Exponential Utility Function

What is the Exponential Utility Function?

In economics and finance, exponential utility refers to a specific form of the Utility Function. This utility function is mainly used for mapping real-world Monetary gain to perceived value. Formally, the exponential utility is given by:

First form of the exponential utility function equation: U(x) equals 1 minus e to the power of negative x over R, where R is the risk tolerance.

Generalized exponential utility function U(x) equals a minus b times e to the power of negative x over R, the form with scaling parameters a and b.

Where R is the Risk Tolerance. x is the real-world value and u(x) is the utility value or perceived value (the value of an outcome in utils). "a" and "b" are essentially scaling parameters. Decision Tree Software can calculate that parameter based on the Minimum and Maximum possible values in the decision context, which is collected from the user.

If you represent real-world money gain on the X-axis and your level of satisfaction on the Y-axis (in terms of 0 to 1, where 0 means no satisfaction and 1 means the highest satisfaction), then the Exponential Utility function will look like this:

Plot of an exponential utility function curve showing concave shape that models risk-averse behavior, with the risk tolerance parameter R controlling concavity.

This utility function is concave, and so it can be used to model risk aversion. In such a utility function, R, the Risk tolerance parameter determines how concave the utility function is, which in turn reflects how risk-averse the decision-maker is. In the above chart, we used the Risk Tolerance value (R) = 1000. Greater values of R (Risk Tolerance) make the exponential utility function flatter, while smaller values make it more concave or more risk-averse. When your risk tolerance is infinite, the above function becomes a straight line equation. Thus, if you are less risk-averse, (you can tolerate more risk), you would assess a greater value for R to obtain a flatter utility function. If you are less tolerant of risk, you would assess a smaller R and have a more curved utility function.

The exponential utility function is mainly used to measure the utility of monetary gain where there is a chance of losing money.

From where this utility function comes from

Well, obviously, this function was not derived. The human brain and behavior are more complex than such a modeling function. It is just an idea that, as such function is concave, it can be used to model the behavior of a risk-averse decision-maker. If you look into this utility function (

Exponential utility function formula U(x) equals 1 minus e to the power of negative x over R, repeated inline as the equation is referenced again in the derivation.

), you will notice that, as x increases, U(x) approaches 1, which means the highest utility. The utility of zero in this equation, U(0), is equal to 0. That makes sense as someone who does not get anything, his utility is 0, right?. Then notice that the utility for any negative x (being in debt) is negative, which makes perfect sense too. So, we can whimsically use this utility function, but of course, there is no scientific ground that proves that human attitude perfectly follows this utility function.

How can the Risk Tolerance (R) be determined?

Approximate method

The Risk Tolerance 'R' has a very intuitive interpretation that makes its assessment very easy.

Consider a decision situation where you can choose to play a lottery or not to play a lottery. If you do not play the lottery, you won't gain or lose anything. But, if you choose to play the lottery, then you can win X with a 50% probability and lose X/2 with a 50% probability. This situation can be depicted by the following decision tree.

Decision tree with a 'Play Lottery' branch (fifty-fifty win X or lose X over 2) and a 'Do not play lottery' branch with payoff zero, used to elicit the certainty equivalent.

Now, you can ask yourself a question, if this X is 1000$, will you play the lottery, or won't play the lottery? (Remember the chance of losing 500$ in that case). If yes, then ask again, what about 2000$, or 10,000$ or more. The more the number you set as a winning number, half of that number can be a loss too. For example, if you set X = 50,000$, then 50% chance that you will win 50,000$ and 50% chance that you will lose 50,000$/2 = 25,000$. Are you willing to play such a lottery where you can lose 25,000$ in the hope of winning 50,000$? If so, that winning value of 50,000 is the value that you can use for R in your exponential utility function. If you think, you can afford to lose even more for the hope of gaining more, i.e., you can lose 100,000$ for the hope of gaining 200,000$. If so, then your Risk Tolerance value can be 200,000. In a word, think about the highest number that can be used for X so that you will be motivated to play the lottery.

Exact method (Certainty Equivalent Based)

Ok, we have demonstrated, how quickly we can approximate the value of the Risk Tolerance "R" by answering a question asked by the above decision tree. Now, let's see how we can get an Exact value of R mathematically. Consider a decision tree again, where, instead of answering the highest value of a winning payoff, you can answer the Certainty Equivalent of a given lottery. That means, say you can make a decision about playing a lottery with a 50% chance of winning the value of X and a 50% chance of losing a value Y. Or you can choose to receive a confirmed (Certain) amount of money, which can be very less than the possible highest outcome of the lottery. This certain amount of payoff is called "Certainty Equivalent".

Anyway, let's consider the following decision tree. You have two options. One option is to play a lottery where you can Win "W" with a 50% probability, and you can lose "L" with a 50% probability. Another option is to receive a certain amount of money without playing the lottery. For example, Your friend is asking if you will like to sell that lottery for a certain amount of money. How much money you will ask for selling that lottery? Say, you asked "CE". Then, that CE is your certainty equivalent.

Decision tree contrasting a Play Lottery branch (fifty-fifty W or L outcomes) with a Sell the lottery branch yielding the Certainty Equivalent CE.

Now, let's see how we can use the concept of Certainty Equivalent to calculate an exact risk tolerance of an Exponential Utility Function.

According to the theory of Expected Utility, and Von Neumann-Morgenstern utility theorem, if you can define a utility function, then your Expected Utility for the given gamble will be equal to your Certainty Equivalent. Because you will be indifferent between two options only if the Expected Utility of the tow options are the same. Here two options are either play the lottery or not to play the lottery for a certain amount of money.

The probabilities of winning and losing are both 0.5.

So, let's formulate the equation, where the EU stands for Expected Utility.

Expected utility of playing the lottery: 0.5 times U(W) plus 0.5 times U(minus L), derived from expected utility theory.

According to the theory of expected utility, the Expected utility of playing the lottery is equal to the expected utility of the Certainty Equivalent (CE).

Equation stating that the expected utility of playing the lottery equals the utility of the certainty equivalent, the fundamental link used to solve for risk tolerance.

Our Utility Function is the Exponential Utility Function which is

Exponential utility function U(x) equals 1 minus e to the power of negative x over R, restated before substituting into the expected-utility equality to solve for R.

So, lets plugin this function to the above equation, after simplifying, we get,

Implicit equation linking the exponential risk tolerance R, win W, loss L, and certainty equivalent CE: (e^(-W/R) + e^(L/R)) / 2 equals e^(-CE/R).

Here, W is the Winning amount from the lottery, L is the loss amount from the lottery, and CE is the certainty equivalent. All of these 3 values are constant. The only variable is R (Risk Tolerance). In order to find that value of R, the above equation needs to be solved. Solving such an equation is not very straight forward. It is a little complicated. But, when you have our Decision Analysis Software (Decision Tree Software or Rational Will), you won't have to worry about solving such an equation by yourself. Our software will solve that for you and tell you the exact value of "R". We will show you how to do that on this page, please keep continue reading.

Calculating Certainty Equivalent

So far, we have developed an equation for finding Risk Tolerance. We can use the same equation to find the Certainty Equivalent of an Exponential Utility Function if all W, L, and R are known. In that case, it will be very easy to solve the equation. Because the left-hand side will just become a constant value. Just take the natural log of both sides and simplify as shown here.

Closed-form equation that expresses the certainty equivalent of a fifty-fifty lottery given an exponential utility function and its risk tolerance R.

Scaling Parameters

In real-life scenarios, you may want a utility function equation where the maximum payoff from an investment or lottery will yield the highest utility value (i.e. 1) and the minimum payoff (or loss) will give the lowest utility value. By introducing 2 parameters "a" and "b", the exponential utility function can be scaled such that,

That means, for different lottery or different scenarios, the value of "a" and "b" will be different.

How to find out the value of "a" and "b"?

Say, your highest utility is 1. For the given investment, your highest possible gain can be H and the lowest possible gain (or loss) can be L.

Then, your utility function with scaling parameters "a" and "b" can be mathematically derived as

Algebraic solution for the scaling parameters a and b of an exponential utility function in terms of the highest gain H and lowest loss L.

Say, in a lottery, you can gain a maximum of 1000$, and lose 500$. Then, setting these values, you can get "a" and "b" as

Numeric example evaluating the scaling parameters a and b for a lottery with maximum gain 1,000 dollars and maximum loss 500 dollars.

In the above equation, R is the Risk Tolerance, as usual. When you use the SpiceLogic Decision Analysis Software (Decision Tree Software or Rational Will), these scaling parameters will be evaluated automatically based on your other inputs.

Marginal Utility

Marginal Utility is a measure that indicates how much a person's utility changes by a little change of payoff. Mathematically, if you differentiate the Utility Function U(x) with respect to the payoff x, then you get the Marginal Utility Function.

Here, the "b" is a scaling parameter, and R is the Risk Tolerance, x is the real-life payoff and U(x) is the utility value for the given payoff x.

Risk Aversions

Risk Aversion is a mathematical function that indicates how risk-averse a decision-maker is. The risk aversion function can be derived from the Utility function. As we explained in the Utility Function chapter that, the absolute risk aversion is

Absolute risk aversion formula: minus U double-prime of x divided by U prime of x, applied to any utility function U(x).

and the relative risk aversion is

Relative risk aversion formula: minus x times U double-prime of x divided by U prime of x, applied to any utility function U(x).

If we apply these operations on a scaled Utility Function equation, we get,

Absolute risk aversion of the exponential utility function, simplifying to the constant 1 over R, confirming exponential utility models constant absolute risk aversion.

Relative risk aversion of the exponential utility function, expressed as x over R, showing how relative risk aversion grows linearly with wealth.

Notice that, the absolute risk aversion of an exponential utility function is a constant (1/R), that is irrespective of wealth. Therefore, the exponential utility function is most appropriate for people whose risk attitude does not change according to the amount of wealth they have. Many individuals might be less risk-averse if they had more wealth, which is the idea of the Bernoulli Utility Function.

Modeling Example

Say, you want to invest in a business where you hope to make a lifetime revenue of 1,000$. But, you also fear that your initial investment of 200$ can be lost with a 50/50 chance. You need to make a decision about that investment. Should you invest or not? You decided to use an Exponential Utility Function to map your monetary gain to a perceived satisfaction. Why? Does not that more money means more satisfaction? Maybe, but more money comes with more risks too. So, you may need to be satisfied as soon as you get a certain target revenue. That's why a utility function makes a big sense.

If you are using the SpiceLogic Decision Tree Analyzer software then you will be greeted with the following screen. If you are using Rational Will software, click the "Decision Tree" button from the home screen to get to this view. Click the button "Set up Criteria".

SpiceLogic Decision Tree Analyzer start page with the 'Set up Criteria Now' button highlighted as the entry point, alongside Decision Node, Chance Node, Markov Model, and Machine Learning options.

Once you click that button, you will be asked, if you want to use a regular single/multiple criteria analysis or Cost-Effectiveness analysis. Choose the first option.

Payoff type selection dialog asking the user to choose between regular single or multiple criteria analysis and a cost-effectiveness analysis.

Then you will be presented with the following screen. Select "Maximize" and enter "Revenue" as shown below.

Criterion setup screen with Maximize selected and 'Revenue' entered as the objective name for the new decision tree analysis.

Then click the "Proceed" button. You will be asked about the type of criterion. Select "Numerical Type". Please remember that, in order to use a Utility function, you need to use the Numerical type criterion.

Criterion type selection dialog offering Subjective Type or Numerical Type for the Revenue criterion; the Numerical Type is required to attach a utility function.

Then you will be asked about the minimum, maximum payoff range from the investment. Enter Minimum = -200 (as you may lose an investment of 200$) and maximum = 1000.

Numerical criterion details for Maximize Revenue: unit dollar, minimum value minus 200, maximum value 1000, with the 'I want to use a utility function' checkbox ticked.

Click Proceed. As you have checked the box "I want to use a utility function...", you will be presented with a utility function editor. Select the "Exponential Utility Function" button.

Utility function editor in Decision Tree Analyzer with the Exponential Utility Function button highlighted, ready to auto-generate the curve with calculated scaling parameters.

In the above screenshot, you can see that an exponential utility function is automatically evaluated for you with calculated scaling parameters.

How Scaling parameters are calculated

You may be curious to know, in the generated utility function, from where these scaling parameters 109.98 and -73.72 come from. The scaling parameters are calculated such that, the maximum payoff will result in the highest utility value which can be 1 or 100, depending on the preference. The lowest payoff will result in the lowest utility value which can be 0, or -1, or -100, depending on the preferences. The preference can be specified from the ribbon as shown here.

Preference unit selector inside the utility function editor with the 0 to 100 scale chosen for the exponential utility examples.

Let's set the utility value scale as 0 to 100. It is up to you but the following examples are used based on 0 to 100 range.

Specifying Risk Tolerance

The most important parameter in the Exponential Utility function is 'Risk Tolerance'. You can directly enter the Risk Tolerance value here, as shown in the following screenshot.

Risk Tolerance input field in the Exponential Utility editor where the user can type R directly to control the concavity of the utility curve.

If you are completely risk-neutral, your Risk Tolerance value should be infinity. You can set that by clicking this radio button.

Risk-neutral utility function plot produced by selecting the infinite Risk Tolerance radio button, yielding a straight line through the payoff range.

Eliciting Risk Tolerance

As we have explained, there are two ways to elicit the Risk Tolerance value (R) of an exponential utility function. Approximate method and exact method. An approximate method is easy to understand and use but as you have our software at hand, why don't you start with the Exact method as all complicated calculations are done by the software.

Exact method

In order to use the Exact method to elicit your Risk Tolerance, click the "Elicit" link as shown below.

Link in the Exponential Utility editor that opens the risk-tolerance elicitation window, offering both the exact and the approximate elicitation methods.

Once you click that link, you will see the following window. Where the "Exact Risk Tolerance" method is checked by default. You will see that a decision tree is already populated with the Best Payoff = Maximum value of the criterion and Worst Payoff = Minimum Value of the criterion, which was entered by you previously. Of course, you can change those numbers using the given sliders. Once you completed setting up the Best Payoff and Worst Payoff, Enter a Certainty Equivalent value using the slider as shown here. You will see a risk tolerance value is calculated instantly.

Exact Risk Tolerance elicitation window in Decision Tree Analyzer showing a pre-populated decision tree where the user enters their certainty equivalent for the lottery.

If you click OK, you will see the calculated risk tolerance is passed to the Exponential Utility function editor.

Auto-generated exponential utility function in the editor after the elicited risk tolerance was passed back from the elicitation dialog.

Approximate method

You can also check how the approximate method can be helpful to find your Risk Tolerance "R". Click the radio button "Approximate Risk Tolerance" as you can see in the following screenshot. There, you will see a decision tree where the Winning amount is set as the maximum value of the criterion entered by you. The losing amount is auto-calculated as half of the winning amount. Using the slider, think about the highest number for what, you are willing to play the lottery given that, half of that winning amount can be lost by 50% chance as well.

Approximate Risk Tolerance elicitation window in Decision Tree Analyzer using a simpler decision tree to derive R without solving the exact equation.

Viewing other derivatives of the generated utility function.

Not only the generated exponential utility function, but you can also see various derivatives of the utility function from the Decision Analysis software (Decision Tree Software, or Rational Will). You will find a set of radio buttons at the bottom of the chart as shown in the following screenshot. Say, you click the radio button for "Marginal Utility", you will see the generated Marginal Utility Function equation, along with a plot as shown here.

Marginal Utility Function view in the utility function editor, selected via the bottom radio buttons that also expose absolute and relative risk aversion plots.

You can find a similar view for Absolute Risk Aversion and Relative Risk Aversion as well.

Finally, model the Decision Tree

Click Ok in your Objective editor when you are done refining your utility function. Then, you will be taken to the Objectives manager page. Click the "Work on Decision Tree" button.

Objectives list page with Maximize Revenue defined and the 'Work on Decision Tree' button highlighted, used to proceed to the tree designer after configuring the utility function.

Then, click the "Decision Node" button to create your decision tree that starts with a decision node.

Decision Node button on the decision tree start screen, used to create a tree whose root is a decision node.

Then, create a decision tree like this. If you are not familiar with how to create the decision tree in our decision tree software, please visit the getting started page. From that page, you will know how to set a payoff to a node.

Sample investment decision tree with 'Invest' and 'Do not invest' branches under the root decision node, used in the exponential utility worked example.

Set payoff 1000$ for the Best outcome, -200$ for the Worst outcome, and set 0 for the 'Do not invest' node.

Finally, when all node payoffs are set, the decision tree will look like this.

Final decision tree after all node payoffs were set, with utility values calculated by the exponential utility function shown on each node.

Note that, the numbers in utils shown on the Decision Tree nodes, are actually calculated utility values. You need to enter the real payoff and then the utility values are calculated based on the utility function. For example, the node "Do not Invest" is showing 86.55 Utils. Why? Because, the exponential utility function we modeled, the utility value for 0 is 86.55. As the real payoff for "do not invest" is 0, the above utility is calculated and shown. For the Worst outcome node, we entered the value = -200, which was evaluated as 0 utility. And we entered the payoff value for the Best Outcome node as 1000, which was evaluated as 100 Utils according to the generated exponential utility function.

Also, notice that the Expected Utility is calculated and shown over the node. As you can see, the expected utility for the "Invest" node is shown as 50 Utils, which is less than the option "Do not invest", therefore, the Node "Do not Invest" is shown highlighted with green color, indicating the recommended strategy.

Click the Utils link on any node, you will see the payoff editor opens up. From there, you can see the payoff and the utility function plot. In that plot, you can also see a green vertical line that indicates where your utility stands in the plot based on the currently set payoff. The line moves as you change the payoff instantly.

Finally, we hope, you enjoy the Decision Analysis software (Decision Tree Analyzer or Rational WIll) while modeling your exponential utility function.