# Randomized Estimation of the Net Present Value of a Residential Housing Development
## 2 Materials and Methods
### Randomization of the Primary Initial Data
#### 2.1.3 Randomized Total Cost and Overall Revenue of the Residential Housing Development
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{s}\) & \(\mathbf{a_{i}}\)\(\in\)\(\mathbf{A^{s}}\) & \(\mathbf{a_{i}}\)\(\in\)\(\mathbf{A}\) & \(E[\mathbf{K_{i}}]\) & \(\mathbf{K_{i}}\) & \(\hat{\mathbf{K_{i}}}\) & \(\overline{\mathbf{K_{i}}}\) & \(\mathbf{p_{i}}\) & \(\overline{\mathbf{p_{i}}}\) \\ \hline \multirow{8}{*}{\(s1\)} & \(a_{1}\) & \(a_{1}\) & 98,379 & 24,888 & 98,242 & 172,415 & 0.75 & 0.76 \\  & \(a_{2}\) & \(a_{2}\) & 11,966,298 & 2,676,893 & 12,083,779 & 20,785,778 & 0.78 & 0.72 \\  & \(a_{3}\) & \(a_{3}\) & 3,371,615 & 670,741 & 3,438,474 & 5,805,051 & 0.80 & 0.69 \\  & \(a_{4}\) & \(a_{4}\) & 6,252,564 & 1,467,082 & 6,287,495 & 10,898,324 & 0.77 & 0.73 \\  & \(a_{5}\) & \(a_{5}\) & 3,312,396 & 753,640 & 3,340,232 & 5,759,812 & 0.77 & 0.72 \\ \hline \multicolumn{8}{c}{Sum} & 25,001,252 & & & & & \\ \hline \multirow{8}{*}{\(s2\)} & \(a_{1}\) & \(a_{6}\) & 184,487 & 35,565 & 188,573 & 317,067 & 0.81 & 0.68 \\  & \(a_{2}\) & \(a_{7}\) & 185,745 & 38,808 & 188,573 & 321,366 & 0.79 & 0.70 \\  & \(a_{3}\) & \(a_{8}\) & 185,745 & 38,469 & 188,573 & 321,706 & 0.80 & 0.71 \\  & \(a_{4}\) & \(a_{9}\) & 554,405 & 109,297 & 565,720 & 954,256 & 0.81 & 0.69 \\  & \(a_{5}\) & \(a_{10}\) & 557,234 & 115,973 & 565,720 & 964,552 & 0.80 & 0.71 \\  & \(a_{6}\) & \(a_{11}\) & 185,745 & 38,469 & 188,573 & 321,706 & 0.80 & 0.71 \\ \hline \multicolumn{8}{c}{Sum} & 1,853,360 & & & & \\ \hline \multirow{8}{*}{\(s3\)} & \(a_{1}\) & \(a_{12}\) & 174,540 & 34,671 & 177,799 & 301,370 & 0.81 & 0.70 \\  & \(a_{2}\) & \(a_{13}\) & 9,210,002 & 1,941,568 & 9,334,462 & 15,980,599 & 0.79 & 0.71 \\  & \(a_{3}\) & \(a_{14}\) & 3,054,443 & 609,852 & 3,111,487 & 5,270,859 & 0.80 & 0.69 \\ \cline{1-1}  & \(a_{4}\) & \(a_{15}\) & 14,350,624 & 2,745,932 & 14,668,440 & 24,684,051 & 0.81 & 0.68 \\ \cline{1-1}  & \(a_{5}\) & \(a_{16}\) & 3,070,001 & 653,412 & 3,111,487 & 5,320,643 & 0.79 & 0.71 \\ \cline{1-1}  & \(a_{6}\) & \(a_{17}\) & 3,470,568 & 752,171 & 3,511,536 & 6,025,093 & 0.79 & 0.72 \\ \cline{1-1}  & \(a_{7}\) & \(a_{18}\) & 872,698 & 174,954 & 888,996 & 1,505,249 & 0.80 & 0.69 \\ \cline{1-1}  & \(a_{8}\) & \(a_{19}\) & 436,349 & 87,477 & 444,498 & 752,624 & 0.80 & 0.69 \\ \hline \multicolumn{8}{c}{Sum} & 34,639,225 & & & & & \\ \hline \multirow{8}{*}{\(s4\)} & \(a_{1}\) & \(a_{20}\) & 85,936 & 14,188 & 88,900 & 145,831 & 0.84 & 0.64 \\  & \(a_{2}\) & \(a_{21}\) & 152,267 Calculated in the above way, the expected total cost \(E[K]\) and the expected overall revenue \(E[D]\) represent only a particular possible case. This means that values \(E[K]\) and \(E[D]\) describe only this one case of the project implementation. The range of probable changings of these values is interesting. These variations were measured by using coefficients of the risk of total cost \(p(k)\) and the risk of overall revenue \(p(d)\). Values of these quantities were calculated assuming student's t-distribution functions for the random variable \(E[K]\) of total cost and for the random variable \(E[D]\) of overall revenue. In accordance with the results of comprehensive analysis, the risk measure of total cost is the probability \(p(k)\) that the actual total cost \(\mathcal{K}=E(K)\) of the project is lesser than k. The risk measure of overall revenue is the probability \(p(d)\) that the actual overall revenue \(\mathcal{D}=E(D)\) of the project is greater than d. Values of \(p(k)\) and \(p(d)\), according to the comparative values k and d, can be calculated as follows:  
1. Cost risk (contingency) of the project \(p(k)\): \[p(k)=P[E(K)\leq k]=Z\Bigg{[}\frac{k-E(K)}{\sqrt{\sigma^{2}(K)}}\Bigg{]};\quad p (k)\in[0,1]\] (13) \(p(k)\)--means probability that the real total cost \(\mathcal{K}=E(K)\) should be less than \(k\). 2. Revenue risk (contingency) of the project \(p(d)\): \[p(d)=P[E(D)\geq d]=1-P[E(D)\leq d]=Z\Bigg{[}\frac{d-E(D)}{\sqrt{\sigma^{2}(D)} }\Bigg{]};\quad p(d)\in[1,0]\] (14) \(p(d)\)--means probability that the real overall revenue \(\mathcal{D}=E(D)\) should be greater than \(d\).