How To Calculate Certainty Equivalent From Utility Function?

Assume you’ve recently completed your studies at a university. I was thinking about starting a new business and came up with a fantastic concept. You’ve put together a business proposal. Your business requires a $50,000 investment, according to your plan. A profit of 400,000 dollars can be expected in a year. However, there’s a 50/50 risk you’ll lose all of your money and get nothing. Your company proposal is so strong that one of your buddies has offered to buy it from you for $5,000. The condition is that you will not carry out the plan; instead, your friend will. Your friend will put his money on the line and take a chance. Simply put, you will receive $5,000 in exchange for giving up the business chance.

What are your thoughts? Will you accept $5,000 in exchange for the business opportunity? If yes, then this 5,000$ is the Certainty Equivalent for this uncertain opportunity in economic terms, as shown in the decision tree below.

As a result, the amount of money that is comparable in your mind to a specific situation including uncertainty is known as the Certainty Equivalent (or risk). Someone would prefer accept a guaranteed return than take a chance on a higher but unknown return.

Your insurance payment or extended warranty on something very expensive to fix is a practical example of certainty equivalent. Let’s say you bought a used automobile and are concerned that if something goes wrong with it (engine or transmission failure) during the next five years, it will cost you $10,000. The dealer is requesting that if you pay only $2,000 for an extended warranty, they will reimburse the cost in the worst-case scenario. So, you’re not sure whether the automobile will break down or not. If it does not break down, your 2,000-dollar down deposit will be forfeited. However, if anything breaks down that will cost $10,000 to repair, your profit will be 8,000 dollars. It’s obvious that it’s similar to gambling. If you agree to spend 2000 dollars for the peace of mind of knowing that you are covered in the worst-case situation, your certainty equivalent for this gamble is 2000 dollars.

Calculate Certainty Equivalent

Let’s return to your first company strategy. Let’s say you’re not convinced by your friend’s $5,000 selling price for your business plan. However, you are unsure of the appropriate price for selling your business idea. Maybe you should ask for more than $5,000? How are you going to figure it out? Is there a mathematical approach to find that number?

We’ve learnt that a utility function converts a real-world value into a level of satisfaction. If you can model your utility function for monetary gain from a risky investment (like this one), you can compute your certainty equivalent using that utility function.

Let’s say your utility function is U(X), where x is the monetary benefit from a risky investment in the real world.

You may compute the Expected Utility of the aforementioned investment using that utility function. The weighted sum of the utility value, where the weight indicates the probability, is the expected utility of a random variable, as shown by the following statement.

So, whatever utility function you use, you pass that real-world number of 400,000 and -50,000 to it. As a result, you’ll get a figure for Expected Utility E, like the one displayed here.

After you’ve calculated the Expected Utility, you’ll need to compute the Utility Function’s Inverse Function.

If you’re unfamiliar with the concept of Inverse Function, here’s a quick primer. Consider the function f(x) = y. You get diverse output y for various input x, right? The inverse function of f(x) is a function that determines for what input value “x” a particular “y” value will be obtained.

If a function is, for example, its inverse function is. So, if we want to know what value of x we can get f(x) or y = 9, we may use this inverse function to find out. We obtain 3 when we punch in the value “9” to this inverse function. So now we know that we can get a value for f(x) = 9 for input “3.”

Let’s now concentrate on our Expected Utility Function. The inverse function of the Utility function, where the value of X equals the Expected Utility, is the certainty equivalent. If CE stands for the Certainty Equivalent, then

A Visual Demonstration

If you’re familiar with SpiceLogic Rational Will or Decision Tree Software, you’ll have no trouble creating such utility functions. The SpiceLogic Decision Tree program is depicted in the above screenshot.

Now, we can observe from the following chart that the utility function value = 0.5 is obtained when the input value of the Utility function is around 60,000$. As a result, we have,

As a result, your Certainty Equivalent for this risky investment should be 60,000 dollars, not 5,000 dollars. You should offer your friend 60,000 dollars in exchange for selling the business plan based on the utility function provided.

It can be simple to derive a function for Certainty Equivalent if your Utility Function is a specified mathematical function with an inverse function that is also a mathematical function. If your utility function is an Exponential Utility Function, for instance:

Where R stands for Risk Tolerance and x stands for Payoff. Then, instance, in a particular investment or lottery, you have a 0.5 chance of winning ‘W’ and a 0.5 chance of losing ‘L.’

But, if you have our Decision Analysis Software (Decision Tree Software or Rational Will), you won’t have to worry about any of these calculations since the software will do them for you effortlessly utilizing all of the information you provide, as outlined in the following sections.

Using the Decision Tree Software for Certainty Equivalent Calculation

When you click that button, you’ll be asked whether you want to perform a traditional single/multiple criterion analysis or a Cost-Effectiveness analysis. The first choice should be chosen.

Then select “Proceed” from the drop-down menu. You’ll be asked what kind of criterion you’re using. Choose “Numerical Type.” Please keep in mind that you must use the Numerical type requirement to use a Utility function.

Then, from this choice context, you’ll be asked to enter the Minimum and Maximum feasible values for “Money in the bank.” Enter -50,000 as the minimum and 400,000 as the maximum. Set the Unit to $ (or any other currency you like). The calculation is unaffected by the unit value. It is solely for the sake of exhibition.)

The utility function editor will appear once you click to proceed. You can define a utility function in a variety of ways using the utility function editor. The visual drag and move builder, as seen here, is the simplest method.

To add a utility point, double-click on the Chart Panel. You can move any utility point in any direction by dragging it with your mouse. Finally, as indicated in the screenshot, add some utility points and move them about.

Then press the next button. If you want to add another criterion, you will be prompted to do so. Simply select “No” and you will be directed to the Decision Tree’s home page.

You’ll be brought to the Decision Tree diagram page after clicking the “Decision Node” button. This article will teach you how to utilize the Decision Tree tool. Make a Decision Tree like the one in the image below.

We’ve already mentioned that you can alter the Diagram’s appearance. Here’s a brief tip for you.

In the same way, give the Worst-Case a prize of $50,000. Select the node “Sell out the plan” and connect the prize of $5,000 dollars that your friend proposed. Then click the “Options Analyzer” tab and enlarge it. You’ll find a section called “Metric Selection” in the Options Analyzer. That tab should be expanded. Check the “Certainty Equivalent” checkbox in the Metric Selection tab.

The “Certainty Equivalent” number for the “Execute Business Plan” option in the Certainty Equivalent Chart is 60,037$, which is similar to what we saw in the Utility chart.

We can observe from the Certainty Equivalent Chart that the value for “Executing Business Plan” is significantly greater than the value for “Selling Out the Business at 5000$.” As a result, you should not sell out the business plan if you stick to the same utility function. If you can receive nearly 60,000 dollars for the plan, you can sell it.

What is the formula for calculating the certainty equivalent?

How to Use the Certainty Equivalent in Practice The risk premium is determined by subtracting the risk-free rate from the risk-adjusted rate of return. By summing the probability-weighted dollar values of each predicted cash flow, the expected cash flow is computed.

What is the lottery L’s certainty equivalent?

The level of money for which the decision maker is indifferent between the money and the gamble in a choice between the two is the certainty equivalent of a gamble or lottery. Decision makers’ attitudes toward risk are determined using certainty equivalents, which are subsequently represented in the shape of their utility functions. Ordering a set of alternatives can also be done using certainty equivalents. Minimum selling price, maximum buying price, and cash equivalent are all examples of operationalizations of certainty equivalents that have been used in the literature. However, due to income effects, buying and selling prices may be theoretically different.

The expected utility of the gamble must be equal to the value of the certainty equivalent. With this in mind, the relationship between a gamble’s certainty equivalent (CE) and expected value (EV) can show the decision maker’s risk attitude. If CE is successful,

What is the value of certainty?

  • For each conceivable outcome x, the von Neumann-Morgenstern utility function assigns a value v(x), and the average value of v is the value the person assigns to the dangerous outcome. People value risk based on its predicted utility, according to this idea.
  • Although there are various experiment-based criticisms of the theory, the von Neumann approach is the most often used model of risk behavior.
  • Concavity is comparable to a negative second derivative for smoothly differentiable functions.
  • Risk-averse people have von Neumann-Morgenstern utility functions that are concave.
  • The certainty equivalent of a gamble is a sum of money that has the same utility as the bet’s random payoff. If the person is risk averse, the certainty equivalent is smaller than the predicted outcome.
  • The difference between the expected payment and the certainty equivalent is known as the risk premium.

How do you calculate the utility function’s expected utility?

The same general formula that you use to determine expected value is used to calculate expected utility. You multiply probabilities and utility amounts instead of multiplying probabilities and monetary quantities. That is, a gamble’s anticipated utility (EU) equals likelihood x number of usefuls. As a result, EU(A)=80.

What is the best way to tell if a utility function is risk-averse?

We introduced the concept of an expected utility function in the previous section, and explained how people optimize their expected utility when presented with a decision involving known probabilities. As a result, an expected utility function for a gamble g looks like this:

The Bernoulli utility function, u(ai), is the utility function over the outcomes.

This function was a strictly concave logarithmic function in Bernoulli’s formulation, therefore the decision-expected maker’s utility from a gamble was less than its expected value. Because most people are risk averse and prefer a more certain conclusion to a less certain one, Bernoulli anticipated this would hold true. At first glance, this appears to be a plausible assumption.

Is this, however, always the case? Most of us get insurance for our automobiles, homes, and other valuables. The logic is simple: even if the chances of losing or injuring the insured object are tiny, the potential loss is so enormous that most individuals would rather spend a small amount of money for certainty than lose a large quantity with a slight chance of losing it.

There is, however, another side to this: insurance firms. How come it’s logical to sell insurance if it’s logical for people to buy it? Insurance companies, after all, are in the business to make money. How is the insurance business better off accepting these modest amounts and consenting to risk a large loss if we as people are better off paying comparatively small, fixed amounts at regular intervals (as an insurance premium) than risking a large loss? If insurance purchasers are risk averse, what about insurance sellers?

Clearly, Bernoulli’s assumption is not always correct. We can characterize people’s attitudes toward risk considerably more clearly using the von Neumann-Morgenstern framework, without making any assumptions about their conduct.

Different Risk Attitudes

Based on the shape of their respective Bernoulli utility functions, we can divide people’s attitudes toward risk into three types. Let’s take a look at a simple coin toss wager that pays out $10 if the coin lands heads and $20 if the coin lands tails. Of course, the expected value of this wager is: (0.5 * 10) + (0.5 * 20) = $15.

1. Risk-Averse: A person is considered to be risk-averse if the expected value of a bet is greater than the expected utility of the gamble itself. This is a more detailed explanation of Bernoulli’s concept.

A concave Bernoulli utility function, like a logarithmic function, captures risk-averse behavior. For the aforementioned bet, a risk-averse person with a Bernoulli utility function of u(w) = log(w), where w is the outcome, would have an expected utility of:

2. Risk-loving: A person is considered to be risk-loving if their predicted utility from a bet is smaller than their expected utility from the gamble itself. It should be noted, however, that this does not represent typical gaming behavior seen in casinos around the world. A true risk-taker, by this definition, would be willing to risk all of their assets, all they had, on a single throw of the dice.

A convex Bernoulli utility function, such as an exponential function, represents risk-taking behavior. For the aforementioned bet, a risk-taking individual with a Bernoulli utility function of u(w) = w2 would have an expected utility of:

3. Risk-neutral: A person is said to be risk-neutral if their utility of the expected value of a bet is exactly equal to their expected benefit from the gamble itself. In practice, the majority of financial institutions invest in a risk-neutral manner.

A linear Bernoulli function captures risk-neutral behavior. For the bet described above, a risk-neutral person with a Bernoulli utility function of the form u(w) = 2w would have an expected utility of:

The insurance firm in the preceding case is risk-averse, but insurance purchasers are risk-averse. The insurance business makes money because the premiums it collects are greater than the loss’s projected value. See the Applications section for a more extensive explanation of how insurance companies work.

See the section under Advanced Topics for a thorough discussion on determining the degree of risk aversion. See the risk-aversion section under Experiments for a description of risk-aversion experiments.

A Note On The Certainty Equivalent

The certainty equivalent of any gamble g is the amount of money, referred to as CE, that is supplied for certain and provides the consumer with the same utility as the gamble.

Similarly, any gamble’s risk premium is the difference between the gamble’s expected value and its certainty equivalent, i.e.

As a result, a risk-averse person’s certainty equivalent will be less than the gamble’s expected value, resulting in a positive risk-premium. Simply put, risk-averse persons require an additional incentive to encourage them to take on the gamble’s risk.

A risk-neutral consumer will pay no risk premium and have a certainty equal to the gamble’s predicted value. A risk-loving customer, on the other hand, will pay a negative risk premium since she will need an extra incentive to take the expected value rather than the hazardous gamble, and her certainty equivalent will be higher than the gamble’s expected value.

What is the capital budgeting certainty equivalent approach?

The decision-making process is always forward-looking, in the sense that we analyze options and make decisions about our future actions. All of the factors we examine when making future decisions are fraught with danger and uncertainty. Financial decision-making is likewise future-oriented and entails a high level of uncertainty, and such judgments are always a risk-reward trade-off. The concept of certainty equivalent is significant in capital budgeting and is a critical method in financial decision-making.

In the context of capital budgeting, the Certainty Equivalent Approach (CEA) addresses risk concerns by expressing risky future cash flows in terms of certain cash flows that investors will accept today. For assessing risk, the Certainty Equivalent is critical.

Investors clearly expect a return on their investment that is proportional to the risk they take, i.e., the higher the risk, the larger the expected return on that investment.

Uncertain cash flows are turned to certain cash flows by multiplying them by the certainty coefficient, which is the probability of occurrence. The certainty coefficient ranges from 0 to 1.

The table below shows cash inflows over the last five years. The certainty coefficient for cash flows is also provided, which represents the possibility of cash flows occurring.

The initial investment cost is Rs. 300,000. In addition, the annual discount rate is 9%. Determine the NPV and examine it using a certainty equivalent technique.

The present value of all cash inflows minus the initial cost of investing equals the Net Present Value (NPV).

The NPV is positive in this case, indicating that the project is financially viable. As a result, the proposal is approved.

The internal rate of return (IRR) in this case is 12.57 percent, which is higher than the discount rate of 9%. As a result, the proposal is approved.

What is the utility function that should be expected?

“Projected utility” is an economic phrase that describes the utility that an entity or the overall economy is expected to achieve in a variety of situations. The weighted average of all conceivable outcomes under particular circumstances is used to determine expected utility.

What are the benefits of employing the certainty equivalent method?

By reference to the certainty equivalent bond, investors can be confident in the returns of similar other bonds.