Risk management


Published on

Risk research for defense procurement

Published in: Economy & Finance, Business
1 Like
  • Be the first to comment

No Downloads
Total Views
On Slideshare
From Embeds
Number of Embeds
Embeds 0
No embeds

No notes for slide

Risk management

  1. 1. Risk Management for Forest Rangers 1
  2. 2. Jeran Binning 2
  3. 3. ‗‗AGAINST THE GODS• The story that I have to tell is marked all the way through by a persistent tension between those who assert that the best decisions are based on quantification and numbers, determined by the patterns of the past, and those who base their decisions on more subjective degrees of belief about the uncertain future. This is a controversy that has never been resolved.‘• — FROM THE INTRODUCTION TO ‗‗AGAINST THE GODS: THE REMARKABLE STORY OF RISK,‘‘ BY PETER L. BERNSTEIN 3
  4. 4. Risk Wei-jan: Chinese for opportunity through danger As long as we wish for safety, we will have difficulty pursuing what matters. - Peter Block Risk has a double-edged nature. Risk can cut, risk can heal. - James Neill 4
  5. 5. Risk 5
  6. 6. John Maynard Keynes• ―There is no harm in being sometimes wrong- especially if one is promptly found out. 6
  7. 7. Predictive ToolsValue at Risk VaR 7
  8. 8. Value at Risk VaR• David Einhorn, who founded Greenlight Capital, a prominent hedge fund, wrote not long ago that VaR was ―relatively useless as a risk-management tool and potentially catastrophic when its use creates a false sense of security among senior managers and watchdogs. This is like an air bag that works all the time, except when you have a car accident.”• NY Times Magazine pp27 4 January 2009 8
  9. 9. Wall Street Journal• Any of these metrics that work in a typical oscillating market…are not working right now,‖ Mr. Rueckert said.• Among the other indicators that aren‘t working: 10-day, 50-day and 200-day moving averages, the put/call ratio and the idea of ―capitulation.‖• ―Capitulation‖ is the concept that stocks require a purgative, high-volume plunge to mark the bottom of the bear market. Guess what: the stock market has seen little other than purgative and high-volume plunges since the failure of Lehman Brothers hit the tape on Sept. 15, and there‘s no sign of a bottom yet.• In an attempt to debunk ―the capitulation myth,‖ Mr. Rueckert, of Birinyi Associates, found an item in the New York Times three days after the bottom of the 1982 bear market that promised the end would take the form of a ―crushing…swift plunge.‖ According to his analysis, bear markets usually end with a whimper rather than a bang.• To date, the best predictor of a market turn was probably an email that circulated among Wall Street traders on Oct. 27, the day of the interim bottom. The analysis was based on lunar cycles, a cornerstone of astrology.• Wall Street Journal January 5th 2009 9
  10. 10. Probabilities/Risk • The major mistake that people make is that they are not very good at dealing with a lot of uncertainty. • So, rather than a rational assessment of data and probabilities, they like stories and they make decisions based more on mental images rather than a sober assessment of their portfolio and how a particular stock fits into it."•James Scott is Managing Director of Global Public Markets for General Motors Asset Management and a member of its Management and Investment Committees. Before joining GMAM, he was President of QuantitativeManagements Associates, a subsidiary of Prudential Financial. Prior to that, Mr. Scott was a Professor at Columbia Business School.Mr. Scott holds a B.A. from Rice University and a Ph.D. in economics from Carnegie Mellon University. He serves as an Associate Editor of the Financial Analysts Journal and the Journal of Investment Management, as aDirector of the Institute for Quantitative Research in Finance and as Research Director of the Heilbrunn Center for Graham and Dodd Investing at Columbia Business School. 10
  11. 11. John Maynard Keynes Risk vs Uncertainty• By ―uncertain‖ knowledge … I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty…. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention…. About these matters, there is no scientific basis on which to form any calculable probability whatever. We simply do not know! 11
  12. 12. Predictive Tools • Prediction is very difficult, especially about the future.’ - Niels Bohr – Physicist (1885-1962) “It is tough to make predictions, especially about the future.” - Yogi Berra, Baseball Savant 12
  13. 13. Predicting the Future• Presidents, Movies, and Influenza• Such markets have been created to predict the next president, Hollywood blockbusters, and flu outbreaks. The newest prediction market, launched in February 2008, focuses on predicting future events in the tech industry, such as whether Yahoo! will accept Microsofts acquisition. But Ho and his co-author, Kay-Yut Chen, a principal scientist at Hewlett-Packard Laboratories, believe that prediction markets also are well-suited to forecasting demand for new product innovations, particularly in the high-tech arena. H-P tested prediction markets to forecast sales of several existing and new products and found that six of eight prediction markets were more accurate than official forecasts. "Prediction markets work because you get a lot of people and ask them to put their money where their mouth is," Chen says. Based on their analysis of several existing prediction markets, Ho and Chen provide a step- by-step guide for firms on how to create a prediction market. They suggest recruiting at least 50 participants and providing a strong monetary incentive to promote active trading. Ho and Chen recommend average compensation of at least $500 for each participant. The firm then creates ten different forecasts – either according to sales or units sold – and gives each participant a set number of shares and cash to trade, buy, and sell, according to their beliefs about which forecast is most accurate. After a product is launched and sales are observed, participants who own shares in the prediction that matches actual sales receive $1 a share. 13
  14. 14. Cal Berkeley • One novel way to improve such forecasts is a prediction market, says Teck-Hua Ho, the Haas Schools William Halford Jr. Family Professor of Marketing. Ho recently coauthored an article titled "New Product Blockbusters: The Magic and Science of Prediction Markets" in the 50th anniversary issue of the Haas School business journal, California Management Review. A prediction market is an exchange in which participants vote on a possible outcome by buying and selling shares that correspond to a particular forecast, similar to trading in the stock market. Shares in a forecast that participants believe is most likely trade for a higher price than shares in a less likely scenario. "The key idea behind a prediction market is pooling the knowledge of many people within a company," Ho says. "Its a very powerful tool for firms with many different pockets of expertise or a widely dispersed or isolated workforce." 14
  15. 15. Phony Forecasting (or Nerds and Herds)• Extremistan might not be so bad if you could predict when outliers would occur and what their magnitude might be. But no one can do this precisely.• Consider hit movies: Screenwriter William Goldman is famous for describing the ―secret‖ of Hollywood hits: “Nobody can predict one”.• Similarly, no one knew whether a book by a mother on welfare about a boy magician with an odd birthmark would flop or make the author a billionaire.• Stock prices are the same way. Anyone who claims to be able to predict the price of a stock or commodity years in the future is a charlatan.• Yet the magazines are filled with the latest ―insider‖ advice about what the market will do. Ditto for technology.• Do you know what the ―next big thing‖ will be? No. No one does. Prognosticators generally miss the big important events – the black swans that impel history. 15
  16. 16. Astrology Astrology AstrologyUniversum - C. Flammarion, Holzschnitt, Paris 1888, Kolorit : Heikenwaelder Hugo, Wien 1998 16
  17. 17. Physics Envy• Social scientists have suffered from physics envy, since physics has been very successful at creating mathematical models with huge predictive value.• In financial economics, particularly in a field called risk management, the predictive value of the models is no different from astrology. Indeed it resembles astrology (without the elegance).• They give you an ex-post ad-hoc explanation.• Nassim Taleb 17
  18. 18. POLICY ANALYSIS MARKET Strategic Insight DARPAs Policy Analysis Market for Intelligence: Outside the Box or Off the Wall? by Robert Looney Sept 2003• Although the Policy Analysis Market appears to be a dead issue, it did break new ground in the countrys search for better intelligence. The PAM idea embodied a solid body of theory and proven empirical capability. While one can quibble about how closely PAM markets would approximate the efficient market hypothesis, there is no doubt trading on many future events would come close enough to provide valuable intelligence. Thus, while it was a public relations disaster, some version of the program will likely be introduced on a restricted basis, perhaps along the lines suggested above, in an attempt to better tap the countrys disperse knowledge base, human insight, and analytical expertise. This solution is far from perfect, not allowing realization of the full potential of the program.• Lou Dobbs (2003), has perhaps best summed up this unfortunate episode: ―We will never know if the Policy Analysis Market would have been successful. But if there were even a small chance that it could have been a useful tool, there should be, at a minimum, further discussion of the idea. This is, after all, not a matter of just partisan politics but one of national security. And forcing the resignations of those involved with the planning is a strong deterrent to progressive thinking, of which we have no surplus.‖ 18
  19. 19. POLICY ANALYSIS MARKET• Poindexter also faced immense criticism from the media and politicians about the Policy Analysis Market project, a prediction market that would have rewarded participants for accurately predicting geopolitical trends in the Middle East. This was portrayed in the media as profiting from the assassination of heads of state and acts of terrorism due to such events being mentioned on illustrative sample screens showing the interface.• The controversy over the futures market led to a Congressional audit of the Information Awareness Office in general, which revealed a fundamental lack of privacy protection for American citizens.• Funding for the IAO was subsequently cut and Poindexter retired from DARPA on August 12, 2003. Wikipedia 19
  20. 20. Analysis of DOD Major Defense Acquisition Program Portfolios ( FY 2008 dollars) •Source: GAO analysis of DOD data. FY 2000 FY 2005 FY 2007Number of Programs 95 75 91•Total planned commitments $790 B $1.5 T $1.6 T•Commitments outstanding $380 B $887 B $858 B•Portfolio performance•Change RDT&E costs from first estimate 27% 33% 40%•Change acquisition cost from first estimate 6% 18% 26%•Estimated total acquisition cost growth $42 B $202 B $295 BPrograms with = >25% increase in Program Acquisition Unit Cost 37% 44% 44%•Ave schedule delay delivering initial capability 16 mos 17 mos 21 mos 20
  21. 21. DoD Risk Definition “A measure of future uncertainties in achieving program goals and objectives within defined cost, schedule and performance constraints.”Each risk event has three components:− A future root cause;− The probability of the future root cause occurring; and− The consequence / impact if the root cause occurs.
  22. 22. Risk Identification• After establishing the context, the next step in the process of managing risk is to identify potential risks. Risks are about events that, when triggered, cause problems. Hence, risk identification can start with the source of problems, or with the problem itself.• Source analysis Risk sources may be internal or external to the system that is the target of risk management. Examples of risk sources are: stakeholders of a project, employees of a company or the weather over an airport.• Problem analysis Risks are related to identified threats. For example: the threat of losing money, the threat of abuse of privacy information or the threat of accidents and casualties. The threats may exist with various entities, most important with shareholders, customers and legislative bodies such as the government.• When either source or problem is known, the events that a source may trigger or the events that can lead to a problem can be investigated. For example: stakeholders withdrawing during a project may endanger funding of the project; privacy information may be stolen by employees even within a closed network; lightning striking a Boeing 747 during takeoff may make all people onboard immediate casualties.• The chosen method of identifying risks may depend on culture, industry practice and compliance. The identification methods are formed by templates or the development of templates for identifying source, problem or event. Common risk identification methods are: – Objectives-based risk identification Organizations and project teams have objectives. Any event that may endanger achieving an objective partly or completely is identified as risk. – Scenario-based risk identification In scenario analysis different scenarios are created. The scenarios may be the alternative ways to achieve an objective, or an analysis of the interaction of forces in, for example, a market or battle. Any event that triggers an undesired scenario alternative is identified as risk - see Futures Studies for methodology used by Futurists. – Taxonomy-based risk identification The taxonomy in taxonomy-based risk identification is a breakdown of possible risk sources. Based on the taxonomy and knowledge of best practices, a questionnaire is compiled. The answers to the questions reveal risks. Taxonomy-based risk identification in software industry can be found in CMU/SEI-93-TR-6.• Common-risk Checking In several industries lists with known risks are available. Each risk in the list can be checked for application to a particular situation. An example of known risks in the software industry is the Common Vulnerability and Exposures list found at http://cve.mitre.org• Risk Charting This method combines the above approaches by listing Resources at risk, Threats to those resources Modifying Factors which may increase or reduce the risk and Consequences it is wished to avoid. Creating a matrix under these headings enables a variety of approaches. One can begin with resources and consider the threats they are exposed to and the consequences of each. Alternatively one can start with the threats and examine which resources they would affect, or one can begin with the consequences and determine which combination of threats and resources would be involved to bring them about. 22
  23. 23. Assessment• Once risks have been identified, they must then be assessed as to their potential severity of loss and to the probability of occurrence. These quantities can be either simple to measure, in the case of the value of a lost building, or impossible to know for sure in the case of the probability of an unlikely event occurring. Therefore, in the assessment process it is critical to make the best educated guesses possible in order to properly prioritize the implementation of the risk management plan.• The fundamental difficulty in risk assessment is determining the rate of occurrence since statistical information is not available on all kinds of past incidents.• Furthermore, evaluating the severity of the consequences (impact) is often quite difficult for immaterial assets. Asset valuation is another question that needs to be addressed. Thus, best educated opinions and available statistics are the primary sources of information.• Nevertheless, risk assessment should produce such information for the management of the organization that the primary risks are easy to understand and that the risk management decisions may be prioritized. Thus, there have been several theories and attempts to quantify risks. Numerous different risk formulae exist, but perhaps the most widely accepted formula for risk quantification is:• Rate of occurrence multiplied by the impact of the event equals risk frequency x impact = risk 23
  24. 24. Assessment• Later research has shown that the financial benefits of risk management are less dependent on the formula used but are more dependent on the frequency and how risk assessment is performed.• In business it is imperative to be able to present the findings of risk assessments in financial terms. Robert Courtney Jr. (IBM, 1970) proposed a formula for presenting risks in financial terms.• The Courtney formula was accepted as the official risk analysis method for the US governmental agencies. The formula proposes calculation of ALE (annualized loss expectancy) and compares the expected loss value to the security control implementation costs (cost-benefit analysis). 24
  25. 25. Potential risk treatments• Once risks have been identified and assessed, all techniques to manage the risk fall into one or more of these four major categories:• Avoidance (elimination) AVOID• Reduction (mitigation) / CONTROL• Retention (acceptance and budgeting) / ACCEPTANCE• Transfer (insurance or hedging) / TRANSFER• Ideal use of these strategies may not be possible. Some of them may involve trade-offs that are not acceptable to the organization or person making the risk management decisions. 25
  26. 26. Risk avoidance• Includes not performing an activity that could carry risk. – Examples:• not buying a property or business in order to not take on the liability that comes with it.• not flying in order to not take the risk that the airplane were to be hijacked. 26
  27. 27. Risk reduction– Involves methods that reduce the severity of the loss or the likelihood of the loss from occurring. Examples include sprinklers designed to put out a fire to reduce the risk of loss by fire. This method may cause a greater loss by water damage and therefore may not be suitable. Halon fire suppression systems may mitigate that risk but the cost may be prohibitive as a strategy.– Modern software development methodologies reduce risk by developing and delivering software incrementaly. Early methodologies suffered from the fact that they only delivered software in the final phase of development; any problems encountered in earlier phases meant costly rework and often jeopardized the whole project. By developing in iterations, software projects can limit effort wasted to a single iteration.– Outsourcing could be an example of risk reduction if the outsourcer can demonstrate higher capability at managing or reducing risks. In this case companies outsource only some of their departmental needs. For example, a company may outsource only its software development, the manufacturing of hard goods, or customer support needs to another company, while handling the business management itself. This way, the company can concentrate more on business development without having to worry as much about the manufacturing process, managing the development team, or finding a physical location for a call center. 27
  28. 28. • Involves accepting the loss when it occurs. True self insurance falls in this category.Risk retention Risk retention is a viable strategy for small risks where the cost of insuring against the risk would be greater over time than the total losses sustained. – All risks that are not avoided or transferred are retained by default. – This includes risks that are so large or catastrophic that they either cannot be insured against or the premiums would be infeasible • War is an example since most property and risks are not insured against war, so the loss attributed by war is retained by the insured. • Also any amounts of potential loss (risk) over the amount insured is retained risk. This may also be acceptable if the chance of a very large loss is small or if the cost to insure for greater coverage amounts is so great it would hinder the goals of the organization too much. 28
  29. 29. Risk transfer• Means causing another party to accept the risk, typically by contract or by hedging.• Insurance is one type of risk transfer that uses contracts.• Other times it may involve contract language that transfers a risk to another party without the payment of an insurance premium. – Liability among construction or other contractors is very often transferred this way.• On the other hand, taking offsetting positions in derivatives is typically how firms use hedging to financially manage risk. 29
  30. 30. Sunk Cost• In economics and business decision-making, sunk costs are costs that cannot be recovered once they have been incurred. Sunk costs are sometimes contrasted with variable costs, which are the costs that will change due to the proposed course of action, and which are costs that will be incurred if an action is taken. In microeconomic theory, only variable costs are relevant to a decision. Economics proposes that a rational actor does not let sunk costs influence ones decisions, because doing so would not be assessing a decision exclusively on its own merits. The decision-maker may make rational decisions according to their own incentives; these incentives may dictate different decisions than would be dictated by efficiency or profitability, and this is considered an and distinct from a sunk cost problem.• For example, when one pre-orders a non-refundable and non-transferable movie ticket, the price of the ticket becomes a sunk cost. Even if the ticket-buyer decides that he would rather not go to the movie, there is no way to get back the money he originally paid. Therefore, the sunk cost of the ticket should have no bearing on the decision of whether or not to actually go to the movie. In other words, it is a fallacy to conclude that he should go to the movie so as to avoid "wasting" the cost of the ticket.• While sunk costs should not affect the rational decision makers best choice, the sinking of a cost can. Until you commit your resources, the sunk cost becomes known as an avoidable fixed cost, and should be included in any decision making processes. If the cost is large enough, it could potentially alter your next best choice, or opportunity cost. For example, if you are considering pre-ordering movie tickets, but havent actually purchased them yet, the cost to you remains avoidable. If the price of the tickets rises to an amount that requires you to pay more than the value you place on them, the cost should be figured into your decision- making, and you should reallocate your resources to your next best choice. 30
  31. 31. Opportunity Lost?– Avoidance may seem the answer to all risks, but avoiding risks also means losing out on the potential gain that accepting (retaining) the risk may have allowed.– Not entering a business to avoid the risk of loss also avoids the possibility of earning profits. 31
  32. 32. Portfolio Investment Management• Large-scale Defense infrastructure modernization programs such as Global Combat Support have complex inter-dependencies and long-time horizons that render fully informed investment decisions difficult to achieve before substantial, and unrecoverable, resources are committed. (sunk cost) – However complex these decisions, they, nonetheless, can be decomposed along three basic dimensions: – Uncertainty – Timing – Irreversibility• These primary parameters define the value of investment options available to a firm,regardless of whether it is in the public or private sector.R Suter Managing Uncertainty and Risk in Public Sector Investments, Richard Suter, Information Technology Systems, Inc., R Consulting A paperpresented at the 4th Annual Acquisition Research Symposium, Graduate School of Business & Public Policy, Naval Postgraduate School 32
  33. 33. Level of Activity over Life Cycle Monitoring and ControlLevel of Activity Execute Plan Close Initiate Start Finish Time Average Duty Cycle for DOD systems is ten years 33
  34. 34. System of Systems Engineering SoSe• System of Systems Engineering (SoSE) methodology is heavily used in Department of Defense applications, but is increasingly being applied to non-defense related problems such as architectural design of problems in air and auto transportation, healthcare, global communication networks, search and rescue, space exploration and many other System of Systems application domains. SoSE is more than systems engineering of monolithic, complex systems because design for System-of-Systems problems is performed under some level of uncertainty in the requirements and the constituent systems, and it involves considerations in multiple levels and domains (as per [1]and [2]). Whereas systems engineering focuses on building the system right, SoSE focuses on choosing the right system(s) and their interactions to satisfy the requirements.• System-of-Systems Engineering and Systems Engineering are related but different fields of study. Whereas systems engineering addresses the development and operations of monolithic products, SoSE addresses the development and operations of evolving programs. In other words, traditional systems engineering seeks to optimize an individual system (i.e., the product), while SoSE seeks to optimize network of various interacting legacy and new systems brought together to satisfy multiple objectives of the program. SoSE should enable the decision-makers to understand the implications of various choices on technical performance, costs, extensibility and flexibility over time; thus, effective SoSE methodology should prepare the decision-makers for informed architecting of System-of- Systems problems.• Due to varied methodology and domains of applications in existing literature, there does not exist a single unified consensus for processes involved in System-of-Systems Engineering. One of the proposed SoSE frameworks, by Dr. Daniel A. DeLaurentis, recommends a three- phase method where a SoS problem is defined (understood), abstracted, modeled and analyzed for behavioral patterns. 34
  35. 35. 35
  36. 36. • Complex System of systems 36
  37. 37. 37
  38. 38. Complex system of systems• Difficulty with System of systems? The technical complexity The programmatic complexity of integrating software intensive systems The absence of accurate cost information at the onset of major systems/ software Programs 38
  39. 39. Portfolio Investment Management-UncertaintyUnfortunately, algorithms capable of modeling the effects of these variables are relatively few, especially for the uncertainty and irreversibility of investment decisions (Dixit & Pyndik, 1994, p. 211).For large-scale information Technology (IT) modernization programs, there are at least three sources of uncertainty— and, thus, risk The technical complexity The programmatic complexity of integrating software intensive systems The absence of accurate cost information at the onset of major systems/ software Programs• Software-intensive systems are particularly sensitive to the systematic underestimation of risk, primarily because the level of complexity is hard to manage, let alone comprehend. Investment in software-intensive systems tends to be irreversible because it is spent on the labor required to develop the intellectual capital embedded in software. The outcome of software development is almost invariably unique, a one-of-kind artifact—despite the numerous efforts to develop reusable software. Unlike physical assets ,the salvage value of software is zero because no benefit is realized until the system is deployed; and that labor, once invested, is unrecoverable. One result is an (implicit) incentive to continue projects that have little chance of success, despite significant cost overruns, schedule delays. R Suter Managing Uncertainty and Risk in Public Sector Investments, Richard Suter, Information Technology Systems, Inc., R Consulting A paper presented at the 4th Annual Acquisition Research Symposium, Graduate School of Business & Public Policy, Naval Postgraduate School 39
  40. 40. 40
  41. 41. UncertaintyFor large-scale information Technology (IT) modernization programs, there are at least three sources of uncertainty—and, thus, risk The technical complexity The programmatic complexity of integrating software intensive systems The absence of accurate cost information at the onset of major systems/ software Programs 41
  42. 42. Technological maturity 42
  43. 43. Common Software Risks that affect cost & schedule 43
  44. 44. Better Methods of Analyzing Cost Uncertainty Can Improve Acquisition Decision making • Cost estimation is a process that attempts to forecast the future expenditures for some capital asset, hardware, service, or capability. Despite being a highly quantitative field, cost estimation and the values it predicts are uncertain. An estimate is a possible or likely outcome, but not necessarily the outcome that will actually transpire. This uncertainty arises because estimators do not have perfect information about future events and the validity of assumptions that underpin an estimate. • Uncertainty may result from an absence of critical technical information, • the presence of new technologies or • approaches that do not have historical analogues for comparison, • the evolution of requirements, or • changes in economic conditions. • The Office of the Secretary of Defense and the military departments have historically underestimated and under funded the cost of buying new weapon systems (e.g., by more than 40 percent at Milestone II). • Much of this cost growth is thought to be the result of unforeseen (but knowable) circumstances when the estimate was developed. In the interest of generating more informative cost estimates, the Air Force Cost Analysis Agency and the Air Force cost analysis community want to formulate and implement a cost uncertainty analysis policy. • To help support this effort, RAND Project AIR FORCE (PAF) studied a variety of cost uncertainty assessment methodologies, examined how these methods and policies relate to a total portfolio of programs, and explored how risk information can be communicated to senior decision makers in a clear and understandable way. 44
  45. 45. •Project Air Force (USAF Rand Project) recommends that any cost uncertainty analysis policy reflect the following:• A single uncertainty analysis method should not be stipulated for all • circumstances and programs.• It is not practical to prefer one specific cost uncertainty analysis methodology in all cases. Rather, the policy should offer the flexibility to use different assessment methods. These appropriate methods fall into three classes: historical, sensitivity, and probabilistic. Moreover, a combination of methods might be desirable and more effective in communicating risks to decision makers.• • A uniform communications format should be used. PAF (USAF Rand Project) suggests a basic three-point format consisting of low, base, and high values as a minimum basis for displaying risk analysis. The advantages of such a format are that it is independent of the method employed and that it can be easily communicated to decision makers.• A record of cost estimate accuracy should be tracked and updated • periodically. Comparing estimates with final costs will enable organizations to identify areas where they may have difficulty estimating and sources of uncertainty that were not adequately examined.• • Risk reserves should be an accepted acquisition and funding practice.• Establishing reserves to cover unforeseen costs will involve a cultural change within the Department of Defense and Congress. The current approach of burying a reserve within the elements of the estimate makes it difficult to do a retrospective analysis of whether the appropriate level of reserve was set, and to move reserves, when needed, between elements of a large program.• Effective cost uncertainty analysis will help decision makers understand the nature of potential risk and funding exposure and will aid in the development of more realistic cost estimates by critically evaluating program assumptions and identifying technical issues. RAND 45
  46. 46. Uncertainty 46
  47. 47. COST ESTIMATING CHALLENGESDeveloping a good cost estimate requires stable program requirements, access todetailed documentation and historical data, well-trained and experienced cost analysts, arisk and uncertainty analysis, the identification of a range of confidence levels, andadequate contingency and management reserves. Cost estimating is nonetheless difficult in the best of circumstances. It requires both science and judgment. And, since answers are seldom—if ever—precise, the goal is to find a ―reasonable‖ answer. However, the cost estimator typically faces many challenges in doing so. These challenges often lead to bad estimates, which can be characterized as containing poorly defined assumptions, OMB first issued the Capital Programming Guide as a Supplement to the 1997 version of Circular A-11,• Part 3, still available on OMB‘s Web site at http://www.whitehouse.gov/omb/circulars/a11/cpgtoc.html.• Our reference here is to the 2006 version, as we noted in the preface: Supplement to Circular A-11, Part 7,• available at http://www.whitehouse.gov/omb/circulars/index.html. 47
  48. 48. John Wilder Tukey• "An appropriate answer to the right problem is worth a good deal more than an exact answer to an approximate problem." 48
  49. 49. Creating a range around a cost estimate 49
  50. 50. The absence of accurate cost information at the onset of major systems/ software ProgramsMeasures of uncertainty for cost/schedule estimates and the rate at which that uncertainty declines are a keyconcern—because, they govern whether and to what extent confidence can be placed in cost and scheduleestimates. The key to overcoming initial estimate uncertainty is the capability to harness and toapply information as it becomes available, thus, enabling a Firm to capturethe time value of that information.Indeed, where IT infrastructure modernization projects are supported by a strong quality-assurance, systems-engineeringculture (e.g., have institutionalized best-practice regimes such as the CMMI, 6-Sigma, Agile Methods are likely to quicklyreduce estimate errors incurred at project start-up. Firms without that culture tend to have limited informationefficiency. (Drawing an analogy to thermo-dynamic systems, such firms constitute highlydissipative systems in that they exhibit a high degree of entropy, which takes the form ofinformation disorganization).Unfortunately, traditional methods of discounting investment risk such as Net Present Value (NPV) do not account forirreversibility and uncertainty. In part, this is due to the fact that NPV computes the value of a portfolio of investments asthe maximized mean of discounted cash flows on the assumption that the risk to underlying investment options canbe replicated by assets in a financial market.NPV also implicitly assumes that the value of the underlying asset is known andaccurate at the time the investment decision is made.These assumptions seldom apply for large-scale infra-modernization programs, ineither the public or the private sector. In addition, NPV investment is undertaken when thevalue of a unit of capital is at least as large as its purchase and installation costs. But, thiscan be error prone since opportunity costs are highly sensitive to the uncertaintysurrounding the future value of the project due to factors such as the riskiness of future cashflows. These considerations also extend to econometric models, which excludeirreversibility, the incorporation of which transforms investment models into non-linearequations (Dixit & Pindyck, 1994, p. 421). Nonetheless, irreversibility constitutes both anegative opportunity cost and a lost-option value that must be included in the cost ofinvestment.R Suter Managing Uncertainty and Risk in Public Sector Investments, Richard Suter, Information Technology Systems, Inc., R Consulting A paper presented at the 4thAnnual Acquisition Research Symposium, Graduate School of Business & Public Policy, Naval Postgraduate School 50
  51. 51. Cost Estimating Process 51
  52. 52. Risk Assessment on Costs: A Cost Probability Distribution COMBINED COST MODELING AND TECHNICAL RISK Cost = a + bXc COST MODELING UNCERTAINTY CostEstimate Historical data point $ Cost estimating relationship TECHNICAL RISK Standard percent error bounds Cost Driver (Weight) Input variable Jeff Kline, Naval Postgraduate School 52 52
  53. 53. COST ESTIMATING METHODOLOGY TIME OF USEGROSS ESTIMATES DETAILED ESTIMATES PARAMETRIC ACTUAL (Program A B Initiation) C IOC FOC Concept Technology System Development Production & Operations &Refinement Development & Demonstration Deployment Support Concept Design FRP Decision Readiness LRIP/IOT&E Decision Review Review Pre-Systems Acquisition Systems Acquisition Sustainment EXPERT OPINION ANALOGY ENGINEERING 53
  54. 54. SOFTWARE DEVELOPMENT CONE OF UNCERTAINTY All software projects are subject to inherent errors in early estimates. The Cone of Uncertainty represents the best-case reduction in estimation error and improvement in predictability over the course of a project. Skillful project leaders treat the cone as a fact of life and plan accordingly. 4X Project predictability and control are attainable only through 2X active, skillful, and continuous efforts that force the cone to narrow. The cone represents the best case; results canRemaining variability in easily be worse. project scope 1.5X 1.25X 1.0X 0.8X 0.67X Estimates are possible anywhere in the cone, but 0.5X organizational commitments tied to project completion should not be made until about here – and only if work has been done to narrow the cone. 0.25X Square Peg in a Round Hole Initial Marketing Detailed Project Concept Approved Requirements Detailed Tech Design Complete Product Complete Requirements Complete Definition Complete Source: Construx, Bellevue WA
  55. 55. Software Cost Estimating• All commercial models (COCOMO II, SEER-SEM, and Price-S) are productivity- based models, and basically based on the same equation: Labor Rate ($/hr) * Software Size/ Productivity.• Maximize use Of actual data for Labor Rate, Productivity, Size.• Good source for productivity rates: http://www.stsc.hill.af.mil/CrossTalk/2002/03/reifer.html• COCOMO II does not capture requirement analysis and government V&V.• As man-effort increases, schedule and productivity decreases. However, cost increases and possible rework. 3• Schedule rule of thumb: Time ~ 3.67* Effort• CAUTIONS: – Code Re-use Lowers Cost, Modification Increases Cost • Per OSD/ CAIG: modified code, with more than 25% of the lines changed or added, is considered new code. (based on NASA Study) • with SEER-SEM cost of 99% Modified Code < Cost of New Code – Analogies: Don’t treat non-similar languages as equivalent Example in PLCCE: SLOC= C + C++ + IDL + JAVA + XML
  56. 56. Cost Risk Analysis The process of quantifying uncertainty in a cost estimate.• By definition a point estimate is precisely wrong – Assessment of risk is not evident in a point estimate – The influence of variables may not be understood by the decision maker• Cost risk predicts cost growth.• Cost risk = cost estimating risk + schedule risk+ technical risk + change in requirements/ threat• Risk analysis adjusts the cost estimate to provide decision makers an understanding of funding risks. 1 0.12 0.9 0.1 0.8 0.7 0.08 0.6 0.5 0.06 0.4 0.04 0.3 0.2 0.02 0.1 0 0 Probability Density Function PDF Cumulative Density Function CDF 56
  57. 57. Simplified Cost Risk Simulation Model If no actual data available Methodology perform the following steps Basis of Estimate Schedule Assign Risk Producibility to Each Element: Reliability None, Low, MedInfluencedby Complexity High, etc. Technology Statusavailabilityof actual Assess Riskdata Categories Assign Risk Limits toor expert For Data Inputs statistical distributionopinion By WBS (e.g. + X; -X to +Y, etc.) 8 7 6 Total Cost PDF 5 4 3 2 Select 1 0 Run statistical 1 4 7 3 6 9 2 5 8 1 4 7 3 6 9 1.2 1.5 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.3 1.3 1.4 1.4 1.4 1.5 1.5 1.5 1 0.9 Model distribution 0.8 0.7 0.6 0.5 0.4 0.3 CDF 0.2 0.1 0 Input PDFs
  58. 58. Example 1 Total Software Cost Estimate schedule slide also 100% schedule risk 80% KTR EAC 3/07 82% Complete 60%Risk PM Estimate/ICE 4/05 40% 20% Contract 6/05 & PM Estimate 11/05 0% $0 $20,000 $40,000 $60,000 $80,000 $100,000 $BY05K
  59. 59. Example (Cont’d) Pre & Post Software Contract Data 350000 30 01-6 ICE 4/05 New 300000 SLOC 25 250000 20 Dollars in MillionsSLOC in Units 200000 Offeror SLOC Estimate 6/05 15 with 38% Reuse 150000 Code PM SLOC 10 100000 Estimate 4/05 Software Metrics Report (SLOC) with 76% Reuse Code Ktr EAC 5 50000 0 0 R R 05 06 10 6 07 07 06 06 06 06 06 07 07 11 5 05 12 5 12 6 06 11 6 00 0 0 00 PD 0 CD 20 20 20 20 20 20 20 20 20 20 20 20 /20 /20 /20 /20 /2 /2 3/ 9/ 2/ 4/ 2/ 4/ 5/ 7/ 8/ 1/ 3/ 4/ 10 Date
  60. 60. Example (Cont’d) Schedule Risk Software Development Schedule Months01-6 ICE (4/05) COCOMO Equation 25 NCCA Equation 30PM Estimate (4/05) 18Contract - Initial (6/05) 18Contract - Current (3/07) (82% Complete) 3101-6 ICE - Current (3/07) 35
  61. 61. The Refining of a Life Cycle Cost EstimateLCCE Cost Estimating Uncertainty MS A MS B MS C Concept Trades Ktr Selection Design Reviews Production AOA Test & Eval / Design Mods CARD Logistics Program / System Evolution 61
  62. 62. 62
  63. 63. DIFFERENTIATING COST ANALYSIS AND COST ESTIMATINGCost analysis, used to develop cost estimates for such things as hardware systems,automated information systems, civil projects, manpower, and training, can be defined as1. the effort to develop, analyze, and document cost estimates with analyticalapproaches and techniques;2. the process of analyzing and estimating the incremental and total resourcesrequired to support past, present, and future systems—an integral step in selecting alternatives; and3. a tool for evaluating resource requirements at key milestones and decision points in the acquisition process.Cost estimating involves collecting and analyzing historical data and applying quantitative models, techniques, tools, and databases to predict a program‘s future cost.More simply, cost estimating combines science and art to predict the future cost of something based on known historical data that are adjusted to reflect new materials, technology, software languages, and development teams.Because cost estimating is complex, sophisticated cost analysts should combine concepts from such disciplines as accounting, budgeting, computer science, economics, engineering, mathematics, and statistics and should even employ concepts from marketing and public affairs. And because cost estimating requires such a wide range of disciplines, it is important that the cost analyst either be familiar with these disciplines or have access to an expert in these fields. 63
  64. 64. 64
  65. 65. Jackson Lears‘s analyzed why the dominant American ―culture of control‖ denies the importance of luck • Drawing on a vast body of research, Lears ranges through the entire sweep of American history as he uncovers the hidden influence of risk taking, conjuring, soothsaying, and sheer dumb luck on our culture, politics, social lives, and economy.T.J. Jackson Lears “Something for Nothing” (2003) 65
  66. 66. Illusion of Control• In a series of experiments, Ellen Langer (1975) demonstrated first the prevalence of the illusion of control and second, that people were more likely to behave as if they could exercise control in a chance situation where ―skill cues‖ were present. By skill cues, Langer meant properties of the situation more normally associated with the exercise of skill, in particular the exercise of choice, competition, familiarity with the stimulus and involvement in decisions.• One simple form of this fallacy is found in casinos: when rolling dice in craps, it has been shown that people tend to throw harder for high numbers and softer for low numbers.• Under some circumstances, experimental subjects have been induced to believe that they could affect the outcome of a purely random coin toss. Subjects who guessed a series of coin tosses more successfully began to believe that they were actually better guessers, and believed that their guessing performance would be less accurate if they were distracted. 66
  67. 67. Critque of Taleb• Talebs point is rather that most specific forecasting is pointless, as large, rare and unexpected events (which by definition could not have been included in the forecast) will render the forecast useless.• However, as Black Swans can be both negative and positive, we can try to structure our lives in order to minimize the effect of the negative Black Swans and maximize the impact of the positive ones. I think this is excellent advice on how to live ones life and seems to be equivalent, for example, to the focus on downside protection (rather than upside potential) that has led to the success of the value approach to investing. 67
  68. 68. Risk = Variance • Risk: Well, it certainly doesnt mean standard deviation. People mainly think of risk in terms of downside risk. They are concerned about the maximum they can lose. So thats what risk means. • In contrast, the professional view defines risk in terms of variance, and doesnt discriminate gains from losses. There is a great deal of miscommunication and misunderstanding because of these very different views of risk. Beta does not do it for most people, who are more concerned with the possibility of loss • Daniel KahnemanDaniel Kahneman is the Eugene Higgins Professor of Psychology at Princeton University) and Professor of Public Affairs at Woodrow Wilson School. Kahneman was born in Israel and educated at the Hebrew University in Jerusalem before taking his PhD at the University of California. He was the joint Nobel Prize winner for Economics in 2002 for his work on applying cognitive behavioural theorie to decision making in economics . 68
  69. 69. Cicero Born: January 3, 106 B.C.E. Arpinum, Latinum Died: December 7, 43 B.C.E. Formiae, Latinum Roman orator and writer Marcus Tullius Cicero ―Probability is the very guide of life.‖• Pp 31 The Drunkards Walk 69
  70. 70. Probability• “ in no other branch of mathematics is it so easy to blunder as in probability theory.” – Martin Gardiner, ―Mathematical Games," Scientific American, October 1959 pp 180-182 70
  71. 71. The Monte Hall problem• Probability Theory The Monte Hall problem, birthday pairings, counting principles, conditional probability and independence, Bayes Rule, random variables and their distributions, Gamblers Ruin problem, random walks, and Markov chains. 71
  72. 72. • Display aircraft movement 72
  73. 73. 73
  74. 74. Probability Theory• Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion.• Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.• As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. 74
  75. 75. 75
  76. 76. 0 random variablesProbability Density • In mathematics, random variables are used in the studyFunction PDF of chance and probability. They were developed to assist in the analysis of games of chance, stochastic events, and the results of scientific experiments by capturing only the mathematical properties necessary to answer probabilistic questions. Further formalizations have firmly grounded the entity in the theoretical domains of mathematics by making use of measure theory. • Fortunately, the language and structure of random variables can be grasped at various levels of mathematical fluency. Set theory and calculus are fundamental. • Broadly, there are two types of random variables — discrete and continuous. Discrete random variables take on one of a set of specific values, each with some probability greater than zero. Continuous random variables can be realized with any of a range of values (e.g., a real number between zero and one), and so there are several ranges (e.g. 0 to one half) that have a probability greater than zero of occurring. • A random variable has either an associated probability distribution (discrete random variable) or probability density function (continuous random variable). 76
  77. 77. Probability Density Function NEED A BETTER DEFINITION• it shows the probability density function (pdf) of a non-linear communications channel - i.e. the embedded output of a 2D system. It has been estimated by using a characteristic function estimator (the characteristic function is the Fourier transform of the pdf so by estimating the characteristic function you can get an estimate of the pdf by an inverse FFT). 77
  78. 78. Game theory Is a branch of applied mathematics that is used in the social sciences (most notably economics), biology, engineering, political science, computer science (mainly for artificial intelligence), and philosophy. Game theory attempts to mathematically capture behavior in strategic situations, in which an individuals success in making choices depends on the choices of others. While initially developed to analyze competitions in which one individual does better at anothers expense (zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several criteria. Today, ―game theory is a sort of umbrella or ‗unified field‘ theory for the rational side of social science, where ‗social‘ is interpreted broadly, to include human as well as non-human players (computers, animals, plants)‖ (Aumann 1987).• Traditional applications of game theory attempt to find equilibria in these games— sets of strategies in which individuals are unlikely to change their behavior. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide. This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.• Although some developments occurred before it, the field of game theory came into being with the 1944 book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game theorists have won Nobel prizes in economics, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. 78
  79. 79. Itōs lemma• In mathematics, Itōs lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is the stochastic calculus counterpart of the chain rule in ordinary calculus and is best memorized using the Taylor series expansion and retaining the second order term related to the stochastic component change. The lemma is widely employed in mathematical finance.• Itōs lemma is the version of the chain rule or change of variables formula which applies to the Itō integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous d- dimensional semimartingale X = (X1,…,Xd) and twice continuously differentiable function f from Rd to R, it states that f(X) is a semimartingale an• This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [Xi,Xj ]. The formula can be generalized to non-continuous semimartingales by adding a pure jump term to ensure that the jumps of the left and right hand sides agree (see Itōs lemma). 79
  80. 80. EVENT• In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is infinite, most notably when the outcome is a real number. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see §2, below).• A simple example• If we assemble a deck of 52 playing cards and no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each individual card is a possible outcome. An event, however, is any subset of the sample space, including any single-element set (an elementary event, of which there are 52, representing the 52 possible cards drawn from the deck), the empty set (which is defined to have probability zero) and the entire set of 52 cards, the sample space itself (which is defined to have probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include:• A Venn diagram of an event. B is the sample space and A is an event. By the ratio of their areas, the probability of A is approximately 0.4.• "Red and black at the same time without being a joker" (0 elements),• "The 5 of Hearts" (1 element),• "A King" (4 elements),• "A Face card" (12 elements),• "A Spade" (13 elements),• "A Face card or a red suit" (32 elements),• "A card" (52 elements).• Since all events are sets, they are usually written as sets (e.g. {1, 2, 3}), and represented graphically using Venn diagrams. Venn diagrams are particularly useful for representing events because the probability of the event can be identified with the ratio of the area of the event and the area of the sample space. (Indeed, each of the axioms of probability, and the definition of conditional probability can be represented in this fashion.) 80
  81. 81. EVENT (continued)• Events in probability spaces• Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions, such as the normal distribution the sample space is the set of real numbers or some subset of the real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers badly-behaved sets, such as those which are nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under countable unions and intersections. The most natural choice is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice.• In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected σ-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the σ-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest will be elements of the σ-algebra. 81
  82. 82. Law of Large Numbers• was first described by Jacob Bernoulli. It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoullis Theorem" (not to be confused with the Law in Physics with the same name.)• In 1835, S.D. Poisson further described it under the name "La loi des grands nombres" ("The law of large numbers").[3] Thereafter, it was known under both names, but the "Law of large numbers" is most frequently used.• After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli and Kolmogorov. These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law. These forms do not describe different laws but instead refer to different ways of describing the mode of convergence of the cumulative sample means to the expected value, and the strong form implies the weak. 82
  83. 83. Law of Large Numbers• Both versions of the law state that the sample average converges to the expected value• where X1, X2, ... is an infinite sequence of i.i.d. random variables with finite expected value; – E(X1)=E(X2) = ... = µ < ∞.• An assumption of finite variance Var(X1) = Var(X2) = ... = σ2 < ∞ is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway. This assumption is often used because it makes the proofs easier and shorter.• The difference between the strong and the weak version is concerned with the mode of convergence being asserted.• The weak law• The weak law of large numbers states that the sample average converges in probability towards the expected value.• Interpreting this result, the weak law essentially states that for any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value, that is, within the margin.• Convergence in probability is also called weak convergence of random variables. This version is called the weak law because random variables may converge weakly (in probability) as above without converging strongly (almost surely) as below.• A consequence of the weak LLN is the asymptotic equipartition property.• The strong law• The strong law of large numbers states that the sample average converges almost surely to the expected value• That is, the proof is more complex than that of the weak law. This law justifies the intuitive interpretation of the expected value of a random variable as the "long-term average when sampling repeatedly".• Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law.• The strong law of large numbers can itself be seen as a special case of the ergodic theorem. 83
  84. 84. Bayesian Analysis• Bayesian inference uses aspects of the scientific method, which involves collecting evidence that is meant to be consistent or inconsistent with a given hypothesis. As evidence accumulates, the degree of belief in a hypothesis ought to change. With enough evidence, it should become very high or very low. Thus, proponents of Bayesian inference say that it can be used to discriminate between conflicting hypotheses: hypotheses with very high support should be accepted as true and those with very low support should be rejected as false. However, detractors say that this inference method may be biased due to initial beliefs that one holds before any evidence is ever collected. (This is a form of inductive bias).• Bayesian inference uses a numerical estimate of the degree of belief in a hypothesis before evidence has been observed and calculates a numerical estimate of the degree of belief in the hypothesis after evidence has been observed. (This process is repeated when additional evidence is obtained.) Bayesian inference usually relies on degrees of belief, or subjective probabilities, in the induction process and does not necessarily claim to provide an objective method of induction. Nonetheless, some Bayesian statisticians believe probabilities can have an objective value and therefore Bayesian inference can provide an objective method of induction 84
  85. 85. The Reverend Thomas Bayes, F.R.S. --- 1701?-1761 Bayes‘ EquationTo convert the Probability of event A given event B tothe Probability of event B given event A, we use Bayes’theorem. We must know or estimate the Probabilities ofthe two separate events. Pr (A|B) Pr (B) Pr(B|A) = Pr (A) Pr (A) = Pr(A|B)Pr(B) + Pr(A|B)Pr(B) Law of Total Probability 85 85
  86. 86. Bayesian Analysis – Example of Bayesian search theory• In May 1968 the US nuclear submarine USS Scorpion (SSN-589) failed to arrive as expected at her home port of Norfolk Virginia. The US Navy was convinced that the vessel had been lost off the Eastern seaboard but an extensive search failed to discover the wreck. The US Navys deep water expert, John Craven USN, believed that it was elsewhere and he organized a search south west of the Azores based on a controversial approximate triangulation by hydrophones. He was allocated only a single ship, the Mizar, and he took advice from a firm of consultant mathematicians in order to maximize his resources. A Bayesian search methodology was adopted. Experienced submarine commanders were interviewed to construct hypotheses about what could have caused the loss of the Scorpion.• The sea area was divided up into grid squares and a probability assigned to each square, under each of the hypotheses, to give a number of probability grids, one for each hypothesis. These were then added together to produce an overall probability grid. The probability attached to each square was then the probability that the wreck was in that square. A second grid was constructed with probabilities that represented the probability of successfully finding the wreck if that square were to be searched and the wreck were to be actually there. This was a known function of water depth. The result of combining this grid with the previous grid is a grid which gives the probability of finding the wreck in each grid square of the sea if it were to be searched.• This sea grid was systematically searched in a manner which started with the high probability regions first and worked down to the low probability regions last. Each time a grid square was searched and found to be empty its probability was reassessed using Bayes theorem. This then forced the probabilities of all the other grid squares to be reassessed (upwards), also by Bayes theorem. The use of this approach was a major computational challenge for the time but it was eventually successful and the Scorpion was found about 740 kilometers southwest of the Azores in October of that year.• Suppose a grid square has a probability p of containing the wreck and that the probability of successfully detecting the wreck if it is there is q. If the square is searched and no wreck is found, then, by Bayes theorem, the revised probability of the wreck being in the square is given by XXXXXXXXX 86
  87. 87. Stochastic• Stochastic is synonymous with "random." The word is of Greek origin and means "pertaining to chance" (Parzen 1962, p. 7).• It is used to indicate that a particular subject is seen from point of view of randomness.• Stochastic is often used as counterpart of the word "deterministic," which means that random phenomena are not involved.• Therefore, stochastic models are based on random trials, while deterministic models always produce the same output for a given starting condition. 87
  88. 88. Randomness 88
  89. 89. Stochastic modeling• "Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. The random variation is usually based on fluctuations observed in historical data for a selected period using standard time- series techniques. Distributions of potential outcomes are derived from a large number of simulations (stochastic projections) which reflect the random variation in the input(s).• Its application initially started in physics (sometimes known as the Monte Carlo Method). It is now being applied in engineering, life sciences, social sciences, and finance. 89
  90. 90. • Valuation• Like any other company, an insurer has to show that its assets exceeds its liabilities to be solvent. In the insurance industry, however, assets and liabilities are not known entities. They depend on how many policies result in claims, inflation from now until the claim, investment returns during that period, and so on.• So the valuation of an insurer involves a set of projections, looking at what is expected to happen, and thus coming up with the best estimate for assets and liabilities, and therefore for the companys level of solvency.• Deterministic approach The simplest way of doing this, and indeed the primary method used, is to look at best estimates. The projections in financial analysis usually use the most likely rate of claim, the most likely investment return, the most likely rate of inflation, and so on. The projections in engineering analysis usually use both the mostly likely rate and the most critical rate. The result provides a point estimate - the best single estimate of what the companys current solvency position is or multiple points of estimate - depends on the problem definition. Selection and identification of parameter values are frequently a challenge to less experienced analysts. The downside of this approach is it does not fully cover the fact that there is a whole range of possible outcomes and some are more probable and some are less.• Stochastic modeling• A stochastic model would be to set up a projection model which looks at a single policy, an entire portfolio or an entire company. But rather than setting investment returns according to their most likely estimate, for example, the model uses random variations to look at what investment conditions might be like.• Based on a set of random outcomes, the experience of the policy/portfolio/company is projected, and the outcome is noted. Then this is done again with a new set of random variables. In fact, this process is repeated thousands of times.• At the end, a distribution of outcomes is available which shows not only what the most likely estimate, but what ranges are reasonable too.• This is useful when a policy or fund provides a guarantee, e.g. a minimum investment return of 5% per annum. A deterministic simulation, with varying scenarios for future investment return, does not provide a good way of estimating the cost of providing this guarantee. This is because it does not allow for the volatility of investment returns in each future time period or the chance that an extreme event in a particular time period leads to an investment return less than the guarantee. Stochastic modeling builds volatility and variability (randomness) into the simulation and therefore provides a better representation of real life from more angles. 90
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.