19 Risk Attitudes
For more information on these topics, see Allen, Doherty, Weigelt, and Mansfield Chapter 14: Risk Analysis.
19.1 Practice Questions
19.2 Classwork 19: Risk Attitudes
A startup founder has utility function \(U = \sqrt{W}\) where \(W\) is wealth in thousands of dollars. Their current wealth is $100K, so their current utility is \(\sqrt{100} = 10\). They face three investment opportunities:
- Safe project: Guaranteed return of $20K
- Moderate risk: 60% chance of $50K return, 40% chance of $-20K return
- High risk: 40% chance of $150K return, 60% chance of $-50K return
Calculate the expected monetary value of each option.
Show that the expected utilities for each option are 10.95, 10.93, and 10.57.
Which should they choose and why?
How does this illustrate the risk-return tradeoff?
Portfolio Effects: Suppose if you invest in a startup, you will have:
- 50% chance of 5x return ($2M investment turns into $10M)
- 50% chance of total loss ($2M investment turns into $0)
If you have $2M to invest, calculate the expected value if you invest in a startup. Then calculate the expected value if you invest in 2 independent startups with $1M each. You should find that the expected value is the same.
Consider investing in the two independent startups: what are the probabilities that both fail, both succeed, and one succeeds and one fails?
If you have the utility function \(U(W) = \sqrt{W}\) where \(W\) is your wealth in thousands of dollars, should you put all your money into one startup, or invest into two? Show your work and use your answer to the previous question.
How does this relate to portfolio diversification?
Insurance Markets: A homeowner has:
- House value: $500K
- Other wealth: $100K
- Probability of losing house to fire: 1% per year
- Utility function: \(U= \ln(W)\) where \(W\) is wealth in thousands
An insurance company offers full coverage for an annual premium of \(P\).
Show that the actuarially fair premium is $5K/year (the amount the insurance company can charge so that they break even on average).
Show that the maximum the homeowner would pay is around $10.7K/year because the homeowner is indifferent between having $600K in wealth and bearing the risk, or having \(589.3K\) for sure.
If the insurance company has 10,000 similar but independent policies, their portfolio is diversified and they can charge anything between $5K and $10.6548K and expect to make a profit. This demonstrates gains from trade: both parties benefit from the transaction.