Financial Risk Assessment and Expected Return Calculations

Expected Returns and Risk Analysis

Suppose you won the lottery and had two options: (1) receiving $0.5 million or (2) taking a gamble in which, at the flip of a coin, you receive $1 million if a head comes up but receive zero if a tail comes up.

Lottery Decision Analysis

• a) What is the expected value of the gamble?
  ($1 million)(0.5) + ($0)(0.5) = $0.5 million
• b) Would you take the sure $0.5 million or the gamble?
  You would probably take the sure $0.5 million.
• c) If you chose the sure $0.5 million, would that indicate that you are a risk averter or a risk seeker?
  Risk averter.

Investment Scenarios: Bonds vs. Stocks

Suppose the payoff was actually $0.5 million and that was the only choice. You now face the choice of investing it in a U.S. Treasury bond that will return $537,500 at the end of a year or a common stock that has a 50-50 chance of being worthless or worth $1,150,000 at the end of the year.

Detailed Investment Questions

1. The expected profit on the T-bond investment is $37,500. What is the expected dollar profit on the stock investment?
   ($1.15 million)(0.5) + ($0)(0.5) = $575,000, resulting in an expected profit of $75,000.
2. The expected rate of return on the T-bond investment is 7.5%. What is the expected rate of return on the stock investment?
   $75,000 / $500,000 = 15%.
3. Would you invest in the bond or the stock? Why?
   This depends on the individual's degree of risk aversion.
4. Exactly how large would the expected profit or the expected rate of return have to be on the stock investment to make you invest in the stock, given the 7.5% return on the bond?
   Again, this depends on the person.
5. How might your decision be affected if, rather than buying one stock for $0.5 million, you could construct a portfolio consisting of 100 stocks with $5,000 invested in each?
   Each of these stocks has the same return characteristics as the single stock (a 50-50 chance of being worth zero or $11,500 at year-end). Would the correlation between returns on these stocks matter? Explain.
   
   The situation would be unchanged if the stocks' returns were perfectly positively correlated. Otherwise, the stock portfolio would have the same expected return as the single stock (15%) but a lower standard deviation. If the correlation coefficient (ρ) between each pair of stocks was negative one, the portfolio would be virtually riskless. Since ρ for stocks is generally in the range of +0.35, investing in a portfolio of stocks would definitely be an improvement over investing in the single stock.