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Decision trees are a great tool when dealing with probabilities. In this lesson, we'll use them to calculated a formula for risk-weighted returns.

**When to use decision trees**

- Problems that have a series of sequential decisions combined with chance events

- The probability of the chance events is roughly known

- The decisions within the tree are not quantitatively complex

**Decision tree elements**

- **Decision node** A point where a decision is taken

- **Chance event nodel** A point where chance event occurs

- **Connecting line** Represents logical sequence between nodes

**Process for calculating decision trees**

**1** Draw decision tree, thinking out the logical sequence of events

**2** Fill in the probabilities associated with each chance node branch

**3** Calculate the pay-off for end of each branch

**4** Calculate **Expected Value (EV)** at each chance node

**5** Define decision route through the tree

For more complex problems such as this one, it can often take multiple steps to diagram the problem correctly. In this lesson, I am going to use decision trees to complete the first part of the problem diagram. In the previous lesson, we mentioned that the project would need to return five times the initial investment over 10 years. Although you have tried to explain to Supra's executives that the time value of money and NPV is a better metric, they are wedded to an old rule of thumb which has served the company well over the past 30 years. The equation below summarizes our problem, however, this case introduces another element of complexity and that is the probability of success. So how do we deal with this? Well, whenever you see probabilities in a problem you should think of expected values and decision trees. These two concepts work very well together for multistage problems than involve probabilities. In a decision tree, we typically have three different shapes. The first is a rectangle which is called a decision node and unsurprisingly is a point where decision is taken. The second is a circle which represents a chance event node. And the third is a straight line which connects nodes in a logical sequence. Let's start to look at decision trees with a simple example. A dice in a casino is about to be rolled once. If you roll an even number you receive $10 dollars. If you roll an odd number we lose $8 dollars and the question is, do we play? 16 00:01:39.26 --> 00:01:45.06 Let's start our decision tree with our decision node which simply says, do we want to play? Here we have two options, we can say No, which costs us $0 dollars or we can say Yes. If we say Yes, this creates a chance event node. We have a 50% percent chance of making $10 dollars and a 50% percent chance of losing $8 dollars. At this stage in the decision tree, we calculate the expected value of these two branches. So 10 by 50% percent is equal to $5 dollars and minus 8 by 50% percent is equal to minus 4. 5 minus 4 is plus 1 and so the expected value of the Yes option is plus 1, which is greater than 0, so we should play the game. Now that we know how decision trees work, let's create one for our case study. The decision node will be, do we invest in the reference plant and if we say No, it costs 0$ dollars. If we say yes, it costs $15 million to build the plant. We then encounter a chance event node. It is a 15% percent chance of success, the value of which we still don't know. If we calculate the expected value from our chance event node, we'll get 15% percent multiplied by the cash flows from the commercial plant which we don't yet know, plus 85% percent by 0, which we can discard.

And then we'll add this to the initial cost of the reference plant which is minus $15 million by a 100% percent. This leaves us with a final equation for the project cash flows which is equal to minus $15 million plus 15% percent multiplied by the commercial plant cash flows. Although this is quite a simple decision tree with only two steps, you can see the value of drawing out decisions in this way and how it's much better than trying to calculate this equation in your head. For more complex decision trees with multiple branches and multiple steps, the concept becomes even more valuable. Now that we have a succinct equation and isolated commercial plant cash flows as the variable to be solved, we'll use a process diagram in the next lesson to lay out the model for this output value.

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