## 1 Introduction
It is obviously a good idea to conduct an appropriate cost-benefit analysis in this context. In addition to the aforementioned CPM, PERT or CCPM scheduling methods, there are many others that also find practical application, such as Line of Balance (LOB) [22; 23; 24; 25; 26], Q Scheduling [27; 28], Resource Oriented Scheduling [29; 30], Last Planner System [31], or a widespread schedule visualisation method such as Gantt Charts [32; 33]. There are also numerous studies that facilitate accurate analysis and prediction of project delay risks. These include Bayesian networks [34; 35], decision trees and Bayesian classification algorithms [36], Naive Bayesian (machine learning models) [36], Bayesian Belief Network [37], logistic regression [38], Markov Dynamic PERT Model [39] and Correlated Schedule Risk Analysis (CSRA) [20]. It should be noted here that understanding schedules is an important thing that often determines project success [2; 3; 40], therefore the more tools and research findings to help understanding in the area of project scheduling, the better. Thus, in the context of some studies mentioned above, it is possible to determine/quantif/ model the probability of project schedule delays using various research methods [36; 37; 38]. On the other hand, when analysing the literature on the subject, it is hard not to notice the lack of appropriate methods that would allow a comprehensive (from the risk control perspective) assessment of the risk of delays of a construction project, taking particular account of individual stages of the project. It would be good if such a method accounted for dependencies between individual stages of the project, and even allowed for conducting appropriate stress-tests in the context of potential results of changes in the schedules. A method of this kind exists and works very well in the financial world. It is known as the Value at Risk (VaR) measure. VaR is a statistic that quantifies the extent of possible financial losses in a firm, portfolio or position over a specified time period. It is a measure that allows to assess the probability of losing a specific value of an investment portfolio and to simulate the impact of changes in the external environment on the value of the entire portfolio. In addition, since many interesting solutions are created by transferring already existing solutions from other fields, it is very likely that applying a solution similar to VaR would work well with construction projects. To this end, individual stages of the project can be perceived as a set/collection of stages/activities and can be treated similarly to an investment portfolio. Instead of VaR in this study we will use Time-at-Risk (TaR), which is constructed similarly to VaR, although its interpretation is slightly different. TaR represents a certain quantile for a given probability distribution, so it is similar to VaR; however, TaR measures the magnitude of risk as time (the time until an adverse event occurs) rather than value (loss amount). This measure has already been analysed in other research papers, but in a slightly different context [41; 42]. TaR can potentially allow construction practitioners not only to understand the impact of postponing activities/stages/phases, but also to visualise how frequently such changes are likely to occur. More importantly, the Time-at-Risk approach applied in construction projects can potentially give construction firms and project managers a high degree of confidence that they are adequately equipped to withstand schedule changes with minimal losses. In this study, we present a proprietary model of the investment process and show how the implementation of its individual stages/phases, as well as the entire project, can be evaluated in the context of the covariance matrix between time schedules of all its stages and how their appropriate schedule timing can affect the risk of extending the implementation time of the entire project.