# Randomized Estimation of the Net Present Value of a Residential Housing Development
## 2 Materials and Methods
### Randomization of the Primary Initial Data
#### 2.1.1 Estimation of the Impact of Disturbances on the Task's Costs and the Revenues
The probabilistic coefficients of cost optimism \(\underline{p}_{i}\), \(i=1,2,\ldots,i^{a}\) reflect likely decreased costs \(K_{i}\) of tasks \(a_{i}\in A\) due to the probable improvement of implementation conditions. The probabilistic coefficients of costs pessimism \(\overline{p}_{i}\), \(i=1,2,\ldots,i^{a}\) reflect likely increased costs \(K_{i}\) of tasks \(a_{i}\in A\) due to the probable degradation of implementation conditions. Average random impact of disturbances \(e_{i,j},\ i=1,2,\ldots,i^{a},\ j=1,2,\ldots,j^{i}\) on costs \(K_{i}\) of tasks \(a_{i}\in A\) can be calculated by using formulas:  
1. Average probability \(r_{i},\ \text{and average severity}\ c_{i}\) of disturbances \(e_{i,j}\in E,i=1,2,\ldots,i^{a},\ j=1,2,\ldots,j^{i}\), which may randomly change costs \(K_{i}\in\mathcal{K}\) of the implementation of tasks \(a_{i}\in A\): \[r_{i}=\left(\sum_{j=1}^{j=j}r_{i,j}\right)\left(j^{i}\right)^{-1};\quad c_{ i}=\left(\sum_{j=1}^{j=j}c_{i,j}\right)\left(j^{i}\right)^{-1}\] (1)
2. Probabilistic coefficients of cost optimism \(\underline{p}_{i}\in[0,1]\) and cost pessimism \(\overline{p}_{i}\in[0,1]\), \(i=1,2,\ldots,i^{a}\): \[\underline{p}_{i}=1-r_{i}c_{i};\quad\overline{p}_{i}=1-(1-r_{i})(1-c_{i})\] (2)
* coefficients \(\underline{p}_{i}=1\) indicate extremely favorable conditions for carrying out tasks \(a_{i}\in A\) and possible maximum reduction of cost \(K_{i}\in\mathcal{K}\) and \(\underline{p}_{i}=0\) points of an opposite case;
* coefficients \(\overline{p}_{i}=1\) indicate extremely difficult conditions for carrying out tasks \(a_{i}\in A\) and possible maximum increase of costs \(K_{i}\in\mathcal{K}\) and \(\overline{p}_{i}=0\) points of an opposite case. Similarly, the probabilistic coefficients of revenue optimism \(\overline{p}_{i}\ i=1,2,\ldots,i^{b}\) reflect the likely increase of revenues \(D_{i}\) paid in tranches \(b_{i}\in B\) due to the probable improvement of business and payment conditions. The probabilistic coefficients of revenue pessimism \(\underline{p}_{i},\ i=1,2,\ldots,i^{b}\) reflect the likely decrease of revenues \(D_{i}\) paid in tranches \(b_{i}\in B\) due to the probable degradation of business and payment conditions. Average random impact of disturbances \(e_{i,j},\ i=1,2,\ldots,i^{b},\ j=1,2,\ldots,j^{i}\) on revenues tranches \(D_{i}\) payment can be calculated by using formulas:  
\[r_{i}=\left(\sum_{j=1}^{j=j^{i}}r_{i,j}\right)\left(j^{i}\right)^{-1};\quad c _{i}=\left(\sum_{z=1}^{j=j^{i}}c_{i,j}\right)\left(j^{i}\right)^{-1} \tag{3}\]1. Probabilistic coefficients of revenue pessimism \(\underline{p}_{i}=[0,1]\) and revenue optimism \(\overline{p}_{i}=[0,1]\), \(i=1,2,\ldots,i^{\underline{p}}\): \[\underline{p}_{i}=1-(1-r_{i})(1-c_{i});\quad\overline{p}_{i}=1-r_{i}c_{i}\] (4) * the coefficient \(\underline{p}_{i}=1\) indicates extremely difficult conditions for carrying out tranches \(b_{i}\in B\) payments and possible maximum decrease of revenues \(D_{i}\in\mathcal{D}\) and \(\underline{p}_{i}=0\) points of an opposite case; * the coefficient \(\overline{p}_{i}=1\) indicates extremely favorable conditions for carrying out tranches \(b_{i}\in B\) payments and possible maximum increase of revenues and \(\overline{p}_{i}=0\) points of an opposite case. Data depicted above allow comprehensively described conditions of the project implementation. In case of the residential housing development costs, the probability \(r_{i,j}\) and severity \(c_{i,j}\) as well as coefficients of optimism \(\underline{p}_{i}\) and pessimism \(\overline{p}_{i}\) have been estimated for all disturbances \(e_{ij}\) that may disrupt execution of tasks \(a_{i}\) that is for \(s=1\), \(2\), \(3\), \(4\), \(i=\overline{1,22}\) and \(j=\overline{1,6^{1}}\), \(j=\overline{1,5^{2}}\), \(j=\overline{1,5^{3}}\), \(j=\overline{1,5^{4}}\).