You are presented with two investment opportunities.
I. The cost is $30,000 to be paid today. In each of the coming three years, you will obtain $19,000
with 60% chance and $4,000 with 40% chance, beginning from a year from today.
II. The cost is $30,000 to be paid today. With probability p, you will obtain $10,000 in each of the
coming 4 years, and with probability (1-p), you will obtain $10,000 in each of the coming 5 years.
a) If the market rate is 10%, and if your choice depends only on the expected value of either option
(you do not base your decision on the riskiness of either option),what should p be for you to be
indifferent between two options?
b) Suppose that p=0.2 and you pick option II. How much would you be willing to pay now to get paid
exactly 5 years? (That is, how much would you pay to avoid the uncertainty?)
Uber is a ridesharing service headquartered in San Francisco, which operates in multiple
international cities. Currently Uber has $20 million and has the following pattern of potential cash flow
with its planned investment in a new fleet of cars, minivans, and pickups to be used by its customers.
If the company can borrow and lend at 10%, should it invest on a big fleet or small fleet today?
(Answer by drawing and solving a decision tree. Identify each decision and chance node properly)
Here are the specifics of each option
Big fleet: costs $200 million today. The investment results in success with probability 0.60 and in
failure with probability 0.40. If the system is successful, it will produce $120 million per year for the
following 5 years beginning a year from today, and $20 million per year otherwise.
Small Fleet: costs $20 million today. The investment results in success with probability 0.60 and in
failure with probability 0.40. If the system is successful, it will produce $10 million per year for the
following 5 years ($0 otherwise).
One year after the investment (after the first cash flow is realized and uncertainty about
success/failure is resolved), Uber may choose to expand and invest $320 million on a bigger fleet
whose projected cash flow is as described above.
Remember that, for this problem, we assumed no
time value of money. Assume also that you are not required by law to buy car insurance. That is, buying car insurance is a
purely economic decision.
You are considering buying car insurance for the coming year. Whether or not you buy insurance, you
have the following probability distribution over the car accident damages.
with 90% chance you will have no accident
with 7% chance you will have a small accident with $300 worth of damage
with 3% chance, you will have a big accident with $13,000 worth of damage.
The terms of the insurance: Your deductible is $500.
Extension: Assume that if you do not buy insurance and have a big accident, there is a a 60% chance
that your father will pay the damages for you (that is, your payment is $0).
What is the risk of being self-insured (measured by standard deviation,?)?