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Lecture 2Introduction to decision theory (cont’d)
Francesco Feri
For of a lottery q, the risk premium 𝑅 𝒒 is defined as
𝑅 𝒒 = 𝐸 𝒒 − 𝐶𝐸 𝒒
where 𝐶𝐸 𝒒 is the certainty equivalent wealth defined as
𝑢 𝐶𝐸 𝒒 = 𝑈 𝒒
Interpretation:
the risk premium 𝑅 𝒒 is the amount of money that an agent is willing to pay toeliminate the risk.
Risk aversion and insurance
Example.
Person A has to “play” the following lottery 𝒒 = 100 $, 0.5; 64 $, 0.5 .
Assume that his utility function is 𝑢 𝑥 = 𝑥
Compute the risk premium 𝑅 𝒒 .
𝑢 𝐶𝐸 𝒒 = 𝑈 𝒒
𝐶𝐸 𝒒 = 0.5 100 + 0.5 64 = 9
𝐶𝐸 𝒒 = 81
𝐸 𝒒 = 100 ∙ 0.5 + 64 ∙ 0.5 = 82
𝑅 𝒒 = 𝐸 𝒒 − 𝐶𝐸 𝒒 = 82 − 81 = 1 $
Person B utility function is 𝑢 𝑥 = 𝑥.Which is the minimum price that person Awill accept in order to sell the lottery?
Answer: 81 $
Is convenient for person B to buy the lottery?𝑢 𝑟 = 0.5 ∙ 100 + 0.5 ∙ 64 = 82 > 𝑢 81 = 81Answer: yes
Selling the lottery for 81 $ is equivalent to hold the lottery and pay 1 $ to playerB that agrees to pay 18 $ in the bad state (when the outcome is 64 $) and toreceive 18 $ in the good state (when the outcome is 100 $)
For 1 $ player B bears all the risk
Questions set 5
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Consider a person with a current wealth of $ 1.000.000 who face the prospect
of a 25% chance of loosing 190.000 (partial destruction of his house). Assume
that this person has the following utility function: 𝑢 𝑥 = 𝑥.
Compute the maximum premium insurance he is willing to pay in order to get a
full insurance.
Answer:
This person faces the following prospect: 𝑟 = 1.000.000, 0.75, 810.000, 0.25
His utility without insurance is:
𝑈 𝑟 = 1.000.000 ∙ 0.75 + 810.000 ⋅ 0.25 = 750 + 225 = 975
The certainty equivalent gives us the sure amount that gives him the same utility level
𝐶𝐸 = 975 → CE = 950.625
Then the maximum premium insurance that he is willing to pay is
1.000.000 − 950.625 = 49.375
From the side of the insurance company (assuming that it is risk neutral) a fair insurance premium is:
190.000 ∙ 0.25 = 47.500
Note:
1. taking the risk the insurance company faces a lottery (−190.000, 0.25))
2. The fair insurance premium is the minimum price the company accepts to bear the risk
Then the risk premium is 49.375 – 47.500 = 1.875
𝐸 𝑟 = 1.000.000 ∙ 0.75 + 810.000 ∙ 0.25 = 952.500
Note 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝐸 𝑟 − 𝐶𝐸 = 952.500 − 950.625 = 1.875
1. Arrow-Pratt measure of absolute risk-aversion:
𝑟 𝑊 = −𝑢" 𝑊
𝑢′ 𝑊
2. Arrow-Pratt-De Finetti measure of relative risk-aversion orcoefficient of relative risk aversion
𝑟𝑟 𝑊 = −𝑊 ∙ 𝑢" 𝑊
𝑢′ 𝑊
Measuring Risk aversion
8
The Arrow-Pratt measure of absolute risk-aversion
𝑟 𝑊 = −𝑢" 𝑊
𝑢′ 𝑊
is proportional to the amount an individual will pay to avoid a fair gamble.
• Let h be the winnings from a fair bet
E(h) = 0
• Let p be the size of the insurance premium that would make the individual exactly indifferent between taking the fair bet h and paying p with certainty to avoid the gamble
E[U(W + h)] = U(W - p)
9
• We now need to expand both sides of the equation using Taylor’s series
• Because p is a fixed amount, we can use a simple linear approximation to the right-hand side
𝑈(𝑊 − 𝑝) = 𝑈(𝑊) − 𝑝𝑈’(𝑊) + ℎ𝑖𝑔ℎ𝑒𝑟 𝑜𝑟𝑑𝑒𝑟 𝑡𝑒𝑟𝑚𝑠
• For the left-hand side, we need to use a quadratic approximation to allow for the variability of the gamble (h)
𝐸[𝑈(𝑊 + ℎ)] = 𝐸[𝑈(𝑊) − ℎ𝑈’(𝑊) +ℎ2
2𝑈”(𝑊)
+ higher order terms
𝐸[𝑈(𝑊 + ℎ)] = 𝑈(𝑊) − 𝐸(ℎ)𝑈’(𝑊) +𝐸 ℎ2
2𝑈”(𝑊)
+ higher order terms
10
• Remembering that E(h)=0, dropping the higher order terms, and substituting k for
E(h2)/2, we get
𝑈 𝑊 − 𝑝𝑈’ 𝑊 ≅ 𝑈 𝑊 + 𝑘𝑈"(𝑊)
𝑝 ≅ −𝑘𝑈"(𝑊)
𝑈’ 𝑊
11
Risk Aversion and WealthIt is not necessarily true that risk aversion declines as wealth increases
• diminishing marginal utility would make potential losses less serious for high-wealth individuals
• however, diminishing marginal utility also makes the gains from winning gambles less attractive
• the net result depends on the shape of the utility function
Example 1
If utility is quadratic in wealth
𝑈(𝑊) = 𝑎 + 𝑏𝑊 + 𝑐𝑊2 where b > 0 and c < 0
Arrow Pratt’s risk aversion measure is
𝑟 𝑊 = −𝑈"(𝑊)
𝑈’ 𝑊= −
2𝑐
𝑏 + 2 𝑐 𝑊
Risk aversion increases as wealth increases
12
Example 2
If utility is logarithmic in wealth
𝑈 𝑊 = ln 𝑊 where W > 0
Arrow Pratt’s risk aversion measure is
𝑟 𝑊 = −𝑈"(𝑊)
𝑈’ 𝑊=1
𝑊
Risk aversion decreases as wealth increases
Example 3
If utility is exponential
𝑈 𝑊 = −𝑒−𝐴𝑊= −exp(−𝐴𝑊) where A is a positive constant
Pratt’s risk aversion measure is
𝑟 𝑊 = −𝑈"(𝑊)
𝑈’ 𝑊=𝐴2𝑒−𝐴𝑊
𝐴𝑒−𝐴𝑊= 𝐴
Risk aversion is constant as wealth increases
13
Relative Risk Aversion
It seems unlikely that the willingness to pay to avoid a gamble is independent of wealth
A more appealing assumption may be that the willingness to pay is inversely proportional to wealth
Arrow-Pratt-De Finetti measure of relative risk-aversion or coefficient ofrelative risk aversion
𝑟𝑟 𝑊 = 𝑊 ∙ 𝑟 𝑊 = −𝑊 ∙ 𝑢" 𝑊
𝑢′ 𝑊
14
Example
• The power utility function
𝑈(𝑊) = 𝑊𝑅/𝑅 for R < 1, 0
exhibits diminishing absolute relative risk aversion
𝑟 𝑊 = −𝑈"(𝑊)
𝑈’ 𝑊= −𝑅 − 1 𝑊𝑅−2
𝑊𝑅−1= −𝑅 − 1
𝑊
but constant relative risk aversion
𝑟𝑟 𝑊 = 𝑤 𝑟 𝑊 = − 𝑅 − 1 = 1 − 𝑅
Type of Risk-Aversion Example of utility functions
Increasing absolute risk-aversion
𝑢 𝑤 = −𝑒−𝑤2
Constant absolute risk-aversion
𝑢 𝑤 = −𝑒−𝑐∙𝑤
Decreasing absolute risk-aversion
𝑢 𝑤 = ln𝑤
Type of Risk-Aversion Example of utility functions
Increasing relative risk-aversion
𝑤 − 𝑐𝑤2
Constant relative risk-aversion
ln(𝑤)
Decreasing relative risk-aversion −𝑒2∙𝑤
−12
Methods for reducing risks• We have seen that risk-averse people will avoid gambles and other risky situations if
possible.
• Often it is impossible to avoid risk entirely.
• Our analysis thus far implies that people would be willing to pay something to at least reduce these risks if they cannot be avoided entirely.
• Four methods
• Insurance
• Diversification
• Flexibility
• Information
Insurance
• A risk-averse person would always want to buy fair insurance to cover any risk he orshe faces.
• No insurance company could afford to stay in business if it offered fair insurance
Several factors make insurance difficult or impossible to provide. Apart from the natureand type of risk we pay attention to the two following factors:
1) Adverse selection problem. Individuals may know more about the likelihood that they will suffer a loss than the insurance company. Only the ‘‘worst’’ customers (those who expect larger or more likely losses) may end up buying an insurance policy.
2) Moral hazard problem. Another problem is that having insurance may make customers less willing to take steps to avoid losses, for example, driving more recklessly with auto insurance.
Diversification
By spreading risk around, it may be possible to reduce the variability of an outcomewithout lowering the expected payoff.
The most familiar setting in which diversification comes up is in investing in financialproducts
Example
a person has wealth W to invest.
This money can be invested in two independent risky assets, 1 and 2, which have equalexpected values (𝜇1 = 𝜇2) and equal variances (𝜎1
2 = 𝜎22).
Undiversified portfolio(UP):
It includes just one of the assets and earns an expected return of 𝜇𝑈𝑃 = 𝜇1 = 𝜇2 andwould face a variance of 𝜎𝑈𝑃
2 = 𝜎12 = 𝜎2
2
Diversified portfolio (DP):
Let 𝛼1 be the fraction invested in the first asset and 𝛼2 = 1 − 𝛼1 in the second.
The expected return does not depend on the allocation across assets and is the same asfor either asset alone:
𝜇𝐷𝑃 = 𝛼1𝜇1 + 1 − 𝛼1 𝜇2 = 𝜇1 = 𝜇2 = 𝜇𝑈𝑃
The variance will depend on the allocation between the two assets:
𝜎𝐷𝑃2 = 𝛼1
2𝜎12 + 1 − 𝛼1
2𝜎22 = (1 − 2𝛼1 + 2𝛼1
2) 𝜎12 < 𝜎1
2
Then in the DP the risk (variance) is reduced with respect to UP and expected returnsare equal
If the expected return from one of the assets is higher than the other, thendiversification results in a lower expected return. But the benefits from risk reductioncan be great enough that a risk-averse investor would be willing to put some share ofwealth into the asset with the lower expected return
Flexibility
In some situations where a decision cannot be divided, diversification cannot be used.When shopping for a car, a consumer cannot combine the attributes from one modelwith those of another.
The consumer could be unsure of her future needs, i.e. which characteristics will bemore useful in the future.
In these cases the consumer can obtain some of the benefit of diversification by makingflexible decisions: Flexibility allows the person to adjust in the future the initial decision.
Example of flexible decision:
I would prefer a car that can use two type of fuel, gasoline and gas. By this decision inthe future I can use the more convenient fuel. To have this flexibility I am willing to paysome additional cost
Flexibility is related to the concept of options
We distinguish two types of option
• Financial option contract. A financial option contract offers the right, but not the obligation, to buy or sell an asset (such as a share of stock) during some future period at a certain price.
• Real option. A real option is an option arising in a setting outside of financial markets
In the previous example (car using two types of fuel) the option is the fuel that can be used
Three fundamental attributes of the options.
1. they specify the underlying transaction
2. they specify a period over which the option may be exercised.
3. the option contract specifies a price.
A Model of real options
Let 𝑥 embody all the uncertainty in the economic environment. Variable 𝑥 might reflect the price of fossil fuels relative to biofuels.
𝑥 is a random variable ( ‘‘state of the world’’) that can take on possibly many different values.
The individual has some 𝑛 of choices currently available.
Let 𝐴𝑖(𝑥) be the payoffs provided by choice i. Then the payoff of each choice depends on the future (and unknown) value of 𝑥
Without flexibility the individual will choose the option with the higher expected utility
A real option allows to make the choice according to the realized value of 𝑥
Suppose that there are two choices, 1 and 2, where 𝐴1 𝑥 and 𝐴2(𝑥) are the payoffs.
Without flexibility the individual has to choose one alternative: the future utility isdescribed by one of the curves on the right panel.
With flexibility the individual can use choice 1 if 𝑥 < 𝑥’ and choice 2 if 𝑥 > 𝑥’. In thiscase the utility is given by the bold line
Value of the flexibility
With no flexibility the individual will choose the best future alternative. Then her future
utility is given by :
With flexibility, the individual can choose the alternative conditional on the realized value of
x. Then her future utility is:
Suppose F is a fee that has be paid in order to have the possibility to choose the best
alternative after x has be realized.
The individual would be willing to pay the fee as long as
The real option’s value is the highest 𝐹 for which the above inequality is satisfied (it holds
with equality)
Discrete variable
𝑥1, 𝑥2, 𝑥3 have respectively probability 𝑝1, 𝑝2, 𝑝3
𝐸(𝑥) = 𝑝1𝑥1 + 𝑝2𝑥2 + 𝑝3𝑥3
𝜎2 = 𝑝1 𝑥1 − 𝐸(𝑥)2 + 𝑝2 𝑥2 − 𝐸(𝑥)
2 + 𝑝3 𝑥3 − 𝐸(𝑥)2
With 𝑛 possible realizations
𝐸(𝑥) =
𝑖=1
𝑛
𝑝𝑖𝑥𝑖
𝜎2 =
𝑖=1
𝑛
𝑝𝑖 𝑥𝑖 − 𝐸(𝑥)2
Continuous variable
𝐸(𝑥) = −∞
∞
𝑥 𝑓 𝑥 𝑑𝑥
𝜎2 = −∞
∞
𝑥 − 𝐸(𝑥) 2𝑓 𝑥 𝑑𝑥
