Section 5.4 Annuities, Present Value, and Amortization

Size: px
Start display at page:

Download "Section 5.4 Annuities, Present Value, and Amortization"

Transcription

1 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 (at the same nterest rate) n order to produce A dollars n n perods. Smlarly, the present value of an annuty s the amount that must be deposted today (at the same compound nterest rate as the annuty) to provde all the payments for the term of the annuty. It does not matter whether the payments are nvested to accumulate funds or are pad out to dsperse funds; the amount needed to provde the payments s the same n ether case. We begn wth ordnary annutes. EXAMPLE: Your rch aunt has funded an annuty that wll pay you $1500 at the end of each year for sx years. If the nterest rate s 8%, compounded annually, fnd the present value of ths annuty. Soluton: Look separately at each payment you wll receve. Then fnd the present value of each payment the amount needed now n order to make the payment n the future. The sum of these present values wll be the present value of the annuty, snce t wll provde all of the payments. To fnd the frst $1500 payment (due n one year), the present value of $1500 at 8% annual nterest s needed now. Accordng to the present-value formula for compound nterest wth A = 1500, =.08, and n = 1, ths present value s A P = (1+) = 1500 n (1+.08) = = 1500( ) $ Ths amount wll grow to $1500 n one year. For the second $1500 payment (due n two years), we need the present value of $1500 at 8% nterest, compounded annually for two years. The present-value formula for compound nterest (wth A = 1500, =.08, and n = 2) shows that ths present value s A P = (1+) = 1500 n (1+.08) = = ( ) $ Less money s needed for the second payment because t wll grow over two years nstead of one. A smlar calculaton shows that the thrd payment (due n three years) has present value $1500( ). Contnue n ths manner to fnd the present value of each of the remanng payments, as summarzed n the Fgure below. 1

2 The left-hand column of the Fgure above shows that the present value s = 1500( ) Now apply the algebrac fact that (1) to the expresson n parentheses (wth x = 1.08 and n = 6). It shows that the sum (the present value of the annuty) s [ ] [ ] = 1500 = $ Ths amount wll provde for all sx payments and leave a zero balance at the end of sx years (gve or take a few cents due to roundng to the nearest penny at each step). The Example above s the model for fndng a formula for the future value of any ordnary annuty. Suppose that a payment of R dollars s made at the end of each perod for n perods, at nterest rate per perod. Then the present value of ths annuty can be found by usng the procedure n that Example, wth these replacements: The future value n the Example above s the sum n equaton (1), whch now becomes P = R[(1+) 1 +(1+) 2 +(1+) (1+) ] Apply the algebrac fact n the box above to the expresson n brackets (wth x = 1+). Then we have [ ] [ ] 1 (1+) 1 (1+) P = R = R (1+) 1 The quantty n brackets n the rght-hand part of the precedng equaton s sometmes wrtten a n (read a-angle-n at ). So we can summarze as follows. 2

3 CAUTION: Do not confuse the formula for the present value of an annuty wth the one for the future value of an annuty. Notce the dfference: The numerator of the fracton n the present-value formula s 1 (1+), but n the future-value formula, t s (1+) n 1. EXAMPLE: Jm Rles was n an auto accdent. He sued the person at fault and was awarded a structured settlement n whch an nsurance company wll pay hm $600 at the end of each month for the next seven years. How much money should the nsurance company nvest now at 4.7%, compounded monthly, to guarantee that all the payments can be made? Soluton: The payments form an ordnary annuty. The amount needed to fund all the payments s the present value of the annuty. Apply the present-value formula wth R = 600, n = 7 12 = 84, and =.047/12 (the nterest rate per month). The nsurance company should nvest [ ] [ ] 1 (1+) 1 (1+.047/12) 84 P = R = 600 = $42, /12 EXAMPLE: To supplement hs penson n the early years of hs retrement, Ralph Taylor plans to use $124,500 of hs savngs as an ordnary annuty that wll make monthly payments to hm for 20 years. If the nterest rate s 5.2%, how much wll each payment be? Soluton: The present value of the annuty s P = $124, 500, the monthly nterest rate s =.052/12, and n = = 240 (the number of months n 20 years). Solve the present-value formula for the monthly payment R: [ ] 1 (1+) P = R [ ] 1 (1+.052/12) ,500 = R.052/12 R = 124, 500 [ ] 1 (1+.052/12) 240 = $ /12 Taylor wll receve $ a month (about $10,026 per year) for 20 years. EXAMPLE: Surnder Snah and Mara Gonzalez are graduates of Kenyon College. They both agree to contrbute to an endowment fund at the college. Snah says he wll gve $500 at the end of each year for 9 years. Gonzalez prefers to gve a sngle donaton today. How much should she gve to equal the value of Snahs gft, assumng that the endowment fund earns 7.5% nterest, compounded annually? Soluton: Snah s gft s an ordnary annuty wth annual payments of $500 for 9 years. Its future value at 7.5% annual compound nterest s [ ] [ ] [ ] (1+) n 1 (1+.075) S = R = 500 = 500 = $ We clam that for Gonzalez to equal ths contrbuton, she should today contrbute an amount equal to the present value of ths annuty, namely, [ ] [ ] [ ] 1 (1+) 1 (1+.075) P = R = 500 = 500 = $

4 To confrm ths clam, suppose the present value P = $ s deposted today at 7.5% nterest, compounded annually for 9 years. Accordng to the compound nterest formula, P wll grow to A = P(1+) n = (1+.075) 9 = $ the future value of Snah s annuty. So at the end of 9 years, Gonzalez and Snah wll have made dentcal gfts. The Example above llustrates the followng alternatve descrpton of the present value of an accumulaton annuty. Corporate bonds, whch were ntroduced n Secton 5.1, are routnely bought and sold n fnancal markets. In most cases, nterest rates when a bond s sold dffer from the nterest rate pad by the bond (known as the coupon rate). In such cases, the prce of a bond wll not be ts face value, but wll nstead be based on current nterest rates. The next example shows how ths s done. EXAMPLE:A15-year$10,000 bondwtha5%couponratewasssuedfveyearsagoandsnow beng sold. If the current nterest rate for smlar bonds s 7%, what prce should a purchaser be wllng to pay for ths bond? Soluton: Accordng to the smple nterest formula, the nterest pad by the bond each half-year s I = Prt = 10, = $250 Thnk of the bond as a two-part nvestment: The frst s an annuty that pays $250 every sx months for the next 10 years; the second s the $10,000 face value of the bond, whch wll be pad when the bond matures, 10 years from now. The purchaser should be wllng to pay the present value of each part of the nvestment, assumng 7% nterest, compounded semannually. The nterest rate per perod s =.07/2, and the number of sx-month perods n 10 years s n = 20. So we have: Present value of annuty [ ] 1 (1+) P = R [ ] 1 (1+.07/2) 20 = /2 = $ Present value of $10,000 n 10 years P = A(1+) = 10,000(1+.07/2) 20 = $ So the purchaser should be wllng to pay the sum of these two present values: $ $ = $

5 Loans and Amortzaton If you take out a car loan or a home mortgage, you repay t by makng regular payments to the bank. From the bank s pont of vew, your payments are an annuty that s payng t a fxed amount each month. The present value of ths annuty s the amount you borrowed. EXAMPLE: Chase Bank n Aprl 2013 advertsed a new car auto loan rate of 2.23% for a 48-month loan. Shelley Fasulko wll buy a new car for $25,000 wth a down payment of $4500. Fnd the amount of each payment. (Data from: Soluton: After a $4500 down payment, the loan amount s $20,500. Use the present value formula for an annuty, wth P = 20,500, n = 48, and =.0223/12 (the monthly nterest rate). Then solve for payment R. [ ] 1 (1+) P = R [ ] 1 ( /12) 48 20,500 = R.0223/12 R = 20, 500 [ ] 1 ( /12) 48 = $ /12 A loan s amortzed f both the prncpal and nterest are pad by a sequence of equal perodc payments. The perodc payment needed to amortze a loan may be found, as n the Example above, by solvng the present-value formula for R. EXAMPLE: In Aprl 2013, the average rate for a 30-year fxed mortgage was 3.43%. Assume a down payment of 20% on a home purchase of $272,900. (Data from: Fredde Mac.) (a) Fnd the monthly payment needed to amortze ths loan. Soluton: Thedownpayments.20(272,900) = $54,580. Thus,theloanamountP s$272,900 $54,580 = $218,320. We can now apply the formula n the precedng box, wth n = 12(30) = 360 (the number of monthly payments n 30 years), and monthly nterest rate =.0343/12. R = P (218,320)(.0343/12) = = $ (1+) 1 ( /12) 360 Monthly payments of $ are requred to amortze the loan. 5

6 (b) After 10 years, approxmately how much s owed on the mortgage? Soluton: You may be tempted to say that after 10 years of payments on a 30-year mortgage, the balance wll be reduced by a thrd. However, a sgnfcant porton of each payment goes to pay nterest. So, much less than a thrd of the mortgage s pad off n the frst 10 years, as we now see. After 10 years (120 payments), the 240 remanng payments can be thought of as an annuty. The present value for ths annuty s the (approxmate) remanng balance on the mortgage. Hence, we use the present-value formula wth R = , =.0343/12, and n = 240: [ ] 1 ( /12) 240 P = = $168, (.0343/12) So the remanng balance s about $168, The actual balance probably dffers slghtly from ths fgure because payments and nterest amounts are rounded to the nearest penny. The Example above, part (b), llustrates an mportant fact: Even though equal payments are made to amortze a loan, the loan balance does not decrease n equal steps. The method used to estmate the remanng balance n the Example works n the general case. If n payments are needed to amortze a loan and x payments have been made, then the remanng payments form an annuty of n x payments. So we apply the present-value formula wth n x n place of n to obtan ths result. Amortzaton Schedules The remanng-balance formula s a quck and convenent way to get a reasonable estmate of the remanng balance on a loan, but t s not accurate enough for a bank or busness, whch must keep ts books exactly. To determne the exact remanng balance after each loan payment, fnancal nsttutons normally use an amortzaton schedule, whch lsts how much of each payment s nterest, how much goes to reduce the balance, and how much s stll owed after each payment. EXAMPLE: Beth Hll borrows $1000 for one year at 12% annual nterest, compounded monthly. (a) Fnd her monthly payment. Soluton: Apply the amortzaton payment formula wth P = 1000, n = 12, and monthly nterest rate =.12/12 =.01. Her payment s R = P 1 (1+) = 1000(.01) = $ (1+.01) 12 (b) After makng fve payments, Hll decdes to pay off the remanng balance. Approxmately how much must she pay? 6

7 (b) After makng fve payments, Hll decdes to pay off the remanng balance. Approxmately how much must she pay? Soluton: Apply the remanng-balance formula just gven, wth R = 88.85, =.01, and n x = 12 5 = 7. Her approxmate remanng balance s [ ] [ ] 1 (1+) (n x) 1 (1+.01) 7 B = R = = $ (c) Construct an amortzaton schedule for Hll s loan. Soluton: An amortzaton schedule for the loan s shown n the table below. It was obtaned as follows: The annual nterest rate s 12% compounded monthly, so the nterest rate per month s 12%/12 = 1% =.01. When the frst payment s made, one month s nterest, namely,.01(1000) = $10, s owed. Subtractng ths from the $88.85 payment leaves $78.85 to be appled to repayment. Hence, the prncpal at the end of the frst payment perod s = $921.15, as shown n the payment 1 lne of the table. When payment 2 s made, one month s nterest on the new balance of $ s owed, namely,.01(921.15) = $9.21. Contnue as n the precedng paragraph to compute the entres n ths lne of the table. The remanng lnes of the table are found n a smlar fashon. Note that Hll s remanng balance after fve payments dffers slghtly from the estmate made n part (b). The fnal payment n the amortzaton schedule n the last Example, part (c), dffers from the other payments. It often happens that the last payment needed to amortze a loan must be adjusted to account for roundng earler and to ensure that the fnal balance wll be exactly 0. 7

8 Annutes Due We want to fnd the present value of an annuty due n whch 6 payments of R dollars are made at the begnnng of each perod, wth nterest rate per perod, as shown schematcally n the Fgure on the rght. The present value s the amount needed to fund all 6 payments. Snce the frst payment earns no nterest, R dollars are needed to fund t. Now look at the last 5 payments by themselves n the Fgure on the rght. If you thnk of these 5 payments as beng made at the end of each perod, you see that they form an ordnary annuty. The money needed to fund them s the present value of ths ordnary annuty. So the present value of the annuty due s gven by [ ] Present value of the ordnary 1 (1+) 5 R+ = R+R annuty of 5 payments Replacng 6 by n and 5 by n 1, and usng the argument just gven, produces the general result that follows. EXAMPLE: The Illnos Lottery Wnner s Handbook dscusses the optons of how to receve the wnnngs for a $12 mllon Lotto jackpot. One opton s to take 26 annual payments of approxmately $461,538.46, whch s $12 mllon dvded nto 26 equal payments. The other opton s to take a lump-sum payment (whch s often called the cash value ). If the Illnos lottery commsson can earn 4.88% annual nterest, how much s the cash value? Soluton: The yearly payments form a 26-payment annuty due. An equvalent amount now s the present value of ths annuty. Apply the present-value formula wth R = 461,538.46, =.0488, and n = 26: [ ] 1 (1+) (n 1) P = R+R [ ] 1 ( ) 25 = 461, , = $7,045, The cash value s $7,045,

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

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

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

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

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

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

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

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

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

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

9 Arithmetic and Geometric Sequence

9 Arithmetic and Geometric Sequence AAU - Busness Mathematcs I Lecture #5, Aprl 4, 010 9 Arthmetc and Geometrc Sequence Fnte sequence: 1, 5, 9, 13, 17 Fnte seres: 1 + 5 + 9 + 13 +17 Infnte sequence: 1,, 4, 8, 16,... Infnte seres: 1 + + 4

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

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

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

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

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

3. Present value of Annuity Problems

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 = -

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solution of Algebraic and Transcendental Equations

Solution of Algebraic and Transcendental Equations CHAPTER Soluton of Algerac and Transcendental Equatons. INTRODUCTION One of the most common prolem encountered n engneerng analyss s that gven a functon f (, fnd the values of for whch f ( = 0. The soluton

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

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

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

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

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

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

Notes on Engineering Economic Analysis

Notes on Engineering Economic Analysis College of Engneerng and Computer Scence Mechancal Engneerng Department Mechancal Engneerng 483 lternatve Energy Engneerng II Sprng 200 umber: 7724 Instructor: Larry Caretto otes on Engneerng Economc nalyss

More information

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA ) February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

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

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

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

Solution : (a) FALSE. Let C be a binary one-error correcting code of length 9. Then it follows from the Sphere packing bound that.

Solution : (a) FALSE. Let C be a binary one-error correcting code of length 9. Then it follows from the Sphere packing bound that. MATH 29T Exam : Part I Solutons. TRUE/FALSE? Prove your answer! (a) (5 pts) There exsts a bnary one-error correctng code of length 9 wth 52 codewords. (b) (5 pts) There exsts a ternary one-error correctng

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

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

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

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

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

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) 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 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

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

Nordea G10 Alpha Carry Index

Nordea G10 Alpha Carry Index Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 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 information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

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

Multivariate EWMA Control Chart

Multivariate EWMA Control Chart Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant

More information

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

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

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

Section 5.1 Simple Interest and Discount

Section 5.1 Simple Interest and Discount Section 5.1 Simple Interest and Discount DEFINITION: Interest is the fee paid to use someone else s money. Interest on loans of a year or less is frequently calculated as simple interest, which is paid

More information

Moment of a force about a point and about an axis

Moment of a force about a point and about an axis 3. STATICS O RIGID BODIES In the precedng chapter t was assumed that each of the bodes consdered could be treated as a sngle partcle. Such a vew, however, s not always possble, and a body, n general, should

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 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 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

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

= 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 =

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

9.1 The Cumulative Sum Control Chart

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

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

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

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

More information

Recurrence. 1 Definitions and main statements

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.

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

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

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

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

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

Capital asset pricing model, arbitrage pricing theory and portfolio management

Capital asset pricing model, arbitrage pricing theory and portfolio management Captal asset prcng model, arbtrage prcng theory and portfolo management Vnod Kothar The captal asset prcng model (CAPM) s great n terms of ts understandng of rsk decomposton of rsk nto securty-specfc rsk

More information

Ping Pong Fun - Video Analysis Project

Ping Pong Fun - Video Analysis Project Png Pong Fun - Vdeo Analyss Project Objectve In ths experment we are gong to nvestgate the projectle moton of png pong balls usng Verner s Logger Pro Software. Does the object travel n a straght lne? What

More information

Section C2: BJT Structure and Operational Modes

Section C2: BJT Structure and Operational Modes Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

The example below solves a system in the unknowns α and β:

The example below solves a system in the unknowns α and β: The Fnd Functon The functon Fnd returns a soluton to a system of equatons gven by a solve block. You can use Fnd to solve a lnear system, as wth lsolve, or to solve nonlnear systems. The example below

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

Generator Warm-Up Characteristics

Generator Warm-Up Characteristics NO. REV. NO. : ; ~ Generator Warm-Up Characterstcs PAGE OF Ths document descrbes the warm-up process of the SNAP-27 Generator Assembly after the sotope capsule s nserted. Several nqures have recently been

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

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

1. Measuring association using correlation and regression

1. 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 information

Traffic-light a stress test for life insurance provisions

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

More information

Question 2: What is the variance and standard deviation of a dataset?

Question 2: What is the variance and standard deviation of a dataset? Queston 2: What s the varance and standard devaton of a dataset? The varance of the data uses all of the data to compute a measure of the spread n the data. The varance may be computed for a sample of

More information

greatest common divisor

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

More information

EXPLORATION 2.5A Exploring the motion diagram of a dropped object

EXPLORATION 2.5A Exploring the motion diagram of a dropped object -5 Acceleraton Let s turn now to moton that s not at constant elocty. An example s the moton of an object you release from rest from some dstance aboe the floor. EXPLORATION.5A Explorng the moton dagram

More information