A Master Time Value of Money Formula. Floyd Vest


 Toby Rich
 3 years ago
 Views:
Transcription
1 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. (See Formula 7 below. See the Appendx n the TI83 (p. A55) or TI84 manuals. The manuals are almost dentcal for fnance. You can download parts of a manual at The followng s a dervaton of the TVM Formula for Future Value (FV) wth examples and exercses. You Try It #1 Check the Appendx, Fnancal Functons n the TI83/84 manual and fnd the Tme Value of Money Formula for FV. Compare t to Formula 7 n ths artcle. Compound Interest Formula. Frst we need to derve the Compound Interest Formula, whch s (1) FV = PV(1 + ), where FV s the future value, PV s the present value, s the nterest rate per compoundng perod, and s the number of compoundng perods. We are usng the termnology n the TI83 and TI84 manuals. Example 1 For an example we calculate the FV for PV = $100 nvested for ten years at the annual nomnal rate of 4%, compounded quarterly. The rate per compoundng perod s = 0.04 = (annual nomnal rate) (number of compoundng perods per year). s the 4 number of compoundng perods, whch s 10 4 = 40 quarterly perods. So FV = ! $ # 4 % & = $ The nvestment of $100 grew to $ n ten years. You Try It #2 For Example 1 above, what s the Future Value f 4% s compounded daly? Money Market Funds usually compound daly. A Master Tme Value of Money Formula Sprng,
2 For the dervaton, consder that compound nterest earns nterest on nterest as recorded n the followng table: Begnnng PV Interest FV at end of perod Perod 1 PV (PV) PV(1 + ) Perod 2 PV(1 + ) [PV(1 + )] PV(1 + ) PV(1 + ) = PV(1 + ) 2 Perod 3 PV(1 + ) 2 [PV(1 + ) 2 ] PV(1 + ) 3 Perod PV(1 + ) 1 [PV(1 + ) 1 ] PV(1 + ) We conclude, from the last lne of the table, the Compound Interest Formula for FV to be FV = PV(1 + ) as descrbed above. Future Value of an Ordnary Annuty. To contnue the dervaton of the TVM Formula, we need the formula for the Future Value of an Ordnary Annuty of payments (PMT), each drawng compound nterest at the rate per perod as llustrated on the followng tmelne PMT PMT PMT PMT PMT By the Compound Interest Formula, the FV of the Ordnary Annuty s the accumulaton of PMTs and the nterest on each: FV = PMT(1 + ) 1 + PMT(1 + ) 2 + PMT(1 + ) PMT(1 + ) 1 + PMT. Multplyng through by (1 + ) we get (1 + )FV = PMT(1 + ) + PMT(1 + ) 1 + PMT(1 + ) PMY(1 + ) 2 + PMT(1 + ). ext we subtract to get (1 + )FV FV = PMT(1 + ) PMT, wth ntermedate terms subtractng out. Solvng for FV we have (2) FV = PMT (1 ) + 1 as the formula for the Future Value of an Ordnary Annuty where PMT s the payments over perods, each drawng compound nterest at the rate per perod. For an Ordnary Annuty, the PMTs occur at the end of perods. A Master Tme Value of Money Formula Sprng,
3 Example 2 Consder payments of $500 = PMT nvested at the end of each year for = 30 years at the rate = 6% = 0.06 per year. The Future Value of the savngs s FV = 500 ( )30!1% $ ' = $39, # 0.06 & The amount nvested was = $15,000 n PMTs to accumulate to $39, You Try It #3 For Example 2 above, what s the Future Value of $500 per month for thrty years n a retrement nvestment program? If there s also a PV nvested at tme 0 on the tme lne, then (3) FV = PMT (1 ) PV(1 + ). See the Exercses for an example. Future Value of an Annuty Due. Consder payments PMT occurrng at the begnnng of each perod as llustrated on the followng tmelne: PMT PMT PMT PMT PMT In ths case each PMT draws compound nterest for an addtonal perod and accumulates to the FV so (1 + ) 1 (4) FV = (1 + ) PMT + PV (1 + ) for an Annuty Due. For the Master TVM Formula, the followng formula s used for two cases: k = 0 f end of perod payments, k = 1 f begnnng of perod payments, wth G = 1 + (k), where s the nterest rate per perod, to gve (5) (1 + ) 1 FV = G PMT + PV (1 + ) ote that for end of perod payments, G = 1 + (0) = 1 gvng Formula 3; and for begnnng of perod payments G = 1 + 1() = 1 + gvng Formula 4. A Master Tme Value of Money Formula Sprng,
4 Algebrac manpulaton s conducted on Formula 5 to get the TVM Formula for FV to be (6) FV = PMT G PMT G (1 ) PV The algebra s left as an exercse. Cash Flow Sgn Conventons. One property of the fnal Master TVM Formula s that sgn changes are made to accommodate Cash Flow Sgn Conventons. For example consder a Cash Flow tmelne wth PV and PMTs negatve and FV postve FV PV PMT PMT PMT PMT PMT In an nvestment of PV and PMTs, money gong out s consdered negatve, and ndcated under the tmelne, and FV comng n at the end s postve, and ndcated wth an upward arrow on the tmelne. Master TVM Formula. To accommodate ths sgn conventon, we wll n Formula 6 change PMT to PMT and PV to PV to get Formula 7. PMT (7) FV = G PMT G (1 ) PV + +, where we have an annuty wth payments PMT, nterest rate per perod of, perods, and a PV at tme zero. G s descrbed above. Ths Master TVM Formula for FV s n the manuals for the TI83 and TI84. Fnancal functons on other calculators may have dfferent sgn conventons. (See the TIBA35, TIBAII, and HP19B.) Example 3 We wll do an example for Formula 7. Consderng the above tmelne, we nvest PV = $100,000, PMT of $10,000 at the end of each year for = 25 years n a retrement program averagng 6% per year. otce the sgn conventons. What s the FV of the savngs program? Substtutng we have G = 1 snce we have end of perod PMTs. FV =!10, ! ( ) 25!100,000 +!10,000 % $ # 0.06 '. FV = $977, & You Try It #4 For Example 3 above, nvestments earn 6% compounded monthly and the famly has $200,000 n nvestments and are savng $500 at the end of each month for the next 25 years. How much s n ther retrement fund? A Master Tme Value of Money Formula Sprng,
5 To get the TVM Formulas for PV and PMT from Formula 7, you smply use algebra to solve. To solve for, you can use logarthms. These solutons are left as exercses. To solve for for an Annuty requres an teratve program. See the references for some teratve calculator programs. To get the Compound Interest Formula from Formula 7, smply let PMT = 0, and remember the sgn changes. For the Compound Interest Formula, you can solve algebracally for. The concept of dscountng and PVOA. From the Compound Interest Formula, FV = PV(1 + ) we get PV = FV(1 + I). Ths s referred to as dscountng FV to get PV. For an Ordnary Annuty, we can dscount each of the PMTs to get a PV of an Ordnary Annuty (PVOA) as the sum, PVOA = 1 PMT (1 + ) 2 + PMT (1 + ) 3 + PMT (1 + ) ( 1) + + PMT (1 + ) + PMT (1 + ). It left as an exercse to show that the PVOA of an Ordnary Annuty s (8) PVOA = PMT 1 (1 ) +, and to derve a formula for the Present Value of an Annuty Due (PVAD). The concept of dscountng s mportant because t comes up n loans, mortgages, long term fnancal plannng, nflaton, bond prcng, dscountng Cash Flows to Intal Equty or et Present Value n busness plannng, Yeld to Maturty (YTM), Internal Rate of Return (IRR), other applcatons. ote here that the term PV has taken on two meanngs. Examples for Exercses The followng s an example of the format of some of the applcaton problems n the Exercses requestng formulas, answers, commentary, tmelnes, knowns, unknowns, varables, formulas, Fnancal Functons, code, and summary. Instructons: Solve the followng problems wth formulas, a scentfc calculator pad, lst and label knowns, unknowns, draw a tme lne. Use any fnancal formulas that are convenent. Then, solve wth Fnancal Functons on a calculator or computer. See the calculator manual for the requred code. The problems are taken from Chapter 14 of the TI83 and TI84 manuals. Wrte code and commentary. Identfy the output, and put on unts. Summarze the answer to the problem. Problem: Consder a problem on p. 143 of the TI83 manual. A car costs $9000. You can afford monthly payments of $250 a month for four years. What (APR) annual percentage rate wll make ths possble? A Master Tme Value of Money Formula Sprng,
6 Answer: From the problem, = 4 12 = 48 months. PV = $9000. PMT = $250 a month. FV = 0. P/Y = 12. Pmt:End snce we have an ordnary annuty. The queston s, what s I%? Usng Formula 8 for the PV of an Ordnary Annuty, we have 48 1 (1 + ) 9000 = 250, where s the monthly rate as a decmal and APR = 12. There s no closed form formula for solvng for for an annuty. TI83 code and commentary for the problem: 2 nd Fnance Enter (To select TVM Solver.) ()250 Enter 0 Enter 12 Enter Enter (To select Pmt:End for payments.) Press (To hghlght the I% entry.) Press Alpha Solve (To solve for I%.) (You see a cursor on I%.) You read I% = 14.90%. Summary: Wth an APR of I% 14.90%, the purchaser can afford the car. (I% = 100() (the number of compoundng perods per year).) A Master Tme Value of Money Formula Sprng,
7 Exercses For some of the followng exercses, solve wth mathematcs of fnance formulas and wth fnancal functons on a calculator. Use any mathematcs of fnance formula that s convenent. For all problems, show all your work, label all nputs, show formulas, label all answers, and summarze. For the basc mathematcs of fnance formulas and ther dervatons, see Luttman or Kastng n Unt 1 of ths course. 1. At what annual nterest rate, compounded monthly, wll a depost of $1250 accumulate to $2000 n seven years? See p. 143 of the TI83 Manual. Consder Formula 9 below.! (9) FV = PV 1+ r $ # k % &, where r s the annual nomnal rate compounded k tmes per year. s the number of compoundng perods. I% s a percent. r = I % For a mortgage of $100,000 at 18% per year for 30 years, what s the monthly payment? See p. 144 n the TI 83 Manual. 3. For a mortgage of $100,000 at 8.5% per year for 30 years, what s the monthly payment? See p. 146 of the TI 83 Manual. 4. For a 30year, 11% mortgage wth $1000 a month payments, what s the amount of the mortgage? See p. 147 of the TI 83 Manual. 5. On p. 143 of the TI 83 Manual, one problem s I% = 6%, PV = 9000, PMT = 350, FV = 0, P/Y = 3, what s? Do ths problem. 6. Set up the Cash Flow equaton for the second tmelne on p. 148 and verfy the answers. The npv (et Present Value) at 6% = $ means the sum of the cash flows dscounted at 6% gves $ The (Internal Rate of Return) rr = 27.88% makes the npv equal to zero. The Internal Rate of Return s the rate that dscounts the remanng cash flows to the ntal equty CF0. To do ths calculaton wth a scentfc calculator pad, store the 1 + n STO and use RCL. 7. By algebra, derve Formula 6 from Formula Derve from Formula 3, the TVM Formula n the Appendx of the TI 83 Manual for, # PMT! FV & ln % PMT + PV ( whch s = $ ' for G = 1. ln(1+ ). A Master Tme Value of Money Formula Sprng,
8 9. Derve from Formula 3, the TVM Formula n the Appendx of the TI 83 Manual for PMT 1 PMT PV wth G = 1, whch s PV = FV. Indcate the requred (1 + ) sgn changes to satsfy the TVM sgn conventons for cash flows. 10. Derve from Formula 3, the TVM Formula n the Appendx of the TI 83 Manual for PV + FV PMT wth G = 1, whch s PMT = PV + (1 + ) 1 and ndcate the requred sgn changes to satsfy the TVM sgn conventons for cash flows. You may want to use the PV (1 + ) PV dentty = PV + (1 + ) 1 (1 + ) What do you change n Formula 7 to get the Compound Interest Formula? Make changes and show the formula. 12. Derve the Formula for the Sum of an Ordnary Annuty from Formula 7. What strange results do you get and what sgn change s requred? 13. Derve a formula for the Present Value of an Annuty Due. 14. Wrte a paragraph wth cash flow tmelnes dscussng some of the entres and sgn changes requred n the TVM Solver and requred by the TVM formulas. 15. Consder a fund of PV dollars that provdes annual wthdrawals of PMTs for years. Draw a cash flow tmelne wth sgn conventons and gve the sgns for PV and PMTs. Gve a formula for ths fund and wthdrawals. 16. (a) Consder a loan of PV dollars that s repad wth PMTs. Draw a cash flow tmelne wth sgn conventons and gve the sgn changes for the varables. Gve a formula for ths loan and payments. (b) otce that fnancal varables have multple meanngs. It s ths multvalency and abstractness that gves the power and generalty to mathematcs. In some publcatons, even more generc fnancal terms such as Rent (R), Prncple (P), and Sum (S) are used. Sometmes appled problems n fnance requre formulas that are not among the common ones. Then you need to derve your own formula. For an example of such a dervaton, derve a formula for the Sum S n of the frst n terms of a geometrc sequence wth frst term a, common rato r, second term ar and nth term ar n 1. Use the technque used to derve Formula 2 above. 17. A smplfcaton of a formula for n the Appendx to the TI83 Manual s ln(1 ) = e + 1. Prove ths formula. Use the defnton of ln(1 + ). 18. Some problems requre two tmelnes. A person makes a debt of $2000 that must be pad back n ten years at 5% nterest compounded semannually. To accumulate money A Master Tme Value of Money Formula Sprng,
9 to pay the debt, after one year they start savng each year wth the last depost made when the debt s due. The nvestment pays 4% compounded annually. How much should they save each year? 19. From Money magazne, June 2011, p. 65: You don t start savng untl age 45, but then dutfully nvest $10,000 a year untl 65, assumng annualzed 8% returns. At age 65 you ll have $460,000. Your brother starts at 35, saves $10,000 a year but quts at age 45. At age 65 he has $675,000. Make the mstake of startng to save late, and you ll have to work hard to make up the error. Problem: Do the calculatons to get ther numbers. Keep tryng and you ll get them. Explan Money magazne s calculatons. 20. As ths course progresses, accumulate at least twenty reasons for knowng the fnancal mathematcs and calculator sklls to complement the Fnancal Functons on a calculator. Items such as The Fnancal Functons on a calculator are referred to as, black boxes. 21. From Forbes, May 9, 2011: In comparng ETFs (Exchange Traded Funds), t s mportant to calculate the effect on earnngs of expense ratos charged for managng the nvestment. Vanguard MSCI Emergng Market (VWO, 49) has an expense rato of 22 bass ponts. BlackRock s Shares charges 69 bass ponts. If your $10,000 earns 9% before expenses, the dfference n expense ratos wll balloon nto a dfference n fnal amount of $16,000 n your pocket n Over 30 years that dfference of half a percentage pont compounds to 13%. Problem: See f you can get these numbers. If you keep tryng, you wll fgure t out. What lesson dd you learn about nvestng? Dd you compound wth ( ) or ( )( )? Whch s correct? How much dfference does t make? A Master Tme Value of Money Formula Sprng,
10 Sde Bar otes Load Funds and oload Funds. From USA Today, May 27, 2011: Let s look at the Amercan Funds Growth Fund of Amerca. The A shares have ganed 4.17% a year for the last decade. To put ths n perspectve: A $10,000 nvestment n the fund would have ganed $5,046 n 10 years. If you pad the fund s maxmum 5.75% sales charge, however, your total return was 3.55%,. Your gan has now shrunk to $4,174. You owe taxes on the dstrbutons. your after tax return would be 3.21%. And f you had sold the fund after 10 years and pad taxes on you gans, you d be left wth a 2.96% average annual gan. So you nvested $10,000 n a taxable account, pad the sales charge, pad taxes on dstrbutons and gans. Your $10,000 s now $13,387. At that rate, you d double your money n about 24 years. But you can reduce your cost. One easy way of course, s to buy a noload fund. You pay no commsson. If you had the money nvested n a Roth IRA or 401k, t would be aftertax money and there would be no ncome taxes. See IRS Publcaton 590. For a noload fund, see for the Vanguard Wellngton Fund, whch s the naton s oldest balanced fund. 6.20% n the last 10 years. Expense rato 0.34%. Some salesmen of load funds wll tell you that you wll pay ndrectly a commsson on a noload fund. How s that wth a 0.34% expense rato on the noload fund, and a 0.69% expense rato on the load fund. Problem: See f your can reproduce the above calculatons. You can do most of them. A Master Tme Value of Money Formula Sprng,
11 References Most of the followng references are cted n the annotated bblography for ths course, and provde dervatons, exercses, and calculator programs for practce wth the fnancal formulas, calculator Fnancal Functons, and graphs. Vest, Floyd Tme, Money, & Polynomals, HMAP PullOut, Consortum 37. Vest, Floyd, What Can a Fnancal Calculator Do for a Mathematcs Teacher or Student? Journal of Computers n Mathematcs and Scence Teachng, pp. 6778, Fall Vest, Floyd, Computerzed Busness Calculus Usng Calculators, Journal of Computers n Mathematcs and Scence Teachng, Summer,1991. Vest, Floyd, Tm and Tom s Fnancal Adventure, HMAP PullOut, Consortum 39. Vest, Floyd, Makng Money Wth Algebra, HMAP PullOut, Consortum 33. A Master Tme Value of Money Formula Sprng,
12 Answers to You Try Its! You Try It #2 FV = $ # 365 % & 3650 = $ (! # You Try It #3 FV = 500 ) $ % & '1 = $502,257.49,! You Try It #4 FV = 200, $ # 12 % & 892, , = $1,239, (! # ) $ % & '1 =, A Master Tme Value of Money Formula Sprng,
13 Answers to Exercses 1. To solve the problem wth a formula, use the Compound Interest Formula 9 gven above: Substtutng n Formula 9, we have 2000 = r 84! $ # 12% & snce there are 7 12 = 84 compoundng perods. Usng algebra and a scentfc calculator pad to solve for r, we have r = = 6.733% as the annual nomnal rate. For the fnancal calculator code and explanaton for the TVM Solver, see page 143 of the TI 83 Manual, or see the TI 84 Manual. 2. We wll use Formula 8 above. Substtutng we get 1! !360 ( % + # $ 12 & ' 100,000 = PMT ), Solvng and usng a scentfc calculator pad gves PMT = $ as the monthly payment. We could have used Formula 3 above wth FV = 0 and PMT would have come out negatve. For a fnancal calculator code, see p. 144 of the TI 83 Manual, or see the TI 84 Manual, for the TVM Solver. 3. For ths problem, the TI83 Manual presents the tvm_pmt fnancal functon, whch s menu tem 2 under 2 nd Fnance. Code and comments: Put the knowns n the TVM Solver. Put 0 n PMT. Code: 2 nd Fnance (to tvm_pmt ) Enter Enter (On the home screen you read The monthly payment s $ ) 4. Substtutng nto Formula 6 and calculatng wth a scentfc calculator pad gves PV = $105,006.35, as the amount of the mortgage. The TI 83 Manual presents the tvm_pv [(,I%,PMT,FV,P/Y,C/Y)] functon for ths problem. On the home screen, you store wth the STO> key, 360 n, 11 n I%, ()1000 n PMT, 0 n FV, 12 n P/Y and calculate PV. Code and comments: (On the home screen.) nd Fnace > Enter (To Sto 360 n.) Alpha : 11 2 nd Fnace > (To I%) Enter Alpha : () nd Fnace > (To PMT.) Enter Alpha : 12 2 nd Fnance > (To P/Y.) Enter Enter 2 nd Fnace (To tvm_pv.) Enter Enter (You read ) The amount of the mortgage s $105, (It s more convenent to enter values drectly nto the TVM Solver when possble.) The above code ddn t store values to the TVM Solver. 5. Substtutng nto Formula 8 above and usng logs to solve, you get = payments. The TVM Solver gves the same answer. A Master Tme Value of Money Formula Sprng,
14 11. To get the Compound Interest Formula FV = PV(1 + ) from Formula 7, set PMT = 0, G = 1 and make a sgn change. 12. To get the Formula for the Sum of an Ordnary Annuty from Formula 7, set PV = 0 and G = 1, and make a sgn change to get Formula To get a Formula for the Present Value of an Annuty Due, use Formula 8 and snce n 1 (1 + ) PMTs are at the begnnngs of perods, use PVAD = PMT (1 + ). 15. For the fund, PV s negatve and PMT s money comng n and s postve. 16. For the loan, PV s money comng n and s postve, PMTs are money gong out and are negatve. 18. They should save $ at the end of each year for ten years. A Master Tme Value of Money Formula Sprng,
15 Teachers otes The TVM Formulas. Ths artcle presents dervatons of the basc mathematcs of fnance formulas found n Luttman and Kastng n Unt 1 of ths course, but n the form of the TVM Formulas n the appendx to the TI83/84. These formulas utlzed tme lne sgn conventons for varables. For example for a loan, PV s postve snce t s thought of as money comng n, and PMT s negatve snce t s thought of as money gong out. All of ths s explaned n the artcle. TI 83/84 Fnancal Functons are ndcated n some the exercses. Students can use other calculators f they wsh. Instructons for use of the TI83/84 are nclude. A lfetme fle on fnancal mathematcs. Have students set up and mantan a lfetme fle on personal fnance and fnancal mathematcs. In the future, ths artcle wll help students use ther TI83/84 n personal fnance. Students should be encouraged to buy ther own calculators. They wll last them a lfetme. Addtonal examples and exercses. The artcles n the references provde addtonal examples and exercses whch students can do wth ther graphng calculators and fnancal calculators. Some gve programs for solvng for n annuty formulas, as well as assocated graphs. A Master Tme Value of Money Formula Sprng,
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 informationTime 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 informationSimple 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 information10.2 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 informationUsing 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 informationSection 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 informationSection 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 informationSection 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 informationSection 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 informationAn 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 informationLecture 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 information8.4. 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 information1. 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 informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI30X 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 informationLecture 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 annutymmedate, and ts present value Study annutydue, and
More informationThursday, 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 informationSolution: 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 informationTexas Instruments 30Xa Calculator
Teas Instruments 30Xa Calculator Keystrokes for the TI30Xa 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 informationLevel Annuities with Payments Less Frequent than Each Interest Period
Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Symoblc approach
More informationFinancial 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 informationTime 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 informationFinite 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 informationMathematics 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 informationCompound Interest: Further Topics and Applications. Chapter 9
92 Compound Interest: Further Topcs and Applcatons Chapter 9 93 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 informationIn 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 informationFINANCIAL 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 information10. (# 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 informationEXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A CALCULATOR
EXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A 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 informationMathematics 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 informationIn 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 information0.02t 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 informationFINANCIAL 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 informationIDENTIFICATION 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 information3. Present value of Annuity Problems
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 = 
More informationMathematics 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 informationIntrayear Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error
Intrayear 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 informationAn 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 informationProfessor Iordanis Karagiannidis. 2010 Iordanis Karagiannidis
Fnancal Modelng Notes Basc Excel Fnancal Functons Professor Iordans Karagannds Excel Functons Excel Functons are preformatted formulas that allow you to perform arthmetc and other operatons very quckly
More informationInterest 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 (nm) months, startng n m months: Example: Depostor wants to fx rate
More informationChapter 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 informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationMultiple 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 informationGraph 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 informationANALYSIS 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 informationInterest 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 informationOn 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 informationTuition Fee Loan application notes
Tuton Fee Loan applcaton notes for new parttme EU students 2012/13 About these notes These notes should be read along wth your Tuton Fee Loan applcaton form. The notes are splt nto three parts: Part 1
More informationSolutions 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 informationChapter 15: Debt and Taxes
Chapter 15: Debt and Taxes1 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 informationJoe 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 informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationA) 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 informationSmall 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 informationSome Mathematics of Investing in Rental Property. Floyd Vest
Some Mathematics of Investing in Rental Property Floyd Vest Example 1. In our example, we will use some of the assumptions from Luttman, Frederick W. (1983) Selected Applications of Mathematics of Finance
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationTVM Appendix B: Using the TI83/84. Time Value of Money Problems on a Texas Instruments TI83 1
Before you start: Time Value of Money Problems on a Texas Instruments TI83 1 To calculate problems on a TI83, you have to go into the applications menu, the blue APPS key on the calculator. Several applications
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationClassic 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 informationAnswer: 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 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationNumber 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 informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 200502 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 148537801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationAmeriprise 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 informationChapter 15 Debt and Taxes
hapter 15 Debt and Taxes 151. 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 informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationYIELD CURVE FITTING 2.0 Constructing Bond and Money Market Yield Curves using Cubic BSpline and Natural Cubic Spline Methodology.
YIELD CURVE FITTING 2.0 Constructng Bond and Money Market Yeld Curves usng Cubc BSplne and Natural Cubc Splne Methodology Users Manual YIELD CURVE FITTING 2.0 Users Manual Authors: Zhuosh Lu, Moorad Choudhry
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationNasdaq 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 information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationThe values in the TVM Solver are quantities involved in compound interest and annuities.
Texas Instruments Graphing Calculators have a built in app that may be used to compute quantities involved in compound interest, annuities, and amortization. For the examples below, we ll utilize the screens
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (InClass) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationTIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY
TIME VALUE OF MONEY PROBLEM #4: PRESENT VALUE OF AN ANNUITY Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction In this assignment we will discuss how to calculate the Present Value
More informationMultiplePeriod Attribution: Residuals and Compounding
MultplePerod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens
More informationADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc.
ADVA FINAN QUAN 00 Vault Inc. VAULT GUIDE TO ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS Copyrght 00 by Vault Inc. All rghts reserved. All nformaton n ths book s subject to change wthout notce. Vault
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More information( ) Homework Solutions Physics 8B Spring 09 Chpt. 32 5,18,25,27,36,42,51,57,61,76
Homework Solutons Physcs 8B Sprng 09 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat = E. Please refer to Fgure EX32.5. We wll choose a clockwse
More informationTrafficlight a stress test for life insurance provisions
MEMORANDUM Date 006097 Authors Bengt von Bahr, Göran Ronge Traffclght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationHomework Solutions Physics 8B Spring 2012 Chpt. 32 5,18,25,27,36,42,51,57,61,76
Homework Solutons Physcs 8B Sprng 202 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat =. Please refer to Fgure EX32.5. We wll choose a clockwse
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationStock Profit Patterns
Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationTHE VALUE OF MONEY PROBLEM #3: ANNUITY. Professor Peter Harris Mathematics by Dr. Sharon Petrushka. Introduction
THE VALUE OF MONEY PROBLEM #3: ANNUITY Professor Peter Harris Mathematics by Dr. Sharon Petrushka Introduction Earlier, we explained how to calculate the future value of a single sum placed on deposit
More informationSUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.
SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976761000
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationFormula of Total Probability, Bayes Rule, and Applications
1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.
More informationSection 5.1  Compound Interest
Section 5.1  Compound Interest Simple Interest Formulas If I denotes the interest on a principal P (in dollars) at an interest rate of r (as a decimal) per year for t years, then we have: Interest: Accumulated
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationfirst complete "prior knowlegde"  to refresh knowledge of Simple and Compound Interest.
ORDINARY SIMPLE ANNUITIES first complete "prior knowlegde"  to refresh knowledge of Simple and Compound Interest. LESSON OBJECTIVES: students will learn how to determine the Accumulated Value of Regular
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationAryabhata 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 informationLIFETIME 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) 3575200 Fax: (617) 3575250 www.ersalawyers.com
More information