1. Rate of Return and Risk

2. How much should you be willing to pay for $750 now?
Obviouly $750.
Suppose somebody offers you a gamble of $500 or $1,000, how much would you be willing to pay for that? The answer depends on your utility function or your attitude towards risk

3. Bentham Resurrected

4. For $750 now you are willing to pay $750

5. How Much Would You Be Pay for an even chance of $500 and $1,000

6. Less than $750

7. And for an even chance of $100 and $1400 even less

8. Risk Premium Say your prices for the immediate gambles are as follows:
$750, with certainty-----$750
Even Chances:
$500/$ $700
$100/$ $600

9. Suppose 1 year risk free rate is 10%
In order to get $750 one year from now you would have to pay today $750/1.1=$682
The $500/$1,000 is equivalent to $700 with certainty, for which you would have to pay $700/1.1=$636. The expected value of your gamble is $750, and therefore the expected rate of return is ($750/$636)-1=17.9%
The risk premium for this particular gamble is 7.9%

10. For second gamble risk premium is 15%.
Conclusion: The greater the standard deviation, the larger is the premium

11. Diversification with Negative Correlation
(500,1000) and (100,1400) have the same average return but the second gamble is riskier. Would you ever have it in your portfolio? Suppose the correlation is -1, then a .5;.5 portfolio has the payoffs (550,950). It has the same average but lower risk.

12. Payoffs and Risks Gamble 1 Gamble 3 (.5 of each) 500 100 950 1000 1400
550
Average
750
Variance
62500
422500
40000
S.D.
250
650
200

13. Creating a Risk Free Portfolio from these Two Assets
We know that whatever weights are put on each asset, the mean will be $750. So we need to solve for:
500weight+(1-weight)1400=750
Or Weight=650/900=~.72222….
CHECK that this yields a payment of 750 also with (1,000,100)

14. Risk of a Two-Assets Portfolio
Var(X+Y)=VarX+VarY+2Cov(X,Y)=
=VarX+VarY+ 2sdX*sdY*Correlation
So negative correlation decreases risk.

15. Negative Correlation Commands a Premium
Even though the second gamble is riskier, if I already have the first asset I would be willing to pay more for the second asset than the price calculated above because its addition will lower my risk. In fact, it could reduce my risk to ZERO. How much more?
Instead of treating this as a theoretical question we will treat it as an empirical question.

16. An Investor Will Always Want to Be on the… Market Line
The market line, see graph below, is a line that connects the risk free asset with the market portfolio.
The Market Portfolio
Of all the possible portfolios that includes also a risk free asset there will be one risky portfolio that, when connected to the risk free asset, yields the highest possible line. All investors will choose it, and this will become the market basket.

17. An investor can choose any point on the market line
Therefore, each share will lie on the market line, its position determined by its risk level

18. The Market Line

19. The risk premium for a stock is calculated empirically relative to the Market Risk Premium
Run a regression of the returns of the stock on the returns of the market. If the slope, BETA, comes out to be .27 this means that the risk premium for this stock would be only .27 of the market risk premium.

20. Rate of Return for a Utility
The Government should pay the utility a rate of return that is consistent with its BETA.