Risk analysis formulas

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Formula for establishing the adjusted estimated value

Let E_g=\,\! the value, initially estimated by the grantee, for a particular item—with all contingency removed.

Let E_p=\,\! adjustments to the grantee’s estimate made by the PMOC.

Let E_a=\,\! The total adjusted estimate, such that:

E_a = E_g + E_p\,\! (0.1)

Method for establishing the optimistic price for an item

Let P_{10} =\,\! a value that represents a 10% likelihood that actual prices received will not exceed (i.e., there is a 90% chance that the value will be exceeded by an actual price).

Assume that the total adjusted estimate (see equation (1.1)) is the optimistic value[1], such that:

P_{10} = E_a \,\! (0.2)

Method for establishing the range of likely values

Assume that the receipt of pricing will follow a lognormal[2] distribution; i.e., if many prices were taken for the item in question (say, bids for a particular SCC line), then the bids would be more likely to have high-cost extremes than low-cost extremes.

Let P_{90} =\,\! a value that represents a 90% likelihood that actual prices will not exceed.

Let \beta =\,\! a factor[3] (called the “Risk factor”) which, when multiplied by P_{10}, yields P_{90}:

P_{90} = \beta P_{10}\,\! (0.3)

Method for establishing the central value and spread of the normalized cumulative distribution function

By definition, a normal distribution function is symmetrical about its mean (\mu). Since the values P_{10} and P_{90} are values on a lognormal distribution, then, also by definition, the values ln(P_{10}) and ln(P_{90}) are values on a normal distribution, and are symmetrically distanced (by 40% each) from the mean (also the center) of the normal curve. The mean, then is the average between the two values, or:

\mu = {{\ln (P_{10} ) + \ln (P_{90} )} \over 2}\,\! (0.4)

Substituting equation (1.3) into equation (1.4) yields:

\mu = {{\ln (P_{10} ) + \ln (P_{90} )} \over 2} = {{\ln (P_{10} ) + \ln (\beta ) + \ln (P_{90} )} \over 2} = \ln (P_{10} ) + {{\ln (\beta )} \over 2} (0.5)

To determine the standard deviation (\sigma) of this normal distribution, it is helpful to determine the “z” value for a standard normal[4] distribution at the 10% probability level. This value may be found using readily available tables, or through using spreadsheet software, such as Excel[5]. The value thus determined is z_{10\% } = - 1.2816; since the cost values here are all positive, the absolute value of this figure is used in the following equations. The formula to transform a value from a normal curve (x) into a standardized normal value (z) is:

z = \frac{{(x - \mu )}}{\sigma } (0.6)

Re-arranging equation (1.6) to solve for \sigma\,\!, yields:

\sigma = \frac{{(x - \mu )}}{z} (0.7)

And, substituting z_{10\% } = - 1.2816 and ln(P_{90})\,\!, equation (1.7) becomes:

\sigma = \frac{{(\ln (P_{10} ) - (\ln (P_{10} ) + \frac{{\ln (\beta )}} {2})}} {{1.2816}} = \frac{{\ln (\beta )}} {{2*1.2816}} = \frac{{\ln (\beta )}} {{2.5632}} (0.8)

Thus, the strong influence of the value \beta becomes apparent in both equations (1.5) and (1.8).

Method for establishing the central value and spread of the lognormal cumulative distribution function

The following formulas are available in probability texts to determine the mean (m) and the standard deviation (s) of the lognormal curve:

m = e^{\mu + 0.5\sigma ^2 } (0.9)

And

s = \sqrt {e^{2 + \mu + \sigma ^2 } (e^{\sigma ^2 } - 1)} (0.10)

Where the parameter is the mean determined in equation (1.5) and \sigma\,\! is the standard deviation determined in equation (1.8).

Summary of key equations

E_a = E_g + E_p\,\!
P_{10} = E_a \,\!
P_{90} = \beta P_{10}\,\!
\mu = \ln (P_{10} ) + {{\ln (\beta )} \over 2}
\sigma = \frac{{\ln (\beta )}}{{2.5632}}
m = e^{\mu + 0.5\sigma ^2 }
s = \sqrt {e^{2 + \mu + \sigma ^2 } (e^{\sigma ^2 } - 1)}

Notes

  1. ↑ This value is established at the “10% likely” point, as an anecdotal, observed acknowledgement of the tendency toward optimism in grantee’s estimates.
  2. ↑ A lognormal distribution is a distribution of values <math>(x)</math> that, when the natural log of the values <math>ln(x) are distributed, the resulting distribution would follow a normal distribution.
  3. ↑ Beta (<math>\beta</math>) values have been empirically identified to vary by phase of work, as follows: 1) during preliminary engineering, <math>2.0 \le \beta \le 3.0</math> 2.0) during final engineering, <math>1.5 \le \beta \le 2.5</math> 3.0) during construction, <math>1.25 \le \beta \le 2.0</math>.
  4. ↑ A standard normal distribution is one in which the mean of the distribution is zero (&lt;math&gt;0&lt;/math&gt;) and the standard distribution is one (&lt;math&gt;1&lt;/math&gt;). The formula that transforms normal values into standard normal “&lt;math&gt;z&lt;/math&gt;” values (a “&lt;math&gt;z&lt;/math&gt;” value represents the number of standard deviations that a value lies from the mean for a given probability) is &lt;math&gt;z = {{(x - \mu )} \over \sigma }&lt;/math&gt;, where the values on the right-hand side of the equation are values from the original normal distribution.
  5. ↑ The following Excel formula will yield the appropriate value: =NORMSINV(10%).
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