Question 1

A risk premium is:

Accepted Answer

A risk premium is the extra amount of money that an individual expects to receive in order to take on a risk. In the case of lotteries or uncertain outcomes, the risk premium is the amount necessary to make an individual indifferent between the lottery and a sure thing. Choice A is describing a premium for insurance, not a risk premium. Choice B describes the cost of insurance, which is not the same as a risk premium. Choice D is describing the difference between expected value and variance, which is not the definition of a risk premium. A risk premium is the extra compensation necessary to make an individual willing to take on a risky investment or situation rather than choosing a guaranteed payoff. The risk premium is the difference between the expected value of a risky option, such as a lottery or investment, and the guaranteed payoff of a sure thing. Therefore, choice C is the correct answer as it defines risk premium accurately.

Question 2

A decision maker can be described with utility that is only a function of income. If this function is linear, the decision maker is:

Accepted Answer

If the utility function is linear, it means that for every additional unit of income, the same amount of utility will be gained. Therefore, the decision maker is indifferent to the level of risk involved in a particular choice, making them risk-neutral. If the utility function is linear, this means that the decision maker places the same value on each additional unit of income. This indicates that the decision maker is risk-neutral, meaning they are indifferent between two options with different payoffs and have no preference for risk.

Question 3

A decision maker can be described with utility which is only a function of income and which exhibits diminishing marginal utility of income. This decision maker is:

Accepted Answer

The diminishing marginal utility of income implies that as income increases, the additional satisfaction derived from each additional unit of income decreases. This suggests that the decision maker is risk-averse, as they would be willing to give up a certain amount of income in order to avoid a certain level of risk. Since the decision maker exhibits diminishing marginal utility of income, they are risk-averse, meaning they are willing to give up a certain amount of income in order to reduce their risk.

Question 4

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is , what is the risk premium associated with this choice?

Accepted Answer

The risk premium is the difference between the expected value of the lottery and the guaranteed payoff. The expected value of the lottery is 0.3 * $30 + 0.7 * $80 = $65. Therefore, the risk premium is $65 - $65 = $0.

Question 5

A decision maker has a utility function U = 10I. This decision maker is:

Accepted Answer

The utility function U = 10I is a linear function, meaning that the decision maker has a constant level of marginal utility for any increase in income. This indicates that the decision maker is risk-neutral because they do not value gains or losses differently. Regardless of the outcome, their utility will increase by a constant amount, which means they are indifferent between choices with different expected values. The utility function U = 10I implies that the decision maker only cares about the monetary outcome (I) and has a linear utility function. Since the coefficient of I is positive and constant (10), the decision maker is risk-neutral. This means that the decision maker chooses between alternatives on the basis of their expected monetary values alone, without regard to the degree of uncertainty involved in their outcomes.

Question 6

An insurance company that sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities should expect to:

Accepted Answer

When an insurance company sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities, it is essentially spreading the risk across many policyholders. The premiums collected should, on average, cover the claims paid out, leading the company to break even, assuming efficient management and accurate risk assessment.

Question 7

Which of the following statements is correct for a decision maker facing a choice between a sure thing and a lottery when the sure thing has the expected payoff of the lottery?

Accepted Answer

Risk-averse decision makers require a positive risk premium to bear risk because they are more concerned about potential losses than potential gains. They will only take on risk if they expect to be compensated for it with a higher expected payoff than the sure thing. Risk-loving decision makers, on the other hand, are willing to bear risk without a positive risk premium because they are more focused on the potential gains rather than potential losses. Risk-neutral decision makers do not have a preference between risk and certainty and will only require a risk premium if the expected payoff of the lottery is lower than the expected payoff of the sure thing. The characteristics of the lottery can affect the size of the risk premium, but ultimately it is the individual decision maker's utility function that determines their preference for risk. Risk-averse decision makers are those who prefer a certain payoff over the uncertain outcome of a lottery with the same expected payoff. Therefore, they would require a positive risk premium to bear risk. Risk-loving and risk-neutral decision makers may not require a risk premium, or may even be willing to pay a premium for the opportunity to take on risk. The risk premium may depend on the characteristics of the lottery, but it is ultimately determined by the decision maker's preference for risk.

Question 8

Consider an insurance policy with $15,000 worth of coverage. If there is a 10% chance the owner of the policy will file a claim for the $15,000 (and a 90% chance they will not file a claim), a fair price for this policy is:

Accepted Answer

The expected value of the policy (the amount the insurance company can expect to pay out on average) is:
(0.10 × $15,000) + (0.90 × $0) = $1,500
Therefore, a fair price for the policy would be $1,500. Option B is the only choice that matches this value.

Question 9

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is $U = \sqrt { I }$ what is the risk premium associated with this choice?

Accepted Answer

Calculating the expected utility of the lottery:

EU(lottery) = (0.3 * sqrt(30)) + (0.7 * sqrt(80)) 
EU(lottery) = 1.732

Utility of the guaranteed payoff: 
U(guaranteed) = sqrt(65) 
U(guaranteed) = 8.06

The risk premium is the difference between the guaranteed payoff and the expected utility of the lottery:

RP = U(guaranteed) - EU(lottery) 
RP = 6.33 - 1.732 
RP = 4.60

Since the risk premium is positive, it means the decision maker is risk-averse and would prefer the guaranteed payoff over the lottery. Therefore, the best choice would be the guaranteed payoff of $65.

Question 10

$$\text { A decision maker has a utility function } U = \sqrt { I } \text {. This decision maker is: }$$

Accepted Answer

The utility function is concave, which means that the decision maker exhibits decreasing marginal utility of income. This is a characteristic of risk-averse individuals who prefer a certain outcome to a risky one with the same expected value. The decision maker's utility function is U = sqrt(I), which implies that the marginal utility of income decreases as income increases. In other words, as the decision maker gets richer, the additional utility they receive from an extra dollar of income diminishes. This is a characteristic of risk aversion, as risk-averse decision makers are willing to pay to avoid the possibility of losing money even if the potential gains are high. Thus, the decision maker is risk-averse.

Question 11

A fairly-priced insurance policy is one in which:

Accepted Answer

A fairly-priced insurance policy is one where the premium equals the expected value of the promised payment, covering the expected loss without additional profit.

Question 12

Would you expect an insurance company in the "real world" to sell an insurance policy for exactly the "fairly-priced" level as defined in the text?

Accepted Answer

Insurance companies have to make a profit to cover their operational costs and investments. Therefore, they would not sell an insurance policy at exactly the "fairly-priced" level, as there would be no expected value of profits for the insurance company. This is because the expected value of profits for the insurance company would be zero, and the insurance company would not be able to sustain itself in the long run. Insurance companies are in the business of making a profit, and therefore selling a policy at the "fairly-priced" level, which includes all expected costs, would not generate any profit for the company. Therefore, it is unlikely that an insurance company in the "real world" would sell a policy at this level.

Question 13

$$\text { A decision maker has a utility function } U = I ^ { 2 } + 500 \text {. This decision maker is: }$$

Accepted Answer

A decision maker is considered risk-loving if their utility function is convex, meaning it increases at an increasing rate with wealth. The given utility function $U = I^2 + 500$ is a convex function because its second derivative with respect to income (I), which is $2$, is positive. This indicates that the decision maker's satisfaction increases at an increasing rate as their income increases, characteristic of risk-loving behavior.

Question 14

Suppose a decision maker has a utility function $$U = \sqrt { I }$$ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

Accepted Answer

The expected value of the lottery is:
(0.3)(30) + (0.7)(80) = 24 + 56 = 80
The expected utility of the lottery is:
√(24) + √(56) = 4.9 + 7.5 = 12.4
Therefore, the best choice is to select the lottery, as it has the highest expected utility.

Question 15

Lotteries A and B have the same expected value, but B has larger variance. Which of the following statements is true, all else equal?

Accepted Answer

If the decision maker is risk averse, they will require a higher risk premium for taking on the larger variance associated with lottery B. Therefore, option D is the correct answer. Option A is incorrect because if the decision maker is risk averse, they would prefer the lottery with a smaller variance (lottery A) regardless of the risk premium. Option B is incorrect because the risk premium is not determined by the decision maker's risk neutrality. Option C is also incorrect because risk-loving decision makers do not necessarily require a positive risk premium for both lotteries.

Question 16

A risk premium, RP, can be computed with the following formula, where I1 and I2 are the two payoffs to a lottery, with probabilities p and (1-p), respectively:

Accepted Answer

The correct formula for computing a risk premium involves the expected utility of the outcomes, where $U$ represents the utility function, $I_1$ and $I_2$ are the payoffs, $p$ is the probability of $I_1$, and $EV$ is the expected value of the lottery. The risk premium, $RP$, is the difference between the expected value of the lottery and the certain amount that gives the same utility as the lottery's expected utility. Therefore, option C correctly represents the calculation involving utilities and the risk premium.

Question 17

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.)Your utility function is $U = \sqrt { I }$ ,where I is income. If this policy is priced at $40, what is the change in your expected utility if you purchase the policy rather than no insurance?

Accepted Answer

Without insurance:
- Expected utility: √10000 = 100
- Expected utility in case of accident: √8100 = 90

With insurance:
- Expected utility: √9960 = 99.799
- Expected utility in case of accident: √8100 = 90

Therefore, the change in expected utility is:

ΔU = EU with insurance - EU without insurance
ΔU = 99.799 - 100
ΔU = -0.201

So, the change in expected utility is negative, which means that purchasing the insurance decreases your expected utility. Therefore, the best choice is not to purchase the insurance.

Question 18

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.)Your utility function is $U = \sqrt { I }$ ,where I is income. What is the fair price of this policy?

Accepted Answer

The expected cost of the insurance policy is: (0.1)(1,900) = $190. 
Without insurance, the expected utility of income is: (0.9)(√(10,000)) + (0.1)(√(8,100)) ≈ 94.24.
With insurance, the expected utility of income is: (0.9)(√(8,100)) + (0.1)(√(8,100)) ≈ 89.99.
Therefore, the expected utility of income with insurance is lower than the expected utility of income without insurance. This means that buying the insurance policy is not the best choice.

Question 19

Consider a fairly-priced insurance policy that fully indemnifies the purchaser against their loss. This insurance policy would most likely be purchased by:

Accepted Answer

A fully indemnifying insurance policy would be most likely purchased by a risk-averse decision maker, who is more concerned about avoiding losses than potentially gaining from risky outcomes. This type of policy provides complete protection against loss, which is attractive to individuals who prefer a guaranteed outcome rather than taking on risk. Risk-loving decision-makers would be more likely to take on risk, while risk-neutral decision makers would be indifferent towards the fully indemnifying policy as they are not primarily concerned with risk. Therefore, the best choice is C - a risk-averse decision maker. A fully indemnifying insurance policy implies that the purchaser will not bear any risk of loss. As a result, risk-averse decision makers, who seek to avoid risk as much as possible, are likely to be the most interested buyers. Risk-neutral decision makers are indifferent to risk and therefore may not see the need to purchase such a policy, while risk-loving decision makers are unlikely to be interested in forgoing potential gains in exchange for complete protection against loss.

Question 20

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.)Your utility function is $U = \sqrt { I }$ ,where I is income. What is the most you would be willing to pay for this policy (rather than no insurance)?

Accepted Answer

The most one would be willing to pay for insurance is the difference between the expected utility without insurance and the utility of the expected income with insurance. Without insurance, the expected utility is $0.1 \times \sqrt{8100} + 0.9 \times \sqrt{10000}$. With insurance, if you pay a premium $P$, your income will be $10000 - P$ with certainty, so your utility will be $\sqrt{10000 - P}$. Setting these equal and solving for $P$ gives the maximum premium you would be willing to pay. Calculating directly might be complex without the exact math, but the principle is to find $P$ that equates the expected utility without insurance to the utility with insurance. Given the options, $C$ $199 is the closest to representing the maximum amount one would rationally pay for insurance to match the expected utility without insurance.

Quiz 15: Risk and Information

A risk premium is:

A decision maker can be described with utility that is only a function of income. If this function is linear, the decision maker is:

A decision maker can be described with utility which is only a function of income and which exhibits diminishing marginal utility of income. This decision maker is:

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is , what is the risk premium associated with this choice?

A decision maker has a utility function U = 10I. This decision maker is:

An insurance company that sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities should expect to:

Which of the following statements is correct for a decision maker facing a choice between a sure thing and a lottery when the sure thing has the expected payoff of the lottery?

Consider an insurance policy with $15,000 worth of coverage. If there is a 10% chance the owner of the policy will file a claim for the $15,000 (and a 90% chance they will not file a claim) , a fair price for this policy is:

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is $U = \sqrt { I }$ what is the risk premium associated with this choice?

$\text { A decision maker has a utility function } U = \sqrt { I } \text {. This decision maker is: }$

A fairly-priced insurance policy is one in which:

Would you expect an insurance company in the "real world" to sell an insurance policy for exactly the "fairly-priced" level as defined in the text?

$\text { A decision maker has a utility function } U = I ^ { 2 } + 500 \text {. This decision maker is: }$

Suppose a decision maker has a utility function $U = \sqrt { I }$ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

Lotteries A and B have the same expected value, but B has larger variance. Which of the following statements is true, all else equal?

A risk premium, RP, can be computed with the following formula, where I1 and I2 are the two payoffs to a lottery, with probabilities p and (1-p) , respectively:

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is $U = \sqrt { I }$ ,where I is income. What is the fair price of this policy?

Consider a fairly-priced insurance policy that fully indemnifies the purchaser against their loss. This insurance policy would most likely be purchased by:

Quiz 15: Risk and Information

A risk premium is:

A decision maker can be described with utility that is only a function of income. If this function is linear, the decision maker is:

A decision maker can be described with utility which is only a function of income and which exhibits diminishing marginal utility of income. This decision maker is:

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is , what is the risk premium associated with this choice?

A decision maker has a utility function U = 10I. This decision maker is:

An insurance company that sells fairly-priced insurance policies to a large number of individuals with similar realized accident risk probabilities should expect to:

Which of the following statements is correct for a decision maker facing a choice between a sure thing and a lottery when the sure thing has the expected payoff of the lottery?

Consider an insurance policy with $15,000 worth of coverage. If there is a 10% chance the owner of the policy will file a claim for the $15,000 (and a 90% chance they will not file a claim) , a fair price for this policy is:

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is U=IU = \sqrt { I }U=I​ what is the risk premium associated with this choice?

A decision maker has a utility function U=I. This decision maker is: \text { A decision maker has a utility function } U = \sqrt { I } \text {. This decision maker is: } A decision maker has a utility function U=I​. This decision maker is:

A fairly-priced insurance policy is one in which:

Would you expect an insurance company in the "real world" to sell an insurance policy for exactly the "fairly-priced" level as defined in the text?

A decision maker has a utility function U=I2+500. This decision maker is: \text { A decision maker has a utility function } U = I ^ { 2 } + 500 \text {. This decision maker is: } A decision maker has a utility function U=I2+500. This decision maker is:

Suppose a decision maker has a utility function U=IU = \sqrt { I }U=I​ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

Lotteries A and B have the same expected value, but B has larger variance. Which of the following statements is true, all else equal?

A risk premium, RP, can be computed with the following formula, where I1 and I2 are the two payoffs to a lottery, with probabilities p and (1-p) , respectively:

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is U=IU = \sqrt { I }U=I​ ,where I is income. What is the fair price of this policy?

Consider a fairly-priced insurance policy that fully indemnifies the purchaser against their loss. This insurance policy would most likely be purchased by:

A decision-maker is faced with a choice between a lottery with a 30% chance of a payoff of $30 and a 70% chance of a payoff of $80, and a guaranteed payoff of $65. If the decision maker's utility function is $U = \sqrt { I }$ what is the risk premium associated with this choice?

$\text { A decision maker has a utility function } U = \sqrt { I } \text {. This decision maker is: }$

$\text { A decision maker has a utility function } U = I ^ { 2 } + 500 \text {. This decision maker is: }$

Suppose a decision maker has a utility function $U = \sqrt { I }$ and is faced with a lottery where there is a 30% chance of earning $30 and a 70% chance of earning $80. What is the expected utility of this lottery?

Your current disposable income is $10,000. There is a 10% chance you will get in a serious car accident, incurring damage of $1,900. (There is a 90% chance that nothing will happen.) Your utility function is $U = \sqrt { I }$ ,where I is income. What is the fair price of this policy?