Using a decision tree in risk analysis

Using a decision tree in risk analysis

Expected value

The expected value of any outcome is the provisional profit of that outcome multiplied by the probability of it taking place

Decision making at any level is affected by possible outcomes. In terms of business planning, different outcomes will lead to different profits. The expected value of any outcome is the provisional profit of that outcome multiplied by the probability of it taking place. For example, if the provisional profit anticipated for a particular outcome is $100,000, and the probability of that outcome occurring is 0.5, or 50 percent, the expected value is $50,000, that is, $100,000 multiplied by 0.5. The best decision is the one that will lead to the highest expected value. If a decision could lead to two or more possible outcomes, the expected value is the sum of the expected values of these two outcomes.

Decision trees

A series of choices based on the expected values of different outcomes can be represented graphically as a decision tree. These are constructed from left to right. A line represents the central decision to be made. Where this encounters a possible divergence of outcome the line branches into two. Each of these new lines represents what would happen if that eventuality were to ensue. These new lines will split in turn anywhere there is more than one outcome.

Scenario

This concept is best understood using a concrete scenario. Say, for example, that your company Imagenie is trying to decide whether or not to develop a graphics software package. Your competitor, InterSwift, may develop a similar product, which could obviously have a negative impact on revenue. Another important consideration is how much testing is done on the product. Experience has shown that the profit made from the product is directly proportional to the amount of testing. However, the amount of testing done will also depend on whether InterSwift decides to develop similar software, as an overly long testing time would be a liability with a direct competitor to your product racing to market.

You have had figures for the probabilities of InterSwift's making particular decisions, and for provisional profits for each outcome. The accuracy of the decision tree is critically dependent on the accuracy of these probabilities. If they are not correct, no amount of decision tree analysis will yield a correct expectation value.

The probability that InterSwift will develop similar software is 0.6, or 60 percent. Therefore the probability that they develop such software is 0.4, or 40 percent. The sum of all probabilities for all the possible outcomes to an event must be equal to 1 or 100 percent. Here the decision tree branches into two.

If InterSwift does develop similar software, Imagenie must decide whether to opt for premium testing, standard testing, or lite testing, so at this point the tree must branch into three. Since this is a decision that the company will make, there is no probability associated with it. However, InterSwift also has a choice about which testing to use, and their choice will affect the profitability of Imagenie's software. Therefore each choice branch must be further split into three branches, based on the probabilities that have been calculated for InterSwift's choices of testing.

options	InterSwift response
	Premium testing		Standard testing		Lite testing
	Probability	Payoff($)	Probability	Payoff ($)	Probability	Payoff ($
Premium testing	0.5	40,000	0.4	60,000	0.1	30,000
Standard testing	0.2	20,000	0.6	40,000	0.2	50,000
Lite testing	0.1	30,000	0.2	50,000	0.7	90,000

In the table above, the probabilities and the provisional profit associated with each possible outcome are displayed. These will make up the end of this particular decision tree. The expected values for the decision tree are calculated moving from right to left. Therefore you start by multiplying each provisional profit in the table by its relevant probability value. This will give you an individual expected value for each of InterSwift's choices in respect of testing. To get the expected value for each decision that Imagenie can make in respect of testing, simply add the values of the three relevant expected values together.

 

Premium testing	Standard testing	Lite testing
0.5 x 40,000 = 20,000	0.2 x 20,000 = 4000	0.1 x 30,000 = 3000
0.4 x 60,000 = 24,000	0.6 x 40,000 = 24,000	0.2 x 50,000 = 10,000
0.1 x 30,000 = 3000	0.2 x 50,000 = 10,000	0.7 x 90,000 = 63,000
Expected value = $47,000	Expected value= $38,000	Expected value = $76,000

From the calculations above it is clear that the best course of action for Imagenie to take in the event that InterSwift does go ahead and develop similar software is to opt for lite testing as this option has the highest expected value by quite a significant margin.

By adding these figures in the last row together and multiplying them by 0.6 - the probability of InterSwift developing similar software - you can get the expected value if InterSwift does develop it. The sum of the three is $161,000, which, multiplied by 0.6, gives $96,600. By now you are back down to only two branches. You now need to look at what the expected value would be if InterSwift decided not to develop. The provisional profits for the three types of testing if InterSwift does decide to develop are as follows:

• premium testing: $1,800,000

• standard testing: $1,500,000

• lite testing: $120,000

With these figures determined by a decision, that decision will be in line with the highest profit, not probability. Therefore you take the figure of $1.8m and multiply it by 0.4, to give $720,000.

If the probabilities and provisional profits are not reliable then the expected values will also be unreliable

The cost of this project is $100,000. Looking at the decision tree it can be seen that this project would be profitable only if InterSwift did not decide to develop. The provisional profit of $96,600 that would result if they did develop is less than the cost of running the project. This means that there is a 60 percent chance of the project running at a loss. On the other hand there is a 40 percent chance of the project netting $640,000 profit. Whether to proceed with this risk is a matter for management. It is important to remember that the values are only as accurate as the figures used to obtain them in the first place, so that if the probabilities and provisional profits are not reliable then the expected values will also be unreliable. With good figures, however, a decision tree can clarify much in terms of risk management and decision making.