Exam FM Formula Summary Version no driver 11/14/2006

Transcription

1 Exam FM Formula Summary Verso 2.01 o rver 11/14/2006 Itroucto Sce ASM oes ot have a formula summary, I ece to comple oe to use as I starte workg o ol test questos. I the terest of other actuaral stuets, I thought I woul share the results. A few otes: 1. Ths set of formulas s mostly erve from the 3r eto of the ASM maual for Exam FM/2. As a referece, t oes ot attempt to recreate the methos presete the ASM maual a skps may of the ecessary techques for usg these formulas to solve certa types of problems. I partcular you wll otce that there are o formulas from chapters 2 a 8, a very lttle from chapter Sce the syllabus for the exam wll chage after the November 2006 sttg, ths complato wll ot be complete for exams gve 2007 a beyo, but t ca probably be use as a startg pot for future exam takers. 3. I may have msstate some of the explaatos of the formulas ether through lack of uerstag or aequate keyboar/tex sklls. Please let me kow f you f errors ths ocumet a I wll attempt to correct them. Also ote that some formulas have o explaato, a are tee to show ettes a useful relatoshps betwee terms that have bee efe prevously. 4. Ths summary s meat as a referece. You o t ee to memorze all of these formulas to o well o the exam. I fact, most of them ca be easly erve from oe aother. As you work problems, some of these formulas wll become seco ature. For some of the problems where these formulas may work, you may prefer workg from frst prcples or a termeate ervato. Mykek has suggeste that you oly ee to kow fve formulas for the 2006 exam: Arthmetcally creasg & ecreasg auty, geometrcally creasg auty, prcple repa at tme t, a the prce of a bo. As you lear the materal you wll fgure out what works for you. 1

2 Chapter 1 Bascs: a (t) : accumulato fucto. Measures the amout a fu wth a vestmet of 1 at tme 0 at the e of year t. a (t) a (t 1) : amout of growth year t. t = a(t) a(t 1) a(t 1) : rate of growth year t, also kow as the effectve rate of terest year t. A (t) = ka (t) : ay accumulato fucto ca be multple by a costat (usually the prcpal amout veste) to obta a result specfc to the amout veste. Commo Accumulato Fuctos: a (t) = 1 + t : smple terest. t a (t) = (1 + j ) : varable terest. j=1 a (t) = (1 + ) t : compou terest. Preset Value a Dscoutg: P V = 1 a(t) = 1 (1+) t = (1 + ) t = v t : amout you must vest at tme 0 to get 1 at tme t. t = a(t) a(t 1) a(t) : effectve rate of scout year t. Some Useful Relatoshps: 1 = v = = 1+ = v 1 Nomal Iterest a Dscout: (m) a (m) are the symbols for omal rates of terest compoue m-thly. ( ) m 1 + = 1 + (m) m 2

3 ( ) (m) = m (1 + ) 1 m 1 ( ) m 1 = 1 (m) m ( ) (m) = m 1 (1 ) 1 m Force of Iterest: δ t = 1 a(t) a (t) = e R t 0 δrr t a (t) = tla (t) : efto of force of terest. If the Force of Iterest s Costat: a (t) = e δt P V = e δt δ = l (1 + ) Chapter 3: Autes: a = 1 v = v + v v : PV of a auty-mmeate. ä = 1 v = 1 + v + v v 1 : PV of a auty-ue. ä = (1 + ) a = 1 + a 1 s = (1+) 1 = (1 + ) 1 + (1 + ) : AV of a auty-mmeate (o the ate of the last epost). s = (1+) 1 = (1 + ) + (1 + ) (1 + ) : AV of a auty-ue (oe pero after the ate of the last epost). s = (1 + ) s = s +1 1 a m = a + v a + v 2 a + + v (m 1) a 3

4 Perpetutes: lm a 1 v = lm = 1 = v + v2 + = a : PV of a perpetuty-mmeate. 1 v lm = lm = 1 ä = 1 + v + v2 + = ä : PV of a perpetuty-ue. ä a = 1 1 = 1 Chapter 4: m-thly Autes & Perpetutes: a (m) ä (m) s (m) s (m) lm a(m) = 1 v (m) = (m) a = s (m) 1 a : PV of a -year auty-mmeate of 1 per year payable m-thly stallmets. = 1 v (m) = (m) a = s (m) 1 a : PV of a -year auty-ue of 1 per year payable m-thly stallmets. = (1+) 1 (m) = (1+) 1 (m) : AV of a -year auty-mmeate of 1 per year payable m-thly stallmets. : AV of a -year auty-ue of 1 per year payable m-thly stallmets. 1 v = lm = 1 (m) stallmets. : PV of a perpetuty-ue of 1 per year payable m-thly stall- ä(m) 1 v lm = lm = 1 (m) mets. ä (m) a (m) = 1 1 = 1 (m) (m) m = a(m) (m) = ä(m) (m) : PV of a perpetuty-mmeate of 1 per year payable m-thly Cotuous Autes: Sce lm m a(m) lm m (m) = lm m (m) = δ, = lm m 1 v (m) pa cotuously. = 1 v δ = a = δ a : PV of a auty (mmeate or ue) of 1 per year Paymets Arthmetc Progresso: I geeral, the PV of a seres of paymets, where the frst paymet s P a each atoal paymet creases by Q ca be represete by: a v A = P a + Q = P v + (P + Q) v 2 + (P + 2Q) v (P + ( 1) Q) v 4

5 Smlarly: a v Ä = P ä + Q s S = P s + Q : AV of a seres of paymets, where the frst paymet s P a each atoal paymet creases by Q. s S = P s + Q (Ia) = (Is) = ä v : PV of a auty-mmeate wth frst paymet 1 a each atoal paymet creasg by 1; substtute for eomator to get ue form. s : AV of a auty-mmeate wth frst paymet 1 a each atoal paymet creasg by 1; substtute for eomator to get ue form. (Da) = a : PV of a auty-mmeate wth frst paymet a each atoal paymet ecreasg by 1; substtute for eomator to get ue form. (Ds) = (1+) s : AV of a auty-mmeate wth frst paymet a each atoal paymet ecreasg by 1; substtute for eomator to get ue form. (Ia) = 1 = creasg by 1. : PV of a perpetuty-mmeate wth frst paymet 1 a each atoal paymet (Iä) = : PV of a perpetuty-ue wth frst paymet 1 a each atoal paymet creasg by (Ia) + (Da) = ( + 1) a Atoal Useful Results: P + Q 2 (Ia) (m) : PV of a perpetuty-mmeate wth frst paymet P a each atoal paymet creasg by Q. ä v = : PV of a auty-mmeate wth m-thly paymets of 1 (m) m the frst year a each atoal year creasg utl there are m-thly paymets of m the th year. May Go Have Mercy o Your Soul: ( I (m) a ) (m) = ä(m) v 1 : PV of a auty-mmeate wth paymets of (m) m 2 of the frst year, 2 m 2 creasg utl there s a paymet of m m 2 at the e of the frst mth at the e of the seco mth of the frst year, a each atoal paymet at the e of the last mth of the th year. ( Ia a v = ) δ : PV of a auty wth cotuous paymets that are cotuously creasg. Aual rate of paymet s t at tme t. 5

6 0 0 f (t) v t t : PV of a auty wth a cotuously varable rate of paymets a a costat terest rate. f (t) e R t 0 δrr t : PV of a auty wth a cotuously varable rate of paymet a a cotuously varable rate of terest. Paymets Geometrc Progresso: 1 ( 1+k 1+ ) k : PV of a auty-mmeate wth a tal paymet of 1 a each atoal paymet creasg by a factor of (1 + k). Chapter 5: Deftos: R t : paymet at tme t. A egatve value s a vestmet a a postve value s a retur. P () = v t R t : PV of a cash flow at terest rate. Chapter 6: Geeral Deftos: R t = I t + P t : paymet mae at the e of year t, splt to the terest I t a the prcple repa P t. I t = B t 1 : terest pa at the e of year t. P t = R t I t = (1 + ) P t 1 + (R t R t 1 ) : prcple repa at the e of year t. B t = B t 1 P t : balace remag at the e of year t, just after paymet s mae. O a Loa Beg Pa wth Level Paymets: I t = 1 v t+1 : terest pa at the e of year t o a loa of a. P t = v t+1 : prcple repa at the e of year t o a loa of a. B t = a t : balace remag at the e of year t o a loa of a, just after paymet s mae. 6

7 For a loa of L, level paymets of L a B t by L a, e B t = L a a t etc. wll pay off the loa years. I ths case, multply I t, P t, a Skg Fus: P MT = L + L s j to the leer a : total yearly paymet wth the skg fu metho, where L s the terest pa L s j s the epost to the skg fu that wll accumulate to L years. s the terest rate for the loa a j s the terest rate that the skg fu ears. L = (P MT L) s j Chapter 7: Deftos: P : Prce pa for a bo. F : Par/face value of a bo. C : Reempto value of a bo. r : coupo rate for a bo. g = F r C : mofe coupo rate. : yel rate o a bo. K : PV of C. : umber of coupo paymets. G = F r : base amout of a bo. F r = Cg Determato of Bo Prces: P = F ra + Cv = Cga + Cv : prce pa for a bo to yel. P = C + (F r C) a = C + (Cg C) a : Premum/Dscout formula for the prce of a bo. P C = (F r C) a = (Cg C) a : premum pa for a bo f g >. C P = (C F r) a = (C Cg) a : scout pa for a bo f g <. 7

8 Bo Amortzato: Whe a bo s purchase at a premum or scout the fferece betwee the prce pa a the reempto value ca be amortze over the remag term of the bo. Usg the terms from chapter 6: R t : coupo paymet. I t = B t 1 : terest eare from the coupo paymet. P t = R t I t = (F r C) v t+1 = (Cg C) v t+1 ( wrte ow ) or P t = I t R t = (C F r) v t+1 = (C Cg) v t+1 ( wrte up ). : ajustmet amout for amortzato of premum : ajustmet amout for accumulato of scout B t = B t 1 P t : book value of bo after ajustmet from the most recet coupo pa. Prce Betwee Coupo Dates: For a bo sol at tme k after the coupo paymet at tme t a before the coupo paymet at tme t + 1: B f t+k = B t (1 + ) k = (B t+1 + F r) v 1 k has o the sale of the bo. : flat prce of the bo, e the moey that actually exchages B m t+k = Bf t+k kf r = B t (1 + ) k kf r : market prce of the bo, e the prce quote a facal ewspaper. Approxmatos of Yel Rates o a Bo: F r+c P 2 (P +C) : Bo Salesma s Metho. Prce of Other Securtes: P = F r : prce of a perpetual bo or preferre stock. P = D k : theoretcal prce of a stock that s expecte to retur a ve of D wth each subsequet ve creasg by (1 + k), k <. 8

9 Chapter 9: Recogto of Iflato: = r 1+r : real rate of terest, where s the effectve rate of terest a r s the rate of flato. Metho of Equate Tme a (Macauley) Durato: t = = tr t t=1 t=1 R t tv t R t t=1 v t R t t=1 : metho of equate tme. : (Macauley) urato. Volatlty a Mofe Durato: P () = v t R t : PV of a cash flow at terest rate. v = P () P () = v = 1+ : volatlty/mofe urato. = (1 + ) P () P () : alterate efto of (Macauley) urato. Covexty a (Regto) Immuzato: c = P () P () : covexty To acheve Regto mmuzato we wat: 1. P () = 0 2. P () > 0 9