# Institute of Actuaries of India Subject CT1 Financial Mathematics

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1 Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios

2 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i fiacial mathematics ad its simple applicatios. Liks to other subjects Subject CT2 Fiace ad Fiacial Reportig: develops the use of the asset types itroduced i this subject. Subject CT4 Models: develops the idea of stochastic iterest rates. Subject CT5 Cotigecies: develops some of the techiques itroduced i this subject i situatios where cashflows are depedet o survival. Subject CT7 Busiess Ecoomics: develops the behaviour of iterest rates. Subject CT8 Fiacial Ecoomics: develops the priciples further. Subjects CA1 Actuarial Risk Maagemet CA2 Model Documetatio Aalysis ad Reportig ad the Specialist Techical ad Specialist Applicatios subjects: use the priciples itroduced i this subject. Objectives O completio of the subject the traiee actuary will be able to: (i) Describe how to use a geeralised cashflow model to describe fiacial trasactios. 1. For a give cashflow process state the iflows ad outflows i each future time period ad discuss whether the amout or the timig (or both) is fixed or ucertai. 2. Describe i the form of a cashflow model the operatio of a zero coupo bod a fixed iterest security a idex-liked security cash o deposit a equity a iterest oly loa a repaymet loa ad a auity certai. (ii) Describe how to take ito accout the time value of moey usig the cocepts of compoud iterest ad discoutig. 1. Accumulate a sigle ivestmet at a costat rate of iterest uder the operatio of: simple iterest compoud iterest 2. Defie the preset value of a future paymet. 3. Discout a sigle ivestmet uder the operatio of simple (commercial) discout at a costat rate of discout. Page 2 Istitute ad Faculty of Actuaries

3 Subject CT1 Fiacial Mathematics Core Techical 4. Describe how a compoud iterest model ca be used to represet the effect of ivestig a sum of moey over a period. (iii) Show how iterest rates or discout rates may be expressed i terms of differet time periods. 1. Derive the relatioship betwee the rates of iterest ad discout over oe effective period arithmetically ad by geeral reasoig. 2. Derive the relatioships betwee the rate of iterest payable oce per effective period ad the rate of iterest payable p times per time period ad the force of iterest. 3. Explai the differece betwee omial ad effective rates of iterest ad derive effective rates from omial rates. 4. Calculate the equivalet aual rate of iterest implied by the accumulatio of a sum of moey over a specified period where the force of iterest is a fuctio of time. (iv) (v) Demostrate a kowledge ad uderstadig of real ad moey iterest rates. Calculate the preset value ad the accumulated value of a stream of equal or uequal paymets usig specified rates of iterest ad the et preset value at a real rate of iterest assumig a costat rate of iflatio. 1. Discout ad accumulate a sum of moey or a series (possibly ifiite) of cashflows to ay poit i time where: the rate of iterest or discout is costat the rate of iterest or discout varies with time but is ot a cotiuous fuctio of time either or both the rate of cashflow ad the force of iterest are cotiuous fuctios of time 2. Calculate the preset value ad accumulated value of a series of equal or uequal paymets made at regular itervals uder the operatio of specified rates of iterest where the first paymet is: deferred for a period of time ot deferred (vi) Defie ad use the more importat compoud iterest fuctios icludig auities certai. 1. Derive formulae i terms of i v d δ i (p) ad d (p) for a s s ( p ) p ( a s ) a ad s. ( p a ) ( p ) s a Istitute ad Faculty of Actuaries Page 3

4 Subject CT1 Fiacial Mathematics Core Techical 2. Derive formulae i terms of i v d δ i (p) ad d (p) for m a ad. m a ( p ) m a m a ( p) m a 3. Derive formulae i terms of i v δ a ad a for ( the respective deferred auities. Ia ) ( ) Ia ( Ia ) ( Ia ) ad (vii) Defie a equatio of value. 1. Defie a equatio of value where paymet or receipt is certai. 2. Describe how a equatio of value ca be adjusted to allow for ucertai receipts or paymets. 3. Uderstad the two coditios required for there to be a exact solutio to a equatio of value. (viii) Describe how a loa may be repaid by regular istalmets of iterest ad capital. 1. Describe flat rates ad aual effective rates. 2. Calculate a schedule of repaymets uder a loa ad idetify the iterest ad capital compoets of auity paymets where the auity is used to repay a loa for the case where auity paymets are made oce per effective time period or p times per effective time period ad idetify the capital outstadig at ay time. (ix) Show how discouted cashflow techiques ca be used i ivestmet project appraisal. 1. Calculate the et preset value ad accumulated profit of the receipts ad paymets from a ivestmet project at give rates of iterest. 2. Calculate the iteral rate of retur implied by the receipts ad paymets from a ivestmet project. 3. Describe payback period ad discouted payback period ad discuss their suitability for assessig the suitability of a ivestmet project. 4. Determie the payback period ad discouted payback period implied by the receipts ad paymets from a ivestmet project. 5. Calculate the moey-weighted rate of retur the time-weighted rate of retur ad the liked iteral rate of retur o a ivestmet or a fud. (x) Describe the ivestmet ad risk characteristics of the followig types of asset available for ivestmet purposes: fixed iterest govermet borrowigs fixed iterest borrowig by other bodies Page 4 Istitute ad Faculty of Actuaries

5 Subject CT1 Fiacial Mathematics Core Techical idex-liked govermet borrowigs shares ad other equity-type fiace derivatives (xi) Aalyse elemetary compoud iterest problems. 1. Calculate the preset value of paymets from a fixed iterest security where the coupo rate is costat ad the security is redeemed i oe istalmet. 2. Calculate upper ad lower bouds for the preset value of a fixed iterest security that is redeemable o a sigle date withi a give rage at the optio of the borrower. 3. Calculate the ruig yield ad the redemptio yield from a fixed iterest security (as i 1.) give the price. 4. Calculate the preset value or yield from a ordiary share ad a property give simple (but ot ecessarily costat) assumptios about the growth of divideds ad rets. 5. Solve a equatio of value for the real rate of iterest implied by the equatio i the presece of specified iflatioary growth. 6. Calculate the preset value or real yield from a idex-liked bod give assumptios about the rate of iflatio. 7. Calculate the price of or yield from a fixed iterest security where the ivestor is subject to deductio of icome tax o coupo paymets ad redemptio paymets are subject to the deductio of capital gais tax. 8. Calculate the value of a ivestmet where capital gais tax is payable i simple situatios where the rate of tax is costat idexatio allowace is take ito accout usig specified idex movemets ad allowace is made for the case where a ivestor ca offset capital losses agaist capital gais. (xii) Calculate the delivery price ad the value of a forward cotract usig arbitrage free pricig methods. 1. Defie arbitrage ad explai why arbitrage may be cosidered impossible i may markets. 2. Calculate the price of a forward cotract i the absece of arbitrage assumig: o icome or expediture associated with the uderlyig asset durig the term of the cotract a fixed icome from the asset durig the term a fixed divided yield from the asset durig the term. Istitute ad Faculty of Actuaries Page 5

6 Subject CT1 Fiacial Mathematics Core Techical 3. Explai what is meat by hedgig i the case of a forward cotract. 4. Calculate the value of a forward cotract at ay time durig the term of the cotract i the absece of arbitrage i the situatios listed i 2 above. (xiii) Show a uderstadig of the term structure of iterest rates. 1. Describe the mai factors ifluecig the term structure of iterest rates. 2. Explai what is meat by the par yield ad yield to maturity. 3. Explai what is meat by derive the relatioships betwee ad evaluate: discrete spot rates ad forward rates cotiuous spot rates ad forward rates 4. Defie the duratio ad covexity of a cashflow sequece ad illustrate how these may be used to estimate the sesitivity of the value of the cashflow sequece to a shift i iterest rates. 5. Evaluate the duratio ad covexity of a cashflow sequece. 6. Explai how duratio ad covexity are used i the (Redigto) immuisatio of a portfolio of liabilities. (xiv) Show a uderstadig of simple stochastic models for ivestmet returs. 1. Describe the cocept of a stochastic iterest rate model ad the fudametal distictio betwee this ad a determiistic model. 2. Derive algebraically for the model i which the aual rates of retur are idepedetly ad idetically distributed ad for other simple models expressios for the mea value ad the variace of the accumulated amout of a sigle premium. 3. Derive algebraically for the model i which the aual rates of retur are idepedetly ad idetically distributed recursive relatioships which permit the evaluatio of the mea value ad the variace of the accumulated amout of a aual premium. 4. Derive aalytically for the model i which each year the radom variable (1 + i) has a idepedet log-ormal distributio the distributio fuctios for the accumulated amout of a sigle premium ad for the preset value of a sum due at a give specified future time. 5. Apply the above results to the calculatio of the probability that a simple sequece of paymets will accumulate to a give amout at a specific future time. END OF SYLLABUS Page 6 Istitute ad Faculty of Actuaries

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