# 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
They also depend on financial market stability during the project implementation and revenue payment. Taking all mentioned terms, the impact of disturbances was projected in accordance with likely prices on domestic and foreign property market, as well as probable conditions and quality of business environment. In the considered situation, the set \(B\) of tranches \(b_{i}\) may be disturbed by relevant set \(E^{b}\) of disturbances \(e_{i,j}\), \(E^{b}=\left\{e_{i,j}:i=1,2,\ldots,i^{b},j=1,2,\ldots,j^{b}\right\}\). In the result, revenues \(D_{i}\) paid in tranches \(b_{i}\) may randomly change. In the analyzed case, the sets \(E^{b}\) as well as disturbances \(e_{i,j}\) that belong to them are relatively independent of each other. Each set \(E^{b}\) of disturbances \(e_{i,j}\) correspond with a particular tranche \(b_{i}\). This means that the execution of each tranche \(b_{i}\in B\) may be randomly affected by each threat \(e_{i,j}\in E^{b}\). For the residential housing development sets \(E^{b}=\left\{e_{i,j}:i=\overline{1,4},j=\overline{1,5}\right\}\), of disturbances \(e_{i,j}\) related to sets \(B=\left\{b_{i}:i=\overline{1,4}\right\}\) of tranches \(b_{i}\), values have been identified as follows:  
1. Price decline of apartments \(j=1\): \(\bullet\) direct decrease of revenues. 2. Decline (slack) in sale of apartments \(j=2\): \(\bullet\) increase of interest costs and rise of loan repayment; \(\bullet\) possible disruptions of accounting liquidity and rise of various activities costs. 3. Slackening in sale of apartments \(j=3\): \(\bullet\) additional cost of maintenance of unsold apartments (at the expense of the developer). 4. Tighter credit policies by banks (financial restriction of investment) \(j=4\): \(\bullet\) increase of interest costs and rise of loan repayment; \(\bullet\) possible disruptions of accounting liquidity and rise of various activities costs. 5. Changing of economic and business conditions and deterioration of payment terms \(j=5\): \(\bullet\) possible direct changes of any costs of employment, core workers and auxiliary workers, work of machines, and auxiliary equipment, as well as goods and services. The random disturbances of costs and revenues reflect the influence of likely conditions of tasks execution and tranches payment. In the various random situations, disturbances can arise with different probabilities, and they can have distinct severity on the project implementation (see [PERSON], [PERSON], [PERSON], [PERSON], 2014 [21]). This means that random implementation conditions directly decide the probability of occurrence \(r_{i,j}\) and severity \(c_{i,j}\) of disturbances \(e_{i,j}\). Disturbances of costs \(e_{i,j},\ i=1,2,\ldots,i^{s},\ j=1,2,\ldots,j^{s}\) that belong to the subsets \(E^{s}\), or disturbances \(e_{i,j},\ i=1,2,\ldots,i^{a},\ j=1,2,\ldots,j^{i}\) that belong to the set E, may emerge with probability \(r_{i,j}\in[0,1]\) and severity \(c_{i,j}[0,1]\), and they may variously impact the execution of tasks \(a_{i}\in A\). The disturbances of revenues \(e_{i,j},\ i=1,2,\ldots,i^{b},\ j=1,2,\ldots,j^{b}\), or \(j=1,2,\ldots,j^{i}\) that belong to the set \(E^{b}\) may emerge with probability \(r_{i,j}\in[0,1]\) and severity \(c_{i,j}[0,1]\) and may variously impact payment of tranches \(b_{i}\in B\). The probability and severity of the disturbances for costs and revenues of the residential housing development should be estimated by construction and financial experts based on their knowledge, experience, and even intuition. Based on that evaluation, the influences of disturbances on project costs and revenues can be estimated. Depending on values of the disturbances' probability and severity, costs of task execution and revenues paid in tranches may change in varying degree. Degree of average random changes of costs and revenues can be estimated by the use of probabilistic coefficients of optimism and probabilistic coefficients of pessimism for costs and revenues, respectively.