Veröffentlicht auf. Februar 2009

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1 F 721 Veöentlicht au Febua 29 COST OF CPITL ND VLUTION WITH IPERFECT DIVERSIFICTION ND UNSYSTETIC RISKS Ein Sevice von: FutueValue Goup G eail: Kontakt@FutueValue.de Intenet:

2 COST OF CPITL ND VLUTION WITH IPERFECT DIVERSIFICTION ND UNSYSTETIC RISKS D. Wene Gleissne, aco Wolum 1,2 FutueValue Goup G, Obee Gäten 18, 7771 Leinelden-Echtedingen, Gemany Tel , Fax , 1 D. Wene Gleissne is chaiman o FutueValue Goup G, Leinelden-Echtedingen, head o isk eseach o ash GmbH, as well as a visiting lectue at the Euopean Business School, the Univesity o Desden and othe institutes. aco Wolum is a senio analyst at FutueValue Goup G. 2 Fo a detailed pesentation, see Gleissne/Wolum, Die Emittlung von Eigenkapitalkosten nicht bösenoientiete Untenehmen ssessing the Equity Capital Cost o Non-Listed Companies, Finanz Betieb 9/28, pp The pesent aticle ollows this text. page 1 o 12

3 I Intoduction and Statement o the Poblem n essential challenge in detemining entepise values lies in deining appopiate isk adequate discount ates, which in the case o an unleveed im can be intepeted as an appoximation to equity capital cost. In spite o numeous estictive assumptions and unsatisactoy empiical esults the Capital sset Picing odel CP will continue to be the dominating method o detemining the cost o equity capital used in pactice o the oeseeable utue. ccoding to the unconditional CP the ollowing equation holds: 1 e E = + β wheeby Cov, ρ 2 β = = 2 with Expected etun om a isk-caying company cost o equity capital E e β Retun om a isk-ee investment govenment bond Expected value o maket etun, i.e. o a potolio o all isk-caying investment possibilities maket potolio Systematic isk o equity capital o investment Cov, Covaiance between etun om investment and maket potolio Standad deviation o maket etun Standad deviation o etun om investment ρ Coelation between etun om investment and maket potolio. Especially with the evaluation o non-listed medium-sized companies, the ollowing poblems and signiicant estictions petaining to the applicability o the CP must be taken into account when detemining cost o capital. 1. Homogeneity o expectations and planning consistency. Given the eality that inomation is distibuted asymmetically, how should the individual state o inomation be taken into account when detemining subjective decision values? 3 nd how should capital cost be kept consistent with the isks that ae explicitly o implicitly taken into account when evenues cash lows ae being planned? 2. Divesiication. How should non-divesiied idiosyncatic isks in capital cost and evaluation be taken into account when the evaluato does not have a peectly divesiied potolio and also cannot obtain one? 3. Risk measues. When detemining cost o capital and company value, what ae the consequences o applying measues o isk othe than the beta acto and standad deviations o the CP because in an impeect capital maket a ceditos ae subject to inancing estictions and/o b the evaluato is applying a saety ist appoach and thus wants to limit the extent o downside isks such as the pobability o insolvency 4? 3 see atschke/bösel, Untenehmensbewetung: Funktionen ethoden Gundsätze, 25. Company Evaluation: Functions ethods Pinciples 4 This can deinitely be economically ational with an impeectly divesiied potolio see 2, such as is geneally ound at medium-sized companies, even in the sense o expected utility theoy. Fo example, c Ke- page 2 o 12

4 This pape deals with all o these aspects, wheeby, o the sake o simplicity, only a 1-peiodmodel without taxes is examined. Chapte II shows how the expected values o etuns and the isk measue ae deived consistent to planning om the company s i.e. the evaluato s inomation egading the uncetain etuns to be evaluated. Chapte III then shows how a eplication model can be used to take impeect divesiication into account. The equity capital cost ate is calculated o an abitay degee o divesiication and it is shown how a company s capital cost can be deived om the pobability distibution o its uncetain etuns eanings, athe than om its etuns expessed as a elative change in value. It is then shown how this appoach can be genealised by utilizing isk measues othe than standad deviations and the beta acto based on them. Such isk measues ae helpul with non-nomal distibutions o etuns. II Subjective State o Inomation and Capital Cost consistent with Planning The valuev CF, as also the subjective use o a etuns seies CF company value, is dependent on the pobability distibution and the isk that this distibution implicitly descibes. One can take isks into account by means o an inteest suchage on the inteest o a isk-ee in the discount ate, o by means o a deduction o isk investment om the expected value o the etuns seies E CF. 5 π = λce R CF With the deduction o isk, cetainty equivalents CE CF ae calculated. Cetainty equivalents ae to be discounted with the isk-ee inteest ate base ate. E 3 CF E CF CE CF E CF λ CE R CF V CF = = = = λ R CF ' s RS With the so-called isk suchage method, which dominates in pactice, the isk-ee inteest ate is inceased by a isk suchage s when the value o the etuns seies CF is detemined. The isk suchage can be descibed as a poduct o the isk set, measued by a suitable isk measue R CF ' 6, and the pice pe unit isk λ RS. ins/smith/smith, Oppotunity Cost o Capital o Ventue Capital Investos and Entepeneus, JoFQ 2/24 pp = π CE CF E CF and uthe to such isk value models Sain/Webe, Risk-Value odels, EJoOR 7/1993 pp R CF is a isk measue that is standadised to the amounts o the etuns, o example as opeationalised by 6 the expected value o the value. It is to be intepeted as a isk measue o a etuns distibution. R CF = R CF V CF implies that λrs = λ CE. page 3 o 12

5 With the CP as in equation 1, it ollows that 4 e s = β and β can theeoe be intepeted in the CP 7 as the isk set R CF ' and the dieence between the maket etun and the isk-ee inteest as the maket pice o the isk λ RS. In pinciple, any evaluation that takes account o isk, i.e. detemination o a value highe than that yielded by the cetainty equivalents pinciple in dependence on the individual utility unctions, is possible. In pactice, howeve, most o the time λ CE is taken as a maket pice o the isk isk pemium o capital maket data. The isk measue R CF can likewise be detemined on the basis o histoical capital maket data. This, o example, is the oute most equently taken in pactice with CP, with the beta acto as the isk measue. Howeve, when detemining subjective decision values it is advantageous to use the supeio data o a isk analysis e.g. with a due diligence, which leads to a desciption o the etuns to be evaluated by means o a suitable stochastic pocess. The evaluatos have an inomation advantage inside inomation with espect to the capital maket, which should consequently and consistent to planning - be utilized by ational evaluatos in the couse o the evaluation. Speciically, this means that capital cost ates have to be deived on the basis o intenal planning- and isk inomation e.g. ove the aggegated oveall isk exposue. This is mandatoy when thee is no capital maket data at all, as is the case with non-listed companies. Risk inomation that shows the causes and extent o possible deviations om plan plays a key ole as pivate inomation o the planne o evaluato. Hee, building up on the identiied and evaluated isks, the evaluation elevant oveall isk exposue, which is captued by the isk measue, is detemined with the aid o aggegation, in the context o the planning 8. When this is done, the isks, which can be systematic o non-divesiied, unsystematic isks, and thei stochastic intedependencies, such as, o example, coelations, ae integated into the company planning based on the evaluation and, by means o a simulation, a epesentative andom sample o isk-elated possible utue scenaios o the company is calculated. The established ealisations o the taget vaiable CF e.g. poit give ise to aggegated equency distibutions 9 that allow conclusions petaining to the extent o isk-elated losses. In this manne, abitay isk measues R CF can be calculated and, o example, the equity 7 The CP cetainty equivalent is ρ E CF V CF = e, see Spemann, Valuation: Gundlagen modene Untenehmensbewetung, 24 Valuation: Fundamentals o oden Company Evaluation. 8 Regading methodology, o the utilization o onte-calo simulations o evaluations with capital maket impeections see: Gleissne, Kapitalkosten - de Schwachpunkt bei de Untenehmensbewetung, Capital Cost the Weak Spot with Evaluating Companies FB 4/25 pp as well as Gleissne, Gundlagen des Risikomanagements im Untenehmen, Vahlen, 28 Fundamentals o Risk anagement in a Company, see also Gleissne, Value-Based Copoate Risk anagement, in: Fenkel/Hommel/Rudol Ed. Risk anagement: Challenge and Oppotunity, xel Spinge, 24, pp These ae descibed by a lage numbe o calculated individual scenaios. Hee, in contast to capital maket equilibium models o peect makets e.g. CP, it is systematic isks and not divesiied unsystematic isks that ae elevant. This is to be explained, o example, by the cost o bankuptcy. C Foot/Schastein/Stein, Famewok o Risk anagement, Havad Business Review, Nov.-Dec. 1994, pp page 4 o 12

6 capital equiements o coveing isk so that a pedeined, goal ating dependent insolvency pobability p is not exceeded is deived. E CF and R CF ae theeoe evidently consistently deived om the pobability distibution CF, wheeby divesiication advantages and othe capital assets o the maket potolio can be taken into account c chapte III.1. The possibility o modelling almost all pobability distibutions and inte-tempoal dependencies o multi-peiodic etuns seies e.g. autoegessive pocesses o even GRCH-models is the big advantage when one caies out such a simulation-based evaluation. In this case, the typical estiction to matingale pocesses is not equied. 1 Cash low oecast o the company: E EZ CF ɶ Cash low- Estimate o the capital maket Risk estimate o the capital maket: Risk measue ß Risk pemium o the capital maket aket pice: Estimate o the values as seen by the capital maket Poblems with use o the CPodel: Does the capital maket know the isks as well as the company management? e only systematic isks elevant? Is histoical capital-maket data epesentative o the utue? Cash low-oecast o the company What must be done i no capital maket data exists? Risk estimate o the company s isk measue RZ R ɶ CF,, e.g. EC-need Decision value undamental Solution: oe isk accoding to planning leads to highe capital costs and alling value Risk pemium om economic oecast o the capital maket Figue 1: Capital maket-oiented evaluation vs. simulation-based evaluation consistent with planning 1 C Gleissne/Wolum, Simulationsbasiete Bewetung und Exit Peis-Schätzung bei PE-Gesellschaten, & Review, FCHVERLG de Velagsguppe Handelsblatt, Düsseldo, 7/28, pp Simulation-based Evaluation and Exit Pice Estimation with PE-companies, & Review, PUBLISHING HOUSE o the publishing goup Handelsblatt, Düsseldo, 7/28, as well as Gleissne/Kamaas/Wolum, Simulationsbasiete Bewetung von kquisitionszielen und Beteiligungen, in: Gleissne/Schalle Hsg., Pivate Equity Beuteilungsund Bewetungsveahen von Kapitalbeteiligungsgesellschaten, Weinheim, 28. Simulation-based Evaluation o cquisition Objectives and Paticipations, in: Gleissne/Schalle editos, Pivate Equity ppaisal and Evaluation Pocedues o Equity Investment Companies. page 5 o 12

7 III Calculation o Cost o Capital with Impeect Divesiication This section shows the way in which impeect divesiication is elevant o the cost o capital and value and also explains how these can be detemined. III.1 Geneal Replication ppoach o Detemining Risk-djusted Cost o Equity Capital III.1.a Uncetain Retuns as the Stating Point o Calculating Cost o Capital The stating point o the one-peiod obsevation 11 is uncetain etuns that ae expected in the utue CF. 12 In ode to detemine the value o these etuns, a eplication that peseves expected values and is adequate o the isks is caied out. Fo this pupose, two investment possibilities have to be pesent: an investment in the maket potolio with an uncetain etun ɶ and a isk-ee investment with inteest ate. Now select the amounts x and y o investment capital so that the isk o investing x in the maket potolio and investing y isk-ee with inteest ate equals the isk o the uncetain etun CF, wheeby isk is measued by an appopiate isk measue R CF. Fo example, such a isk measue could be the standad deviation, the value at isk, the conditional value at isk, o even an LP-measue. 5 R CF = R x ɶ + y The eplication should peseve expected values, which means that the expected value o the epayments o the investment into the maket potolio and into the isk-ee investment should equal the expected value E CF o the etuns seies CF. This is achieved by investing capital amount y in the isk-ee investment. 11 Fo an extension o a simple eplication appoach to seveal peiods, see Spemann, Valuation: Gundlagen modene Untenehmensbewetung, 24, pp Valuation: Fundamentals o oden Company Evaluation. 12 This etuns seies not only chaacteises the etun lows om the opeational business in the peiod, t= 1 peiod CF V 1 CF, t= 1 unde consideation, it also contains the value o attainable pice o the company at the end o. I only one peiod is actually being consideed, then this value will be zeo. I, howeve, the investment being consideed is still ecoveable ate the peiod being consideed, then one assumes that it will, o can, be sold at the end o the peiod. This can make it necessay to distinguish between the undamental value, o the detemination o which a ecusive evaluation is necessay so as to guaantee consistency with planning, and the attainable sales pice. The latte can, o example, be estimated with the aid o a multiplie model in that a pepetual annuity is detemined on the basis o the uncetain value o the opeational etuns and has discounting inteest i ɶ t = 1 that is deived om capital maket data and is thus likewise uncetain om today s point o view. CF, t= 1 V 1 CF, t= 1 = = mɶ CF, t= 1, so iɶ t= 1 CF = CF, t= 1 + W CF, t= 1 = CF, t= 1 + mɶ CF, t= 1 = CF, t= 1 + m ɶ. 1 1 page 6 o 12

8 6 E CF = E x ɶ + E y = x E ɶ + y With abitage eedom, the value o the uncetain etuns seies CF is the sum o the investments x and y. = + 7 V CF x y The next steps equie additional knowledge o the isk measue in use. I the standad deviation is taken as the isk measue 13, then the value o the etuns seies CF 14 is CF E CF E ɶ ɶ V Z = x + y = 8 ɶ Then the weighted aveage cost o capital expessed as expected etun om the uncetain investment tuns out to be 15 CF e E CF W CF = = 1 = + 9 ɶ E E = W CF ɶ CF = + ɶ E E CF ɶ CF E ɶ Hee, the etun o investment and the maket potolio ae assumed to be completely coelated, so that ρ = 1, which says that the company in question is its only asset, whence divesiication eects can be neglected. III.1.b Extension: Consideation o Coelations In geneal, howeve, the assumption ρ = 1 will not apply and thus divesiication options will be available. This is to be taken into account when evaluating the uncetain etuns. In this egad, though, only the non-divesiiable shae o the isk the systematic isks o the etuns is elevant o the evaluation inasmuch as unsystematic isks ae assumed to be divesiiable within the context o the suiciently lage total potolio The ollowing equation holds in geneal o standad deviation: + = a CF b a CF. 14 Fo the deivation see Gleissne/Wolum, ssessing the Equity Capital Cost o Non-Listed Companies, in: Finanz Betieb 9/28, pp I the value o the etuns seies o capital asset is zeo, then the expected etun is ininite. 16 C Spemann, Valuation: Gundlagen modene Untenehmensbewetung, 24 Valuation: Fundamentals o oden Company ssessment page 7 o 12

9 Nomally, the coelation coeicient o Bavais and Peason is used to measue the stochastic dependence between two isks o the undelying distibutions 17. Let ρ denote the coelation coeicient between the etuns CF om the capital asset and the maket potolio. Then the standad deviation o the etuns seies CF is educed in its elevance o the evaluation by being multiplied by ρ 18 so that the esulting value o the etuns seies CF is 1 V CF ρ CF E CF E ɶ ɶ = ccodingly, the weighted aveage cost o capital is ρ CF e E CF V CF = = 1 = + 11 ɶ E E = V CF ɶ ρ CF = + ɶ E E CF ɶ ρ CF E ɶ I the etuns om the investment and the maket etun stochastic ae independent o each othe, so that ρ =, then the equity capital cost ate is, as expected, identical with the isk-ee inteest ate, inasmuch as all isks ae then divesiiable and thus assumed not to be elevant o the evaluation. Compaison o the evaluation equation 1 with the CP s cetainty equivalent equation see ootnote 7 shows that these ae only in ageement unde the ollowing conditions: - The isk measue used is the standad deviation - Only the non-divesiied isks ae evaluated. - The state o inomation is symmetic, i.e. the etuns evaluated ae assumed to be assessed by the capital maket in the same way as by company planning CF =. III.1.c Extension: Potolio Examination and Impeect Divesiication Evaluation o the uncetain etuns seies CF was undetaken in Chapte III.1a o its own sake. Thus it was assumed that the investo holds no othe assets in his potolio o that the amount o isk elevant to evaluating the etuns in question is not inluenced by the emainde o his potolio. On the contay, chapte III.1.b assumes that the emaining potolio maket potolio investment is so lage that only systematic isks ae elevant o evaluation. 17 The poblem with using this coelation coeicient is that it only captues linea stochastic dependencies. The ank coelation coeicient, by compaison, only captues monotonic dependencies. Theoetically, copulae, on the othe hand, captue nealy all dependencies. 18 This is evident om the equation o the beta acto β ρ =. page 8 o 12

10 Hee, we assume that an investo has just invested a pat o his own assets in his own company and that he also has an abitaily lage potolio with value P. Fo the sake o simplicity, this potolio is assumed to be composed o one investment in the maket potolio shae in potolio a and one isk-ee investment shae in potolio 1-a. Thus the expected value o the investo s potolio P' at the end o the peiod in question is ɶ ɶ 12 E P' = E P a + P 1 a + CF = P a E + P 1 a + E CF Taking the standad deviation as the isk measue leads to ɶ 13 ɶ P ' = P a + P 1 a + CF = P a + CF Let ρ denote the coelation coeicient between the etuns seies Z ɶ and the maket potolio. Then the esulting standad deviation is 14 P ' = P a ɶ + 2ρ P a ɶ CF + CF Now let the potolio be evaluated by means o eplication once again. The value o the complete potolio esults in V = P ' ' P ' ɶ ɶ E P ' E = = ɶ 2 ɶ ɶ P a + ρ P a CF + CF E P E ɶ The divesiication advantage in this potolio is now teated as pat o the etuns om the company. Divesiication has a isk-loweing eect and thus inceases value; theeoe, value additivity does not apply. Thus the value o the company tuns out to be: 16 V CF ρ P a ɶ + ɶ P a CF + CF E CF P a ɶ E ɶ = ccodingly, the weighted aveage cost o capital is 17 e E CF E = = 1 = V CF P a ɶ + 2ρ ɶ + ɶ P a CF CF P a = + ɶ E E CF ɶ ɶ + 2ρ ɶ + ɶ ɶ P a P a CF CF P a E Let us now conside an investo who has invested capital P o 1. altogethe, placing one hal o it a =.5 in the maket potolio and the othe hal in a isk-ee investment. Let the E ɶ o 9% and a stan- maket etun have the nomal distibution with an expected value 19 C Gleissne/Wolum, Die Emittlung von Eigenkapitalkosten nicht bösenoientiete Untenehmen ssessing the Equity Capital Cost o Non-Listed Companies, Finanz Betieb 9/28, pp page 9 o 12

11 ɶ o 3%. ssume uthe that the isk-ee inteest is 5%. In addition, suppose this investo has a company o his own with an expected etun o E CF = 1. dad deviation at time t = 1 2 and a standad deviation CF o 3. Finally, let the coelation ρ between etun om the investo s company and maket etun be.75. Then the standad deviation o the investo s complete potolio is calculated in accodance with 14 as P a ɶ CF + = 1,,75,3 + 2,5 1,,75, = 456 Thus the value o the etuns seies o the capital asset company is calculated at 456 1, 1,,75,9,5,3 W CF = = 923,5 Hence the weighted aveage cost o capital in this example is E e 1, = = 1 = 8,3% 923 Finally, the next section shows how this appoach can be genealised to accommodate isk measues othe than the beta acto and the standad deviation which it is based on hee. The beta acto is only an appopiate measue o isk i a nomal distibution can be assumed. In paticula, standad deviation is not an appopiate measue o isk when etuns seies distibutions ae asymmetic o stongly ached because then the extent o isk may well, unde cetain cicumstances, be signiicantly undeestimated. Fo a isk to be evaluated, both possible positive as well as possible negative impacts have to be captued wheeby the latte have an even stonge inluence on evaluation, accoding to psychological eseach, than is shown by the utility theoy see the pospect theoy. Due to the special impotance o possible losses, so-called down-side isk measues, which ae designed to captue the possible scope o negative deviations, ae used in addition when oveall isk exposue is being descibed. Two such isk measues ae value at isk 21 and conditional value at isk 22. Use o them is sensible when the isks ae not symmetic and losses equie special attention. III.1.d Extension o the odel to bitay Risk easues The eplication equation o geneal isk measues R CF is given in Chapte III.3.a equation 5 as ollows: 18 R CF = R x ɶ E ɶ + E CF 2 This expected etun can, o example, be estimated by means o a multiplie model c ootnote 12. With, t= 1 CF =1 and mɶ =9 the ollowing aises as a esult: =, t= 1 ɶ CF CF m =1,. α 21 The value at isk at conidence level α is deined as P CF VaR CF = α, with < 1 22 The conditional value at isk at conidence level α is deined as CVaR CF = E[ < -VaR CF CF CF ]. a a < α. page 1 o 12

12 I the isk measue is known, this equation can be solved o x and thus be evaluated. Hee, howeve, one has to distinguish between position-dependent measues o isk, such as the value at isk and the conditional value at isk, and position-independent measues o isk, such as the standad deviation and the deviation value at isk. Since eal wold etuns can equently not be descibed by means o a nomal- o lognomal distibution e.g. due to at tails, position-dependent isk measues ae inceasing in impotance. In the ollowing, only position-independent isk measues such as standad deviation and DVaR 23 ae teated in geate detail. ccodingly, let R a + bcf = br CF. Then equation 18 simpliies to 2 R CF = x R ɶ Hence this isk measue is position-independent and can be egaded as a measue o planning cetainty o, equivalently, o the extent o possible plan deviations om the expected value. The value obtained ate suitable tansomations is 21 W CF R CF E CF E ɶ R ɶ = Thus the weighted aveage cost o equity capital, expessed as the expected etun o the entepeneu, is 22 R CF e E CF W CF = = 1 = + ɶ E E = W CF R ɶ R CF = + ɶ E E CF R ɶ R CF E ɶ IV Summay and Outlook ltogethe, it is shown that in an impeect, incomplete maket the cost o equity capital and values, especially o non-listed companies o o individual business aeas o listed companies can be ascetained with the speciic estictions and the inomation state o the evaluato being taken into account. With the aid o a simple and obust eplication model, we have shown how the company value decision values and the cost o capital discount ate can be computed. The computation stats with the pobability distibution o the uncetain etuns 23 The deviation value at isk o elative value at isk is deined = + α as DVaR CF E CF VaR CF. α 24 See the isk measues axiom system o Rockaella/Uyasev/Zabaankin, Deviation easue in Risk nalysis and Optimization, Reseach Repot, 22. page 11 o 12

13 seies CF to be evaluated, which is consistently tansomed to the expected value E CF and a isk measue R CF that need not be the standad deviation but can also, o example, be the value at isk. This consistency between the expected value o etuns as deived om planning and the isk measue is not given i the latte as o example in pactical utilization o the CP is detemined om capital maket data but stict inomation eiciency in the sense o Fama 197 is not assumed at the same time. 25 This pape has also shown which consequences o value and cost o capital esult om coelation o the cash lows to be evaluated with the evaluato s emaining asset positions e.g. its investment o the maket potolio and an impeect divesiication. Impeect divesiication is chaacteistic o the popietos o numeous non-listed companies especially in small and medium-sized businesses and implies highe cost o capital and hence a highe isk-adjusted etun as a esult o the geate isks. These highe cost o the evaluato s capital ae to be taken into account duing evaluation, especially when detemining subjective decision values, inasmuch as capital maket impeections geneally make it impossible to ealise peectly divesiied potolios. ll in all, the appoach poposed hee shows how evaluation o companies that elects isks appopiately is diectly possible on the basis o company planning that ceates tanspaency o isks while taking account o capital maket impeections. 25 C Fama, Eicient Capital akets: Review o Theoy and Empiical Wok, JoF 25/197 pp page 12 o 12