Intuitive Bayesian methods for portfolio selection – Part II Bayes and Jeffrey

Bayes’ theorem in its common form describes the way in which one’s beliefs about observing ‘A’ are updated by having observed ‘B’. Bayes’ theorem relates the conditional and marginal probabilities of events A and B, where B has a non-vanishing probability.

Each term in Bayes’ theorem has a conventional name:

P(A) is the prior probability or marginal probability of A. It is “prior” in the sense that it does not take into account any information about B.

P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.

P(B|A) is the conditional probability of B given A.

P(B) is the prior or marginal probability of B, and acts as a normalizing constant.

Bayesian belief updating is the model we use for learning. We in effect already use it when we sit in meetings, discussing best options, as we will have individually modified belief over time as we receive new information – the problem is that it is difficult for others to see what evidence corroborates this belief, which opens up the door for our cognitive biases and simple heuristics.

Jeffrey’s Rule

During product selection and development we acquire and learn new information, which allows us to update our belief about how to make future investment. However, we know that some information is of a higher quality e.g. let’s say two people make exactly the same statement; one is a lead customer and the other is a stranger on the street, we know which is of a higher quality with a higher information content. Bayes’ rule relies on learning a definitive new truth to revise our belief. Most new knowledge we acquire during product development cannot be classed as definitively true e.g. one customer may say one thing and another may say something totally different. Jeffrey’ rule allows us to deal with opinion, rumor and weakly supporting evidence.

We can formulate a partition of hypothesis Ho and ~Ho

Ho = We will sell 10 products to customer x this year

We are at a trade show talking to a distributor who tells us he has heard that customer x is currently trialing our competitors products. We will call this new piece of evidence E

E = Customer x is currently trialing our competitors products

Before we had heard this we may have been quite bullish about the prospects of selling to customer x because we have had several meetings where they expressed interest and have been talking about using some demo equipment.

Pr(Ho)=0.8

However if it is true that customer x is currently trialing the competitor products then I figure that is bad news as they need to commit resource to testing and are further down the line with our competitors.

Pr(Ho/E)=0.1

If what I’ve heard is not true then I have no other reason to revise my prior belief

Pr(Ho/~E)=0.8

I represent my belief in light of the new rumor as Pr*, so that Pr*(Ho) stands for my belief in Ho in light of the new information E.

When talking to the distributor he can’t remember who he heard it from but is pretty sure that he is right. I might assign a probability that the information is right to 0.75.

Pr*(E)=0.75    Pr*(~E)=0.25

Jeffrey’s revision of Bayes’ rule is reminiscent of the rule for total probability

Pr*(Ho) =Pr(Ho/E)Pr*(E)+Pr(Ho/~E)Pr*(~E)

Jeffrey tells us to conclude that Pr*(Ho)=0.275. Before we heard the rumor we thought it was quite probable that we would sell to customer x, but things are looking a bit more bleak.

Dashboard representation

We can put together a dashboard that allows a user to start with a prior belief and update using Jeffrey’s rule. Two sliders are used to input Pr(Ho/E) and Pr*(E). The numeric inputs are augmented with descriptive labels.

Examples

If we receive a new piece of information that definitively refutes our hypothesis, but we know the source is completely unreliable then we would have no reason to update our belief e.g. if a stranger in the street says he wouldn’t buy our chemical detection equipment, this has no relevance or impact on my belief that the US Army will.

If we receive a new piece of information that we know is definitely true but is doesn’t add much to support our hypothesis then our posterior belief will be unchanged. For example, two people from one company tell me a piece of information separately. When I hear it from the first person I update my belief accordingly, when I hear it for the second time is gives me no new knowledge even though I believe the source completely.

Potential problems with the application of Jeffrey’s rule

Prior Belief

We can look at what happens if we start out with very different prior beliefs. If we are rationally updating with new evidence and agree on the impact and quality we should eventually converge on a common belief.

Evidence	Pr(Ho/~E)	Pr(Ho/E)	Pr*(E)	Updated
0	1.00	0.16	0.23	0.81
1	0.81	0.11	0.44	0.50
2	0.50	0.64	0.51	0.57
3	0.57	0.90	0.69	0.80
4	0.80	0.16	0.04	0.78
5	0.78	0.74	0.38	0.76
6	0.76	0.62	0.40	0.71

Table 1 Change in belief from a starting belief of 1

Evidence	Pr(Ho/~E)	Pr(Ho/E)	Pr*(E)	Updated
0	0.00	0.16	0.23	0.04
1	0.04	0.11	0.44	0.07
2	0.07	0.64	0.51	0.36
3	0.36	0.90	0.69	0.74
4	0.74	0.16	0.04	0.72
5	0.72	0.74	0.38	0.73
6	0.73	0.62	0.40	0.68

Table 2 change in belief from a starting belief of 0

The tables above and graph below illustrate the sequential application of Jeffrey’s rule. We start with differing prior beliefs and as new evidence arrives we update our belief. The dataset for Pr(Ho/E) and Pr*(E) are randomly generated number between 0 and 1. We can see that after 3-4 pieces of evidence we are starting to converge on a common belief. While not rigorous, inspection of simulated cases supports the idea that beliefs will converge irrespective of the staring belief.

Applying the principle of insufficient reason to prior belief

What happens if we start with no evidence at all for a hypothesis? We may be inclined to say that there is nothing to choose between the alternatives, true or false, so they should be treated as equally probable- this is the principle of insufficient reason or the principle of indifference. However we can look at a simple example; I state a hypothesis, “your car is red”. Initially without any evidence it doesn’t seem that the partition “your car is red” and “your car is not red” would have an equal probability.

In most business examples I can think of it is usually more likely for a specific hypothesis to be false; “this product will be successful” vs “this product will fail”. There are usually many more ways to fail than to be successful. We may be happy to assign a personal probability to the prior belief as opposed to assuming indifference. However this may allow certain hypothesis an ‘easy ride’ without forcing us to find evidence to corroborate or falsify. I prefer to operate the maxim ‘guilty until proven innocent’; assume the hypothesis is false until proven otherwise. This forces me to find evidence so I can justify my belief position – just because I think it is obvious that something is true doesn’t mean that others do. If I already have a high prior belief it should be easy for me to find the supporting evidence. This also means that I will be operating conservatively in the early stages as my belief is ‘dragged down’ by the memory of initial belief up to the point of convergence.

Order of discovery

It would also seem intuitively obvious that the order in which we uncover new evidence should make no difference to our eventual beliefs. We have generated 20 discrete pieces of evidence and updated belief at each stage. We have then reordered the evidence (re-sampling without replacement) and calculate the new belief trajectory. Interestingly we can have marked differences in belief at the end of the process. The results are presented without further discussion, but this may pose a significant problem in the application of this belief updating methodology.

The above re-sampling example assumes that we would actually assign the same ‘marginal belief change’ irrespective of the order of discovery. This may not be a valid assumption and we can look at an example from history. In 1818 Siméon Poisson deduced from Augustin Fresnel’s theory the necessity of a bright spot at the centre of the shadow of a circular opaque obstacle. With his counterintuitive result Poisson hoped to disprove the wave theory; however Dominique Arago experimentally verified the prediction and today the demonstration goes by the name “Poisson’s (or Arago’s) spot.” Since the spot occurs within the geometrical shadow, no particle theory of light could account for it, and its discovery in fact provided weighty evidence for the wave nature of light, much to Poisson’s chagrin. If I believed in the corpuscular theory of light I would be extremely surprised to see a Poisson spot. However once I have seen it and adjusted my belief accordingly, seeing it again would only have a very small impact on my belief; the new experiment contains very little information. This is the same as saying that the marginal belief change for a particular piece of evidence depends on my current belief and the history of how I arrived here. It doesn’t therefore seem valid to resample, as we deal with marginal change in belief, not absolute values as new evidence arrives.

Intuitive Bayesian methods for portfolio selection – Part I Background

Introduction

Disruptive platform technologies usually have a broad base of application. During early stage development, before there is a developed market, the selection of a particular product is usually a ‘high risk, low data’ decision. There are a large number of unknowns, both the known unknowns and the unknown unknowns; we seek the resolve these over time. In this type of situation it is difficult to make the initial portfolio selection decision and to effectively monitor the resolution of uncertainty, and determine the ultimate ‘chance of success’ for the product.

Problems in portfolio selection and project monitoring

The portfolio selection process, even when highly structured, often reduces to persuasion by advocates and champions. When a lot of data is being presented it is easy to forget ‘how we arrived’ at a particular position, assigning a higher importance to things that we heard recently (or long ago, depending on how your mind works). Soaring rhetoric can outweigh sober analysis and dispassionate appraisal of risk. It can be difficult to judge the ‘quality’ of a piece of information, which may find itself as a lynchpin in an argument to take a particular course of action. With a lot of unknowns it can be difficult to formulate go/no-go metrics and not relax the criteria when you get to the decision point.

Cognitive biases

The field of behavior economics examines some of the less rational beliefs of Homo economicus. Work by Tversky and Kahneman illustrate cases of overconfidence in our abilities, the desire to go with the herd and a propensity for rolling rationalization. Here is a list of cognitive biases that you can easily imagine arise in portfolio selection processes.

Objectives

1. Develop a simple methodology and toolset that allows us to :-
   
2. Reduce complex business decisions to specific and testable hypothesis, which can be definitively refuted.
   
3. Systematically revise our ‘belief’ in a hypothesis as we receive new information.
   
4. Integrate new information of many types and forms, of varying degrees of ‘quality’.
   
5. Maintain a history of how we arrived at a particular belief to provide an ‘audit trial’ or ‘memory’ to support future decisions and actions.
   
6. Integrate and logically connect hypothesis to create a ‘belief network’ that supports complex decision making.
   
7. Avoid cognitive biases and increase objectivity
   

Logic and Probability

There are three main modes of argument, deduction, induction and abduction (inference to best explanation IBE). Inductive logic analyses risky arguments using probability ideas. There are however different interpretations of what ‘a probability is’.

Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment’s outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency “in the long run” of outcomes.

Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual’s degree of belief in a statement, given the evidence.

Logical Probability is thought of as a logical relation between a hypothesis and the evidence for it. J.M. Keynes and Rudolf Carnap both favored a logical theory of probability. Personal probabilities are a private matter, they are up to the individual and anything goes so long as be basic rules of coherency are obeyed. Logical probability maintains that there are uniquely correct, uniquely rational judgments of the probability of a hypothesis in the light of evidence.

For the purposes of decision making in a business context there are very few cases where a Frequentists approach can be used. We tend to use the Bayesian notion of probability where belief allows us to make investment decisions.

It is plausible to connect personal degrees of belief and personal betting rates

You would not pay more than $1 to win $2 on the flip of a coin. If you have some domain specific business knowledge that allows you to exploit an opportunity, your betting rate would be markedly different from someone without that knowledge. During product development as uncertainty is resolved our beliefs are updated and we revise the level of investment we are willing to make. People have always used this ‘managerial flexibility’ and there is now a move to formalize this type of ‘real option’ thinking in investment and portfolio selection.

Verificationism and Falsifiability

There are two common problems in portfolio decision making, how do we extrapolate experience to the future? And how can we provide definitive go/no-go criteria when we do not know the problem well? The former is the problem of induction, and is the question of whether inductive reasoning leads to truth. That is, what is the justification for presupposing that a sequence of events in the future will occur as it always has in the past (for example, that the laws of physics will hold as they have always been observed to hold). If we cannot assume uniformity of nature for physical laws we definitely cannot do so in a business context where we know that the landscape changes very quickly.

Often a go/no-go criteria is framed in a way that allows it to get out of jail down the line. A criteria such as, “show interest from a customer” is quite broad. If in a month’s time if we hear a statement “Fred and Jeff seem quite interested”, this adds practically no new useful knowledge upon which to base a decision – “A difference that makes no difference is no difference”. It also allows us to introduce an ad hoc revisions to ‘pass’ the criteria. If we set criteria such as “one sale made by the end of the quarter”, then we have something that is definitively testable. This is a criterion that puts itself at risk, which can be refuted or falsified – falsification adds new knowledge as it allows us to eliminate options and make definite investment decisions i.e. don’t invest. Falsifiability was put forward as solution to the problem of induction by Karl Popper.

This is related to the Logical Positivist view of the verifiability theory of meaning: the meaning of a sentence consists in its method of verification. In other words, if a sentence or statement has no possible method of verification, it has no meaning. It is pointless to make a go/no-go goal such as, “demonstrate our value proposition and facilitate end to end knowledge transfer”, as there is no possible way to test this and it therefore falls into the category of a nonsensical statement (also known as bullshit bingo).

Intuitive Bass Diffusion

There is an excellent post by Mathias, which rephrases the Bass diffusion curve in more accessible terms and language. You can view his posting for the full explanation. I’ve taken the Excel file and created an interactive dashboard from it.

Click here to launch the Dashboard

Interactive Gompertz Model

In high tech start-ups the development cycle can last for a period of several years. We can capture new product introduction where the pre-revenue start-up phase is anticipated to be long using a Gompertz curve. There is a full description on Wikipedia.

Sales Function

Cumulative Sales Function

m	500	Ultimate market potential (m)
b	0.4	Scale Parameter (b)
η	30	Shape Parameter (n)

Click here to launch Interactive Gompertz Model

The Xcelcius dashboard allows users to interactively vary the parameters of the model. This is useful when doing ‘what if’ analysis during product portfolio planning stage. The sales profile is more realistic and can be embedded into the interactive portfolio or we can create a Monte Carlo income statement with distributions to describe uncertainty.

Interactive Net Present Value (NPV)

Net present value (NPV) is defined as the present value of net cash flows. It is a standard method for using the time value of money to appraise long-term projects. See full description at Wikipedia.

Each cash inflow/outflow is discounted back to its present value (PV). Then they are summed. Therefore

Where

t – the time of the cash flow

N – the total time of the project

r – the discount rate (the rate of return that could be earned on an investment in the financial markets with similar risk)

C_t – the net cash flow (the amount of cash) at time t

Interactive NPV Calculation – Click here to launch NPV chart

The following dashboard allows you to change the discount rate and see the time value of a dollar over a period of 5 years.

At a 10% discount rate a dollar in year 5 is worth 62 cents in today’s money

At a 50% discount rate a dollar in year 5 is worth 13 cents in today’s money

We can also demonstrate an arbitrary project with an initial cash outlay and an increasing yearly income stream

At a higher discount rate of 50% the future income is heavily discounted and the overall project NPV is significantly reduced.

Index of Interactive Dashboards

Interactive Portfolio Model – Posting / Dashboard

Product Requirements Capture and Competitive Benchmarking – Posting / Dashboard

Economic Value Model – Posting / Dashboard

Bass Diffusion Model – Posting / Dashboard

Value at Risk – Posting / Dashboard

Net Present Value – Posting / Dashboard

Gompertz Model – Posting / Dashboard

Intuitive Bass Diffusion – Posting / Dashboard

 

Interactive Bass Diffusion Model

Click here to launch Interactive Bass Diffusion Model

The Bass diffusion model was developed by Frank Bass and describes the process how new products get adopted as an interaction between users and potential users. The model is widely used in forecasting, especially product forecasting and technology forecasting. Click here for the full description on Wikipedia.

The function describing Sales S(t) is given by

m	500	Ultimate market potential
p	0.02	Coefficient of imitation
q	0.4	Coefficient of innovation

Click here to launch Interactive Bass Diffusion Model

The Xcelcius dashboard allows users to interactively vary the parameters of the diffusion model. This is useful when doing ‘what if’ analysis during product portfolio planning stage. The sales profile is more realistic and can be embedded into the interactive portfolio or we can create a Monte Carlo income statement with distributions to describe uncertainty in ultimate market potential (m), coefficient of imitation (p) and coefficient of innovation (q).

Market Requirement and Product Fit

In a high value technology product there are typical three types of stakeholder that have separate concerns who each ’speak a different language’. These types of buyer are highlighted in Crossing the Chasm. The economic buyer, technical buyer and user.

When building a sales proposition and message it is common to make the mistake of trying to force your perception of benefits, which may not be shared by your customer. It is also easy to convolve messages, which may not be relevant to the buyer you are talking to, for example, focusing on price or ROI with a technical buyer.

We want to deliberately and separately address the concerns of the technical buyer and the economic buyer. We can use an interactive dashboard to capture market requirements and support the sales process from the technical buyer perspective.

Product Requirements Capture and Competitive Benchmarking – Click here to launch the product dashboard

Instead of pitching a solution to a perceived problem we can work with a technical buyer to capture their ‘total’ requirement and the key points of pain. The feature or performance space has many dimensions, for example, size, weight, speed etc. Different requirements will also have different levels of importance, some are essential and some are ‘nice to haves’. With the dashboard the technical buyer can work though the feature list and select their target requirement and associated importance. They build up a map of the total requirement, which is then plotted on a spider diagram. They can drill down in a gap analysis chart to compare specific features across multiple products.

Target real needs in sales process – Instead of ranting about ‘guessed’ requirements, you allow the customer to fully articulate their need. There will be fewer objections as you spend more time understanding the problem instead of pitching a boxed solution. As they work through the requirements and weighting you can engage them to better understand the reasoning and problem implications. For example, if speed is an essential requirement you may drill down to find that the implication of delay is under utilisation of another costly resource etc. This implication knowledge will be valuable in future sales where there are common problem sets.

Map the need to your product – There is no algorithm for sales! However if the customer builds up the map of total requirement and there is a good match with your product this can be compelling when making the decision. In addition the document is permanent; often you can bounce between objections on specific features, with this representation you keep a global focus. The technical buyer can forward this dashboard to colleagues and they will have an audit trail of the decision process.

Competitive Benchmarking – once the requirement has been mapped you can compare it not only to your own offering but also competitive products. Ideally you are demonstrating significant and compelling differentiation. Even if other products are comparable in meeting the requirement you are acting more in the role of an honest information broker as opposed to just hawking goods by saying anything. This may be valuable in building trust in the customer relationship, which is often a larger determining factor in the sales process.

Interactive and Engaging – Most people have sat through interminable power point pitches. The dashboard changes the process from a passive information dump to an interactive conversation. Also dashboards are new and different; people find them fun and ‘every little helps’ when you are trying to get the product and company noticed in the crowd.

Market Analysis – In addition to using a dashboard in a sales context they can also be used when doing market analysis. They are a lot more interesting than a questionnaire. I’ve previously described two principle uncertainties in new product introduction 1)’Likelihood of market entry’- a binary event; whether we can enter the market at all with a product and subsequently 2) ‘Market penetration and growth rate’ –the income over time for a product. The risks can be split into further levels of granularity. The requirements capture can be used to make a detailed assessment of likelihood of market entry. If your product has a very poor fit with requirement then the ability to enter the market at all is greatly diminished. For example if the ‘fit metric’ was 20% we could use this directly in our Monte Carlo portfolio analysis as the threshold for market entry. You can also spot requirement trends that can have a strong influence on the direction of your technology roadmap.

Interactive Economic Value Models – Click here to launch the value model

The language of an economic buyer is very different from a technical buyer. They use words like, cost savings, ROI, payback period, utilisation, efficiency, payment terms etc. Again we often fall into the trap of guessing points of pain instead of listening to the customer who actually feels them. We can use interactive dashboards to capture costs and metrics on the fly and automatically calculate saving, ROI etc.

We don’t talk about technical advantages at this stage; we assume we have addressed the needs of technical buyer at the level of requirements capture and we are now speaking to the person who writes the cheques (in reality we probably speak to both in a parallel fashion).

The example value model above was developed for the Lonestar chemical process monitor. A factory processing an arbitrary number of samples will have a proportion of faulty goods. If these are not detected then the faulty goods can be shipped to customers. The question is whether we can save money by employing a detection system to identify the out of spec goods. The user can input the number of samples they process and how many defects occur. They can then input the associated costs, for example if a batch of out of spec goods was shipped then there may be a direct monetary cost in refunding / paying compensation as well as potential for significant brand damage. If there are false alarms then we may end up scraping product which was actually good. The model takes these inputs and calculates the cost of not using detection vs the cost of using a detection system. They get metrics including ROI and payback period.

 

 

Monte Carlo Portfolio Optimisation

Download the excel model here

The portfolio representation described previously included project costs and returns that were purely deterministic. In reality there were be large uncertainties in both, for example a 6 month project overrun will increase overall project costs and is also likely to reduce returns. We can use Monte Carlo analysis to model the effect of these uncertainties on the discrete portfolio options.

A real model can have different layers of granularity and detail, specific to the individual projects. For example, uncertain returns will be dependent on volumes of sales over time, pricing, competition etc. To illustrate the principle we use a simple model where the individual project are independent and returns and costs are triangular distributions in which we specify the minimum, maximum and average.

Project		Return			Cost
-	Min	Average	Max	Min	Average	Max
A	$5.00	$8.00	$9.00	$3.80	$4.00	$6.00
B	$0.50	$2.50	$5.00	$0.80	$1.10	$1.50
C	$1.00	$4.00	$4.80	$1.00	$1.30	$2.00
D	$1.00	$3.00	$6.00	$0.70	$0.75	$1.30
E	$1.50	$2.00	$3.00	$1.00	$1.25	$2.00
F	$0.30	$1.60	$1.90	$1.70	$2.00	$2.40

Simulating Uncertain Portfolio Returns and Costs

When we run a simulation we resample from the distributions and realise a particular outcome. Below are two simulated instances; we can run a full simulation of thousands of samples with a package such as @Risk or the Solver risk optimiser.

In this instance Project A has a return of $7.15M and cost of $5.24M, Project D had a return of $3.95M and cost of $0.78M; they are independent so the portfolio selection of Project A and D has an overall return of $11.4M and cost of $6.02M. This turns out to be the optimum portfolio for this level of investment.

In this instance Project A has a return of $7.36M and cost of $3.91M, Project D had a return of $2.41M and cost of $0.94M; they are independent so the portfolio selection of Project A and D has an overall return of $9.77 and cost of $4.85M. In this instance the combination of A and D is no longer the optimum portfolio for this level of investment.

What becomes clear is that when we add uncertainty there will not necessarily be a unique optimum that we can select. The optimum portfolio can be different but we don’t know how the uncertainties will be resolved in advance.

Visualisation of Portfolio Uncertainties

When we run the simulation each discrete portfolio point will have an associated cost and return distribution. We can add this distribution information to the portfolio representation by adding errors bars. The y error bars are the 95% limits of the distribution of return and the x error bars are the 95% limits of the distribution of cost. We now have representation that embeds risk, which has a very high information density to help us make optimum portfolio decisions with regard to return, cost and risk.

Optimisation of an Uncertain Portfolio

MinMax Rules

The uninitiated executive may have no knowledge of a probability distribution or understand how to use it. What are more common are questions like, “well what is the worst case?”, “what happens if the costs are high and the return is lower than we expect?” We can add extra interactivity to the dash board to allow the executive to explore these possibilities and see how that may affect the optimum portfolio selection as a function of their own appetite for risk.

The first representation is what we have already seen previously; we plot the average cost and return and select the optimum as normal.

We now re-plot the portfolio to show the worst case of maximum cost and minimum return. In most cases there would be a degree of interdependency in cost and return and we may have to re-simulate to obtain this plot. One approach would be to extract the 5% and 95% percentiles for return and cost respectively.

The best case would be the maximum return for minimum cost.

Between these extremes we can continuously vary the plot of return and cost. It is also very straightforward to add likelihood figures to the plots to show the executive that the extreme values are less likely than the average values; they are getting all the information about the underlying distribution but the representation and output is more accessible.

What is apparent is that the pareto efficient frontier will change. The optimum portfolio will be different for those who have a differing preference for risk. Some of the portfolio options may have a very large upside but also a large downside; if I was risk averse and was operating a max cost minimum return decision rule this portfolio option would be plotted in the lower region of the chart and would not be an optimum choice for me given the actual portfolio and my risk appetite. The reverse would be true if I was risk seeking. We have a representation that allows us to revise the optimal profile as a function of risk behaviour.

Semantic Zooming

Each portfolio combination point has an associated cost and return distribution. Points in the lower region of the plot are clearly suboptimal, the returns are low and costs are high, however with points close to each other on Pareto frontier it can sometimes be difficult to decide which one is better given their distribution profiles.

We can use a drill down or semantic zoom to look at distributions of adjacent points on the efficient frontier.

There are a few interesting situations where it is difficult to decide the optimum; typically the resolution boils down to a question of appetite for risk. In the example below the portfolio combinations A and B are very similar; the costs are comparable and the average returns are almost the same. However when we drill down to the return distribution we can see that even though the mean is similar there is a big difference in range of possible returns. Portfolio A has both a higher upside and lower downside; if I was risk averse I may prefer portfolio B, which has a slightly lower return but with more chance of hitting.

Interactive Value at Risk (VAR) chart

The VAR chart is useful in the sense that it describes values such as NPV a distribution as opposed to a single point estimate. No amount of modelling will ever fully anticipate what lady fortune has in store but it is still worth trying to gain a deeper understanding of risk drivers so you can make better decisions or create managerial options that help to exploit future uncertainty.

In the vein of trying to make the concept generally accessible to the uninitiated, busy executive we can add some useful refinements. The general shape of the chart itself conveys some information quickly e.g. if it is bimodal.

Personally I have these specific problems with the representation

1. A cumulative plot is not as intuitive as a probability density function – especially if the distribution is bimodal or more complex (in a cumulative plot people look at the flat region of constant probability and get confused, whereas when you see two humps in a pdf it is obvious that some values are not allowed)
   
2. When I want to extract numeric values (in a presentation when showing power point slides) it is quite tiresome to try and read off the chart axis every time; especially if it is interval data
   
3. Not everyone is familiar with likelihood or probability but odds on they’ll be familiar with the concept of odds.
   

Interactive VAR Chart – Click here to launch VAR chart

The output of a Monte Carlo model will be table of numbers describing the distribution. We can easily take these values and create an interactive VAR chart dashboard.

1. The user still gains the same insight from the overall shape
   
2. They can control a lower limit slider that allows them to look at the downside part of the distribution to answer questions such as “what are the odds we’ll lose money overall on this project”
   
3. A second slider allows them to get upside and interval information e.g. “what are the odds the NPV will be between $5m and $10m”.
   

Some of the newer Monte Carlo packages (Risk Solver at www.solver.com) allow you to generate these charts in near real time; however with Xcelcius I can embed this chart into power point and pdf documents and distribute the model.