Subject CT1 Financial Mathematics Core Technical Core Reading

Transcription

1 Subject CT1 Fiacial Mathematics Core Techical Core Readig for the 2016 exams 1 Jue 2015

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3 Copyright i this Core Readig is the property of the Istitute ad Faculty of Actuaries who are the sole distributors. Core readig is iteded for the exclusive use of the purchaser ad the Istitute ad Faculty of Actuaries do ot permit it to be used by aother party, copied, electroically trasmitted or published o a website without prior permissio beig obtaied. Legal actio will be take if these terms are ifriged. I the case of a member of the Istitute ad Faculty of Actuaries, we may seek to take discipliary actio through the Discipliary Scheme of the Istitute ad Faculty of Actuaries. These coditios remai i force after the Core Readig has bee superseded by a later editio.

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5 SUBJECT CT1 CORE READING Cotets Accreditatio Itroductio Uit 1 Uit 2 Uit 3 Uit 4 Uit 5 Uit 6 Uit 7 Uit 8 Uit 9 Uit 10 Uit 11 Uit 12 Uit 13 Uit 14 Geeralised cashflow model The time value of moey Iterest rates Real ad moey iterest rates Discoutig ad accumulatig Compoud iterest fuctios Equatios of value Loa schedules Project appraisal Ivestmets Elemetary compoud iterest problems The No Arbitrage assumptio ad Forward Cotracts Term structure of iterest rates Stochastic iterest rate models Syllabus with cross referecig to Core Readig

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7 SUBJECT CT1 CORE READING Accreditatio The Istitute ad Faculty of Actuaries would like to thak the umerous people who have helped i the developmet of this material ad i the previous versios of Core Readig. The followig book has bee used as the basis for several Uits: A itroductio to the mathematics of fiace. McCutcheo, J. J.; Scott, W. F. Heiema, ISBN: X, by permissio of the authors who are the holders of copyright of the book. All rights reserved.

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9 CORE READING Itroductio The Core Readig maual has bee produced by the Istitute ad Faculty of Actuaries. The purpose of the Core Readig is to assist i esurig that tutors, studets ad examiers have clear shared appreciatio of the requiremets of the syllabus for the qualificatio examiatios for Fellowship of the Istitute ad Faculty of Actuaries. The maual supports coverage of the syllabus i helpig to esure that both depth ad breadth are re-eforced. I examiatios studets will be expected to demostrate their uderstadig of the cocepts i Core Readig. Examiers will have this Core Readig maual whe settig the papers. I preparig for examiatios studets are recommeded to work through past examiatio questios ad will fid additioal tuitio helpful. The maual will be updated each year to reflect chages i the syllabus, to reflect curret practice ad i the iterest of clarity.

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11 2016 Geeralised cashflow model Subject CT1 UNIT 1 GENERALISED CASHFLOW MODEL Syllabus objective (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 idexliked security, cash o deposit, a equity, a iterest oly loa, a repaymet loa, ad a auity certai. 1 Cashflow process The practical work of the actuary ofte ivolves the maagemet of various cashflows. These are simply sums of moey, which are paid or received at differet times. The timig of the cashflows may be kow or ucertai. The amout of the idividual cashflows may also be kow or ukow i advace. From a theoretical viewpoit oe may also cosider a cotiuously payable cashflow. For example, a compay operatig a privately owed bridge, road or tuel will receive toll paymets. The compay will pay out moey for maiteace, debt repaymet ad for other maagemet expeses. From the compay s viewpoit the toll paymets are positive cashflows (i.e. moey received) while the maiteace, debt repaymets ad other expeses are egative cashflows (i.e. moey paid out). Similar cashflows arise i all busiesses. I some busiesses, such as isurace compaies, ivestmet icome will be received i relatio to positive cashflows (premiums) received before the egative cashflows (claims ad expeses). Where there is ucertaity about the amout or timig of cashflows, a actuary ca assig probabilities to both the amout ad the existece of a cashflow. I this Subject we will assume that the existece of the future cashflows is certai. 2 Examples of cashflow scearios I this sectio some simple examples are give of practical situatios which are readily described by cashflow models. 2.1 A zero-coupo bod The term zero-coupo bod is used to describe a security that is simply a cotract to provide a specified lump sum at some specified future date. For the ivestor there is a egative cashflow at the poit of ivestmet ad a sigle kow positive cashflow o the specified future date. Istitute ad Faculty of Actuaries Uit 1, Page 1

12 Subject CT1 Geeralised cashflow model A fixed iterest security A body such as a idustrial compay, a local authority, or the govermet of a coutry may raise moey by floatig a loa o the stock exchage. I may istaces such a loa takes the form of a fixed iterest security, which is issued i bods of a stated omial amout. The characteristic feature of such a security i its simplest form is that the holder of a bod will receive a lump sum of specified amout at some specified future time together with a series of regular level iterest paymets util the repaymet (or redemptio) of the lump sum. The ivestor has a iitial egative cashflow, a sigle kow positive cashflow o the specified future date, ad a series of smaller kow positive cashflows o a regular set of specified future dates. 2.3 A idex-liked security With a covetioal fixed iterest security the iterest paymets are all of the same amout. If iflatioary pressures i the ecoomy are ot kept uder cotrol, the purchasig power of a give sum of moey dimiishes with the passage of time, sigificatly so whe the rate of iflatio is high. For this reaso some ivestors are attracted by a security for which the actual cash amout of iterest paymets ad of the fial capital repaymet are liked to a idex which reflects the effects of iflatio. Here the iitial egative cashflow is followed by a series of ukow positive cashflows ad a sigle larger ukow positive cashflow, all o specified dates. However, it is kow that the amouts of the future cashflows relate to the iflatio idex. Hece these cashflows are said to be kow i real terms. Note that i practice the operatio of a idex-liked security will be such that the cashflows do ot relate to the iflatio idex at the time of paymet, due to delays i calculatig the idex. It is also possible that the eed of the borrower (or perhaps the ivestors) to kow the amouts of the paymets i advace may lead to the use of a idex from a earlier period. 2.4 Cash o deposit If cash is placed o deposit, the ivestor ca choose whe to disivest ad will receive iterest additios durig the period of ivestmet. The iterest additios will be subject to regular chage as determied by the ivestmet provider. These additios may oly be kow o a day-to-day basis. The amouts ad timig of cashflows will therefore be ukow. 2.5 A equity Equity shares (also kow as shares or equities i the UK ad as commo stock i the USA) are securities that are held by the owers of a orgaisatio. Equity shareholders ow the compay that issued the shares. For example if a compay issues 4,000 shares ad a ivestor buys 1,000, the ivestor ows 25 per cet of the compay. I a small Uit 1, Page 2 Istitute ad Faculty of Actuaries

13 2016 Geeralised cashflow model Subject CT1 compay all the equity shares may be held by a few idividuals or istitutios. I a large orgaisatio there may be may thousads of shareholders. Equity shares do ot ear a fixed rate of iterest as fixed iterest securities do. Istead the shareholders are etitled to a share i the compay s profits, i proportio to the umber of shares owed. The distributio of profits to shareholders takes the form of regular paymets of divideds. Sice they are related to the compay profits that are ot kow i advace, divided rates are variable. It is expected that compay profits will icrease over time. It is therefore expected also that divideds per share will icrease though there are likely to be fluctuatios. This meas that i order to costruct a cashflow schedule for a equity it is ecessary first to make a assumptio about the growth of future divideds. It also meas that the etries i the cashflow schedule are ucertai they are estimates rather tha kow quatities. I practice the relatioship betwee divideds ad profits is ot a simple oe. Compaies will, from time to time, eed to hold back some profits to provide fuds for ew projects or expasio. Compaies may also hold back profits i good years to subsidise divideds i years with poorer profits. Additioally, compaies may be able to distribute profits i a maer other tha divideds, such as by buyig back the shares issued to some ivestors. Sice equities do ot have a fixed redemptio date, but ca be held i perpetuity, we may assume that divideds cotiue idefiitely (uless the ivestor sells the shares or the compay buys them back), but it is importat to bear i mid the risk that the compay will fail, i which case the divided icome will cease ad the shareholders would oly be etitled to ay assets which remai after creditors are paid. The future positive cashflows for the ivestor are therefore ucertai i amout ad may eve be lower, i total, tha the iitial egative cashflow. 2.6 A auity certai A auity certai provides a series of regular paymets i retur for a sigle premium (i.e. a lump sum) paid at the outset. The precise coditios uder which the auity paymets will be made will be clearly specified. I particular, the umber of years for which the auity is payable, ad the frequecy of paymet, will be specified. Also, the paymet amouts may be level or might be specified to vary for example i lie with a iflatio idex, or at a costat rate. The cashflows for the ivestor will be a iitial egative cashflow followed by a series of smaller regular positive cashflows throughout the specified term of paymet. I the case of level auity paymets, the cashflows are similar to those for a fixed iterest security. From the perspective of the auity provider, there is a iitial positive cashflow followed by a kow umber of regular egative cashflows. I Subject CT5, Cotigecies, the theory of this Subject will be exteded to deal with auities where the paymet term is ucertai, that is, for which paymets are made oly so log as the auity policyholder survives. Istitute ad Faculty of Actuaries Uit 1, Page 3

14 Subject CT1 Geeralised cashflow model A iterest-oly loa A iterest-oly loa is a loa that is repayable by a series of iterest paymets followed by a retur of the iitial loa amout. I the simplest of cases, the cashflows are the reverse of those for a fixed iterest security. The provider of the loa effectively buys a fixed iterest security from the borrower. I practice, however, the iterest rate eed ot be fixed i advace. The regular cashflows may therefore be of ukow amouts. It may also be possible for the loa to be repaid early. The umber of cashflows ad the timig of the fial cashflows may therefore be ucertai. 2.8 A repaymet loa (or mortgage) A repaymet loa is a loa that is repayable by a series of paymets that iclude partial repaymet of the loa capital i additio to the iterest paymets. I its simplest form, the iterest rate will be fixed ad the paymets will be of fixed equal amouts, paid at regular kow times. The cashflows are similar to those for a auity certai. As for the iterest-oly loa, complicatios may be added by allowig the iterest rate to vary or the loa to be repaid early. Additioally, it is possible that the regular repaymets could be specified to icrease (or decrease) with time. Such chages could be smooth or discrete. It is importat to appreciate that with a repaymet loa the breakdow of each paymet ito iterest ad capital chages sigificatly over the period of the loa. The first repaymet will cosist almost etirely of iterest ad will provide oly a very small capital repaymet. I cotrast, the fial repaymet will cosist almost etirely of capital ad will have a small iterest cotet. E N D Uit 1, Page 4 Istitute ad Faculty of Actuaries

15 2016 The time value of moey Subject CT1 UNIT 2 THE TIME VALUE OF MONEY Syllabus objectives (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. 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. 1 The idea of iterest Iterest may be regarded as a reward paid by oe perso or orgaisatio (the borrower) for the use of a asset, referred to as capital, belogig to aother perso or orgaisatio (the leder). Whe the capital ad iterest are expressed i moetary terms, capital is also referred to as pricipal. The total received by the leder after a period of time is called the accumulated value. The differece betwee the pricipal ad the accumulated value is called the iterest. Note that we are assumig here that o other paymets are made or icurred (e.g. charges, expeses). If there is some risk of default (i.e. loss of capital or o-paymet of iterest) a leder would expect to be paid a higher rate of iterest tha would otherwise be the case. Aother factor that may ifluece the rate of iterest o ay trasactio is a allowace for the possible depreciatio or appreciatio i the value of the currecy i which the trasactio is carried out. This factor is very importat i times of high iflatio. We will ow cosider two types of iterest withi the framework of a savigs accout. Istitute ad Faculty of Actuaries Uit 2, Page 1

16 Subject CT1 The time value of moey Simple iterest The essetial feature of simple iterest is that iterest, oce credited to a accout, does ot itself ear further iterest. Suppose a amout C is deposited i a accout that pays simple iterest at the rate of i 100% per aum. The after years the deposit will have accumulated to: Whe is ot a iteger, iterest is paid o a pro-rata basis. 1.2 Compoud (effective) iterest C(1 + i) (1.1) The essetial feature of compoud iterest is that iterest itself ears iterest. Suppose a amout C is deposited i a accout that pays compoud iterest at the rate of i 100% per aum. The after years the deposit will have accumulated to: 1.3 Accumulatio factors C(1 i) (1.2) For t 1 t 2 we defie A(t 1, t 2 ) to be the accumulatio at time t 2 of a ivestmet of 1 at time t 1. The umber A(t 1, t 2 ) is ofte called a accumulatio factor, sice the accumulatio at time t 2 of a ivestmet of C at time t 1 is, by proportio: CA(t 1, t 2 ) (1.3) A ( ) is ofte used as a abbreviatio for the accumulatio factor A(0, ). 1.4 The priciple of cosistecy Now let t 0 t 1 t 2 ad cosider a ivestmet of 1 at time t 0. The proceeds at time t 2 will be A(t 0, t 2 ) if oe ivests at time t 0 for term t 2 t 0, or A(t 0, t 1 ) A(t 1, t 2 ) if oe ivests at time t 0 for term t 1 t 0 ad the, at time t 1, reivests the proceeds for term t 2 t 1. I a cosistet market these proceeds should ot deped o the course of actio take by the ivestor. Accordigly, we say that uder the priciple of cosistecy: A(t 0, t ) = A(t 0, t 1 ) A(t 1, t 2 )... A(t 1, t ) (1.4) Uit 2, Page 2 Istitute ad Faculty of Actuaries

17 2016 The time value of moey Subject CT1 2 Preset values It follows by formula 1.2 that a ivestmet of C (1 i) (2.1) at time 0 (the preset time) will give C at time 0. This is called the discouted preset value (or, more briefly, the preset value) of C due at time 0. We ow defie the fuctio 1 v 1 i (2.2) It follows by formulae 2.1 ad 2.2 that the discouted preset value of C due at time 0 is: Cv (2.3) 3 Discout rates A alterative way of obtaiig the discouted value of a paymet is to use discout rates. 3.1 Simple discout As has bee see with simple iterest, the iterest eared is ot itself subject to further iterest. The same is true of simple discout, which is defied below. Suppose a amout C is due after years ad a rate of simple discout of d per aum applies. The the sum of moey required to be ivested ow to amout to C after years (i.e. the preset value of C) is C(1 d) (3.1) I ormal commercial practice, d is usually ecoutered oly for periods of less tha a year. If a leder bases his short-term trasactios o a simple rate of discout d the, i retur for a repaymet of X after a period t (t < 1) he will led X(1 td) at the start of the period. I this situatio, d is also kow as a rate of commercial discout. Istitute ad Faculty of Actuaries Uit 2, Page 3

18 Subject CT1 The time value of moey Compoud (effective) discout As has bee see with compoud iterest, the iterest eared is subject to further iterest. The same is true of compoud discout, which is defied below. Suppose a amout C is due after years ad a rate of compoud (or effective) discout of d per aum applies. The the sum of moey required to be ivested ow to accumulate to C after years (i.e. the preset value of C) is 3.3 Discout factors C(1 d) (3.2) I the same way that the accumulatio factor A ( ) gives the accumulatio at time of a ivestmet of 1 at time 0, we defie v ( ) to be the preset value of a paymet of 1 due at time. Hece: 1 v ( ) A ( ) (3.3) 4 Effective rates of iterest ad discout Effective rates are compoud rates that have iterest paid oce per uit time either at the ed of the period (effective iterest) or at the begiig of the period (effective discout). This distiguishes them from omial rates where iterest is paid more frequetly tha oce per uit time. We ca demostrate the equivalece of compoud ad effective rates by a alterative way of cosiderig effective rates. 4.1 Effective rate of iterest A ivestor will led a amout 1 at time 0 i retur for a repaymet of (1 i) at time 1. Hece we ca cosider i to be the iterest paid at the ed of the year. Accordigly i is called the rate of iterest (or the effective rate of iterest) per uit time. So deotig the effective rate of iterest durig the th period by i, we have i A ( ) A ( 1) A ( 1) Uit 2, Page 4 Istitute ad Faculty of Actuaries

19 2016 The time value of moey Subject CT1 If i is the compoud rate of iterest, we have: 1 (1 i) (1 i) i (1 i) 1 i 1 (1 i) Sice this is idepedet of, we see that the effective rate of iterest is idetical to the compoud rate of iterest we met earlier 4.2 Effective rate of discout We ca thik of compoud discout as a ivestor ledig a amout (1 d ) at time 0 i retur for a repaymet of 1 at time 1. The sum of (1 d) may be cosidered as a loa of 1 (to be repaid after 1 uit of time) o which iterest of amout d is payable i advace. Accordigly d is called the rate of discout (or the effective rate of discout) per uit time. We ca also show that the effective rate of discout is idetical to the compoud rate of discout we met earlier. 5 Equivalet rates Two rates of iterest ad/or discout are equivalet if a give amout of pricipal ivested for the same legth of time produces the same accumulated value uder each of the rates. Comparig formulae (2.3) ad (3.2), we see that: Ad from (2.2) ad (5.1) we obtai the rearragemets: v1 d (5.1) d iv (5.2) ad: i d (5.3) 1 i Recall that d is the iterest paid at time 0 o a loa of 1, whereas i is the iterest paid at time 1 o the same loa. If the rates are equivalet the if we discout i from time 1 to time 0 we will obtai d. This is the iterpretatio of equatios (5.2) ad (5.3). E N D Istitute ad Faculty of Actuaries Uit 2, Page 5

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21 2016 Iterest rates Subject CT1 UNIT 3 INTEREST RATES Syllabus objective (iii) Show how iterest rates or discout rates may be expressed i terms of differet time periods. 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. 1 Nomial rates of iterest ad discout Recall from Uit 2 that effective rates of iterest ad discout have iterest paid oce per measuremet period, either at the ed of the period or at the begiig of the period. Nomial is used where iterest is paid more (or less) frequetly tha oce per measuremet period. 1.1 Nomial rates of iterest We deote the omial rate of iterest payable p times per period by i referred to as the rate of iterest covertible pthly or compouded pthly.. This is also A omial rate of iterest per period, payable pthly, i, is defied to be a rate of iterest of i p applied for each pth of a period. For example, a omial rate of iterest of 6% p.a. covertible quarterly meas a iterest rate of % per quarter. Hece, by defiitio, i is equivalet to a pthly effective rate of iterest of i p. Therefore the effective iterest rate i is obtaied from: i 1i 1 p p (3.1) Istitute ad Faculty of Actuaries Uit 3, Page 1

22 Subject CT1 Iterest rates 2016 Note that i (1) i. The treatmet of problems ivolvig omial rates of iterest (or discout) is almost always cosiderably simplified by a appropriate choice of the time uit. By choosig the basic time uit to be the period correspodig to the frequecy with which the omial rate of iterest is covertible, we ca use i pas the effective rate of iterest per uit time. For example, if we have a omial rate of iterest of 18% per aum covertible mothly, we should take oe moth as the uit of time ad 1½% as the rate of iterest per uit time. 1.2 Nomial rates of discout We deote the omial rate of discout payable p times per period by d. This is also referred to as the rate of discout covertible pthly or compouded pthly. A omial rate of discout per period payable pthly, of d p applied for each pth of a period. Hece, by defiitio, d, is defied as a rate of discout d is equivalet to a pthly effective rate of discout of Therefore the effective discout rate d is obtaied from: d p. d 1d 1 p p (3.2) Note that (1) d d. 2 The force of iterest 2.1 Derivatio from omial iterest covertible pthly We assume that for each value of i there is umber,, such that: lim i p ( p ) is the omial rate of iterest per uit time covertible cotiuously (or mometly). This is also referred to as the rate cotiuously compouded. We call it the force of iterest. Uit 3, Page 2 Istitute ad Faculty of Actuaries

23 2016 Iterest rates Subject CT1 Euler s rule states that: x lim 1 e Applyig this to the right-had-side of (3.1) gives: x i lim 1 p p p e ( ) i Hece: 1i e (3.3) Sice 1 i v(1 ), we have: v e (3.4) From equatio (3.4) we have: t t t v ( e ) e Hece, the discout factor for a force of iterest is: v ( ) e 2.1 Derivatio from omial discout covertible pthly It ca also be show that: lim d p ( p ) However, d teds to this limit from below whereas i teds to this limit from above. Hece, we have: (2) (3) (3) (2) d d d i i i Istitute ad Faculty of Actuaries Uit 3, Page 3

24 Subject CT1 Iterest rates Relatioships betwee effective, omial ad force of iterest 3.1 A alterative way of cosiderig omial iterest covertible pthly Recall that effective iterest i ca be thought of as iterest paid at the ed of the period. Hece, a ivestor ledig a amout 1 at time 0 receives a repaymet of (1 i) at time 1. Similarly, omial iterest covertible pthly ca be thought of as the total iterest per uit of time paid o a loa of amout 1 at time 0, where iterest is paid i p equal istalmets at the ed of each pth subiterval (i.e. at times 1 p,2 p,3 p,,1). Sice i is the total iterest paid ad each iterest paymet is of amout i ( ) accumulated value at time 1 of the iterest paymets is: p p the the ( p1) p ( p2) p i i i (1 i) (1 i) i p p p Hece: 1 p i p(1 i) A alterative way of cosiderig omial discout covertible pthly Recall that effective discout d ca be thought of as iterest paid at the start of the period. Hece, a ivestor ledig a amout 1 at time 0 receives a repaymet of 1 at time 1, but d is paid at the start so a sum of (1 d) is let at time 0. Similarly, d is the total amout of iterest per uit of time payable i equal istalmets at the start of each pth subiterval (ie at times 0,1 p, 2 p,,( p 1) p). As a cosequece the preset value at time 0 of the iterest paymets is: 1 p ( p1) p d d d (1 d) (1 d) d p p p Hece: d p1 (1 d) 1 p Uit 3, Page 4 Istitute ad Faculty of Actuaries

25 2016 Iterest rates Subject CT1 3.3 A alterative way of cosiderig force of iterest Now is the total amout of iterest payable as a cotiuous paymet stream, ie a amout dt is paid over a ifiitesimally small period dt at time t. As a cosequece the accumulated value at time 1 of these iterest paymets is: 1 (1 i) 0 1t dt which, by symmetry, is equal to: Hece: 1 (1 i) t dt i 0 l(1 i) or e 1 i It is essetial to appreciate that, at force of iterest per uit time, the five series of paymets illustrated i Figure below all have the same value. 0 1 p 2 p 3 p... p 1 p 1 time (1) d (2) d p d p d p d p... d p (3) i p i p i p... i p i p equivalet paymets (4) i (5) Figure Equivalet paymets Istitute ad Faculty of Actuaries Uit 3, Page 5

26 Subject CT1 Iterest rates Force of iterest as a fuctio of time 4.1 Formal defiitio The force of iterest is the istataeous chage i the fud value, expressed as a aualized percetage of the curret fud value. So the force of iterest at time t is defied to be: Vt () t V t where V t is the value of the fud at time t ad Vt is the derivative of V t with respect to t. Hece: d () t lv t dt Itegratig this from t 1 to t 2 gives: V t2 t2 t2 ( tdt ) lvt lvt lv l t1 2 t1 t V 1 t1 V V t2 () t dt t2 t1 t1 e Hece: 1 2 t2 t At (, t ) e 1 () t dt Uit 3, Page 6 Istitute ad Faculty of Actuaries

27 2016 Iterest rates Subject CT1 4.2 Relatioship to costat force of iterest For the case whe the force of iterest is costat,, betwee time 0 ad time, we have: 0 A(0, ) e e dt Hece: Therefore: as before. (1 i) e (1 i) 4.3 Applicatios of force of iterest e Although the force of iterest is a theoretical measure it is the most fudametal measure of iterest (as all other iterest rates ca be derived from it). However, sice the majority of trasactios ivolve discrete processes we ted to use other iterest rates i practice. It still remais a useful coceptual ad aalytical tool ad ca be used as a approximatio to iterest paid very frequetly, e.g. daily. E N D Istitute ad Faculty of Actuaries Uit 3, Page 7

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29 2016 Real ad moey iterest rates Subject CT1 UNIT 4 REAL AND MONEY INTEREST RATES Syllabus objective (iv) Demostrate a kowledge ad uderstadig of real ad moey iterest rates. 1 Defiitio of real ad moey iterest rates Accumulatig a ivestmet of 1 for a period of time t from time 0 produces a ew total accumulated value A(0, t), say. Typically the ivestmet of 1 will be a sum of moey, say 1 or $1 or 1 Euro. I this case, if we are give the iformatio o the iitial ivestmet of 1 i the specified currecy, the period of the ivestmet, ad the cash amout of moey accumulated, the the uderlyig iterest rate is termed a moey rate of iterest. More geerally, give ay series of moetary paymets accumulated over a period, a moey rate of iterest is that rate which will have bee eared so as to produce the total amout of cash i had at the ed of the period of accumulatio. I practice, most such accumulatios will take place i ecoomies subject to iflatio, where a give sum of moey i the future will have less purchasig power tha at the preset day. It is ofte useful, therefore, to recosider what the accumulated value is worth allowig for the erodig effects of iflatio. Returig to the iitial example above, suppose the accumulatio took place i a ecoomy subject to iflatio so that the cash A(0, t) is effectively worth oly A*(0, t) after allowig for iflatio, where A*(0, t) < A(0, t). I this case, the rate of iterest at which the origial sum of 1 would have to be accumulated to produce the sum A* is lower tha the moey rate of iterest. The sum A*(0, t) is referred to as the real amout accumulated, ad the uderlyig iterest rate, reduced for the effects of iflatio, is termed a real rate of iterest. More geerally, give ay series of moetary paymets accumulated over a period, a real rate of iterest is that rate which will have bee eared so as to produce the total amout of cash i had at the ed of the period of accumulatio reduced for the effects of iflatio. Uit 11 of this Subject will describe ways of calculatig real rates of iterest give the moey rates of iterest (ad vice versa). Istitute ad Faculty of Actuaries Uit 4, Page 1

30 Subject CT1 Real ad moey iterest rates Deflatioary coditios The above descriptios assume that the iflatio rate is positive. Where the iflatio rate is egative, termed deflatio the above theory still applies ad A*(0, t) > A(0, t), givig rise to the coclusio that the real rate of iterest i such circumstaces would be higher tha the moey rate of iterest. As might be expected, where there is o iflatio A*(0, t) = A(0, t) ad the real ad moey rates of iterest are the same. 3 Usefuless of real ad moey iterest rates We assume here that we have a positive iflatio rate. Which of the two rates of iterest, real or moey, is the more useful will deped o two mai factors: the purpose to which the rate will be put whether the uderlyig data has or has ot already bee adjusted for iflatio. The purpose to which the rate will be put Geerally, where the actuary is performig calculatios to determie how much should be ivested to provide for future outgo, the first step will be to determie whether the future outgo is real or moetary i ature. The type of iterest rate to be assumed would the be, respectively, a real or a moetary rate. For example, first suppose that a actuary was asked to calculate the sum to be ivested by a perso aged 40 to provide a roud-the-world cruise, whe the perso reaches 60, ad where the perso says the cruise costs 25,000. Uless the perso has, for some reaso, already made a allowace for iflatio i suggestig a figure of 25,000 the that amout is probably today s cost of the cruise. I this case, the actuary would be wise to assume (checkig his uderstadig with the perso) a iflatio rate ad this could be achieved by assumig a real rate of iterest. As a alterative example, suppose that a perso has a mortgage of 50,000 to be paid off i twety years time. Here, the party which grated the mortgage would cotractually be etitled to oly 50,000 i twety years time. Accordigly, i workig out how much should be ivested to repay the outgo i this case, a moey rate of iterest would be assumed. Whether the uderlyig data has or has ot already bee adjusted for iflatio I the first example above, we see that the data may already have bee adjusted for iflatio ad i that case it would ot be appropriate to allow for iflatio agai. A moey rate would the be assumed. Uit 4, Page 2 Istitute ad Faculty of Actuaries

31 2016 Real ad moey iterest rates Subject CT1 More geerally i actuarial work, the ature of the data provided must be uderstood before choosig the type ad amout of assumptios to be made. E N D Istitute ad Faculty of Actuaries Uit 4, Page 3

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33 2016 Discoutig ad accumulatig Subject CT1 UNIT 5 DISCOUNTING AND ACCUMULATING Syllabus objective (v) 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 Real rates of iterest are dealt with i Uit Preset values of cashflows I may compoud iterest problems oe must fid the discouted preset value of cashflows due i the future. It is importat to distiguish betwee (a) discrete ad (b) cotiuous paymets. 1.1 Discrete cashflows The preset value of the sums < t ) is, c, c,..., c due at times t 1, t 2,..., t (where 0 t 1 < t 2 <... t1 t2 t ct 1 v(t 1 ) + c v(t 2 ) c v( t ) = t 2 t c v(t j ) (1.1.1) t j1 j Istitute ad Faculty of Actuaries Uit 5, Page 1

34 Subject CT1 Discoutig ad accumulatig 2016 If the umber of paymets is ifiite, the preset value is defied to be ct j1 v(t j ) (1.1.2) provided that this series coverges. It usually will i practical problems. 1.2 Cotiuously payable cashflows (paymet streams) Suppose that T > 0 ad that betwee times 0 ad T a ivestor will be paid moey cotiuously, the rate of paymet at time t beig (t) per uit time. What is the preset value of this cashflow? I order to aswer this questio it is essetial to uderstad what is meat by the rate of paymet of the cashflow at time t. If M(t) deotes the total paymet made betwee time 0 ad time t, the by defiitio, j (t) = M (t) for all t (1.2.1) The, if 0 < T, the total paymet received betwee time ad time is M() M() = M () tdt = (t)dt (1.2.2) Thus the rate of paymet at ay time is simply the derivative of the total amout paid up to that time, ad the total amout paid betwee ay two times is the itegral of the rate of paymets over the appropriate time iterval. Betwee times t ad t + dt the total paymet received is M(t + dt) M(t). If dt is very small this is approximately M(t)dt or (t)dt. Theoretically, therefore, we may cosider the preset value of the moey received betwee times t ad t + dt as v(t)(t)dt. The preset value of the etire cashflow is obtaied by itegratio as If T is ifiite we obtai, by a similar argumet, the preset value T 0 v(t)(t)dt (1.2.3) 0 v(t)(t)dt (1.2.4) By combiig the results for discrete ad cotiuous cashflows, we obtai the formula c t v(t) + 0 v(t)(t)dt (1.2.5) Uit 5, Page 2 Istitute ad Faculty of Actuaries

35 2016 Discoutig ad accumulatig Subject CT1 for the preset value of a geeral cashflow (the summatio beig over those values of t for which c t, the discrete cashflow at time t, is o-zero). So far we have assumed that all paymets, whether discrete or cotiuous, are positive. If oe has a series of icome paymets (which may be regarded as positive) ad a series of outgoigs (which may be regarded as egative) their et preset value is defied as the differece betwee the value of the positive cashflow ad the value of the egative cashflow. 2 Valuig cashflows Cosider times t 1 ad t 2, where t 2 is ot ecessarily greater tha t 1. The value at time t 1 of the sum C due at time t 2 is defied as: (a) If t 1 t 2, the accumulatio of C from time t 2 util time t 1 ; or (b) If t 1 < t 2, the discouted value at time t 1 of C due at time t 2. I both cases the value at time t 1 of C due at time t 2 is 2 C exp t t1 () tdt (2.1.1) 2 1 (Note the covetio that, if t 1 > t 2, () tdt= (t)dt.) Sice t t 1 t t 2 t 2 (t)dt = 2 t1 0 t (t)dt 1 0 t (t)dt it follows immediately from equatio that the value at time t 1 of C due at time t 2 is vt ( 2) C (2.1.2) vt ( ) The value at a geeral time t 1 of a discrete cashflow of c t at time t (for various values of t) ad a cotiuous paymet stream at rate (t) per time uit may ow be foud, by the methods give i sectio 1, as 1 vt () vt () c t () t dt (2.1.3) vt ( 1) vt ( 1) Istitute ad Faculty of Actuaries Uit 5, Page 3

36 Subject CT1 Discoutig ad accumulatig 2016 where the summatio is over those values of t for which c t 0. We ote that i the special case whe t 1 = 0 (the preset time), the value of the cashflow is c t vt () (t)v(t)dt (2.1.4) where the summatio is over those values of t for which c t 0. This is a geeralisatio of formula to cover the past as well as preset or future paymets. If there are icomig ad outgoig paymets, the correspodig et value may be defied, as i sectio 1, as the differece betwee the value of the positive ad the egative cashflows. If all the paymets are due at or after time t 1, their value at time t 1 may also be called their discouted value, ad if they are due at or before time t 1, their value may be referred to as their accumulatio. It follows that ay value may be expressed as the sum of a discouted value ad a accumulatio. This fact is helpful i certai problems. Also, if t 1 = 0 ad all the paymets are due at or after the preset time, their value may also be described as their (discouted) preset value, as defied by formula It follows from formula that the value at ay time t 1 of a cashflow may be obtaied from its value at aother time t 2 by applyig the factor v(t 2 )/v(t 1 ), i.e. Value at time t Value at time t vt ( ) = of cashflow of cashflow vt ( 1) (2.1.5) or Value at time t Value at time t of cashflow of cashflow vt ( ) = vt ( ) (2.1.6) Each side of equatio is the value of the cashflow at the preset time (time 0). I particular, by choosig time t 2 as the preset time ad lettig t 1 = t, we obtai the result: Value at time t Value at the preset 1 = of cashflow time of cashflow vt () (2.1.7) These results are useful i may practical examples. The time 0 ad the uit of time may be chose so as to simplify the calculatios. 3 Iterest icome Cosider ow a ivestor who wishes ot to accumulate moey but to receive a icome while keepig his capital fixed at C. If the rate of iterest is fixed at i per time uit, ad if the ivestor wishes to receive icome at the ed of each time uit, it is clear that the Uit 5, Page 4 Istitute ad Faculty of Actuaries

37 2016 Discoutig ad accumulatig Subject CT1 icome will be ic per time uit, payable i arrear, util such time as the capital is withdraw. However, if iterest is paid cotiuously with force of iterest () t at time t the the icome received betwee times t ad t dt will be C () t dt. So the total iterest icome from time 0 to time T will be: T I( T) C ( t) dt 0 If the ivestor withdraws the capital at time T, the preset values of the icome ad capital at time 0 are, ad respectively. Sice C 0 T (t)v(t)dt (3.1.2) Cv(T) (3.1.3) T 0 (t)v(t)dt (t) exp t 0 () s ds dt = 0 T = t exp 0 ( s) ds = 1 v(t) T 0 we obtai C = C 0 T (t)v(t)dt + Cv(T) (3.1.4) as oe would expect by geeral reasoig. So far we have described the differece betwee moey retured at the ed of the term ad the cash origially ivested as iterest. I practice, however, this quatity may be divided ito iterest icome ad capital gais, the term capital loss beig used for a egative capital gai. E N D Istitute ad Faculty of Actuaries Uit 5, Page 5

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39 2016 Compoud iterest fuctios Subject CT1 UNIT 6 COMPOUND INTEREST FUNCTIONS Syllabus objective (vi) Defie ad use the more importat compoud iterest fuctios icludig auities certai. 1. Derive formulae i terms of i, v,, d,, i (p) ad d (p) for,,,,,,,, ad a s a s a s a s a s. 2. Derive formulae i terms of i, v, d,, i (p) ad d (p) ) for a, a, a, a m m m m ad. m a 3. Derive formulae i terms of i, v,,, a ad a for ( Ia ), ( Ia ), ( Ia ), ( Ia ) ad the respective deferred auities. 1 Auities certai: preset values ad accumulatios 1.1 Aual paymets Cosider a series of paymets, each of amout 1, to be made at time itervals of oe uit, the first paymet beig made at time t paymet t t + 1 t + 2 t t + 1 t + time Such a sequece of paymets is illustrated i the diagram above, i which the rth paymet is made at time t + r. Istitute ad Faculty of Actuaries Uit 6, Page 1

40 Subject CT1 Compoud iterest fuctios 2016 The value of this series of paymets oe uit of time before the first paymet is made is deoted by a. Clearly, if i = 0, the a = ; otherwise a = v + v 2 + v v = = v(1 v ) 1 v 1 v 1 v 1 = 1 v i (1.1.1) If = 0, a is defied to be zero. Thus a is the value at the start of ay period of legth of a series of paymets, each of amout 1, to be made i arrear at uit time itervals over the period. It is commo to refer to such a series of paymets, made i arrear, as a immediate auity certai ad to call a the preset value of the immediate auity certai. Whe there is o possibility of cofusio with a life auity (i.e. a series of paymets depedet o the survival of oe or more huma lives), the term auity may be used as a alterative to auity certai. The value of this series of paymets at the time the first paymet is made is deoted by a. If i = 0, the a = ; otherwise a = 1 + v + v v 1 = 1 v 1 v = 1 v d (1.1.2) Thus a is the value at the start of ay give period of legth of a series of paymets, each of amout 1, to be made i advace at uit time itervals over the period. It is commo to refer to such a series of paymets, made i advace, as a auity due ad to call a the preset value of the auity due. Uit 6, Page 2 Istitute ad Faculty of Actuaries

41 2016 Compoud iterest fuctios Subject CT1 It follows directly from the above defiitios that a = (1 ia ) ad that, for 2, (1.1.3) a = 1 a 1 The value of the series of paymets at the time the last paymet is made is deoted by s. The value oe uit of time after the last paymet is made is deoted by s. If i = 0 the s = s = ; otherwise s = (1 + i) 1 + (1 + i) 2 + (1 + i) = (1 + i) a = (1 i ) 1 i (1.1.4) ad s = (1 + i) + (1 + i) 1 + (1 + i) (1 + i) = (1 + i) a = (1 i ) 1 d (1.1.5) Thus s ad s are the values at the ed of ay period of legth of a series of paymets, each of amout 1, made at uit time itervals over the period, where the paymets are made i arrear ad i advace respectively. Sometimes s ad s are called the accumulatio (or the accumulated amout) of a immediate auity ad a auity due respectively. Whe = 0, s ad s are defied to be zero. It is a immediate cosequece of the above defiitio that s = (1 is ) ad that s 1 = 1 s or s = 1 1 s (1.1.6) Istitute ad Faculty of Actuaries Uit 6, Page 3

42 Subject CT1 Compoud iterest fuctios Cotiuously payable auities Let be a o-egative umber. The value at time 0 of a auity payable cotiuously betwee time 0 ad time, where the rate of paymet per uit time is costat ad equal to 1, is deoted by a. Clearly a = 0 e t dt = 1 e = 1 v (if 0) (1.2.1) Note that a is defied eve for o-itegral values of. If = 0 (or, equivaletly, i = 0), a is of course equal to. Sice equatio may be writte as a = it follows immediately that, if is a iteger, i 1 v i a = i a (if 0) (1.2.2) The accumulated amout of such a auity at the time the paymets cease is deoted by s. By defiitio, therefore, Hece s = 0 e (t) dt. s = (1 + i). a Uit 6, Page 4 Istitute ad Faculty of Actuaries

43 2016 Compoud iterest fuctios Subject CT1 If the rate of iterest is o-zero, 1.3 Auities payable pthly s = (1 i ) 1 i =. s If p ad are positive itegers, the otatio a is used to deote the value at time 0 of a level auity payable pthly i arrear at the rate of 1 per uit time over the time iterval [0, ]. For this auity the paymets are made at times 1/p, 2/p, 3/p,..., ad the amout of each paymet is 1/p. By defiitio, a series of p paymets, each of amout i (p) /p i arrear at pthly subitervals over ay uit time iterval, has the same value as a sigle paymet of amout i at the ed of the iterval. By proportio, p paymets, each of amout 1/p i arrear at pthly subitervals over ay uit time iterval, have the same value as a sigle paymet of amout i/i (p) at the ed of the iterval. Cosider ow that auity for which the preset value is a. The remarks i the precedig paragraph show that the p paymets after time r 1 ad ot later tha time r have the same value as a sigle paymet of amout i/i (p) at time r. This is true for r = 1, 2,...,, so the auity has the same value as a series of paymets, each of amout i/i (p), at times 1, 2,...,. This meas that a = i a i (1.3.1) A alterative approach, from first priciples, is to write a = p 1 v p t1 t/ p = = 1/ p 1 v (1 v ) p 1/ p 1 v 1 v 1/ p p[(1 i) 1] Istitute ad Faculty of Actuaries Uit 6, Page 5

44 Subject CT1 Compoud iterest fuctios 2016 = 1 v i (1.3.2) which cofirms equatio Likewise we defie a to be the preset value of a level auity due payable pthly at the rate of 1 per uit time over the time iterval [0, ]. (The auity paymets, each of amout 1/p, are made at times 0, 1/p, 2/p,..., (1/p).) By defiitio, a series of p paymets, each of amout d (p) /p, i advace at pthly subitervals over ay uit time iterval has the same value as a sigle paymet of amout i at ed of the iterval. Hece, by proportio, p paymets, each of amout 1/p i advace at pthly subitervals, have the same value as a sigle paymet of amout i/d (p) at the ed of the iterval. This meas (by a idetical argumet to that above) that a = d i a (1.3.3) Alteratively, from first priciples, we may write a = p 1 v p t1 ( t1)/ p = 1 v d (1.3.4) (o simplificatio), which cofirms equatio Note that a = v 1/p a (1.3.5) ( p ) each expressio beig equal to (1 v ). i Note that, sice lim p i (p) = lim p d (p) = it follows immediately from equatio ad that lim p a = lim p ( p ) a = a Uit 6, Page 6 Istitute ad Faculty of Actuaries

45 2016 Compoud iterest fuctios Subject CT1 Similarly, we defie s ad s to be the accumulated amouts of the correspodig pthly immediate auity ad auity due respectively. Thus ( p ) s = (1 + i) a = (1 + i) i a i (by 1.3.1) = i i s (1.3.6) Also ( p ) s = (1 + i) a = (1 + i) i a d (by 1.3.3) = d i s (1.3.7) The above proportioal argumets may be applied to other varyig series of paymets. Cosider, for example, a auity payable aually i arrear for years, the paymet i the tth year beig x t. The preset value of this auity is obviously a = 1 t x t vt (1.3.8) Cosider ow a secod auity, also payable for years with the paymet i the tth year, agai of amout x t, beig made i p equal istalmets i arrear over that year. If a (p) deotes the preset value of this secod auity, by replacig the p paymets for year t (each of amout x t /p) by a sigle equivalet paymet at the ed of the year of amout x t [i/i (p) ], we immediately obtai where a is give by equatio above. a (p) = i i a Istitute ad Faculty of Actuaries Uit 6, Page 7

46 Subject CT1 Compoud iterest fuctios Auities payable pthly where p < 1 I sectio 1.3 the symbol a was itroduced. Ituitively, with this otatio oe cosiders p to be a iteger greater tha 1 ad assumes that the product.p is also a iteger. (This, of course, will be true whe itself is a iteger, but oe might for example, have p = 4 ad = 5.75 so that.p = 23.) The a deotes the value at time 0 of.p paymets, each of amout 1/p, at times 1/p, 2/p,..., (p)/p. From a theoretical viewpoit it is perhaps worth otig that whe p is the reciprocal of a iteger ad.p is also a iteger (e.g. whe p = 0.25 ad = 28), a still gives the value at time 0 of.p paymets, each of amout 1/p, at times 1/p, 2/p,..., (p)/p. For example, the value at time 0 of a series of seve paymets, each of amout 4, at times (0.25) 4, 8, 12,..., 28 may be deoted by a 28. It follows that this value equals. This last expressio may be writte i the form 28 1 v 4 (0.25).[(1 i) 1] 4 1 v 4. (1 i) 1 i i a = 28 s4 1.5 No-iteger values of Let p be a positive iteger. Util ow the symbol a has bee defied oly whe is a positive iteger. For certai o-itegral values of the symbol a has a ituitively obvious iterpretatio. For example, it is ot clear what meaig, if ay, may be give to (4) a 23.5, but the symbol a 23.5 ought to represet the preset value of a immediate auity of 1 per aum payable quarterly i arrear for 23.5 years (i.e. a total of 94 quarterly (2) paymets, each of amout 0.25). O the other had, a has o obvious meaig. Uit 6, Page 8 Istitute ad Faculty of Actuaries

47 2016 Compoud iterest fuctios Subject CT1 Suppose that is a iteger multiple of 1/p, say = r/p, where r is a iteger. I this case we defie a to be the value at time 0 of a series of r paymets, each of amout 1/p, at times 1/p, 2/p, 3/p,..., r/p =. If i = 0, the clearly a =. If i 0, the a = 1 p (v1/p + v 2/p + v 3/p v r/p ) (1.5.1) = = 1 1/ p 1 v v p 1 v r/ p 1/ p / 1 r p 1 v p 1/ p (1 i) 1 Thus 1 v if i 0 i a = (1.5.2) if i = 0 Note that, by workig i terms of a ew time uit equal to 1/p times the origial time uit ad with the equivalet effective iterest rate of i (p) /p per ew time uit, we see that a at rate i = 1 ap p at rate i (p) /p (1.5.3) This formula is useful whe i (p) /p is a tabulated rate of iterest. Note that the defiitio of a give by equatio is mathematically meaigful for all o-egative values of. For our preset purpose, therefore, it is coveiet to adopt equatio as a defiitio of a for all. If is ot a iteger multiple of 1, there is o uiversally p recogised defiitio of a. For example, if = 1 + f, where 1 is a iteger multiple of 1/p ad 0 < f < 1/p, some writers defie a as a + fv. 1 Istitute ad Faculty of Actuaries Uit 6, Page 9

48 Subject CT1 Compoud iterest fuctios 2016 With this alterative defiitio (2) a = (2) a ¼v which is the preset value of a auity of 1 per aum, payable half-yearly i arrear for 23.5 years, together with a fial paymet of 0.25 after years. Note that this is ot equal to the value obtaied from defiitio If i 0, we defie for all o-egative a ( p ) = (1 + i) 1/p 1 v a = d s ( p ) = (1 + i) (1 i) 1 a = i s ( p ) = (1 + i) (1 i) 1 a = d If i = 0, each of these last three fuctios is defied to equal. (1.5.4) Wheever is a iteger multiple of 1/p, say = r/p, the,, a s s are values at differet times of a auity certai of r paymets, each of amout 1/p, at itervals of 1/p time uit. As before, we use the simpler otatios a, a, s ad s to deote (1) a, (1) a, s (1) s respectively, thus extedig the defiitio of a etc. to all o-egative values of. It is a trivial cosequece of our defiitios that the formulae (1), ad i a = ( p ) a i i a = a p d i s = ( p ) s i i s = s p d ( ) ( ) (1.5.5) (valid whe i 0) ow hold for all values of. Uit 6, Page 10 Istitute ad Faculty of Actuaries