Efficient Project Portfolio as a tool for Enterprise Risk Management

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1 Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27

2 Effcent Proect Portfolo as a tool for Enterprse Rsk Management Abstract Rsks of dfferent types are embedded nto every busness process and every busness actvty no matter what the busness of the organzaton. At the same tme, today all organzatons seekng for sustanable growth smultaneously manage a number of proects: endeavors that are even rsker than ther usual daly routne. Examples are proects for the development of a new product or servce, nvestment actvtes, mplementaton of an nformaton system, enterng a new market, re-brandng and many others. All proects of an organzaton consttute a proect portfolo. Snce proects are realzed wthn one organzaton, they are subect to all types of rsks that the organzaton has. And each proect at the same tme has ts own rsks that appear from the proect s scope and uncertantes. Proect Portfolo captures all the rsks that the organzaton s subect to along wth the rsks of the proects. These rsks nteract wth and nfluence each other, and have a strong mpact on the proect portfolo realzaton and organzaton s overall busness performance. That s why a tool for managng proect portfolo rsks should be an ndspensable part of an Enterprse Rsk Management system. A tool for managng organzaton s proect portfolo rsks s ntroduced n the paper. Ths tool s based on the approaches of H. Markowtz Portfolo Theory wth the man dea of the organzaton s portfolo rsk dversfcaton. The H. Markowtz theory was developed to help an nvestor buld an effcent nvestment portfolo (portfolo of securtes). However today we would hardly fnd organzaton that nvests only n equty market. Organzatons nvest n securtes and at the same tme nvest n proects of dfferent types. In the report we ntroduce a generalzaton of H. Markowtz Theory and apply t to proect portfolos (that could contan nvestments n securtes). The dea remans the same: we suggest that analyzng the proftablty (utlty) and rsks of every proect n the portfolo s mportant but nsuffcent; the correlatons of the proect s rsks and the nfluence that the proect has on rsks of other proects and proect portfolo should be consdered n order to buld an effcent proect portfolo. An effcent proect portfolo term s ntroduced n the presentaton. The term means a proect portfolo that s bult n order to acheve the organzaton s strategc goals wth mnmal rsks under the condtons of lmted resources. 2

3 We also consder a mathematcal model for buldng an effcent proect portfolo along wth an algorthm based on the model. In order to create the model, we have to consder a number of tasks that are descrbed n the paper. We present the formalzaton of the characterstcs of a proect and specfy the dfferences between proect portfolo and portfolo of securtes. That makes t clear that the dfferences between defnng and formalzng rsks of a securty and rsks of a proect are substantal. A new approach for defnng a proect rsk should be offered. The descrpton of ths approach follows; a way to determne a proect portfolo rsk s also presented. Based on ths approach to quantfyng the rsks of the proects, we offer a new concept of the nterference of proects wthn a portfolo. The nterference can be determned through the creaton of a rsk matrx of a smultaneous realzaton of proects. The descrpton of the model for buldng an effcent proect portfolo, based on Markowtz theory, completes the theoretcal part of the presentaton. Then the algorthm s ntroduced along wth the results of ts approbaton n one of Russan fnancal nsttutons. We beleve that the approach descrbed n the paper could help organzatons buld balanced and effcent proect portfolos, thus mnmzng the rsks. That, n turn, s a necessary condton for achevng strategc obectves and sustanable growth. Keywords: proect portfolo management, proect management, modern portfolo theory 3

4 . INTRODUCTION In the paper we ntroduce an approach to manage the organzaton s rsks whch s based on Proect Management Methodology, Modern Portfolo Theory and Rsk Management Technques. Nowadays all organzatons that are seekng for growth and sustanable compettve advantage need to realze dfferent proects. Proect s an endeavor that s unque and lmted n tme. Examples are proects for the development of a new product or servce, mplementaton of an nformaton system, enterng a new market, re-brandng and many others. Innovaton and nvestment, actvtes that have a very strong nfluence on busness performance and yeld compettveness of an organzaton on mcro level (and economc growth on macro level) are proects by defnton. All proects of an organzaton consttute a proect portfolo. Hence rsks of a proect portfolo should be managed and mtgated. Snce proects are realzed wthn one organzaton, they are subect to all types of rsks that the organzaton has. And each proect at the same tme has ts own rsks that appear from the proect s scope and uncertantes. Proect management methodology provdes tools for managng one proect at tme. Busness success however, depends on the characterstcs of the whole portfolo of proects that are under realzaton. Proect Portfolo Management mples ) the analyss of all proects wthn a portfolo as a whole; and that makes t possble to perform 2) the analyss of rsks that result from smultaneous realzaton of proects (that mples the analyss of proect s nterdependences). Consderaton of all these factors leads to buldng a balanced proect portfolo and provdes strateges for rsk hedgng. The concept of Proect Portfolo Management that s presented s based on the approaches of Modern Portfolo Theory whch was ntroduced by Harry Markowtz n 952. Models, methods, and approaches of Modern Portfolo Theory proved to be successful when appled to a stock market. The nature of nvestment n a proect portfolo remans the same: nvestor wants to maxmze return and mnmze the rsk. The only dfference s that he or she nvests n a wder range of actvtes. Models of Modern Portfolo Theory can t be appled to proects wthout modfcaton. For example, the characterstcs of an asset wthn a portfolo of securtes can be defned by ts expected return and standard devaton of return (whch s treated as a rsk of a securty). That doesn t work wth proects: the fact that the cumulatve rsk of a proect s not formalzed makes t mpossble to apply the concepts of the Modern Portfolo Theory to a proect portfolo. 4

5 Before we ntroduce the model, some conceptual aspects of proect portfolo management need to be underlned: Proect Portfolo Management s closely related to the strategc plannng process. Effectve strategc plannng process and systematc achevement of strategc goals are key factors necessary for organzaton s development, compettveness and growth. Organzatons now apply dfferent strategc management technques (e.g. Balanced Scorecard). These technques usually are helpful n determnaton of the organzaton s strategy. But they don t provde any tools for ts realzaton. Proect Portfolo may serve as a tool for achevng strategc goals. We can determne proects that must be realzed n order to acheve each goal. In other words, for a set of strategc obectves we determne a correspondng set of proects or a proect portfolo: Pcture. Strategc obectves and a proect portfolo Proect portfolo s an essental result of strategc plannng process, even f the proect portfolo s not recognzed as such. Every proect wthn a portfolo can be treated as an asset whch has a number of characterstcs. We can see the parallel wth portfolo of securtes: all proects wthn a portfolo nfluence each other; proect portfolo s somethng that can be characterzed wth rsk and return. The task s to buld a portfolo that gans maxmum return wth mnmal rsk. The applcaton of ths concept thus can provde strateges for strategc rsk hedgng. Wthout gong nto formal mathematcal defnton of an effcent proect portfolo, we can say 5

6 that an effcent proect portfolo s a portfolo of proects that ) comples wth strategc obectves of an organzaton and 2) has a Pareto optmal rsk-return rato. Proect Portfolo realzaton s a tool that leads to systematc achevement of strategc goals wth a mnmum level of rsk. Investment n an asset on a stock market s an endeavor that s unque and lmted n tme. Hence t can be consdered a proect. So a generalzaton of Modern Portfolo Theory and ts applcaton to proects could help sgnfcantly mtgate organzatons rsks whch leads to a better busness dversfcaton. 2. MODEL FOR BUILDING AN EFFICIENT PROJECT PORTFOLIO a. Dfferences between a Portfolo of Proects and a Portfolo of Securtes In order to generalze the classcal model of Modern Portfolo Theory thus makng t applcable to the case when we have proects nstead of securtes we need to perform a formalzaton of the man dfferences between two types of portfolos. There are sgnfcant dfferences n characterstcs of a proect portfolo and a portfolo of securtes. The man dfferences are descrbed n Table : Aspect Securty Portfolo Proect Portfolo For what purpose does To gan maxmum return. To realze organzaton s strategy an nvestor buld a portfolo? What s an effcent portfolo? A portfolo that yelds maxmum Besdes classcal rsk-return rato return wth a gven level of rsk or we can search for portfolos wth mnmum rsk wth a gven level of optmal ratos of dfferent return. characterstcs. What parameters can Return, rsk, correlaton wth other A lot of parameters: mnmum be used to formalze assets. requred nvestment, duraton, the characterstcs of an The characterstcs are requred human resources etc. obect wthn a mathematcally formalzed. The characterstcs are not portfolo? formalzed. What resources do we Mostly fnancal (that can be Fnancal and human resources. need n order to buld a portfolo? borrowed). Human resources usually are unque and n case of ther nsuffcency can t be borrowed. Who determnes the It can be determned by nvestor, The proect s duraton s determned duraton of a portfolo snce usually nvestors work on a by ts lfecycle and scope specfcs. 6

7 realzaton? lqud market. Investor can hardly nfluence How can we determne nterdependences between obects wthn a portfolo? Do we have any hstorcal nformaton on characterstcs of obects that consttute portfolo Interdependences are determned by correlaton of random varables that characterze returns of assets wthn a portfolo. Yes, statstcal nformaton on return of an asset. proect s duraton. Interdependences are determned by the rsks of proects smultaneous realzaton (ths approach s descrbed below). Only nformaton on analogcal proects that were realzed n the past. Table. The dfferences between a proect portfolo and a securty portfolo. The maor dfference that we can see n the table s that the characterstcs of a proect are not formalzed. Formalzaton s necessary n order to apply the concepts of MPT to proects. b. Formalzaton of proect s characterstcs Based on the results of the analyss of Modern Portfolo Theory, Rsk Management Theory, Proect Management Methodology, Fnancal Management we can pck out the followng parameters that characterze a proect. Let us consder P as a proect that s realzed n organzaton and that s a part of a proect portfolo. For each proect P we can use the followng characterstcs (presented n Table 2): Characterstcs Descrpton. Mnmum Investment Requred 2. Net Present Value (NPV). 3. Human Resources Requred W - mnmum amount of nvestment requred for the realzaton of the proect. NPV - Net Present Value of the proect. NPV s a random varable snce ts value depends on the proect rsks. We wll treat an expected NPV as a bass, whch should be calculated under the assumpton of favorable realzaton of a proect (when no rsks occurred). Very often organzatons can t realze proects due to the lack of competent staff. That s why we ntroduce the parameter HR - human resourced requred for proect. 4. Proect s Duraton tme T expected duraton of proect. 7

8 5. Oblgatory feature Oblgatory are proects that shall be ncluded nto portfolo 6. Strategc Obectves Complance Index 7. Cumulatve rsk of a Proect 8. Correlaton wth other proects 9. Human resources development ndex. Busness Process Improvement Index. Organzaton s mage mprovement ndex. due to the regulatory or normatve requrements (f Obl =, a proect shall be ncluded nto the portfolo, f Obl =, a proect may be or may not be ncluded nto portfolo). Str - strategc obectves complance ndex. The complance to strategc obectves s one of the key success factors of a proect (s measured n ponts, from to ). Cumulatve rsk of a proect s a random varable U (ω) (full descrpton wll be provded later n the paper). Infl Correlaton wth other proects can be determned by = { Infl } a vector that shows proect s nfluence on other proects. Ths fgure captures the extent to whch the proect s nterestng for organzaton from the perspectve of Human Resources development (s measured n ponts, from to ). Nowadays a lot of proects are realzed wth the purposes of mprovng the qualty of busness processes of an organzaton. The return from proects of ths type can hardly be measured wth fnancal fgures, hence the nfluence of ths knd of proects on organzaton should be expressed n Busness Performance Improvement Index (s measured n ponts, for to ). Ths parameter shows the extent to whch the realzaton of a proect enhances the mage of an organzaton (s measured n ponts, from to ). Detaled descrpton and formalzaton of proect s rsk along wth the approach to measure nterdependences of proects wthn a portfolo wll be ntroduced later n the paper. So, proect P can be descrbed by the followng vector of varables (parameters): P = { W, NPV, HR, T, U ( ω),{ Infl }, Obl, Str, Hrd, m, Bp} Net Present Value and Return on Investment can be treated as maor characterstcs of a proect snce they show ts economc effect. At the same tme we should note that there are a lot of proects that are not of nvestment proects type. For these proects NPV does not work. There are some proects that NPV can hardly be calculated at all. 8

9 We suggest that a more comprehensve characterstc should be used - usablty of a proect. Usablty captures the aspects of proects that can t be taken nto account when calculatng NPV. We can pck out the key factors that determne the usablty of a proect:. Fnancal result (whch s measured wth NPV); 2. Human resources development; 3. Image Improvement; 4. Busness Process Improvement. Utlty functon looks lke the followng: and s measured n ponts. Y(P)=F(NPV,HRd,Im,BPI) The use of the utlty functon makes t possble to evaluate proects and to take nto account the key factors of usablty of the proect rather than NPV only. c. Proect s partcular and cumulatve rsk In ths secton we consder the smplest case when NPV only s consdered as a factor of usablty of the proect. Let s assume, that for each proect wthn a portfolo, we can determne a set of rsky events ω ω,..., ω, 2 K. The rsk of a proect can be measured by the mpact that certan rsky event has on t. The mpact that s assocated wth the rsky event ω can be measured as a part U (ω) of the expected NPV (whch s calculated under the assumpton that no rsky event occurs). We consder the mpacts of rsky events to be multplcatve: f the rsky event ω occurs and other rsky events don t, nvestor gets ( U ( ω)) NPV nstead of an expected NPV. For example, f the rsky event ω results n the rse of nterest rate, and that, n turn, causes 3% loss for the proect, then U ( ω) =. 3 and ( U ( ω)) =. 7. ( U ( ω)) can also be negatve; that means that the rsks lead to losses durng proect mplementaton. We treat rsk of a proect as a random varable, whch s defned on the set of elementary rsky events Ω characterzng the mpact of the rsky event on the proect. In the smplest case, the random varable s determned by the probablty P (ω) of the rsky event and by the magntude of the mpact U (ω) that the partcular rsky event has on the proect. Below s the probablty dstrbuton for the random varable Partcular Rsk of a Proect : 9

10 Name of the rsk Probablty Probablty dstrbuton of a partcular proect rsk The probablty of the The probablty that the rsk occurrence rsky event wll not occur P (ω) P( ω) Impact U (ω) In order to generalze Markowtz theory we need to formalze the rsk of the proect n such a way that t allows us to assocate each proect wth a random varable that characterzes ts rsk. In other words, we need to formalze the cumulatve rsk of a proect. If we have two or more rsky events that happen smultaneously, the magntude of the cumulatve rsk depends on the rsks nterference type. Let s denote by ω an event symbolzng that both rsks ω and. We consder rsks ω and ω to be addtve, f U ( ω ) = U ( ω ) + U ( ω ). ω occur smultaneously. Ths group of rsks usually contans such events as delays n supples: equpment, materals or servces. That results n addtonal costs (we need to fnd new supplers, etc). If we have two supplers that delay the shpment smultaneously, that wll result n proect losses that are equal to the sum of the mpacts from the frst suppler and from the second one. We suppose that ndependent rsks have such a property. 2. We consder rsks ω and ω to mutually strengthen each other, f U ( ω ) = α ( U ( ω ) + U ( ω )) α >, and mutually mtgate each other, f α <. Most of the proect s rsks belong to ths group. Realzaton of two events of dfferent types usually leads to more substantal losses, than the sum of losses assocated wth two rsks that occur separately. 3. If two events ω and ω occur smultaneously and the mpact that the event ω has on the proect makes meanngless the consderaton of the mpact of the event ω, we say that rsk ω absorbs rsk ω. In ths case U ( ω ) = U ( ω ) = max{ U ( ω ), U ( ω )}. We also call ths type of rsks absorbent rsks. In ths report we wll consder the stuaton that s determned by the followng assumpton.

11 Assumpton. In a group of dependent rsks (the group of dependent rsks s comprsed by the three groups as descrbed above), the probablty that three or more rsks wll occur smultaneously can be consdered neglgble and equal to zero. Under Assumpton, let s now consder an arbtrary event ω *. As t was ndcated above, each proect has a correspondng set of rsky events ω ω,..., ω, 2 K. Among these rsks, we choose those that are ndependent. In other words, the realzaton of these rsks doesn t depend on the realzaton of other rsks. These are usually rsks that are external for the proect: changes n nterest rate, unfavorable weather condtons, etc. Wthout loss of generalty we consder that these rsks are numbered respectvely,, К. At the same tme, we group dependent rsks n such a way that we could treat rsks from dfferent groups as ndependent. Agan, wthout loss of generalty we consder that there s only one group of dependent rsks. All the constructons that are presented below can be broadened to the case wth a larger number of groups. To sum up, we have К ndependent rsky events ω, ω2,..., ωk and a group of rsks ω, ω 2,..., ω K + K K that are nterdependent. Let s assume that there s only one rsk occurrng n + the group of dependent rsks ω n (n>k). We denote the probablty of ths event as P n. Thus the probablty of the event Р (ω*) s determned by the equalty * P ω ) = P( ω ) P( ω )... P( ω ) ( 2 K P n Let the set of all possble events of such type be referred to Ω *. When the elementary rsky event ω ** mples the smultaneous occurrence of two dependent rsks ( ωk and ω m, ( k, m > K)) we have ** P ( ω ) = P( ω) P( ω2 )... P( ω K ) P k, m We wll use the symbol Ω ** n order to denote ths set of events. Thus, denotng by ω the event when no rsks occur, we can determne a random varable characterzng the mpact of all possble rsky events on the proect. Ths random varable s treated as a cumulatve rsk of a proect. The determnaton of ths varable allows us to compare dfferent proect on the bass of the rsk crteron. The dstrbuton functon for the random varable U (ω) s presented below:

12 Dstrbuton functon for the varable Cumulatve rsk of a proect U K U l + l= U n K U k + U k, m k= K ω ω Ω * ** ω Ω ω d. Cumulatve proect rsk determnaton algorthm Let s consder a bank whch strategc goal s to enter new geographcal areas thus t s launchng proects for openng new offces. Openng a new offce s a classc example of a proect. It s subect to the followng rsks (as they are presented n the Table below): Rsks of the proect for openng of a new offce ω P( ω ) U (ω). Market Rsk (Demand Volatlty) Delays n Equpment Supples...3 Uncertantes that result from the possble changes n regulatory requrements..2.4 Operatonal rsks.2.5 Here we have a lst of partal proect rsks. A proect may have any number of partcular rsks. Our knowledge of partcular rsks s very mportant however t s not enough f we want to compare proects. In order to do that we need to calculate the cumulatve rsk of the proect on the bass of the model that was descrbed above. Frst of all, we need to determne dependent and ndependent rsks. Wthout gong nto detals, we consder rsks ω and ω 4 to be ndependent. Accordng to the approach descrbed earler, we determne the elementary events of the Ω set, the probabltes of the elementary events and the mpacts assocated wth these events. Whle determnng the probablty of the elementary events we take nto account rsks nterference (we consder rsks ω 2 and ω 3 to strengthen each other). As a result, we have the followng table that descrbes the set of the elementary rsky events (we note the elementary event of the Ω set as α :

13 Event Probab Descrpton Impact lty α.499 None of rsky events occurred α.5 Market rsk (Demand Volatlty),5 2 α. Delays n Equpment Supples, 3 α. Uncertantes that result from the 4 Regulatory Requrements,3 α.2 Operatonal Rsks,5 5 α.5 Market rsk (Demand Volatlty) and 6 Delays n Equpment Supples α.5 Market rsk (Demand Volatlty) and 7 Uncertantes that result from the,6,8 Regulatory Requrements α. Market rsk (Demand Volatlty) and 8 Operatonal Rsks α. Delays n Equpment Supples and 9 Uncertantes that result from the Regulatory,55,6 Requrements α.2 Delays n Equpment Supples and Operatonal Rsks α. Uncertantes that result from the Regulatory Requrements and Operatonal,5,35 Rsks The probablty dstrbuton functon looks lke the followng:

14 ,6,5,4,3 Ряд,2,,2,4,6,8 The expected value of the cumulatve rsk of the proect s equal to,946. e. Interference determnaton of Proects wthn a Portfolo. Smultaneous Realzaton of Proects Rsk Matrx. The concept that we suggest mples that the nterference of proects A and B can be determned through the extent to whch the smultaneous realzaton of proects changes ther respectve cumulatve rsks. If two proects are beng realzed smultaneously and ther rsks reman the same as f they were realzed separately, these two proects are ndependent. If the rsks ether decrease or ncrease, these proect are dependent. In order to formalze the nterference (nterdependence) of proects wthn a portfolo, we should determne the rsk matrx of a proects smultaneous realzaton. A rsk of a smultaneous realzaton of proect and proect ρ can be determned by experts and s a value n the nterval [-;]. If ρ s equal to zero, the proects are ndependent, f ρ =, the proects are mutually exclusve, and f ρ = the proects can be consdered complementary. Determnaton of ρ brngs us closer to the orgnal Markowtz model snce ρ s an analogue of the correlaton coeffcent. Determnaton of the rsk matrx of proects s smultaneous realzaton s necessary n order to create an adequate model for buldng effcent proect portfolo smply because the nterference and nterdependence of proects wthn a portfolo has a strong mpact on portfolo rsk. The ρ matrx s determned wth the help of the expert udgments approach. These values gve us an opportunty to determne the covaraton values and consequently determne the rsk of proect portfolo.

15 f. Model for buldng an effcent proect portfolo Let proect portfolo contan L proects that consttute a set J={,., L }. If NPV s an expected (desgned) net present value of the proect, and U (ω) - s a coeffcent of the cumulatve rsk of proect, then the net present value of the portfolo s a random varable that can be defned as a sum: NPV = U (ω ) NPV P I By ω here we denote a random rsky event of all the proects wthn a portfolo. Detaled probablty descrpton of such a random varable s complcated enough due to the huge number of dfferent combnatons of rsks. At the same tme, we don t need t n order to calculate the portfolo expected NPV ( E[ NPVP ]) and standard devaton σ [ NPVP ]. All we need to do s to determne the probabltes of a smultaneous occurrence of rsky events of dfferent proects α k and α m, where ndexes and denote the proect ndces and k and m ndces of rsky events of respectve proects. Symbol ω s substtuted wth α to stress that here we also consder the composte rsks, that unte two elementary rsky events. We wll note these probabltes as P, k, m. portfolo. In accordance wth the characterstcs of the expected value functon we have J E [ NPV ] = U ( α ) NPV = E[ U ( α )] NPV P E [ NPV ] P that s determned by ths equaton s treated as net present value of a proect Accordng to the concepts of the Modern Portfolo Theory, we wll treat the standard devaton of the net present value of proect portfolo as a rsk of a proect portfolo. In order to determne σ [ NPVP ] let us denote q J a vector, whch contans values of NPV, wth =,..., L, ( J ). Covaraton coeffcents of respectve random varables are determned by the followng equatons,, J : cov( U ( α ), U ( α )) = k, m P, k, m J { U ( α ) E[ U ]}{ U k ( α In order to calculate the proect portfolo rsk let us remember that cov( U ( ω), U ρ = σ σ Rsk of a proect portfolo hence can be presented as ( ω)) m ) E[ U ]})

16 2 σ =, J NPV NPV cov( U ( α ), U ( α )) In other symbols we have 2 σ [ NPV ] = q V q p k Wth V denotng the covaraton matrx for a set of random varables U k ( α ) wth ndexes k J that corresponds to a gven proect portfolo, where σ = σ ( U ( α )). Evolvng the Markowtz approach, we wll call a proect portfolo Р* : J*={ *,., L* } effcent when we can t fnd any other portfolo Р : J={,., М } wth: one nequalty beng strct. T E[ NPV P ] E[ NPV P* ], σ [ NPV P ] σ [ NPV P* ], The defnton shows the unmprovablty of the effcent proect portfolo. Let us consder a problem of constructng a proect portfolo n a smplest case, wth NPV as a measure for proect s effectveness and when we have a probablty space of the rsky events. In a more general case, t s approprate to use the usablty functon as a tool for estmatng the proect s effectveness. The approaches and calculatons wll reman the same. g. Algorthm for buldng an effcent proect portfolo Let s assume that the organzaton has determned:. Strategc goals represented as a vector S = { S, S 2,..., S n}, where S s a strategc goal of the organzaton. 2. A budget B that organzaton s eager to nvest n proects and 3. HR an overall quantty of human resources that can work wth proects (measured n man-days ). 4. Rsks that may mpact the organzaton. Let us assume that the organzaton has determned a set P of proects that mght be nterestng for organzaton. For each proect, accordng to Table 2, we determne followng characterstcs:. Expected NPV; 2. Cumulatve rsk of a proect U (ω). 3. Strategc goals complance ndex Str ; 4. Mnmum resources requred W ; 5. Human resources requred HR ; Let s assume that all human resources are of one type. In a more general case, we should dvde the human resources n classes accordng to ther competence level.

17 6. Oblgatory feature of a proect, a vector that s comprsed of numbers and Obl = {,,,,}, where k means that proect k s oblgatory and should be ncluded n the proect portfolo no matter what, proect may be or may not be ncluded nto the proect portfolo. 7. A probablty space of rsky events of proects. The problem can be formulated as follows: to buld an effcent proect portfolo wthn the gven constrants. That means a portfolo that has a maxmum level of complance to strategc goals of organzaton and can t be mproved n terms of return and rsk. Frst of all, we have to determne a subset of feasble portfolos. As t was stated above, we assocate a proect portfolo wth a vector that contans numbers and, where coordnate wth a value of corresponds to the case when proect s ncluded nto the portfolo. When coordnate has a value of - t means that the proect s not ncluded nto the portfolo. The set of feasble proects portfolos P p s fnte and lmted by the organzaton s budget and human resources constrants. A set of feasble proect portfolos has the followng characterstcs: W B HR HR Pp P Oblgatory proects are ncluded nto portfolo by default. Thus an effcent proect portfolo s selected from a fnte set of feasble portfolos and that can be done by a fnte algorthm. We should select portfolos that have a maxmum level of complance to strategc goals of organzaton from a set P p of feasble proects. We wll denote ths set of portfolos as P s. Ths set can be characterzed wth followng equatons: W B HR HR Ps Pp Str( J J Ps ) > Str( J J Ps ) Then for each portfolo from the set P s we need to determne NPV and portfolo rsk U (ω) the way t was descrbed above. Thus we have a set of effcent proect portfolos P*={J k } that meet the followng constrants:

18 W B HR HR P Ps * NPV ( J k J k P ) NPV ( J n J * * σ ( J k J k P ) σ ( J n J n P ) n * P ) The proposed model and algorthm were approbated n several fnancal and nvestment organzatons. 3. APPROBATION OF THE MODEL FOR BUILDING AN EFFICIENT PROJECT PORTFOLIO Below s an example of the approbaton of the model. The approbaton was performed at Bank24.ru (one of Russan banks) n September 26. The set of possble proects looked the followng way: Table 3 Intal set of proects Название проекта HRd NPV Im Bp сумма Y(P) Implementaton of ISO 27:25 Informaton Securty Management System Implementaton of the leasng servce Implementaton of the factorng servce Openng of a new offce Implementaton of a corporate governance system управления 6 Implementaton of a budgetng system 6 6

19 In order to determne the cumulatve rsk of each proect we construct a probablty dstrbuton functon as t was descrbed above. followng: Then we consder the resources constrants and select the set of feasble proects: Portfolo Structure Portfolo Budget Strategc goals complance ndex {,,,,,} {,,,,,} {,,,,,} {,,,,,} {,,,,,} {,,,,,} {,,,,,} 4 35 In ths sample the Rsk Matrx of Smultaneous Realzaton of Proects looks lke the ρ =,,,2,2,,,,,, Now we calculate the characterstcs of each portfolo of the gven set: Portfolo Structure Expected Portfolo Rsk Usablty of Portfolo {,,,,,} 93,39 9,7 2 {,,,,,} 86,23 8,76 3 {,,,,,} 92,56 9,89 4 {,,,,,} 73,28 8,32 5 {,,,,,} 86,37 8,59 7 {,,,,,} 7,86 8,78 4 {,,,,,} 63,99 8,78 The set of Effcent Portfolos can be presented the followng way: ,5 9 9,5

20 Pcture 2. Effcent Proect Portfolos Thus, under the gven budget and human resources constrants we got 3 effcent proect portfolos:. - {,,,,,}; {,,,,,}; {,,,,,}. The fact that portfolo s effcent can be easly explaned. It doesn t nclude the proect, whch has a large amount of rsk wth the smallest usablty. That s why the excluson of ths proect from the portfolo make the latter effcent. Portfolo 4 s effcent snce t doesn t contan 2 proects that ncrease each other s rsks under the assumpton of smultaneous realzaton. The excluson of proect for mplementaton of the nformaton securty management system n portfolo 5 makes t effcent snce ths proect has a very large amount of rsk. 2

21 4. CONCLUSION The methods for proect portfolo management that are presented n the paper gve organzatons an opportunty to consder the rsk-return rato when constructng a proect portfolo. That s necessary n order to make decsons that lead to a sustanable compettve advantage and growth. These methods broaden the classcal model of Modern Portfolo Theory and make t easly applcable by every organzaton, not only by nvestors workng on a stock market. The mportance of strategc rsk management s recognzed worldwde. An organzaton can use all knds of approaches n order to develop and mplement ts strategy. An approach when a strategy of organzaton s supposed to be mplemented through the realzaton of a proect portfolo s ntroduced n the paper. Strategc plannng process essentally results n a set of proects or a proect portfolo. Hence ths set of proects absorbs strategc rsks of the organzaton. The use of proect portfolo management based of the methods of Modern Portfolo Theory leads to a better understandng of organzaton s strategc rsks and helps to mtgate these rsks. The methods for buldng an effcent portfolo of proects can be used as a tool for the mplementaton of a strategy wth mnmum level of rsk wth thus leadng to compettveness, dversfcaton and growth. 2

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