Decision Making Under Uncertainty

Dr. Michael Rees, Head of DMUU Training Team

Dr. Michael Rees

How is quantitative modelling applied to today's risk analysis problems in finance, insurance, oil and gas, manufacturing, and more? Monte Carlo Simulation is the key. In this blog, we'll discuss the latest tips and techniques to help you judge which risks to take and which to avoid, allowing for the best decision making under uncertainty.

Announcing @RISK 5.5 and DecisionTools Suite 5.5!

Posted Wednesday, July 1, 2009 by DMUU Training Team

Building on the worldwide success of @RISK 5.0 and DecisionTools Suite 5.0, Palisade is pleased to announce the version 5.5 release of these best-selling risk and decision analysis tools.

New @RISK 5.5 - risk analysis with Monte Carlo simulation

Current @RISK 5.0 users will benefit from faster simulations — 2x to a remarkable 20x times faster than before — as well as new scatter plots from scenario analyses, a freehand distribution artist, and an Excel-style Insert Function dialog and graphs. @RISK 5.5 brings a range of new features to improve your analysis, save time, and encourage systematic adoption of risk analysis across your organization.

@RISK 5.5 is the best Monte Carlo simulation package available today, blending high-powered analysis with highly intuitive ease-of-use. The bottom line for you is a better understanding of what could happen and how likely it is to happen. Applications include value-at-risk, design and analysis of experiments, discounted cash flow analysis, exploration and production, option valuation, and more.

@RISK 5.5 has been fully translated into Spanish, German, French, Portuguese and Japanese.

» What's New in @RISK 5.5
» Download a free trial version

Giving Kurtosis a Workout

Posted Monday, June 15, 2009 by DMUU Training Team

Kurtosis is a statistical measure of a random process that is often used, but perhaps less widely understood. This blog mentions a couple of key issues and misunderstandings about kurtosis in a risk assessment model.

A high kurtosis figure is sometimes described as being associated with a distribution that has “fat tails”. However, by simply overlaying two Normal distributions with the same mean but different standard deviations (e.g. using @RISK to do so), it is visually clear that the distribution with the larger standard deviation has the “fatter tails”. However, every Normal distribution has a kurtosis of 3 (sometimes “excess kurtosis” is referred to, whereby any base calculation has three subtracted from it; this is the case when using the Excel KURT function to calculate kurtosis, for example), so the kurtosis figure does not pick up the idea that one of the distributions has more weight in the tail.

In fact, kurtosis is a simultaneous measure of the “peakedness” of a distribution and the extent to which it has “fat tails”; the Normal distribution with the larger standard deviation will have fatter tails, but will also be less peaked, and in terms of how kurtosis is calculated, these effects balance out. Kurtosis is then a bit like going for a workout, where you are required to push weights in a central direction whilst keeping your elbows up!

Another important aspect of kurtosis that is little appreciated is that the kurtosis of a binomial distribution (e.g. modelling an event risk that may or may not happen with a certain probability) increases as the probability of the event decreases. In this sense, distributions with high kurtosis figures are perhaps most easily understood as ones relating to events of low probability but high impact.

Such topics are very easy to explore with @RISK (risk analysis Monte Carlo software add-in to Excel), where the visual ability to view overlays, combined with the use of the RiskTheo functions to obtain in Excel the numerical values of statistics associated with distributions allows for a powerful environment to rapidly address such issues.

Rethinking Monte Carlo Simulation for Retirement Planning

Posted Friday, June 12, 2009 by DMUU Training Team

A recent article entitled “When Monte Carlo analysis meets a black swan” in Investment News addresses the criticisms Monte Carlo simulation has received for “missing the meltdown.” The author, Moshe A. Milevsky , notes that people typically seek a single number “answer” from a Monte Carlo simulation, such as the probability of meeting a single retirement planning goal. Milevsky points out that many Monte Carlo software packages do not include sensitivity or scenario analyses to drill down and determine which variables are really driving the risk inherent in the results. He proposes what amounts to a stress test – simulating what could happen under likely scenarios, and simulating again under 1-in-100 chance “black swan” disastrous scenarios. Milevsky wraps up by saying, “Instead of condemning Monte Carlo analyses for missing the meltdown, let's properly harness the full power of stochastic methods to give us tools that provide clear utility.”

I believe Milevsky makes a great point, focusing on the modeling practices rather than the tools themselves in this case. Monte Carlo simulation tools are very important for applications like retirement planning, but even the best hammer can’t help an unskilled carpenter. Tools like @RISK include sensitivity and scenario analysis, enabling easy implementation of tests under different scenarios for portfolio value, inflation, longevity, or any combination of these.

Randy Heffernan
Vice President

Using Named Ranges in Excel: Some Comments

Posted Tuesday, June 9, 2009 by DMUU Training Team

An earlier blog on Best Practice Principles in Excel Modelling generated quite some interest, as well as demand for more details on some of the points made, especially those concerning the use of named ranges risk asssessment models in Microsoft Excel. In the earlier posting, I had simply stated that (in my opinion): “Named ranges should be used highly selectively but not excessively”. Here I will expand a little more; the topic itself can be a subject of quite animated discussion within the risk analysis modelling community, with a wide set of opinions expressed. The points I make below are therefore simply my view of the topic.

In my view, named ranges are indispensible in some types of modelling situations. The most frequent of these in my experience are:

When writing VBA code (macros) that refer to ranges in the workbook (as such code almost always would do at some point), the use if names provides a much more robust way of creating flexible code, rather than referring to the range using cell references.
For general Excel modelling, it can be useful to name a small set of key ranges, so that the F5 key or the name box can be used to rapidly navigate around the model.
Where the model process is not required as a process to experiment with or modify a model, but is purely required to implement a known situation which will never be changed. However, much Excel modelling involves the process of experimenting with different approaches, and the use of named ranges in such cases can create extra complexity.

Some disadvantages of using named ranges include: that their use too early on in the risk analysis modelling process can create cumbersome structures, that it can be easy to create models with far too many names that then become poorly labelled, and the possibility to inadvertently create links between models. The management of names (such as their deletion and their scope) has traditionally been cumbersome in Excel. It is important to note that Excel’s 2007 Name Manager has radically reduced some of these disadvantages (this change being one of the most important improvements made to Excel when moving from Excel 2003 to 2007, in my opinion).

This set of points is by no means complete; a deeper discussion of modelling in Excel, including robust and readily understandable risk analysis models or option valuation and price forecasting, is contained within my book Financial Modelling in Practice.

Dr. Michael Rees
Director of Training and Consulting

Don’t Blame the Math

Posted Thursday, June 4, 2009 by DMUU Training Team

A recent article in Bank Investment Consultant criticized the risk analysis method of Monte Carlo simulation for not taking into account extreme events like the stock market crash. According the article, a Morningstar executive states that the “bell-shaped curve that Monte Carlo simulations use” artificially assigns the probability of what happened as zero. Furthermore, the Retirement Income Industry Association calls for an update to Monte Carlo software simulators to include “a larger number of scenarios that assume greater volatility.”

These arguments demonstrate a fundamental lack of understanding of what Monte Carlo simulation is. The underlying assumption that Monte Carlo simulation itself is somehow to blame fails to recognize that Monte Carlo simulation is simply a mathematical technique that takes into account many different possible scenarios – but only within boundaries set by the user. You can’t change the underlying math behind these “what-if” calculations.

When modelers set up retirement planning or financial models, the people doing the modeling must make assumptions about the likelihood of different things happening – like the market crashing, for example. People may make those assumptions based on historical evidence or expert opinion, but it’s people who make those assumptions – not Monte Carlo simulation software. People then enter their assumptions into a Monte Carlo simulation model, setting up probability distributions to reflect their chosen likelihoods of occurrence. If the assumed volatility is insufficient, that is the fault of the modelers, not the simulation itself.

In addition, Monte Carlo simulation does not always “use” a bell-shaped curve. Uncertainty can be modeled with dozens of different probability distributions, many of them not bell-shaped. And the call to include more scenarios and volatility can easily be met by existing Monte Carlo software such as @RISK. It’s simply up the user to change the model parameters to look at more possible outcomes. The Monte Carlo simulation package won’t fight it.

It’s disappointing to see esteemed financial organizations such as Morningstar and the Retirement Income Association missing the point. Calling for changes to Monte Carlo simulation itself is not only impossible but fails to recognize the problems with modeling practices that led everyone to miss the crash.

Randy Heffernan
Vice President

Sources of Skewness in Risk Modelling

Posted Monday, June 1, 2009 by DMUU Training Team

The topic of skewness of an uncertain variable is perhaps one of the most fundamental in risk assessment modelling. When it is believed that a (continuous) process is symmetric, the choice of distributions to use to represent that process is generally of less consequence than when the uncertainty is asymmetric. For example, a symmetric Triangular, PERT, and Normal distribution (with appropriately selected parameters e.g. so that the means and standard deviations for each are the same) will be broadly similar; of course there are some differences, but they are generally at the margin and of little significance in many practical risk analysis modelling situations for general business purposes (though such differences can still be important in cases where extremely accurate models are required).

Here, I briefly mention some sources of skewness that arise in real-life processes, or in the associated modelling of risk:

Multiplicative processes. A process in which random variables are multiplied will create a skewed output, tending to a Lognormal distribution when many such independent variables are multiplies, and often approximated by such a distribution in any case. Such process arise in cost budgeting (e.g. the total cost estimate as the product of an uncertain volume, unit cost, and perhaps a duration), in asset price forecasting (% changes to asset values over several periods work in a multiplicative sense), and in oil reservoir modelling (uncertain reserve estimation volume estimate being the product of uncertain spatial dimensions and some additional other factors, i.e. for exploration and production).
Compound processes with event risks. When taking a pragmatic modelling approach in cost budgeting (e.g. using a Triangular distribution), one often simply assumes that the cost distribution of an item is asymmetric; that is we assume that (for unspecified reasons) the costs are more likely to be over the base estimate than below it. Often part of the underlying reason is the presence of event risks in the situation, where the occurrence of a specific event creates an additional (perhaps uncertain) set of costs in addition to a (perhaps uncertain and symmetric) base cost.
Parameter estimation for small sample sizes. When estimating a probability from a set of observations (for example 5 occurrences of an event during 100 periods, or trials), one sometimes takes the “maximum likelihood” approach (i.e. assume 5%) or otherwise assumes that there is a distribution of possible probabilities (such as a Beta distribution). Either way, for small sample sizes, the distribution of the uncertainty of the true probability is not symmetric. Examples of this were given in the earlier blog about the difficulties in estimating the probability of low probability events.

Dr. Michael Rees
Director of Training and Consulting

Fed Uses Monte Carlo Simulation for Stress Test

Posted Friday, May 29, 2009 by DMUU Training Team

The U.S. Federal Reserve recently released the results of a comprehensive assessment of the financial conditions of the nation's 19 largest banks, which hold two-thirds of American economic assets. This “stress test” was designed to determine the capital buffers required for the banks to withstand losses and maintain lending even in worsening economic conditions. Officially called the Supervisory Capital Assessment Program (SCAP), the test identified the potential losses, resources available to absorb losses, and resulting capital buffer needed.

Monte Carlo simulation was used to determine the potential losses from further defaults on loans. According to Federal Reserve Chairman Ben Bernanke, “The assessment program was a forward-looking, ‘what-if’ exercise.”

Monte Carlo simulation is one of the most widely used methods of stress testing for capital and operations risk, according to Investopedia. It takes into account variables such as interest rates, lending requirements, and unemployment. As any @RISK software user will tell you, this type of sophisticated simulation can be accomplished easily within the Microsoft Excel environment. The result of a Monte Carlo software simulation is a look at a whole range of possible outcomes, including the probabilities they will occur -- a valuable tool when stress testing.

Randy Heffernan
Vice President

Eating Time and Economic Growth: The Stability of Correlation Coefficients

Posted Tuesday, May 26, 2009 by DMUU Training Team

I have been meaning to write about the stability of correlation coefficients for a while, and have been spurred into action by a recent article in the International Herald Tribune (“Eat Quickly, for the Economy’s Sake,” May 8, 2009). The article discusses the relationship between economic growth and time spent eating, based on a recent study published by the OECD, which looked at a variety of social indicators in the 22 countries surveyed. The article finds a negative correlation between eating time and economic growth, with the U.S., Canada and Mexico being the countries in which people spend the least time eating (and with higher economic growth), and France the country where most time is spent eating (“to no one’s surprise,” according to the author).

Since I was keen to try to replicate the results of the study for myself, I downloaded the eating survey data from the OECD web-site, but then for some reason was unable to find the same economic growth figures used in the article, and so searched for similar data myself, with the result that the time period covered in measuring economic growth was different to that used in the article… the key point being that I ended up in my analysis with a positive (+16%) correlation coefficient between eating time and economic growth, rather than a negative one (…maybe a bit of relaxation does help creativity?...back to the land of dreams…).

In fact, the reality is that correlation coefficients measured between data samples are not particularly stable until the data sets become very large. The @RISK (risk analysis using Monte Carlo simulation in Excel) tool can be used to explore this issue. For example, two data sets of a given size (e.g. 22 points) containing independent distributions can be set up, and for each random sampling of the points (each iteration of a simulation), the correlation coefficient between the sample points drawn from independent distributions calculated, with this correlation coefficient being set as the output call for the simulation. Having done this, a few observations and notes can be made from such reflections and calculations:

Data sets containing only two points each will have a correlation coefficient which is either 100% or minus 100% (except in the rare case where two data points have exactly the same value), as the points are ordered either both low-high, or one is low-high and the other high-low. This already provides some intuition as to the lack of stability of the coefficient.
The correlation coefficient as measured from two independent samples of 22 data points has a standard deviation of around 20% (and a mean of zero); in roughly one-third of cases, the measured correlation coefficient between these two independent samples will be more than 20% (in either the positive of negative direction)
About 100 data points are required in each set before the standard deviation of the correlation coefficient drops to below 10%
The inverse square-root law applies: a doubling of the sample size reduces the standard deviation of the correlation coefficient to approximately 71% of the value with a smaller sample size.

Referring back to the original article, my own feeling is therefore that the relationship discussed is driven by the inherent uncertainty in dealing with small sample sizes. As the article’s author states: “Such correlations may be nothing but coincidence, of course.” As we saw when @RISK was applied to the problem, any statistical analysis can be subject to these traps.

Undaunted, the author continues: “But if the data are genuine, a contribution to world growth is rendered by any institution that enables people to eat rapidly and gain weight. Take a bow, McDonalds.”

Dr. Michael Rees
Director of Training and Consulting

Wall Street Journal Confuses Monte Carlo Simulation with Models

Posted Thursday, May 21, 2009 by DMUU Training Team

The recent Wall Street Journal article “Odds-On Imperfection: Monte Carlo Simulation” asserts that Monte Carlo simulation did not predict the market crash, and cites a chorus of critics calling for a fix to the technique. The article equates the technique of Monte Carlo simulation with the models that are using it – two very different things. For instance, the article states, “These models were supposed to help quantify and manage the risks of mortgage-backed securities, credit-default swaps and other complex instruments. But given the events of the past couple of years, it appears that the models often gave big institutions, as well as small investors, a false sense of security.”

This is true – the models for decision making under uncertainty gave a false sense of security. But that’s because the assumptions underlying the risk analysis models were flawed, not because the technique of Monte Carlo simulation was problematic. Monte Carlo simulation is simply a mathematical technique that recalculates many different possible scenarios – but only within boundaries set by the user. You can’t change the underlying math behind these “what-if” calculations.

The article comes close to making this distinction in one sentence: “Critics emphasize that the problem isn't Monte Carlo itself, but the assumptions that go into it.” It then goes on to describe efforts by firms to include “fatter tail” distributions that more accurately reflect the probability of extreme events occurring as an effort to improve Monte Carlo simulation. Tools like @RISK (risk analysis software add-in for Microsoft Excel) allow complete control over the definition of many dozens of distribution types, enabling users to create as fat a tail as they want. While these efforts make sense, it should be made clear that these are changes to underlying model assumptions, not changes to Monte Carlo simulation itself. To equate Monte Carlo simulation as a technique with the probability distributions people decide to use is to equate a carpenter’s choice of nails with his hammer.

Finally, the article cites the need to run tens or hundreds of thousands of scenarios, instead of just 100 or 1000. This too is user-defined, and tools like @RISK can run as many scenarios as desired.

Randy Heffernan
Vice President

Modal Forecasting

Posted Friday, May 15, 2009 by DMUU Training Team

An article in last week’s International Herald Tribune leads me to post this item in order to reinforce with a practical example some earlier postings about reasons to perform risk analysis.

The article states: “Wall Street has a notoriously bad forecasting record. It almost always predicts that the economy will grow by something like 3 percent a year, which happens to be correct most of the time… Amazingly enough, Wall Street’s consensus forecast has failed to predict a single recession in the past 30 years.”

Amazingly enough? Not really. One of the problems that Wall Street (and perhaps the majority of other forecasters) has to deal with is that their audience expects (and often demands) a single point forecast; many investors and decision-makers seem to prefer the false security attached to a static forecast.

When it comes to defending Wall Street, don’t count me in. However, it is unavoidable that if one is required to present a single static forecast, then the presentation of what one considers to be the most likely is the only logical (apparently non-biased) response. With a recession occurring on average say once every 10 years, such scenarios will simply be excluded from a static forecast. Of course, risk modelling would allow the possibility and consequences of a recession to be included (thereby giving a lower or less optimistic estimate of base case earnings – say the average, rather than the most likely).

What is “consensus”? The range of analyst estimates making up a consensus forecast is therefore something close to the distribution of the mode of the outcome (rather than the distribution of the outcome).

Such biases in static forecasting are unavoidable, and will only disappear when the clients of such forecasters are willing (or demand) a proper uncertainty analysis around forecasts, something that is happening more and more, but still has a long way to go. Of course, the use of @RISK (risk analysis using Monte Carlo simulation) and the DecisionTools Suite (complete toolkit for decision making under uncertainty) provide an ideal way to implement risk analysis within the Excel environment that is typically used for forecasting.

Dr. Michael Rees
Director of Training and Consulting

Portfolio Optimization III: Efficient Frontier based on Huang Litzenberger

Posted Tuesday, May 5, 2009 by DMUU Training Team

In the second entry of this series, we mentioned the existence of an analytic method to calculate the efficient frontier of a portfolio. Here we provide the formulae for this method.

As for other methods to calculate the efficient frontier, this method requires knowledge of the variance-covariance matrix of the set of assets, as well as the expected return and standard deviation (volatility) of the asset returns. The variance-covariance matrix is of course itself generally calculated from the standard deviations of the returns of individual assets and the correlation matrix between those returns.

Denoting the vector of expected returns by e, the variance-covariance matrix by V, and its inverse by VINV, and letting U denote the unit vector (consisting of ones, with the same dimension as the number of assets), then the vectors of weights (g and h, below) have returns of 0% and 100% respectively and the optimal portfolio is the vector g+rh, for any given desired level of return, r.

(For simplicity of presentation in the above formula, where the transposes of vectors are required for the matrix multiplication, these have not been shown.)

Since analytic formula allow the easy implementation of sensitivity analysis, we show in the chart below the position of the optimal curve when it is assumed that equity returns are lower than historic figures case and with higher volatilities (the raw data is not shown here, but the portfolio is assumed to consist of bills, bonds, and equities).

As stated earlier, the advantages of such analytic methods include their computational efficiency, and the ease with which sensitivity analysis can be conducted (for example, that associated with parameter uncertainty). The main disadvantage is the inability to include constraints in the analysis.

In fact, there is a large set of real-life optimization problems for which analytic methods cannot readily be applied. Moreover, many problems in real-life cannot be adequately described from a risk-reward perspective in terms which involve only the expected return and standard deviation. For this reason, tools such as Solver, and Palisade’s Evolver (for genetic algorithm optimization in Microsoft Excel) and RISKOptimizer (Monte Carlo software with genetic algorithm optimization) may need to be applied in a large set of real-life cases.

Dr. Michael Rees
Director of Training and Consulting

Why Use Risk Analysis? Part V

Posted Friday, May 1, 2009 by DMUU Training Team

The final entry about reasons to perform risk analysis is really a summary in which we say that the process of risk modelling forces a more structured and rigorous model building and decision evaluation process. Such a process forces consideration of those factors that would lead to a change in an output (and hence require a understanding of the dynamics and relationships in a situation). It also makes explicit those areas where people may be making different assumptions, particularly since static models are much more arbitrary in their choice of variables to include (Event risks? Risk mitigation measures?) as well as in their choice of variables (which, in a static model, may not map with the true drivers of uncertainty in the situation).

As a result, risk models have many advantages, including:

They place any reference or base static case within the true range of possible outcomes, and hence are less biased.
They establish the average outcome (the average being a key reference quantity that is not known when building static models in general).
They allow consideration of output variability in decision-making; such considerations are required in non-financial decision-making, as well as when incorporating risk preferences into financial decision-making.
They transfer some decision-making (and modelling) responsibility toward the decision-maker (and not just the modeller); it is no longer sufficient or credible to blame the modeller when the true outcome is different to that which the model predict.
It allows for a wider set of variables (correctly) to be included in the mode.
Monte Carlo simulation allows modelling aspects, such as correlation and other dependencies, probabilities, and contingent claims can be modelled, whereas this is not generally possible in static modelling.

For more about decision making under uncertainty, see the previous posts in the "Why Use Risk Analysis?" series:
» Part I
» Part II
» Part III
» Part IV

Dr. Michael Rees
Director of Training and Consulting

Why Use Risk Analysis? Part IV

Posted Tuesday, April 28, 2009 by DMUU Training Team

In the first three entries of this series (see links below), we discussed the use of risk analysis to:

establish the average outcome, the average being a crucial reference quantity in decision-making in financial contexts (but a value that is not shown in a static model generally), and
establish the variability of the outcome, such variability being of high importance in both non-financial and financial decision making in practice, and
create a revised and more complete model including event risks, risk mitigation measures, and implied optimization of risk management.

Here we point out that risk modelling using Monte Carlo simulation can be used to capture aspects of modelling that are hard or impossible to capture using other techniques. For example:

Dependency relationships, such as correlated sampling and parameter-dependency between distributions
Simulation establishes the probabilities of outcomes (not just their possibility, as would a sensitivity analysis), it allows for the simultaneous variation of three or more risk factors (whereas Excel DataTables do not), and hence can deal with the large number of possible combinations for input variables.
Some situations are inherently stochastic, and cannot be modeled using sensitivity analysis; these include especially the modelling of contingent claims (e.g. options, real options, profit share agreements, incentive schemes penalty clauses) etc.

The DecisionTools Suite is a complete toolkit for decision making under uncertainty, including @RISK risk analysis software.

See the previous posts in the "Why Use Risk Analysis?" series:
» Part I
» Part II
» Part III

Dr. Michael Rees
Director of Training and Consulting

Portfolio Optimization II: An Overview of Markowitz-related Approaches

Posted Thursday, April 23, 2009 by DMUU Training Team

This entry follows on from an earlier one describing optimal portfolio selection for portfolios where the expected return and standard deviation are sufficient to describe the decision-makers’ risk profile (i.e. the criteria to be used in deciding what is meant by “optimal”).

There are several ways to calculate this optimal or “efficient” frontier:

Use Excel’s Solver. This would require the use of a matrix multiplication in Excel to determine, for any trial set of portfolio weights, the standard deviation of the resulting portfolio (taking into account the correlation matrix i.e. using matrix multiplication of the vector of standard deviation and the correlation matrix to determine the variance-covariance matrix). This will generate a single optimal portfolio for any given return (i.e. that portfolio with the minimum standard deviation).
Use Palisade’s Evolver (genetic algorithm optimization add-in for Microsoft Excel statistics). This is analogous to the use of Solver, requiring the use of a matrix multiplication in Excel and returning a single optimal portfolio for any given return (i.e. that portfolio with the minimum standard deviation).
Use Palisade’s RISKOptimizer (Monte Carlo simulation with genetic algorithm optimization software add-in for Excel). This requires essentially no use or knowledge of the mathematics of portfolio analysis (specifically of the matrix calculations to determine a variance-covariance matrix), because the software automatically deals with correlations between asset returns and calculates (by simulation) the standard deviation of any given portfolio. In this sense, it is the easiest method to implement, but one pays the price in terms of computation time.
Repeated application of the above methods. Each of the above methods will each generate a single optimal point (e.g. a specific portfolio with a minimum standard deviation, for a single assumed level of return), so the frontier itself needs to be calculated by repetitive calculation of several points. For example VBA code can be used to repeatedly run Solver (see my book Financial Modelling in Practice for an example).
Finally, the less well known analytic method developed by Huang and Litzenberger (H-L) can be applied. This allows the frontier to be determined analytically, which means that it is generally by far the quickest computation method. It essentially involves matrix inversion of the variance-covariance matrix followed by some straightforward vector multiplications. I will provide an example in a later posting. As well as being computationally quick, the method facilitates the exploration of parameter uncertainty, which is a topic barely addressed in much of the traditional literature; that is, to what extent is the composition of an optimal portfolio altered by the fact that historic returns and standard deviations themselves can only be measured within some confidence interval. Since H-L allows direct determination of the optimal portfolio, a sensitivity analysis (or full risk analysis) as parameter values vary is straightforward to conduct. The main disadvantage of this method is that it cannot be used when additional constraints wish to be applied to the situation, such as that all weights are positive with no short positions in the portfolio (whereas Solver, Evolver or RISKOptimizer can generally still be used when such constraints are to apply).

Dr. Michael Rees
Director of Training and Consulting

Portfolio Optimization I: An Introduction to Markowitz-related Approaches

Posted Monday, April 20, 2009 by DMUU Training Team

The topic of the selection and weighting of assets (or projects) associated with an optimal portfolio is a large and complex one. For example in general business applications, the determination of the optimal set of companies or subsidiaries that a holding company or conglomerate should own may depend on a large number of factors (including basic risk-return measures, but perhaps also the timing and risk of future cash flows from each business, exchange rate effects, the choice of reporting currency and so on).

In general, the need to reflect specific aspects of such situations (as well as the risk profile of decision-makers) may require the use of tools which readily allow many risk measures and heuristic methods to be easily implemented. Palisade’s Evolver (statistical analysis using genetic algorithm optimization in Microsoft Excel) and RISKOptimizer (Monte Carlo software with genetic algorithm optimization) belong to this set.

However, there is a special class of optimization problem which is standard in financial theory. This problem consists of the determination of an optimal portfolio of a set of financial assets, where the expected return and volatility, and correlation between the assets’ returns are known (or assumed).

As a reminder, the graphs that result for the return of a two asset portfolio with various asset weightings and correlation coefficients between them are shown in the chart below. In a multi-asset situation (more than two assets), theses lines cannot be directly calculated, but are determined as the result of an optimization calculation.

Note that a key (often unstated) assumption in this approach is that the decision-makers’ risk profile can be described by these and no other metrics, so that such lines do indeed describe what a decision-maker would consider to be optimal. (As far as I am aware, there is as such no requirement for an assumption that asset returns are normally distributed. Such an assumption is more restrictive, but which would then imply that the portfolio’s returns are also normally distributed, and so the mean and standard deviation of the portfolio can be used to determine any other relevant risk metrics).

In a subsequent entry, we provide an overview of methods to find this efficient frontier.

Dr. Michael Rees
Director of Training and Consulting

Why Use Risk Analysis? Part III

Posted Thursday, April 16, 2009 by DMUU Training Team

This blog is a continuation of the series (started a couple of months ago) which discusses different reasons to perform a risk analysis. In the first two blogs, I discussed the use of risk analysis to a) establish the average outcome, the average being a crucial reference quantity in decision-making in financial contexts, and b) establish the variability of the outcome, such variability being of high importance in both non-financial and financial decision making in practice. Risk analysis with Monte Carlo simulation can be accomplished directly in your Excel models with Palisade's @RISK software add-in.

Here I wish to make a third point: risk analysis will generally lead to a revised (and more correct) definition of the model. It will at the same time allow any base (reference) case to be presented within the context of the full range of possible outcomes.

Examples of where differences arise in the two modelling philosophies include: 1. A situation in which there is an event risk with say 40% probability of occurrence would include this risk variable in the risk model, but typically not in a static model (its occurrence not being the most likely outcome). 2. A risk model would generally contain risk mitigation measures that make sense to conduct when the consequences of risky outcomes are considered; however, such line items may not form part of the static analysis.

The variables in the risk model will be an augmented set compared to the static mode, and ultimately, a risk model represents the residual uncertainty once all measures deemed to be appropriate have been implemented; in this sense risk modelling is inherently a process of optimization.

For more about optimization with Monte Carlo software in Excel, see RISKOptimizer.

Dr. Michael Rees
Director of Training and Consulting

Is Norway’s Pension Fund Adequately Diversified? Part II

Posted Tuesday, April 7, 2009 by DMUU Training Team

In an earlier blog I allowed myself some raw speculation as to whether holistic risk management thinking is being adequately applied when it comes to the Norwegian government’s management of the state pension fund. This fund represents one of the world’s largest exercises in risk analysis in “retirement planning.” The Fund invests the oil wealth generated in the country in a mix of global equities and % bonds, and whose performance is essentially currently measured against a global benchmark portfolio of bonds and equities.

I specifically asked the question as to whether the fund should be devoting far more significant efforts to invest in non-traditional assets, as a way to mitigate some potential scenarios that could adversely affect both equity and bond investments. Investments that could potentially mitigate some of these scenarios could perhaps include very large positions in alternative energy technologies, and I noted however that although the costs and risks of such an investment strategy could be large (particularly as the scenario which it mitigates may never materialize), it could nevertheless be a prudent one, given the already very large fund that already exists for a small population base of about 5m people. Could it be so bad if 5%-10% of that fund were invested in such technologies (with such investments arguably supporting some of the fund’s other goals – such as ethical or social investments)?

It is therefore with interest that the Financial Times reported last Saturday that the Norwegian government is planning to review the operations of its sovereign wealth fund after it lost €75bn ($100bn, £68bn) on investments last year. I await eagerly the results of this review, specifically of course to see whether such fundamentally new investment strategies will be implemented.

Dr. Michael Rees
Director of Training and Consulting

Best Practices in Risk Modelling

Posted Wednesday, April 1, 2009 by DMUU Training Team

The recent blog positing on best practices in Excel modelling could be thought of as providing a reasonable and robust set of principles for building static Excel models. When building simulation models for risk analysis in Excel (for instance, with @RISK Monte Carlo software), some other points are worthy of consideration:

A risk model may need to be built at an appropriate level of detail. A model which is too detailed will be more complex to add risk distributions to and will require more effort to capture the dependencies between the larges number of variables. In many practical cases, key dependencies will simply not be captured, and the result will have an excessively narrow range (for additive type models e.g. cost budgeting) or an excessively wide range (for subtractive models e.g. profit as the difference between uncertain revenues and costs).
The inclusion or not of event risks. Generically, a static model of a situation in which there are event risks (e.g. something adverse happening in 20% of cases in a reserve estimation model) would not include such a risk as a line item (since the most likely outcome is its non-occurrence), whereas a risk model would.
The prioritisation of event risks to include may be non trivial, and depend on the decision maker's risk profile (i.e. tolerance and decision-making criteria), as well as the potential total number of event risks under consideration. For example, in a retirement planning model where a decision is to be based on the P90 outcome (i.e. the worst or best 10% of cases) it would be more important to include an event with an impact of 1 (with 100% probability) than an event with impact of 100 with 1% probability, as this latter event in isolation would have very little effect on the P90 of the output distribution (Were decisions to be made on the P99.5 value, we would have a different situation, of course.).
The use of DataTables will generally slow down simulation models, as the tables need to be recalculated at every iteration. DataTables may be used when building to model and as an error-checking tool (TopRank may also be used, to check that errors are zero across a range of scenarios), but may need to be removed before running the final simulation model.
The real challenges in risk modelling boil down to those related to model formulation and decision-making; that is: the selection of variables, capturing the true dynamic of the situation in the model, choice of distributions, capturing dependencies correctly etc. (so come to a training course!)
Other principles of model implementation (as discussed in the earlier blog) are essentially identical.

Dr. Michael Rees
Director of Training and Consulting

Further Successes with DecisionTools Suite training in Europe

Posted Wednesday, March 25, 2009 by DMUU Training Team

Following the introduction of the DecisionTools Suite 5.0 in 2008, we modified the structure of the public training courses, which consist of a two-day @RISK course and a consecutive one day course on the other DTS products. Although this has required a major commitment from customers in terms of the time away from their office, it appears that the decision to structure the course in this way has worked well in Europe. First, it does allow customers to explicitly choose whether they wish to learn only @RISK, or all the other products, or only the non-@RISK products. Second, there are certainly enough problem solving applications that can be addressed with the software to use the time productively in a value-added way. Third, we generally run the courses on a Tuesday through Thursday, so that people can be in their offices at the beginning and end of the week. By staying in touch with their offices during the course (e.g. using the e-mail facilities provided to customers for our courses based in the London office), most participants are able to take the time out without excessive disruption to their work. All the customers that I spoke to that have attended all three days have reinforced the feeling that this three-day approach is most appropriate, value-added, and allows customers to sign up for only those parts of the course that they feel most useful, whether it be pure @RISK, the non-@RISK DTS products, or the full Suite.

More details of the courses can be found on the Palisade web-site. As a reminder of the contents of the non-@RISK products (day 3), we cover TopRank as a tool to audit and conduct sensitivity analysis on general Excel models, PrecisionTree to build decision trees for making decisions under uncertainty. We then move on to optimization modelling using genetic algorithm optimization methods of Evolver and RISKOptimizer. Finally, we use StatTools to conduct a variety of statistical analysis and NeuralTools to perform predictive analysis based on neural network logic. As always, the course is built around practical examples and applications.

View the training schedules:

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Dr. Michael Rees
Director of Training and Consulting

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