3. Present value of Annuity Problems

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "3. Present value of Annuity Problems"

Transcription

1 Mathematcs of Fnance The formulae 1. A = P(1 +.n) smple nterest 2. A = P(1 + ) n compound nterest formula 3. A = P(1-.n) deprecaton straght lne 4. A = P(1 ) n compound decrease dmshng balance 5. P = - present value of annuty 6. A = - future value of annuty 7. A = - future value of annuty one nstalment does not earn nterest Use for snkng fund. 8. B n = - m s remanng perod or payments all payments equal. 9. P = - d s deferred perod A. Calculaton Problems 1. Use the formula A = P(1 +.n) to fnd 1.1 A f P = R5000 ; = 10% pa; n= 6 years 1.2 P f A = R10000 ; = 9% pa; n = 5 years 1.3 f A = R20000; P = R10000 ; n = 10years 1.4 n f A = R15000; P = R10000; = 7 years 2. Use the formula A = P(1 + ) n to fnd 2.1 A f P = R5000 ; = 10% pa compounded monthly; n= 6 years 2.2 P f A = R10000 ; = 9% pa compounded quarterly; n = 5 years 2.3 f A = R20000; P = R10000 ; n = 10years; nterest compounded monthly 2.4 n f A = R15000; P = R10000; = 7 years; nterest compounded quarterly 2.5 A f P = R60000 ; = 12%pa compounded daly; n = 5 years 3. Use the formula A = to fnd 3.1 A f x = R3500; = 10% pa compounded monthly; n = 10 years. 3.2 x f A = R500000; = 13% pa compounded monthly; n = 8 years 3.3 n f A = R600000; = 12%; x = R10000 ; nterest compounded monthly. 4. Use the formula A = to fnd 4.1 A f x = R3500; = 10% pa compounded monthly; n = 10 years. 4.2 x f A = R500000; = 13% pa compounded monthly; n = 8 years 4.3 n f A = R600000; = 12%; x = R10000 ; nterest compounded monthly. 5. Use the formula P = 5.1 P f x = R2000; = 8% pa compounded monthly and n = 10 years 5.2 P f x = R50000; = 10% pa compounded annually; n = 6 years 5.3 x f P = R ; = 9% pa compounded monthly; n = 20 years 5.4 n f P = R ; = 8,3% pa compounded monthly; n = 18 years.

2 6. Determne the effectve rate of nterest for the year f the nomnal rate of nterest s % pa compounded monthly % pa compounded monthly % pa compounded daly % pa compounded daly % pa compounded quarterly 7. Determne the nomnal rate of nterest pa f the effectve rate of nterest s % pa; compounded monthly ,4 % pa ; compounded monthly ,8% pa compounded daly ,2% pa compounded quarterly 8. Determne the effectve rate of nterest for 6 months f 8.1 the nomnal rate of nterest s 13,6% pa compounded monthly 8.2 the nomnal rate of nterest s 11,4% pa compounded monthly. B. Problems on Mathematcs of Fnance 1. Mrs Khumalo wanted to nvest a certan amount of money to ensure that her two chldren, ages 3 years and 5 years and whose brthday fall on the same day, are guaranteed a sum of R each when they each reach 19. She was gven an nterest rate of 9,6% compounded monthly for the gven perod. Determne the amount she must nvest now n order to acheve her objectve. 2. Mr Zee saved a certan amount so that on hs 40 th brthday he would receve R and on hs 50 th brthday he would receve R He s 24 years old when he dd hs nvestment at a rate of nterest of 15% pa compounded monthly. What must he nvest now? 3. Present value of Annuty Problems 3.1 Mr Y took out a loan for a certan amount. He pad ths loan off n 10 equal monthly nstalments of R1000 each at an nterest rate of 12% pa compounded annually. Determne the amount of the loan taken f the frst payment starts at the end of the frst year.

3 Frst Prncples: - P = (1,12) (1,12) (1,12) (1,12) -10 P = (1,12) -1 [ 1 + (1,12) -1 + (1,12) (1,12) -10 ] 1 + (1,12) -1 + (1,12) (1,12) - 9 = a[r n 1] = [(1,12) 1 ] 10 1 r 1 (1,12) -1 1 P = 1000 (1,12) -1 [(1,12) -10 1] (1,12) -1 1 = 1000 [(1,12) -10 1] 1,12[(1,12) -1 1] = 1000 [(1,12) -10 1] 1 1,12 = 1000 [1 (1,12) -10 ] 0,12 = R5 650,22 P = x [1 (1 + ) -n ] 3.2 A homeowner takes a loan of R The nterest s 9,8% p.a. compounded monthly. The loan s to be repad monthly n 240 nstalments wth the frst payment starts at the end of the frst month Determne the monthly nstalment Determne total amount pad Calculate the actual nterest If nflaton s calculated at a rate of 9% p.a. compounded annually determne the new value of the house after 10 years Dscusson Bascally all loans are deferred payments. No one takes out a loan and pay an amount mmedately. Suppose the borrower wants to pay R5 000 mmedately then that person should take out a loan for R The formula s for one deferred payment. P= x [1 (1 + ) -n ] = x [1 (1 + 9,8 1200) -240 ] 9, x = R3807,24 (nstalment s actually R3808, not less) A = 3808 x 240 = R = x. n P = 3808 x = R New Value = ( ) 10 = R

4 3.3 A homeowner takes a loan of R The nterest s 8,9% p.a. compounded monthly. The loan s to be repad monthly n 240 nstalments wth the frst payment starts at the end of the frst month Determne the monthly nstalment Determne total amount pad Calculate the actual nterest If nflaton s calculated at a rate of 8% p.a. compounded annually determne the new value of the house n 20 years tme. 3.4 A small busness enterprse decded to take a loan for R at a low nterest of 6% p.a. compounded monthly for 5 years Determne the monthly nstalment Determne the total nterest pad on ths loan Determne the new monthly nstalment f the frst payment s deferred for 6 months and the total nterest pad. Dscusson Let us consder a smpler problem:- A loan of R was to be repad n yearly nstalments at a rate of 10% p.a compounded annually. Two cases:- (a) Payment to commence 1 year after the loan was awarded. (b) Payment to commence 3 years after the loan s awarded. Determne the nstalment n each case. (a) Standard formula:- P= x [1 (1 + ) -n ] = x [1 ( ) -10 ] x = R (b) Let us go back to frst prncples P = x (1,10) -3 + x (1,10) -4 + x (1,10) x (1,10) -12 = x(1,10) -2. (1,10) -1 [1 +(1,12) -1 + (1,12) (1,12) -10 ] The only dfference s the addtonal factor of (1,10) -2 P = (1,1) -2.x [1-(1+) -10 ] x = R19693 (c) New formula: P = (1+) -d.x [1-(1+) -n ] d s deferred perod = deferred per A homeowner takes a home loan of R Ths amount s to be repad n monthly nstalments over 20 years at a rate of 9.6% p.a. compounded monthly. Calculate the monthly nstalments f the frst payment s made at the end of the 6 months.

5 3.6 A shopkeeper takes a loan and pays t off n 12 equal yearly nstalments of R startng at the end of the 4 th year at an nterest rate of 10% p.a. compounded annually. How much can she borrow? Hnt: Remember one postponed payment (standard) s (1 + ) -1. In ths case we start wth (1 + ) -4. P = (1 + ) (1 + ) (1 + ) A company takes out a loan for R at an nterest rate of 7,5% pa compounded monthly for 10 years Determne the monthly nstalment How long wll t take to pay of the loan f they double the nstalment? 3.8 Jojo nvests R wth a bank that agreed to gve hm a return of 18% pa compounded monthly. How much wll Jojo receve every month for 10 years? 3.9 Patence nvested R wth a bank that agreed to gve her 15% pa compounded monthly on ths nvestment. What wll her balance be after 5 years f she s gven a monthly annuty of R6000? 4. Future Value of Annuty 4.1 Shrley nvested R5000 per month startng on 1 May 2000 earnng nterest at a rate of 17% pa compounded monthly. What can Shrley expect on 30 Aprl 2012? 4.2 Trevor nvested R per annum startng on 1 July 2003 earnng nterest at a rate of 15,8%pa compounded monthly. What s the total amount that Trevor receved on 30 June 2011? 4.3 Dudu wanted to become a mllonare n 8 years tme. The bank was very happy wth her commtment to save and offered her a whoppng 20% pa return compounded monthly. She nvested her money at the end of each month startng on 31 August At what date can she expect to be a mllonare? What must her monthly nvestment be to acheve ths objectve? 4.4 Randy took out an nsurance polcy whch was fxed at R500 per month. The guaranteed nterest rate was 12% pa compounded annually. What wll Randy s polcy be worth n 30 years tme?

6 5. General Problems 5.1 A loan of R was taken by a homeowner. Interest was to be pad at 9.5%p.a. compounded monthly. The repayment perod s 20 years. 5.1 Calculate the monthly nstalment. 5.2 Calculate the balance on the loan after (a) 5 years (b) 10 years 5,2 ABC Incorporated nvested a certan amount n a bank earnng nterest at a rate of 12,6% p.a compounded monthly. They ntend wthdrawng R9 000 per month for 10 years. How much must they nvest now to ensure that ths wthdrawal can be made? 5.3 Mrs Wse nvested a certan amount n a bank earnng nterest at a rate of 15% p.a compounded monthly. She ntend wthdrawng R per month for 10 years. How much must they nvest now to ensure that ths wthdrawal can be made? 5.4 A father decded to buy a house for hs famly for R He agreed to pay monthly nstalments of R10000 on a loan whch ncurred nterest at a rate of 14% pa compounded monthly. The frst payment was made at the end of the frst month Show that the loan would be pad off n 234 months Suppose the father encountered unexpected problems and was unable to pay any nstalments at the end of the 120 th ; 121 st ; 122 nd ; 123 rd months. At the end of the 124 th month he ncreased hs payment so as to stll pay off the loan n 234 months by 111 equal monthly payments. Calculate the value of ths new nstalment. 5.5 A company purchased a buldng for R They took out a loan for ths amount at an nterest rate of 6% per annum compounded monthly. They pad a fxed amount of R40000 per month How many payments must be made? Determne the fnal payment What wll the balance be after 36 months? 5.6 Snkng Fund A company purchased a new vehcle for R The company wshed to replace the vehcle n 5 years tme. The rate of deprecaton s 10% p.a on dmnshng balance. It s envsaged that a new vehcle wll apprecate n value at a rate of 11% p.a. Calculate (a) the resdual value of the vehcle n 5 years tme. (b) the prce of the new vehcle n 5 years tme. (c) the value that the snkng fund must attan n 5 years tme. (d) the monthly payments nto the snkng fund ( payment commenced 1 month after the snkng fund s set up) at a rate of 13,6%p.a compounded monthly.

7 A borehole machne presently cost R It deprecates n value at a rate of 12 % on a reducng balance. A new borehole machne wll be requred n 5 years tme. The cost of the borehole machne s expected to grow at a rate of 10% p.a. compound nterest. Determne (a) scrap value of old borehole machne. (b) cost of new borehole machne (c) (d) and hence the value that the snkng fund must attan. equal monthly nstalments whch wll start n 6 months tme and fnsh n 5 years tme. Interest s at 11,2% p.a. compounded monthly A car presently cost R It deprecates at a rate of 10% p.a. on a reducng balance. A new car wll be needed n 6 years tme. The cost of the new car s expected to grow by 8% p.a. compounded annually. Determne:- (a) the scrap value of the car to the nearest R. (b) the cost of the new car to the nearest R. (c) (d) hence the value that the snkng fund must attan. the equal monthly nstalments whch wll start n 6 months tme and fnsh n 5 years tme. Interest s calculated at 14% p.a. compounded monthly 6. Suppose an amount of R3000 s wthdrawn every 6 months startng n one year s tme and fnsh 6 months before the purchase. What wll the new nstalment be for and 5.6.3?

8 HIRE PURCHASE 1. Mr X purchase a car to the value of R He takes out a loan for ths amount. He repays ths loan at a rate of 12% p.a. compounded monthly over 48 months. Calculate hs monthly nstalments. Rather than take out a bank loan the purchaser sgned a H.P agreement because the H.P charges were 9%p.a. smple nterest. Fnd 1.1 the new nstalments. 1.2 nomnal rate of nterest 1.3 effectve rate of nterest 2. Applances at home cost a new homeowner R The owner pays a depost of 20% and pays the balance over 36 months n equal monthly nstalments. 2.1 Fnd the nstalments f the borrower borrows the outstandng amount and agrees to pay nterest at a rate of 11% p.a. compounded monthly. 2.2 Suppose the homeowner borrows the outstandng balance and agrees to pay the H.P at a rate of 10% p.a. smple nterest. Fnd new nstalments nomnal rate of nterest effectve rate of nterest. 3. Student loans are beng offered by a leadng bankng nsttuton to tertary students. It offers loans of up to R The nterest s fxed at 10% of the amount. A further condton s that ths loan be pad over 10 months. Student A takes a loan of R Student B takes a loan of R Determne the monthly repayment for student A and B. 3.2 Calculate the nomnal and effectve rate of nterest p.a. 3.3 Suppose a parent was offered these loans and pad at 12% p.a. compounded monthly over one year. Calculate the monthly nstalment and the actual nterest pad on these loans. Comment on these two optons and how would you advse.

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

FINANCIAL MATHEMATICS

FINANCIAL MATHEMATICS 3 LESSON FINANCIAL MATHEMATICS Annutes What s an annuty? The term annuty s used n fnancal mathematcs to refer to any termnatng sequence of regular fxed payments over a specfed perod of tme. Loans are usually

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

0.02t if 0 t 3 δ t = 0.045 if 3 < t

0.02t if 0 t 3 δ t = 0.045 if 3 < t 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual effectve

More information

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual e ectve

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

10.2 Future Value and Present Value of an Ordinary Simple Annuity

10.2 Future Value and Present Value of an Ordinary Simple Annuity 348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

More information

Section 2.2 Future Value of an Annuity

Section 2.2 Future Value of an Annuity Secton 2.2 Future Value of an Annuty Annuty s any sequence of equal perodc payments. Depost s equal payment each nterval There are two basc types of annutes. An annuty due requres that the frst payment

More information

Section 2.3 Present Value of an Annuity; Amortization

Section 2.3 Present Value of an Annuity; Amortization Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1,000.00

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

More information

Texas Instruments 30Xa Calculator

Texas Instruments 30Xa Calculator Teas Instruments 30Xa Calculator Keystrokes for the TI-30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check

More information

1. Math 210 Finite Mathematics

1. Math 210 Finite Mathematics 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value 8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at

More information

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143 1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly

More information

Chapter 4 Financial Markets

Chapter 4 Financial Markets Chapter 4 Fnancal Markets ECON2123 (Sprng 2012) 14 & 15.3.2012 (Tutoral 5) The demand for money Assumptons: There are only two assets n the fnancal market: money and bonds Prce s fxed and s gven, that

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

Mathematics of Finance

Mathematics of Finance CHAPTER 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty; Amortzaton Revew Exercses Extended Applcaton: Tme, Money, and Polynomals Buyng

More information

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution.

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution. ACTS 408 Instructor: Natala A. Humphreys SOLUTION TO HOMEWOR 4 Secton 7: Annutes whose payments follow a geometrc progresson. Secton 8: Annutes whose payments follow an arthmetc progresson. Problem Suppose

More information

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

More information

Mathematics of Finance

Mathematics of Finance 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car

More information

Compound Interest: Further Topics and Applications. Chapter 9

Compound Interest: Further Topics and Applications. Chapter 9 9-2 Compound Interest: Further Topcs and Applcatons Chapter 9 9-3 Learnng Objectves After letng ths chapter, you wll be able to:? Calculate the nterest rate and term n ound nterest applcatons? Gven a nomnal

More information

Mathematics of Finance

Mathematics of Finance Mathematcs of Fnance 5 C H A P T E R CHAPTER OUTLINE 5.1 Smple Interest and Dscount 5.2 Compound Interest 5.3 Annutes, Future Value, and Snkng Funds 5.4 Annutes, Present Value, and Amortzaton CASE STUDY

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

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

Chapter 15 Debt and Taxes

Chapter 15 Debt and Taxes hapter 15 Debt and Taxes 15-1. Pelamed Pharmaceutcals has EBIT of $325 mllon n 2006. In addton, Pelamed has nterest expenses of $125 mllon and a corporate tax rate of 40%. a. What s Pelamed s 2006 net

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns

More information

A Master Time Value of Money Formula. Floyd Vest

A Master Time Value of Money Formula. Floyd Vest A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes.

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative. Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

More information

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000 Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from

More information

On some special nonlevel annuities and yield rates for annuities

On some special nonlevel annuities and yield rates for annuities On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes

More information

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars

More information

Solutions to First Midterm

Solutions to First Midterm rofessor Chrstano Economcs 3, Wnter 2004 Solutons to Frst Mdterm. Multple Choce. 2. (a) v. (b). (c) v. (d) v. (e). (f). (g) v. (a) The goods market s n equlbrum when total demand equals total producton,.e.

More information

Stress test for measuring insurance risks in non-life insurance

Stress test for measuring insurance risks in non-life insurance PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance

More information

Uncrystallised funds pension lump sum payment instruction

Uncrystallised funds pension lump sum payment instruction For customers Uncrystallsed funds penson lump sum payment nstructon Don t complete ths form f your wrapper s derved from a penson credt receved followng a dvorce where your ex spouse or cvl partner had

More information

Nasdaq Iceland Bond Indices 01 April 2015

Nasdaq Iceland Bond Indices 01 April 2015 Nasdaq Iceland Bond Indces 01 Aprl 2015 -Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Small pots lump sum payment instruction

Small pots lump sum payment instruction For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested

More information

Properties of American Derivative Securities

Properties of American Derivative Securities Capter 6 Propertes of Amercan Dervatve Securtes 6.1 Te propertes Defnton 6.1 An Amercan dervatve securty s a sequence of non-negatve random varables fg k g n k= suc tat eac G k s F k -measurable. Te owner

More information

ANALYSIS OF FINANCIAL FLOWS

ANALYSIS OF FINANCIAL FLOWS ANALYSIS OF FINANCIAL FLOWS AND INVESTMENTS II 4 Annutes Only rarely wll one encounter an nvestment or loan where the underlyng fnancal arrangement s as smple as the lump sum, sngle cash flow problems

More information

Uncrystallised funds pension lump sum

Uncrystallised funds pension lump sum For customers Uncrystallsed funds penson lump sum Payment nstructon What does ths form do? Ths form nstructs us to pay the full penson fund, under your non-occupatonal penson scheme plan wth us, to you

More information

Documentation about calculation methods used for the electricity supply price index (SPIN 35.1),

Documentation about calculation methods used for the electricity supply price index (SPIN 35.1), STATISTICS SWEDEN Documentaton (6) ES/PR-S 0-- artn Kullendorff arcus rdén Documentaton about calculaton methods used for the electrct suppl prce ndex (SPIN 35.), home sales (HPI) The ndex fgure for electrct

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

FINANCE m e p STRAND: FINANCE Unit 2 Simple and Compound Interest TEXT Section

FINANCE m e p STRAND: FINANCE Unit 2 Simple and Compound Interest TEXT Section CMM Subject Support Strand: FINANCE Unit 2 Simple and Compound Interest: Text m e p STRAND: FINANCE Unit 2 Simple and Compound Interest TEXT Contents Section 2.1 Simple Interest 2.2 Compound Interest 2.3

More information

Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

More information

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? Real-Tme Systems Laboratory Department of Computer

More information

Ameriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration

Ameriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration CED0105200808 Amerprse Fnancal Servces, Inc. 70400 Amerprse Fnancal Center Mnneapols, MN 55474 Incomng Account Transfer/Exchange/ Drect Rollover (Qualfed Plans Only) for Amerprse certfcates, Columba mutual

More information

AS 2553a Mathematics of finance

AS 2553a Mathematics of finance AS 2553a Mathematcs of fnance Formula sheet November 29, 2010 Ths ocument contans some of the most frequently use formulae that are scusse n the course As a general rule, stuents are responsble for all

More information

Interest Rate Futures

Interest Rate Futures Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon

More information

IS-LM Model 1 C' dy = di

IS-LM Model 1 C' dy = di - odel Solow Assumptons - demand rrelevant n long run; assumes economy s operatng at potental GDP; concerned wth growth - Assumptons - supply s rrelevant n short run; assumes economy s operatng below potental

More information

LIFETIME INCOME OPTIONS

LIFETIME INCOME OPTIONS LIFETIME INCOME OPTIONS May 2011 by: Marca S. Wagner, Esq. The Wagner Law Group A Professonal Corporaton 99 Summer Street, 13 th Floor Boston, MA 02110 Tel: (617) 357-5200 Fax: (617) 357-5250 www.ersa-lawyers.com

More information

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,

More information

HSC Mathematics General II

HSC Mathematics General II HSC Mathematics General II Term 2 Week 7 Student name:. Class code:.. Teacher name:. DUX 1 TERM 2 WEEK 7 LOANS AND ANNUITIES FLAT RATE LOANS A flat rate loan is a loan where simple (flat) interest is charged

More information

Study on Model of Risks Assessment of Standard Operation in Rural Power Network

Study on Model of Risks Assessment of Standard Operation in Rural Power Network Study on Model of Rsks Assessment of Standard Operaton n Rural Power Network Qngj L 1, Tao Yang 2 1 Qngj L, College of Informaton and Electrcal Engneerng, Shenyang Agrculture Unversty, Shenyang 110866,

More information

MGF 1107 Spring 11 Ref: 606977 Review for Exam 2. Write as a percent. 1) 3.1 1) Write as a decimal. 4) 60% 4) 5) 0.085% 5)

MGF 1107 Spring 11 Ref: 606977 Review for Exam 2. Write as a percent. 1) 3.1 1) Write as a decimal. 4) 60% 4) 5) 0.085% 5) MGF 1107 Spring 11 Ref: 606977 Review for Exam 2 Mr. Guillen Exam 2 will be on 03/02/11 and covers the following sections: 8.1, 8.2, 8.3, 8.4, 8.5, 8.6. Write as a percent. 1) 3.1 1) 2) 1 8 2) 3) 7 4 3)

More information

www.gov.uk/studentfinance 2016/17

www.gov.uk/studentfinance 2016/17 www.gov.uk/studentfnance SECTION 1 WHAT SUPPORT CAN YOU GET? FEES, LOANS, GRANTS & MORE *Fgures shown n ths secton are based on the 2015/16 student fnance polcy and may change SECTION 1 TUITION FEES AND

More information

Interest Rate Forwards and Swaps

Interest Rate Forwards and Swaps Interest Rate Forwards and Swaps Forward rate agreement (FRA) mxn FRA = agreement that fxes desgnated nterest rate coverng a perod of (n-m) months, startng n m months: Example: Depostor wants to fx rate

More information

Multiple discount and forward curves

Multiple discount and forward curves Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

Ch 3 Understanding money management

Ch 3 Understanding money management Ch 3 Understanding money management 1. nominal & effective interest rates 2. equivalence calculations using effective interest rates 3. debt management If payments occur more frequently than annual, how

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

Certificate No. 68613082 ONTARIO COURT (PROVINCIAL DIVISION) - versus - PAULO RAPOSO TRANSCRIPT OF PROCEEDINGS

Certificate No. 68613082 ONTARIO COURT (PROVINCIAL DIVISION) - versus - PAULO RAPOSO TRANSCRIPT OF PROCEEDINGS Certfcate No. 686182 ONTARIO COURT (PROVINCIAL DIVISION) HER MAJESTY THE QUEEN - versus - PAULO RAPOSO TRANSCRIPT OF PROCEEDINGS Heard before The Honourable Mr. Justce D. Cooper at Hamlton, Ontaro on Aprl

More information

14.74 Lecture 5: Health (2)

14.74 Lecture 5: Health (2) 14.74 Lecture 5: Health (2) Esther Duflo February 17, 2004 1 Possble Interventons Last tme we dscussed possble nterventons. Let s take one: provdng ron supplements to people, for example. From the data,

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

STATE OF RHODE ISLAND AND PROVIDENCE PLANTATIONS

STATE OF RHODE ISLAND AND PROVIDENCE PLANTATIONS STATE OF RHODE SLAND AND PROVDENCE PLANTATONS Department of Administration DVSON OF PURCHASES One Capitol Hill Providence, R 02908-5855 Tel: (401) 574-8100 Fax: (401) 574-8387 Website: www.purchasing.ri.gov

More information

Trivial lump sum R5.0

Trivial lump sum R5.0 Optons form Once you have flled n ths form, please return t wth your orgnal brth certfcate to: Premer PO Box 2067 Croydon CR90 9ND. Fll n ths form usng BLOCK CAPITALS and black nk. Mark all answers wth

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

DISCLOSURES I. ELECTRONIC FUND TRANSFER DISCLOSURE (REGULATION E)... 2 ELECTRONIC DISCLOSURE AND ELECTRONIC SIGNATURE CONSENT... 7

DISCLOSURES I. ELECTRONIC FUND TRANSFER DISCLOSURE (REGULATION E)... 2 ELECTRONIC DISCLOSURE AND ELECTRONIC SIGNATURE CONSENT... 7 DISCLOSURES The Dsclosures set forth below may affect the accounts you have selected wth Bank Leum USA. Read these dsclosures carefully as they descrbe your rghts and oblgatons for the accounts and/or

More information

10/19/2011. Financial Mathematics. Lecture 24 Annuities. Ana NoraEvans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.

10/19/2011. Financial Mathematics. Lecture 24 Annuities. Ana NoraEvans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia. Math 40 Lecture 24 Autes Facal Mathematcs How ready do you feel for the quz o Frday: A) Brg t o B) I wll be by Frday C) I eed aother week D) I eed aother moth Aa NoraEvas 403 Kerchof AaNEvas@vrga.edu http://people.vrga.edu/~as5k/

More information

Semiconductor sensors of temperature

Semiconductor sensors of temperature Semconductor sensors of temperature he measurement objectve 1. Identfy the unknown bead type thermstor. Desgn the crcutry for lnearzaton of ts transfer curve.. Fnd the dependence of forward voltage drop

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006

Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006 Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,

More information

Section 8.3 Notes- Compound Interest

Section 8.3 Notes- Compound Interest Section 8.3 Notes- Compound The Difference between Simple and Compound : Simple is paid on your investment or principal and NOT on any interest added Compound paid on BOTH on the principal and on all interest

More information

Effective December 2015

Effective December 2015 Annuty rates for all states EXCEPT: NY Prevous Index Annuty s effectve Wednesday, December 7 Global Multple Index Cap S&P Annual Pt to Pt Cap MLSB Annual Pt to Pt Spread MLSB 2Yr Pt to Pt Spread 3 (Annualzed)

More information

REQUIRED FOR YEAR END 31 MARCH 2015. Your business information

REQUIRED FOR YEAR END 31 MARCH 2015. Your business information REQUIRED FOR YEAR END 31 MARCH 2015 Your busness nformaton Your detals Busness detals Busness name Balance date IRD number Contact detals - to ensure our records are up to date, please complete the followng

More information

Chapter 28 Time Value of Money

Chapter 28 Time Value of Money Chapter 28 Time Value of Money Lump sum cash flows 1. For example, how much would I get if I deposit $100 in a bank account for 5 years at an annual interest rate of 10%? Let s try using our calculator:

More information

Account Transfer and Direct Rollover

Account Transfer and Direct Rollover CED0105 Amerprse Fnancal Servces, Inc. 70100 Amerprse Fnancal Center Mnneapols, MN 55474 Account Transfer and Drect Rollover Important: Before fnal submsson to the Home Offce you wll need a Reference Number.

More information

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)

More information

Module 1: Corporate Finance and the Role of Venture Capital Financing TABLE OF CONTENTS

Module 1: Corporate Finance and the Role of Venture Capital Financing TABLE OF CONTENTS Module 1: Corporate Finance and the Role of Venture Capital Financing Time Value of Money 1.0 INTEREST THE COST OF MONEY 1.01 Introduction to Interest 1.02 Time Value of Money Taxonomy 1.03 Cash Flow Diagrams

More information

Effective September 2015

Effective September 2015 Annuty rates for all states EXCEPT: NY Lock Polces Prevous Prevous Sheet Feld Bulletns Index Annuty s effectve Monday, September 28 Global Multple Index Cap S&P Annual Pt to Pt Cap MLSB Annual Pt to Pt

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

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

With compound interest you earn an additional $128.89 ($1628.89 - $1500).

With compound interest you earn an additional $128.89 ($1628.89 - $1500). Compound Interest Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle

More information

What is important to understand about Compounded Interest?

What is important to understand about Compounded Interest? Use your COMPINT program. Interest Worksheet: COMPINT: Stands for compounded interest. You simply put money in an account in one lump sum and it accrues interest over a period of time. The interest could

More information

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the Bond futures INRODUCION Bond futures contracts are futures contracts that allow nvestor to buy n the future a theoretcal government notonal bond at a gven prce at a specfc date n a gven quantty. Compared

More information

Chapter 3 Understanding Money Management. Nominal and Effective Interest Rates Equivalence Calculations Changing Interest Rates Debt Management

Chapter 3 Understanding Money Management. Nominal and Effective Interest Rates Equivalence Calculations Changing Interest Rates Debt Management Chapter 3 Understanding Money Management Nominal and Effective Interest Rates Equivalence Calculations Changing Interest Rates Debt Management 1 Understanding Money Management Financial institutions often

More information

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1 (4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information