# Randomized Estimation of the Net Present Value of a Residential Housing Development
## 2 Materials and Methods
### Randomization of the Primary Initial Data
#### 2.1.2 Randomized Costs of Tasks and Revenues of Tranches  
In reality, costs of task execution and revenue tranches payment are random variables of costs of carrying out tasks and random variables of revenue tranches payment. Such quantities (see, e.g., [PERSON] and [PERSON] 2014 [24]) should be described by means of appropriate probability density functions or probability distribution functions. Unfortunately, these functions in an analyzed situation are unknown or are virtually impossible to directly elaborate. In that case, probabilistic characteristic of costs and revenues as random variables have been developed by the use of the data randomization method. With this end in view, values of primary initial costs \(K_{i},i=1,2,\ldots,i^{\underline{p}}\) and primary initial revenues \(D_{i},i=1,2,\ldots,i^{\underline{p}}\) after additional analysis have been verified as the most probable values of random costs \(\hat{K}_{i},i=1,2,\ldots,i^{\underline{p}}\) of tasks \(a_{i}\) execution and the most probable random revenues \(\underline{O}_{i},i=1,2,\ldots,i^{\underline{p}}\) paid in tranches \(b_{i}\). Next, using coefficients of optimism \(\underline{p}_{i}\) and pessimism \(\overline{p}_{i}\), the probable bottom boundaries \(\underline{K}_{i}\) and the probable upper boundaries \(\overline{K}_{i}\) of costs \(K_{i}\) of tasks \(a_{i}\) have been calculated. Similarly, using coefficients of optimism \(\underline{p}_{i}\) and pessimism \(\overline{p}_{i}\), the probable bottom boundaries \(\underline{D}_{i}\) and the probable upper boundaries \(\overline{D}_{i}\) of revenues \(D_{i}\) paid in tranches \(b_{i}\) have been calculated. Based on these values (see e.g., [PERSON], [PERSON] 2016 [25]), using PERT-beta probability distribution function and simplified formulas, the expected values \(E[K_{i}]\), \(i=1,2,\ldots,i^{\underline{p}}\), of random variables of costs \(K_{i}\) and the expected values \(E[D_{i}],i=1,2,\ldots,i^{\underline{p}}\), of random variables of revenues \(D_{i}\) can be calculated reliably enough for practice. These values, for the residential housing development, have been estimated as follows:  
1. The most probable costs \(\hat{K}_{i}\), \(i=\overline{1,22}\), of tasks \(a_{i}\) execution--primary initial costs verified by experts due to the predicted random implementation conditions.
2. Probable bottom boundaries of costs \(\underline{K}_{i}=\hat{K}_{i}-\underline{p}_{i}\hat{K}_{i}\), \(i=\overline{1,22}\) of tasks \(a_{i}\) execution.
3. Probable upper boundaries of costs \(\overline{K}_{i}=\hat{K}_{i}+\overline{p}_{i}\hat{K}_{i}\), \(i=\overline{1,22}\) of tasks \(a_{i}\) execution.
4. Expected costs \(E[K_{i}]\), \(i=\overline{1,22}\) of tasks \(a_{i}\) execution: \[E[K_{i}]=\underline{K_{i}+4\hat{K}_{i}+\overline{K}_{i}}\] (5)5. Variance of costs \(\sigma^{2}[K_{i}]\), \(i=\overline{1,22}\) of tasks \(a_{i}\) execution: \[\sigma^{2}[K_{i}]=\left(\frac{\overline{K}_{i}-\underline{K}_{i}}{6}\right)^{2}\] (6) 6. The most probable revenues \(\hat{D}_{i}\), \(i=1,2,3,4\), paid in tranches \(b_{i}\)--primary initial revenues verified by experts due to the predicted random implementation conditions. 7. Probable bottom boundaries of revenues \(\underline{D}_{i}=\hat{D}_{i}-\underline{p}_{i}\hat{D}_{i}\), \(i=1,2,3,4\), paid in tranches \(b_{i}\). 8. Probable upper boundaries of revenues \(\overline{D}_{i}=\hat{D}_{i}+\overline{p}_{i}\hat{D}_{i}\), \(i=1,2,3,4\), paid in tranches \(b_{i}\). 9. Expected revenues \(E[D_{i}]\), \(i=1,2,3,4\), paid in tranches \(b_{i}\): \[E[D_{i}]=\frac{\underline{D}_{i}+4\hat{D}_{i}+\overline{D}_{i}}{6}\] (7) 10. Variance of revenues \(\sigma^{2}[D_{i}]\), \(i=1,2,3,4\), paid in tranches \(b_{i}\): \[\sigma^{2}[D_{i}]=\left(\frac{\overline{D}_{i}-\underline{D}_{i}}{6}\right)^{2}\] (8)  
Estimated in this way, values of the quantities were used for calculations of the randomized project total cost and overall revenue.