The Value of a Gamble is in the Eye of the Beholder

If you had to pay for the opportunity to participate in a gamble in which there is a fifty percent chance you’ll win $20,000 and a fifty percent chance you’ll lose $10,000 would pay and participate?  If so, how much would you be willing to pay?  The expected value of this gamble is ($20,000 * 0.5) + (–$10,000 * 0.5) = $5,000 so if you’re risk neutral, i.e. willing to “play the long run averages,” you would be willing to pay $5,000 for this opportunity.

Despite the fact that this gamble’s expected value is positive, I’m confident many of you wouldn’t be willing to pay anything close to $5,000 for this opportunity.  In fact, some of you probably wouldn’t be willing to pay any amount for this opportunity.  If after thinking about it you would be willing to pay some amount less than $5,000 to participate you can consider yourself risk averse.  Don’t worry, it’s perfectly normal.

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Everyday Decisions and Your Attitude toward Risk

While the gamble we’ve been discussing is just a simple example you can use to acknowledge and assess your attitude toward risk, the concept has important implications for a variety of decisions you make on a regular basis.   For example, think about a specific stock in your investment portfolio.  First you buy the stock and then you monitor it to see whether its value increases or decreases (with some probability).  In this situation, you’re essentially entering into a gamble like the one above.  You’re choosing to pay a specific amount of money for the opportunity to participate in a gamble in which you could gain additional money with some probability or lose your money with some (hopefully smaller) probability.

You face similar decision situations when deciding whether to purchase a home, start a business, or switch jobs.  Of course these situations may be more complicated than the example, but it is relatively easy to add more decision nodes and chance nodes to the decision tree to make it more realistic.  The important thing to remember is that the values in the decision tree should reflect your degree of risk aversion so that your analysis and decision will appropriately respect your feelings about risk.

Assessing Your Degree of Risk Aversion Using Certainty Equivalents

To incorporate your attitude toward risk into the decision tree, you need to focus on the chance nodes in your tree and derive a certainty equivalent for each one.  A certainty equivalent is an amount of money that would make you indifferent between, on the one hand, receiving that amount of money and walking away from your risky opportunity and, on the other, turning down the money to keep your risky opportunity.  Let’s say you have an 80 percent chance of winning $500,000 and a 20 percent chance of winning $0.  If someone offered to pay you $5,000 for this gamble would you accept the money and walk away from the opportunity?  No way, the person is offering far too little.  If the person offered you $400,000 would you accept and walk away?  Absolutely, you’d probably jump out of your seat to accept the riskless cash.  Now somewhere in between $5,000 and $400,000 there is an amount where you would be indifferent between forgoing the riskless cash and keeping your risky opportunity and accepting the cash and walking away.  This amount, whatever it is for you, is your certainty equivalent for the gamble.  This measure of your attitude toward risk is what you need to calculate for each of the chance nodes in your decision tree.

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Should I Apply for the Grant?

Let’s go through a quick example.  You’re trying to decide whether to apply for a grant worth $210,000.  Because of the application fee and time (i.e. salary costs) it takes you and your coworkers to write the grant, applying for the grant costs you $10,000.  Suppose there is a 75 percent chance you’ll receive the grant and earn a net benefit of $200,000 (i.e. $210,000 – $10,000).  There is a 25 percent chance you won’t receive the grant and your net return will be –$10,000 (i.e. $0 – $10,000).  The expected value of this chance event is ($200,000 * 0.75) + (–$10,000 * 0.25) = $150,000 – $2,500 = $147,500.  There is an amount of money somewhere between $0 and the expected value of $147,500 that would make you indifferent between accepting the riskless cash and walking away from the grant opportunity and forgoing the cash and keeping the grant opportunity.  This amount, a measure of your attitude toward risk, is your certainty equivalent for the grant gamble.  To decide whether to apply for the grant, you would determine this amount and compare it to the value of the alternative “do not apply for the grant”.  We can assume the value of that alternative is $0 since presumably it wouldn’t change your current net monetary position.  If your certainty equivalent for the alternative “apply for the grant” is greater than the value of the alternative “do not apply for the grant” you should apply for the grant.

Make It a Helpful Habit

As you can see, incorporating your attitude toward risk into a decision tree adds an additional step in the process of evaluating your decision situation; however, once you become comfortable with the procedure you can sketch decision trees and adjust the values relatively quickly.  There are certainly times when in-depth analysis of a decision situation would be overkill, but if the situation is complex or the possible consequences are important to you take the time to add a bit more formality to your evaluation to ensure you properly understand the situation and make a decision that respects your healthy aversion to risk.

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