Data Aggregation and Reduction Project A stochastic is a probability process. Therefore, a stochastic process is adequately explainable using probability. Many situations in life have an element of chance, from war disease and business, chance plays a significant role. Insight into the underpinnings of chance is therefore imperative in decision making or understanding the possible outcomes of certain events. A stochastic process can assume many forms, but the overriding feature is that the process is unpredictable, and a number of options are available as to the occurrence of an event, for instance, war, disease, market dynamics, insurance claims and so on.
The purpose of this report is to identify a stochastic process, and analyze it using stochastic tools to arrive at conclusions about the process. Introduction: Stochastic Process Included (Business) The stochastic process considered in this case analytical report is commerce. In business environment, numerous factors affect the business operations and by extension the profitability of a business. Many of these factors, although they involve decision-making processes of the management, require detailed quantitative analysis with the use of statistical tools. In business, cash presents numerous challenges to the management, as it is one area, which has to depend on probability of future business trends given past experiences.
Inappropriate levels of liquid cash by a firm works to the detriment of the firm. Too much cash means lost opportunity as the cash cannot earn a return, too little cash on the other hand may put the firm in liquidity problems, and failure to meet business obligations may likewise push away clients and creditors whose demands remain unmet. To avoid either situation, an organization needs to take a position, based on firm’s policy, on the optimum levels of cash flow necessary to mitigate losses by having too much or too little cash.
Change in cash flow position in a business on a daily basis is therefore an important aspect that most management staff would like to handle with the help of more precise and detailed information. Statistical tools can help in the achievement of this endeavor as cash flow undulations depend largely on chance. This report looks at how stochastic tools can aid to make predictions that are more accurate on the expected changes in cash flow changes within an organization to minimize loss in revenue. Data: Relevant Data Values For a cash flow scenario, a number of data values are of great relevance.
The most important values in our case are the cash outflows and the cash inflows, because it is from this pair of value that we can compute the daily change in the cash flow position of the firm. Therefore, a new value is introduced, a derivative of the two daily cash flow values: the daily change in cash flow levels.
The daily change in cash flows is our most important figure in the analysis, since from it we will be able to make analysis of the situation to arrive at a solution that is more fitting for the case, and can be applied by the management in making cash flow decisions in future. Other values of note are the number of says the independent values upon which the other figures depend. The case A company, during trading over a period i, receives cash revenues ri and spends ci so that the cash flow position keeps changing according to the relation Zi = ri - ci The following are the values of Zi, changes in cash flow, over a period of 20 days in $ 234 -435 267 -567 -200 100 498 344 256 0 144 88 98 -10 -243 45 57 30 68 125 Discussion Markov Process To calculate the probability that change cash flow, Zi will be negative on the day following the 20th day From the data, there is a (5/20)*100 = 20% of a negative cash flow change, and an 80% chance that the cash flow will be positive.
The transition matrix that a negative change in cash flow will be as follows To Positive Negative From Positive 0.8 0.2 Negative 0.4 0.6 Solution The initial system is S1 = [0.2, 0.8], The transition matrix, T = | 0.8 0.2 | |0.4 0.6 | On the 21st day, the system will be S1xT = S2 [0.48, 0.52] On the 21st day, the probability of having a negative cash flow will be 48%, On the 22nd day, the system will be S2xT = S3 [0.592, 0.408] On the 22nd day, the probability for a negative change in cash flow will be 59.2% Gambler’s Ruin By modeling the case to a Gambler’s ruin scenario, where the days are taken as the fortune, and reporting a negative cash flow change as going broke, with the 20th day probability of a positive and negative cash flow change as 80% and 20% respectively We take day 20 as our starting point, 1, 21st day as 2, and so on. If the firm at day 20 has an 80% chance of reporting a positive cash flow change, What is the probability that the firm will report a positive cash flow on day 21, (2), and 22nd day (3)? Solution p=80/100 = 0.8, q=1-0.8 = 0.2, and q/p=0.2/0.8=1/4 Probability of reporting a positive cash flow in the first day (21st) P= 1-(1/4)1 1-(1/4)2 = 0.75/0.9375 = 0.8 Probability of reporting a positive cash flow change 22nd day P= 1-(1/4)1 1-(1/4)3 =0.75/0.984375 = 0.76 Poisson Process From the data, the rate λ of having a negative cash flow change is ¼ P(n) = e- λt(tλ)n n! If the rate of having a zero cash flow is 1 in 20 days, Then the probability of having one zero in the 20-day period was λ = (1/20) λt = (1/20)*20 = 1.0 Therefore the probability of a zero cash flow change on the 21st day is n = 1 λt = (1/20)*1 = 1/20 = 0.05 P(n)=e- λt(tλ)n n! P(n)=e- 0.05(0.05)1 1!
= 0.0475, 4.75% chance of having a zero cash flow change Conclusions Predicting an organization’s cash flow position with accuracy is a daunting task with many uncertainties. However, the implications of cash flow levels within an organization are dire, and could affect gravely the smooth running of the organization’s cash operations, or even its profitability.
With help of statistical tools, an organization can decipher and even have some form of control over the cash changes in an organization. Considering that the information has its basis on probability, the level of certainty is indeterminate be assured, but it can be improved, or at least understood. Through Markov analysis, we see that the possibility of having a negative cash flow, given that usual operation procedures remain constant is unlikely to happen with a 48%.
However, on the day after, the probability of getting a negative cash flow increases to 59.2%. Because of these figures, the management can alter its withdrawals for immediate cash requirements. When the probability is 48%, the company can decrease its cash at hand reserves, and when the likelihood is high, the company can prepare to avert the same by increasing the cash reserves. Poisson is useful in analysis of situations where the likeliness of an event is significantly low. For instance, a zero change in cash flow occurs rarely and through the Poisson process, we can establish that the likeliness of having a zero change in cash flow in the first day of the new period will be a lowly 4.75%. From the analysis done with Gambler’s ruin, we the examine the possibility of the company having a positive change in cash flow after a prior positive change in cash flow.
On the day after the period in the data, the probability of achieving a positive cash flow change is positive, at 80%, a figure consistent with the expected probability of having a positive change in cash flow for a new 20 period.
On the second day of the new 20-day period, given the first day had a positive change, the probably should decrease, and the probability of 76% validates this expectation. References Sigman, K. 2009. Gambler’s Ruin Problem. Columbia University. Retrieved on February 14, 2012 from http: //www. columbia. edu/~ks20/stochastic-I/stochastic-I-GRP. pdf Beasley, J. 2011. Operations Research. Brunel. Retrieved on February 14, 2012 from http: //people. brunel. ac. uk/~mastjjb/jeb/or/moremk. html Jacod, J and Protter P. 2004. Probability Essentials. Springer.