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 These coditios remai i force after the Core Readig has bee superseded by a later editio.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

49 2016 Compoud iterest fuctios Subject CT1 1.6 Perpetuities We ca also cosider a auity that is payable forever. This is called a perpetuity. For example, cosider a equity that pays a divided of 10 at the ed of each year. Equities are covered i more detail i Uit 10 Sectio 3. A ivestor who purchases the equity pays a amout equal to the preset value of the divideds. The preset value of the divideds is: v10v 10v This ca be summed usig the formula for a ifiite geometric progressio: v 10 10v10v 10v 1 v i Recall the formula for the preset value of a auity of 10 p.a. that cotiues for years: 10a 10(1 v ) i We have let i this expressio i order to arrive at the formula 10 i. Note that this formula oly holds whe i is positive. I geeral: Perpetuity The preset value of paymets of 1 p.a. payable at the ed of each year forever is 1 i. This preset value is writte as a, i.e. 1 a i. The preset value of paymets of 1 p.a. payable at the start of each year forever is 1 d. This preset value is writte as a 1, i.e. a d. Istitute ad Faculty of Actuaries Uit 6, Page 11

50 Subject CT1 Compoud iterest fuctios 2016 Perpetuities payable pthly The preset value of paymets of 1 p.a. payable i istalmets of 1 p pthly time period forever is: at the ed of each a i 1 The preset value of paymets of 1 p.a. payable i istalmets of 1 p pthly time period forever is: at the start of each a 1 d 2 Deferred auities 2.1 Aual paymets Suppose that m ad are o-egative itegers. The value at time 0 of a series of paymets, each of amout 1, due at times (m + 1), (m + 2),..., (m + ) is deoted by (see the figure below). m a paymet m m + 1 m m + time Such a series of paymets may be cosidered as a immediate auity, deferred for m time uits. Whe > 0, m a = v m+1 + v m+2 + v m v m+ (2.1.1) = (v + v 2 + v v m+ ) (v + v 2 + v v m ) = v m (v + v 2 + v v ) Uit 6, Page 12 Istitute ad Faculty of Actuaries

51 2016 Compoud iterest fuctios Subject CT1 The last two equatios show that m a a a (2.1.2) = m m = v m a (2.1.3) Either of these two equatios may be used to determie the value of a deferred immediate auity. Together they imply that We may defie the correspodig deferred auity due as am = a m + vm a (2.1.4) = vm a (2.1.5) m a 2.2 Cotiuously payable auities If m is a o-egative umber, we use the symbol to deote the preset value of a cotiuously payable auity of 1 per uit for time uits, deferred for m time uits. Thus m a m a = m m e t dt = e m 0 e s ds = 0 m e t dt 0 m e t dt Hece m a a a (2.2.1) = m m = v m a (2.2.2) Istitute ad Faculty of Actuaries Uit 6, Page 13

52 Subject CT1 Compoud iterest fuctios Auities payable pthly The preset values of a immediate auity ad a auity due, payable pthly at the rate of 1 per uit time for time uits ad deferred for m time uits, are deoted by ad m a ( p ) m a = v m a = vm a (2.3.1) respectively. 2.4 No-iteger values of We may also exted the defiitios of m a ad m a to all values of by the formulae ad so m a ( p ) m a = v m a = vm a m a m a = a m am = a m a m (2.4.1) (2.4.2) 3 Varyig auities 3.1 Aual paymets For a auity i which the paymets are ot all of a equal amout it is a simple matter to fid the preset (or accumulated) value from first priciples. Thus, for example, the preset value of such a auity may always be evaluated as i1 X v where the ith paymet, of amout X i, is made at time t i. i t i Uit 6, Page 14 Istitute ad Faculty of Actuaries

53 2016 Compoud iterest fuctios Subject CT1 I the particular case whe X i = t i = i the auity is kow as a icreasig auity ad its preset value is deoted by ( Ia ). Thus ( Ia ) = v + 2v 2 + 3v v (3.1.1) Hece (1 + i) ( Ia ) = 1 + 2v + 3v v 1 By subtractio, we obtai iia ( ) = 1 + v + v v 1 v = a v so ( Ia ) = a v i (3.1.2) The preset value of ay auity payable i arrear for time uits for which the amouts of successive paymets form a arithmetic progressio ca be expressed i terms of a ad ( Ia. If the first paymet of such a auity is P ad the secod paymet is (P + Q), ) the tth paymet is (P Q) + Qt, the the preset value of the auity is (P Q) a Q( Ia) (3.1.3) Alteratively, the preset value of the auity ca be derived from first priciples. The otatio ( Ia is used to deote the preset value of a icreasig auity due payable ) for time uits, the tth paymet (of amout t) beig made at time t 1. Thus ( Ia ) = 1 + 2v + 3v v 1 = (1 + i) ( Ia ) (3.1.4) = 1 + a1 ( Ia) 1 (3.1.5) Istitute ad Faculty of Actuaries Uit 6, Page 15

54 Subject CT1 Compoud iterest fuctios Cotiuously payable auities For icreasig auities which are payable cotiuously it is importat to distiguish betwee a auity which has a costat rate of paymet r (per uit time) throughout the rth period ad a auity which has a rate of paymet t at time t. For the former the rate of paymet is a step fuctio takig the discrete values 1, 2,... For the latter the rate of paymet itself icreases cotiuously. If the auities are payable for time uits, their preset values are deoted by ( Ia ) ad ( Ia ) respectively. Clearly ad ( Ia ) = ( r1 r v t dt) r1 ( Ia ) = 0 t v t dt r ad it ca be show that ad ( Ia ) = a v (3.2.1) ( Ia ) = a v (3.2.2) The preset values of deferred icreasig auities are defied i the obvious maer: for example, m ( Ia ) = v m ( Ia ) E N D Uit 6, Page 16 Istitute ad Faculty of Actuaries

55 2016 Equatios of value Subject CT1 UNIT 7 EQUATIONS OF VALUE Syllabus objective (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. 1 The equatio of value ad the yield o a trasactio a t 1, a t 2,.., Cosider a trasactio that provides that, i retur for outlays of amout time t 1, t 2,... t, a ivestor will receive paymets of b t 1, b t 2,..., bt at these times respectively. (I most situatios oly oe of at r ad rate of iterest does the series of outlays have the same value as the series of receipts? At force of iterest the two series are of equal value if ad oly if at at bt r will be o-zero.) At what force or tr a e = r1 tr tr be t r r1 (1.1.1) This equatio may be writte as r1 tr tr c e = 0 (1.1.2) where c t r = b tr a tr is the amout of the et cashflow at time t r. (We adopt the covetio that a egative cashflow correspods to a paymet by the ivestor ad a positive cashflow represets a paymet to the ivestor.) Equatio 1.1.2, which expresses algebraically the coditio that, at force of iterest, the total value of the et cashflows is 0, is called the equatio of value for the force of iterest implied by the trasactio. If we let e = 1 + i, the equatio may be writte as c (1 ) r = 0 (1.1.3) t r1 r t i Istitute ad Faculty of Actuaries Uit 7, Page 1

56 Subject CT1 Equatios of value 2016 The latter form is kow as the equatio of value for the rate of iterest or the yield equatio. Alteratively, the equatio may be writte as r ct v = 0 r r1 I relatio to cotiuous paymet streams, if we let 1 (t) ad 2 (t) be the rates of payig ad receivig moey at time t respectively, we call (t) = 2 (t) 1 (t) the et rate of cashflow at time t. The equatio of value (correspodig to equatio 1.1.2) for the force of iterest is t 0 (t) e t dt = 0 (1.1.4) Whe both discrete ad cotiuous cashflows are preset, the equatio of value is r1 ad the equivalet yield equatio is tr tr t 0 () c e t e dt = 0 (1.1.5) 0 r c (1 i) ( t) (1 + i) t dt = 0 (1.1.6) t r1 r t For ay give trasactio, equatio may have o roots, a uique root, or several roots. If there is a uique root, 0 say, it is kow as the force of iterest implied by the 0 trasactio, ad the correspodig rate of iterest i 0 = e 1 is called the yield per uit time. (Alterative terms for the yield are the iteral rate of retur ad the moeyweighted rate of retur for the trasactio.) Thus the yield is defied if ad oly if equatio has precisely oe root greater tha 1 ad, whe such a root exists, it is the yield. The aalysis of the equatio of value for a give trasactio may be somewhat complex depedig o the shape of the fuctio f(i) deotig the left had side of equatio However, whe the equatio f(i) = 0 is such that f is a mootoic fuctio, its aalysis is particularly simple. The equatio has a root if ad oly if we ca fid i 1 ad i 2 with f(i 1 ) ad f(i 2 ) of opposite sig. I this case, the root is uique ad lies betwee i 1 ad i 2. By choosig i 1 ad i 2 to be tabulated rates sufficietly close to each other, we may determie the yield to ay desired degree of accuracy. Uit 7, Page 2 Istitute ad Faculty of Actuaries

57 2016 Equatios of value Subject CT1 0 It should be oted that, after multiplicatio by (1 i) t, equatio takes the equivalet form t0 tr ct (1 i) r r1 = 0 (1.1.7) This slightly more geeral form may be called the equatio of value at time t 0. It is of course directly equivalet to the origial equatio (which is ow see to be the equatio of value at time 0). I certai problems a particular choice of t 0 may simplify the solutio. 2 Ucertai paymet or receipt If there is ucertaity about the paymet or receipt of a cashflow at a particular time, allowace ca be made i oe of two ways: apply a probability of paymet/receipt to the cashflow at each time use a higher rate of discout 2.1 Probability of cashflow The probability of paymet/receipt ca be allowed for by adaptig the earlier equatios. For example, equatio ca be revised to produce: r1 0 p c (1 i) r p( t) ( t) (1 + i) t dt = 0 (2.1.1) t r t r t where p ad p( t ) represet the probability of a cashflow at time t. t r r Where the force of iterest is costat, ad we ca say that the probability is itself i the form of a discoutig fuctio, the equatio ca be geeralised as: r1 tr tr tr e 0 c e (t) e t e t dt = 0 (2.1.2) where is a costat force, rather tha rate, of the probability of a cashflow at time t. These probabilities of cashflows may ofte be estimated by cosideratio of the past experiece of similar cashflows. For example, this approach is used to assess the probabilities of cashflows that are depedet o the survival of a life this is the theme of Subjects CT4, Models ad CT5, Cotigecies. I other cases, there may be lack of data from which to determie a accurate probability for a cashflow. Istead a more approximate probability, or likelihood, may be determied after careful cosideratio of the risks. Istitute ad Faculty of Actuaries Uit 7, Page 3

58 Subject CT1 Equatios of value 2016 I some cases, it may be spurious to attempt to determie the probability of each cashflow ad so more approximate methods may be justified. Wherever the ucertaity about the probability of the amout or timig of a cashflow could have sigificat fiacial effect, a sesitivity aalysis may be performed. This ivolves calculatios performed usig differet possible values for the likelihood ad the amouts of the cashflows. Alteratively a stochastic approach could be used to idicate possible outcomes (see Uit 14 ad Subject CT4, Models). 2.2 Higher discout rate As the discoutig fuctios ad the probability fuctios i equatios ad are both depedet o time, they ca be combied ito a sigle time depedet fuctio. I cases where there is isufficiet iformatio to objectively produce the probability fuctios, this combied fuctio ca be viewed as a adjusted discoutig fuctio that makes a implicit allowace for the probability of the cashflow. Where the probability of the cashflow is a fuctio that is of similar form to the discoutig fuctio, the combiatio ca be treated as if a differet discout rate were beig used. For example, equatio becomes: r1 t 0 r ct e (t) e 't dt = 0 r where ' = + The revised force of discout is therefore greater tha the actual force of discout as must be positive i order to give a probability betwee 0 ad 1. It ca therefore be show that the rate of discout that is effectively used is greater tha the actual rate of discout before the implicit allowace for the probability of the cashflow. E N D Uit 7, Page 4 Istitute ad Faculty of Actuaries

59 2016 Loa schedules Subject CT1 UNIT 8 LOAN SCHEDULES Syllabus objective (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. 1 Itroductio A very commo trasactio ivolvig compoud iterest is a loa that is repaid by regular istalmets, at a fixed rate of iterest, for a predetermied term. Cosider a very simple example. Assume a bak leds a idividual 1,000 for three years, i retur for three paymets of X, say, oe at the ed of each year. The bak will charge a effective rate of iterest of 7% per aum. The equatio of value for the trasactio gives: 1000 = Xa 3 X = So the borrower pays at times t = 1, 2 ad 3 i retur for the loa of 1,000 at time 0. These three paymets cover both the iterest due ad the 1,000 capital. It is helpful to see how this works i detail: At time 1 the iterest due o the loa of 1000 is 70. The total paymet made is This leaves that is available to repay some of the capital. The capital outstadig after this is the ( ) = At time 2 the iterest due is ow oly 7% of = 48.22, as the borrower does ot pay iterest o the capital that is already repaid, oly o the amout outstadig. This leaves ( ) = available to repay capital. The capital outstadig after this is the ( ) = Fially, at time 3 the iterest due is 7% of = 24.93, leavig = available to pay the outstadig sum of , ad the capital is precisely repaid. Istitute ad Faculty of Actuaries Uit 8, Page 1

60 Subject CT1 Loa schedules 2016 Oe importat poit is that each repaymet must pay first for iterest due o the outstadig capital. The balace is the used to repay some of the capital outstadig. Each paymet therefore comprises both iterest ad capital repaymet. It may be ecessary to idetify the separate elemets of the paymets for example if the tax treatmet of iterest ad capital differs. Notice also that, where repaymets are level, the iterest compoet of the repaymet istalmets will decrease as capital is repaid, with the cosequece that the capital paymet will icrease. 2 Calculatig the capital outstadig Let L t be the amout of the loa outstadig at time t = 0, 1,...,, immediately after the repaymet at t. The repaymets are assumed to be i regular istalmets, of amout X t at time t, t = 1, 2, 3,...,. (Note that we are ot assumig all istalmets are the same amout.) Let i be the effective rate of iterest, per time uit, charged o the loa. Let f t be the capital repaid at t, ad let b t be the iterest paid at t, so that X t = f t + b t. The equatio of value for the loa at time 0 is: L 0 = X 1 v + X 2 v X v We ca fid the loa outstadig at t prospectively or retrospectively. 2.1 Prospective loa calculatio Cosider the loa trasactios at time, which is the ed of the cotract term. After the fial istalmet of capital ad iterest the loa is exactly repaid. So the fial istalmet, X must exactly cover the capital that remais outstadig after the istalmet paid at 1, together with the iterest due o that capital. That is: b = il 1 ; f = L 1 so that X = il 1 + L 1 = (1 + i) L 1 L 1 = X v Similarly, at ay time t + 1, t 2 we kow that the capital repaid is L t L t+1, so that the istalmet X t+1 is: X t+1 = il t + (L t L t+1 ) L t = (L t+1 + X t+1 ) v Uit 8, Page 2 Istitute ad Faculty of Actuaries

61 2016 Loa schedules Subject CT1 Similarly, L t+1 = L t+2 + X t+2 v, ad workig forward, successively substitutig for L t+r util we get to L = 0, we get: L t = (L t+1 + X t+1 ) v = ((L t+2 + X t+2 ) v + X t+1 ) v = X t+1 v + X t+2 v 2 + L t+2 v 2 = X t+1 v + X t+2 v 2 + X t+3 v 3 + L t+3 v 3 = = X t+1 v + X t+2 v 2 + X t+3 v X v t This gives the prospective method for calculatig the loa outstadig. What this equatio tells us is that, for calculatig the loa outstadig immediately after the repaymet at t, say, we have: Prospective Method: The loa outstadig at time t is the preset (or discouted) value at time t of the future repaymet istalmets. Note carefully the coditio for this method the preset value must be calculated at a repaymet date. 2.2 Retrospective loa calculatio At t = 1 the iterest due is b 1 = il 0, so the capital repaid is f 1 = X 1 il 0, leavig a loa outstadig of: L 1 = L 0 (X 1 il 0 ) = L 0 (1 + i) X 1 I geeral, at time t 1 the iterest due is b t = il t1, leavig capital repaid at t of X t il t1, givig L t = L t1 (1 + i) X t Istitute ad Faculty of Actuaries Uit 8, Page 3

62 Subject CT1 Loa schedules 2016 Similarly, L t1 = L t2 (1 + i) X t1 ad, workig back from t to 0 we have: L t = L t1 (1 + i) X t = (L t2 (1 + i) X t1 ) (1 + i) X t = L t2 (1 + i) 2 X t1 (1 + i) X t = L 0 (1 + i) t (X 1 (1 + i) t1 + X 2 (1 + i) t X t1 (1 + i) + X t ) This gives the retrospective method of calculatig the outstadig loa. This may be described i words as: Retrospective Method: The loa outstadig at time t is the accumulated value at time t of the origial loa less the accumulated value at time t of the repaymets to date. Both of these approaches are very useful i calculatig the capital outstadig at ay time. Neither result actually depeds o the iterest rate beig costat. It may be useful to work through the equatios assumig the iterest charged o the loa i year r 1 to r is i r, say. 3 Calculatig the iterest ad capital elemet of the repaymets Give the outstadig capital at ay time we ca calculate the iterest ad capital elemet of ay istalmet. For example, cosider the istalmet X t at time t. We ca calculate the iterest elemet cotaied i this paymet by calculatig the loa outstadig immediately after the previous istalmet, at t 1, L t1. The iterest due o capital of L t1 for oe uit of time at effective rate i per time uit is il t1, ad this is the iterest paid at t. The capital repaid may be foud usig X t il t1, or by L t1 L t. Similarly, it is a simple matter to calculate the iterest paid ad capital repaid over several istalmets. For example, cosider the five istalmets from t + 1 to t + 5, iclusive. The loa outstadig immediately before the first istalmet is L t. The loa outstadig after the fifth istalmet is L t+5. The total capital repaid is therefore L t L t+5. The total capital ad iterest paid is X t+1 + X t X t+5. Hece, the total iterest paid is t5 b k = (X t+1 + X t X t+5 ) (L t L t+5 ). k t 1 Uit 8, Page 4 Istitute ad Faculty of Actuaries

63 2016 Loa schedules Subject CT1 4 The loa schedule The loa paymets ca be expressed i the form of a table, or schedule, as follows. Year Loa Istalmet Iterest Capital Loa r r + 1 outstadig at r + 1 due repaid outstadig at r at r + 1 at r + 1 at r L 0 X 1 il 0 X 1 il 0 L 1 = L 0 (X 1 il 0 ) t t + 1 L t X t+1 il t X t+1 il t L t+1 = L t (X t+1 il t ) 1 L 1 X il 1 X il 1 0 With spreadsheet software it is a simple matter to costruct the etire schedule for ay loa. 5 Repaymet istalmets payable more frequetly tha aually Most loas will be repaid i quarterly, mothly or weekly istalmets. No ew priciples are ivolved where paymets are made more frequetly tha aually, but care eeds to be take i calculatig the iterest due at ay istalmet date. If the rate of iterest used is effective over the same time uit as the frequecy of the repaymet istalmets, the the calculatios proceed exactly as above, with the time uit redefied appropriately. For the case where the iterest is expressed as a effective aual rate, with repaymet istalmets payable pthly, we have the equatio of value for the loa, give repaymets of 3 X t at time t = 1, 2,,...,, p p p L 0 = X 1/p v 1/p + X 2/p v 2/p + X 3/p v 3/p X v It is easy to show that the two basic priciples for calculatig the loa outstadig hold whe repaymets are more frequet tha aual. That is, the loa outstadig at ay repaymet date, immediately after a istalmet has bee paid, may still be calculated as Istitute ad Faculty of Actuaries Uit 8, Page 5

64 Subject CT1 Loa schedules 2016 the preset value of the remaiig repaymet istalmets, or as the accumulated value of the origial loa less the repaymets made to date. Prospectively: L t = X t+1/p v 1/p + X t+2/p v 2/p X v t Retrospectively: L t = L 0 (1 + i) t t t p p p 1 2 t 1 p p p ( X (1 i) X (1 i)... X (1 i) X ) t Give a aual effective rate of iterest of i, the effective rate of iterest over a period 1 p is (1 + i) 1/p 1, which is equal to i (p) /p. The iterest due at t + 1, give capital p outstadig of L t at some repaymet date t, is therefore b t+1/p = ((1 + i) 1/p 1) L t. The capital repaid at t + 1 is the p f t+1/p = X ((1 + i) 1/p 1) L = 1 t 1 t p Xt p i p L t 6 Cosumer credit: flat rates ad APRs Where the borrower is a idividual, borrowig from a istitutio such as a bak, it is commo to use the flat rate of iterest as a measure of the iterest charge. The flat rate of iterest is defied as the total iterest paid over the whole trasactio, per uit of iitial loa, per year of the loa. For example, if a loa of L 0 is repaid over two years by level mothly istalmets of amout X, the the total capital ad iterest paid is 24X. The total capital must be the amout of the origial loa, so the total iterest paid is 24X L 0. This gives the flat rate of iterest per aum 24X L0 F = 2L 0 The flat rate is a very simple calculatio that igores the details of the gradual repaymet of capital over the term of a loa. Flat rates are oly useful for comparig loas of equal term. Two loas of differet terms calculated usig the same effective rate of iterest will have differet flat rates. Sice the flat rate igores the repaymet of capital over the term of the loa, it will be cosiderably lower tha the true effective rate of iterest charged o the loa. Uit 8, Page 6 Istitute ad Faculty of Actuaries

65 2016 Loa schedules Subject CT1 To esure that cosumers ca make iformed judgemets about the iterest rates charged, leders are required (i most circumstaces) to give iformatio about the effective rate of iterest charged. I the UK this is i the form of the Aual Percetage Rate of charge, or APR, which is defied as the effective aual rate of iterest, rouded to the earer 1/10th of 1%. E N D Istitute ad Faculty of Actuaries Uit 8, Page 7

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67 2016 Project appraisal Subject CT1 UNIT 9 PROJECT APPRAISAL Syllabus objective (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 timeweighted rate of retur ad the liked iteral rate of retur o a ivestmet or a fud. 1 Itroductio Suppose a ivestor is cosiderig the merits of a ivestmet or busiess project. The ivestmet or project will ormally require a iitial outlay ad possibly other outlays i future, which will be followed by receipts, although i some cases the patter of icome ad outgo is more complicated. The cashflows associated with the ivestmet or busiess veture may be completely fixed (as i the case of a secure fixed-iterest security maturig at a give date) or they may have to be estimated. The estimatio of the cash iflows ad outflows associated with a busiess project usually requires cosiderable experiece ad judgemet. All the relevat factors (such as taxatio ad ivestmet grats) ad risks (such as costructio delays) should be cosidered by the actuary, with assistace from experts i the relevat field (e.g. civil egieerig for buildig projects). The idetificatio ad assessmet of the risks may be doe usig the Risk Aalysis ad Maagemet for Projects (RAMP) approach for risk aalysis ad maagemet that has bee developed by, ad published o behalf of, the actuarial ad civil egieerig professios. Cosiderable ucertaity will exist i the assessmet of may of the risks, so it is prudet to perform calculatios o more tha oe set of assumptios, e.g. o the basis of optimistic, average, ad pessimistic forecasts respectively. More complicated techiques (usig statistical theory) are available to deal with this kid of ucertaity. Precisio is ot attaiable i the estimatio of cashflows for may busiess projects ad hece extreme accuracy is out of place i may calculatios. Istitute ad Faculty of Actuaries Uit 9, Page 1

68 Subject CT1 Project appraisal 2016 Net cashflow c t at time t (measured i suitable time uits) is c t = cash iflow at time t cash outflow at time t (1.1.1) If ay paymets may be regarded as cotiuous the (t), the et rate of cashflow per uit time at time t, is defied as (t) = 1 (t) 2 (t) (1.1.2) where 1 (t) ad 2 (t) deote the rates of iflow ad outflow at time t respectively. 2 Fixed iterest rates 2.1 Net preset values Havig ascertaied or estimated the et cashflows of the ivestmet or project uder scrutiy, the ivestor will wish to measure its profitability i relatio to other possible ivestmets or projects. I particular, the ivestor may wish to determie whether or ot it is prudet to borrow moey to fiace the veture. Assume for the momet that the ivestor may borrow or led moey at a fixed rate of iterest i per uit time. The ivestor could accumulate the et cashflows coected with the project i a separate accout i which iterest is payable or credited at this fixed rate. By the time the project eds (at time T, say), the balace i this accout will be c t (1 + i) Tt + 0 T (t) (1 + i) Tt dt (2.1.1) where the summatio exteds over all t such that c t 0. The preset value at rate of iterest i of the et cashflows is called the et preset value at rate of iterest i of the ivestmet or busiess project, ad is usually deoted by NPV(i). Hece NPV(i) = c t (1 + i) t + 0 T (t) (1 + i) t dt (2.1.2) (If the project cotiues idefiitely, the accumulatio is ot defied, but the et preset value may be defied by equatio with T =.) If (t) = 0, we obtai the simpler formula where v = (1 + i) 1. NPV(i) = c t v t (2.1.3) Uit 9, Page 2 Istitute ad Faculty of Actuaries

69 2016 Project appraisal Subject CT1 Sice the equatio NPV(i) = 0 (2.1.4) is the equatio of value for the project at the preset time, the yield i 0 o the trasactio is the solutio of this equatio, provided that a uique solutio exists. It may readily be show that NPV(i) is a smooth fuctio of the rate of iterest i ad that NPV(i) c 0 as i. 2.2 Iteral rate of retur I ecoomics ad accoutacy the yield per aum is ofte referred to as the iteral rate of retur (IRR) or the yield to redemptio. The latter term is frequetly used whe dealig with fixed-iterest securities, for which the ruig (or flat ) yield is also cosidered. The practical iterpretatio of the et preset value fuctio NPV(i) ad the yield is as follows. Suppose that the ivestor may led or borrow moey at a fixed rate of iterest i 1. Sice, from equatio 2.1.2, NPV(i 1 ) is the preset value at rate of iterest i 1 of the et cashflows associated with the project, we coclude that the project will be profitable if ad oly if NPV(i 1 ) > 0 (2.2.1) Also, if the project eds at time T, the the profit (or, if egative, loss) at that time is (by expressio 2.1.1) NPV(i 1 )(1 + i 1 ) T (2.2.2) Let us ow assume that, as is usually the case i practice, the yield i 0 exists ad NPV(i) chages from positive to egative whe i = i 0. Uder these coditios it is clear that the project is profitable if ad oly if i 1 < i 0 (2.2.3) i.e. the yield exceeds that rate of iterest at which the ivestor may led or borrow moey. May projects will eed to provide a retur to shareholders ad so there will ot be a specific fixed rate of iterest that has to be exceeded. Istead a target, or hurdle, rate of retur may be set for assessig whether a project is likely to be sufficietly profitable. Istitute ad Faculty of Actuaries Uit 9, Page 3

70 Subject CT1 Project appraisal Accumulated value The accumulated value, at time T, of a cashflow ca be expressed as: A(T) = c t (1 + i) Tt + 0 T (t) (1 + i) Tt dt (2.3.1) 2.4 The compariso of two ivestmet projects Suppose ow that a ivestor is comparig the merits of two possible ivestmets or busiess vetures, which we call projects A ad B respectively. We assume that the borrowig powers of the ivestor are ot limited. Let NPV A (i) ad NPV B (i) deote the respective et preset value fuctios ad let i A ad i B deote the yields (which we shall assume to exist). It might be thought that the ivestor should always select the project with the higher yield, but this is ot ivariably the best policy. A better criterio to use is the profit at time T (the date whe the later of the two projects eds) or, equivaletly, the et preset value, calculated at the rate of iterest i 1 at which the ivestor may led or borrow moey. This is because A is the more profitable veture if NPV A (i 1 ) > NPV B (i 1 ) (2.4.1) Uit 9, Page 4 Istitute ad Faculty of Actuaries

71 2016 Project appraisal Subject CT1 The fact that i A > i B may ot imply that NPV A (i 1 ) > NPV B (i 1 ) is illustrated i Figure Although i A is larger tha i B, the NPV(i) fuctios cross over at i. It follows that NPV B (i 1 ) > NPV A (i 1 ) for ay i 1 < i, where i is the cross-over rate. There may eve be more tha oe cross-over poit, i which case the rage of iterest rates for which project A is more profitable tha project B is more complicated. Example Figure Ivestmet compariso A ivestor is cosiderig whether to ivest i either or both of the followig loas: Loa A Loa B For a purchase price of 10,000, the ivestor will receive 1,000 per aum payable quarterly i arrear for 15 years. For a purchase price of 11,000, the ivestor will receive a icome of 605 per aum, payable aually i arrear for 18 years, ad a retur of his outlay at the ed of this period. The ivestor may led or borrow moey at 4% per aum. Would you advise the ivestor to ivest i either loa, ad, if so, which would be the more profitable? Istitute ad Faculty of Actuaries Uit 9, Page 5

72 Subject CT1 Project appraisal 2016 Solutio We first cosider loa A: NPV A (i) = 10,000 + (4) 1, 000a 15 ad the yield is foud by solvig the equatio NPV A (i) = 0, or 5.88%. For loa B we have NPV B (i) = 11, a ,000v 18 (4) a 15 = 10, which gives i A ad the yield (i.e. the solutio of NPV B (i) = 0) is i B = 5.5%. The rate of iterest at which the ivestor may led or borrow moey is 4% per aum, which is less tha both i A ad i B, so we compare NPV A (0.04) ad NPV B (0.04). Now NPV A (0.04) = 1,284 ad NPV B (0.04) = 2,089, so it follows that, although the yield o loa B is less tha o loa A, the ivestor will make a larger profit from loa B. We should therefore advise him that a ivestmet i either loa would be profitable, but that, if oly oe of them is to be chose, the loa B will give the higher profit. The above example illustrates the fact that the choice of ivestmet depeds very much o the rate of iterest i 1 at which the ivestor may led or borrow moey. If this rate of iterest were 5¾%, say, the loa B would produce a loss to the ivestor, while loa A would give a profit. 3 Differet iterest rates for ledig ad borrowig We have assumed so far that the ivestor may borrow or led moey at the same rate of iterest i 1. I practice, however, the ivestor will probably have to pay a higher rate of iterest (j 1, say) o borrowigs tha the rate (j 2, say) he receives o ivestmets. The differece j 1 j 2 betwee these rates of iterest depeds o various factors, icludig the credit-worthiess of the ivestor ad the expese of raisig a loa. The cocepts of et preset value ad yield are i geeral o loger meaigful i these circumstaces. We must calculate the accumulatio of et cashflows from first priciples, the rate of iterest depedig o whether or ot the ivestor s accout is i credit. I may practical problems the balace i the ivestor s accout (i.e. the accumulatio of et cashflows) will be egative util a certai time t 1 ad positive afterwards, except perhaps whe the project eds. Uit 9, Page 6 Istitute ad Faculty of Actuaries

73 2016 Project appraisal Subject CT1 I some cases the ivestor must fiace his ivestmet or busiess project by meas of a fixed-term loa without a early repaymet optio. I these circumstaces the ivestor caot use a positive cashflow to repay the loa gradually, but must accumulate this moey at the rate of iterest applicable o ledig, i.e. j Payback periods I may practical problems the et cashflow chages sig oly oce, this chage beig from egative to positive. I these circumstaces the balace i the ivestor s accout will chage from egative to positive at a uique time t 1, or it will always be egative, i which case the project is ot viable. If this time t 1 exists, it is referred to as the discouted payback period (DPP). It is the smallest value of t such that A(t) 0, where A(t) = c s (1 + j 1 ) ts t + 0 (s) (1 + j 1 ) ts ds (3.1.1) st Note that t 1 does ot deped o j 2 but oly o j 1, the rate of iterest applicable to the ivestor s borrowigs. Suppose that the project eds at time T. If A(T) < 0 (or, equivaletly, if NPV(j 1 ) < 0) the project has o discouted payback period ad is ot profitable. If the project is viable (i.e. there is a discouted payback period t 1 ) the accumulated profit whe the project eds at time T is 1 P = A(t 1 ) (1 j2) T t c t (1 + j 2 ) Tt tt 1 T t1 (t) (1 + j 2 ) Tt dt (3.1.2) This follows sice the et cashflow is accumulated at rate j 2 after the discouted payback period has elapsed. If iterest is igored i formula (i.e. if we put j 1 = 0), the resultig period is called the payback period. However, its use istead of the discouted payback period ofte leads to erroeous results ad is therefore ot to be recommeded. The discouted payback period is ofte employed whe cosiderig a sigle ivestmet of C, say, i retur for a series of paymets each of R, say, payable aually i arrear for years. The discouted payback period t 1 years is clearly the smallest iteger t such that A*(t) 0, where A*(t) = C(1 + j 1 ) t + i.e. the smallest iteger t such that Rs t at rate j 1 (3.1.3) Ra t C at rate j 1 (3.1.4) Istitute ad Faculty of Actuaries Uit 9, Page 7

74 Subject CT1 Project appraisal 2016 The project is therefore viable if t 1, i which case the accumulated profit after years is clearly 1 P = A*(t 1 ) (1 j ) t Rst at rate j 2 (3.1.5) Measuremet of ivestmet performace It is ofte ecessary to be able to measure the ivestmet performace of a fud (for example a pesio fud, or the fuds of a isurace compay) over a period. 4.1 Moey Weighted Rate of Retur Oe measure of the performace is the yield eared o the fud over the period. For example, cosider a fud with value F 0 at time 0, with et cashflows Ct k at times t 1, t 2,..., t ad fud value F T at time T t, the the equatio of value, equatig values at time T, is Tt 1 2 Tt F 0 (1 + i) T Ct (1 i) Ct (1 i)... Ct (1 i) = F T where i is the effective aual rate of iterest eared by the fud i the iterval [0, T]. I this equatio of value the left had side is the value at time T of the fud at the start of the period plus or mius all the cashflows received or paid out i the iterval. The yield eared o the fud is also called the moey weighted rate of retur (MWRR). As a measure of ivestmet performace the moey weighted rate of retur is ot etirely satisfactory, as it is sesitive to the amouts ad timig of the et cashflows. If, say, we are assessig the skill of the fud maager, this is ot ideal, as the fud maager does ot cotrol the timig or amout of the cashflows he or she is merely resposible for ivestig the positive et cashflows ad realisig cash to meet the egative et cashflows. 4.2 Time Weighted Rate of Retur Defie F 0, F T, ad Ct k as above, ad let C 0 be the cashflow (if ay) at time t = 0. I additio let Ft k be the amout of the fud just before the cashflow due at time t k, so that the amout of the fud just after the receipt of the et cashflow due at time t k is Ftk Ct. k The the Time Weighted Rate of Retur (TWRR) is i per aum, where (1 + i) T Ft 1 Ft2 Ft 3 FT =... F C F C F C F C 0 0 t1 t1 t2 t2 t t Tt Uit 9, Page 8 Istitute ad Faculty of Actuaries

75 2016 Project appraisal Subject CT1 Each factor o the right had side gives the proportioate icrease i the fud betwee cashflows. The product of these factors gives the otioal accumulatio factor for a sigle ivestmet of 1 at time t = 0, ivested util time T. Usig the TWRR elimiates the effect of the cashflow amouts ad timig, ad therefore gives a fairer basis for assessig the ivestmet performace for the fud. The disadvatages of both the time weighted ad moey weighted rates of retur are that the calculatio requires iformatio about all the cashflows of the fud durig the period of iterest. I additio, the TWRR requires the fud values at all the cashflow dates. A disadvatage of the MWRR is that the equatio may ot have a uique solutio or ideed ay solutio. If the fud performace is reasoably stable i the period of assessmet the the TWRR ad the MWRR will give similar results. 4.3 Liked Iteral Rate of Retur If the rate of retur o a fud is measured over a series of itervals (0, t 1 ), (t 1, t 2 ), (t 2, t 3 ),..., (t 1, t ), such that the aual effective rate of iterest eared by a fud i the iterval (t r1, t r ) is i r (where i 1 is the aual rate eared i (0, t 1 )) the the Liked Iteral Rate of Retur is i per aum, where t t t t t t t t (1 i) = (1 i ) (1 i ) (1 i )... (1 i ) The liked iteral rate of retur will be equal to the TWRR if the sub itervals (t r1, t r ) are the same i each calculatio. I practice, the yields i r may be calculated by approximate methods, ad the, if the sub itervals used are sufficietly short, the liked iteral rate of retur will be close to the TWRR. E N D Istitute ad Faculty of Actuaries Uit 9, Page 9

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77 2016 Ivestmets Subject CT1 UNIT 10 INVESTMENTS Syllabus objective (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 idex-liked govermet borrowigs shares ad other equity-type fiace derivatives 1 Fixed iterest govermet borrowigs 1.1 Fixed iterest govermet bods A govermet or govermet body may raise moey by floatig a loa o a stock exchage. The terms of the issue are set out by the borrower ad ivestors may be ivited to subscribe to the loa at a give price (called the issue price), or the issue may be by teder, i which case ivestors are ivited to omiate the price that they are prepared to pay ad the loa is the issued to the highest bidders, subject to certai rules of allocatio. The aual iterest payable to each holder, which is ofte but ot ivariably payable halfyearly, is foud by multiplyig the omial amout of his holdig N by the rate of iterest per aum D, which is geerally called the coupo rate. The moey payable at redemptio is calculated by multiplyig the omial amout held N by the redemptio price R per uit omial (which is ofte quoted per cet i practice). If R = 1 the stock is said to be redeemable at par; if R > 1 the stock is said to be redeemable above par or at a premium; ad if R < 1 the stock is said to be redeemable below par or at a discout. The redemptio date is the set date o which the redemptio moey is due to be paid. Some bods have variable redemptio dates, i which case the redemptio date may be chose by the borrower (or perhaps the leder) as ay iterest date withi a certai period, or ay iterest date o or after a give date. I the latter case the stock is said to have o fial redemptio date, or to be udated. Some baks allow the iterest ad redemptio proceeds to be bought ad sold separately, effectively creatig bods with o coupo ad bods redeemable at zero. The coupo rate, redemptio price ad term to redemptio of a fixed iterest security serve to defie the cash paymets promised to a tax-free ivestor i retur for his purchase price. If the ivestor is subject to taxatio, appropriate deductios from the cashflow must be made. For example, if a ivestor is liable to icome tax at rate t 1 o the iterest paymets, the aual icome after tax will be (1 t 1 )DN. I most developed ecoomies, bods issued by the govermet form the largest, most importat ad most liquid part of the bod market. Ivestors ca therefore deal i large quatities with little (or o) impact o the price. Bods issued by the govermets of Istitute ad Faculty of Actuaries Uit 10, Page 1

78 Subject CT1 Ivestmets 2016 developed coutries i their domestic currecy are the most secure log-term ivestmet available. However, this security together with the low volatility of retur relative to other log-term ivestmets should lead to a low expected retur, though this will be compesated for to a extet by very low dealig costs. Relative to iflatio, however, the icome stream may be volatile. Some govermets therefore issue bods that provide iterest ad redemptio paymets that are liked to a iflatio idex. However, idexatio will eed to be based o the movemet of the iflatio idex with a time lag to allow for publicatio of the idex figure ad the eed to calculate moetary amouts of coupos i advace. There is effectively o iflatio protectio durig the lag period. 1.2 Govermet bills Govermet bills are short-dated securities issued by govermets to fud their short-term spedig requiremets. They are issued at a discout ad redeemed at par with o coupo. They are mostly deomiated i the domestic currecy, although issues ca be made i other currecies. The yield o govermet bills is typically quoted as a simple rate of discout for the term of the bill. For example, a 3-moth bill may be quoted as beig offered at a discout of 2%. This would mea that the iitial ivestmet required to buy the bill would be 2% less tha the paymet 3 moths later. Govermet bills are absolutely secure ad ofte highly marketable, despite ot beig quoted. They are ofte used as a bechmark risk-free short-term ivestmet. 2 Fixed iterest borrowig by other bodies 2.1 Characteristics of corporate debt Corporate bods are, i may ways, similar to covetioal govermet bods i their characteristics. Here the debt is issued by a compay rather tha a govermet. The major differeces betwee corporate bods ad govermet bods are: Corporate bods are usually less secure tha govermet bods. The level of security depeds o the type of bod, the compay which has issued it, ad the term. Corporate bods are usually less marketable tha govermet bods, maily because the sizes of issues are much smaller. Uit 10, Page 2 Istitute ad Faculty of Actuaries

79 2016 Ivestmets Subject CT1 2.2 Debetures Debetures are part of the loa capital of compaies. The term loa capital usually refers to log-term borrowigs rather tha short-term. The issuig compay provides some form of security to holders of the debeture. Debeture stocks are cosidered more risky tha govermet bods ad are usually less marketable. Accordigly the yield required by ivestors will be higher tha for a comparable govermet bod. 2.3 Usecured loa stocks Usecured loa stocks are issued by various compaies. They are usecured holders rak alogside other usecured creditors. Yields will be higher tha o comparable debetures issued by the same compay, to reflect the higher default risk. 2.4 Eurobods Eurobods are a form of usecured medium or log-term borrowig made by issuig bods which pay regular iterest paymets ad a fial capital repaymet at par. Eurobods are issued ad traded iteratioally ad are ofte ot deomiated i a currecy ative to the coutry of the issuer. Eurobods are issued by large compaies, govermets ad supra-atioal orgaisatios. They are usually usecured. Yields deped upo the issuer (ad hece risk) ad issue size (ad hece marketability), but will typically be slightly lower tha for the covetioal usecured loa stocks of the same issuer. The features of Eurobods vary a lot more tha traditioal bod issues. I the absece of ay full-blow govermet cotrol, issuers have bee free to add ovel features to their issues. They do this to make them appeal to differet ivestors. 2.5 Certificates of deposit A certificate of deposit is a certificate statig that some moey has bee deposited. They are issued by baks ad buildig societies. Terms to maturity are usually i the rage 28 days to 6 moths. Iterest is payable o maturity. The degree of security ad marketability will deped o the issuig bak. There is a active secodary market i certificates of deposit. 3 Shares ad other equity-type borrowig 3.1 Ordiary shares Ordiary shares also called equities are securities, issued by commercial udertakigs ad other bodies, which etitle their holders to receive all the et profits of Istitute ad Faculty of Actuaries Uit 10, Page 3

80 Subject CT1 Ivestmets 2016 the compay after iterest o loas ad fixed iterest stocks has bee paid. The cash paid out each year is called the divided, the remaiig profits (if ay) beig retaied as reserves or to fiace the compay s activities. Ordiary shares are the pricipal way i which compaies i may coutries are fiaced. They offer ivestors high potetial returs for high risk, particularly risk of capital losses. Ordiary shares are the lowest rakig form of fiace issued by compaies. Divideds are ot a legal obligatio of the compay but are paid at the discretio of the directors. The iitial ruig yield o ordiary shares is low but divideds should icrease with iflatio ad real growth i a compay s earigs. The expected overall future retur o ordiary shares ought to be higher tha for most other classes of security to compesate for the greater risk of default, ad for the variability of returs. The retur o ordiary shares is made up of two compoets, the divideds received ad ay icrease i the market price of the shares. Marketability of ordiary shares varies accordig to the size of the compay but will be better tha for the loa capital of the same compay if: the bulk of the compay s capital is i the form of ordiary shares the loa capital is fragmeted ito several differet issues ivestors buy ad sell ordiary shares more frequetly tha they trade i loa capital, perhaps because the residual ature of ordiary shares makes them more sesitive to chages i ivestors views about a compay Ordiary shareholders get votig rights i proportio to the umber of shares held, so shareholders may have the ability to ifluece the decisios take by the directors ad maagers of the compay. 3.2 Preferece shares Preferece shares are less commo tha ordiary shares. Assumig that the compay makes sufficiet profits, they offer a fixed stream of ivestmet icome. The ivestmet characteristics are ofte more like those of usecured loa stocks tha ordiary shares. The crucial differece betwee preferece shares ad ordiary shares is that preferece share divideds are limited to a set amout which is almost always paid. Preferece shareholders rak above ordiary shareholders (both for divideds ad, usually, o widig up), ad oly get votig rights if divideds are upaid or if there is a matter which directly affects the rights of preferece shareholders. Preferece divideds, like ordiary divideds, are oly paid at the directors discretio, but o ordiary divided ca be paid if there are ay outstadig preferece divideds. I Uit 10, Page 4 Istitute ad Faculty of Actuaries

81 2016 Ivestmets Subject CT1 most cases preferece shares are cumulative, which meas that upaid divideds are carried forward. I a give compay, the risk of preferece shareholders ot gettig their divideds is greater tha the risk of loa stockholders ot beig paid, but less tha the risk of ordiary shareholders ot beig paid. For all ivestors, the expected retur o preferece shares is likely to be lower tha o ordiary shares because the risk of holdig preferece shares is lower. Preferece shares rak higher o a widig-up, ad the level of icome paymets is more certai. 3.3 Property Marketability of preferece shares is likely to be similar to loa capital marketability. There are may differet types of properties available for ivestmet, for example: offices, shops ad idustrial properties (e.g. warehouses, factories). The retur from ivestig i property comes from retal icome ad from capital gais, which may be realised o sale. Property is a real ivestmet ad as such rets ad capital values might be expected to icrease broadly with iflatio i the log term, which makes the returs from property similar i ature to those from ordiary shares. However, either retal icome or capital values are guarateed ad there ca be cosiderable fluctuatios i capital values i particular, i real ad omial terms. Retal terms are specified i lease agreemets. Typically, it is agreed that rets are reviewed at specific itervals such as every three or five years. The ret is chaged, at a review time, to be more or less equal to the market ret o similar properties at the time of the review. Some leases have clauses which specify upward-oly adjustmets of rets. The followig characteristics are particular to property ivestmets: (a) large uit sizes, leadig to less flexibility tha ivestmet i shares (b) each property is uique, so ca be difficult to value. Valuatio is expesive, because of the eed to employ a experieced surveyor (c) the actual value obtaiable o sale is ucertai: values i property markets ca fluctuate just as stock markets ca (d) buyig ad sellig expeses are higher tha for shares ad bods (e) et retal icome may be reduced by maiteace expeses (f) there may be periods whe the property is uoccupied, ad o icome is received Marketability is poor because each property is uique ad because buyig ad sellig icur high costs. The ruig yield from property ivestmets will ormally be higher tha that for ordiary shares. The reasos for this are: 1. divideds usually icrease aually, whereas rets are reviewed less ofte 2. property is much less marketable 3. expeses associated with property ivestmet are much higher 4. large, idivisible uits of property are much less flexible Istitute ad Faculty of Actuaries Uit 10, Page 5

82 Subject CT1 Ivestmets o average, divideds will ted to icrease more rapidly tha rets, as divideds beefit from returs arisig from the retetio of profits ad their reivestmet withi the compay. 4 Derivatives 4.1 Futures A derivative is a fiacial istrumet with a value depedet o the value of some other, uderlyig asset. A futures cotract is a stadardised, exchage tradable cotract betwee two parties to trade a specified asset o a set date i the future at a specified price. Fiacial futures are based o a uderlyig fiacial istrumet, rather tha a physical commodity. They exist i four mai categories: bod futures short iterest rate futures stock idex futures currecy futures Each party to a futures cotract must deposit a sum of moey kow as margi with the clearig house. Margi paymets act as a cushio agaist potetial losses which the parties may suffer from future adverse price movemets. Whe the cotract is first struck, iitial margi is deposited with the clearig house. Additioal paymets of variatio margi are made daily to esure that the clearig house s exposure to credit risk is cotrolled. This exposure ca icrease after the cotract is struck through subsequet adverse price movemets Bod futures For delivery, the cotract requires physical delivery of a bod. If the cotract were specified i terms of a particular bod the it would be possible simply to deliver the required amout of that stock. If the cotract is specified i terms of a otioal stock the there eeds to be a likage betwee it ad the cash market. The bods which are eligible for delivery are listed by the exchage. The party deliverig the bod will choose the stock from the list which is cheapest to deliver. The price paid by the receivig party is adjusted to allow for the fact that the coupo may ot be equal to that of the otioal bod which uderlies the cotract settlemet price Short iterest rate futures The way that the quotatio is structured meas that as iterest rates fall the price rises, ad vice versa. The price is stated as 100 mius the 3-moth iterest rate. For example, with a iterest rate of 6.25% the future is priced as Uit 10, Page 6 Istitute ad Faculty of Actuaries

83 2016 Ivestmets Subject CT1 The cotract is based o the iterest paid o a otioal deposit for a specified period from the expiry of the future. However o pricipal or iterest chages hads. The cotract is cash settled. O expiry the purchaser will have made a profit (or loss) related to the differece betwee the fial settlemet price ad the origial dealig price. The party deliverig the cotract will have made a correspodig loss (or profit) Stock idex futures The cotract provides for a otioal trasfer of assets uderlyig a stock idex at a specified price o a specified date Currecy futures 4.2 Optios Swaps The cotract requires the delivery of a set amout of a give currecy o the specified date. A optio gives a ivestor the right, but ot the obligatio, to buy or sell a specified asset o a specified future date. A call optio gives the right, but ot the obligatio, to buy a specified asset o a set date i the future for a specified price. A put optio gives the right, but ot the obligatio, to sell a specified asset o a set date i the future for a specified price. A America style optio is a optio that ca be exercised o ay date before its expiry. A Europea style optio is a optio that ca be exercised oly at expiry. A swap is a cotract betwee two parties uder which they agree to exchage a series of paymets accordig to a prearraged formula. I the most commo form of iterest rate swap, oe party agrees to pay to the other a regular series of fixed amouts for a certai term. I exchage, the secod party agrees to pay a series of variable amouts based o the level of a short-term iterest rate. Both sets of paymets are i the same currecy. The fixed paymets ca be thought of as iterest paymets o a deposit at a fixed rate, while the variable paymets are the iterest o the same deposit at a floatig rate. The deposit is purely a otioal oe ad o exchage of pricipal takes place. A currecy swap is a agreemet to exchage a fixed series of iterest paymets ad a capital sum i oe currecy for a fixed series of iterest paymets ad a capital sum i aother. The swap will be priced so that the preset value of the cashflows is slightly egative for the ivestor ad positive for the issuig orgaisatio. The differece represets the price Istitute ad Faculty of Actuaries Uit 10, Page 7

84 Subject CT1 Ivestmets 2016 that the ivestor is prepared to pay for the advatages brought by the swap o the oe had, ad the issuer s expected profit margi o the other. Each couterparty to a swap faces two kids of risk: Market risk is the risk that market coditios will chage so that the preset value of the et outgo uder the agreemet icreases. The market maker will ofte attempt to hedge market risk by eterig ito a offsettig agreemet. Credit risk is the risk that the other couterparty will default o its paymets. This will oly occur if the swap has a egative value to the defaultig party so the risk is ot the same as the risk that the couterparty would default o a loa of comparable maturity. 4.3 Covertibles Covertible forms of compay securities are, almost ivariably, usecured loa stocks or preferece shares that covert ito ordiary shares of the issuig compay. The covertible will have a stated aual iterest paymet. The date of coversio might be a sigle date or, at the optio of the holder, oe of a series of specified dates. The characteristics of a covertible security i the period prior to coversio are a cross betwee those of fixed iterest stock ad ordiary shares. As the likely date of coversio (or ot) gets earer, it becomes clearer whether the covertible will stay as loa stock or become ordiary shares. As this happes, its behaviour becomes closer to that of the security ito which it coverts. Covertibles geerally provide higher icome tha ordiary shares ad lower icome tha covetioal loa stock or preferece shares. There will geerally be less volatility i the price of the covertible tha i the share price of the uderlyig equity. From the ivestor s poit of view, covertible securities offer the opportuity to combie the lower risk of a debt security with the potetial for large gais of a equity. The price paid for this is a lower ruig yield tha o a ormal loa stock or preferece share. The optio to covert will have time value, which will be reflected i the price of the stock. E N D Uit 10, Page 8 Istitute ad Faculty of Actuaries

85 2016 Elemetary compoud iterest problems Subject CT1 UNIT 11 ELEMENTARY COMPOUND INTEREST PROBLEMS Syllabus objective (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 idexliked 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. This uit also deals with real rates of iterest as required i syllabus objective (iv). Istitute ad Faculty of Actuaries Uit 11, Page 1

86 Subject CT1 Elemetary compoud iterest problems Fixed iterest securities As i other compoud iterest problems, oe of two questios may be asked: (1) What price P per uit omial, should be paid by a ivestor to secure a et yield of i per aum? (2) Give that the ivestor pays a price P per uit omial, what et yield per aum will be obtaied? 1.1 Price to be paid or yield to be obtaied The price, P, to be paid to achieve a yield of i per aum is equal to: Preset value, at rate Preset value, at rate P = of iterest i per aum, + of iterest i per aum, of et iterest paymets of et capital paymets (1.1) 1.2 No tax The yield available o a stock that ca be bought at a give price, P, ca be foud by solvig equatio (1.1) for the et yield i. If the ivestor is ot subject to taxatio the yield i is referred to as a gross yield. The yield o a security is sometimes referred to as the yield to redemptio or the redemptio yield to distiguish it from the flat (or ruig) yield, which is defied as D/P, the ratio of the coupo rate to the price per uit omial of the stock. Cosider a year fixed iterest security which pays coupos of D per aum, payable pthly i arrear ad has redemptio amout R. The price of this bod, at a effective rate of iterest i per aum, with o allowace for tax (i.e. i represets the gross yield) is: P Da Rv at rate i per aum (1.2) Note: Oe could also work with a period of half a year. The correspodig equatio of value would the be D 2 P a Rv at rate iwhere (1 i) 1 i Uit 11, Page 2 Istitute ad Faculty of Actuaries

87 2016 Elemetary compoud iterest problems Subject CT1 1.3 Icome tax Suppose a ivestor is liable to icome tax at rate t 1 o the coupos, which is due at the time that the coupos are paid. The price, P, of this bod, at a effective rate of iterest i per aum, where i ow represets the et yield, is ow: 1 P (1 t ) Da Rv at rate i per aum (1.3) It is possible i some coutries that the tax is paid at some later date, for example at the caledar year ed. This does ot cause ay particular problems as we follow the usual procedure idetify the cashflow amouts ad dates ad set out the equatio of value. For example, suppose that icome tax o the bod is paid i a sigle istalmet, due, say, k years after the secod half-yearly coupo paymet each year. The the equatio of value for a give et yield i ad price (or value) P is, immediately after a coupo paymet, k 1 P Da Rv t Dv a Other arragemets may be dealt with similarly from first priciples. 1.4 Capital gais tax If the price paid for a bod is less tha the redemptio (or sale price if sold earlier) the the ivestor has made a capital gai. Capital gais tax is a tax levied o the capital gai. I cotrast to icome tax, this tax is ormally payable oce oly i respect of each disposal, at the date of sale or redemptio. Istitute ad Faculty of Actuaries Uit 11, Page 3

88 Subject CT1 Elemetary compoud iterest problems Capital gais test Cosider a year fixed iterest security which pays coupos of D per aum, payable pthly i arrear ad has redemptio amout R. A ivestor, liable to icome tax at rate t 1, purchases the bod at price P. If R P the there is a capital gai ad from (1.3), we have: 1 R (1 t ) Da Rv 1 v R(1 v ) (1 t1 ) D i D R(1 t1 ) i D i (1 t1) R (1.4) If the ivestor is also subject to tax at rate t 2 (0 < t 2 < 1) o the capital gais, the let the price payable, for a give et yield i, be P. D (1 ) the there is a capital gai. At the redemptio date of the loa there is R therefore a additioal liability of t 2 ( R P). If ( p i ) t1 I this case: 1 2 P (1 t ) Da Rv t ( R P ) v at rate i per aum (1.5) Note that if a stock is sold before the fial maturity date, the capital gais tax liability will i geeral be differet, sice it will be calculated with referece to the sale proceeds rather tha the correspodig redemptio amout. If ( p ) D i (1 t1 ) the there is o capital gai ad o capital gais tax liability due at R redemptio. Hece P P i (1.3). (We are assumig that it is ot permissible to offset the capital loss agaist ay other capital gai: see example 1.4.2) Fidig the yield whe there is capital gais tax A ivestor who is liable to capital gais tax may wish to determie the et yield o a particular trasactio i which he has purchased a loa at a give price. Oe possible approach is to determie the price o two differet et yield bases ad the estimate the actual yield by iterpolatio. This approach is ot always the quickest method. Sice the purchase price is kow, so too is the amout of the capital gais tax, ad the et receipts for the ivestmet are thus kow. I this situatio oe may more Uit 11, Page 4 Istitute ad Faculty of Actuaries

89 2016 Elemetary compoud iterest problems Subject CT1 easily write dow a equatio of value which will provide a simpler basis for iterpolatio, as illustrated by the ext example. Example A loa of 1,000 bears iterest of 6% per aum payable yearly ad will be redeemed at par after te years. A ivestor, liable to icome tax ad capital gais tax at the rates of 40% ad 30% respectively, buys the loa for 800. What is his et effective aual yield? Solutio Note that the et icome each year of 36 is 4.5% of the purchase price. Sice there is a gai o redemptio, the et yield is clearly greater tha 4.5%. The gai o redemptio is 200, so that the capital gais tax payable will be 60 ad the et redemptio proceeds will be 940. The et effective yield p.a. is thus that value of i for which 800 = 36 a v 10 If the et gai o redemptio (i.e. 140) were to be paid i equal istalmets over the teyear duratio of the loa rather tha as a lump sum, the et receipts each year would be 50 (i.e ). Sice 50 is 6.25% of 800, the et yield actually achieved is less tha 6.25%. Whe i = 0.055, the right-had side of the above equatio takes the value , ad whe i = 0.06 the value is By iterpolatio, we estimate the et yield as i = = The et yield is thus 5.84% per aum. Alteratively, we may fid the prices to give et yields of 5.5% ad 6% per aum. These prices are ad , respectively. The yield may the be obtaied by iterpolatio. However, this alterative approach is somewhat loger tha the first method Offsettig capital losses agaist capital gais Util ow we have cosidered the effects of capital gais tax o the basis that it is ot permitted to offset capital gais by capital losses. I some situatios, however, it may be permitted to do so. This may mea that a ivestor, whe calculatig his liability for capital gais tax i ay year, is allowed to deduct from his total capital gais for the year the total of his capital losses (if ay). If the total capital losses exceed the total capital gais, o credit will geerally be give for the overall et loss, but o capital gais tax will be payable. Istitute ad Faculty of Actuaries Uit 11, Page 5

90 Subject CT1 Elemetary compoud iterest problems 2016 Example Suppose that i a particular tax year a ivestor sold the followig two assets, both of which were purchased some years ago. Asset A Sold for 1,865 (Purchase price 1,300) Asset B Sold for 500 (Purchase price 900) The sale of asset A produces a capital gai of 565 while for asset B there is a capital loss of 400. I the absece of both the right to offset losses agaist gais ad of idexatio the sales of these two assets lead to a overall capital gai of 565. (The loss of 400 is simply regarded as a zero capital gai.) If the offsettig of losses agaist gais is allowed, the sales of these two assets lead to a overall capital gai of 165 (i.e ) The idexatio of capital gais Whe a asset is sold at a profit, it may be permitted to reduce the capital gai for taxatio purposes by determiig the amout of the capital gai ot by referece to the actual purchase price but by referece to a (greater) otioal purchase price. The otioal purchase price is the actual purchase price icreased i lie with a approved idex. This practice is geerally referred to as the idexatio of gais. Example I example 1.4.2, suppose that idexatio is also allowed ad that over the period durig which the ivestor owed asset A the approved idex icreased by 18%. I this case the otioal purchase price of asset A is 1,534 (i.e. 1, ) ad the capital gai arisig o the sale of this asset is reduced to 331 (i.e. 1,865 1,534). If o offsettig of losses is allowed, this last amout will be the total capital gai arisig from the sale of the two assets. If, however, offsettig is permitted, there is zero overall capital gai sice the loss of 400 o the sale of asset B exceeds the (reduced) gai arisig from the sale of asset A. It is perhaps worth poitig out that idexatio of losses is ot usually permitted. Thus, for example, if over the period durig which the ivestor owed asset B the approved idex icreased by 12%, the loss o the sale of this asset is still cosidered to be 400. It is ot take as 508 (i.e ). The priciples used here ca be applied to other ivestmets that are subject to such tax. Uit 11, Page 6 Istitute ad Faculty of Actuaries

91 2016 Elemetary compoud iterest problems Subject CT1 1.5 Optioal redemptio dates Sometimes a security is issued without a fixed redemptio date. I such cases the terms of issue may provide that the borrower ca redeem the security at the borrower s optio at ay iterest date o or after some specified date. Alteratively, the issue terms may allow the borrower to redeem the security at the borrower s optio at ay iterest date o or betwee two specified dates (or possibly o ay oe of a series of dates betwee two specified dates). The latest possible redemptio date is called the fial redemptio date of the stock, ad if there is o such date, the the stock is said to be udated. It is also possible for a loa to be redeemable betwee two specified iterest dates, or o or after a specified iterest date, at the optio of the leder, but this arragemet is less commo tha whe the borrower chooses the redemptio date. A ivestor who wishes to purchase a loa with redemptio dates at the optio of the borrower caot, at the time of purchase, kow how the market will move i the future ad hece whe the borrower will repay the loa. The ivestor thus caot kow the yield which will be obtaied. However, by usig (1.4) the ivestor ca determie either: (1) The maximum price to be paid, if the et yield is to be at least some specified value; or (2) The miimum et yield the ivestor will obtai, if the price is some specified value. Cosider a fixed iterest security which pays coupos of D per aum, payable pthly i arrear ad has redemptio amout R. The security has a outstadig term of years, which may be chose by the borrower subject to the restrictio that 1 2. (We assume that 1 ad 2 are iteger multiples of 1/p.) Suppose that a ivestor, liable to icome tax at rate t 1, wishes to achieve a et aual yield of at least i. It follows from equatios (1.3) ad (1.4) that if ( p ) D i (1 t1 ) the the purchaser will R receive a capital gai whe the security is redeemed. From the ivestor s viewpoit, the sooer a capital gai is received the better. The ivestor will therefore obtai a greater yield o a security which is redeemed first. So to esure the ivestor receives a et aual yield of at least i the they should assume the worst case result: that the redemptio moey is paid as late as possible, i.e. 2. Similarly if ( p ) D i (1 t1 ) the there will be a capital loss whe the security is redeemed. R The ivestor will wish to defer this loss as log as possible, ad will therefore obtai a greater yield o a security which is redeemed later. So to esure the ivestor receives a et aual yield of at least i the they should assume the worst case result: that the redemptio moey is paid as soo as possible, i.e. 1. Istitute ad Faculty of Actuaries Uit 11, Page 7

92 Subject CT1 Elemetary compoud iterest problems 2016 Fially, if ( p ) D i (1 t1 ) the there is either a capital gai or a capital loss. So it will R make o differece to the ivestor whe the security is redeemed. The et aual yield will be i irrespective of the actual redemptio date chose. Suppose, alteratively, that the price of the loa is give. The miimum et aual yield is obtaied by agai assumig the worst case result for the ivestor. So if: (a) P < R, the the ivestor receives a capital gai whe the security is redeemed. The worst case is that the redemptio moey is repaid at the latest possible date. If this does i fact occur, the et aual yield will be that calculated. If redemptio takes place at a earlier date, the et aual yield will be greater tha that calculated. (b) P > R, the the ivestor receives a capital loss whe the security is redeemed. The worst case is that the redemptio moey is repaid at the earliest possible date. The actual yield obtaied will be at least the value calculated o this basis. (c) P = R, the the ivestor receives either a capital gai or a capital loss. The et aual yield is i, where ( p ) D i (1 t1 ), irrespective of the actual redemptio date R chose. Note that a capital gais tax liability does ot chage ay of this. For example, a ivestmet which has a capital gai before allowig for capital gais tax must still have a et capital gai after allowig for the capital gais tax liability, so that the worst case for the ivestor is still the latest redemptio. However, i some cases, for example if the redemptio price varies, the simple strategy described above will ot be adequate, ad several values may eed to be calculated to determie which is lowest. 2 Ucertai icome securities Securities with ucertai icome iclude: 1. Equities, which have regular declaratios of divideds. The divideds vary accordig to the performace of the compay issuig the stocks ad may be zero. 2. Property which carries regular paymets of ret, which may be subject to regular review. 3. Idex-liked bods which carry regular coupo paymets ad a fial redemptio paymet, all of which are icreased i proportio to the icrease i a relevat idex of iflatio. For all of these ivestmets ivestors may be iterested i calculatig the yield for a give price, or the price or value of the security for a give yield. I order to calculate the value or the yield it is ecessary to make assumptios about the future icome. Uit 11, Page 8 Istitute ad Faculty of Actuaries

93 2016 Elemetary compoud iterest problems Subject CT1 Give the ucertai ature of the future icome, oe method of modellig the cashflows is to assume statistical distributios for, say, the iflatio or divided growth rate. I this course however we will make simpler assumptios for example that divideds icrease at a costat rate. It is importat to recogise that modellig radom variables determiistically, igorig the variability of the paymets ad the ucertaity about the expected growth rate, is ot adequate for may purposes ad stochastic methods will be required. I all three cases, usig this determiistic approach meas that we estimate the future cashflows ad the solve the equatio of value usig the estimated cashflows. Idex-liked bods differ slightly from the other two i that the icome is certai i real terms. These are therefore covered separately, i sectio Equities Give determiistic assumptios about the growth of divideds, we ca estimate the future divideds for ay give equity, ad the solve the equatio of value usig estimated cashflows for the yield or the price or value. So, let the value of a equity just after a divided paymet be P, ad let D be the amout of this divided paymet. Assume that divideds grow i such a way that the divided due at time t is estimated to be D t. We geerally value the equity assumig divideds cotiue i perpetuity, ad without explicit allowace for the possibility that the compay will default ad the divided paymets will cease. I this case, assumig aual divideds, t1 t t i P Dv where i is the retur o the share, give price P. If we assume a costat divided growth rate of g, say, the D D(1 g) t ad t P = Da i 1 g i where i = 1 1 D(1 g) P i g At certai times close to the divided paymet date the equity may be offered for sale excludig the ext divided. This allows for the fact that there may ot be time betwee the sale date ad the divided paymet date for the compay to adjust its records to esure the buyer receives the divided. A equity which is offered for sale without the ext divided is called ex-divided or xd. The valuatio of ex-divided stocks requires o ew priciples. Istitute ad Faculty of Actuaries Uit 11, Page 9

94 Subject CT1 Elemetary compoud iterest problems Property The valuatio of property by discoutig future icome follows very similar priciples to the valuatio of equities. Both require some assumptio about the icrease i future icome; both have icome which is related to the rate of iflatio (both property rets ad compay profits will be broadly liked to iflatio, over the log term); i both cases we use a determiistic approach. The major differeces betwee the approach to the property equatio of value, compared with the equity equatio of value, are (1) property rets are geerally fixed for a umber of years at a time ad (2) some property cotracts may be fixed term, so that after a certai period the property icome ceases ad owership passes back to the origial ower (or aother ivestor) with o further paymets. Let P be the price immediately after receipt of the periodic retal paymet. Let m be the frequecy of the retal paymets each year. We estimate the future cashflows, such that Dt m is the retal icome at time t, t 1, 2,... m m If the rets cease after some time the clearly D t = 0 for t >. The the equatio of value is: P = 1 P Dkmv k1 m k m 3 Real rates of iterest The idea of a real rate of iterest, as distict from a moey rate of iterest, was itroduced i Uit 4. Ways of calculatig real rates of iterest will ow be examied. 3.1 Iflatio adjusted cashflows The real rate of iterest of a trasactio is the rate of iterest after allowig for the effect of iflatio o a paymet series. The effect of iflatio meas that a uit of moey at, say, time 0 has differet purchasig power tha a uit of moey at ay other time. We fid the real rate of iterest by first adjustig all paymet amouts for iflatio, so that they are all expressed i uits of purchasig power at the same date. As a simple example, cosider a trasactio represeted by the followig paymet lie: Time: 0 1 Paymet: That is, for a ivestmet of 100 at time 0 a ivestor receives 120 at time 1. Uit 11, Page 10 Istitute ad Faculty of Actuaries

95 2016 Elemetary compoud iterest problems Subject CT1 The effective rate of iterest o this trasactio is clearly 20% per aum. The real rate of iterest is foud by first expressig both paymets i uits of the same purchasig power. Suppose that iflatio over this oe year period is 5% per aum. This meas that 120 at time 1 has a value of 120/1.05 = i terms of time 0 moey uits. So, i real terms, that is, after adjustig for the rate of iflatio, the trasactio is represeted as: Time: 0 1 Paymet: Hece, the real rate of iterest is %. 3.2 Calculatig real yields usig a iflatio idex Where the rates of iflatio are kow (that is, we are lookig back i time at a trasactio that is complete) we may adjust paymets for the rate of iflatio by referece to a relevat iflatio idex. For example, assume we have a iflatio idex, Q(t k ) at time t k, ad a paymet series as follows: Time, t: Paymet: Q(t) Clearly the rate of iterest o this trasactio is 8%. Istitute ad Faculty of Actuaries Uit 11, Page 11

96 Subject CT1 Elemetary compoud iterest problems 2016 Now we ca chage all these amouts ito time 0 moey values by dividig the paymet at time t by the proportioal icrease i the iflatio idex from 0 to t. For example the iflatio-adjusted value of the paymet of 8 at time 1 is 8 Q(1)/Q(0). The series of paymets i time 0 moey values is the as follows: Time, t: Paymet: This gives a yield equatio for the real yield: v i vi vi 3 = 0 where i is the real rate of iterest which ca be solved usig umerical methods to give i = 2.63%. I geeral, the real yield equatio for a series of cashflows { Ct, C,..., }, 1 t C 2 t give associated iflatio idex values {Q(0), Q(t 1 ), Q(t 2 ),..., Q(t )} is, usig time 0 moey uits: Q(0) tk Ct v k i = 0 Qt ( ) k1 Ctk tk vi = 0 k 1 Qt ( ) k k The secod equatio here, i which all terms are divided by Q(0), demostrates that the solutio of the yield equatio is idepedet of the date the paymet uits are adjusted to. 3.3 Calculatig real yields give costat iflatio assumptios If we are cosiderig future cashflows, the actual iflatio experiece will ot be kow, ad some assumptio about future iflatio will be required. For example, if it is assumed that a costat rate of iflatio of j per aum will be experieced, the a cashflow of, say, 100 due at t has value 100(1 + j) t i time 0 moey values. So, for a fixed et cashflow series { C t k }, k = 0, 1, 2,...,, assumig a rate of iflatio of j per aum, the real, effective rate of iterest, i, is the solutio of the real yield equatio: k1 k k Ct v k j vi t = 0 t Uit 11, Page 12 Istitute ad Faculty of Actuaries

97 2016 Elemetary compoud iterest problems Subject CT1 We also kow that the effective rate of iterest with o iflatio adjustmet which may be called the moey yield to distiguish from the real yield, is i where k Ct v k i = 0 k1 t So the relatioship betwee the real yield i, the rate of iflatio j ad the moey yield i is v i = v j v i i = 1 i j j I some cases a combiatio of kow iflatio idex values ad a assumed future iflatio rate may be used to fid the real rate of iterest. Coversely, if we kow the real yield i which we have obtaied from a equatio of value usig iflatio-adjusted cashflows the we ca calculate the moey yield as follows: i = i + j(1 + i) 3.4 Paymets related to the rate of iflatio Some cotracts specify that the cashflows will be adjusted to allow for future iflatio, usually i terms of a give iflatio idex. The idex-liked govermet security is a example. The actual cashflows will be ukow util the iflatio idex at the relevat dates are kow. The cotract cashflows will be specified i terms of some omial amout to be paid at time t, say c t. If the iflatio idex at the base date is Q(0) ad the relevat value for the time t paymet is Q(t) the the actual cashflow is Qt () C t = c t Q(0) It is easy to show that if the real yield i is calculated by referece to the same iflatio idex as is used to iflate the cashflows, the i is the solutio of the real yield equatio: Q(0) tk Ct v k i = 0 Qt ( ) k1 k k1 tk t v k i c = 0 I other words we ca solve the yield equatio usig the omial amouts. However, it is ot always the case that the idex used to iflate the cashflows is the same as that used to calculate the real yield. For example the idex-liked UK govermet security has coupos iflated by referece to the iflatio idex value 3 moths before the Istitute ad Faculty of Actuaries Uit 11, Page 13

98 Subject CT1 Elemetary compoud iterest problems 2016 paymet is made. The real yield, however, is calculated usig the iflatio idex at the actual paymet dates. 3.5 The effects of iflatio Cosider the simplest situatio, i which a ivestor ca led ad borrow moey at the same rate of iterest i 1. I certai ecoomic coditios the ivestor may assume that some or all elemets of the future cashflows should icorporate allowaces for iflatio (i.e. icreases i prices ad wages). The extet to which the various items i the cashflow are subject to iflatio may differ. For example, wages may icrease more rapidly tha the prices of certai goods, or vice versa, ad some items (such as the icome from retcotrolled property) may ot rise at all, eve i highly iflatioary coditios. The case whe all items of cashflow are subject to the same rate of escalatio j per time e uit is of special iterest. I this case we fid or estimate c t ad e (t), the et cashflow ad the et rate of cashflow allowig for escalatio at rate j per uit time, by the formulae e c t = (1 + j) t c t (3.1) e (t) = (1 + j) t (t) (3.2) where c t ad (t) are estimates of the et cashflow ad the et rate of cashflow respectively at time t without ay allowace for iflatio. It follows that, with allowace for iflatio at rate j per uit time, the et preset value of a ivestmet or busiess project at rate of iterest i is NPV j () i t t t t t 0 c (1 j) (1 i) ( t)(1 j) (1 i) dt t t t 0 0 (3.3) 0 c (1 i ) ( t)(1 i ) dt where: 1 i 1 i j 0 1 or: i 0 i 1 j j (3.4) If j is ot too large, oe sometimes uses the approximatio i0 i j (3.5) Uit 11, Page 14 Istitute ad Faculty of Actuaries

99 2016 Elemetary compoud iterest problems Subject CT1 These results are of cosiderable practical importace, because projects which are apparetly uprofitable whe rates of iterest are high may become highly profitable whe eve a modest allowace is made for iflatio. It is, however, true that i may vetures the positive cashflow geerated i the early years of the veture is isufficiet to pay bak iterest, so recourse must be had to further borrowig (uless the ivestor has adequate fuds of their ow). This i itself does ot udermie the profitability of the project, but the ivestor would require the agreemet of his ledig istitutio before further loas could be obtaied ad this might cause difficulties i practice. 4 Idex-liked bods Idex-liked bod cashflows are described i Uit 10. The coupo ad redemptio paymets are icreased accordig to a idex of iflatio. Give simple assumptios about the rate of future iflatio it is possible to estimate the future paymets. Give these assumptios we may calculate the price or yield by solvig the equatio of value usig the estimated cashflows. For example, let the omial aual coupo rate for a -year idex-liked bod be D per 1 omial face value with coupos payable half-yearly, ad let the omial redemptio price be R per 1 omial face value. We assume that paymets are iflated by referece to a idex with base value Q(0), such that the coupo due at time t years is D Qt () 2 Q(0) The the equatio of value, give a effective (moey) yield of i per aum, ad a preset value or price P per 1 omial at issue or immediately followig a coupo paymet, is P = ( k k D Q ) 2 2 Q( ) vi R vi 2 Q(0) Q(0) 2 k1 We estimate the ukow value of Q(t) usig some assumptio about future iflatio ad usig the latest kow value which may be Q(0). For example, assume iflatio icreases at rate j t per aum i the year t 1 to t, the we have Q(½) = Q(0). (1 + j 1 ) ½ Q(1) = Q(0). (1 + j 1 ) Q(1½) = Q(0). (1 + j 1 ) (1 + j 2 ) ½ Q(2) = Q(0). (1 + j 1 ) (1 + j 2 ) etc. Istitute ad Faculty of Actuaries Uit 11, Page 15

100 Subject CT1 Elemetary compoud iterest problems 2016 It is importat to bear i mid that the idex used may ot be the same as the actual iflatio idex value at time t that oe would use, for example, to calculate the real (iflatio-adjusted) yield. I the case of UK idex-liked bods, the paymets are icreased usig the idex values from 3 moths before the paymet date. Real yields would be calculated usig the iflatio idex values Pt the paymet date. Like equities, idex-liked bods (ad fixed iterest bods) may be offered for sale ex-divided. No ew priciples are ivolved i the valuatio of ex-divided idexliked bods. E N D Uit 11, Page 16 Istitute ad Faculty of Actuaries

101 2016 The No Arbitrage assumptio ad Forward Cotracts Subject CT1 UNIT 12 THE NO ARBITRAGE ASSUMPTION AND FORWARD CONTRACTS Syllabus objective (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 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. 1 The No Arbitrage assumptio 1.1 Itroductio Arbitrage i fiacial mathematics is geerally described as a risk-free tradig profit. More accurately, a arbitrage opportuity exists if either (a) a ivestor ca make a deal that would give her or him a immediate profit, with o risk of future loss; or (b) a ivestor ca make a deal that has zero iitial cost, o risk of future loss, ad a o-zero probability of a future profit. The cocept of arbitrage is very importat because we geerally assume that i moder developed fiacial markets arbitrage opportuities do ot exist. This assumptio is referred to as the No Arbitrage assumptio, ad is fudametal to moder fiacial mathematics. If we assume that there are o arbitrage opportuities i a market, the it follows that ay two securities or combiatios of securities that give exactly the same paymets must have Istitute ad Faculty of Actuaries Uit 12, Page 1

102 Subject CT1 The No Arbitrage assumptio ad Forward Cotracts 2016 the same price. This is sometimes called the Law of Oe Price. The ideas are demostrated i the followig example. Example 1 Cosider a very simple securities market, cosistig of two securities, A ad B. At time t = 0 the prices of the securities are P0 A ad P 0 B respectively. The term of both the securities is 1 year. At t = 1 there are two possible outcomes. Either the market goes A B up, i which case security A pays P1 ( u) ad B pays P1 ( u), or it goes dow, with A paymets P1 ( d ) ad P B 1 ( d) respectively. Ivestors ca buy securities, i which case they pay the time 0 price ad receive the time 1 icome, or they ca sell securities, i which case they receive the time 0 price ad must pay the time 1 outgo. Now, assume first that we have the followig paymet table: Time 0 price Market goes up Market goes dow Security: P 0 P 1 (u) P 1 (d) A B There is a arbitrage opportuity here. A ivestor could buy oe uit of security B ad sell two uits of security A. This would give icome at time 0 of 12 from the sale of security A ad a outgo of 11 from the purchase of security B which gives a et icome at time 0 of 1. At time 1 the outgo due o the portfolio of 2 uits of security A exactly matches the icome due from security B, whether the market moves up or dow. Thus, the ivestor makes a profit at time 0, with o risk of a future loss. It is clear that ivestmet A is uattractive compared with ivestmet B. This will cause pressure to reduce the price of A ad to icrease the price of B, as there will be o demad for A ad a excessive demad for B. Ultimately we would achieve balace, whe P0 A = P0 B / 2, whe the arbitrage opportuity is elimiated, ad the prices are cosistet. Uit 12, Page 2 Istitute ad Faculty of Actuaries

103 2016 The No Arbitrage assumptio ad Forward Cotracts Subject CT1 Aother example is give i the followig table: Time 0 price Market goes up Market goes dow Security: P 0 P 1 (u) P 1 (d) A B A arbitrage opportuity exists, as a ivestor could buy oe uit of A ad sell oe uit of B. The et icome at time 0 is 0, as the icome from the sale of B matches the outgo o the purchase of A. At time 1 the et icome is 0 if the market goes up, ad 1 if the market goes dow. So, for a zero ivestmet, the ivestor has a possibility of makig a profit (assumig the probability that the market goes dow is ot zero) ad o possibility of makig a loss. With these prices, ivestors will aturally choose to buy ivestmet A ad will wat to sell ivestmet B. This will put pressure o the price of A to icrease, ad o the price of B to A B decrease. The arbitrage opportuity is elimiated whe P P. 1.2 Why do we assume No Arbitrage? 0 0 The No Arbitrage assumptio is very simple ad very powerful. It eables us to fid the price of complex istrumets by replicatig the payoffs. This meas that if we ca costruct a portfolio of assets with exactly the same paymets as the ivestmet that we are iterested i, the the price of the ivestmet must be the same as the price of the replicatig portfolio. I practice, i the major developed securities markets arbitrage opportuities, whe they do arise are very quickly elimiated as ivestors spot them ad trade o them. Such opportuities are so fleetig i ature, accordig to the empirical evidece, that it is sesible, realistic ad prudet to assume that they do ot exist. We also assume here that there are o trasactio costs or taxes associated with buyig, sellig or holdig assets. These are also idealised assumptios, but they eable us to develop a methodology that may be adapted to deal with these istitutioal features if ecessary. The No Arbitrage assumptio will be used extesively i Subject CT8, Fiacial Ecoomics. I this subject we itroduce the ideas i the cotext of Forward Cotracts. Istitute ad Faculty of Actuaries Uit 12, Page 3

104 Subject CT1 The No Arbitrage assumptio ad Forward Cotracts Forward cotracts 2.1 Itroductio A forward cotract is a agreemet made at some time t = 0, say, betwee two parties uder which oe agrees to buy from the other a specified amout of a asset (deoted S) at a specified price o a specified future date. The ivestor agreeig to sell the asset is said to hold a short forward positio i the asset, ad the buyer is said to hold a log forward positio. Let S r be the price of the uderlyig asset (for example, a uit of equity stock) at time r. The price will ot geerally be predictable for example, we may cotract to buy shares i a compay i 6 moths time. We kow what the curret price of the shares is, but the price will vary more or less cotiuously, so we do ot kow with certaity what the share price will be at ay future date. Let K be the price agreed at time t = 0 to be paid at time t = T, called the forward price. t = 0 is the time the forward cotract is agreed, T is the time the cotract matures (that is, whe the sale actually happes). We also assume there is a kow force of iterest that is available o a risk-free ivestmet over the term of the cotract. This is kow as the risk-free force of iterest. At time 0 whe the agreemet is made o moey chages hads (except possibly a good faith deposit we will igore this here). The price K agreed at time t = 0 is determied such that the value of the forward cotract at the time t = 0 is zero. The forward cotract will geerally have o-zero value at time T; if K > S T the the seller receives K for a asset worth (at that time) S T, ad has made a profit at time T of K S T. Similarly, if K < S T the the buyer has paid K for a asset worth S T, givig the buyer a profit at time T of S T K. 2.2 Calculatig the forward price for a security with o icome Oe importat questio is how to determie the forward price, K. This is the price agreed at time t = 0 but ot actually paid util the cotract eds, at t = T. The price S r is ucertai for all r > 0. However, usig the o arbitrage assumptio we ca fid the forward price without havig to make ay assumptio about the statistical properties of the process S r. Istead, we ca use a replicatio argumet. We assume at this stage that there are o paymets or costs associated with holdig the stock. Uit 12, Page 4 Istitute ad Faculty of Actuaries

105 2016 The No Arbitrage assumptio ad Forward Cotracts Subject CT1 Cosider the followig two ivestmet portfolios: Portfolio A: Eter a forward cotract to buy oe uit of a asset S, with forward price K, maturig at time T; simultaeously ivest a amout Ke T i the risk-free ivestmet. Portfolio B: Buy oe uit of the asset, at the curret price S 0. At time t = 0 the price of Portfolio A is Ke T for the risk-free ivestmet; recall that the price of a forward cotract is zero. The price of Portfolio B is S 0. At time t = T the cashflows for Portfolio A are: A amout K is received from the risk-free ivestmet (Ke T ivested at force of iterest for T years gives a accumulated value of K). The same amout K is paid o the forward cotract. Receive 1 uit of asset S. The payout from Portfolio B is oe uit of asset S. Now, the future cashflows of portfolio A are idetical to those of portfolio B both give a et portfolio of oe uit of the uderlyig asset S. The o arbitrage assumptio states that whe the future cashflows of two portfolios are idetical, the price must also be the same that is: Ke T = S 0 K = S 0 e T The o arbitrage assumptio gives the price for the forward cotract with o eed for ay model of how the asset price S t will actually move over the term of the cotract. 2.3 Calculatig the forward price for a security with fixed cash icome Assume ow that at some time t 1, 0 t 1 < T, the security uderlyig the forward cotract provides a fixed amout c to the holder. For example, if the security is a govermet bod, there will be fixed coupo paymets due every six moths. Now cosider the followig two portfolios: Portfolio A: Eter a forward cotract to buy oe uit of a asset S, with forward price K, maturig at time T; simultaeously ivest a amout Ke T + ce t 1 i the risk-free ivestmet. Portfolio B: Buy oe uit of the asset, at the curret price S 0. At time t 1 ivest the icome of c i the risk-free ivestmet. Istitute ad Faculty of Actuaries Uit 12, Page 5

106 Subject CT1 The No Arbitrage assumptio ad Forward Cotracts 2016 At time t = 0 the price of Portfolio A is Ke T + ce t 1 for the risk-free ivestmet, ad zero for the forward cotract. The price of Portfolio B is S 0. At time t = T the payout from Portfolio A is: Icome of K + ce ( Tt 1 ) from the risk-free ivestmet; Outgo of K o the forward cotract. Receive 1 uit of asset, value S T. The et portfolio at T is oe uit of the asset S plus ce ( Tt 1 ) uits of the risk-free security. The payout from Portfolio B is oe uit of asset, value S T plus ce ( Tt 1 ) uits of the riskfree security, from the ivested coupo paymet. The et cashflows of portfolio A at time T are idetical to those of portfolio B both give a et portfolio of oe uit of the uderlyig asset S plus ce ( Tt 1 ) uits of the risk-free security. Usig the o arbitrage assumptio the prices must also be the same that is: Ke T + ce t 1 = S 0 K = S 0 e T ce T t ( 1 ) For a log forward cotract o a fixed iterest security there may be more tha oe coupo paymet. It is easy to adapt the above method to allow for this. If we let I deote the preset value at time t = 0 of the fixed icome paymets due durig the term of the forward cotract, the the forward price at time t = 0 per uit of security S is K = (S 0 I) e T 2.4 Calculatig the forward price for a security with kow divided yield Let D be the kow divided yield per aum. We assume that divideds are received cotiuously, ad are immediately reivested i the security of S. If we start with oe uit of the security at time t = 0, the accumulated holdig at time T would be e DT uits of the security. This is because the umber of uits owed is cotiuously compoudig at rate D per aum for T years. If istead of 1 uit at time t = 0 we hold e DT uits, reivestig the divided icome, at time T we would hold e DT e DT = 1 uit of the security. Uit 12, Page 6 Istitute ad Faculty of Actuaries

107 2016 The No Arbitrage assumptio ad Forward Cotracts Subject CT1 Now cosider the followig two portfolios: Portfolio A: Eter a forward cotract to buy oe uit of a asset S, with forward price K, maturig at time T; simultaeously ivest a amout Ke T i the risk-free ivestmet. Portfolio B: Buy e DT uits of the asset S, at the curret price S 0. Reivest divided icome i the security S immediately it is received. At time t = 0 the price of Portfolio A is Ke T for the risk-free ivestmet, ad zero for the forward cotract. The price of Portfolio B is e DT S 0. At time t = T the cashflows of Portfolio A are: A amout K is received from the risk-free ivestmet. Outgo K is paid o the forward cotract. Receive 1 uit of asset S. The et portfolio at T is oe uit of the asset S. The payout from Portfolio B is oe uit of the asset S. The et cashflows of portfolio A at time T are idetical to those of portfolio B both give a et portfolio of oe uit at the uderlyig asset S. Usig the o arbitrage assumptio the prices must also be the same that is: Ke T = S 0 e DT K = S 0 e (D)T It is simple to adjust the portfolios to get the forward price if the divideds are paid discretely. The importat priciple for this case ad the kow icome case is that, if the icome is proportioal to the uderlyig security, S, we assume the icome is reivested i the security. If the icome is a fixed amout regardless of the price of the security at the paymet date, the we assume it is ivested i the risk-free security. This is because whe the paymet is proportioal to the stock price (e.g. divideds) we kow how may uits of stock they will purchase, but we do ot kow how much cash is paid (as the stock price is ukow). So we ca predict the amout of stock held at the ed if we assume reivestmet i the stock. With a cash paymet o the other had, we would ot kow how much stock could be bought, but we do kow how much the cash would accumulate to at the risk-free force of iterest. Assumig divideds are reivested i the security, but cash is ivested at the risk-free (ad kow) force of iterest eables us to predict the fial portfolio without requirig ay iformatio about the price of the asset S durig the course of the cotract. Istitute ad Faculty of Actuaries Uit 12, Page 7

108 Subject CT1 The No Arbitrage assumptio ad Forward Cotracts Hedgig Hedgig is a geeral term which describes the use of fiacial istrumets (icludig stocks, bods, forward cotracts ad more complex fiacial cotracts such as optios) to reduce or elimiate a future risk of loss. A ivestor who agrees to sell a asset at a give price i a forward cotract eed ot hold the asset at the start of the cotract. However, by the ed of the cotract he or she must ow the asset ready to sell uder the terms of the forward cotract. If the ivestor waits util the ed of the cotract to buy the asset S the risk exists that the price will rise above the forward price, ad they will have to pay more tha the forward price K tha they receive for the asset. O the other had, if they buy the asset at the start of the forward cotract, ad hold it util the cotract matures, there is a risk that the price will have falle, ad they have paid more tha they eeded to. To hedge the risk the ivestor could borrow a amout Ke T at the risk-free force of iterest, buy the asset S at the start of the cotract, at the price S 0, ad hold it util it is to be haded over at time T. The price of this hedge portfolio is Ke T + S 0 = 0. We are assumig here that there is o iterest or divided icome associated with the asset S. At time T the ivestor owes K that is exactly covered by the forward price received at T. He or she also owes oe uit of asset S uder the forward cotract, which is also paid from the hedge portfolio. This way, if the ivestor holds the hedge portfolio he or she is certai ot to make a loss o the forward cotract. There is also o chace of makig a profit. This is called a static hedge sice the hedge portfolio, which cosists of the asset to be sold plus the borrowed risk-free ivestmet, does ot chage over the term of the cotract. For more complex fiacial istrumets, such as optios, the hedge portfolio is more complex, ad requires (i priciple) cotiuous rebalacig to maitai. This is called a dyamic hedge. 2.6 The value of a forward cotract With o iterest or divided icome Cosider a forward cotract agreed at time t = 0, with a forward price K 0, for oe uit of security S. The maturity date of the cotract is time T. At the start of the cotract the value, to buyer ad seller of the asset S, is 0. At the maturity date the value of the cotract to the seller of the asset is K 0 S T ad to the buyer is (K 0 S T ). It is of iterest to fid the value of the cotract at itermediate times. Uit 12, Page 8 Istitute ad Faculty of Actuaries

109 2016 The No Arbitrage assumptio ad Forward Cotracts Subject CT1 Suppose at time r > 0 a ivestor holds a log forward cotract that is, holds a cotract agreeig to buy a asset S at T > r at a price agreed at time t = 0 of K 0. The ivestor wats to kow the value of this cotract durig the term at time r. Cosider the followig two portfolios purchased at time r: Portfolio A: Cosists of the existig log forward cotract (bought at time 0) with curret value V l. Ivest K 0 e (Tr) at time r i the risk-free ivestmet for T r years. Portfolio B: Buy a ew log forward cotract at time r, with maturity at T, forward price K r = S r e (Tr). The price of a forward cotract at issue is zero. Also, ivest K r e (Tr) i the risk-free ivestmet for T r years. The price of portfolio A at r is V l + K 0 e (Tr). The price of portfolio B at r is K r e (Tr). The payout from portfolio A at T is oe uit of the asset S; the ivestmet of K 0 e (Tr) accumulates to K 0, which matches the outgo o the forward cotract. The payout from portfolio B at T is also oe uit of the asset S; agai, the risk-free ivestmet accumulates to K r, which meets the forward price required. By the o arbitrage assumptio we have: V l + K 0 e (Tr) = K r e (Tr) V l = (K r K 0 ) e (Tr) We may substitute K r ad K 0 to get the value of the forward cotract i terms of the asset price, V l = S r S 0 e r which gives the value of the log forward cotract at time r. The value of a short forward cotract may be determied by similar argumets, to be V s = S 0 e r S r, that is V s = V l. The value of forward cotracts where there is some iterest or divided icome associated with the uderlyig asset may be determied easily usig similar argumets. Istitute ad Faculty of Actuaries Uit 12, Page 9

110 Subject CT1 The No Arbitrage assumptio ad Forward Cotracts Note We have simplified the calculatios by assumig that the risk-free force of iterest is idepedet of the time or duratio of the ivestmet. I fact, as show i Uit 13 there is a term structure to iterest rates that is, the iterest rate eared o a ivestmet depeds o both the time a sum is ivested ad o the legth of time for which it is ivested. The results above may be adjusted to allow for this, replacig with the appropriate spot or forward force of iterest. E N D Uit 12, Page 10 Istitute ad Faculty of Actuaries

111 2016 Term structure of iterest rates Subject CT1 UNIT 13 TERM STRUCTURE OF INTEREST RATES Syllabus objective (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. 1 Itroductio So far i this course it has geerally bee assumed that the iterest rate i or force of iterest eared o a ivestmet are idepedet of the term of that ivestmet. I practice the iterest rate offered o ivestmets does usually vary accordig to the term of the ivestmet. It is ofte importat to take this variatio ito cosideratio. I ivestigatig this variatio we make use of uit zero coupo bod prices. A uit zero coupo bod of term, say, is a agreemet to pay 1 at the ed of years. No coupo paymets are paid. It is also called a pure discout bod. We deote the price at issue of a uit zero coupo bod maturig i years by P. Istitute ad Faculty of Actuaries Uit 13, Page 1

112 Subject CT1 Term structure of iterest rates Discrete time 2.1 Discrete time spot rates The yield o a uit zero coupo bod with term years, y, is called the -year spot rate of iterest. Usig the equatio of value for the zero coupo bod we fid the yield o the bod y from P = 1 (1 y ) (1 + y ) = P 1 Sice rates of iterest differ accordig to the term of the ivestmet, i geeral y s y t for s t. Every fixed iterest ivestmet may be regarded as a combiatio of (perhaps otioal) zero coupo bods. For example, a bod payig coupos of D every year for years, with a fial redemptio paymet of R at time may be regarded as a combied ivestmet of zero coupo bods with maturity value D, with terms of 1 year, 2 years..., years, plus a zero coupo bod of omial value R with term years. Defiig v y t = (1 + y t ) 1, the price of the bod is: A = D(P 1 + P P ) + RP = D( v + y 1 2 v y v y ) + Rv y This is actually a cosequece of o arbitrage ; the portfolio of zero coupo bods has the same payouts as the fixed iterest bod, ad the prices must therefore be the same. The variatio by term of iterest rates is ofte referred to as the term structure of iterest rates. The curve of spot rates {y t } is a example of a yield curve. 2.2 Discrete time forward rates The discrete time forward rate, f t,r, is the aual iterest rate agreed at time 0 for a ivestmet made at time t > 0 for a period of r years. That is, if a ivestor agrees at time 0 to ivest 100 at time t for r years, the accumulated ivestmet at time t + r is 100(1 + f t,r ) r Forward rates, spot rates ad zero-coupo bod prices are all coected. The accumulatio at time t of a ivestmet of 1 at time 0 is (1 + y t ) t. If we agree at time 0 to ivest the amout (1 + y t ) t at time t for r years, we will ear a aual rate of f t,r. So we kow that 1 ivested for t + r years will accumulate to (1 + y t ) t (1 + f t,r ) r. But we also Uit 13, Page 2 Istitute ad Faculty of Actuaries

113 2016 Term structure of iterest rates Subject CT1 kow from the (t + r) spot rates that 1 ivested for t + r years accumulates to (1 + y t+r ) t+r, ad we also kow from the zero coupo bod prices that 1 ivested for t + r years 1 accumulates to Pt r. Hece we kow that (1 + y t ) t (1 + f t,r ) r = (1 + y t+r ) t+r = from which we fid that 1 Pt r (1 + f t,r ) r = (1 ytr ) (1 y ) t tr t = P t Pt r so that the full term structure may be determied give the spot rates, the forward rates or the zero coupo bod prices. Oe-period forward rates are of particular iterest. The oe-period forward rate at time t (agreed at time 0) is deoted f t = f t,1. We defie f 0 = y 1. Comparig a amout of 1 ivested for t years at the spot rate y t, ad the same ivestmet ivested 1 year at a time with proceeds reivested at the appropriate oe-year forward rate, we have (1 + y t ) t = (1 + f 0 ) (1 + f 1 ) (1 + f 2 )... (1 + f t1 ) 3 Cotiuous time rates 3.1 Cotiuous time spot rates Let P t be the price of a uit zero coupo bod of term t. The the t-year spot force of iterest is Y t where P t = Yt e t Y t = 1 t log P t This is also called the cotiuously compouded spot rate of iterest or the cotiuoustime spot rate. Y t ad its correspodig discrete aual rate y t are coected i the same way as ad i; a ivestmet of 1 for t years at a discrete spot rate y t accumulates to (1 + y t ) t Yt ; at the cotiuous time rate it accumulates to e t ; these must be equal, so Y y t = e t Cotiuous time forward rates The cotiuous time forward rate F t,r is the force of iterest equivalet to the aual forward rate of iterest f t,r. Istitute ad Faculty of Actuaries Uit 13, Page 3

114 Subject CT1 Term structure of iterest rates 2016 A 1 ivestmet of duratio r years, startig at time t, agreed at time 0 t accumulates usig the aual forward rate of iterest to (1 + f t,r ) r at time t + r. Ftr, r Usig the equivalet forward force of iterest the same ivestmet accumulates to e. Hece the aual rate ad cotiuous-time rate are related as F tr, f t,r = e 1 The relatioship betwee the cotiuous time spot ad forward rates may be derived by cosiderig the accumulatio of 1 at a cotiuous time spot rate of Y t for t years, followed by the cotiuous time forward rate of F t,r for r years. Compare this with a ivestmet of 1 at a cotiuous time spot rate of Y t+r for t + r years. The two ivestmets are equivalet, so the accumulated values must be the same. Hece ty t rf tr, e e = e ( tr) Y t r ty t + rf t,r = (t + r) Y t+r F t,r = ( t r ) Yt r tyt r Also, usig Y t = 1 t log P t, we have F t,r = 1 log r P t Pt r 3.3 Istataeous forward rates The istataeous forward rate F t is defied as F t = lim r0 F t,r The istataeous forward rate may broadly be thought of as the forward force of iterest applyig i the istat of time t t + t. F t = 1 Pt lim log r P r0 t r (1) log Pt r log Pt = lim (2) r0 r Uit 13, Page 4 Istitute ad Faculty of Actuaries

115 2016 Term structure of iterest rates Subject CT1 d = log P t (3) dt = 1 d P t (4) P dt t We also fid, by itegratig both sides of (3) ad usig the fact that P 0 = 1 (as the price of a uit zero coupo bod of term zero years must be 1), that Note Fds P t = 0 s e t We have described i this uit the iitial term structure, where everythig is fixed at time 0. I practice the term structure varies rapidly over time, ad the 5-year spot rate tomorrow may be quite differet from the 5-year spot rate today. I more sophisticated treatmets we model the chage i term structure over time. I this case all the variables we have used, i.e. P t y t f t,r Y t F t,r eed aother argumet, v, say, to give the startig poit. For example, y v,t would be the t-year discrete spot rate of iterest applyig at time v; F v,t,r would be the force of iterest agreed at time v, applyig to a amout ivested at time v + t for the r-year period to time v + t + r. Istitute ad Faculty of Actuaries Uit 13, Page 5

116 Subject CT1 Term structure of iterest rates Theories of the term structure of iterest rates 4.1 Itroductio Some examples of typical (spot rate) yield curves are give below. Figure 1: Decreasig yield curve I Figure 1 the log-term bod yields are lower tha the short-term bods. Sice price is a decreasig fuctio of yield, a iterpretatio is that log-term bods are more expesive tha short-term bods. There are several possible explaatios for example it is possible that ivestors believe that they will get a higher overall retur from log-term bods, despite the lower curret yields, ad the higher demad for log-term bods has pushed up the price, which is equivalet to pushig dow the yield, compared with shortterm bods. Other explaatios for differet yield curve shapes are give below. Figure 2: Icreasig yield curve I Figure 2 the log-term bods are higher yieldig (or cheaper) tha the short-term bods. Uit 13, Page 6 Istitute ad Faculty of Actuaries

117 2016 Term structure of iterest rates Subject CT1 Figure 3: Humped yield curve I Figure 3 the short-term bods are geerally cheaper tha the log bods, but the very short rates (with terms less tha 1 year) are lower tha the 1 year rates. The three most popular explaatios for the fact that iterest rates vary accordig to the term of the ivestmet are: 1. Expectatios Theory 2. Liquidity Preferece 3. Market Segmetatio Expectatios Theory The relative attractio of short ad loger term ivestmets will vary accordig to expectatios of future movemets i iterest rates. A expectatio of a fall i iterest rates will make short-term ivestmets less attractive ad loger term ivestmets more attractive. I these circumstaces yields o short-term ivestmets will rise ad yields o log-term ivestmets will fall. A expectatio of a rise i iterest rates will have the coverse effect. I Figure 1 it appears that the demad for log-term bods may be greater tha for short, implyig a expectatio that iterest rates will fall. By buyig log-term bods ivestors ca cotiue gettig higher rates after a future fall i iterest rates, for the duratio of the log bod. I Figure 2 the demad is higher for short-term bods perhaps idicatig a expectatio of a rise i iterest rates. Liquidity Preferece Loger dated bods are more sesitive to iterest rate movemets tha short dated bods. It is assumed that risk averse ivestors will require compesatio (i the form of higher yields) for the greater risk of loss o loger bods. This might explai some of the excess retur offered o log-term bods over short-term bods i Figure 2. Istitute ad Faculty of Actuaries Uit 13, Page 7

118 Subject CT1 Term structure of iterest rates 2016 Market Segmetatio Bods of differet terms are attractive to differet ivestors, who will choose assets that are similar i term to their liabilities. The liabilities of baks, for example, are very short-term (ivestors may withdraw a large proportio of the fuds at very short otice); hece baks ivest i very short-term bods. May pesio fuds have liabilities that are very log-term, so pesio fuds are more iterested i the logest dated bods. The demad for bods will therefore differ for differet terms. The supply of bods will also vary by term, as govermets ad compaies strategies may ot correspod to the ivestors requiremets. The market segmetatio hypothesis argues that the term structure emerges from these differet forces of supply ad demad. These theories are covered i more detail i Subject CA1, Core Applicatios Cocepts. 4.2 Yields to maturity The yield to maturity for a coupo payig bod (also called the redemptio yield) has bee defied as the effective rate of iterest at which the discouted value of the proceeds of a bod equal the price. It is widely used, but has the disadvatage that it depeds o the coupo rate of the bod, ad therefore does ot give a simple model of the relatioship betwee term ad yield. I the UK, yield curves plottig the average (smoothed) yield to maturity of coupo payig bods are produced separately for low coupo, medium coupo ad high coupo bods. 4.3 Par yields The -year par yield represets the coupo per 1 omial that would be payable o a bod with term years, which would give the bod a curret price uder the curret term structure of 1 per 1 omial, assumig the bod is redeemed at par. That is, if yc is the -year par yield, 1 = (yc ) 2 3 y 1 y 2 y 3 y y ( v v v... v ) 1v The par yields give a alterative measure of the relatioship betwee the yield ad term of ivestmets. The differece betwee the par yield rate ad the spot rate is called the coupo bias. 5 Duratio, covexity ad immuisatio I this sectio we cosider simple measures of vulerability to iterest rate movemets. For simplicity we assume a flat yield curve, ad that whe iterest rates chage, all chage by the same amout, so that the curve stays flat. A flat yield curve implies that y t = f t,r = i for all t, r ad Y t = F t,r = F t = for all t, r. Uit 13, Page 8 Istitute ad Faculty of Actuaries

119 2016 Term structure of iterest rates Subject CT1 5.1 Iterest rate risk Suppose a istitutio holds assets of value V A, to meet liabilities of value V L. Sice both V A ad V L represet the discouted value of future cashflows, both are sesitive to the rate of iterest. We assume that the istitutio is healthy at time 0 so that curretly V A V L. If rates of iterest fall, both V A ad V L will icrease. If rates of iterest rise the both will decrease. We are cocered with the risk that followig a dowward movemet i iterest rates the value of assets icreases by less tha the value of liabilities, or that, followig a upward movemet i iterest rates the value of assets decreases by more tha the value of the liabilities. I order to examie the impact of iterest rate movemets o differet cashflow sequeces we will use chages i the yield to maturity to represet chages i the uderlyig term structure. This is approximately (but ot exactly) the same as assumig a costat movemet of similar magitude i the oe-period forward rates. 5.2 Effective duratio Oe measure of the sesitivity of a series of cashflows, to movemets i the iterest rates, is the effective duratio (or volatility). Cosider a series of cashflows{ Ct k } for k = 1, 2,...,. Let A be the preset value of the paymets at rate (yield to maturity) i, so that A = k1 C tk v tk i The the effective duratio is defied to be v(i) = 1 d A = Adi A (5.2.1) A = 1 tk 1 Ct t k k vi t k1 k Ct v k i k1 This is a measure of the rate of chage of value of A with i, which is idepedet of the size of the preset value. Equatio assumes that the cashflows do ot deped o the rate of iterest. For a small movemet i iterest rates, from i to i +, the relative chage i value of the preset value is approximately v(i) so the ew preset value is approximately A(1 v(i)). Istitute ad Faculty of Actuaries Uit 13, Page 9

120 Subject CT1 Term structure of iterest rates Duratio Aother measure of iterest rate sesitivity is the duratio, also called Macauley Duratio or discouted mea term. This is the mea term of the cashflows { C t k }, weighted by preset value. That is, at rate i, the duratio of the cashflow sequece { C t k } is = tk tk Ct v 1 k i k tk Ct v k i k1 Comparig this expressio with the equatio for the effective duratio it is clear that = (1 + i) v(i) Aother way of derivig the Macauley duratio is i terms of the force of iterest, : = 1 d Ad A = di d v(i) i = e 1 di d = e = e v(i) = (1 + i) v(i) The equatio for i terms of the cashflows k respect to, recallig that v t i = e t k. Ct k may be foud by differetiatig A with The duratio of a -year coupo payig bod, with coupos of D payable aually, redeemed at R, is = DIa ( ) Da Rv Rv The duratio of a -year zero coupo bod of omial amout 100, say, is = 100 v 100v = Uit 13, Page 10 Istitute ad Faculty of Actuaries

121 2016 Term structure of iterest rates Subject CT1 Note that aother defiitio of duratio exists; the modified duratio. This is covered i more detail i Subjects ST5 ad ST6 (Fiace ad Ivestmet Specialist Techical A ad B) but ca be expressed i terms of the Macauley Duratio as i 1 p Where i (p) ad p are as defied i Uit Covexity The covexity of the cashflow series { C t k } is defied as c(i) = 2 1 d A di 2 A = A A = 1 tk 2 ( 1) Ct t k k tk vi t t1 k Ct v k i k1 Combiig covexity ad duratio gives a more accurate approximatio to the chage i A followig a small chage i iterest rates. For small Ai ( ) Ai ( ) A = A 1 i A + ½ 2 A 2 i A 2 (i) + 2 ½ c(i) Covexity gives a measure of the chage i duratio of a bod whe the iterest rate chages. Positive covexity implies that (i) is a decreasig fuctio of i. This meas, for example, that A icreases more whe there is a decrease i iterest rates tha it falls whe there is a icrease of the same magitude i iterest rates. Istitute ad Faculty of Actuaries Uit 13, Page 11

122 Subject CT1 Term structure of iterest rates Immuisatio Cosider a fud with asset cashflows { A t k } ad liability cashflows { L t k }. Let V A (i) be the preset value of the assets at effective rate of iterest i ad let V L (i) be the preset value of the liabilities at rate i; let v A (i) ad v L (i) be the volatility of the asset ad liability cashflows respectively, ad let c A (i) ad c L (i) be the covexity of the asset ad liability cashflows respectively. At rate of iterest i 0 the fud is immuised agaist small movemets i the rate of iterest of if ad oly if V A (i 0 ) = V L (i 0 ) ad V A (i 0 + ) V L (i 0 + ). The cosider the surplus S(i) = V A (i) V L (i). From Taylor s theorem: S(i 0 + ) = S(i 0 ) + S(i 0 ) S(i 0 ) +... Cosider the terms o the right had side. We kow that S(i 0 ) = 0. The secod term, S(i 0 ), will be equal to zero for ay values of (positive or egative) if ad oly if S(i 0 ) = 0, that is if VA (i 0 ) = VL (i 0 ). This is equivalet to requirig that v A (i) = v L (i) or (equivaletly) that the duratios of the two cashflow series are the same. 2 I the third term, is always positive, regardless of the sig of. Thus, if we esure that 2 S(i 0 ) > 0, the the third term will also always be positive. This is equivalet to requirig that VA (i 0 ) > VL (i 0 ), which is equivalet to requirig that c A (i) > c L (i). For small the fourth ad subsequet terms i the Taylor expasio will be very small. Hece, give the three coditios above, the fud is protected agaist small movemets i iterest rates. This result is kow as Redigto s immuisatio after the British actuary who developed the theory. The coditios for Redigto s immuisatio may be summarised as follows: 1. V A (i 0 ) = V L (i 0 ) that is, the value of the assets at the startig rate of iterest is equal to the value of the liabilities. 2. The volatilities of the asset ad liability cashflow series are equal, that is, v A (i 0 ) = v L (i 0 ). Uit 13, Page 12 Istitute ad Faculty of Actuaries

123 2016 Term structure of iterest rates Subject CT1 3. The covexity of the asset cashflow series is greater tha the covexity of the liability cashflow series that is, c A (i 0 ) > c L (i 0 ). I practice there are difficulties with implemetig a immuisatio strategy based o these priciples. For example the method requires cotiuous rebalacig of portfolios to keep the asset ad liability volatilities equal. There may be optios or other ucertaities i the assets or i the liabilities, makig the assessmet of the cashflows approximate rather tha kow. Assets may ot exist to provide the ecessary overall asset volatility to match the liability volatility. Despite these problems, immuisatio theory remais a importat cosideratio i the selectio of assets. E N D Istitute ad Faculty of Actuaries Uit 13, Page 13

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125 2016 Stochastic iterest rate models Subject CT1 UNIT 14 STOCHASTIC INTEREST RATE MODELS Syllabus objective (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. 1 A itroductio to stochastic iterest rate models 1.1 Prelimiary remarks Fiacial cotracts are ofte of a log-term ature. Accordigly, at the outset of may cotracts there may be cosiderable ucertaity about the ecoomic ad ivestmet coditios which will prevail over the duratio of the cotract. Thus, for example, if it is desired to determie premium rates o the basis of oe fixed rate of iterest, it is early always ecessary to adopt a coservative basis for the rate to be used i ay calculatios. A alterative approach to recogisig the ucertaity that i reality exists is provided by the use of stochastic iterest rate models. I such models o sigle iterest rate is used. Variatios i the rate of iterest are allowed for by the applicatio of probability theory. Possibly oe of the simplest models is that i which each year the rate of iterest obtaied is idepedet of the rates of iterest i all previous years ad takes oe of a fiite set of Istitute ad Faculty of Actuaries Uit 14, Page 1

126 Subject CT1 Stochastic iterest rate models 2016 values, each value havig a costat probability of beig the actual rate for the year. Alteratively, the rate of iterest may take ay value withi a specified rage, the actual value for the year beig determied by some give probability desity fuctio. 1.2 A itroductory example At this stage we cosider briefly a elemetary example, which although ecessarily artificial provides a simple itroductio to the probabilistic ideas implicit i the use of stochastic iterest rate models. Suppose that a ivestor wishes to ivest a lump sum of P ito a fud which grows uder the actio of compoud iterest at a costat rate for years. This costat rate of iterest is ot kow ow, but will be determied immediately after the ivestmet has bee made. The accumulated value of the sum will, of course, be depedet o the rate of iterest. I assessig this value before the iterest rate is kow, it could be assumed that the mea iterest rate will apply. However, the accumulated value usig the mea rate of iterest will ot equal the mea accumulated value. I algebraic terms: P(1 + k (i j p j )) P j1 k (p j (1 + i j ) ) j1 where i j is the jth of k possible rates of iterest p j is the probability of the rate of iterest i j. 1.3 Idepedet aual rates of retur I our previous example the effective aual rate of iterest was fixed throughout the duratio of the ivestmet. A more flexible model is provided by assumig that over each sigle year the aual yield o ivested fuds will be oe of a specified set of values or lie withi some specified rage of values, the yield i ay particular year beig idepedet of the yields i all previous years ad beig determied by a give probability distributio. Measure time i years. Cosider the time iterval [0, ] subdivided ito successive periods [0, 1], [1, 2],..., [ 1, ]. For t = 1, 2,..., let i t be the yield obtaiable over the tth year, i.e. the period [t 1, t]. Assume that moey is ivested oly at the begiig of each year. Let F t deote the accumulated amout at time t of all moey ivested before time t ad let P t be the amout of moey ivested at time t. The, for t = 1, 2, 3,..., F t = (1 + i t )(F t1 + P t1 ) (1.3.1) Uit 14, Page 2 Istitute ad Faculty of Actuaries

127 2016 Stochastic iterest rate models Subject CT1 It follows from this equatio that a sigle ivestmet of 1 at time 0 will accumulate at time to S = (1 + i 1 )(1 + i 2 )... (1 + i ) (1.3.2) Similarly a series of aual ivestmets, each of amout 1, at times 0, 1, 2,..., 1 will accumulate at time to A = (1 + i 1 )(1 + i 2 )(1 + i 3 )... (1 + i ) + (1 + i 2 )(1 + i 3 )... (1 + i ) +... (1.3.3)... + (1 + i 1 ) (1 + i ) + (1 + i ) Note that A ad S are radom variables, each with its ow probability distributio fuctio. For example, if the yield each year is 0.02, 0.04, or 0.06 ad each value is equally likely, the value of S will be betwee 1.02 ad Each of these extreme values will occur with probability (1/3). I geeral, a theoretical aalysis of the distributio fuctios for A ad S is somewhat difficult. It is ofte more useful to use simulatio techiques i the study of practical problems. However, it is perhaps worth otig that the momets of the radom variables A ad S ca be foud relatively simply i terms of the momets of the distributio for the yield each year. This may be see as follows. Momets of S From equatio we obtai (S ) k = (1 + i t t1 )k ad hece k ES [ ] = = E (1 it ) t1 k E[(1 + i t ) k ] (1.3.4) t1 Istitute ad Faculty of Actuaries Uit 14, Page 3

128 Subject CT1 Stochastic iterest rate models 2016 sice (by hypothesis) i 1, i 2,..., i are idepedet. Usig this last expressio ad give the momets of the aual yield distributio, we may easily fid the momets of S. For example, suppose that the yield each year has mea j ad variace s 2. The, lettig k = 1 i equatio 1.3.4, we have E[S ] = = E[1 + i t ] t1 (1 + E[i t ]) t1 = (1 + j) (1.3.5) sice, for each value of t, E[i t ] = j. With k = 2 i equatio we obtai 2 ES [ ] = = E[1 + 2i t + i t 2 ] t1 (1 + 2E[i t ] + 2 Ei t t1 [ ]) = (1 + 2j + j 2 + s 2 ) (1.3.6) sice, for each value of t, 2 t Ei [ ] = (E[i t ]) 2 + var[i t ] = j 2 + s 2 The variace of S is var[s ] = 2 ES [ ] (E[S ]) 2 = (1 + 2j + j 2 + s 2 ) (1 + j) 2 (1.3.7) from equatios ad These argumets are readily exteded to the derivatio of the higher momets of S i terms of the higher momets of the distributio of the aual rate of iterest. Uit 14, Page 4 Istitute ad Faculty of Actuaries

129 2016 Stochastic iterest rate models Subject CT1 Momets of A It follows from equatio (or from equatio 1.3.1) that, for 2, A = (1 + i )(1 + A 1 ) (1.3.8) The usefuless of this equatio lies i the fact that, sice A 1 depeds oly o the values i 1, i 2,..., i 1, the radom variables i ad A 1 are idepedet. (By assumptio the yields each year are idepedet of oe aother.) Accordigly, equatio permits the developmet of a recurrece relatio from which may be foud the momets of A. We illustrate this approach by obtaiig the mea ad variace of A. Let ad let = E[A ] m = 2 EA [ ] Sice A 1 = 1 + i 1 it follows that 1 = 1 + j (1.3.9) ad m 1 = 1 + 2j + j 2 + s 2 (1.3.10) where, as before, j ad s 2 are the mea ad variace of the yield each year. Takig expectatios of equatio 1.3.8, we obtai (sice i ad A 1 are idepedet) = (1 + j)(1 + 1 ) 2 (1.3.11) This equatio, combied with iitial value 1, implies that, for all values of, = s at rate j (1.3.12) Thus the expected value of A is simply s, calculated at the mea rate of iterest. Istitute ad Faculty of Actuaries Uit 14, Page 5

130 Subject CT1 Stochastic iterest rate models 2016 Sice by takig expectatios we obtai, for 2, 2 A = (1 + 2i + i 2 )(1 + 2A A 1 ) m = (1 + 2j + j 2 + s 2 )( m 1 ) (1.3.13) As the value of 1 is kow (by equatio ), equatio provides a recurrece relatio for the calculatio successively of m 2, m 3, m 4,... The variace of A may be obtaied as var[a ] = 2 EA [ ] (E[A ]) 2 = m 2 (1.3.14) I priciple the above argumets are fairly readily exteded to provide recurrece relatios for the higher momets of A. Example A compay cosiders that o average it will ear iterest o its fuds at the rate of 4% p.a. However, the ivestmet policy is such that i ay oe year the yield o the compay s fuds is equally likely to take ay value betwee 2% ad 6%. For both sigle ad aual premium accumulatios with terms of 5, 10, 15, 20, ad 25 years ad sigle (or aual) ivestmet of 1, fid the mea accumulatio ad the stadard deviatio of the accumulatio at the maturity date. (Igore expeses.) Solutio The aual rate of iterest is uiformly distributed o the iterval [0.02, 0.06]. The correspodig probability desity fuctio is costat ad equal to 25 (i.e. 1/( )). The mea aual rate of iterest is clearly j = 0.04 ad the variace of the aual rate of iterest is s 2 = 1 4 (.06.02) We are required to fid E[A ], (var[a ]) 1/2, E[S ], ad (var[s ]) 1/2 for = 5, 10, 15, 20, ad 25. Uit 14, Page 6 Istitute ad Faculty of Actuaries

131 2016 Stochastic iterest rate models Subject CT1 Substitutig the above values of j ad s 2 i equatios ad 1.3.7, we immediately obtai the results for the sigle premiums. For the aual premiums we must use the recurrece relatio (with 1 = s 1 at 4%) together with equatio The results are summarised i table It should be oted that, for both aual ad sigle premiums, the stadard deviatio of the accumulatio icreases rapidly with the term. Table Accumulatios for example Term (years) Sigle ivestmet 1 Aual ivestmet 1 Mea accumulatio ( ) Stadard deviatio ( ) Mea accumulatio ( ) Stadard deviatio ( ) The log-ormal distributio I geeral a theoretical aalysis of the distributio fuctios for A ad S is somewhat difficult, eve i the relatively simple situatio whe the yields each year are idepedet ad idetically distributed. There is, however, oe special case for which a exact aalysis of the distributio fuctio for S is particularly simple. Suppose that the radom variable log(1 + i t ) is ormally distributed with mea ad variace 2. I this case, the variable (1 + i t ) is said to have a log-ormal distributio with parameters ad 2. Equatio is equivalet to log S = log(1 + i t ) t1 The sum of a set of idepedet ormal radom variables is itself a ormal radom variable. Hece, whe the radom variables (1 + i t ) (t 1) are idepedet ad each has a log-ormal distributio with parameters ad 2, the radom variable S has a log-ormal distributio with parameters ad 2. Istitute ad Faculty of Actuaries Uit 14, Page 7

132 Subject CT1 Stochastic iterest rate models 2016 Sice the distributio fuctio of a log-ormal variable is readily writte dow i terms of its two parameters, i the particular case whe the distributio fuctio for the yield each year is log-ormal we have a simple expressio for the distributio fuctio of S. Similarly for the preset value of a sum of 1 due at the ed of years = V = (1 + i 1 ) 1... (1 + i ) 1 : logv = log(1 + i 1 ) log(1 + i ) Sice, for each value of t, log(1 + i t ) is ormally distributed with mea ad variace 2, each term o the right had side of the above equatio is ormally distributed with mea ad variace 2. Also the terms are idepedetly distributed. So, logv is ormally distributed with mea ad variace 2. That is, V has log-ormal distributio with parameters ad 2. By statistically modellig V it is possible to aswer questios such as: to a give poit i time, for a specified cofidece iterval, what is the rage of values for a accumulated ivestmet what is the maximum loss which will be icurred with a give level of probability Although outside of the scope of this Subject, it is iterestig to ote that such techiques may be exteded readily to model ad predict the behaviour of portfolios of ivestmets. These techiques are referred to as Value at Risk or VaR methods ad are covered i more detail i subjects CT8, Fiacial Ecoomics ad ST6 Fiace & Ivestmet Specialist Techical B. Oe possible defiitio of Value at Risk is a portfolio s maximum loss from a adverse market movemet, withi a specified cofidece iterval ad over a defied period of time. As with all statistical modellig techiques, the results of VaR ca oly be as good as the statistical model of the performace of the uderlyig ivestmets. I all ivestmet markets, eve seemigly efficiet oes, it cotiues to prove very difficult to choose a reliable statistical model which is robust over eve short periods of time. E N D Uit 14, Page 8 Istitute ad Faculty of Actuaries

133 Subject CT1 Fiacial Mathematics Core Techical Syllabus with Cross referecig to the Core Readig for the 2016 exams 1 Jue 2015

134 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. (Uit 1) 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. (Uit 2) 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. Page 2 Istitute ad Faculty of Actuaries

135 Subject CT1 Fiacial Mathematics Core Techical 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. (Uits 2 ad 3) 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) Demostrate a kowledge ad uderstadig of real ad moey iterest rates. (Uit 4) (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. (Uit 5) 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 Istitute ad Faculty of Actuaries Page 3

136 Subject CT1 Fiacial Mathematics Core Techical (vi) Defie ad use the more importat compoud iterest fuctios icludig auities certai. (Uit 6) 1. Derive formulae i terms of i, v,, d,, i (p) ad d (p) for a, s, s, ( p ), p ( a s ), a ad s. ( p a ), ( p ), s a, 2. Derive formulae i terms of i, v,, d,, i (p) ad d (p) for, m a ad. m a ( p ), m a, m a m a 3. Derive formulae i terms of i, v,,, a ad a for ( the respective deferred auities. Ia ), ( ) Ia, ( Ia ), ( Ia ) ad (vii) Defie a equatio of value. (Uit 7) 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. (Uit 8) 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. (Uit 9) 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. Page 4 Istitute ad Faculty of Actuaries

137 Subject CT1 Fiacial Mathematics Core Techical 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: (Uit 10) fixed iterest govermet borrowigs fixed iterest borrowig by other bodies idex-liked govermet borrowigs shares ad other equity-type fiace derivatives (xi) Aalyse elemetary compoud iterest problems. (Uit 11) 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. Istitute ad Faculty of Actuaries Page 5

138 Subject CT1 Fiacial Mathematics Core Techical (xii) Calculate the delivery price ad the value of a forward cotract usig arbitrage free pricig methods. (Uit 12) 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. 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. (Uit 13) 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. (Uit 14) 1. Describe the cocept of a stochastic iterest rate model ad the fudametal distictio betwee this ad a determiistic model. Page 6 Istitute ad Faculty of Actuaries

139 Subject CT1 Fiacial Mathematics Core Techical 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 Istitute ad Faculty of Actuaries Page 7