Risk policy provides a way to make logical, consistent trade-offs between risk and value.

Daniel Bernoulli established a mathematical foundation for *utility functions * in 1738. "Utility" is a term economists and psychologists use to describe value. A utility function, or curve, translates an objective measure, say $ or
*NPV*
$, into how the person *feels* about that quantity.

Bernoulli surveyed people and found that, unsurprisingly, they are risk-averse. He posited that one's risk tolerance is approximately proportional to wealth. No surprise: wealthy people are more able to tolerate more risk.

While there are competing formulas for utility functions, the "exponential utility function" (**EUF**) is the most practical. It is named for the exponential term in the formula. EUF is the easiest and most consistent expression of risk attitude and, hence, risk policy.

I recommend this equation form:

where:

U = |
utility value in risk-neutral dollars (RN$) (in the U.S. and other countries using the $ symbol) |

x = |
objective outcome value, usually
NPV
$ |

r = |
risk tolerance coefficient, typically on the order of 1/5 of the decision maker’s or organization’s net worth |

There are at least two other EUFs that are equivalent but have less meaningful units.

An aside: The EUF equation, above, is similar to the exponential equation for present value discounting that appears in many textbooks..

Here is an EUF scaled for
*r*
= $5,000:

**Exponential Utility Function.** This
concave function
represents a complete risk policy if the

outcomes are objectively valued in
*NPV*
or
*NPV*-equivalents.
The 45° line is for reference

and is the utility function for a risk-neutral person. The red arrows show determining

the utility of $4,000 *NPV*. The Excel equation is =5000*(1-EXP(-4000/5000)), which returns RN$2,753.

An aside: There are other possible utility functions. Mathematicians can be very creative and elaborate. Reseachers have *elicited* utility preferences with a step-sequence of lotteries or by looking at historical decisions. Most produce abysmal curves—not very workable, in my opinion.

Only an EUF has:

- The same buy and sell values. The asset or project value doesn't suddenly change when you make a deal.
- The risk-attitude-adjusted value is the
*certain equivalent*(*CE*, described in the next section). This EUF has the*delta property*:

Adding $*x*to all the outcomes increases the*CE*by $*x*. Similarly, subtracting $*x*from all the outcomes reduces the*CE*by $*x*.

Though not unique to EUF, it is convenient that a positive outcome has positive utility, and a negative outcome has a negative utility.

*r*
is just a scaling factor that determines the curvature of the utility function. On average, people with 10 times as much wealth have
*r*
's 10 times as high. Conceptually, Warren Buffet and I have the same utility curve shape. Mr. Buffet's utility function just has much larger numbers on his *x*- and *y*-axis scales.

The EUF has utility units in **"risk-neutral dollars"** (**RN$**). Here are examples relating utility units to dollars:

- For small values, numerically, U(
*x*) =*x*, though in different units. - RN$100k feels 100k times better than $1.
- RN$100k feels as good as -RN$100k feels bad:

A decision maker with this*r*would be indifferent about a coin toss with outcomes RN$100k and -RN$100k. - The $RN units are more-pleasing. For outcomes small compared to
*r*, the utility value U(*x*) (though $RN units) is about the same as $*x*. The utility value diminishes for large gains (*law of diminishing marginal utility*) and magnifies for large losses.

Expected value utility is the probability-weighted utility outcome. I conform to common practice and call this *expected utility* with the abbreviation *EU*.

The ** expected utility decision rule** is: Choose the alternative with the highest

Though this is sufficient for decision making, I recommend one more step: Convert *r*s to *certain equivalents* (*CE*s). The *CE* is the cash-in-hand value of a risk.

**The Analysis Process with Utility.**

**Here are the steps:**

- Calculate the
*NPV*s for possibles outcomes. - Calculate
*EMV*s for each alternative for comparison. - Convert
*NPV*s to utility units using the decision maker's utility function. - For each alternative, calculate
*EU*. - Convert
*EU*s to*CE*s (using the formula below). - Choose the alternative with the highest
*CE*.

The inverse equation, translating *EU* into
*CE*
is:

The meaning of
*CE*:

- If an alternative has no uncertainty, then its
*CE*is the*NPV*. - 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:

- The
*CE*is the same, whether buying or selling the risky asset. The asset’s value doesn’t suddenly change as soon as you close a transaction. - Adding $
*x*to all the outcomes increases the*CE*by $*x*. Similarly, subtracting $*x*from all the outcomes decreases the*CE*by $*x*. This is thethat is unique to EUF. Of course, the**delta property***EMV*decision policy also has this property with the special EUF as a 45° line.

Determining
*r*
is typically through “lottery” experiments. The link takes you to one example.

Utility
Elicitation
Program
is a free (for most users) automated tool to help you self-assess *r*'s through several lottery types. Access UEP at http://www.maxvalue.com/online.htm.

© 2020 John R. Schuyler. www.maxvalue.com