An Introduction to Certificates of Deposit, Bonds, Yield to Maturity, Accrued Interest, and Duration
|
|
- Tracy Todd
- 6 years ago
- Views:
Transcription
1 1 A Itroductio to Certificates of Deposit, Bods, Yield to Maturity, Accrued Iterest, ad Duratio Joh A. Guber Departet of Electrical ad Coputer Egieerig Uiversity of Wiscosi Madiso Abstract A brief itroductio is give to copoud iterest, certificates of deposit, ad bods. The focus is o deteriig a fair price, yield to aturity, accrued iterest, ad duratio. MATLAB code is give to copute the accrued iterest with the 30/360 US ethod, which is used for US corporate bods ad ay US agecy bods. If you fid this writeup useful, or if you fid typos or istakes, please let e kow at Joh.Guber@wisc.edu Cotets 1 Copoud Iterest Daily Copoudig 3 2 Preset Value 3 3 Certificates of Deposit 3 4 Bod Prices A Special Case The Geeral Case 6 5 Yield to Maturity Part Iterpretatio 8 6 Buyig Bods The Price The Accrued Iterest 9 7 Yield to Maturity Part Sesitivity ad Duratio 10 9 Day by Day, or Thirty Days Hath Septeber 12 Refereces 13 Idex 14
2 2 1. Copoud Iterest Recall that if you ivest pricipal A 0 at aual iterest rate r (as a decial 1 ) copouded ties per year for y years, the the aout of oey you will have after y years is A(y) = A 0 (1 + r/) y. I this forula, is called the copoudig frequecy ad has uits of years 1. The reciprocal 1/ is called the copoudig period ad has uits of years. Sice y is easured i years, the product y has o uits. The aual iterest rate r also has uits of years 1 so that the quotiet r/ has o uits. To copute the aout of oey you will have at the tie you receive the kth iterest payet, set y = k/ for k = 1,2,... This results i A(k/) = A 0 (1 + r/) k. For exaple, if iterest is copouded quarterly, the whe you receive the first iterest payet, after three oths (1/4 of a year), you will have After six oths, you will have After ie oths, you will have A(1/4) = A 0 (1 + r/4). A(2/4) = A 0 (1 + r/4) 2. A(3/4) = A 0 (1 + r/4) 3. Sice iterest is paid at ties which are ultiples of 1/, if y is a tie betwee payet dates, say k y < k + 1, the we ca express y i the for y = k + ρ, 0 ρ < 1, where ρ is the fractio of a copoudig period that has passed sice the kth iterest payet. With this otatio, the aout of oey you will have at tie (k + ρ)/ is ( ) k + ρ A = A 0 (1 + r/) k+ρ. 1 For exaple, a 5% aual rate would use r = 0.05.
3 Daily Copoudig For daily copoudig, it is ore coveiet to easure tie i days (see [6] for how to do this). After d days you will have A(d) = A 0 (1 + r/365) d. Iterest eared usig this forula is called exact iterest to distiguish it fro ordiary iterest which arises usig the baker s rule, A(d) = A 0 (1 + r/360) d. Whe bakers talk about iterest, they ea ordiary iterest uless explicitly stated otherwise. 2. Preset Value Suppose that at soe tie y = (k+ρ)/ i the future, you will receive a aout of oey A(y). How uch is it worth today? The aswer is called the preset value, ad it is give by ( ) k + ρ A PV = (1 + r/) k+ρ. Of course, whe A((k+ρ)/) is give by A 0 (1+r/) k+ρ the preset value is siply A Certificates of Deposit Cosider a bak certificate of deposit (CD) i which you ivest pricipal A 0 at aual iterest rate r paid ties a year. Rather tha take the iterest payets ad sped the, you choose to have your iterest added to your CD balace so that you get the beefit of copoudig. Suppose that your CD will ature after / years so that there will be copoudigs. At aturity your CD will be worth A 0 (1 + r/). (1) However, if you wat the curret CD balace before aturity, you will have to pay a pealty. Now suppose that you ivest i your CD today at 11 a, ad as your tur to leave the bak, they aouce that startig at oo, ew CDs will ear aual iterest rate r ew. You retur to the bak at oo ad strike up a coversatio with a potetial ew
4 4 CD custoer waitig i lie. You ask her how uch she will pay you i exchage for your CD that pays the old iterest rate r. If r ew > r, she will pay you less tha A 0, sice otherwise she ca just buy a ew CD directly fro the bak. But what is the fair price you should ask for your CD? A little thought suggests that the fair price is p(r ew ), where p(r ew ) is chose so that if she ivested p(r ew ) i a ew bak CD at the ew rate r ew, the value at aturity would equal that of your CD; i.e., p(r ew ) should solve p(r ew )(1 + r ew /) = A 0 (1 + r/). (2) We coclude that the price should be p(r ew ) = A 0(1 + r/) (1 + r ew /). As expected, if r ew > r, p(r ew ) < A 0, ad if r ew < r, the p(r ew ) > A 0. This illustrates the fact that CD prices ad iterest rates ove i opposite directios. As r ew rages over the iterval (, ), the price p(r ew ) decreases cotiuously fro to 0. Hece, every positive price correspods to a uique value of r ew. This eas that if we kow the aturity value of the CD, the uber of copoudigs, ad the curret price, say p, we ca solve the equatio for the curret CD iterest rate r ew ; i.e., [( aturity value ) 1/ ] r ew = 1. p 4. Bod Prices Cosider a bod with face value F (also called the aturity value or par value) ad aual iterest rate r (called the coupo, coupo rate or oial yield), with coupos paid ties a year. The aout of each iterest payet, or coupo payet, is C := Fr/. (3) 4.1. A Special Case Although it is ot possible i practice, assue that you will deposit each coupo payet i a savigs accout that pays aual rate r ew copouded ties per year. 2 How uch oey will you have if the bod atures upo receipt of the th 2 It would ake ore sese to assue that each coupo payet is ivested i a bak CD at rate r ew that atures whe the bod atures. Sice successive coupo payets will be ivested for shorter ad shorter ties, the successive values of r ew should decrease. However, sice we keep r ew costat i our aalysis, it is sipler to say that the coupo payets are deposited i a savigs accout.
5 5 coupo payet? Whe the bod atures, you get the face value F plus you have your savigs accout, whose value is C(1 + r ew /) 1 +C(1 + r ew /) 2 + +C(1 + r ew /) 0, where the first ter is the result of depositig the first coupo payet i your savigs accout for the reaiig 1 tie copoudig periods, ad the last ter is siply the fial coupo payet, which speds zero tie i your savigs accout. Hece, at aturity, you have 3 A = F +C(1 + r ew /) 1 +C(1 + r ew /) 2 + +C(1 + r ew /) 0 1 = F +C (1 + r ew /) l. (4) If the process just described starts exactly copoudig periods prior to the aturity date, what is a fair price for the bod? Because of our savigs accout assuptio, a potetial buyer could either ivest p(r ew ) i a savigs accout at rate r ew copouded ties per year, leavig the iterest i the bak to copoud, or she could buy the bod for p(r ew ) ad deposit the coupo payets i a savigs accout at rate r ew copouded ties per year. Hece, we ust have (cf. (2)) or p(r ew ) = p(r ew )(1 + r ew /) = A, (5) 1 A (1 + r ew /) = F (1 + r ew /) +C (1 + r ew /) l (1 + r ew /) 1 F = (1 + r ew /) +C 1. (6) (1 + r ew /) l This shows that bod prices ad iterest rates ove i opposite directios. Exaple 1. Cosider two bods with the sae face value F. The first bod was issued five years ago with coupo rate r to ature i te years; hece this bod atures five years fro today. The secod bod is beig issued today with rate r ew, atures i five years, ad sells at par. Use (6) to fid today s price of the first bod if F = $100, r = 2.5%, ad r ew = 2.0%. Solutio. We use the followig MATLAB code to copute (6). 3 The proof of Propositio 2 shows that if r ew = r, the (4) siplifies to A = F(1 + r/), which is the CD aturity value (1) with A 0 replaced by F.
6 6 F = 100; r = 2.5/100; rew = 2.0/100; = 2; = 5*; % five years = 10 coupo payets C = F*r/; theta = 1 + rew/; ueratorvec = [ repat(c,1,) F ]; powers = [ 1: ]; price = su(ueratorvec./theta.ˆpowers) We fid that the price rouds to $ Propositio 2. If r ew is equal to the coupo rate r, the the price (6) is equal to the bod face value F. Proof. First cosider the case r ew = r = 0. The C = Fr/ = 0 o accout of (3), ad the (6) reduces to p(0) = F. It reais to cosider the case r ew = r 0. Put θ := 1+r/ so that (4) becoes By the geoetric series, 1 A = F +C θ l. 1 θ l = 1 θ 1 θ, θ 1. The use the fact that 1 θ = r/. Sice C = Fr/, we fid that A = F + Fr 1 θ r/ = F + F(θ 1) = Fθ. We ca ow write (6) as p(r)θ = Fθ ad the propositio follows. Exaple 3. I the MATLAB code used to solve Exaple 1, if you chage the third lie to rew = r; what value do you obtai for price? 4.2. The Geeral Case Let us repeat the aalysis leadig to (6), but assue that the startig tie is idway betwee coupo payet dates, say a fractio ρ of the copoudig period
7 7 sice the ost recet coupo payet, ad that there payets reaiig. The the forula (4) for A is the sae, but (5) becoes p(r ew )(1 + r ew /) ρ = A because the tie to aturity is o loger, but is a little shorter by the fractio ρ of a copoudig period. It ow follows that 1 F p(r ew ) = (1 + r ew /) ρ +C (1 + r ew /) l (1 + r ew /) ρ 1 F = (1 + r ew /) ρ +C 1 (1 + r ew /) l ρ F = (1 + r ew /) ρ +C 1. (7) (1 + r ew /) k ρ Eve i this slightly ore geeral situatio, bod prices ad iterest rates still ove i opposite directios. Reark. I the ext sectio, we itroduce the yield to aturity, which is defied as the solutio of (7) for r ew whe the left-had side is give. Based o our derivatio of (7), it appears that the yield to aturity depeds o the assuptio that the coupo payets are reivested at rate r ew. However, cosider the followig view suggested by [5]. The first ter o the right i (7) is the preset value of the face value F received at aturity. The fractio C/(1 + r ew /) k ρ is the preset value of the kth coupo payet; i.e., we ca write (7) as PV = PV F + PV Ck. Now there is o assuptio of reivestig the coupo payets at rate r ew. 5. Yield to Maturity Part 1 Suppose I ow the bod described i the previous sectio, ad I ake you the followig offer. If you pay e p today, the I will give you y reaiig iterest payets C whe I receive the, ad I will give you the face value F at aturity. I this offer, there is o etio of a iterest rate, so istead of (7), we cosider the equatio F p = (1 + λ/) ρ +C 1, (8) (1 + λ/) k ρ
8 8 ad try to solve it for λ. The solutio is called the yield to aturity. Observe that the right-had side (RHS) of the equatio as a fuctio of λ is cotiuous ad strictly decreasig o (, ). Sice the RHS teds to ifiity as λ ad the RHS teds to zero as λ, the equatio ca be solved for ay positive, fiite value of p. Exaple 4. I Exaple 1, we showed that the price of the first bod was $ Use (8) with ρ = 0 to obtai the yield to aturity. Solutio. Usig the values of, ueratorvec, ad powers fro the solutio of Exaple 1, we add the followig MATLAB code. phat = ; v g YTM = fzero(g,0.5) % Solve phat = v(labda) What do you expect YTM to be? 5.1. Iterpretatio Let λ deote the solutio of (8), ad ultiply (8) by (1 + λ/) ρ ; i.e., we reverse the steps that led to (7) but replace p(r ew ) with p ad r ew with λ. The p(1 + λ/) ρ 1 = F +C (1 + λ/) l. The left-had side is equal to what you would have if you could ivest p i a CD payig rate λ util the bod atures. The right-had side is equal to what you would have if you bought the bod ad could ivest the coupo payets at rate λ util the bod atures The Price 6. Buyig Bods Whe bods are offered for sale, the price is quoted as a percetage of the face value F that you wat to buy. For exaple, the price ight be P % = , eaig that you pay % of the face value. 4 If you wat to buy this bod with a face value F = $15,000, it will cost you F P % = $15,000 = $15, Equivaletly, P % is the price of a bod with a $100 face value.
9 9 I this case, you pay a preiu of $414.45, which is 2.763% of the face value. Siilarly, if the bod price is P % = , the cost of a $15,000 bod would be F P % = $15, = $14, I this case, you obtai the bod at a discout of = 1.575% The Accrued Iterest If you buy a bod betwee coupo payets dates, whe you get your first coupo payet, oly a fractio of it really belogs to you. For this reaso, at the tie you buy the bod, you pay a total of the bod price plus a portio of your first coupo payet. That portio is called accrued iterest, ad is deoted by AI. It is coputed by solvig the equatio Equivaletly, AI C = ρ. (9) AI = Cρ. Hece, o accout of (3), the accrued iterest AI is proportioal to the face value F. 7. Yield to Maturity Part 2 Fro the discussio i Sectio 6, we should replace p i (8) with F P % AI, which we call the total cost. This results i the forula Dividig by F results i F P % AI = F (1 + λ/) ρ +C P % AI F = 1 (1 + λ/) ρ + C F 1 (1 + λ/) k ρ. 1 (1 + λ/) k ρ. Substitutig C = Fr/ fro (3) ad AI = Cρ fro (9) yields P % r ρ = 1 (1 + λ/) ρ + r 1. (10) (1 + λ/) k ρ
10 10 The value of λ that solves this equatio is the yield to aturity, which, as we would expect, does ot deped o the face value F of the bod. We call (10) the yield to aturity equatio. The left-had side is the total cost per dollar of face value. Cautio. Of all the paraeters i the yield to aturity equatio, the fractio of a copoudig period ρ is the ost difficult to deterie, as explaied i Sectio 9. Fortuately, for a bod with a specific face value F, the seller will provide P %, r,, ad AI. The you ca copute C usig (3) ad the ρ usig (9). 8. Sesitivity ad Duratio Let ϕ(λ) deote the right-had side of (10). This fuctio gives the total cost per dollar of face value as a fuctio of the yield to aturity. How uch does the cost chage if the yield chages fro λ to λ + λ? For sall λ, ϕ(λ + λ) ϕ(λ) ϕ (λ) λ. Sice ϕ is a decreasig fuctio (cf. the discussio below (8)), its derivative is egative. Hece if the yield icreases, the cost decreases, ad vice verse. What we really wat, however, is the percetage chage i cost, ϕ(λ + λ) ϕ(λ) ϕ(λ) 100% ϕ (λ) λ 100%. ϕ(λ) To ake further progress, we eed to copute ϕ (λ). To this ed, put ψ t (λ) := 1 (1 + λ/) t, where we suppress the depedece o. With this otatio, Now observe that ϕ(λ) = (r/) It follows that [ ϕ 1 (λ) = (r/) 1 + λ/ ψ k ρ (λ) + ψ ρ (λ). ψ t t (λ) = (1 + λ/) ψ t(λ). k ρ ψ k ρ(λ) + ρ ] ψ ρ(λ).
11 11 Notice that the fractios (k ρ)/ have uits of years, sice has uits of years 1. Hece, ϕ (λ) has uits of years. The sesitivity is defied as S(λ) := ϕ (λ) ϕ(λ) = λ/ (r/) (r/) k ρ ψ k ρ(λ) + ρ ψ ρ(λ) ψ k ρ (λ) + ψ ρ (λ) ad has uits of years. The Macaulay duratio, or siply the duratio, is D(λ) := (r/) (r/) k ρ ψ k ρ(λ) + ρ ψ ρ(λ) ψ k ρ (λ) + ψ ρ (λ) ad also has uits of years, sice 1 + λ/ has o uits. Clearly, S(λ) = D(λ)/(1 + λ/). O accout of this, the sesitivity is usually called the odified duratio. To put the foregoig all together, suppose the curret total cost per dollar of face value is ϕ(λ), where λ is the curret yield to aturity. If the yield chages to λ + λ, the the percetage cost chage will approxiately be S(λ) λ 100% = D(λ) λ 100%, 1 + λ/,, where D(λ) = (r/) (k ρ)/ ( ρ)/ + (1 + λ/) k ρ (1 + λ/) ρ. ϕ(λ) Exaple 5. Recall the first bod i Exaple 1 with coupo rate r = 2.5% ad curretly priced at $ Lettig λ = r ew = 2.0% deote the curret yield, fid the duratio, sesitivity, ad approxiate price chage if yields icrease back to r = 2.5%. Solutio. Usig the values of, ueratorvec, ad powers fro the solutio of Exaple 1 ad the fuctio v fro Exaple 4, we ca add the followig code to copute the duratio, sesitivity, ad approxiate price chage (we take ρ = 0). 5 5 To uderstad how the code relates to the forula for D(λ), keep i id that ueratorvec, which also occurs i the defiitio of v, cotais the factor F. Hece, i the code stateet that coputes D, the coo factor F cacels.
12 12 u2 = ueratorvec.*[ 1: ]/; labda = rew; Dlabda = r - rew; D = -su(u2./(1+labda/).ˆpowers)/v(labda) S = D/(1+labda/) S*Dlabda*100 We fid that the approxiate price chage is 2.34%; i.e., the price would drop by $ = $2.40, ad so the ew price would be approxiately $ Of course the true ew price will be the face value of $100 sice the coupo rate is 2.5%. I other words, the true price chage would be $2.37 rather tha the approxiate $ Day by Day, or Thirty Days Hath Septeber Sice it was challegig to fid the uber of days betwee two dates before coputers were cooplace, the followig 30/360 US ethod cotiues to be used to copute accrued iterest o US corporate bods ad ay US agecy bods [8]. If the previous coupo payet was o M1/D1/Y 1, ad the settleet date is M2/D2/Y 2 (ad either oth is February; to hadle this case, see [8]), the the uber of days betwee the previous coupo payet date ad the settleet date is approxiated by the followig MATLAB code. 6 if D2==31 && (D1>=30) D2 = 30; ed if D1==31 D1 = 30; ed ApproxNuDays = 360*(Y2-Y1)+30*(M2-M1)+(D2-D1); The ethod also approxiates the uber of days i a year by 360, ad the ratio ApproxNuDays, 360 is used to replace ρ/ i all of the above forulas; i.e., ρ = ApproxNuDays Notice that M2-M1 or D2-D1 ay be egative.
13 13 Refereces [1] B. R. Barish, ECE 601 Fiacial Egieerig Course Notes, [2] E. J. Elto. (1999, Sept.). Yield to aturity, accrued iterest, quoted price, ivoice price, [Olie.] Available: eelto/debt_ist_class/ytm.pdf, accessed Feb. 20, [3] F. J. Fabozzi, Fixed Icoe Securities, New York: Wiley, [4] FDIC, FDIC Law, Regulatios, Related Acts: 6500 Cosuer Fiacial Protectio Bureau, Appedix A to Part 1030 Aual Percetage Yield Calculatio, [Olie.] Available: fdic6500appedixatopart1030, accessed Feb. 25, [5] S. M. Forbes, J. J. Hate, ad C. Paul, Yield-to-aturity ad the reivestet of coupo payets, Joural of Ecooetrics ad Fiace Educatio, vol. 7, o. 1, Suer [6] Math Foru, Calculatig how ay days betwee two dates, [Olie.] Available: accessed Feb. 20, [7] A. Thau, The Bod Book, 3rd ed. New York: McGraw-Hill, [8] Wikipedia, Day cout covetio Wikipedia, The Free Ecyclopedia, [Olie.] Available: oldid= , accessed Feb. 20, 2017.
14 14 Idex 30/360 US ethod, 12 accrued iterest, 9 aual iterest rate, 2 baker s rule, 3 ad ordiary iterest, 3 bod, 4 discout, 9 preiu, 9 certificate of deposit (CD), 3 copoudig daily, 3 frequecy, 2 period, 2 coupo, 4 payet, 4 rate, 4 preiu, 9 preset value, 3, 7 pricipal, 2 sesitivity, 11 total cost, 9 per dollar of face value, 10 yield to aturity, 7, 8, 10 equatio, 10 daily copoudig, 3 discout, 9 duratio Macaulay, 11 odified, 11 exact iterest, 3 face value, 4 geoetric series, 6 iterest accrued, 9 aual rate, 2 exact, 3 ordiary, 3 Macaulay duratio, 11 aturity value, see face value odified duratio, 11 oial yield, 4 ordiary iterest, 3 ad baker s rule, 3 par value, see face value
CHAPTER 2 PRICING OF BONDS
CHAPTER 2 PRICING OF BONDS CHAPTER SUARY This chapter will focus o the time value of moey ad how to calculate the price of a bod. Whe pricig a bod it is ecessary to estimate the expected cash flows ad
More information2.6 Rational Functions and Their Graphs
.6 Ratioal Fuctios ad Their Graphs Sectio.6 Notes Page Ratioal Fuctio: a fuctio with a variable i the deoiator. To fid the y-itercept for a ratioal fuctio, put i a zero for. To fid the -itercept for a
More information43. A 000 par value 5-year bod with 8.0% semiaual coupos was bought to yield 7.5% covertible semiaually. Determie the amout of premium amortized i the 6 th coupo paymet. (A).00 (B).08 (C).5 (D).5 (E).34
More information1 + r. k=1. (1 + r) k = A r 1
Perpetual auity pays a fixed sum periodically forever. Suppose a amout A is paid at the ed of each period, ad suppose the per-period iterest rate is r. The the preset value of the perpetual auity is A
More informationad covexity Defie Macaulay duratio D Mod = r 1 = ( CF i i k (1 + r k) i ) (1.) (1 + r k) C = ( r ) = 1 ( CF i i(i + 1) (1 + r k) i+ k ) ( ( i k ) CF i
Fixed Icome Basics Cotets Duratio ad Covexity Bod Duratios ar Rate, Spot Rate, ad Forward Rate Flat Forward Iterpolatio Forward rice/yield, Carry, Roll-Dow Example Duratio ad Covexity For a series of cash
More informationChapter Four Learning Objectives Valuing Monetary Payments Now and in the Future
Chapter Four Future Value, Preset Value, ad Iterest Rates Chapter 4 Learig Objectives Develop a uderstadig of 1. Time ad the value of paymets 2. Preset value versus future value 3. Nomial versus real iterest
More information1 The Power of Compounding
1 The Power of Compoudig 1.1 Simple vs Compoud Iterest You deposit $1,000 i a bak that pays 5% iterest each year. At the ed of the year you will have eared $50. The bak seds you a check for $50 dollars.
More information2. The Time Value of Money
2. The Time Value of Moey Problem 4 Suppose you deposit $100 i the bak today ad it ears iterest at a rate of 10% compouded aually. How much will be i the accout 50 years from today? I this case, $100 ivested
More informationMS-E2114 Investment Science Exercise 2/2016, Solutions
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 Perpetual auity pays a xed sum periodically forever. Suppose a amout A is paid at the ed of each period, ad suppose the per-period iterest rate
More informationAUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY
AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY Dr. Farha I. D. Al Ai * ad Dr. Muhaed Alfarras ** * College of Egieerig ** College of Coputer Egieerig ad scieces Gulf Uiversity * Dr.farha@gulfuiversity.et;
More informationChapter 4: Time Value of Money
FIN 301 Class Notes Chapter 4: Time Value of Moey The cocept of Time Value of Moey: A amout of moey received today is worth more tha the same dollar value received a year from ow. Why? Do you prefer a
More information1 Savings Plans and Investments
4C Lesso Usig ad Uderstadig Mathematics 6 1 Savigs las ad Ivestmets 1.1 The Savigs la Formula Lets put a $100 ito a accout at the ed of the moth. At the ed of the moth for 5 more moths, you deposit $100
More informationClass Sessions 2, 3, and 4: The Time Value of Money
Class Sessios 2, 3, ad 4: The Time Value of Moey Associated Readig: Text Chapter 3 ad your calculator s maual. Summary Moey is a promise by a Bak to pay to the Bearer o demad a sum of well, moey! Oe risk
More informationChapter Four 1/15/2018. Learning Objectives. The Meaning of Interest Rates Future Value, Present Value, and Interest Rates Chapter 4, Part 1.
Chapter Four The Meaig of Iterest Rates Future Value, Preset Value, ad Iterest Rates Chapter 4, Part 1 Preview Develop uderstadig of exactly what the phrase iterest rates meas. I this chapter, we see that
More informationChapter Six. Bond Prices 1/15/2018. Chapter 4, Part 2 Bonds, Bond Prices, Interest Rates and Holding Period Return.
Chapter Six Chapter 4, Part Bods, Bod Prices, Iterest Rates ad Holdig Period Retur Bod Prices 1. Zero-coupo or discout bod Promise a sigle paymet o a future date Example: Treasury bill. Coupo bod periodic
More informationSTRAND: FINANCE. Unit 3 Loans and Mortgages TEXT. Contents. Section. 3.1 Annual Percentage Rate (APR) 3.2 APR for Repayment of Loans
CMM Subject Support Strad: FINANCE Uit 3 Loas ad Mortgages: Text m e p STRAND: FINANCE Uit 3 Loas ad Mortgages TEXT Cotets Sectio 3.1 Aual Percetage Rate (APR) 3.2 APR for Repaymet of Loas 3.3 Credit Purchases
More informationAPPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES
APPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES Example: Brado s Problem Brado, who is ow sixtee, would like to be a poker champio some day. At the age of twety-oe, he would
More informationClass Notes for Managerial Finance
Class Notes for Maagerial Fiace These otes are a compilatio from:. Class Notes Supplemet to Moder Corporate Fiace Theory ad Practice by Doald R. Chambers ad Nelso J. Lacy. I gratefully ackowledge the permissio
More informationWe learned: $100 cash today is preferred over $100 a year from now
Recap from Last Week Time Value of Moey We leared: $ cash today is preferred over $ a year from ow there is time value of moey i the form of willigess of baks, busiesses, ad people to pay iterest for its
More informationModels of Asset Pricing
APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationModels of Asset Pricing
APPENDIX 1 TO CHAPTER4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationCAPITAL PROJECT SCREENING AND SELECTION
CAPITAL PROJECT SCREEIG AD SELECTIO Before studyig the three measures of ivestmet attractiveess, we will review a simple method that is commoly used to scree capital ivestmets. Oe of the primary cocers
More informationFixed Income Securities
Prof. Stefao Mazzotta Keesaw State Uiversity Fixed Icome Securities Sample First Midterm Exam Last Name: First Name: Studet ID Number: Exam time is: 80 miutes. Total poits for this exam is: 400 poits Prelimiaries
More informationAppendix 1 to Chapter 5
Appedix 1 to Chapter 5 Models of Asset Pricig I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy a asset, we are
More informationCourse FM Practice Exam 1 Solutions
Course FM Practice Exam 1 Solutios Solutio 1 D Sikig fud loa The aual service paymet to the leder is the aual effective iterest rate times the loa balace: SP X 0.075 To determie the aual sikig fud paymet,
More informationModels of Asset Pricing
4 Appedix 1 to Chapter Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationOverlapping Generations
Eco. 53a all 996 C. Sims. troductio Overlappig Geeratios We wat to study how asset markets allow idividuals, motivated by the eed to provide icome for their retiremet years, to fiace capital accumulatio
More informationof Asset Pricing R e = expected return
Appedix 1 to Chapter 5 Models of Asset Pricig EXPECTED RETURN I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy
More informationDr. Maddah ENMG 602 Intro to Financial Eng g 01/18/10. Fixed-Income Securities (2) (Chapter 3, Luenberger)
Dr Maddah ENMG 60 Itro to Fiacial Eg g 0/8/0 Fixed-Icome Securities () (Chapter 3 Lueberger) Other yield measures Curret yield is the ratio of aual coupo paymet to price C CY = For callable bods yield
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS This set of sample questios icludes those published o the iterest theory topic for use with previous versios of this examiatio.
More informationThe University of British Columbia Diploma Program in Urban Land Economics Sample Final Examination BUSI 121 FOUNDATIONS OF REAL ESTATE MATHEMATICS
The Uiversity of British Colubia Diploa Progra i Urba Lad Ecooics Saple Fial Exaiatio BUSI 121 FOUNDATIONS OF REAL ESTATE MATHEMATICS Tie: Date: 3 Hours Saple Fial Exa Istructios This exaiatio cosists
More informationof Asset Pricing APPENDIX 1 TO CHAPTER EXPECTED RETURN APPLICATION Expected Return
APPENDIX 1 TO CHAPTER 5 Models of Asset Pricig I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy a asset, we are
More informationCalculation of the Annual Equivalent Rate (AER)
Appedix to Code of Coduct for the Advertisig of Iterest Bearig Accouts. (31/1/0) Calculatio of the Aual Equivalet Rate (AER) a) The most geeral case of the calculatio is the rate of iterest which, if applied
More informationIntroduction to Financial Derivatives
550.444 Itroductio to Fiacial Derivatives Determiig Prices for Forwards ad Futures Week of October 1, 01 Where we are Last week: Itroductio to Iterest Rates, Future Value, Preset Value ad FRAs (Chapter
More information0.07. i PV Qa Q Q i n. Chapter 3, Section 2
Chapter 3, Sectio 2 1. (S13HW) Calculate the preset value for a auity that pays 500 at the ed of each year for 20 years. You are give that the aual iterest rate is 7%. 20 1 v 1 1.07 PV Qa Q 500 5297.01
More informationChapter 2. Theory of interest
Chapter 2 Theory of iterest Tie alue of oey Cash flow ( 現金流 ) aout of oey receied (+) or paid out (-) at soe tie poit Tie alue of oey whe aluig cash flows i differet tie periods, the iterest-earig capacity
More informationMonetary Economics: Problem Set #5 Solutions
Moetary Ecoomics oblem Set #5 Moetary Ecoomics: oblem Set #5 Solutios This problem set is marked out of 1 poits. The weight give to each part is idicated below. Please cotact me asap if you have ay questios.
More informationChapter 3. Compound interest
Chapter 3 Compoud iterest 1 Simple iterest ad compoud amout formula Formula for compoud amout iterest is: S P ( 1 Where : S: the amout at compoud iterest P: the pricipal i: the rate per coversio period
More informationFixed Income Securities
Prof. Stefao Mazzotta Keesaw State Uiversity Fixed Icome Securities FIN4320. Fall 2006 Sample First Midterm Exam Last Name: First Name: Studet ID Number: Exam time is: 80 miutes. Total poits for this exam
More informationCourse FM/2 Practice Exam 1 Solutions
Course FM/2 Practice Exam 1 Solutios Solutio 1 D Sikig fud loa The aual service paymet to the leder is the aual effective iterest rate times the loa balace: SP X 0.075 To determie the aual sikig fud paymet,
More informationChapter 5: Sequences and Series
Chapter 5: Sequeces ad Series 1. Sequeces 2. Arithmetic ad Geometric Sequeces 3. Summatio Notatio 4. Arithmetic Series 5. Geometric Series 6. Mortgage Paymets LESSON 1 SEQUENCES I Commo Core Algebra I,
More informationELEMENTARY PORTFOLIO MATHEMATICS
QRMC06 9/7/0 4:44 PM Page 03 CHAPTER SIX ELEMENTARY PORTFOLIO MATHEMATICS 6. AN INTRODUCTION TO PORTFOLIO ANALYSIS (Backgroud readig: sectios 5. ad 5.5) A ivestor s portfolio is the set of all her ivestets.
More informationBond Valuation. Structure of fixed income securities. Coupon Bonds. The U.S. government issues bonds
Structure of fixed icome securities Bod Valuatio The Structure of fixed icome securities Price & ield to maturit (tm) Term structure of iterest rates Treasur STRIPS No-arbitrage pricig of coupo bods A
More informationChapter 11 Appendices: Review of Topics from Foundations in Finance and Tables
Chapter 11 Appedices: Review of Topics from Foudatios i Fiace ad Tables A: INTRODUCTION The expressio Time is moey certaily applies i fiace. People ad istitutios are impatiet; they wat moey ow ad are geerally
More informatione-companion ONLY AVAILABLE IN ELECTRONIC FORM
OPERATIONS RESEARCH doi 0.87/opre.090.079ec e-copaio ONLY AVAILABLE IN ELECTRONIC FORM ifors 009 INFORMS Electroic Copaio Equilibriu Capacity Expasio Uder Stochastic Dead Growth by Alfredo Garcia ad Zhijiag
More informationSection 3.3 Exercises Part A Simplify the following. 1. (3m 2 ) 5 2. x 7 x 11
123 Sectio 3.3 Exercises Part A Simplify the followig. 1. (3m 2 ) 5 2. x 7 x 11 3. f 12 4. t 8 t 5 f 5 5. 3-4 6. 3x 7 4x 7. 3z 5 12z 3 8. 17 0 9. (g 8 ) -2 10. 14d 3 21d 7 11. (2m 2 5 g 8 ) 7 12. 5x 2
More informationMark to Market Procedures (06, 2017)
Mark to Market Procedures (06, 207) Risk Maagemet Baco Sumitomo Mitsui Brasileiro S.A CONTENTS SCOPE 4 2 GUIDELINES 4 3 ORGANIZATION 5 4 QUOTES 5 4. Closig Quotes 5 4.2 Opeig Quotes 5 5 MARKET DATA 6 5.
More informationAnomaly Correction by Optimal Trading Frequency
Aomaly Correctio by Optimal Tradig Frequecy Yiqiao Yi Columbia Uiversity September 9, 206 Abstract Uder the assumptio that security prices follow radom walk, we look at price versus differet movig averages.
More informationSolutions to Interest Theory Sample Questions
to Iterest Theory Sample Questios Solutio 1 C Chapter 4, Iterest Rate Coversio After 7.5 years, the value of each accout is the same: 7.5 7.5 0.04 1001 100e 1.336 e l(1.336) 7.5 0.0396 7.5 Solutio E Chapter
More informationFINANCIAL MATHEMATICS
CHAPTER 7 FINANCIAL MATHEMATICS Page Cotets 7.1 Compoud Value 116 7.2 Compoud Value of a Auity 117 7.3 Sikig Fuds 118 7.4 Preset Value 121 7.5 Preset Value of a Auity 121 7.6 Term Loas ad Amortizatio 122
More informationUsing Math to Understand Our World Project 5 Building Up Savings And Debt
Usig Math to Uderstad Our World Project 5 Buildig Up Savigs Ad Debt Note: You will have to had i aswers to all umbered questios i the Project Descriptio See the What to Had I sheet for additioal materials
More informationA New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions
A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhu-qi Lu, Ph.D. Abstract A -valued two-variable truth fuctio is called fuctioally complete,
More informationBinomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge
Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity
More informationFINANCIAL MATHEMATICS GRADE 11
FINANCIAL MATHEMATICS GRADE P Prcpal aout. Ths s the orgal aout borrowed or vested. A Accuulated aout. Ths s the total aout of oey pad after a perod of years. It cludes the orgal aout P plus the terest.
More informationEstimating Proportions with Confidence
Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter
More informationFinancial Analysis. Lecture 4 (4/12/2017)
Fiacial Aalysis Lecture 4 (4/12/217) Fiacial Aalysis Evaluates maagemet alteratives based o fiacial profitability; Evaluates the opportuity costs of alteratives; Cash flows of costs ad reveues; The timig
More informationNotes on Expected Revenue from Auctions
Notes o Epected Reveue from Auctios Professor Bergstrom These otes spell out some of the mathematical details about first ad secod price sealed bid auctios that were discussed i Thursday s lecture You
More informationDate: Practice Test 6: Compound Interest
: Compoud Iterest K: C: A: T: PART A: Multiple Choice Questios Istructios: Circle the Eglish letter of the best aswer. Circle oe ad ONLY oe aswer. Kowledge/Thikig: 1. Which formula is ot related to compoud
More informationThe material in this chapter is motivated by Experiment 9.
Chapter 5 Optimal Auctios The material i this chapter is motivated by Experimet 9. We wish to aalyze the decisio of a seller who sets a reserve price whe auctioig off a item to a group of bidders. We begi
More information1 Basic Growth Models
UCLA Aderso MGMT37B: Fudametals i Fiace Fall 015) Week #1 rofessor Eduardo Schwartz November 9, 015 Hadout writte by Sheje Hshieh 1 Basic Growth Models 1.1 Cotiuous Compoudig roof: lim 1 + i m = expi)
More informationWhen you click on Unit V in your course, you will see a TO DO LIST to assist you in starting your course.
UNIT V STUDY GUIDE Percet Notatio Course Learig Outcomes for Uit V Upo completio of this uit, studets should be able to: 1. Write three kids of otatio for a percet. 2. Covert betwee percet otatio ad decimal
More informationChapter 5 Time Value of Money
Chapter 5 Time Value of Moey 1. Suppose you deposit $100 i a bak that pays 10% iterest per year. How much will you have i the bak oe year later? 2. Suppose you deposit $100 i a bak that pays 10% per year.
More informationMATH : EXAM 2 REVIEW. A = P 1 + AP R ) ny
MATH 1030-008: EXAM 2 REVIEW Origially, I was havig you all memorize the basic compoud iterest formula. I ow wat you to memorize the geeral compoud iterest formula. This formula, whe = 1, is the same as
More informationT4032-MB, Payroll Deductions Tables CPP, EI, and income tax deductions Manitoba Effective January 1, 2016
T4032-MB, Payroll Deductios Tables CPP, EI, ad icome tax deductios Maitoba Effective Jauary 1, 2016 T4032-MB What s ew as of Jauary 1, 2016 The major chages made to this guide sice the last editio are
More informationPension Annuity. Policy Conditions Document reference: PPAS1(6) This is an important document. Please keep it in a safe place.
Pesio Auity Policy Coditios Documet referece: PPAS1(6) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity. These
More informationT4032-ON, Payroll Deductions Tables CPP, EI, and income tax deductions Ontario Effective January 1, 2016
T4032-ON, Payroll Deductios Tables CPP, EI, ad icome tax deductios Otario Effective Jauary 1, 2016 T4032-ON What s ew as of Jauary 1, 2016 The major chages made to this guide sice the last editio are outlied.
More informationEstimating Volatilities and Correlations. Following Options, Futures, and Other Derivatives, 5th edition by John C. Hull. Chapter 17. m 2 2.
Estiatig Volatilities ad Correlatios Followig Optios, Futures, ad Other Derivatives, 5th editio by Joh C. Hull Chapter 17 Stadard Approach to Estiatig Volatility Defie as the volatility per day betwee
More informationA random variable is a variable whose value is a numerical outcome of a random phenomenon.
The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss
More informationBasic formula for confidence intervals. Formulas for estimating population variance Normal Uniform Proportion
Basic formula for the Chi-square test (Observed - Expected ) Expected Basic formula for cofidece itervals sˆ x ± Z ' Sample size adjustmet for fiite populatio (N * ) (N + - 1) Formulas for estimatig populatio
More informationFINM6900 Finance Theory How Is Asymmetric Information Reflected in Asset Prices?
FINM6900 Fiace Theory How Is Asymmetric Iformatio Reflected i Asset Prices? February 3, 2012 Referece S. Grossma, O the Efficiecy of Competitive Stock Markets where Traders Have Diverse iformatio, Joural
More informationStatistics for Economics & Business
Statistics for Ecoomics & Busiess Cofidece Iterval Estimatio Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio How to determie
More informationMath of Finance Math 111: College Algebra Academic Systems
Math of Fiace Math 111: College Algebra Academic Systems Writte By Bria Hoga Mathematics Istructor Highlie Commuity College Edited ad Revised by Dusty Wilso Mathematics Istructor Highlie Commuity College
More information2. Find the annual percentage yield (APY), to the nearest hundredth of a %, for an account with an APR of 12% with daily compounding.
1. Suppose that you ivest $4,000 i a accout that ears iterest at a of 5%, compouded mothly, for 58 years. `Show the formula that you would use to determie the accumulated balace, ad determie the accumulated
More informationAnnual compounding, revisited
Sectio 1.: No-aual compouded iterest MATH 105: Cotemporary Mathematics Uiversity of Louisville August 2, 2017 Compoudig geeralized 2 / 15 Aual compoudig, revisited The idea behid aual compoudig is that
More informationT4032-BC, Payroll Deductions Tables CPP, EI, and income tax deductions British Columbia Effective January 1, 2016
T4032-BC, Payroll Deductios Tables CPP, EI, ad icome tax deductios British Columbia Effective Jauary 1, 2016 T4032-BC What s ew as of Jauary 1, 2016 The major chages made to this guide, sice the last editio,
More informationSubject CT1 Financial Mathematics Core Technical Syllabus
Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig
More informationLecture 4: Probability (continued)
Lecture 4: Probability (cotiued) Desity Curves We ve defied probabilities for discrete variables (such as coi tossig). Probabilities for cotiuous or measuremet variables also are evaluated usig relative
More informationUnbiased estimators Estimators
19 Ubiased estimators I Chapter 17 we saw that a dataset ca be modeled as a realizatio of a radom sample from a probability distributio ad that quatities of iterest correspod to features of the model distributio.
More informationpoint estimator a random variable (like P or X) whose values are used to estimate a population parameter
Estimatio We have oted that the pollig problem which attempts to estimate the proportio p of Successes i some populatio ad the measuremet problem which attempts to estimate the mea value µ of some quatity
More informationWhere a business has two competing investment opportunities the one with the higher NPV should be selected.
Where a busiess has two competig ivestmet opportuities the oe with the higher should be selected. Logically the value of a busiess should be the sum of all of the projects which it has i operatio at the
More informationChapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1
Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for
More informationThe Time Value of Money in Financial Management
The Time Value of Moey i Fiacial Maagemet Muteau Irea Ovidius Uiversity of Costata irea.muteau@yahoo.com Bacula Mariaa Traia Theoretical High School, Costata baculamariaa@yahoo.com Abstract The Time Value
More informationSingle-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P)
Sigle-Paymet Factors (P/F, F/P) Example: Ivest $1000 for 3 years at 5% iterest. F =? i =.05 $1000 F 1 = 1000 + (1000)(.05) = 1000(1+.05) F 2 = F 1 + F 1 i = F 1 (1+ = 1000(1+.05)(1+.05) = 1000(1+.05) 2
More informationSupersedes: 1.3 This procedure assumes that the minimal conditions for applying ISO 3301:1975 have been met, but additional criteria can be used.
Procedures Category: STATISTICAL METHODS Procedure: P-S-01 Page: 1 of 9 Paired Differece Experiet Procedure 1.0 Purpose 1.1 The purpose of this procedure is to provide istructios that ay be used for perforig
More informationACTUARIAL RESEARCH CLEARING HOUSE 1990 VOL. 2 INTEREST, AMORTIZATION AND SIMPLICITY. by Thomas M. Zavist, A.S.A.
ACTUARIAL RESEARCH CLEARING HOUSE 1990 VOL. INTEREST, AMORTIZATION AND SIMPLICITY by Thomas M. Zavist, A.S.A. 37 Iterest m Amortizatio ad Simplicity Cosider simple iterest for a momet. Suppose you have
More informationFirst determine the payments under the payment system
Corporate Fiace February 5, 2008 Problem Set # -- ANSWERS Klick. You wi a judgmet agaist a defedat worth $20,000,000. Uder state law, the defedat has the right to pay such a judgmet out over a 20 year
More informationLecture 16 Investment, Time, and Risk (Basic issues in Finance)
Lecture 16 Ivestmet, Time, ad Risk (Basic issues i Fiace) 1. Itertemporal Ivestmet Decisios: The Importace o Time ad Discoutig 1) Time as oe o the most importat actors aectig irm s ivestmet decisios: A
More informationMath 312, Intro. to Real Analysis: Homework #4 Solutions
Math 3, Itro. to Real Aalysis: Homework #4 Solutios Stephe G. Simpso Moday, March, 009 The assigmet cosists of Exercises 0.6, 0.8, 0.0,.,.3,.6,.0,.,. i the Ross textbook. Each problem couts 0 poits. 0.6.
More informationThe Time Value of Money
Part 2 FOF12e_C03.qxd 8/13/04 3:39 PM Page 39 Valuatio 3 The Time Value of Moey Cotets Objectives The Iterest Rate After studyig Chapter 3, you should be able to: Simple Iterest Compoud Iterest Uderstad
More information1 Estimating sensitivities
Copyright c 27 by Karl Sigma 1 Estimatig sesitivities Whe estimatig the Greeks, such as the, the geeral problem ivolves a radom variable Y = Y (α) (such as a discouted payoff) that depeds o a parameter
More informationFurther Pure 1 Revision Topic 5: Sums of Series
The OCR syllabus says that cadidates should: Further Pure Revisio Topic 5: Sums of Series Cadidates should be able to: (a) use the stadard results for Σr, Σr, Σr to fid related sums; (b) use the method
More informationliving well in retirement Adjusting Your Annuity Income Your Payment Flexibilities
livig well i retiremet Adjustig Your Auity Icome Your Paymet Flexibilities what s iside 2 TIAA Traditioal auity Icome 4 TIAA ad CREF Variable Auity Icome 7 Choices for Adjustig Your Auity Icome 7 Auity
More informationNPTEL DEPARTMENT OF INDUSTRIAL AND MANAGEMENT ENGINEERING IIT KANPUR QUANTITATIVE FINANCE END-TERM EXAMINATION (2015 JULY-AUG ONLINE COURSE)
NPTEL DEPARTMENT OF INDUSTRIAL AND MANAGEMENT ENGINEERING IIT KANPUR QUANTITATIVE FINANCE END-TERM EXAMINATION (2015 JULY-AUG ONLINE COURSE) READ THE INSTRUCTIONS VERY CAREFULLY 1) Time duratio is 2 hours
More informationMAT 3788 Lecture 3, Feb
The Tie Value of Money MAT 3788 Lecture 3, Feb 010 The Tie Value of Money and Interest Rates Prof. Boyan Kostadinov, City Tech of CUNY Everyone is failiar with the saying "tie is oney" and in finance there
More informationChapter 8: Estimation of Mean & Proportion. Introduction
Chapter 8: Estimatio of Mea & Proportio 8.1 Estimatio, Poit Estimate, ad Iterval Estimate 8.2 Estimatio of a Populatio Mea: σ Kow 8.3 Estimatio of a Populatio Mea: σ Not Kow 8.4 Estimatio of a Populatio
More informationAsset Valuation with known cash flows. Annuities and Perpetuities care loan, saving for retirement, mortgage
Asset Valuatio with kow cash flows Auities ad Perpetuities care loa, savig for retiremet, mortgage Simple Perpetuity A perpetuity is a stream of cash flows each of the amout of dollars, that are received
More informationCAPITALIZATION (PREVENTION) OF PAYMENT PAYMENTS WITH PERIOD OF DIFFERENT MATURITY FROM THE PERIOD OF PAYMENTS
Iteratioal Joural of Ecoomics, Commerce ad Maagemet Uited Kigdom Vol. VI, Issue 9, September 2018 http://ijecm.co.uk/ ISSN 2348 0386 CAPITALIZATION (PREVENTION) OF PAYMENT PAYMENTS WITH PERIOD OF DIFFERENT
More informationToday: Finish Chapter 9 (Sections 9.6 to 9.8 and 9.9 Lesson 3)
Today: Fiish Chapter 9 (Sectios 9.6 to 9.8 ad 9.9 Lesso 3) ANNOUNCEMENTS: Quiz #7 begis after class today, eds Moday at 3pm. Quiz #8 will begi ext Friday ad ed at 10am Moday (day of fial). There will be
More informationEVEN NUMBERED EXERCISES IN CHAPTER 4
Joh Riley 7 July EVEN NUMBERED EXERCISES IN CHAPTER 4 SECTION 4 Exercise 4-: Cost Fuctio of a Cobb-Douglas firm What is the cost fuctio of a firm with a Cobb-Douglas productio fuctio? Rather tha miimie
More informationOptimizing of the Investment Structure of the Telecommunication Sector Company
Iteratioal Joural of Ecoomics ad Busiess Admiistratio Vol. 1, No. 2, 2015, pp. 59-70 http://www.aisciece.org/joural/ijeba Optimizig of the Ivestmet Structure of the Telecommuicatio Sector Compay P. N.
More information