Transcription of MATH419: Actuarial Science. Exam-FM Formulas
1 math419 : Actuarial FormulasInterest:sum of geometric seriesSn=a(1 rn)/(1 r) Compound:A(t) =A(0)(1 +i)t=A(0)(1 d) tSimple:A(t) =A(0)(1 +it) v=11+idiscountd= 1 force of interest = ln(1 +i). varying force of interest (t) =dA/dtA(t).separate and integrateA(t) =A(0)e t0 (s)ds. interest earned fromatob=A(b) A(a).Xdeposited ataaccumulated tillbisA(b) =Xe ba (s)dsLevel Annuities:5-button formulaPV=PMTan+Fvn PV immediatean=1 vniPV due an= (1 +i)ancontinuously paidan=an(i ) FVsn= (1 +i)nan=(1+i)n 1i sn=(1+i)n 1dperpetuitya =1i a =1d a(m)nmeansmpayments per year (12)nominal meansi(12)12interest per monthVarying Annuities:CF button, to enter PMTs and frequency. geometric: increasee% per payment, calculate new interest rate11+j=1+e1+i. arithmetic: initP, increaseQ:PV=Pan+Qi(an nvn)Qis negative for be payable - multiply by(i ) ctsly compounding, ctsly payablef(t):PV= n0f(t)vtdt varying force of interest (t):PV= n0f(t)e t0 (r)drdt FV= n0f(t)e nt (r)drdtLoans:AMORT button after entering info into 5-buttons Lis principle,OBtis outstanding balance just after payment att, Itis interest intth payment,Ptis principle repaidtth +It= 1.
2 Prospective:OBt=PMTan t, present value of remaining t+1, retrospective:OBt=L(1 +i)t PMTst, FVloan - FVpayments (1 +i)t 1(PMT Li)Bonds:F= par = face,C= redemption amount,r= coupon rate,i= yield rate. bond pricePV=Fran+Cvn, book value is outstanding balance write down is principal repaid:Pt= (Fr Ci)vn t+1, amortization of bond. premuim= & IRR:CF, NPV, IRR (finds solution closest to zero only). IRR is rate at which PV of flows equals 0, interest rate = cost of capital dollar-weighted:simple interest rate that must have been in effect. solve fori. time-weighted:(b/a)(c/b)(d/c) = 1 +iwhereagrew tob,bgrew tocetc. solve fori. investment year:interest rate depends on when deposited (row). portfolio method:interest rate depends on current year (column). new money rate:investment year rate for money deposited this Rates:(1 +st 1)t 1(1 +ft) = (1 +st) rate:strate for termtstarting at 0.
3 Forward rate:fa,brate for term starting ataand ending 1,t. modified durationDM= dP/diP, equalst/(1 +i) for constantiand termt. duration (Macaulay)D= (1 +i)DM, equalstfor constantiand tPVt PVt asset-liability matching:Asset income equals Liability due at allt. Redington immunization:PVA=PVLati0andPVA> PVLforineari0. duration of assets = duration of liabilitiesdPVAdi=dPVLdi, and convexity of assets>convexity of liabilitiesd2 PVAdi2>d2 PVLdi2. full immunization:Asset income greater than or equal to Liability due for : Derivatives:value determined by price of something :seller. insurance is risk-sharing. Insurance firms use reinsurance to share risk of extreme events. diversifiablerisk is unrelated to other risks and can be does not vanish when shared (it already affects everyone). bid:price can sell at,ask:price can buy for.
4 You always pay more than you get soask> : Forwards and Options:call:right to buy,put:right to sell,forward:obligation. European:exercise at :exercise :exercise specified times between. Option profit = payoff - FV(option price). Options are insurance, strike = : Insurance and Collars:Put-Call ParityC P=FP e rtK prepaid forward priceFP: current price less the PV of dividends. forward priceF: FV of prepaid forward : Hedging reasons to hedge:risk-aversion, distress costs, costly external financing, increase debt capacity, tax. reasons NOT to hedge:transaction costs, bid/ask spread, needs more expertise, regulating, : Forwards and futures cost-of-carry:r , cost of holding long position. futures:-mark-to-market:settled daily so no money is owed. When asset looses value, buyer pays :deposit from both buyer/seller left with broker when buying future, from which dailylosses can be taken.
5 It does earn margin:minimum proportion of the initial margin that must be maintainedthroughout the contract period-on S& P 500:only sold in bundles of 250Ch8: Swapssettles throughout the term. like a set of forwards. prepaid commodity swap:single payment at time 0 equivalent to varying payments commodity swap:swap price is the level paymentXequivalent to varying paymentsXiX1(1 +i1)+X2(1 +i2)2+X3(1 +i3)3+ =X(1 +i1)+X(1 +i2)2+X(1 +i3)3+.. interest rate swap:fixed rateRequivalent to varying rates, wherefiis the forward rate for (1 +i1)+f2(1 +i2)2+f3(1 +i3)3+ =R(1 +i1)+R(1 +i2)2+R(1 +i3)3+.. interest rate swap payment:difference between actual interest payment due and the interest dueaccording to the swap.
