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# How to Use a Decision Tree to Make Tough Choices

May 29, 2016

Q.  I own a successful restaurant, which I plan to sell in the next couple of years. I face a difficult decision between three options. I can only pursue one.

·         Option 1 – I could open a second location. This would cost me about \$100,000. If this location fails, I would lose my investment. If it succeeds, I estimate that I would make an additional \$30,000 before I sell the business and it would increase the price I would get by \$200,000. New ventures are always risky. I estimate the probability of success at 50%.

·         Option 2 – Open a catering business. This would cost me about \$20,000. Again, if the catering business fails, I would lose my investment. If it succeeds, I estimate that I would make an additional \$10,000 before I sell the business and it would increase the price I would get by \$50,000. I am more certain this would succeed, so I would put the chance of success at 80%.

·         Option 3 – Focus on my existing restaurant to improve its profitability and do not pursue any other ventures. By doing this, I might make an additional \$5,000 before I sell and increase the price I get by \$20,000.

Can you help me think about which option is best?

A.    The approach we recommend is a classic decision tree. We’ll walk through each option and calculate the expected value of each.

·         Option 1
o   Failure means losing \$100,000. There is a 50% chance this will happen. Therefore, the expected value of this outcome is -\$50,000 (-\$100,000 X 50%).
o   Success means making \$230,000 (\$30,000 in additional profit prior to sale plus a sales price that is \$200,000 higher). There is a 50% chance this will happen. Therefore, the expected value of this outcome is \$115,000 (\$230,000 X 50%).
o   The expected value of choosing Option 1 is \$65,000 (\$115,000 – \$50,000).

·         Option 2
o   Failure means losing \$20,000. There is a 20% chance this will happen. Therefore, the expected value of this outcome is -\$4,000 (-\$20,000 X 20%).

o   Success means making \$60,000 (\$10,000 in additional profit prior to sale plus a sales price that is \$50,000 higher). There is an 80% chance this will happen. Therefore, the expected value of this outcome is \$48,000 (\$60,000 X 80%).
o   The expected value of choosing Option 2 is \$42,000 (\$48,000 – \$4,000)

·         Option 3 – You don’t assign any probability of failure to this option. Therefore, we will assume that following Option 3 will allow you to capture an additional \$25,000 with certainty (\$5,000 in increased profit before you sell and a sales price that is \$20,000 higher).

We will assume that you have \$100,000 to invest and that if you don’t invest it in your business, you will put it in short-term CDs. Rates on short-term CDs are so low at this point that these earnings are unlikely to influence your outcome, therefore we will ignore this for the purposes of this decision.

At this point, the issue becomes your risk tolerance. If you are not risk averse, you will choose the option with the highest expected value—Option 1 which has an expected value of \$65,000. However, you may be quite reasonably risk adverse. For example, suppose you saved the \$100,000 for your 16-year old daughter’s college education. Although Option 1 has the largest expected value, there is also a 50% chance you will lose your investment. If this would mean that you couldn’t afford to send your child to college, you might well decide to leave the money in CDs and pursue Option 3—focus on improving the operation of your restaurant. Alternatively, if you could afford to lose \$20,000, but losing \$100,000 would cause you severe hardship, Option 2 might look good to you.

Therefore, your risk tolerance will determine which option is best for you. There is no one-size-fits-all best answer. However, using a decision tree to lay out your choices clearly allows you to weigh your options more clearly.