ANNEX 77 FINANCE MANAGEMENT (Workg materal) Chef Actuary Prof. Gada Pettere BTA INSURANCE COMPANY SE 1
FUNDAMENTALS of INVESTMENT I THEORY OF INTEREST RATES 1.1 ACCUMULATION Iterest may be regarded as a reward pad by oe perso or orgazato (the borrower) for the use of a asset, referred to as captal, belogg to aother perso or orgazato (the leder). Whe expressed moetary terms, captal s also referred to as prcpal. Example 1.1 A vestor who had opeed a accout some tme ago wth a tal depost of LVL 100, ad who have made o other paymets to or from the accout, would expect to wthdraw more tha LVL 100 f he were ow to close the accout. Suppose, for example, that he receves LVL 106 o closg hs accout. We shall say LVL 100 tal depost (prcpal) LVL 6 - terest. Iterest usually expressed percetages 6 100 6% 100. The terest s a paymet by the borrower to the vestor for the use of hs captal. We shall deote terest by expressed decmal fracto ad by C tal captal or prcpal. Example 1.2 Suppose you vest LVL 100 a bak accout that pays 8% terest per year. What shall you receve after year? 2
Always the precse codtos of ay trasacto wll be mutually agreed. For example: a) after stated perod the captal may be retured to the leder wth the terest due, b) several terest paymets may be made before the asset s fally retured to the borrower. Smple terest: Uder a smple terest rule, moey vested for a perod dfferet from 1 year accumulates terest proportoal to the total tme of the vestmet. A C (1 t ) Example 1.3 Suppose LVL 900 s deposted a savgs accout, whch pays smple terest at the rate of 5% per aum. Assumg that there are o subsequet paymets to or from the accout, fd the amout fally wthdraw f the accout s closed after a) sx moths, b) te moths, c) oe year, d) three years. Compoud terest: If terest s compouded yearly, the after 1 year, the frst year s terest s added to the orgal prcpal to defe the larger prcpal base for the secod year. Thus durg the secod year, the accour ears terest o terest. Ths s the compoudg effect, whch s cotued year after year. Accumulato uder the compoud terest: A(1) C C C (1 ) A(2) A(1) A(1) A(1) (1 ) C (1 ) A(3) C (1 )... A( t) C (1 ) t 3 2 3
Example 1.5 Suppose LVL 900 s deposted a savgs accout, whch pays compoud terest at the rate of 5% per aum. Assumg that there are o subsequet paymets to or from the accout, fd the amout fally wthdraw f the accout s closed after a) sx moths, b) te moths, c) oe year, d) three years. The seve-te rule Moey vested at 7% per year doubles approxmately 10 years. Also, moey vested at 10% per year doubles approxmately 7 years. 300 250 Compoud terest Smple terest 200 150 100 50 0 0,25 1 1,75 2,5 3,25 4 4,75 5,5 6,25 7 7,75 8,5 9,25 10 Fgure 1. Accumulato of LVL 100 by terest 10%. If terest s compouded oce per year the compoud terest rate s called effectve rate of terest. Ofte terest s pad more frequetly lke oce per year: credts, deposts, ad bods. 4
The aual rate of terest compouded more frequetly lke oce per year s called omal rate of terest. Nomal rate of terest per ut of tme o trasacto s such that the effectve rate of terest for the perod of legth h s h h. We shall cosder oly stuato whe 1 h where p s 4 for quarter, 2 for half year p ad 12 for moth, 365 for day ad we shall deote omal rate of terest ( p). At ( ) 1 p For example, a aual rate of 6% compouded every half year produce come year t moth two years Example 1.7 Suppose LVL 900 s deposted a savgs accout, whch pays aual terest rate of 8% compouded quarterly. Assumg that there are o subsequet paymets to or from the accout, fd the amout fally wthdraw f the accout s closed after a) sx moths, b) e moths, c) oe year, d) three years. Moey vested today leads to creased value the future as a result of terest. The formulas of the prevous secto show how to determe future value. Ths cocept ca be reversed tme to calculate the value that should be assged ow, the preset, to moey that s to be receved at a later tme. Ths value of moey s called preset value. 5
1.2 PRESENT VALUE Example 1.8 How much shall we eed to put depost ow to receve after year LVL 100, f bak guaratees 4% effectve aual terest rate? Example 1.9 How much shall we eed to put depost ow to receve after two years LVL 100, f bak guaratees 4% effectve aual terest rate? PV A (1 ) t Example 1.11 How much shall we eed to put depost ow to receve after twety days LVL 100, f bak guaratees 4% effectve aual terest rate (year-365 days)? 1.3 FUTURE AND PRESENT VALUES OF PAYMENT STREAMS Let us assume that cash arrves or we eed to pay may tmes after fxed perod, whch ca be year, quarter, moth or some aother perod. Every paymet ca be dfferet ad some of them ca be zero. Cash flows ca arrve or have to be pad the begg of perod or the ed. Such cash flows are called streams. Future value of the stream Gve a cash flow stream ( x1,..., x) ad terest rate each perod. Fd the future value of the stream. A x x x 1 ( ) 1(1 ) 2(1 )... (1 ) or 1 2 ( ) 1 (1 ) A x x2(1 )... x The preset value of a geeral cash flow stream lke future value ca also be calculated by cosderg each flow elemet separately. 6
Preset value of a stream Gve a cash flow stream ( x1,..., x) ad terest rate each perod. Fd the preset value of the stream. x x x x PV x... 1 (1 ) (1 ) (1 ) 2 3 4 1 2 3 1 or x1 x2 x3 x PV... 2 3 1 (1 ) (1 ) (1 ) Example 1.13 Cosder a cash flow stream (100,50,30,0,100) whe the perods are years ad the aual terest rate s 8% effectve. Calculate the future value a) after 5 years f paymets are the begg of each year, b) after 5 years f paymets are the ed of each year. Example 1.16 Cosder a cash flow stream (100,50,30,0,100) whe the perods are years ad the aual terest rate s 8% effectve. Calculate the preset value a) f paymets are the begg of each year, b) f paymets are the ed of each year. 2 BASIC COMPOUND INTEREST FUNCTIONS Seres of paymets, each of amout 1, to be made at tme tervals of oe ut are called autes-certa (or smply autes). If frst paymet s made after oe ut of tme auty s called auty arrear or mmedate auty. If frst paymet s made advace ( momet 0) auty s called auty-due. Preset values Oe ut of tme before the frst paymet s made (auty arrear). a / 1 1 (1 ) At the tme the frst paymet s made (auty-due): 7
a / 1 1 (1 ) (1 ) Futures value At the tme the last paymet s made (auty arrear). s / (1 ) 1 Oe ut of tme after the last paymet s made (auty-due). s / (1 ) 1 (1 ) Example 2.1 A loa of LVL 3000 s to be repad by 12 equal aual stalmets. The rate of terest for the trasacto s 6% per aum effectve. Fd the amout of each aual repaymet, assumg that paymets are made (a) arrear ad (b) advace. Example 2.2 A loa of LVL 5000 s to be repad three years by mothly paymets. The rate of terest for the trasacto s 6% per aum effectve. Fd the amout of each aual repaymet, assumg that paymets are made (a) arrear ad (b) advace. Example 2.3 Ivestor has agreed to pay LVL 20 each moth a savg accout 5 years (60 paymets). Bak s usg aual terest rate 4% effectve for ts savg accouts. Fd the accumulated sum of vestor after 5 years from ow f he s gog to start hs paymets (a) from ow the begg of each moth, (b) from ow the ed of each moth. 3 DISCOUNTED CASH FLOW (NET PRESENT VALUE AND YIELDS) 8
3.1 INTERNAL RATE OF RETURN Example 3.1 Gve cash flow stream ( 3,1,1, 3) wth tme terval oe year ad paymets the begg of each year. Wrte the formula for PV of that stream ad sketch the graph showg relatos betwee PV ad effectve aual terest rate. Iterest rate () 0 0.02 0.04 0.06 0.08 0.10 0.12 PV Formal defto Let ( x0, x1, x2,..., x) be a cash flow stream. The the teral rate of retur (or yeld of trasacto) of ths stream s a umber r satsfyg the equato x1 x2 x 0 x0... 2 1 r (1 r) (1 r). Equvaletly, t s a umber r satsfyg 1 1 r v 1 or r 1, v where v satsfes the polyomal equato 2 0 0 1 2... x x v x v x v. Ma theorem of teral rate of retur Suppose the cash flow stream ( x0, x1, x2,..., x ) has x0 0 ad xk 0 for all k, k 1,2,..., wth at least oe term beg strctly postve. The there s a uque postve root to the equato 2 0 x0 x1 v x2 v... x v. 9
Furtermore, f xk 0 (meags that the total amout retured exceeds the tal k 0 vestmet), the the correspodg teral rate of retur 1 r 1 s postve. v 3.2 EVALUATION CRITERIA The vestor always s before the problem whch vestmet project s better to vest. Therefore alteratve possble cash flow streams must be evaluated accordg to a logcal ad stadard crtero. Several dfferet crtera are used practse, but the two most mportat methods are based o preset value ad o teral rate of retur. 3.2.1 NET PRESENT VALUE The vestmet or project wll ormally requre a tal outlay ad possble other outlays future, whch ca be postve ad egatve. These cash flows may be completely fxed or they may have to be estmated. Therefore we shall deote by where c k s cash flow tme k ad xk ck cok that, the expresso et preset value ( NPV ) s used. Example 3.2 (Whch oos to buy?) co k s cash outflow tme k. To emphasze Suppose that you have the opportuty to buy ad plat oos, whch are blossomg after year ad aother oes, whch are blossomg after two years but wth most expesve flowers. Oos caot be used more after cuttg flowers. Both projects requres a tal outlay of moey to purchase ad plat oos. No other cash flows occur utl flowers are cut ad sold. We assume that the cash flow streams wth these two projects are ( 100,300) ad ( 100,0,400). We also assume that terest rate s 7%. Calculate et preset value for both projects ad compare them. 3.2.2 INTERNAL RATE OF RETURN Iteral rate of retur ca also be used to evaluate vestmet projects. The rule s: the hgher teral rate of retur, the better vestmet projects. 10
Example 3.3 (Whch oos to buy, cotued) approach. Evaluate both projects from example 3.2 to use teral rate of retur Approxmate formula for calculato of teral rate of retur P P0 I r P P0 2 where P 0 s the prce of the asset the begg, P s the prce of the asset at tme (after years), I s the average yearly come receved durg years. 3.3 APPLICATIONS AND EXTENSIONS Exercse 3.4 (Cost evaluato) Suppose you have to buy for offce computer techque ad you have to choose betwee two possbltes. Techque A cost LVL 4000 ad expected to have mateace cost LVL 1000 per year the ed of each year of usg ad assumed has useful lfe 4 years wthout salvage value. Techque B cost LVL 6500 ad expected to have mateace cost LVL 1000 per year the ed of each year of usg ad assumed has useful lfe 6 years wthout salvage value. The terest rate s 7%. Prepare oe cycle ad combed cycle aalyss for twelve years. INFLATION Iflato s characterzed by a crease geeral prces wth tme. It ca be descrbed quattatvely terms of a flato rate f. That leads us to defe the real terest rate. It shows how much moey crease realty takg accout a flato: 1 1 1 f 0 or 0 f 1 f 11
Example 3.5 Suppose that flato s 6%, the aual terest rate 10%, ad we have estmated cash flow of our busess real LVL as show the secod colum of table. Calculate the preset value real terms. Year Cash flow (LVL) 0-20000 1 7000 2 7000 3 7000 4 4000 4 RISK AND RETURN 4.1 THE MARKET ENVIRONMENT AND SECURITIES BONDS A bod s a facal cotract. The bod ssuer wll pay the bod s buyer perodc terest ad at the ed of the specfed term the prcpal, also called par value. Bod characterstcs Prcpal Sum at maturty Coupo paymets Fxed perodc terest stalmets Example 4.1 Sketch the dagram of the cash flow of a 4%, sem aual, three-year bod wth par value LVL 1000. 12
Advatages Good sources of curret come. Relatvely safe from large losses Coupo paymets are pad before dvdeds Dsadvatages Potetal proft s lmted. Proft s very sestve to flato. Proft s very sestve to terest rates. STOCKS COMMON STOCK Commo stock (commo shares or equty) represets part owershp a frm. Ower of commo stock has rghts: 1) has rghts to take part maagg of frm, 2) to receve part from proft what s left over after all other clams, 3) to receve part of salvage value of frm after all aother clams are satsfed the case of bakruptcy. Cash dvdeds are part of proft pad to stockholder after other labltes have bee pad. Declarato date s the day whe the board of drectors actually aouces that stockholders o the date of record wll receve dvded. Date of record s the day o whch the stockholder must actually ow the shares to be able to receve the dvded. Ex-dvded date s the frst day o whch, f the stock s purchased, stockholder are o loger allowed to receve the dvded. The paymet day s the day that the compay actually pays dvdeds. (For example, NYSE ex-dvded date s four tradg days before the date of record. The paymet day s about three weeks after the ex-dvded date). Commo stock does ot have a date o whch the cooperato must buy them back. OTHER SECURITIES Preferred stocks, Forwards, Futures, Swaps, Puts, Calls,. 4.2 RATE OF RETURN 13
Why the rate of retur s eeded? Example 4.2 Facal formato LVL about four vestmets s show followg table. Whch vestmet project s the best? Depost Stock Bod Lad Icome I quarter 15 10 0 0 Icome II quarter 15 10 50 0 Icome III quarter 15 10 0 0 Icome IV quarter 15 15 50 0 Begg of year 1000 2000 1000 3000 Ed of year 1000 2100 970 3200 Icome per year Captal ga or loss Total come HPR (%) The past rates of retur are used for several purposes. 1. Measurg hstorcal performace (s possble to compare the rates of retur o alteratve vestmets). 2. Estmatg future rates of retur. 3. Estmatg cost of captal. 4.2.1 SIMPLE RATE OF RETURN (HPR) Ivestors hold securty for a gve perod. The rate of retur measured for ths perod s called the holdg perod retur (HPR). The smple rate of retur measures how much the value of a gve vestmet creases or decreases over a gve perod of tme. 14
r t P P I P t t1 t t1 where Pt 1 s the prce of the asset at tme t 1, P t s the prce of the asset at tme t, I t s the come pad at tme t. Ths formula ft for perods of legth oe year or less. Example 4.3 exercse 4.2. Calculate HPR to use the smple rate of retur for vestmet projects Example 4.4 Let assume you purchased oe share of stock at the begg of the year for LVL 100 per share ad you have receved LVL 6 cash dvded per share the ed of year. The prce of stock the ed of year s LVL 105 per share. Calculate rate of retur f you are gog to sell havg by you share of stock. Example 4.5 Prces ad dvdeds receved each correspodg year are show followg table. Calculate HPR each year assumg that dvdeds are ot vested further ad they are receved the ed of year. Prce market (LVL) Dvded Begg of Year s (LVL) year Ed of year HPR (%) 1 2 3 4 5 2000 2 90 95 2001 4 100 2002 5 110 2003 5 100 2004 4 115 Smple rate of retur formula s created for stuatos whe come s receved at the ed of the perod. If come s pad durg the perod the smple rate of retur has certa lmtatos. 15
If cash flows occurs durg the perod, they must be theoretcally be used to buy addtoal uts of the vestmet (revested). The the most accurate approach s to calculate smple rate of retur for the sub-perod, ad the lk sub-perod returs to get the retur for moth, quarter, sem-aual or aual to use formula: r r1 r2 r (1 ) (1 )... (1 ) 1. It s called lkg method. If the holdg perod s loger tha oe year the aual rate of retur s calculated as teral rate of retur of vestmet: Pk P0 I r Pk P0 2 where P 0 s the prce of the asset the begg, P k s the prce of the asset at tme k (tme whe vestor plas to sell asset), I s the average yearly come receved durg k years. Example 4.6 Ivestor plas after vestg LVL 10000 ew busess to receve three years the ed of each year LVL 2000 but the ed of fourth year after sellg busess LVL 8000. Wrte formula for PV calculato ad calculate Iteral rate of retur for that project. 4.2.2 AVERAGE RATE OF RETURN: THE MEASURE OF PROFITABILITY The past average rate of retur measures the average proftablty of a vestmet. Example 4.7 16
the table. Suppose that you observe the rates of retur o two stocks. They are show Rate of retur (%) Year Stock A Stock B 1-20 2 2 50 24 Whch stock has a better hstorcal retur? Arthmetc average: r r t, t1 Geometrc mea: r, g (1 r,1) (1 r,2)... (1 r, ) 1 1 Example 4.7 Calculate arthmetc ad geometrc mea for rates of retur for stocks example 4.6. Whch oe s better? Example 4.8 Suppose a ut trust bega wth the LVL 100 per 100 uts. At the ed of frst year cost of 100 uts were LVL 50, but the ed of secod year oce more LVL 100. Calculate arthmetc ad geometrc mea of retur ad compare wth real retur. 17
Example 4.9 Calculate the arthmetc ad geometrc aual average rates of retur for the two vestmet fuds. Expla the dfferece betwee the two results. Year Rate of retur (%) Fud A Fud B 1 15 10 2 5 14 3-10 8 4 50 12 Arthmetc mea Geometrc mea 4.3 RISKS OR BONDS AND STOCKS 1. Default rsk Compay ca fal to pay coupo or omal f t has bods or ca have bakruptcy case of stocks. 2. Iterest rate rsk Chage terest rates affects prces of bods ad stocks. 3. Iflato rate rsk Iflato rate rsk or purchasg power rsk affects purchasg power of future cash recepts. 4. Rsk of call May bods cota call provsos. Rsk that bod wll be take back f terest rates goes dow. 5. Lqudty rsk 18
Rsk that vestor wll ot be able sell securty at all or to receve approprate prce. 6. Poltcal ad regulatory rsk Bods ad stocks are also exposed to poltcal ad regulatory rsk. Ths rsk refers to uforesee chages the tax or legal evromet that have mpact o prces. 7. Busess rsk Stocks ad corporate bod prces are flueced greatly by the prosperty of the partcular compay, as well as by ecoomy geeral. 8. Market rsk The prces of all securtes a partcular market ted to move together. 9. Exchage rate rsk Exchage rate rsk s a rsk of exchage rates f vestmet s dfferet curreces. 4.4 THE MEASURES OF RISK Volatlty of past rates of retur also provdes some formato for future. Example 4.10 Suppose that you observe the rates of retur o two stocks. They are show the table. Rate of retur (%) Year Stock A Stock B 1-20 6 2 0 8 3 50 7 19
The dsperso of the past rates of retur aroud the mea measures the hstorcal volatlty related to the correspodg vestmet (or rsk). The ma measure of dsperso s varace or stadard devato. Varace 2 t1 ( r r), t 1 2. Stadard devato. 2 Example 4.11 Calculate the stadard devato of rates of retur of stocks A ad B exercse 4.10. = Hstorcal rsk = Hstorcal average rate - premum of retur Hstorcal rskless terest rate Exercse 4.12 Calculate hstorcal rsk premum for stocks example 4.10 assumg that hstorcal rskless terest rate s 4%. 4.4 THE FUTURE 4.4.1 THE EXPECTED RATE OF RETURN Example 4.13 20
Suppose you have two possbltes to vest for oe year: 1) You ca buy govermet bod wth zero coupo rate ad LVL 1000 face value. The prce of bod s LVL 900. The bod matures exactly oe year from today. 2) You ca buy stock, whch s traded by LVL 900. Suppose o dvdeds are pad, ad the stock prce at the ed of the year s ether LVL 1000 wth a probablty (chace) of ½ or LVL 800 wth a probablty (chace) of ½. Calculate the rates of retur for both vestmets. Certaty s the stuato whch the future value of the asset s kow wth probablty 1. Ucertaty or rsk s the stuato whch there s more tha oe possble future value of the asset. I such case we say that the future value s a radom varable. PROBABILITY Objectve probabltes Calculated from past experece Subjectve probabltes Estmated probabltes The expected rate of retur: E( R) 1 p r Example 4.14 Assume you have three possbltes to vest dfferet securtes show followg table. All vestmets requre the same tal outlay LVL 100000. Securty A Securty B Securty C 21
Rate of Probablty Rate of Probablty Rate of Probablty retur (%) retur (%) retur (%) 4 1-4 ¼ -8 ¼ 0 ¼ 6 ½ 20 ½ 40 ¼ Calculate the expected rate of retur for gve securtes. Suppose that you have to select the vestmet by the expected rate of retur. Whch s better? Do you agree wth results? 4.4.2 THE MEASURES OF RISK (VARIANCE AND STANDARD DEVIATION) Oe of the measures of dsperso aroud the mea s varace. It s ofte used lke measure of rsk (If there are o sgfcat chages of rates of retur tme). Varace: p ( ) 2 r E R. 2 1 Stadard devato:. 2 Example 4.15 Calculate the varace wth both methods ad stadard devato for rates of retur of securtes gve example 4.14. 4.4.3 THE MEAN-VARIANCE CRITERION I a comparso of two vestmets, A ad B, there are sx possbltes: 1. E( R ) E( R ) ad. A B 22 2 2 A B 2 2 2. E( R ) E( R ) ad. A B A B
3. E( R ) E( R ) ad. A A B B 2 2 A B 4. E( R ) E( R ) ad. A B 2 2 A B 5. E( R ) E( R ) ad. A B 2 2 A B 6. E( R ) E( R ) ad. 2 2 A B The Mea-Varace Crtero The MVC s used to select those assets (or portfolos of assets) wth (1) the lowest varace for the same (or hgher) expected retur, or (2) the hghest expected retur for the same (or lower) varace. Example 4.16 Choose the securty example 4.14 to use MVC. Example 4.17 Assume you have three possbltes to vest dfferet securtes show followg table. All vestmets requre the same tal outlay LVL 100. Securty A Securty B Securty C Retur Probablty Retur Probablty Retur Probablty (LVL) (LVL) (LVL) 120 1 110 ½ 100 ½ 130 ½ 140 ½ Expected retur Expected retur Expected retur Varace Varace Varace Calculate the expected retur for gve securtes ad varace of retur. Suppose that you have to select the vestmet by the MVC. Whch s better? Do you agree wth results? 23
ATTITUDES TOWARDS RISK Ivestors who do ot lke rsk (volatlty) are called rsk averters. Ivestors who are rsk eutral completely gore a asset s varace ad make vestmet decsos based oly o the asset s expected rate of retur. Ivestors who lke the rsk (are able to hgher prce for a asset whose varace s larger) are called rsk seekers. 5. BOND VALUATION FIXED-INCOME SECURITIES Savg Deposts Moey Market Istrumets Bod Govermet bod: treasury bll, treasury bod Mortgage bod Preferred stock Covertble bod Qualty ratgs Moody s Stadard & Poor s Hgh grade Aaa AAA Aa AA 24
Medum grade A A Baa BBB Speculatve grade Ba BB B B Default grade Caa CCC Ca CC C C D The assgmet of a ratg class by a ratg orgazato s largely based o the ssuer s facal status as measured by varous facal ratos. For example, the rato of debt to equty, the rato of curret assets to curret labltes, the rato of cash flow to outstadg debt, as well as several others are used. Prce of bod The bd prce s the prce dealers are wllg to pay for the bod. The ask prce s the prce at whch dealers are wllg to sell the bod, ad hece the prce at whch t ca be bought mmedately. Prces are quoted as a percetage of face value the momet o ext coupo paymet date. Therefore bod quotatos gore accrued terest, whch must be added to the prce quoted order to obta the actual amout that must be pad for the bod. Used otatos: N -face value, R - redempto prce per ut of omal expressed as decmal umber, - tme to maturty expressed years, 25
g - rate of coupo expressed as decmal umber, m - frequecy of coupo paymets year, - desred rate of vestmet by vestor, P - calculated prce the momet of ext coupo paymet, AI - accrued terest P0 - bod prce exstg momet, Pk - estmated bod prce after k years, k -years after whc vestor plas sell bod, t1 - umber of days sce the last coupo paymet, t2 - umber of days utll ext coupo paymet, K - umber of coupo paymets from settlemet data to maturty, E - total umber of days betwee two coupo paymets Bod prce formula (called clea prce): m 1 1 1 g N m 1 P0 R N m 1 m m m More precse calculatos g K N R N g t1 P0 m t N 2 t2 K1 k1 E k1 E m E 1 1 m m Yeld: IRR YTM 26
Iteral Rate of Retur = Yeld to Maturty Therefore yeld s root of equato: m 1 1 1 g N m 1 P0 R N m 1 m m m Approxmately t ca be calculated to use the formula: R N P0 gn R N P0 2. Aother yelds: CY (Curret Yeld) u YTC (Yeld to Call) Exercse 5.1 The omal of bod s LVL 1000; t bears terest at 5% per aum, payable aually o 15. Jauary. Its maturty tme s 15 Jauary 2009, ad t s redeemed at par. Calculate clea ad total prce o 3 March 2005, f aual yeld s 4%. Example 5.2 Mortgage bod has omal LVL 10000; t bears terest at 6% per aum, payable quarterly o 15 February, 15 May, 15 August ad 15 November. Its maturty 27
tme s 15 May 2020 ad t s redeemed at par. Calculate total prce o 20 September 2005, f aual yeld s 7%. Example 5.3 Bod has omal LVL 10000; t bears terest at 5% per aum, payable twce per year ad t s redeemed at par after 5 years from ext coupo paymet. Calculate clea ad total prce what vestor ca pay f he wshes to have 4% rate of retur per aum ad 15 days have passed from prevous coupo paymet (167 left utl ext paymet). Calculate YTM prce of bod s LVL 10050. 6 STOCK VALUATIONS Beta as measure of rsk: Market rate of retur s rate of retur of all traded asstes. Rate of retur of dfferet securtes 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Market rate of retur yˆ a b x where b=slope, a=tercept where x s market rate of retur, y - correspodg securty rate of retur. Example 6.1 28
Calculate beta of gve securty. Market rate of retur (%) Securty rate of retur (%) x y 22 18 23 20 24 22 25 26 26 28 Beta s measure of market rsk. Beta Drecto of chages Speed of chages 2 Drecto of chages of securty rate of Twce qucker. 1 retur cocdes wth drecto of chages The same. 0.5 of market rate of retur. Twce slowly 0 Chages do ot have relatoshp. -0.5 Drecto of chages of securty rate of Twce slowly. -1 retur s opposte to drecto of chages The same -2 of market rate of retur. Twce qucker. CAPM (Captal Asset Prcg Model) Necessary rate of retur o asset s rate of retur what would be ecessary wth ts rsk measured wth beta. Rsk-free rate of retur s rate of retur eared wth assets wthout rsk. r r ( r r ) f M f 29
where r - ecessary rate of retur, rf - rskless rate of retur, r M - market rate of retur. Example 6.2 A) Gve that the beta of securty s 1.25, market rate of retur 10% ad rsk free rate of retur 6%, calculate ecessary rate of retur for gve securty. B) Market rate of retur s 13%, rsk free rate of retur s 8% ad beta for dfferet securtes are gve table. Calculate ecessary rate of retur for gve securtes. Securty Beta Necessary rate of retur A 1,4 B 0,8 C -0,9 Used otatos: D 0 -dvded pad curret year, g - dvded growth rate, P - preset value of paymet stream, P0 - curret prce of stock, Pk - prce of stock after k years (estmated), k -tme years whe vestor s plag to sell stock. Example 6.3 30
Market rate of retur ad correspodg rate of retur o stock A s show followg table. Assumg that market rate of retur s plag for ext year 18%, rsk free rate of retur s 4%, just pad dvded s LVL 0.8 ad t s plaed that dvdeds of that compay wll grow by aual rate 0.5 %, calculate 1) beta, 2) ecessary rate of retur o stock, 3) preset value of future paymets, 4) yeld f curret prce of stock s LVL 5.5. To kow that vestor s plag to buy stock above curret prce ad to sell t after 3 years above LVL 6, calculate teral rate of that trasacto. Market rate of retur (%) Stock rate of retur (%) 17 12 20 15 18 15 21 16 23 17 Exercse 6.4 Market rate of retur ad correspodg rate of retur o stock A s show followg table. Assumg that market rate of retur s plag for ext year 22%, rsk free rate of retur s 6%, just pad dvded s LVL 1.8 ad grows of dvdeds are ot plag, calculate 1) beta, 2) ecessary rate of retur o stock, 3) preset value of future paymets, 4) yeld f curret prce of stock s LVL 7. To kow that vestor s plag to buy stock above curret prce ad to sell t after 5 years above LVL 12, calculate teral rate of that trasacto. Market rate of retur (%) Rate of retur o stock (%) 13 12 18 15 18 15 20 16 22 17 31