Presentation on theme: "MANAGEMENTUL PROIECTELOR DE INVESTITII STRAINE DIRECTE"— Presentation transcript:
1MANAGEMENTUL PROIECTELOR DE INVESTITII STRAINE DIRECTE LECT.DR. ROXANA VOICU-DOROBANTU
2STRuctura cursului PARTEA 1: MANAGEMENTUL PROIECTELOR PARTEA 2: INVESTITIILE STRAINE DIRECTE
3Detalii administrative Despre cursDetalii administrativeCursul urmăreşte:să familiarizeze studenţii cu conceptele, metodele şi tehnicile specifice managementului de proiect, cu precădere in contextul proiectelor investiţionale internaţionalesă ofere studenţilor un cadru specific de înţelegere a fenomenului investiţional global, atât din perspectiva investiţiilor străine directe, cât şi a celor de portofoliusă construiască şi să consolideze capacitatea studenţilor de analizare a tendinţelor şi caracteristicilor investiţiilor străine directe şi a celor de portofoliu din perspectiva motivaţiilor, strategiilor şi riscurilor specificeModulul de MPISD se desfăşoară pe parcursul a 7 intâlniri de curs şi 7 intâlniri de seminar pentru fiecare grupă :CURSURI:Vineri, – orele – – sala C1, C2Vineri, – orele – – sala 1201 – C3, C4Vineri, – orele – – sala 1201 – C5, C6Vineri, – orele – – sala 1201 – C7Pentru planificarea seminariilor pe grupe, urmăriţi orarul.EvaluareBibliografieEvaluarea se va face pe baza unui exercitiu de grup, ce va fi predat in ziua examenului. Exercitiul se va realiza in grupe de 5 studenti, repartizati in ordine alfabetica. Tema exercitiului: realizarea unui proiect de investitie straina directa pentru o companie prezenta pe piata din Romania pe o piata internationala. Nu exista restrictii cu privire la forma sau dimensiunea exercitiului predat in scris. Acesta va trebui sa acopere toate informatiile relevante pentru un manager pentru a putea implementa proiectul.B Lientz, K P Rea – International Project Management, Butterworth Hienemann, 2003JC Binder - Global project management : communication, ollaboration and management across borders, Gower, 2007P Dinsmore, J Cabanis-Brewin – The AMA Handbook of project management, AMACOM, 2006B Solnik – International Investments, McGraw-Hill, Reading, MA, 2004A Horobeţ – Managementul riscului în investiţiile internaţionale, Editura AllBeck, Bucureşti, 2005
5STRuctura partii 1 1.1: Definire si Istoric 1.2: Principii fundamentale1.3 Etapa 1: Pregatirea unui proiect1.4 Etapa 2: Planificarea unui proiect1.5. Etapa 3: Implementarea unui proiect1.6. Proiecte globale1.7. ?
6The “Textbook account” of project management Traditional Project Management ModelsDefining, Organizing, Planning, Executing, Monitoring, FinishingPurpose, Scope, Estimating, Budgeting, and DeadlinesWork breakdowns, Dependencies, Key ResourcesScheduling: PERT, CPM, GANNTReporting, milestones, adjustmentsThe “Textbook account” of project management
7Traditional Project Management Preparations for the projectPlanning the ProjectManaging the Project In-ProgressTraditional Project ManagementTraditional PM Focus
8Product vs project life cycles Source: Dinsmore, 2006Product vs project life cycles
9The Idea Behind Formalizing PM… Use structure to combat scale and complexityMake future possibilities more predictable, foresee problems, develop contingency plansThoroughness – Checklists as a way of being sureForces us to confront things that are hard to think about in detailHelpful even when the details don’t unfold if expectedEisenhower: “The plan is nothing. Planning is everything.”The danger…Structure can become an end in itselfObsession with documents, milestonesA distraction or, worse, a constraint on the ability to adjustThe Idea Behind Formalizing PM…
10Strategic model for managing projects Source: Dinsmore, 2006Strategic model for managing projects
11Creating a Project Organization Define who is going to do whatDefine roles and responsibilitiesIdentify people, resources; ensure their commitment to projectIdentify a project leader, specify her/his authority and responsibilitiesImportant questions:Who is the PM? What decisions are within PM’s area of authority? Is this authority sufficient to carry out the project?Who is on team? Full-time or part-time? What are their areas of expertise? Their roles?Who is the project sponsor? Is he or she at sufficiently high level in the organization to provide the project with support and a good chance of success?
12Project management process groups’ interactions Source: Dinsmore, 2006Project management process groups’ interactions
13The project management plan 1. Introduction/overview 2. Mission and objectives 3.Work scope 4. Planning basis: deliverables, requirements, constraints, approaches, assumptions, exclusions 5. Work breakdown structure 6. Organization development plan 7. Resource plan 8. Procurement and logistics plan 9. Logic and schedules 10. Cost estimates, budgets, and financial management 11. Risk analysis and contingency plan 12. Quality and productivity plan 13. Environmental, safety, and health protection plan 14. Security plan 15. Project planning, control, and administration plan 16. Documentation and configuration management plan 17. AppendixThe project management plan
14Defining the Project’s Objectives and Scope Make sure the proposed project is well understood an that all stakeholders agree on what it will accomplishClearly spell out expected outcomes, deliverables, objectivesAgree scope – what’s in, what’s not inDocument agreements formally, in writing, to surface/eliminate ambiguity in different stakeholders’ expectationsImportant questions:What is the scope?What does the project need to accomplish?By when?
15Formal Objective Statement A Formal Objective Statement Short, simple language, unambiguous Should Scope, Resources, and Schedule A Famous Example: “Put a man on the moon and return him safely to Earth by the end of the decade at a cost of $9 billion.” Scope – “Put a man on the moon and return him safely to Earth.” Schedule – “By the end of the decade.” Resources –”At a cost of $9 billion.” The advantage in keeping it short and simple: Longer statements offer greater opportunity for people to come away with different understanding of what the project will accomplish while mistakenly assuming they have reached agreementFormal Objective Statement
16What’s In/Out List for Project Work Products List 1: Things included in the work productList 2: Things excluded from the work productWork Product Example: A Business PlanIs included: 30 pages + 10 pages of appendix, spiral bound, cover sheet with 300 word executive summary, one section must be financial projections, another must spell out marketing plansNot included: Potential customer market segment analysis, a formal presentation
17Setting Up Project Norms Determine how the project will “operate” from day to daySet norms for meetings, updates, communicationEstablish “official” processes for logging, reviewing, updating progress on issuesSet norms and escalation procedures for disagreements and unresolved issuesImportant questions:What do we do when we encounter a new problem?Who do we go to for help in making decisions?How do we check progress on a known issue?
18One of the most useful concrete tools in issue tracking A running list of “open issues” Log issue, assign priority and problem solving owner, current status, resources assigned to address Revisit list on a regular basis (daily, weekly)The Humble Checklist
19It’s a good idea to have a common collection point for all project documentation Objective statements Organization definitions and roles Issue checklists Etc.Project “Workbook”
20The Work Breakdown Structure Need to identify all of the work required by a projectIdentify tasks and sub-tasksAssign “owners” for each taskEstimate how long each task will takeImportant questions:Are all tasks identified?Do all tasks have owners?How long will it take to do each task?
22Work breakdown structure, or the WBS = hierarchical structure defining tasks that can be completed independently of other tasks, facilitating resource allocation, assignment of responsibilities, and measurement and control of the project.Task 1Subtask 1.1Work package 1.1.1Work package 1.1.2Work package 1.1.3Subtask 1.2Work package 1.2.1Work package 1.2.2Task 2Subtask 2.1Work package 2.1.1Work package 2.1.2Work package 2.1.3Work package 2.1.4Subtask 2.2Work package 2.2.1Subtask 2.3Work package 2.3.1Work package 2.3.2
23tasks, sub-tasks, and work packages phases, entries, and activities. Terminology for Different Levelstasks, sub-tasks, and work packagesphases, entries, and activities.Organization by Deliverables or Phasesdeliverables or phases of the project life cycle.higher levels in the structure generally are performed by groups.lowest level in the hierarchy often comprises activities performed by individuals,a WBS that emphasizes deliverables does not necessarily specify activities.Level of Detailfacilitates resource allocation and the assignment of individual responsibilities.! Beware: micro-management OR tasks too large to manage effectively.Defining tasks so that their duration is between several days and a few months works well for most projects.
24The 100% rule states thatthe WBS includes 100% of the work defined by the project scope and captures all deliverables – internal, external, interim – in terms of the work to be completed, including project management.The rule applies at all levels within the hierarchy: the sum of the work at the “child” level must equal 100% of the work represented by the “parent” and the WBS should not include any work that falls outside the actual scope of the project, that is, it cannot include more than 100% of the work…It is important to remember that the 100% rule also applies to the activity level.The work represented by the activities in each work package must add up to 100% of the work necessary to complete the work package.The 100% rule
25You can create a WBS top down or bottom up Top down – Start with largest work groupings and break into smaller and smaller pieces Bottom up – Brainstorm specific low level tasks, group them into larger groupingsTop down vs. Bottom up
26Estimating is done in many ways… Guestimates by task owners Guestimates by a group, including experts who have “done this kind of work before” Other group consensus processes Statistical models that relate “size” of a project result to time and effort estimates Suppose a project is expected to produce a software product with about 60,000 Lines of Code (written software) According to statistical model: 60,000 Lines of Code translates into 200 person months of effort 200 person days of effort = it’ll take 20 people 10 months, or 10 people 20 months (or 2 people 100 months, etc.)
27Frederick Brooks, author of The Mythical Man Month: “Adding more people to a late project makes it later” “Adding more people to a late project helps less than you might think, and it helps less and less the more people you add ” people for 10 months is not the equivalent to 10 people for 20 months (or 2 people for 100 months) – The more people work on a project, the more overhead required to coordinate work (and the less spent on value-adding work)“Brooks’s Law”Overall ProductivityNumber of people
28Developing a Project Schedule Given what you need to do, figure out when things will happen and when you’ll be finished (when you’ll deliver project results) Identify dependencies between tasks (e.g., task A must be done before task B can be started) Use task time estimates and dependencies to create a schedule, typically by generating a “Gantt Chart” Important questions: Have all dependencies been identified? Is there a way to eliminate dependencies? Does estimated completion fit with project objectives?
30Assumptions to create the network diagram: the Critical Path Method (CPM) was developed as a network model for project management.a deterministic method that uses a fixed time estimate for each activitydoes not consider the time variations that can have a great impact on the completion time of a complex project.PERT = a network model that allows for randomness in activity completion times.Assumptions to create the network diagram:an activity is a task that must be performed and an event is a milestone marking the completion of one or more activities.before an activity can begin, all of its predecessor activities must be completed.project network models represent activities and milestones by arcs and nodes.PERT
31Steps in the PERT Planning Process Identify the specific activities and milestones.Determine the proper sequence of the activities.Construct a network diagram.Estimate the time required for each activity.Determine the critical path.Update the PERT chart as the project progresses.Steps in the PERT Planning Process
32Steps in the PERT Planning Process a. Identify Activities and MilestonesThe activities are the tasks required to complete the project.The milestones are the events marking the beginning and end of one or more activities.It is helpful to list the tasks in a table that in later steps can be expanded to include information on sequence and duration.b. Determine Activity SequenceThis step may be combined with the activity identification step since the activity sequence is evident for some tasks.Other tasks may require more analysis to determine the exact order in which they must be performed.3. Construct the Network DiagramYou may use software for this stepSteps in the PERT Planning Process
33Steps in the PERT Planning Process d. Estimate Activity TimesWeeks are a commonly used unit of time for activity completion, but any consistent unit of time can be used.For each activity, the model usually includes three time estimates:Optimistic time - generally the shortest time in which the activity can be completed. It is common practice to specify optimistic times to be three standard deviations from the mean so that there is approximately a 1% chance that the activity will be completed within the optimistic time.Most likely time - the completion time having the highest probability. Note that this time is different from the expected time.Pessimistic time - the longest time that an activity might require. Three standard deviations from the mean is commonly used for the pessimistic time.PERT assumes a beta probability distribution for the time estimates. For a beta distribution, the expected time for each activity can be approximated using the following weighted average:Expected time = ( Optimistic + 4 x Most likely + Pessimistic ) / 6This expected time may be displayed on the network diagram.To calculate the variance for each activity completion time, if three standard deviation times were selected for the optimistic and pessimistic times, then there are six standard deviations between them, so the variance is given by:[ ( Pessimistic - Optimistic ) / 6 ]2Steps in the PERT Planning Process
34Steps in the PERT Planning Process – critical path e. Determine the Critical PathThat sequence of tasks that are the “bottleneck” in the scheduleThe critical path is determined by adding the times for the activities in each sequence and determining the longest path in the project.The critical path determines the total calendar time required for the project.If activities outside the critical path speed up or slow down (within limits), the total project time does not change.The amount of time that a non-critical path activity can be delayed without delaying the project is referred to as slack time.If the critical path is not immediately obvious, it may be helpful to determine the following four quantities for each activity:ES - Earliest Start timeEF - Earliest Finish timeLS - Latest Start timeLF - Latest Finish timeSteps in the PERT Planning Process – critical path
35Steps in the PERT Planning Process – critical path ES - Earliest Start time EF - Earliest Finish timeLS - Latest Start time LF - Latest Finish timeThese times are calculated using the expected time for the relevant activities.The earliest start and finish times of each activity are determined by working forward through the network and determining the earliest time at which an activity can start and finish considering its predecessor activities.The latest start and finish times are the latest times that an activity can start and finish without delaying the project. LS and LF are found by working backward through the network.The difference in the latest and earliest finish of each activity is that activity's slack.The critical path then is the path through the network in which none of the activities have slack.Steps in the PERT Planning Process – critical path
36Critical path - example Source: syque.comCritical path - example
37Steps in the PERT Planning Process – critical path The variance in the project completion time can be calculated by summing the variances in the completion times of the activities in the critical path. Given this variance, one can calculate the probability that the project will be completed by a certain date assuming a normal probability distribution for the critical path. The normal distribution assumption holds if the number of activities in the path is large enough for the central limit theorem to be applied. Since the critical path determines the completion date of the project, the project can be accelerated by adding the resources required to decrease the time for the activities in the critical path. Such a shortening of the project sometimes is referred to as project crashing.Steps in the PERT Planning Process – critical path
38Critical Path “What If” Analysis The results of this process for arriving at a project schedule are often poorly received “That project completion date is far too late!” So a “what if” exercise ensues To shorten a project’s overall duration (to be “done” sooner), you must shorten tasks on the Critical Path (“What if task 14, on the critical path, could be done faster?) or remove tasks from the Critical Path (Does task 15 really depend on task 14 – what if we can, by making an adjustment, do them at the same time, thus moving task 14 off the Critical Path?) Sometimes when you shorten the Critical Path, a different sequence of tasks becomes the new Critical Path (the “bottleneck” shifts) – that is, you’ve shortened the old Critical Path to a point where it is no longer the Critical Path
39Steps in the PERT Planning Process f. Update as Project Progresses Make adjustments in the PERT chart as the project progresses. As the project unfolds, the estimated times can be replaced with actual times. In cases where there are delays, additional resources may be needed to stay on schedule and the PERT chart may be modified to reflect the new situation.Steps in the PERT Planning Process
40Benefits Limitations pert Expected project completion time. Probability of completion before a specified date.The critical path activities that directly impact the completion time.The activities that have slack time and that can lend resources to critical path activities.Activity start and end dates.LimitationsThe activity time estimates - subjective and depend on judgement.Even if the activity times are well-estimated, PERT assumes a beta distribution for these time estimates, but the actual distribution may be different.Even if the beta distribution assumption holds, PERT assumes that the probability distribution of the project completion time is the same as the that of the critical path.Because other paths can become the critical path if their associated activities are delayed, PERT consistently underestimates the expected project completion time.The underestimation of the project completion time due to alternate paths becoming critical is perhaps the most serious of these issues. To overcome this limitation, Monte Carlo simulations can be performed on the network to eliminate this optimistic bias in the expected project completion time.pert
41Critical Path “What If” Analysis The results of this process for arriving at a project schedule are often poorly received “That project completion date is far too late!” So a “what if” exercise ensues To shorten a project’s overall duration (to be “done” sooner), you must shorten tasks on the Critical Path (“What if task 14, on the critical path, could be done faster?) or remove tasks from the Critical Path (Does task 15 really depend on task 14 – what if we can, by making an adjustment, do them at the same time, thus moving task 14 off the Critical Path?) Sometimes when you shorten the Critical Path, a different sequence of tasks becomes the new Critical Path (the “bottleneck” shifts) – that is, you’ve shortened the old Critical Path to a point where it is no longer the Critical Path
42Assigning Project Resources With tasks and dependencies identified, assign people and other resources (e.g., specialized equipment) to tasks Allocate resources across tasks Make sure resources available can cover tasks Make sure key there are no conflicts for key resources (same person/machine can’t be two places at once) Analyze the merits of different ways of allocating resources Re-examine scope, schedule, and resources Important questions: Is work spread reasonably across resources? Should we make adjustments to scope, schedule, or resources? Assigning more resources is one way of shortening the CP
43Key Resources and “Switching Cost” A very common problem: Some resources are over allocated while others are under allocated People working on some tasks are overwhelmed, while others have nothing much to do Over allocated resources end up being asked to multi-task Literally, work on more than one task at once Their time gets allocated in smaller and smaller fractions But there are big hidden costs in doing this… Switching tasks takes effort People work more hours Burn out, morale issues Worst for key peopleValue-add time on each taskNumber of tasks
46Gantt charts illustrate the start and finish dates of the terminal elements and summary elements of a project.Terminal elements and summary elements comprise the work breakdown structure of the project.Some Gantt charts also show the dependency (i.e., precedence network) relationships between activities.Pitfalls:Attempt to define the project work breakdown structure at the same time that they define schedule activities. the WBS should be fully defined to follow the 100% Rule, then the project schedule can be designed.Difficult to grasp for projects with more than 30 activitiesThe equal horizontal dimension of activities does not take into consideration the resource load of each activityGantt charts
47Making TradeoffsWith tasks, CP, and resource assignments in place, you are positioned to make explicit planning tradeoffs Analyze the impact on schedule of different feasible allocations of available resources Determine impact on schedule and budget of adding (or subtracting) resources (keeping in mind Brooks’s Law) Determine impact of changes in project scope Important questions: Are schedule and cost consistent with project objectives? If not, can they be made so? Or should we change the objectives? Can we eliminate dependencies by working differently? Are we trying to do too much?
48Project Structure – How many “chunks”? In considering scope, resource, and schedule tradeoffs, often the question of how you might “chunk” the project comes up What if we divide this project into smaller projects? Can improve fit between available resources and schedule Might allow most important objectives to be delivered sooner, at the cost of waiting longer for some other, less important objectives Can also have budget implications – dividing a project up into smaller chunks can make it appear more expensive Although it can sometimes turn out cheaper if the smaller projects have fewer problems than the big one would have… Which brings us to…
49Planning for Project Risks As much as possible during planning stages of a project, draw attention to project risks and develop plans for managing them. Identify likely sources of risk Develop plans to minimize probability that risks will materialize Might result in new tasks being identified, which will require additional resource requirements Develop plans to minimize impact if risks do materialize Important questions: Have we identified all the risks we can? Which should we worry most about (because the are very likely or would have major impact on the project)?
50Project Structure and Risk There is a relationship between project structure and risk In general, larger projects involve greater risks than a series of smaller projects (smaller projects are less complex and constitute smaller failures even when something does go wrong) An additional advantage in “chunking” large projects into smaller ones: Can incorporate learning from early chunks into later chunks Certain projects have characteristics that make risks harder to anticipate and make smaller chunks more beneficial If project outcomes are difficult to specify in advance (thus must be “discovered” to some extent – such as when a user of a product can’t easily envision how the product might be used differently) If project objectives are likely to change while the project is in progress (due to changes in technology, moves by competitors, etc.)
51Ongoing Status Monitoring The challenge is to keep the PM and project team focused on areas that that provide the best indication of project progress, and to maintain access to high quality information about the progress of the project Important questions: How will status information be collected? What will be monitored and how? Are we confident in the quality of our measures of progress? The problem of open communication: To do a good job of collecting status information, you need people to be really honest about the problems they’re seeing
52“Sweeping things under the rug” A very common sequence of events on projects Team members discover problems that will delay the project PM and senior managers are not happy to hear of the delay and convey the impression (often unintentionally) that the person who discovered the problem is to blame for the delay People get the idea that identification of problems that result in project delays is an unwelcome behavior They start “sweeping things under the rug” After problems accumulate “under the rug” for some time, something goes terribly wrong with the project…
53In progress adjustment and adaptation… In every project, there is a need to adjust the ongoing project to account for unexpected problems (or opportunities) A continuation of the process of making planning tradeoffs Change scope, eliminate dependencies, or add resources Manage “scope creep” – enlarging project scope without really acknowledging that the project is getting bigger, and without assigning additional resources or otherwise accounting (e.g., moving deadline later) for the increase in work that accompanies the increase in scope Important questions: Are mid-project requests for changes really requirements? Should we make adjustments to the project? Should we abandon the project?
54Learning from project experiences Important to capture key learning that will improve management of future projects Conduct “post mortem” meetings Capture and communicate key learning to others on other projects and throughout the larger organization Often not done, so learning gets lost Important questions: What worked? What didn’t? Have we acknowledged and celebrated good work?Learning from project experiences
55The Three Main “Levers” of PM Scope Adjust what is included in the project e.g., decide to drop Feature 7 from the project objectives Resources Adjust who is working on a project e.g., decide to add two more people to the project Schedule Adjust target completion target e.g., decide to allow another month to complete the projectYou decide these things as you plan, then revisit during a project
57Project control cost baseline Develop a project cost control baseline from the project cost estimateImportant questions:How many individual cost elements can the office and field personnel be expected to break actual project costs into for reporting purposes?How many individual cost elements can the project management team effectively review and monitor?How can Pareto’s Law be taken into account—that 80 percent of the total project cost is probably represented by only 20 percent of the cost items?! Beware: cost escalation
58Estimating costs – basic techniques Expert opinionAnalogy - reviewing costs from previous, similar projectsCostproposed = Costanalogy x (Capacityproposed/Capacityanalogy)3. Parametric - an estimate derived from an empirical or mathematical relationship – eg. a multiple regression model that links material cost to floor space to receiving docks in a building4. Cost engineering - detailed cost analysis of individual cost categories at the work package or task level. It is a bottom-up approach.Estimating costs – basic techniques
59Determining earned value Units completed:For example, suppose that 420 linear feet (LF) of 4-inch diameter steel pipe has been installed. If the total project requires 2,100 LF of pipe, then the percentage of completion is 20 percent (420 LF divided by 2,100 LF).2. Incremental milestones:When the work task involves a sequential series of subtasks, the percentage of completion may be estimated by assigning a proportionate percentage to each of the subtasks.For example, the installation of a major item of equipment might be broken down as follows:Construct foundation pad 10 percentSet equipment on foundation 60 percentConnect mechanical piping 75 percentConnect electrical 90 percentPerformance testing and start up 100 percent.Percentage of completion is estimated by determining which of the milestones have been reached.The accuracy of this procedure depends upon a fair allocation of percentage to the subtask in relation to costs.Determining earned value
60Determining earned value 3. Cost to complete:The cost to complete the remaining work for a given task is first estimated. This detailed cost estimate utilizes both the original cost estimate and any historical cost data acquired so far on the project.For example, if the actual cost to date for structural steel erection is $18,500 and theestimated cost to complete the task is $6,500, the percentage of completion is 74%.Earned value = % of completion x original budget cost for the taskDetermining earned value
64Cost control systems Based on the variance in costs, caused by Resources used to accomplish the work have been paid more than was planned.Resources used to accomplish the work have been consumed in more quantity than was planned.Cost control systems
65Cost management indices Implementing changeCost management indices
68Evaluarea opţiunilor Factorii care influenţează preţul unei opţiuni Preţul actual al activului de bazăTimpul până la expirarea opţiuniiPreţul de exercitareRata de dobândă pe piaţăCaracteristicile aleatoare ale preţului activului de bazăPreţurile opţiunilor sunt determinate prin intermediul unor modele care au la bază ipoteze cu privire la procesul stocastic care guvernează preţul activului de bază la expirarea opţiunii.
69Abordările dominantePreţul are o distribuţie lognormală: logaritmul preţului viitor urmează distribuţia normală. Aceasta este ipoteza de bază a modelului de evaluarea a opţiunilor Black-Scholes-Merton.Preţul activului are a distribuţie în timp discret, conform căreia în fiecare perioadă randamentele activului de bază pot lua numai două valori posibile. Aceasta este esenţa modelului binomial.
70Abordările dominante (cont.) Ambele modele (Black-Scholes-Merton şi modelul binomial) pot determina un preţ al opţiunii numaipe baza argumentului lipsei de arbitraj (free lunch).Alte distribuţii necesită ipoteze adiţionale.
71Un model binomial simplu Preţul acţiunii este în prezent $20În trei luni va ajunge fie la $22 fie la $18Preţul acţiunii = $22Preţul acţiunii = $20Preţul acţiunii = $18
72O opţiune CallO opţiune call la 3 luni pe această acţiune are un preţ de exercitare de 21.Preţul acţiunii = $22Preţul opţiunii = $1Preţul acţiunii = $20Preţul opţiunii=?Preţul acţiunii = $18Preţul opţiunii = $0
73Construcţia unui portofoliu fără risc Fie portofoliul: long D acţiuni short o opţiune callPortofoliul este fără risc dacă 22D – 1 = 18D adică D = 0.2522D – 118D
74Evaluarea portofoliului (Rata fără risc este 12%) Portofoliul fără risc este:long 0.25 acţiuni short 1 opţiune callValoarea portofoliului în 3 luni este 22´0.25 – 1 = 4.50Valoarea portofoliului astăzi este 4.5e – 0.12´0.25 =
75Evaluarea unei opţiuni Portofoliullong acţiuni short 1 opţiunevalorează 4.367Valoarea acţiunilor din portofoliu este (= 0.25´20 )Astfel, valoarea opţiunii va fi0.633 (= – )
76Generalizare S0u ƒu S0 ƒ S0d ƒd Un produs derivat durează până la momentul T şi este dependent de preţul acţiuniiS0uƒuS0dƒdS0ƒ
77Portofoliul este fără risc dacă S0uD – ƒu = S0d D – ƒd, adică Generalizare (cont)Să considerăm portofoliul care constă din o poziţie long pe D acţiuni şi o poziţie short pe 1 derivatPortofoliul este fără risc dacă S0uD – ƒu = S0d D – ƒd, adicăS0 uD – ƒuΔS0– fS0dD – ƒd
78Generalizare (cont) Valoarea portofoliului la momentul T este S0u D – ƒuValoarea portofoliului astăzi este(S0u D – ƒu )e–rTO altă expresie pentru valoarea portofoliului de astăzi este S0D – fPrin urmareƒ = S0D – (S0u D – ƒu )e–rT
79ƒ = [ p ƒu + (1 – p )ƒd ]e–rT Înlocuind pe D vom obţine unde Generalizare (cont)Înlocuind pe D vom obţineƒ = [ p ƒu + (1 – p )ƒd ]e–rTunde
80Media randamentelor activului de bază este nerelevantă Când evaluăm o opţiune în funcţie de activul de bază, randamentul aşteptat (mediu) nu apare în analiză
81Evaluarea neutră la risc ƒ = [ p ƒu + (1 – p )ƒd ]e-rTVariabilele p şi (1 – p ) pot fi interpretate ca fiind probabilităţi neutre la risc pentru mişcările în sus şi în josValoarea unui produs financiar derivat este dată de câştigul aşteptat într-o lume neutră la risc, actualizat cu rata fără riscS0uƒuS0dƒdS0ƒp(1 – p )
82Revizitarea exemplului iniţial S0u = 22ƒu = 1pCum p este o probabilitate neutră la risc20e0.12 ´0.25 = 22p + 18(1 – p );p =Sau, putem folosi formulaS0ƒ(1 – p )S0d = 18ƒd = 0