Decision Making Flashcards
(20 cards)
Risk vs uncertainty
Risk is about random processes with known probabilities.
Uncertainty is for random processes with unknown probabilities.
Monotonicity axiom
A lottery with a higher probability on the preferred outcome is preferred.
St. Petersburg Paradox
Consider the following game: a fair coin is flipped until the first heads
shows up. If heads occurs on the k
th flip, you receive $2k. Heads at the
first flip gives just $2, while heads at the second gives $4, heads after three trials gives $8, and so on. For instance, after 19 tails you are a millionaire!
How much would you pay to play this game?
The expected value of the game = ∞
Yet Bernoulli reported that people are only willing to pay around $5 to play.
Expected Utility Property
If a set of lotteries and preferences satisfy the monotonicity and independence axioms, then there is a utility function that satisfies the expected utility property.
i.e., a lottery with outcomes x and y occur with probabilities p and 1-p respectively. A function u has the expected utility property if:
u(lottery) = p * u(x) + (1-p) * u(y)
Risk Averse Individuals
Individuals are risk averse if they prefer a fixed payment to a random payment of equal expected value.
Risk aversion and certainty equivalent
What are the utility equations?
E[U(x)] < U(E[x])
E[U(x)] = U(E[x]- π)
E[x]- π is the certainty equivalent of x
π is the risk premium of x
Risk premium factors:
The risk premium is made of two factors:
- The risk aversion factor (subjective)
- The market variability factor (objective)
Risk aversion factor (Arrow-Pratt)
γ is absolute risk aversion.
This is the measurement of an individual’s risk tolerance at a given level of wealth, i.e., how much is an individual willing to lose in absolute dollar terms at a given level of wealth.
How much utility you lose from a small fixed loss.
γ > 0
The more concave the utility function, the more risk averse the individual.
Relative risk aversion
γx is the relative risk aversion of a utility function.
Relative risk aversion considers how an individual’s risk aversion changes relative to wealth, i.e., how does risk aversion change as wealth increases?
How much utility is lost from a percentage loss of wealth.
Linear Utility Function
U’(x) = const.
U’‘(x)= 0
γ = 0
Risk-neutral investor.
How can average risk aversion across investors be measured in a market, given that risk aversion is a subjective concept?
Option prices measure the implied volatility (i.e., market’s expectation of volatility).
The US stock market implied volatility is measured by the VIX index. This is the price of volatility and informs of γ.
CARA
Constant absolute risk aversion
γ is constant
The degree of risk aversion does not depend on wealth.
CRRA
Constant relative risk aversion
The willingness of an individual to bear risk is the same proportionally, regardless of the level of wealth that they have.
What is the certainty equivalent? [Concept]
The CE denotes the amount that a risk averse investor must receive as a risk-free payoff in order to achieve the same utility than from investing into a risky asset.
The CE measures the extent of risk aversion, the lower it is the more risk averse the investor is.
What does the risk premium measure?
The risk premium is the additional payoff that a risk averse investor requires over the risk-free payoff in order to undertake a risky investment.
The Allais Paradox
Individuals are irrational and violate the independence axiom.
The expected utility theory fails.
Psychologically, the difference between 0% and 1% chance of zero is greater than the difference between an 89% and 90% chance of zero.
Such non-linearity is explained by Prospect Theory with S-shaped subjective probability, or probability weighting.
Independence Axiom
If an agent is indifferent between two risky outcomes x and y, she should also be indifferent between the composition of each of the two risky outcomes with a third outcome that occurs with the same probability p.
The Ellsberg Paradox
People tend to avoid unknown probabilities.
Such choice behaviour is known as ambiguity aversion.
Expected utility theory fails to explain ambiguity aversion.
Investment gains vs losses
Value functions (utility of changes of payoffs) differ for gains vs losses.
Deriving utility from gains has concavity (risk aversion).
Deriving utility from losses has convexity (risk seeking).
Insensitivity to losses is similar to concepts of diminishing marginal returns for gains.
The slope of the value function below zero is steeper than the slope above zero - there is a kink at the origin - individuals hate losing much more than they enjoy winning.
What is the rationale for an optimal investment allocation that is based on expected utility?
The rationale behind using expected utility to determine optimal investment allocation lies in capturing an investor’s risk preferences and making consistent, rational decisions under uncertainty.
