What is Risk Neutrality?
Risk neutrality, as its name would suggest, is a characteristic of an individual who is indifferent to risk. More specifically, risk neutrality refers to a characteristic where an individual is indifferent between a certain outcome and a risky gamble (i.e. a scenario where the specific outcome is not known in advance). For example, let's suppose that an individual was presented with the following two options:
Both options have an expected value of $50, and a risk-neutral individual will be indifferent between the certain $50 and the coin flip.
The Relationship Between Risk Neutrality and Marginal Utility
Risk neutrality results when an individual experiences constant marginal utility over the outcomes being considered. In other words, if an individual finds each incremental unit of the outcome to be exactly as useful or happiness inducing than the one before, then the individual will exhibit risk neutrality.
Graphically, risk neutrality results when an individual has a utility function that is a straight line (and is upward sloping).
This is because marginal utility is represented by the slope of the utility curve, and a utility function that is a straight line has the same slope regardless of the value of the outcome.
Risk Neutrality and the Indifference to Certainty
The shape of the utility curve for a risk-neutral individual explains why the individual is indifferent between a certain outcome and a gamble with an equivalent expected value. The expected utility of the certain outcome is just given by the utility of that outcome, whereas the expected utility of the gamble lies on the line segment between the possible outcomes at the point where the x-coordinate is equal to the expected value of the gamble.
By definition, the individual prefers the option that gives the highest expected utility. The shape of the utility curve implies that the point on the utility curve at the expected value outcome will always be the same as the point on the expected utility line segment at the expected value. Therefore, the risk-neutral individual will always be indifferent between a certain outcome and a risky one with an equivalent expected value.
An individual's certainty equivalent is the certain outcome that would make the individual exactly as happy as the gamble she is presented with. If a risk-neutral individual is indifferent between a certain outcome and a gamble with an equivalent expected outcome, then it stands to reason that a risk-neutral individual's certainty equivalent is equal to the expected value of the relevant gamble.
This can also be seen graphically, since the point on the utility curve (i.e.
the certain outcome) that gives the risk-averse individual the same level of expected utility as the gamble (given on the line segment between the possible outcomes) will be the same as the expected value.
Risk neutrality explains why explains why insurance companies are willing to offer insurance to risk-averse individuals.
To see why, let's again consider the very simple coin flip scenario, but for a risk-averse individual. Because the individual's certainty equivalent is less than $50 (the expected value), a risk-averse individual would pay a premium in order to get $50 for sure rather than the outcome of the coin flip.
Mathematically, the maximum amount that a risk-averse individual would pay is equal to the difference between the expected value of $50 and the certainty equivalent.
Because of this willingness to pay, an insurance company could charge a premium equal to the difference between the expected value of $50 and the certainty equivalent and agree to give the policy holders a guaranteed $50 in return for taking on the gamble. Because the expected value of the gamble is $50, the insurance company will, on average, have enough money from the gambles in order to fulfill the guarantees to policy holders and then have the amount of the premiums left over as profit. The insurance company will be willing to do this because it is just as happy with the gambles as with certainty. The insurance company is likely to have these risk-neutral preferences because it enjoys the benefits of diversification due to the fact that it encounters the gamble many (theoretically independent) times rather than only once.
Risk neutrality, as its name would suggest, is a characteristic of an individual who is indifferent to risk. More specifically, risk neutrality refers to a characteristic where an individual is indifferent between a certain outcome and a risky gamble (i.e. a scenario where the specific outcome is not known in advance). For example, let's suppose that an individual was presented with the following two options:
- Receiving $50 for sure
- Flipping a coin and receiving $100 if the coin comes up heads and $0 if the coin comes up tails
Both options have an expected value of $50, and a risk-neutral individual will be indifferent between the certain $50 and the coin flip.
The Relationship Between Risk Neutrality and Marginal Utility
Risk neutrality results when an individual experiences constant marginal utility over the outcomes being considered. In other words, if an individual finds each incremental unit of the outcome to be exactly as useful or happiness inducing than the one before, then the individual will exhibit risk neutrality.
Graphically, risk neutrality results when an individual has a utility function that is a straight line (and is upward sloping).
This is because marginal utility is represented by the slope of the utility curve, and a utility function that is a straight line has the same slope regardless of the value of the outcome.
Risk Neutrality and the Indifference to Certainty
The shape of the utility curve for a risk-neutral individual explains why the individual is indifferent between a certain outcome and a gamble with an equivalent expected value. The expected utility of the certain outcome is just given by the utility of that outcome, whereas the expected utility of the gamble lies on the line segment between the possible outcomes at the point where the x-coordinate is equal to the expected value of the gamble.
By definition, the individual prefers the option that gives the highest expected utility. The shape of the utility curve implies that the point on the utility curve at the expected value outcome will always be the same as the point on the expected utility line segment at the expected value. Therefore, the risk-neutral individual will always be indifferent between a certain outcome and a risky one with an equivalent expected value.
An individual's certainty equivalent is the certain outcome that would make the individual exactly as happy as the gamble she is presented with. If a risk-neutral individual is indifferent between a certain outcome and a gamble with an equivalent expected outcome, then it stands to reason that a risk-neutral individual's certainty equivalent is equal to the expected value of the relevant gamble.
This can also be seen graphically, since the point on the utility curve (i.e.
the certain outcome) that gives the risk-averse individual the same level of expected utility as the gamble (given on the line segment between the possible outcomes) will be the same as the expected value.
Risk neutrality explains why explains why insurance companies are willing to offer insurance to risk-averse individuals.
To see why, let's again consider the very simple coin flip scenario, but for a risk-averse individual. Because the individual's certainty equivalent is less than $50 (the expected value), a risk-averse individual would pay a premium in order to get $50 for sure rather than the outcome of the coin flip.
Mathematically, the maximum amount that a risk-averse individual would pay is equal to the difference between the expected value of $50 and the certainty equivalent.
Because of this willingness to pay, an insurance company could charge a premium equal to the difference between the expected value of $50 and the certainty equivalent and agree to give the policy holders a guaranteed $50 in return for taking on the gamble. Because the expected value of the gamble is $50, the insurance company will, on average, have enough money from the gambles in order to fulfill the guarantees to policy holders and then have the amount of the premiums left over as profit. The insurance company will be willing to do this because it is just as happy with the gambles as with certainty. The insurance company is likely to have these risk-neutral preferences because it enjoys the benefits of diversification due to the fact that it encounters the gamble many (theoretically independent) times rather than only once.
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