# Decision tree in investment valuation

The investment valuation decision tree is a tool for project evaluation. As a starting point, all the options from which the investor can choose and the eventual scenarios they face are compiled.

The above will be reflected in a graph like the following:

Once the tree is designed, the next step is to calculate the estimated profit in each scenario. Thus, it is determined which alternative offers the most profit.

The decision tree can be a very useful one, but only if the probability of occurrence and the expected return are known exactly under all the assumptions. Otherwise, you will not be sure which is the most optimal path.

Specifically, the decision tree is a visual tool. However, it requires mathematical calculations to arrive at a result, as we will see in an example later.

## Elements of a decision tree in investment valuation

The elements of a decision tree in investment valuation are:

- Decisional knots: They represent the options from which the investor chooses, and do not depend on chance or external factors. For example, a company may have to choose between contracting with supplier A or with supplier B.
- Random knots: These are circumstances that depend on external factors, and not only on the will of the agent. That is, they cannot be controlled. For example, let's imagine three future scenarios for the demand for a product: Optimistic, moderate, and pessimistic. Then, each one is assigned a different probability of occurrence, having to add said percentages whose sum is always one.
- Arches or branches: These are the arrows that graphically join the decisional and / or random knots. Thus, investor decisions can be connected with their possible consequences.

## Decision tree example in investment valuation

Next, let's look at an example of a decision tree. Suppose that a company can choose in period 0 between building its new factory in the north or south. For the first case, the cost is US $ 100,000, while in the second it is US $ 105,000.

In the next period, once the factory is ready and the company is able to start its sales, it must choose between commercial strategy A or B. Also, in both cases, it may face high, moderate or low demand.

So, to evaluate the profitability of the project in each case, we assume an equivalent annual rate of 0.15. Thus, we calculate the net present value (NPV) for all combinations of decisions and scenarios.

In short, twelve possible situations will be presented as we see in the following table.

Case | Decision initial | Strategy | Demand | Probability | FC * Year 1 | FC Year 2 | GO |

1 | North Zone | TO | high | 0,35 | 90 | 150 | 91,68 |

2 | North Zone | TO | Moderate | 0,35 | 30 | 100 | 1,7 |

3 | North Zone | TO | Come down | 0,3 | 10 | 50 | -53,5 |

4 | North Zone | B | high | 0,33 | 110 | 140 | 101,51 |

5 | North Zone | B | Moderate | 0,33 | 40 | 90 | 2,84 |

6 | North Zone | B | Come down | 0,34 | 20 | 45 | -48,58 |

7 | South Zone | TO | high | 0,3 | 85 | 170 | 97,46 |

8 | South Zone | TO | Moderate | 0,4 | 25 | 110 | 8,61 |

9 | South Zone | TO | Come down | 0,3 | 15 | 40 | -61,71 |

10 | South Zone | B | high | 0,31 | 120 | 160 | 120,33 |

11 | South Zone | B | Moderate | 0,38 | 55 | 115 | 29,78 |

12 | South Zone | B | Come down | 0,31 | 25 | 55 | -41,67 |

* FC = Cash flow

For example, let's imagine that the investor chooses the northern zone and strategy A. Therefore, if the demand is high, the NPV of the two years of the project will be:

The decision tree could be graphed as follows:

## Decision tree example output

To obtain a result of the previous example, we find the weighted average of the three scenarios that emerge from each possible combination of zones and strategies. This, considering the probability of each situation.

For example, if the individual chooses the southern zone and strategy A, the expected value of his investment will be:

Expected value = (0.3 * 97.46) + (0.4 * 8.61) + (0.3 * -61.71) = US $ 14.17 thousand

Next, following this methodology, we obtain the following results:

Location | Strategy | Expected value |

North Zone | TO | 16,64 |

North Zone | B | 17,92 |

South Zone | TO | 14,17 |

South Zone | B | 35,7 |

Therefore, the most profitable alternative for the investor is to settle in the southern zone and implement strategy B.