Risky business ECO 61 Microeconomic Analysis Udayan Roy Fall 2008
Choices can have unpredictable results • The possible outcomes of a risky decision are called: – States of nature
Probability • Each state of nature has a probability – Probability is a number between 0 and 1, a fraction – The sum of the probabilities of all states of nature is one (1) – Probability of a state of nature may be its relative frequency if and when the risky decision is repeated a large number of times (objective probability) – Or it might simply be the gut feeling of the person making the risky decision (subjective probability)
Payoffs • The financial reward in each state of nature (to the person who takes a risky decision) is called its payoff • The possible outcomes of a risky decision could also be thought of as the payoffs from the decision (instead of the states of nature) • The full list of all possible payoffs and the probability of each payoff is called the probability distribution of the risky decision
Expected payoff • Let P 1, P 2, … , PN be the N possible payoffs of a risky decision, and • Let π1, π2, … , πN be the probabilities of those payoffs • Then the expected payoff of the risky decision is EP = π1 P 1 + π2 P 2 + … + πN PN • This is what the average payoff would be if the risky decision is, hypothetically, repeated numerous times
Variance: a measure of riskiness • The variance of a risky decision is defined as Variance = π1 (P 1 – EP)2 + π2 (P 2 – EP)2 + … + πN (PN – EP)2 • Note that if the payoffs of a risky decisions, P 1, P 2, … , PN, are all close to the expected payoff, EP, then the variance is small. • And this is precisely a case in which the risk is small, because the payoffs, though unpredictable, are all quite similar • Therefore, the variance is an excellent measure of risk • The square root of the variance is called the standard deviation – It is also a popular measure of the riskiness of a risky decision
Example • New Stuff, Inc. is deciding whether to build a large factory or a small factory for its new product • The risk is two-fold: – A competitor has filed for a patent on a similar product. Royalties must be paid if patent is upheld – The demand for the product may be low or high • Here are the payoffs for New Stuff, Inc.
Example: probability distribution • New Stuff, Inc. ’s payoffs and probability distribution • The expected payoff is what would be the average payoff if, hypothetically, New Stuff, Inc. ’s risky decision is repeated numerous times.
Example: variance and std. dev.
Risk Preferences • One can think of a consumption bundle as a list of the quantities of each good consumed in each possible state of nature – That is, each consumption bundle can be thought of as a lottery ticket • The guaranteed consumption line shows the consumption bundles for which the level of consumption does not depend on the state • For bundles that do not lie on this line, the consumer’s payoff is uncertain – One can compute expected consumption for any particular bundle • A constant expected consumption line shows all risky consumption bundles with the same level of expected consumption 11 -10
Figure 11. 3: Consumption Bundles Example Π, probability of sunny weather, is 2/3. Therefore, 1 – Π is the probability of hurricane. It is 1/3. Exercise: Calculate the expectation and variance of bundles A, B, and C. Could you have calculated Π from the diagram alone? 11 -11
Preferences and Indifference Curves • If one bundle guarantees more of every good than a second bundle, a consumer should prefer the first – Reflects the More-Is-Better Principle – Does not have to guarantee a particular level of consumption • Slope of an indifference curve indicates willingness to shift consumption from one state of nature to another – Depends on the probabilities of the states – Change in probabilities changes slopes of indifference curves 11 -12
Figure 11. 4: Preferences for Risky Consumption Bundles 11 -13
Risk Aversion • A person is risk averse if, in comparing a riskless bundle to a risky bundle with the same level of expected consumption, he prefers the riskless bundle – Risk averse individuals do not avoid risk at all costs – Usually willing to accept some risk provided they receive adequate compensation in the form of higher expected consumption 11 -14
Risk Premium • The certainty equivalent of a risky bundle is the amount of consumption which, if provided with certainty, would make the consumer equally well off – This is the consumer’s willingness to pay for the risky bundle • For a risk-averse person, the certainty equivalent of a risky bundle is always less than expected consumption – Providing the same expected consumption with no risk would make the individual better off • The risk premium of a risky bundle is the difference between its expected consumption and the consumer’s certainty equivalent – Certainty equivalent = expected payoff + risk premium 11 -15
Figure 11. 6: Risk Aversion The lottery B is riskier than the lottery A, and has the same expected consumption. Therefore, B is on a lower indifference curve than A is. This reasoning also shows why any indifference curve must be tangent to the constant expected consumption line at precisely the guaranteed consumption line. 11 -16
Expected Utility Functions • Recall from our discussion of consumer behavior without uncertainty, that a consumer’s preferences can be represented by – A map of indifference curves, or – A utility function • For consumer choice under uncertainty, preferences can be represented by – A map of indifference curves, or – An expected utility function
Expected Utility Functions • An expected utility function: – Assigns a benefit level to each possible state of nature based only on what is consumed – Then takes the expected value of those benefits – It is a weighted average of all possible benefit levels using the probability of each level as its weight • Sample expected utility function: 11 -18
Expected Utility and Risk Aversion • Can determine the consumer’s attitude toward risk from the shaper of her benefit function, W(F): – If W(F) is concave (flattens as F increases), she’s risk averse • Example: W(F) = F 0. 5. – If it’s convex (gets steeper as F increases), she’s risk loving • Example: W(F) = F 5. – If it’s linear, she’s risk neutral • Example: W(F) = 5 F. • The greater the concavity of the benefit function, the greater the risk aversion 11 -19
Figure 11. 10: Expected Utility for a Risk. Averse Consumer 11 -20
The Nature of Insurance • People address a wide range of risks by purchasing insurance policies • An insurance policy is a contract that reduces the financial loss associated with some risky event, such as burglary • The purchaser of an insurance policy is essentially placing a bet • Having paid M, the premium, the policy holder receives B, the benefit, if a loss occurs, for a net gain of B – M • If a loss doesn’t occur the consumer loses the premium M – The probability of this (favorable, to the insurance buyer) outcome is Π – The probability of the unfavorable outcome is 1 – Π 11 -21
The Nature of Insurance • Suppose you will get bundle/lottery A if you have no insurance. • This lottery gives – FS units of food with probability Π, and – FH units of food with probability 1 – Π • So, without insurance, expected consumption is ECA = Π FS + (1 – Π) FH. • After buying insurance, you will get the bundle C, which gives – FS – M units of food with probability Π, and – FH + B – M units of food with probability 1 – Π • So, with insurance, expected consumption is ECC = Π (FS – M) + (1 – Π) (FH + B – M).
The Nature of Insurance • hurricane Guaranteed Consumption Line • • Insurance budget line FH + B – M C FH A FS – M FS • sunny When the insurance contract is actuarially fair, both A and C have the same expected consumption, And, therefore, the insurance budget line lies on the constant expected consumption line through A When the insurance contract is actuarially unfair, the expected consumption of C is less than the expected consumption of A, And, therefore, the insurance budget line lies below the constant expected consumption line through A
Actuarial Fairness • An insurance policy is actuarially fair if its expected net payoff is zero • Actuarial fairness requires: • So an actuarially fair insurance premium equals the promised benefit times the probability of a loss • Insurance policies are usually less than actuarially fair because insurance companies must cover their costs of operation • On average purchasing such a policy reduces the purchaser’s expected consumption 11 -24
Actuarial Fairness • Recall that, without insurance, expected consumption is ECA = Π FS + (1 – Π) FH. • And, with insurance, expected consumption is ECC = Π (FS – M) + (1 – Π) (FH + B – M) = Π FS + Π (– M) + (1 – Π) FH + (1 – Π) (B – M) = ECA + Π (– M) + (1 – Π) (B – M). • Therefore, for actuarially fair insurance, ECC = ECA. • Therefore, A and C must lie on the same constant expected consumption line
Demand for Insurance • Risk-averse consumers are willing to purchase insurance because it cancels out other risks • If the insurance is actuarially fair, a risk averse consumer will purchase full insurance – With full insurance, the promised benefit equals the potential loss – This does not depend on degree of risk aversion • If the insurance is less-than-actuarially fair, the amount of insurance purchased depends on degree of risk aversion – Risk neutral consumer will purchase none 11 -26
Figure 11. 12: Demand for Insurance The probability of sunny weather is Π = 2/3. Without insurance, the outcome is A The insurer promises to pay 3 kg of food in case of a hurricane for every 1 kg of food paid as premium This is actuarially fair insurance , because the insurance company’s expected payoff is zero The consumer’s best choice is B. That is, when actuarially fair insurance is available, the consumer will buy full insurance (and, therefore, end up on the guaranteed consumption line) 11 -27
Figure 11. 13: Demand for Unfair Insurance Now the insurer offers to pay 2 kg of food in case of a hurricane for every 1 kg of food paid as premium Now the insurer’s expected payoff is positive. That is, the insurance contract is actuarially unfair The consumer chooses E, which is not on the guaranteed consumption line. That is, the consumer’s uncertainty will be only partially insured 11 -28
Other Methods of Managing Risk • Four other strategies for managing risk • Object of risk management is to make risky activities more attractive by reducing the potential losses while preserving much of the potential gains • Risk sharing involves dividing a risky prospect among several people • Hedging is the practice of taking on two risky activities with negatively correlated financial payoffs • Diversification is the practice of undertaking many risky activities each on a small scale • People also often try to reduce risk through information acquisition 11 -29
Insurance: actuarially fair and unfair Note that the value of actuarially fair insurance = risk premium.
Figure 11. 15: Risk Sharing • Bundle A is initial lottery, and is riskless • By investing, consumer can move to B with higher expected consumption • Consumer prefers to avoid risk associated with B • With partners to split investment and profits, can reach points on the green line • D is most preferred bundle with risk sharing 11 -31
Hedging • Hedging is the practice of taking on two risky activities with negatively correlated financial payoffs – Two variables are negatively correlated if they tend to move in the opposite direction • Bad news on one investment tends to be offset by good news on the other • Insurance is a form of hedging – Benefit paid by a flood insurance policy is perfectly negatively correlated with a loss from flooding 11 -32
Figure 11. 16: Hedging a Risky Venture Maria earns $600 regardless of the weather. She has the opportunity to buy a sunscreen concession at the local beach for $300. If the weather turns out to be sunny, she will earn $600 profit—revenues of $900 minus cost of $300. Her consumption will, therefore, be $1200. If there’s a hurricane, she will have no sales and she will lose her $300 investment. Her consumption will, therefore, be $300. This is bundle A. The probability of sunny weather is 2/3. Maria gets F if she buys both investments. This shows the benefits of hedging. Also, Maria gets G if she has accurate weather info. The value of such info is $300. Maria also has the opportunity to invest $300 in a portable generator distributorship. For this business, the payoffs are exactly the opposite of the sunscreen concession. This is bundle B. 11 -33
Diversification • Diversification is the practice of undertaking many risky activities, each on a small scale, rather than a few risky activities on a large scale – “Don’t put all your eggs in one basket” – Dividing investments among many activities reduces risk • As correlation between the payoffs on the investments increases, the risk-reducing effect of diversification decreases 11 -34
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