# Level Annuities with Payments Less Frequent than Each Interest Period

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due

2 Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due

3 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

4 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

5 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

6 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

7 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

8 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

9 Symoblc approach In ths chapter we have to dstngush between payment perods and nterest perods Consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k In other words, ths annuty has a payment at the end of each k nterest perods... the effectve nterest rate per nterest perod I... the effectve nterest rate per payment perod,.e., I = (1 + ) k 1 Then, the value at ssuance of ths annuty s a r I and a r I = 1 (1 + I ) r I = 1 (1 + ) rk (1 + ) k 1 = a n s k The accumulated value s s r I = s n s k

10 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

11 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

12 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

13 An Example Fnd an expresson n terms of symbols of the type a n and s n, for the present value of an annuty n whch there are a total of r payments of 1. The frst payment s to be made 7 years from today, and the remanng payments happen at three year ntervals. The present value of ths annuty can be expressed n terms of the annual dscount factor as v 7 + v 10 + v v 3r+4 Calculatng the partal sum of the geometrc seres above, we get v 7 v 3r+7 1 v 3 = (1 v 7 ) + (1 v 3r+7 ) 1 v 3 = 1 v v 3 1 v 3r+7 Caveat: The expresson we obtaned above s not unque! = a 3r+7 a 7 a 3

14 An Example: Unknown fnal payment An nvestment of \$1000 s used to make payments of \$100 at the end of each year for as long as possble wth a smaller fnal payment to be made at the tme of the last regular payment. If nterest s 7% convertble semannually, fnd the number of payments and the amount of the total fnal payment.

15 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = \$10.09 R s s = 1000 (1.035) 36

16 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = \$10.09 R s s = 1000 (1.035) 36

17 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = \$10.09 R s s = 1000 (1.035) 36

18 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = \$10.09 R s s = 1000 (1.035) 36

19 An Example: Unknown fnal payment (cont d) Usng the expresson for the present value of ths annuty, we get the equaton of value at tme an s = 1000 where n denotes the unknown number of regular nterest perods that the annuty lasts. The equaton of value yelds a n = 10 s = We get that n = 36 and that 18 regular payments and an addtonal smaller payment must be made. Let R denote the amount of the smaller fnal payment. Then, the tme n equaton of value reads as Thus, R = \$10.09 R s s = 1000 (1.035) 36

20 Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due

21 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

22 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

23 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

24 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

25 Value at ssuance and accumulated value Agan, consder a basc annuty that lasts for n nterest perods, and has r payments where n = r k for some nteger k Ths annuty has a payment at the begnnng of each k nterest perods Then, the value at ssuance of ths annuty-due s ä r I and ä r I = (1 + I ) a r I = a n a k Smlarly, we get that the accumulated value equals s r I = s n a k Caveat: The above accumulated value s k nterest converson perods after the last payment...

26 An Example: Accumulated value Fnd the accumulated value at the end of four years of an nvestment fund n whch \$100 s deposted at the begnnng of each quarter for the frst two years and \$200 s deposted at the begnnng of every quarter for the second two years. Assume that the fund earns 12% convertble monthly. The rate of nterest s 1% per month. In ths annuty-due, there are 48 nterest perods and each payment perod conssts of 3 nterest coverson perods. So, the accumulated value s 100 s s a = = \$2999 Assgnment: Examples 4.2.9, 12 Problems 4.2.1,3

27 An Example: Accumulated value Fnd the accumulated value at the end of four years of an nvestment fund n whch \$100 s deposted at the begnnng of each quarter for the frst two years and \$200 s deposted at the begnnng of every quarter for the second two years. Assume that the fund earns 12% convertble monthly. The rate of nterest s 1% per month. In ths annuty-due, there are 48 nterest perods and each payment perod conssts of 3 nterest coverson perods. So, the accumulated value s 100 s s a = = \$2999 Assgnment: Examples 4.2.9, 12 Problems 4.2.1,3

28 An Example: Accumulated value Fnd the accumulated value at the end of four years of an nvestment fund n whch \$100 s deposted at the begnnng of each quarter for the frst two years and \$200 s deposted at the begnnng of every quarter for the second two years. Assume that the fund earns 12% convertble monthly. The rate of nterest s 1% per month. In ths annuty-due, there are 48 nterest perods and each payment perod conssts of 3 nterest coverson perods. So, the accumulated value s 100 s s a = = \$2999 Assgnment: Examples 4.2.9, 12 Problems 4.2.1,3

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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,

### Professor 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

### 3. 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 = -

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### 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

### 9.1 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

### = i δ δ s n and PV = a n = 1 v n = 1 e nδ

Exam 2 s Th March 19 You are allowe 7 sheets of notes an a calculator 41) An mportant fact about smple nterest s that for smple nterest A(t) = K[1+t], the amount of nterest earne each year s constant =

### 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

### 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

### Solutions to the exam in SF2862, June 2009

Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

### Calculation 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 two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

### The Cox-Ross-Rubinstein Option Pricing Model

Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

### 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

### 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

### Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

### 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

### 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

### Traffic-light a stress test for life insurance provisions

MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

### 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

### 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

### 2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

### 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,

### The Application of Fractional Brownian Motion in Option Pricing

Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

### 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)

### greatest common divisor

4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

### benefit 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

### Recurrence. 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.

### Future Value of an Annuity

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

### 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

### v 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

### Formula 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.

### 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

### 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.

### Project Networks With Mixed-Time Constraints

Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa

### INSTITUT FÜR INFORMATIK

INSTITUT FÜR INFORMATIK Schedulng jobs on unform processors revsted Klaus Jansen Chrstna Robene Bercht Nr. 1109 November 2011 ISSN 2192-6247 CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL Insttut für Informat

### YIELD CURVE FITTING 2.0 Constructing Bond and Money Market Yield Curves using Cubic B-Spline and Natural Cubic Spline Methodology.

YIELD CURVE FITTING 2.0 Constructng Bond and Money Market Yeld Curves usng Cubc B-Splne and Natural Cubc Splne Methodology Users Manual YIELD CURVE FITTING 2.0 Users Manual Authors: Zhuosh Lu, Moorad Choudhry

### Traffic-light extended with stress test for insurance and expense risks in life insurance

PROMEMORIA Datum 0 July 007 FI Dnr 07-1171-30 Fnansnspetonen Författare Bengt von Bahr, Göran Ronge Traffc-lght extended wth stress test for nsurance and expense rss n lfe nsurance Summary Ths memorandum

### Complex Number Representation in RCBNS Form for Arithmetic Operations and Conversion of the Result into Standard Binary Form

Complex Number epresentaton n CBNS Form for Arthmetc Operatons and Converson of the esult nto Standard Bnary Form Hatm Zan and. G. Deshmukh Florda Insttute of Technology rgd@ee.ft.edu ABSTACT Ths paper

### Interest Rate Fundamentals

Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

### Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall

SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent

### 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

### ( ) 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

### Homework 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

### n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

MATH 16T Exam 1 : Part I (In-Class) 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

### Fast degree elevation and knot insertion for B-spline curves

Computer Aded Geometrc Desgn 22 (2005) 183 197 www.elsever.com/locate/cagd Fast degree elevaton and knot nserton for B-splne curves Q-Xng Huang a,sh-mnhu a,, Ralph R. Martn b a Department of Computer Scence

### Stock 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

### New bounds in Balog-Szemerédi-Gowers theorem

New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

### EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

### 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

### Implementation of Deutsch's Algorithm Using Mathcad

Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

### 8 Algorithm for Binary Searching in Trees

8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

### CHAPTER EVALUATING EARTHQUAKE RETROFITTING MEASURES FOR SCHOOLS: A COST-BENEFIT ANALYSIS

CHAPTER 17 EVALUATING EARTHQUAKE RETROFITTING MEASURES FOR SCHOOLS: A COST-BENEFIT ANALYSIS A.W. Smyth, G. Deodats, G. Franco, Y. He and T. Gurvch Department of Cvl Engneerng and Engneerng Mechancs, Columba

### Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance

Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom

### Addendum to: Importing Skill-Biased Technology

Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our

### Interest Rates and The Credit Crunch: New Formulas and Market Models

Interest Rates and The Credt Crunch: New Formulas and Market Models Fabo Mercuro QFR, Bloomberg Frst verson: 12 November 2008 Ths verson: 5 February 2009 Abstract We start by descrbng the major changes

### LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit

LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE

### Examples of Multiple Linear Regression Models

ECON *: Examples of Multple Regresson Models Examples of Multple Lnear Regresson Models Data: Stata tutoral data set n text fle autoraw or autotxt Sample data: A cross-sectonal sample of 7 cars sold n

### SUPPLIER 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: 976-76-10-00

### A Critical Note on MCEV Calculations Used in the Life Insurance Industry

A Crtcal Note on MCEV Calculatons Used n the Lfe Insurance Industry Faban Suarez 1 and Steven Vanduffel 2 Abstract. Snce the begnnng of the development of the socalled embedded value methodology, actuares

### Construction Rules for Morningstar Canada Target Dividend Index SM

Constructon Rules for Mornngstar Canada Target Dvdend Index SM Mornngstar Methodology Paper October 2014 Verson 1.2 2014 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property

### Supplementary material: Assessing the relevance of node features for network structure

Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada

### 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