Monte Carlo Method for Risk and Contingency Analysis
It is important to note that a project cost contingency is not "fat" that is added to the project budget. The contingency is an allowance that is added for the items that are known to be required, but whose exact costs are uncertain at the time of preparing the estimate. The uncertainty could be because of incomplete engineering and planning, lack of time to get definitive pricing, minor errors and omissions and minor changes within the scope. It is not to cover for gross incompetence, scope changes and "nice-to-haves". The contingency, or at least a major part of it, is expected to be spent in the execution of the project.
Before the project is started it is necessary for management to judge its financial feasibility, and to do this the final costs must be known reasonably accurately. Including the contingency in the costs allows the profitability calculations to be based on costs that can be realistically expected to occur. If the contingency is made too small the project will overrun, and if it is made too large it may decrease the profitability to the point where a potentially profitable project is abandoned.
There are several methods used to estimate the required cost contingency. At the simplest level there is the "10% rule", where the estimate is simply increased by 10%. This disregards the realities of the situation and is not a recommended procedure. A slightly better method is the "expert judgement" method. This relies on the experience and knowledge of the expert and remains subjective and in truth just provides a "gut feel" amount for the contingency.
The better methods rely on estimating the risks involved for sub-sections of the project, and then using some method to combine the risks for the individual sections to get the risk and required contingency for the overall project. This uses the same divide-and-conquer procedure that is used to generate the overall cost, i.e. divide the project up into manageable sections and estimate the costs piece by piece and then combine them. There are various heuristic methods that have been developed by different institutions, but the most widely accepted and fundamentally sound method of combining the individual risks is Monte Carlo simulation.
This article is aimed at describing how to use Monte Carlo simulation to estimate the required cost contingency and only the briefest description will be given here of its inner workings. For those who are interested to know more, there are detailed descriptions and tutorials included in the Help system for the Project Risk Analysis program. A free trial version, including the Help system and example data, can be downloaded by clicking the Download option on the menu above.
In summary, the project is broken down into sub-sections and the cost of each sub-section is described by specifying the range of costs and their distribution over that range. This procedure is described in more detail in the next section. The individual ranges and distributions then have to be combined to give the overall range and distribution for the complete project. The distributions for the individual items give the worst case scenario (as well as the best) for each item but it would be overly conservative to conclude that the worst case for the overall project would be the sum of the individual worst cases. This is because it is extremely unlikely that all the individual items would simultaneously be at their worst cases.
Instead, the project is simulated by running it in the Monte Carlo model which allocates a cost for each section in accordance with the specified range and distribution for that section, and then adds together the costs for each of the sections to get the total project cost. This procedure is repeated many times (typically 10,000 or more) and in each iteration new individual costs are allocated in accordance with the specified range and distribution for that section. This results in slightly different total costs for each iteration.
When all the iterations are complete the total costs from each iteration are plotted on a histogram to give the range and distribution for the total project cost. The distribution of costs for the total project gives a sound basis for estimating the cost contingency required. This will be illustrated with an example later in this article.
The first step is to divide the project up into manageable sections. This is done on the basis of independence and impact. Independence means that the cost of an item can move without other costs automatically moving. It is difficult to obtain total independence between all combinations. For example, piping and vessels are usually considered separately, but both would be affected if there was a shift in the basic material price. So we have to take a practical view of independence and accept some inter-relationships.
Impact means that we should not waste time trying to estimate cost ranges for very small items if the maximum possible variation in the price of that item would have no discernible impact on the overall project cost. The impact is a function of both the cost of the item and its variability. As a rough guide, if the potential variation in the cost of an item is less than 0.3% of the total project cost it can be lumped in with other similar items.
When broken down in this way a project will typically have between 15 and 40 line items. If you use more than this you will run into two problems. Firstly it is unlikely that you will have so many genuinely independent items and you will artificially narrow the risk profile (this is discussed in the Help system referred to in the previous section). Secondly you will be wasting management resources trying to estimate cost ranges for insignificant items. In Risk Management and Cost Control the Pareto Principle (often called the 80/20 Principle) is very important. Look after the high impact items and deal with the rest by exception.
For each cost item we start by estimating the Most Likely Cost. In all project cost estimates at least some effort is made to attach an accuracy to the estimate. A good way to formalize this is to introduce the concept of "range estimating". In using this technique, rather than simply quoting the Most Likely Cost a bit more information is recorded and a Low Cost (or Minimum Cost) and a High Cost (or Maximum Cost) are also noted. The final bit of information that is required is to note how the costs are distributed between the Low Cost and the High Cost.
While there is a whole branch of statistics devoted to probability distributions, we will keep it simple here and illustrate the process with the triangular distribution illustrated in Figure 1 below. Other useful distributions are described in the Help system previously referenced.
In a triangular distribution there is zero probability of the cost being lower than the defined Low Cost. The probability increases linearly from zero at the Low Cost up to a maximum probability at the Most Likely Cost. And then the probability decreases linearly down to zero again at the High Cost.
In this example the Most Likely Cost is $10,000. For the purposes of this example we will say that there is no chance that the cost could be more than 10% below the Most Likely Cost, and we define the Low Cost to be $9,000. In most cases there is more chance of over-spending than of under-spending so we will follow our intuition and say there is no chance of the cost being more than 50% greater than the Most Likely Cost. This makes the High Cost $15,000.
In addition to the triangular distribution, the normal and lognormal distributions are very useful in cost estimating. For more detail see the Project Risk Analysis Help system previously referred to.
This process of defining the Most Likely Cost, the Low Cost, the High Cost and the distribution is repeated for each of the sub-sections of the estimate. The best way to estimate the cost ranges is by examining the track record of your own estimating and project management functions. At the end of a project the actual costs should always be compared with the original estimates and any variations should be explained and note taken of the lessons learned. Variations of 35% on a single category are not uncommon in real projects.
The next best way to set the ranges is to ask those who did the estimates to also estimate the accuracies. Unfortunately, what often happens at finalization meetings where the estimate is reviewed by all concerned, is that those responsible for the individual estimates become very defensive of their own work. It should always be pointed out at the beginning of these meetings that the accuracy of the estimate calculations is only one of the factors that determines the risk involved. Factors such as scope definition, engineering progress and development, price variations, weather conditions and other factors beyond the control of the estimator can have a much larger impact on the risk. The risks due to all of these items have to be considered and it must be made clear that the purpose of the meeting is not to attack any estimator or his work.
A typical project is shown in Figure 2 below. Note that the Expected Cost in the last column has been calculated by the program and is not a required input.
Once all the costs and distributions have been determined, the Monte Carlo simulation can be carried out to determine the overall risk for the combined costs of the project. The number of iterations required makes this process impossible to do by hand and suitable software has to be used. The software package will allow you to set the number of iterations and the number of intervals to show on the histogram, and some other statistical details. If you would like to actually run this example yourself you can download the free trial of the software by clicking on the Download option on the menu above.
The results of the Monte Carlo simulation are usually shown on either a histogram (as shown in Figure 3 below), or as an S-Curve (as shown in Figure 4 below).
In the histogram view, the costs of the many iterations are accumulated into intervals and the number of occurrences counted for each interval. Dividing the number of occurrences in any interval by the total number of iterations gives the probability of the overall project cost being in that interval. Each vertical bar on the histogram represents an interval and shows the probability of the cost falling in that interval. The histogram makes it easy to see the minimum and maximum costs, as well as the most likely cost for the total project.
If the probabilities for each of the intervals are summed together they will come to 100%. Or, by summing the costs from the minimum cost to any chosen value it is possible to get the probability of the total cost being less than or equal to that cost. Adding all these individual probabilities would be a tedious task and the software performs this job for us and displays the results in the well known S-Curve shown in Figure 4 below.
The S-Curve makes it very easy to read off the cost for any selected probability. For example, if we read from the 25% probability on the vertical axis across to the curve and then down to the cost we can see that there is a 25% probability of the cost being 18.5 million or less. However, when evaluating the feasibility of a project we would want to have a better than 25% chance of coming in on or under budget so we would select a probability of 95% or so to estimate the required budget. In this case reading from the 95% point across to the curve and then down to the cost indicates that we would need to set the budget at about 19.6 million to have a 95% chance of coming in on budget.
The normal practice in project cost estimating is to use the sum of the Most Likely Costs of the individual items as the project cost. This was shown to be 18.54 million in Figure 2 above. The difference between this cost and the 95% cost we obtained from the S-Curve (i.e. 19.6 million) is 1.06 million and this is the cost contingency that must be added to the sum of the Most Likely Costs to give the project cost if we want a confidence level of 95% of coming in on budget.
The actual probability that will be used to determine the contingency will depend on the company's attitude to risk. If it was acceptable to have a 90% chance of coming in on budget then the estimated overall cost would drop to 19.4 million and the required contingency would be 0.86 million. On the other hand, if a 99% probability was required the estimated project cost would be 19.91 million and the contingency would be 1.37 million. As would be expected, the more certainty that is required that the project will come in on budget, the higher the required contingency becomes.
To increase your understanding of why cost contingency allowances are required, and why projects seem to go over budget so often, we have prepared another article that discusses why project costs are expected to overrun.