45 $ $ $ $11.21 $ $ $ $9.97 $ $ $ $8.73 $ $ $ $7.49 $ $

14 8 $ $6.25 $ $ $ $5.00 $ $ $ $3.75 $ $ $ $2.50 $ $ For the last paymet, we eed to pay off the outstadig balace of $ This meas the amout of the last paymet applied to the balace must be $ The iterest i the last paymet is Iterest i $ $1.25 Combiig these two amouts gives the amout of the last paymet, Amout of $ $ With these amouts, we ca complete the amortizatio table. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $ $14.92 $ $ $ $13.68 $ $ $ $.45 $ $ $ $11.21 $ $ $ $9.97 $ $ $ $8.73 $ $ $ $7.49 $ $ $ $6.25 $ $ $ $5.00 $ $

15 10 $ $3.75 $ $ $ $2.50 $ $ $ $1.25 $ $0 If we add the iterest amouts, we fid the total amout of iterest paid is $ If we roud the paymet or iterest amouts differetly, the amortizatio table yields differet amouts of iterest. I the ext example, we roud all paymets ad iterest amouts up to the earest pey to see how these chage the total amout of iterest paid. Example 4 Make a Amortizatio Table Suppose Navy Federal Credit Uio rouds all iterest ad paymet amouts up. a. Fid the amortizatio table o a loa of $10,000 amortized at a aual rate of 1.79% over moths with mothly paymets. Solutio The terms of the loa are the same as was described above. If the paymet is rouded up, we still get a paymet of $ Whe we carry out the process described earlier, we get the table below. Number Amout of Iterest i the Amout i Applied to Balace Outstadig Balace at the Ed of the Period 0 $10,000 1 $ $14.92 $ $ $ $13.69 $ $ $ $.45 $ $ $ $11.22 $ $ $ $9.98 $ $ $ $8.74 $ $

16 7 $ $7.50 $ $ $ $6.25 $ $ $ $5.01 $ $ $ $3.76 $ $ $ $2.51 $ $ $ $1.26 $ $0 I this table, several of the paymets iclude a slightly higher amout of iterest. This meas that less of the paymet goes towards the outstadig balace. This amout is made up i the last paymet where $ is paid to brig the balace to zero. This causes the fial paymet to be slightly higher. b. Add the iterest amout i the third colum to fid the total amout of iterest paid. Solutio The sum of the iterest amouts is $ This is slightly higher tha whe iterest amouts are rouded to the earest pey. This is to be expected sice we rouded all iterest amouts up. The paymets ad iterest amouts may be rouded to the earest pey, rouded up to the earest pey, or rouded dow to the earest pey. I all cases, ay discrepacies are made up i the fial paymet. 16

Question 2: How is a loan amortized?

Question 2: How is a loan amortized? Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

FI A CIAL MATHEMATICS

FI A CIAL MATHEMATICS CHAPTER 7 FI A CIAL MATHEMATICS Page Cotets 7.1 Compoud Value 117 7.2 Compoud Value of a Auity 118 7.3 Sikig Fuds 119 7.4 Preset Value 122 7.5 Preset Value of a Auity 122 7.6 Term Loas ad Amortizatio 123

More information

Time Value of Money. First some technical stuff. HP10B II users

Time Value of Money. First some technical stuff. HP10B II users Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

More information

CHAPTER 11 Financial mathematics

CHAPTER 11 Financial mathematics CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

More information

Terminology for Bonds and Loans

Terminology for Bonds and Loans ³ ² ± Termiology for Bods ad Loas Pricipal give to borrower whe loa is made Simple loa: pricipal plus iterest repaid at oe date Fixed-paymet loa: series of (ofte equal) repaymets Bod is issued at some

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

I. Why is there a time value to money (TVM)?

I. Why is there a time value to money (TVM)? Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

How to use what you OWN to reduce what you OWE

How to use what you OWN to reduce what you OWE How to use what you OWN to reduce what you OWE Maulife Oe A Overview Most Caadias maage their fiaces by doig two thigs: 1. Depositig their icome ad other short-term assets ito chequig ad savigs accouts.

More information

FM4 CREDIT AND BORROWING

FM4 CREDIT AND BORROWING FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014 1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value

More information

2 Time Value of Money

2 Time Value of Money 2 Time Value of Moey BASIC CONCEPTS AND FORMULAE 1. Time Value of Moey It meas moey has time value. A rupee today is more valuable tha a rupee a year hece. We use rate of iterest to express the time value

More information

MMQ Problems Solutions with Calculators. Managerial Finance

MMQ Problems Solutions with Calculators. Managerial Finance MMQ Problems Solutios with Calculators Maagerial Fiace 2008 Adrew Hall. MMQ Solutios With Calculators. Page 1 MMQ 1: Suppose Newma s spi lads o the prize of $100 to be collected i exactly 2 years, but

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

Simple Annuities Present Value.

Simple Annuities Present Value. Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.

More information

THE TIME VALUE OF MONEY

THE TIME VALUE OF MONEY QRMC04 9/17/01 4:43 PM Page 51 CHAPTER FOUR THE TIME VALUE OF MONEY 4.1 INTRODUCTION AND FUTURE VALUE The perspective ad the orgaizatio of this chapter differs from that of chapters 2 ad 3 i that topics

More information

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2 TO: Users of the ACTEX Review Semiar o DVD for SOA Exam FM/CAS Exam FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Exam FM (CAS

More information

CDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest

CDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Time Value of Money, NPV and IRR equation solving with the TI-86

Time Value of Money, NPV and IRR equation solving with the TI-86 Time Value of Moey NPV ad IRR Equatio Solvig with the TI-86 (may work with TI-85) (similar process works with TI-83, TI-83 Plus ad may work with TI-82) Time Value of Moey, NPV ad IRR equatio solvig with

More information

A Guide to the Pricing Conventions of SFE Interest Rate Products

A Guide to the Pricing Conventions of SFE Interest Rate Products A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios

More information

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

VALUATION OF FINANCIAL ASSETS

VALUATION OF FINANCIAL ASSETS P A R T T W O As a parter for Erst & Youg, a atioal accoutig ad cosultig firm, Do Erickso is i charge of the busiess valuatio practice for the firm s Southwest regio. Erickso s sigle job for the firm is

More information

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

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

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

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

Choosing a Mortgage FIXED-RATE MORTGAGES. ADJUSTABLE-RATE MORTGAGES (ARMs)

Choosing a Mortgage FIXED-RATE MORTGAGES. ADJUSTABLE-RATE MORTGAGES (ARMs) Choosig A Mortgage Like homes, home mortgages come i all shapes ad sizes: short-term, log-term, fixed, adjustable, jumbo, balloo these are all terms that will soo be familiar to you, if they re ot already.

More information

How to read A Mutual Fund shareholder report

How to read A Mutual Fund shareholder report Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.

More information

CHAPTER 4: NET PRESENT VALUE

CHAPTER 4: NET PRESENT VALUE EMBA 807 Corporate Fiace Dr. Rodey Boehe CHAPTER 4: NET PRESENT VALUE (Assiged probles are, 2, 7, 8,, 6, 23, 25, 28, 29, 3, 33, 36, 4, 42, 46, 50, ad 52) The title of this chapter ay be Net Preset Value,

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

Classic Problems at a Glance using the TVM Solver

Classic Problems at a Glance using the TVM Solver C H A P T E R 2 Classc Problems at a Glace usg the TVM Solver The table below llustrates the most commo types of classc face problems. The formulas are gve for each calculato. A bref troducto to usg the

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

Sole trader financial statements

Sole trader financial statements 3 Sole trader fiacial statemets this chapter covers... I this chapter we look at preparig the year ed fiacial statemets of sole traders (that is, oe perso ruig their ow busiess). We preset the fiacial

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

10.5 Future Value and Present Value of a General Annuity Due

10.5 Future Value and Present Value of a General Annuity Due Chapter 10 Autes 371 5. Thomas leases a car worth $4,000 at.99% compouded mothly. He agrees to make 36 lease paymets of $330 each at the begg of every moth. What s the buyout prce (resdual value of the

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

For customers Key features of the Guaranteed Pension Annuity

For customers Key features of the Guaranteed Pension Annuity For customers Key features of the Guarateed Pesio Auity The Fiacial Coduct Authority is a fiacial services regulator. It requires us, Aego, to give you this importat iformatio to help you to decide whether

More information

Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving

Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig Tax Policy Brach Departmet of Fiace Jue 30, 1998 2 Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig This

More information

_ìóáåö=vçìê=cáêëí= eçãé=áå=`~å~ç~ What Newcomers Need to Know

_ìóáåö=vçìê=cáêëí= eçãé=áå=`~å~ç~ What Newcomers Need to Know _ìóáåö=vçìê=cáêëí= eçãé=áå=`~å~ç~ What Newcomers Need to Kow ^=mi^`b=ql=`^ii=eljbk=^=mi^`b=lc=vlro=ltkk You ve made Caada your ew home ad ow you re lookig for a place of your ow. Buyig a house is a excitig

More information

Current Year Income Assessment Form

Current Year Income Assessment Form Curret Year Icome Assessmet Form Academic Year 2015/16 Persoal details Perso 1 Your Customer Referece Number Your Customer Referece Number Name Name Date of birth Address / / Date of birth / / Address

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

FINANCIAL MATHEMATICS 12 MARCH 2014

FINANCIAL MATHEMATICS 12 MARCH 2014 FINNCIL MTHEMTICS 12 MRCH 2014 I ths lesso we: Lesso Descrpto Make use of logarthms to calculate the value of, the tme perod, the equato P1 or P1. Solve problems volvg preset value ad future value autes.

More information

Information about Bankruptcy

Information about Bankruptcy Iformatio about Bakruptcy Isolvecy Service of Irelad Seirbhís Dócmhaieachta a héirea Isolvecy Service of Irelad Seirbhís Dócmhaieachta a héirea What is the? The Isolvecy Service of Irelad () is a idepedet

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Grow your business with savings and debt management solutions

Grow your business with savings and debt management solutions Grow your busiess with savigs ad debt maagemet solutios A few great reasos to provide bak ad trust products to your cliets You have the expertise to help your cliets get the best rates ad most competitive

More information

Introducing Your New Wells Fargo Trust and Investment Statement. Your Account Information Simply Stated.

Introducing Your New Wells Fargo Trust and Investment Statement. Your Account Information Simply Stated. Itroducig Your New Wells Fargo Trust ad Ivestmet Statemet. Your Accout Iformatio Simply Stated. We are pleased to itroduce your ew easy-to-read statemet. It provides a overview of your accout ad a complete

More information

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place.

PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place. PENSION ANNUITY Policy Coditios Documet referece: PPAS1(7) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity.

More information

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

Institute of Actuaries of India Subject CT1 Financial Mathematics

Institute of Actuaries of India Subject CT1 Financial Mathematics Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i

More information

DC College Savings Plan Helping Children Reach a Higher Potential

DC College Savings Plan Helping Children Reach a Higher Potential 529 DC College Savigs Pla Helpig Childre Reach a Higher Potetial reach Sposored by Govermet of the District of Columbia Office of the Mayor Office of the Chief Fiacial Officer Office of Fiace ad Treasury

More information

Savings and Retirement Benefits

Savings and Retirement Benefits 60 Baltimore Couty Public Schools offers you several ways to begi savig moey through payroll deductios. Defied Beefit Pesio Pla Tax Sheltered Auities ad Custodial Accouts Defied Beefit Pesio Pla Did you

More information

CHAPTER 2. Time Value of Money 6-1

CHAPTER 2. Time Value of Money 6-1 CHAPTER 2 Tme Value of Moey 6- Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 6-2 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Two people, one policy. Affordable long-term care coverage for both.

Two people, one policy. Affordable long-term care coverage for both. Log-term care rider, available o Natiowide YourLife No-Lapse Guaratee SUL II Feature guide LIFE INSURANCE + LONG-ERM CARE wo people, oe policy. Affordable log-term care coverage for both. NAIONWIDE LIFE

More information

Future Value of an Annuity

Future Value of an Annuity Future Value of a Auty After payg all your blls, you have $200 left each payday (at the ed of each moth) that you wll put to savgs order to save up a dow paymet for a house. If you vest ths moey at 5%

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PERFORMANCE COUNCIL (IPC) INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

More information

A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT

A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT INTERNATIONAL JOURNAL OF BUSINESS, 1(1), 1996 ISSN:1083-4346 A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT Wm R McDaiel, Daiel E. McCarty, ad Keeth A. Jessell Whe oe examies stadard fiacial maagemet

More information

Comparing Credit Card Finance Charges

Comparing Credit Card Finance Charges Comparig Credit Card Fiace Charges Comparig Credit Card Fiace Charges Decidig if a particular credit card is right for you ivolves uderstadig what it costs ad what it offers you i retur. To determie how

More information

Finance Practice Problems

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.

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Subject CT5 Contingencies Core Technical Syllabus

Subject CT5 Contingencies Core Technical Syllabus Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value

More information

Enhance Your Financial Legacy Variable Annuity Death Benefits from Pacific Life

Enhance Your Financial Legacy Variable Annuity Death Benefits from Pacific Life Ehace Your Fiacial Legacy Variable Auity Death Beefits from Pacific Life 7/15 20172-15B As You Pla for Retiremet, Protect Your Loved Oes A Pacific Life variable auity ca offer three death beefits that

More information

Working with numbers

Working with numbers 4 Workig with umbers this chapter covers... This chapter is a practical guide showig you how to carry out the types of basic calculatio that you are likely to ecouter whe workig i accoutig ad fiace. The

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

Get advice now. Are you worried about your mortgage? New edition

Get advice now. Are you worried about your mortgage? New edition New editio Jauary 2009 Are you worried about your mortgage? Get advice ow If you are strugglig to pay your mortgage, or you thik it will be difficult to pay more whe your fixed-rate deal eds, act ow to

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

Health and dental coverage that begins when your group health benefits end

Health and dental coverage that begins when your group health benefits end Health ad detal coverage that begis whe your group health beefits ed Uderwritte by The Maufacturers Life Isurace Compay Page 1 of 5 FollowMeTM Health ca be your solutio. Life is full of chages. Some are

More information

Retirement By the Numbers

Retirement By the Numbers Retiremet By the Numbers Fiacial priorities may vary depedig o your age, but every perso wats a successful retiremet. This workbook ca help you estimate your retiremet eeds ad explai the optios available

More information

ECONOMICS. Calculating loan interest no. 3.758

ECONOMICS. Calculating loan interest no. 3.758 F A M & A N H S E E S EONOMS alculatig loa iterest o. 3.758 y Nora L. Dalsted ad Paul H. Gutierrez Quick Facts... The aual percetage rate provides a coo basis to copare iterest charges associated with

More information

Chapter 7. V and 10. V (the modified premium reserve using the Full Preliminary Term. V (the modified premium reserves using the Full Preliminary

Chapter 7. V and 10. V (the modified premium reserve using the Full Preliminary Term. V (the modified premium reserves using the Full Preliminary Chapter 7 1. You are give that Mortality follows the Illustrative Life Table with i 6%. Assume that mortality is uiformly distributed betwee itegral ages. Calculate: a. Calculate 10 V for a whole life

More information

Enhance Your Financial Legacy Variable Annuities with Death Benefits from Pacific Life

Enhance Your Financial Legacy Variable Annuities with Death Benefits from Pacific Life Ehace Your Fiacial Legacy Variable Auities with Death Beefits from Pacific Life 9/15 20188-15C FOR CALIFORNIA As You Pla for Retiremet, Protect Your Loved Oes A Pacific Life variable auity ca offer three

More information

MainStay Funds IRA/SEP/Roth IRA Distribution Form

MainStay Funds IRA/SEP/Roth IRA Distribution Form MaiStay Fuds IRA/SEP/Roth IRA Distributio Form Do ot use for IRA Trasfers or SIMPLE IRA INSTRUCTIONS Before completig this form, please refer to the applicable Custodial Agreemet ad Disclosure Statemet

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Installment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate

Installment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Nia-Nia JI a,*, Yue LI, Dog-Hui WNG College of Sciece, Harbi

More information

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology

More information

Best of security and convenience

Best of security and convenience Get More with Additioal Cardholders. Importat iformatio. Add a co-applicat or authorized user to your accout ad you ca take advatage of the followig beefits: RBC Royal Bak Visa Customer Service Cosolidate

More information

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff, NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical

More information

Discounting. Finance 100

Discounting. Finance 100 Discoutig Fiace 100 Prof. Michael R. Roberts 1 Topic Overview The Timelie Compoudig & Future Value Discoutig & Preset Value Multiple Cash Flows Special Streams of Cash Flows» Perpetuities» Auities Iterest

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

HOSPITAL NURSE STAFFING SURVEY

HOSPITAL NURSE STAFFING SURVEY 2012 Ceter for Nursig Workforce St udies HOSPITAL NURSE STAFFING SURVEY Vacacy ad Turover Itroductio The Hospital Nurse Staffig Survey (HNSS) assesses the size ad effects of the ursig shortage i hospitals,

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

I apply to subscribe for a Stocks & Shares ISA for the tax year 20 /20 and each subsequent year until further notice.

I apply to subscribe for a Stocks & Shares ISA for the tax year 20 /20 and each subsequent year until further notice. IFSL Brooks Macdoald Fud Stocks & Shares ISA Trasfer Applicatio Form IFSL Brooks Macdoald Fud Stocks & Shares ISA Trasfer Applicatio Form Please complete usig BLOCK CAPITALS ad retur the completed form

More information

Chapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization

Chapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization Chapter 3 Mathematcs of Face Secto 4 Preset Value of a Auty; Amortzato Preset Value of a Auty I ths secto, we wll address the problem of determg the amout that should be deposted to a accout ow at a gve

More information

Investing in Stocks WHAT ARE THE DIFFERENT CLASSIFICATIONS OF STOCKS? WHY INVEST IN STOCKS? CAN YOU LOSE MONEY?

Investing in Stocks WHAT ARE THE DIFFERENT CLASSIFICATIONS OF STOCKS? WHY INVEST IN STOCKS? CAN YOU LOSE MONEY? Ivestig i Stocks Ivestig i Stocks Busiesses sell shares of stock to ivestors as a way to raise moey to fiace expasio, pay off debt ad provide operatig capital. Ecoomic coditios: Employmet, iflatio, ivetory

More information