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The exponential utility function (EUF) is the easiest and most-consistent expression of risk attitude and, hence, risk policy. The equation form that I recommend is:
where
x = | outcome value, usually NPV $ (in the U.S. and other countries using the $ symbol) | |
r = | risk tolerance coefficient, typically on the order of 1/5 of the decision maker’s or organization’s net worth (NW) |
I call the utility units "risk neutral dollars" (RN$). Here is an example in relation to actual money: RN$100k feels 100k times better than $1.
The value of a risk is its cash-in-hand value. This value is the certain equivalent (CE). The CE of a risky project or asset is the most you would be willing to pay to acquire it. Or, if you already own the project or asset, CE is the smallest amount for which you would be willing to sell it.
The EUF has constant risk aversion and features:
For extreme outcome values, the exponential shape may not fit the decision maker’s risk preference. Many academics and practitioners advocate for utility functions with decreasing risk-aversion, such as U = Log(ending wealth). The problem is, if we use something other than an EUF, we lose the two highly-desirable properties, above.
This tip demonstrates some extreme situations where the EUF doesn't work well, I hope you enjoy the following thought experiments.
High-payoff lottery tickets provide extreme examples. Wikipedia (checked as writing this Tip) listed nine PowerBall or Mega Millions jackpots where the award cash value was over $300 million (among 17 tickets). These payout amounts are off the utility charts for 99.999% of the world's population. As such, big payout lotteries provide examples of where the EUF degrades.
This page has similar examples, though with a .50 chance of winning.
Several years ago, I had an online discussion with a colleague, Tim Nieman, who owns a nice boat. Let’s presume that, if Tim gains enormous wealth, he will buy a yacht.
Fun facts about Microsoft co-founder Paul Allen’s super yacht.
I proposed this question: What fraction (F) of your net worth (NW) would you be willing to bet on this 50:50 gamble?
Win, gain $100 million
Lose, lose F x NW.
Assuming an EUF shape, these lottery parameters (F and NW), and Tim’s answer, we can easily solve for r using Microsoft^{®} Excel^{®}’s Goal Seek tool.
Tim’s answer was “1/7” of his NW. We set up Excel to find r that makes CE = $0, and we find that Tim’s imputed r = .206 x NW.
I recently presented a similar question to a reasonably wealthy friend. “How much, as a fraction of your net worth, would you willing to risk on a 50:50 gamble with a $1 billion win payoff. He responded:
When the win value is much greater than r, then U(win value) ≈ r. Then, we can solve for r given NW and F. The formula works out to:
r = F x NW / Ln(2) = 1.443 x F x NW
So, his r appears to be about 1.443 x (1/20) x NW = .072 NW.
To explore this extreme investment situation further, I set up a calculation routine to solve for 100 combinations of:
The chart, below, maps r / NW contours as a function of the two parameters.
Contour chart of r’s. This assumes a 50:50 gamble. The Payout Multiple of Bet Amount, y-axis, is the multiple of the bet amount that is the win amount. The Bet Fraction is the fraction of NW the person is willing to risk. The contour values are r / NW. Here is an example at the red dot: For a decision maker willing to bet 0.10 of his or her NW, and when the Payoff multiple of the bet amount is 3, the inferred r is 0.164 x NW. Thus, an investor with NW = $10 million, willing to bet $1 million for a $3 million win amount has an r ≈ $1.64 million.
I’ve worked with the EUF for years. Still, I was a bit surprised with the chart showing how quickly the r / NW contours turn vertical with even modest Payoff Multiples. The perceived value—utility—starts to diminish quickly as the Payoff approaches r. And at a Payoff at 3r is at .95 of its utility ceiling. This isn’t of too much concern in normal investment decisions, because few projects would have success values greater than r. So, except in extreme situations, the EUF works well.
When I was young, there was a popular TV series, The Millionaire. It was about how an ordinary person’s life changed after receiving an unexpected $1 million, tax-free. It was fun to imagine how it would feel to be “the next millionaire.” Mom said that few people would ever be able to consume a million dollars by prudent spending on themselves. She may have been right. $1 million then is about $9 million in 2017 dollars.
With excess wealth, I think most people would shift their thoughts from self-indulgence toward altruism: helping their community, nation, or the world.
Would you risk all your NW for a chance to save, say, 100 lives? Would you risk your life? I would, though this isn’t for everyone. Suppose you have a Chernobyl or Fukushima to clean up, and the cleaners face certain death from radiation. If you’re willing to pay, say, $5 million to my designated causes, then I’ll step up to be one of your cleanup men.
Perhaps the EUF still applies. For self-serving, financial decisions, r is typically on the order of NW/5. But for an ethical risk policy involving human lives, perhaps r should instead be on the order of someone’s life value.
A popular discussion question of ethics in a life-risking decision dilemma is the Trolley Problem.
If you are interested in determining your r for risky economic decisions, I suggest that you check out the online Utility Elicitation Program app. UEP is available (free, except to my competitors) at http://www.maxvalue.com/online.htm
—John Schuyler, December 2017.
Copyright © 2017 by John R. Schuyler. All rights reserved. Permission to copy with reproduction of this notice.