= i δ δ s n and PV = a n = 1 v n = 1 e nδ
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1 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 = K Only the orgnal prncpal K earns nterest from year to year, an nterest accumulate n any gven year oes not earn nterest n future years Usually smple nterest s only apple for tme peros of less than a year 42) Suppose the annuty pays X nstea of $1 for n years Then for an annutymmeate, AV = Xs, PV = Xa For the annuty-ue, AV = X s, PV = Xä Know everythng on the Exam 1 revew Below s new materal 43) Contnuous annutes have a δ n the enomnator Hence for a contnuous annuty of $1 pa contnuously n 1 year, the AV = s = (1 + )n 1 = δ δ s an PV = a = 1 v n = 1 e nδ = δ δ δ a Multply the AV an PV by J for a contnuous annuty of $J pa contnuously n 1 year Use J = Km to approxmate Kms (m) an Kma (m) 44) Formulas for annutes assume that the nterest pero an tme pero are the same So a monthly tme pero nees a monthly nterest rate j 45) ä = (1 + )a = 1 + a n 1 46) s = (1 + )s = s n ) Conser n equally space payments of K If the AV s evaluate mmeately after the nth payment or f the PV s evaluate one tme pero before the frst payment, then the annuty s an annuty-mmeate If the AV s evaluate one pero after the last payment or f the PV s evaluate at the tme of the frst payment, then the annuty s an annuty-ue Hence the labels on the tme agram of an annuty can be c, c+1,, c+n where c s an arbtrary nteger Look for whether the AV s evaluate on the ate of the last epost (annuty-mmeate) or one tme pero after the last epost (annuty-ue) 48) An m-year eferre n-payment annuty-mmeate = an (m + 1)-year eferre n-payment annuty-ue s an annuty wth n payments K where the PV s value m + 1 tme peros before the frst payment The tme pero year can be replace by other tme peros If K = 1, the PV = m a = (m+1) ä = v m a = v m+1 ä = a m+ a m Make a tme agram wth labels 0, 1,, m, m + 1,, m + n wth a 1 over the tmes m + 1,, m + n Then the frst payment s eferre untl tme m + 1, the PV evaluate at tme m s a, the PV evaluate at tme m +1 s ä Hence the PV evaluate at tme 0 s v m a = v m+1 ä 49) m a, m ä, etc, s PV notaton Suppose the PV s value at tme 0 Then go to tme m an pay what the symbol to the rght of the vertcal lne says (for m a, pay an annuty-mmeate for n years wth 1s at tmes m + 1,, m+ n)) If the PV s worth J at tme m, then the PV s v m J at tme 0 50) s = (1 + ) n a snce AV (n) = PV a(n) = PV (1 + ) n a = v n s snce v n = (1 + ) n 51) An nfnte pero annuty that results as n s a perpetuty A perpetutymmeate values the PV 1 pero before the frst payment If the payments are 1, then PV = a = a = 1/ = lm a A perpetuty-ue values the PV at the tme of n 1
2 the frst payment wth PV = ä = ä = 1/ = lm ä = (1 + )a = 1 + a f n payments are 1 The AV of a perpetuty oes not exst 52) Another nterpretaton of a perpetuty s that a epost X wll generate K = X nterest at the en of the year So f the nterest payment s pa to the holer of the annuty at the en of the year, then X s left n the account an wll generate K = X nefntely So X = K/ an f X = an = 7%, the annuty wll pay K = an PV = = 70000a 007 = 70000/ 53) An mthly payable annuty-mmeate makes payments of J = K/m at the en of every 1/m year for n years If K = 1, the AV at the tme of the last payment s AV = s (m) = (1 + )n 1 = s (m) (m) If K = 1, the PV one tme pero ((1/m)th year) before the frst payment s PV = a (m) = 1 vn = a (m) (m) Here s the effectve annual nterest rate, an there are a total of nm payments AV (n) = Jms (m) PV = Jma(m) 54) An mthly payable annuty-ue s smlar to 53) except the AV s evaluate one tme pero ((1/m)th year) after the last payment an the PV s evaluate at the tme of the frst payment If K = 1, AV = s (m) = (1 + )n 1 = s (m) = s (m) (m), whle PV = ä (m) = 1 vn = ä (m) = a (m) (m) 55) The AV of n payments of 1 k peros after the nth epost s AV (n + k) = s (1 + ) k = s n+k s k snce AV (n) = s an AV (n + k) = AV (n)(1 + ) k 56) An n+k annuty mmeate has AV mmeately after the fnal payment of s n+k The frst n payments are worth s at tme n an accumulate to s (1 +) k at tme n+ k whle the fnal k payments accumulate to s k at tme n+k Smlarly, the frst k payments are worth s k at tme k an accumulate to s k (1 + ) n at tme n + k whle the fnal n payments accumulate to s at tme n+k Hence s n+k = s k (1+) n +s = s (1+) k +s k 57) Conser an n + k payment annuty-mmeate wth equally space payments of 1 per pero wth nterest rate 1 per payment pero up to the tme of the nth payment followe by a rate of 2 per payment pero from the nth pero on The AV mmeately after the last payment s AV (n + k) = s 1 (1 + 2 ) k + s k 2 snce the frst n payments are worth s 1 at tme n an accumulate to s 1 (1 + 2 ) k at tme n + k whle the fnal k payments are worth s k 2 at tme n + k The PV one pero before the frst payment s PV (0) = a 1 + v n 1 a k 2 snce the frst n payments have PV (0) = a 1 whle the last k payments have PV (n) = a k 2 whch has a PV at tme 0 of v n 1 a k 2 58) Suppose there are n+k regularly space payment peros wth payments of J for the frst n peros an payments of L for the last k peros The AV rght after the fnal payment s AV = J s (1 + ) k + L s k snce the frst n payments have AV (n) = J s whch accumulates to J s (1+) k at tme n+k an the last k payments have AV = L s k at tme n+k If L > J, an equvalent annuty makes n+k payments of J an k payments of L J at tmes n + 1,, n + k Hence AV = Js n+k + (L J) s k 59) a J a = 1 + v n + v 2n + + v (J 1)n 2
3 60) a a p = ä = a (m) ä p a (m) p = ä(m) ä (m) p = a a p 61) Formulas for a an s ten to hol f bars are ae: so ten to hol for a an s See 50), 55) an 56) 62) Conser an annuty-mmeate wth n payments of J Let AV (n) = M = [ (1 + ) n ] 1 J = Js Then M, J, n, an are varables Gven 3 of the varables, t s M ln(1 + J possble to solve for the fourth Then n = ) ln(1 + ) [ 1 v n] Smlarly PV = PV (0) = L = J = Ja, an gven 3 of the varables, t s L ln(1 J possble to solve for the fourth Then n = ) ln(v) Typcally n wll not be an nteger, for example n = Let m = [n] be the nteger part of n so m = [14207] = 14 Then make m = [n] payments of J an then an atonal payment of X to complete the annuty If X s mae at tme [n], then the fnal payment of X + J s known as a balloon payment Then AV = M = Js m + X an PV = L = Ja m + Xv m Make a tme agram to see these formulas An alternatve s to make payment X one tme pero after the tme pero of the fnal payment of J, so at tme m + 1 Then from a tme agram, PV = L = Ja m + Xv m+1 Often use PV snce the PV s a loan mae at tme 0 that s pa off wth the annuty 63) Referrng to 62), solvng for an unknown nterest rate usually nees a fnancal calculator, or teous tral an error for a multple choce problem On a BA II Plus for an annuty mmeate, f the PV = on 18 equally space payments of 1300, enter 18 then press N, enter then press +/, then press PV, then enter 1300, then press PMT, then press CPT then press I/Y (See E1 revew above 7 for why the PV s entere as a negatve number) Shoul get whch s n percent, so the nterest s ) An mthly perpetuty-mmeate uses a (m) = 1 = lm (m) n a(m) An mthly perpetuty-ue uses ä (m) = 1 = lm (m) n ä(m) 65) s (m) = / (m) an a (m) 1 = a = 1 vn (m) (m) 66) Conser an mthly annuty-mmeate that pays J = K/m every 1/mth year for n years The coeffcent on a (m) s the sum of the payments for each nterest pero (so 12 J f the pero s a year wth 12 monthly payments) So the PV s mja (m) Smlarly the PV of an mthly annuty-ue that pays J = K/m every 1/mth year for n years s PV = mjä (m) So an annuty that pays 100 at the en of each month for 10 years at an effectve annual nterest rate of 5% has PV = 1200a (12) wth J = 100 an m = ) Suppose an annuty pays 1 at tmes 1,, n, but the accumulaton functon s a(t) where nterest vares Want AV at tme t = n Now 1 at tme t = j has AV = a(n)/a(j) at tme n See[ E1 revew 19) Hence the annuty AV mmeately after the fnal payment 1 s AV = a(n) a(1) + 1 a(2) ] a(n) 3
4 68) A common problem uses the AV 1 (n) of one fun at tme n to buy a secon fun that makes k future payments an has a PV 2 (n) at tme n Usually set AV 1 (n) = PV 2 (n) an solve for an unknown See HW4: 3, 4 an HW5: 4 69) Conser an n payment annuty wth an arthmetc progresson as shown below P P+Q P+2Q P+(j-1)Q P+(n-1)Q j n So P s the frst payment an Q s the common fference [ Then the PV one tme a nv n ] pero before the frst payment s PV = A A = Pa + Q, an the AV [ ] mmeately after the last payment of AV = S A = (1 + ) n s n A A = Ps + Q 70) Conser an n payment annuty wth an arthmetc progresson as shown below P P+Q P+2Q P+(j-1)Q P+(n-1)Q j n-1 n Then the PV evaluate on the ate of the frst payment s PV = ÄA = (1 + )A A = [ a nv n ] Pä + Q, an the AV one pero after the last payment s AV = S A = [ ] (1 + ) n s n Ä A = P s + Q 71) An ncreasng annuty-mmeate has P = Q = 1 Then 69) can be shown to have PV one payment before the frst payment of PV = (Ia) = ä nv n an an AV at the tme of the fnal payment of PV = (Is) = s n = s n+1 (n + 1) 72) An ncreasng annuty-ue replaces by n the enomnator of the RHS So the PV at the tme of the frst payment s PV = (Iä) = ä nv n an an AV one tme pero after the fnal payment of PV = (I s) = s n 73) A ecreasng annuty-mmeate has P = n an Q = 1, so the n payments are n, n 1, n 2,, 3, 2, 1 The PV one tme pero before the frst payment s PV = (Da) = n a, an the AV mmeately after the fnal payment s AV = (Ds) = n(1 + ) n s = (1 + ) n (Da) 74) The ue form has AV an PV 1 pero after the mmeate form, so AV ue = (1 + )AV mmeate an PV ue = (1 + )PV mmeate Also AV mmeate = vav ue an PV mmeate = vpv ue For both forms, AV = (1 + ) n PV an PV = v n AV 75) An ncreasng perpetuty-mmeate wth payments 1, 2, 3, at tmes 1, 2, 3, has PV at the tme before the frst payment of PV = (Ia) = 1 =
5 76) An ncreasng perpetuty-ue wth payments 1, 2, 3, at tmes 0, 1, 2, 3, has PV at the tme of the frst payment of PV = (Iä) = 1 2 = lm n (Iä) 77) Although the general P an Q forms 69) an 70) can be use to evaluate an annuty n arthmetc progresson, such an annuty can be wrtten as a combnaton of a ecreasng or ncreasng annuty an a level payment annuty P P+Q P+2Q P+(j-1)Q P+(n-1)Q j n 78) ) Conser PV an suppose Q > 0 Then wrte own Q(Ia) ) The annuty n ) pays Q, Q+Q,, Q+(n 1)Q So a level payments of P Q, so the other term n the combnaton s (P Q)a Hence the PV of the annuty s Q(Ia) + (P Q)a 79) ) Conser PV an suppose Q < 0 Then wrte own Q (Da) ) The annuty n ) pays Q, (n 1) Q,, Q So a level payments of P Q, so the other term n the combnaton s (P Q )a Hence the PV of the annuty s Q (Da) + (P Q )a 80) For AV, replace a by s for the mmeate form So use s an ether (Is) or (Ds) So the AV n 78) s Q(Is) + (P Q)s whle the AV n 79) s Q (Ds) + (P Q )s Smlar formulas work for the PV an AV of the ue form but wth ä replacng a an s replacng s 81) Suppose an ncreasng perpetuty-mmeate has frst payment P an ncreases by Q thereafter Then PV = P + Q n n n n n+1 n+2 82) Conser a perpetuty-mmeate wth payments of 1, 2,, n at the en of each year an then payments of n for each subsequent year Then the PV of the frst n payments one tme pero before the frst payment s (Ia) whle the PV of the remanng payments evaluate at tme n s na whch has PV at tme 0 of v n na Hence the PV of the perpetuty-mmeate one tme pero before the frst payment s PV = (Ia) + v nn = ä 83) Suppose a lener makes a loan of L at nterest rate an nvests the payments K t at nterest rate If F s the AV for the lener at tme n years usng nterest rate, ( ) F 1/n then L(1 + j) n = F an j = 1 s the annual yel rate for the lener L 5
6 84) Conser an annuty that makes a cost of lvng ajustment (COLA) so that there are n payments, the frst payment s 1, an all subsequent payments are (1 + r) tmes the prevous payment Ths annuty s n geometrc progresson wth payments 1, (1 + r), (1 + r) 2,, (1 + r) n 1 wth PV before the frst payment of PV = 1 (1+r 1+ )n r an an AV mmeately after the last payment of AV = PV (1+) n = (1 + )n (1 + r) n r 85) There are many varants of the problem of fnng the AV of nvestments an renveste nterest Use tme agrams of nvestments an of renveste nterest Three examples are a) 83) b) Suppose L s nveste an generates nterest at rate per tme pero whch s renveste at rate j Then L generates nterest L per pero Hence the AV mmeately after the last nterest payment conssts of L an the renveste nterest wth AV = L + Ls j L L L L n n payments nterest 2 3 (n-1) c) Suppose n eposts of 1 generate nterest at rate per tme pero whch are renveste at rate j Then at tme 2 nterest s nveste, at tme 3 nterest on the frst two payments of 1 s 2 et cetera as shown n the above rght tme agram Then the AV mmeately after the nth payment s AV = n + (Is) n 1 j where n s the value of the n payments of 1 an the 2n term s the AV of the renveste nterest 86) Amortzng a loan reuces the outstanng balance of a loan by makng payments that pay nterest an reuce the prncpal Let OB t = B t be the outstanng balance mmeately after the tth payment K t = R t Let I t be the nterest pa at the en of pero t Let PR t = P t be the prncpal repayment at the en of pero t K1 K2Kt K(t+1) Kn K K K K K t t+1 n t t+1 n L OBt L OBt The retrospectve metho says OB t = L(1 + ) t K 1 (1 + ) t 1 K 2 (1 + ) t 2 K t 1 (1+) K t = AV of loan L at tme t AV of the 1st t payments mmeately after tth payment If K t K, then OB t = L(1 + ) t Ks t The prospectve metho says OB t = K t+1 v + K t+2 v K n v n t = PV(t) of the remanng payments mmeately after the tth payment If K t K, then OB t = Ka n t 87) L = K 1 v + K 2 v K n v n If K t K, then L = Ka 88) I t = OB t 1, PR t = K t I t, OB t = OB t 1 PR t, OB 0 = L If K t K, then K = L/a, I t = K(1 v n t+1 ), PR t = Kv n t+1 = PR t 1 (1 + ) = PR 1 (1 + ) t 1 89) The retrospectve metho can be better f L s gven but you nee to fgure out n an the last payment K n The prospectve metho can be better f L s not known 6
7 uraton payment nterest pa prncpal repa outstanng balance t K t I t = ()OB t 1 PR t = K t I t OB t = OB t 1 PR t 0 OB 0 = L 1 K 1 I 1 = ()OB 0 PR 1 = K 1 I 1 OB 1 = OB 0 PR 1 t K t I t = ()OB t 1 PR t = K t I t OB t = OB t 1 PR t n K n I n = ()OB n 1 PR n = K n I n OB n = OB n 1 PR n = 0 total nj=1 K nj=1 j I nj=1 j PR j = L 90) the above table shows an amortzaton scheule Note that the sum of the prncpal repayments s equal to the loan amount L So n j=1 PR j = L Also the outstanng balance at tme n s 0, an L + n j=1 I j = n j=1 (I j + PR j ) = n j=1 K j Also OB t = OB t 1 (1 + ) K t = OB t 1 + I t K t = OB t 1 PR t 91) If payments K t K, then the sum of payments s nk, an the sum of prncpal repayments s n j=1 PR j = Ka = L Also n j=1 I j = nk L 92) A fun esgne to accumulate a specfe amount of money n a specfe amount of tme by makng regular eposts s a snkng fun (SF) There s a) nterest rate on the loan b) nterest rate on the snkng fun j c) a peroc nterest payment I t = L ) a peroc snkng fun epost (SFD) such that L = SFDs j so SFD = L/s j Hence the SF accumulates to L at tme t = n e) Note that the total peroc payment s J = L + SFD, the sum of the quanttes foun n c) an ) 93) You may nee to fn the quanttes n 92) as well as the payment K for an amortze loan that woul pay off the loan L n 92) Then K = L/a See HW7 4 94) If = j then the snkng fun approach an the amortzaton approach are equvalent n that the peroc payments J an K n 92) an 93) satsfy J = K, but often > j an the lener makes more money uner the snkng fun approach Techncally the borrower owns the snkng fun untl tme n, but often the snkng fun s hel by a fnancal nsttute lke a bank Typcally the nterest the borrower can make on the SF satsfes j < 1 95) The result n 94) mples that = + 1 a s 96) The general approach to SF problems s to answer two questons: ) what nterest s pa to the lener (fn L), an ) what s the SFD (= L/s j )? Then the total peroc payment mae by the the borrower to repay the loan s J = L + L [ = L + 1 ] s j s j 97) SF wth nonlevel SFDs: suppose the snkng fun has nterest rates an j, the total annual payment s J the 1st n years an K the next m years The SF accumulates to L at tme n + m, so (J L)s n+m j + (K J L)s m j = L Note that the nterest payment s L an the SFD s J L for years 1,, n an K L for years n+1,, n+m 7
8 K-J-L K-J-L J-L J-L K-L K- L J-L J-L J-L 0 1 n n+1 n+m 0 1 n n+1 n+m Ch 4 98) A bon makes peroc nterest payments Fr for peros 1,, n as well as a payment C at tme n where C s the reempton value of the bon an r s the coupon rate of nterest per tme pero F s the face value of the bon, an = j s the yel rate of nterest per tme pero Note that n s the number of payments Fr Fr Fr Fr+C n-1 n 99) The bon prce P s the PV of future cash flows The coupon rate r s fxe for the lfe of the bon Bons are sol n a bon market an the yel rate j s etermne by market forces Typcally nterest payments are mae twce a year so n = 2Y where Y s the number of years of the bon, but other payment peros are possble When the payment pero s 6 months, often two bon nterest rates r (m) an j (m) are gven as nomnal annual nterest rates compoune semannually where m = 2 Then r = r (2) /2 an j = j (2) /2 In general you may nee to convert a gven nterest rate to the nterest ( ) m1 ( ) m2 rate for the tme pero Note that 1 + (m 1) = 1 + (m 2) = 1 + where s the annual nterest rate Hence f m 1 < m 2 an m 2 /m 1 s a postve nteger, then ( ) ( ) 1 + (m 1) = 1 + (m 2) m2 /m 1 m 1 m 2 100) Let g be the coupon rate apple to C to etermne the amount of the coupon = peroc nterest payment Then Cg = F r an g = F r/c The basc formula for the bon prce s P = Fra j + Cv n j = Cga j + Cv n j The premum scount formula s P = C + (Fr Cj)a j = C + (Cg Cj)a j Let K = Cv n j, then Makeham s formula s P = g j (C Cvn j ) + Cvj n = g j (C K) + K = K + g (C K) See HW7 5 j 101) If F = C then r = g, an the bon s reeemable at par If F = C an r = j, then P = F If g = j, then P = C If g > j then P > C, an the bon s reeemable at a premum wth the premum = P C = (Cg Cj)a j If g < j then P < C an the bon s reeemable at a scount wth the scount = C P = (Cg Cj)a j = (Cj Cg)a j See the premum scount formula n 100) Note that both the premum an the scount are postve To etermne whether the bon s reeemable at a premum or scount, compute g an see whether g > j or g < j If the scount s D = C P then the prce s P = C D If the premum s E = P C, then the prce s P = E + C 102) If the bon s reeemable at par then F = C If no nformaton s gven, then assume F = C an that the nterest payment pero s 6 months Sometmes you are tol that the bon s reeemable at 950 or reeemable at 95% of the face value (so C = 095F) 8 m 1 m 2
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