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 2 1.1 Daily Copoudig 3 2 Preset Value 3 3 Certificates of Deposit 3 4 Bod Prices 4 4.1 A Special Case 4 4.2 The Geeral Case 6 5 Yield to Maturity Part 1 7 5.1 Iterpretatio 8 6 Buyig Bods 8 6.1 The Price 8 6.2 The Accrued Iterest 9 7 Yield to Maturity Part 2 9 8 Sesitivity ad Duratio 10 9 Day by Day, or Thirty Days Hath Septeber 12 Refereces 13 Idex 14
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 1.1. 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 0. 3. 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 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 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 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 $102.37. 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 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 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 $102.37. 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 = 102.37; v = @(labda)su(ueratorvec./(1+labda/).ˆpowers); g = @(labda)phat-v(labda); 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. 6.1. 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 % = 102.763, eaig that you pay 102.763% 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 % 102.763 = $15,000 = $15, 414.45. 100 100 4 Equivaletly, P % is the price of a bod with a $100 face value.
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 % = 98.425, the cost of a $15,000 bod would be F P % 98.425 = $15,000 100 100 = $14,763.75. I this case, you obtai the bod at a discout of 100 98.425 = 1.575%. 6.2. 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 % 100 + AI, which we call the total cost. This results i the forula Dividig by F results i F P % 100 + AI = F (1 + λ/) ρ +C P % 100 + 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 % 100 + r ρ = 1 (1 + λ/) ρ + r 1. (10) (1 + λ/) k ρ
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 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) = 1 1 + 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 $102.37. 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 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 0.0234 $102.37 = $2.40, ad so the ew price would be approxiately $99.97. 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 $2.40. 9. 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. 360 6 Notice that M2-M1 or D2-D1 ay be egative.
13 Refereces [1] B. R. Barish, ECE 601 Fiacial Egieerig Course Notes, 2017. [2] E. J. Elto. (1999, Sept.). Yield to aturity, accrued iterest, quoted price, ivoice price, [Olie.] Available: http://pages.ster.yu.edu/ eelto/debt_ist_class/ytm.pdf, accessed Feb. 20, 2017. [3] F. J. Fabozzi, Fixed Icoe Securities, New York: Wiley, 2008. [4] FDIC, FDIC Law, Regulatios, Related Acts: 6500 Cosuer Fiacial Protectio Bureau, Appedix A to Part 1030 Aual Percetage Yield Calculatio, [Olie.] Available: https://www.fdic.gov/regulatios/laws/rules/6500-3950.htl# fdic6500appedixatopart1030, accessed Feb. 25, 2017. [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 2008. [6] Math Foru, Calculatig how ay days betwee two dates, [Olie.] Available: http://athforu.org/library/drath/view/66857.htl, accessed Feb. 20, 2017. [7] A. Thau, The Bod Book, 3rd ed. New York: McGraw-Hill, 2011. [8] Wikipedia, Day cout covetio Wikipedia, The Free Ecyclopedia, [Olie.] Available: https://e.wikipedia.org/w/idex.php?title=day_cout_covetio& oldid=758083890, accessed Feb. 20, 2017.
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