Probability concept and Decision Tree

Probability concepts:

It is the assumption of probabilities or likelihoods of future events from the analysis of previous experience with the similar project or the market survey. The use of probabilistic information helps to provide management with a range of possible outcomes and the likelihood of achieving different goals under each investment alternative. We consider some of the terms related to probability such as random variable, probability distribution, and cumulative probability distribution. The assignment of probability to the various outcomes of an investment project is generally called as risk analysis. If the probability of an event approaches 0 (zero), the event is increasingly less likely to occur.

Random variable is a variable that can have more than one possible value and its value is unknown until the event occurs but its probability is known in advance. Random variable may be discrete or continuous that discrete has only single countable value whereas continuous random variable may have any value within a certain interval.

Probability distribution is prepared considering a range of probabilities for each feasible outcome exists. It can be graphically represented. It is used to represent the variability of a random variable. This represents the probability of considering or choosing of some value of the certain random variable.

Cumulative Probability distribution is a function that gives the probability that the random variable will reach a value smaller than or equal to some value of x.

For example, demand probability distribution is shown as:

Then, Cumulative distribution function is shown as

Joint, Conditional and Marginal Probabilities

It is likely that the values of some parameters will be dependent on or influenced by the values of others. These dependencies can be commonly expressed in terms of conditional probabilities. An example is product demand which will be probably influenced by unit price.

P(X = x | Y = y) is the conditional probability of observing x on given Y = y.

P(Y = y) is the marginal probability of observing Y = y

Joint Probability is defined as: P(x, y) = P(X = x | Y = y). P(Y = y)

Certainly an important case exist such that X and Y are independent i.e. occurrence of event X does not change the probability of an event Y. Then joint probability is simply, P(x, y) = P(x).P(y)

Example:

The marginal distribution for x can be developed from the joint event by fixing x and summing over y.

Risk Simulation:

It is the process of modeling reality to observe and weigh the likelihood of possible outcomes of a risky undertaking. Many investment situations may not be solved easily by analytical methods especially when many random variables are involved so in this situation we may develop the NPW distribution and make analysis through computer simulation.

The following logical steps are suggested for a computer program that simulates investment scenarios:

Initially, identify all the variables that affect the measure of investment worth and identify the relationship among all the variables representing in the form of equations or series of numerical computations by which we compute the NPW of an investment project.

Monte Carlo Simulation:

It is a specific type of randomized sampling method in which a random sample of outcomes is generated for specified probability distributions of values of random input variables.

One of the way to conduct a risk simulation is to use an Excel-based program such as @RISK. @RISK is an add-in to Microsoft Excel and can be integrated with our spreadsheet. @RISK uses Monte Carlo Simulation to show all possible outcomes. Running an analysis with @RISK involves the below steps:

1. Create a cash flow statement with Excel
2. Define Uncertainty
3. Pick your bottom line to get required output
4. Simulate
5. Analyzing the simulation Result Screen

Simulation Output Analysis:

Through the descriptive statistics and histogram of the values of the output variable, we can determine and analyze the probability distribution of the output variable such as net profit, NPV, IRR, etc.

Decision Tree:

Decision Tree is a technique that can facilitate in making the decision on investment when uncertainty prevails especially when the problem involves a sequence of decisions. Decision tree analysis encompasses the choice of a decision criterion to maximize expected profit. An experiment is conducted if possible and feasible and the prior probabilities of the states of nature are revised on the basis of the experimental results. The probable profit associated with each possible decision is computed and the act will the highest estimated profit is chosen as the optimum action.

Decision Tree includes:

1. More than one stage of alternative selection
2. Estimate economic value for each outcome
3. Selection of an alternative at one stage leads to another stage
4. Probability estimates for each estimate
5. Expected results form a decision at each stage
6. Measure of worth as selection criterion such as Expected value (E)

Components of Decision Tree:

1. The decision tree is constructed left to right and comprises each possible decision and outcome
2. Decision node: A square represents a decision node for making decision by a decision maker
3. Branch: It is a line connecting modes from the left to the right of the diagram
4. Probability node: A circle represents probability node with the possible outcomes and estimated probabilities on the branches

The following information is needed for the evaluation and selection of the alternative.

1. The probability that is estimated must sum to 1 for each set of outcomes (branches) that results from the decision.
2. Economic information for each decision substitute and possible outcome such as initial investment and estimated cash flows.

Procedure of Solving Decision Tree using Present Worth analysis:

1. Start at the top right of the tree. Determine the PW value for each outcome branch
2. Calculate the expected value for each decision alternative.

E (decision) = ∑(outcome estimate) P (outcome)

3. At each decision node, select the best E (decision) value – minimum cost or maximum value (if both cost and revenues are estimated)
4. Continue moving to the left of the tree to the most decision in order to select the best alternative.

BIBLIOGRAPHY:

Chan S.Park, Contemporary Engineering Economics, Prentice Hall, Inc.
E. Paul De Garmo, William G.Sullivan and James A. Bonta delli, Engineering
Economy, MC Milan Publishing Company.
James L. Riggs, David D. Bedworth and Sabah U. Randhawa,Engineering
Economics, Tata MCGraw Hill Education Private Limited.