Understanding Risk
Describing risk
We need three things to define risk in this class:
- Probability
- Expected value
- Preferences (i.e., a utility function)
1. Probability
Definition: The likelihood that a given outcome will occur.
Important to note the timing here…probability applies to an uncertain event that may have several possible outcomes. For example, I may have a heart attack or I may not. Risk Calculator.
2. Expected value
Definition: The probability weighted average of the payoffs (or costs) associated with all possible outcomes.
For two potential outcomes, \(x_{1}\) and \(x_{2}\), with probabilities \(p_{1}\) and \(p_{2}\):
\(E[x] = p_{1}x_{1} + p_{2}x_{2}\)
Example
What is my expected cost?
- Two possible outcomes: heart attack or no heart attack
- 10% chance of having a heart attack
- Cost of $100,000 if I have a heart attack (but I will survive and recover)
Answer
I will incur a cost of $100,000 with 10% probability. So my expected cost is just \(E[cost]=0.1*100,000 =\) 10,000.
3. Preferences
Definition: Preferences take the form of a utility function, \(u(x)\), which tells us how much we benefit get from some consumption bundle, \(x\).
Expected utility combines expected value and utility…
\(E[u(x)] = p_{1}u(x_{1}) + p_{2}u(x_{2})\)
Risk preferences
With probabilities, expected values, and utilities/preferences, we can now measure preferences toward risk.
- Risk averse: We prefer to avoid the risky situation. You would rather have the same (or slightly less) with certainty than a lottery over two risky outcomes.
- Risk neutral: Indifferent between the risky situation or that of certainty.
- Risk loving: Prefer the risky situation.
Risk aversion
Most common assumption is that individuals are risk averse. Mathematically, this follows from diminishing marginal utility.
\(u'(x_{1}) > u'(x_{2})\) for \(x_{1} < x_{2}\)
What does this mean in words?
Risk aversion
In-class Problem: Expected values
An individual starts with a wealth of $100,000. With probability 0.3, they will get sick and incur a cost of $40,000.
- What is this person’s expected cost of illness?
- Assume this individual has a utility function of the form, \(u(w) = w^{0.20}\). What is this person’s expected utility?
- Calculate this person’s utility if they were to incur the expected cost of illness. Is this utility higher or lower than what you found in part (2)?
Why purchase health insurance?
Say your utility function is \(u(w)=\sqrt{w}\) and that you’re starting with \(w=\) $100. I propose a lottery in which I flip a coin…heads you win $20 and tails you lose $20.
What is the expected monetary value of this lottery?
What is your utility at this expected value?
What is the expected utility from this lottery?
Answer
Expected wealth is simply \(\frac{1}{2} \times 80 + \frac{1}{2} \times 120 =\) 100, which yields a utility of \(u(w)=\) 10. But your expected utility is \(\frac{1}{2} \times u(w_{heads}) + \frac{1}{2} \times u(w_{tails}) = \frac{1}{2} \times \sqrt{80} + \frac{1}{2} \times \sqrt{120} =\) 9.95.