Exam FM/2 Interest Theory Formulas

1 exam fm /2 Interest Theory Formulas by (/iropracy This is a collaboration of Formulas for the Interest Theory section of the SOA exam fm / CAS Exam 2. This study sheet is a free non-copyrighted document for students taking exam fm /2. The author of this study sheet is using some notation that is unique so that no designation will repeat. Each designation has only one meaning throughout the sheet. Fundamentals of Interest Theory and Time Value of Money ()niPVFV+=1 ()niFVPV+=1 ()iid+=1 vd =1iddi= ()iv+=11 dv =1ivd= ()ta The amount an initial investment of 1 grows to by time t ()tA The amount an initial investment of ()0A grows to by time t () ( )()ilntteita+ =+=11 ()()()()()ilntteAiAtA+ =+=1 01 0 ) ()t eta = (iln +=1 () nnneiv =+=1 ()()()tatat = ()()taeduu t= 0 ()()() 00 tAeAduu t= Effective Interest rate with nominal rate ()mi convertible m-thly ()11 +=mmmii Effective discount rate with nominal rate ()pd convertible p-thly ()pppdd = 11 Nominal Rate Equivalence ()()ppmm pdmidvei = += ===+111111 Effective annual rateduring the t-th year is given by.)

2 Ti ()()()()()()1111 = ==tAtAtAtatataamountbeginning earnedamountit Note that the t-th year is given by the time period []t,t1 Therefore, the Interest earned during the t-th year is given by: ()()()11 = tAtAitA For equivalent measures of Interest we have the following relationship: ()()()()iii ddd<<<<<<<<2332"" Annuities Annuity Immediate payments are made at the end of the period Annuity Due payments are made at the beginning of the period Annuity Immediate () ()()iiiisnnni|n1111121 +=+++++= " ivvvvanni|n =+++=12 " i|nni|nsva = ()i|nni|nais 1 += Annuity Due ()() ()()diiiisnnni|n111111 +=++++++= " dvvvanni|n =+++= 111 " i|nni|nsva = ()i|nni|nais 1 += Identities for Annuity Immediate and Annuity Due ()|n|n|naiadia 1+== ()|n|n|nsisdis 1+== |n|naa 1 1 += 1 1 =+|n|nss Perpetuity ivvvaai|nni|1lim32 =+++== " daai|nni|1lim == Continuous Annuities i|nni|na i va 1= = ()i|nni|ns i is 11= += =ntindtva0 | ()() = nduu dttpePVt0 0 where ()() 0ntn uduFVep t dt = ()tp = payment function Increasing Annuities Payments are 1, 2.

3 , n ()invaIan|ni|n = ()()()()dnvaIaiIadiaIn|ni|ni|ni|n =+== 1 ()()()insIaiIs|ni|nni|n =+= 1 ()()()()dnsaIiIsdisI|ni|nni|ni|n =+== 1 ()()2 111limiidiIaIai|nni|+=== ()()2 1limdaIaIi|nni|== Decreasing Annuities Payments are n, n-1,.., 2, 1 ()ianDai|ni|n = ()()()()danDaiDadiaDi|ni|ni|ni|n 1 =+== ()()()()isinDaiDsinninnin | | | 11 +=+= ()()()i|nni|naDisD 1 += Present Value of the annuity with terms ()YnXYXYXX1 ,,2 , , +++.. + invaYaXnnin| | Present Value of the perpetuity with terms ..,2 , ,YXYXX++ 2iYiX+ Annuities with Terms in Geometric Progression ()()()121 , ,1 ,1 ,1 +++ Present Value is () () ()()()qivqvqvqvqvVnnnn + = +++ ++ ++ = 111 1 1 101322" Useful Identities |kn|n|knavaa +=+ ()|n|mmnaaivv = ()()()|n|n|nanIaDa 1+=+ |nnaiv 1+= nnnnnnnvvvaaaa+= ==1112| | 2| | 2 ()()()1111112| | 2| | 2++= + +==nnnnnnniiissss If the Interest rate varies: ()()()naaaa|n12111 +++=" ()()()()()()nanaanaanas|n+++="21 If the compounding frequency of the Interest exceeds the payment frequency of k years Use an equivalent Interest rate over k years: ()11 +=kij If the payment frequency exceeds the compounding frequency of the Interest (1) Use an m-thly annuity ()()|nmm|naiia = ()()|nmm|nsiis = ()()|nmm|nadda = ()()| | nmmnsdds= (2) Use an equivalent Interest rate effective over the payment period.

4 ()111 +=mij ()j|nmi|naa = ()j|nmi|nss = ()j|nmi|naa = ()j|nmi|nss = If the payments are mn,,m,m 2 , then the present value is ()()()mni|nmi|ninvaIa = If the payments are 222 2 1mn,,m, , then the present value is ()()()()()mnmi|nmi|nminvaaI = Loan Repayment Amortization Amortization Method when a payment is made, it must be first applied to pay Interest due and then any remaining part of the payment is applied to pay principle Notation L amount of the loan n number of payment periods AP amount of level payment at the end of the period (amortized payment) ()kP loan payment at time k i effective Interest rate per payment period kB balance at time k, balance after k-th payment. Note that LB=0 kP principle paid in payment ()kP kI Interest paid in payment ()kP Useful Equations for Level Payments inAaPL | = inAaLP | = Prospective Method iknAkaPB | = Retrospective Method ()ikAkksPiLB | 1 += (tktkiBB+=+1) and ()tktkiPP+=+1 kkAIPP+= ()111+ = =knAkkvPBiI 1+ = =knAkAkvPIPP Useful Equations for Non-Level Payments ()( )( )nnvPvPvPL+++="221 () ( )()()()kkknnkkkPiBvPvPvPB +=+++= ++11221" 1 =kkBiI ()kkkkkBBIPP = = 1 Loan Repayment Sinking Fund Sinking Fund Loan (SFL) accumulate money in a separate fund by making a payment, in addition to the regular Interest payment, every period.

5 Notation L amount of the loan n number of payment periods i effective Interest rate per payment period by the borrower to the lender j effective Interest rate earned by the borrower in the sinking fund SD periodic sinking fund deposit (SFD), assumed to be level SP periodic outlay by the borrower = Interest payment to lender + SFD kS sinking fund balance after k-th deposit kL net loan balance at time k Useful Equations j|nSsDL = j|nSsLD = j|nSSsLLiDLiP +=+= j|nj|kj|kSkssLsDS = = j|kSksDLL = Net Principal Paid ()1 1 11 += = kSj|kSj|kSkkjDsDsDSS Net Interest Paid j|kSksjDLijSLi 11 = Notes on Loans Amortized Loan over time Interest paid decreases and principal paid increases SFL for each outlay Interest paid to lender is constant Installment Loan over time Interest paid decreases while the principal paid is constant Bonds Bonds Interest bearing securities.

6 Basically loans from lenders perspective Callable Bond a bond that can be paid off (called) before maturity Notation F par value r coupon rate ( Interest rate of bond) Fr coupon amount (payment to lender) C redemption value (usually = F ) n number of coupon periods to maturity P market price of the bond kBV book value of the bond (bond amortized balance after k-th payment) i yield per period to investor at price P ivi+=11 niCvK= Present value of the redemption value CFrg= modified coupon rate Premium If ri> then the bond is priced at a premium. , and is the CP>CP amount of the premium. Premium()inaiCFrCP | = ()()()()iknikkknkkksaPiivvvPCP | 1 | 121111+ +=+++++++++= "" Discount If ri< then the bond is priced at a discount. CP<, and is the PC amount of the discount Discount()inaFriCPC | = Par If ri=the bond is selling at the price CP=we say that it sells at par.

7 Price and Premium-Discount Formula KFraPin+= | ()()inaigCP | 1 += if CF=, then ()()inairFP | 1 += Bond Amortized kniiknkCvFraBV += | ()imkmkmksFriBVBV | 1 += kkPIFr+= ()111 1+ + + = =knknkkviCvFrBViI 11 + + =knknkviCvFrP If , then CF=(tktkiPP+=+1) Write-Up during the first k years (Discount) PBVk Write-Down during the first k years (Premium) kBVP Write-Up/Write-Down in general during time m to time k,()mk> mkBVBV ()1nkkWDFr iC v += ()1nkkWUiC Fr v += Makeham s Formula ()KCigKP += if CF=, then ()KFirKP += Maturity to use in Pricing a Callable Bond Type of Bond Take N Premium Bond Earliest Possible Redemption Date Discount Bond Latest Possible Redemption Date Price Between Payment Dates period bond in the days ofnumber date settlement todatecoupon last from days ofnumber =t Price Plus Accrued Accrued Interest ()tiP+10 ()Frt P ()()FrtiPt += 1 Interest Accrued Accrued Plus Price0 Yield Rate of an Investment Internal Rate of Return (IRR) the rate of Interest at which the present value of all amounts invested is equal to the present value of all the amounts paid back to the investor Internal Rate of Return (IRR) Given investment cash flows , the IRR is a solution for i of the equation nCCCC,,,, ()()()01112210=+++++++ or 02210=++++ Time Weighted Rates of Interest (TWR)

8 KC Contribution at time kt kB Fund value at time before the contribution ktkC is made kj Effective rate over []kktt,1 111 + =+kkkkCBBj TWR ()()()mjjji+ ++=+111121" Dollar Weighted Rates of Interest (DWR) A Initial fund balance B Final fund balance I Interest earned tC Contribution or withdrawal at time t (cash flows) NetC Net contribution =tNetCC ICABNet++= NetCABI = DWR () +=tCAIit1 Term Structure of Interest Rates Spot Rates Denoted by the n-year spot rate ns (1) The annual Interest rate on the n-year Treasury STRIP is called the n-year spot rate, and the series of spot rates over time is called the yield curve. (2) To value a bond, take the present value of each payment at the appropriate yield curve rate and sum the present values.

9 ()()()()()()()()()()()()()12122111221111 1111nnnnniiPPP PPPP sfffssf==+ ++ =++++++++++ "" (3) Once we have found the price of a bond using the yield curve we can find the yield to maturity as the constant yield on the bond at that price. For example Purchasing a bond with coupons has cash flows given by () ( )( )nPPPP , , , , If payments are not level () ( )( )121 , , , the BA-II Plus 1. CF Worksheet Set ()( )( )nPPPP=== =CN , , C02 , C01 , 2. IRR CPT Constant Yield on the Bond = IRR If payments are level () ( )( )121 , , , the BA-II Plus 1. TVM Worksheet Set ()CPnPn=== = FV , PMT ,N ,PV1 2. I/Y CPT Constant Yield on the Bond = I/Y Forward Rates Denoted by nf the year forward rate n The rate agreed upon today for a one-year loan to be made n years in the future ()()11111nnnnnsfs ++=+ ()()()11111nnnnnsfs +=+ + Duration Duration a measure of sensitivity of a financial asset to changes in Interest rates Investment Cash Flows nCCC,,, Investment Price () >=+++=0221tttnnCvCvCvvCiP" Weights for Macaulay Duration () >=+++==0221tttttnntttttCvCvCvCvvCCviPCvw " Macaulay Duration >>= ++ + =002121ttttttnMCvCtvwnwwD" Modified Duration ()MDiiPdidD+= =11 Duration of a Level Payment Investment ()

10 I|ni|naIaD = Macaulay Duration of a coupon bond with face value F and coupon Fr for n periods and redemption value C ()ninninMCvFranCvIaFrD++= | | The Duration of a Zero-Coupon Bond payable in n periods is n Modified Duration of a Portfolio with Investments kXnnTotDWDWDWD+++="2211 where nkkXXXXW+++="21 Convexity and Approximations in Change of Price Convexity and Estimate C =()()221diiPdiP ()()(() ( ))22iiPiiPiPiiP ++ Change in Price ()()()()()()()iiPDiiPiPiPiiPiPiiPP = = += ()()()iiPDiiPiiPP = + 22+C ()22iiP Immunization Notation ()iA Present Value of Assets tA Asset Amount at time t ()iL Present Value of Liabilities tL Liability Amount at time t ()iS Surplus () () ()iLiAiS = Conditions for Immunization To achieve immunization we must have ()0=oiS, ()0= oiS, and ()0> oiS Immunization in terms of duration and convexity we (1) PV Matching ()()ooiLiA= (2) Duration Matching ()()00 iiiLdidiAdid= (3) Greater Convexity for Assets ()()00 2222iiiLdidiAdid> Immunization in terms of the asset and liability amounts at time (1) PV Matching >>=0000ttitttitvLvA (2) Duration Matching >>=0000ttitttitvtLvtA (3) Greater Convexity for Assets >>=020200ttitttitvLtvAt Special Cases Yield Rate Reinvestments Notation y annual yield of total investment (IRR)