Uncertainty – Expected Utility Model, Risk Attitudes
Expected utility from a gamble vs utility from the expected value of the gamble
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Suppose I have £10 and I put all of it on a bet to get £4 if I do not draw a diamond
from a deck of cards.
0.75 chance of having £14, 0.25 chance of having £0.
The expected value of the gamble is £10.50.
The expected utility of the gamble is:
o 0.75 × 𝑢(14) + 0.25 × 𝑢(0)
o E.g., if 𝑢(𝑥) = √𝑥, this is 0.75 × √14 + 0.25 × √0 ≈ 2.81
The utility from the expected value of a gamble is 𝑢(10.5).
o E.g., if 𝑢(𝑥) = √𝑥, this is √10.5 ≈ 3.24
Here, the utility from getting the expected value of a gamble with certainty is higher
than the expected utility from the gamble.
People whose expected utility functions have this property are called risk averse.
Risk Aversion
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The expected utility of the gamble is lower than the utility of the expected value of the
gamble:
𝑢(𝑝1 𝑐1 + 𝑝2 𝑐2 ) > 𝑝1 𝑢(𝑐1 ) + 𝑝2 𝑢(𝑐2 )
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This is the definition of risk aversion.
With general notation:
𝑢(𝐸[𝑐]) > 𝐸[𝑢(𝑐)]
Risk Premium
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Risk premium is the maximum amount a risk-averse person would pay to avoid the
risk (for a particular gamble).
Or you can think of it as what you’d need to be paid to be compensated for taking on
the risk of a particular gamble.
It’s calculated as the difference between the expected value of the gamble and what
you’d be willing to pay for it.
With general notation:
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𝑢−1 (𝐸[𝑢(𝑐)] is called the certainty equivalent (CE) of the gamble. If you had this
amount for sure it would give you the same utility as the expected utility of the
gamble.
So the risk premium can also be written as:
𝐸[𝑐] − 𝐶𝐸
I.e., expected value of gamble minus certainty equivalent.
Risk Loving
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It is also possible to be risk loving, the opposite of risk aversion.
This is preferring gambles over getting their expected value for certain, i.e.,
𝑢(𝑝1 𝑐1 + 𝑝2 𝑐2 ) < 𝑝1 𝑢(𝑐1 ) + 𝑝2 𝑢(𝑐2 )
Risk Aversion and Marginal Utility
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A risk averse utility function is concave.
o There is declining marginal utility of income.
Each extra £1 is worth less the richer you become.
o As a consequence income lost has more impact than income gained.
o A risk averse person would not want to take an actuarially fair gamble (a
gamble where the expected pay-off is zero)
o The level of consumption you need with certainty to make you as well of as a
gamble is sometimes called the certainty equivalent of the gamble, called
𝑐𝑐𝑒
 𝑐𝑐𝑒 is the level of consumption such that 𝑢(𝑐𝑐𝑒 ) = 𝑝1 𝑢(𝑐1 ) + 𝑝2 𝑢(𝑐2 )
o So the risk premium is (𝑝1 𝑐1 + 𝑝2 𝑐2 ) − 𝑐𝑐𝑒
o A concave utility function (i.e., risk aversion) also means the risk premium for
a particular gamble will decrease as your initial level of wealth increases.