Question

Let's say that my utility function over wealth is LaTeX: U=\ln\left(W\right) U = ln ⁡ ( W ) where W is my wealth in dollars. Suppose I currently have $1,000,000 in wealth (oh ye-ah), but my friend Rob offers me an opportunity to invest in his new start-up creating autonomous window-washing robots. [Note: the robots part is a kinda-true story. Ask me sometime!] If the start-up is successful—and we estimate it has a 10 percent chance of success—Rob will pay me $2,000,000. If it fails, however, I am paid nothing. He offers me this opportunity at a price of $100,000 1.What is the expected value of this opportunity? 2.What is the expected *utility* of this opportunity? 3.Would I take Rob's offer given my risk preferences?

Answer 1

For an investment of $100,000, the opportunity pays $2,000,000 (net payoff=2,000,000-100,000=$1,900,000) with a probability of success = 0.10 and pays $0 (net payoff = -$100,000) with a probability of failure= 0.90

1. The expected value of this opportunity is

Ans: the expected value of this opportunity is $100,000

Based on the expected value, I would invest in the opportunity, given that the expected value is positive.

2. My initial Wealth is W=$1,000,000. My utility would be

If the startup is a success, with a probability of 0.1, the net payoff is $1,900,000. My wealth if success would be $1,000,000+1,900,000=$2,900,000. My utility if success would be

If the startup is a failure, with a probability of 0.9, the net payoff is -$100,000. My wealth if failure would be $1,000,000-100,000=$900,000. My utility if failure would be

The expected utility would be

ans: the expected *utility* of this opportunity is 13.8272

3. My utility of wealth while not investing in the opportunity is 13.8155. My expected utility of this opportunity is 13.8272.

Since the expected utility of investing is higher than the current utility, I would take Rob's offer given my risk preference.