Step 3. Calculate MIRR using the formula
4.4 Normal distribution tables and confidence levels
A normal distribution table can also be used to establish confidence levels. In risk analysis, it is usual to assess risk at the 95% confidence level or the 99% confidence level.
A 95% confidence level means that there is a 95% probability.
A 99% confidence level means that there is a 99% probability.
Confidence levels for values within a range
We can use normal distribution tables to calculate, at a given confidence level, that the value of an item will be within a specified range of values above and below the mean.
For example, suppose that a simulation model produces results showing that the EV of the NPV for a project is $125,000 and the standard deviation (project volatility) is
$40,000. We can predict at the 95% confidence level that the NPV of the project will be within a specified range around the EV.
To establish a 95% confidence level for the range around the NPV, we need to identify the range of values below the mean that represent 47.5% of total probabilities and the range of values above the mean that represent 47.5% of total probabilities (since 2 × 47.5% = 95%).
We therefore need to identify the number of standard deviations from the mean that cover 47.5% of all probabilities. From the normal distribution table, we can identify that this is 1.96 standard deviations.
At a 95% confidence level, we can therefore predict that the NPV will be somewhere in the range between 1.96 standard deviations below the mean and 1.96 standard deviations above the mean.
Similarly, at a 99% confidence level, we can predict that the NPV will be somewhere in the range between 2.575 standard deviations below the mean and 2.575 standard deviations above the mean. (This is because there is a probability of 0.4950 that the value will be 2.575 standard deviations below the mean and a probability of 0.4950 that the value will be 2.575 standard deviations above the mean.)
Example
A simulation model has been used to calculate the expected value of the NPV of a project. This is + $150,000. The standard deviation of the project is $55,000.
The variability in the NPV approximates to a normal distribution. (This is usual for the output results from a simulation model.)
At the 95% confidence level, we can predict that the NPV of the project will be in the range of 1.96 standard deviations above and below the mean. 1.96 × $55,000 =
$107,800. At the 95% confidence level, we can therefore predict that the NPV will be somewhere in the range $42,800 to $257,800.
At the 99% confidence level, we can predict that the NPV of the project will be in the range of 2.57 standard deviations above and below the mean. 2.57 × $55,000 =
$141,350. At the 99% confidence level, we can therefore predict that the NPV will be somewhere in the range + $8,650 to $291,350.
Confidence levels for values above or below a specified amount
Confidence levels can also be established to identify the probability that the actual value will be above or below a specified amount. For example, we can establish at the 95% or 99% level of confidence that a value will be more than $X or less than $X (or not more than $X or not less than $X).
With this type of calculation we are looking at only one side of the normal distribution table (and carrying out a ‘one-tailed test’). This differs from confidence levels for a range above and below the mean, where we are looking at both sides of the normal distribution, above and below the mean.
Mean
95%
confidence level
5% probability
The diagram shows that if we want to establish a 95% confidence level for a value that is above the mean, we need to calculate the number of standard deviations above the mean for which there is a 45% probability. There is a 50% probability that the value will be less than the mean, so taken together we have a 95% probability level.
From the normal distribution table, we can find that 0.4500 of probabilities are within 1.645 standard deviations of the mean, on one side of the normal distribution table.
At the 95% level of confidence we can therefore state that the value will be less than an amount equal to 1.645 standard deviations below the mean. We can also state that at the 95% confidence level, the value will not exceed an amount that is more than 1.645 standard deviations above the mean.
Example
A simulation model has been used to calculate the expected value of the NPV of a project. This is + $70,000. The standard deviation of the project is $28,000.
The variability in the NPV approximates to a normal distribution.
At the 95% confidence level, we can therefore predict that the NPV will be not less than $46,060 (1.645 × $28,000) below the EV of $70,000. This means that at the 95%
confidence level the NPV will be not less than $23,940.
To establish the 99% confidence level, we can establish that 0.4900 of probabilities are between the men and about 2.33 standard deviations of the mean. At the 99%
confidence level, we can therefore predict that the NPV will be not less than $65,240 (2.33 × $28,000) below the EV of $70,000. This means that at the 99% confidence level the NPV will be not less than + $4,760.