Risk and Return: Capital Market Theory
To find the expected return from James Fromholtz’s investment opportunity, we will use equation 7-3:
where i indexes the various states of nature that are possible. We can picture the states of nature for James’s opportunity as:
Despite the symmetrical appearance of the graph, the outcomes are not symmetrical: There are many more outcomes that are positive than negative. Only the 100% return (probability 5%) is negative; 95% of the weight of the distribution is positive. We could still have a negative expected return if the magnitude of the negative return were large enough to overwhelm the other possible outcomes. However, that won’t happen here, since the +100% return, which also has a 5% chance of occurring, “balances” our one negative outcome. A.
Applying equation 7-3 to our probability distribution of returns, we have: E(r) (0.05) (100%) (0.45) (35%) (0.45) (5%) (0.05) (100%) 18%.
We can see these calculations in the spreadsheet below. Note that the probabilities must sum to 1(100%).
The expected return for this investment is positive, as it will be for all investments on an ex ante basis—people wouldn’t invest if they expected to lose money! (This is what distinguishes investing from gambling.) However, just because the expected return is positive does not mean that I would necessarily invest. The expected return must be sufficient to compensate me for the risk that I bear. Knowing that there is a possibility of a negative outcome is not sufficient as a measure of risk. Therefore, I can’t say whether or not I’d invest in this opportunity—I’d need more information. Part of what I’d need to know we will find out in Problem 8-2 (but again, that won’t be enough!). 8-2.
To find the standard deviation of the probability distribution given in Problem 8-1, we will use equation 7-5:
where I indexes the various states of nature. For James Fromholtz’s opportunity, we have:
We can see the calculations more clearly from the spreadsheet below:
(In equation 7-5, the variance is the quantity under the radical sign; the standard deviation is then the square root of the variance: 8-3.
Mary Guilott is considering creating an equally weighted portfolio of stocks A and B. (“Equally weighted” means that each asset has the same weight, equal to where n is the number of assets. Since we have two assets here, an equally weighted portfolio puts 50% of the total investment into each stock.) To find the expected return of this portfolio, we use equation 8-1:
where i indexes the assets included in the portfolio.
Since Mary is considering two assets, whose expected returns are 15% and 10%, we have: E(rportfolio) (0.50) (15%) (0.50) (10%) 12.5%.
These calculations are detailed in the spreadsheet below:
The expected return is exactly between the two assets’ returns, since the portfolio is equally weighted. Note that we did not use the correlation coefficient here: The expected return on a portfolio is not affected by the assets’ correlation. We will therefore not need to change the expected return as we explore various correlation values below. B.
To find the portfolio’s standard deviation, we first find the variance as:
then we find the standard deviation as the square root of this variance: portfolio . (This is what is done in one step in the text’s equation 8-3.)
Thus, for Mary’s portfolio, we have:
This result will change if we change the correlation coefficient. The lower is , the lower will be the standard deviation of the portfolio. This is the benefit of diversification. We can see this in the table and chart below:
When the correlation coefficient is 1 (that is, when assets A and B are perfectly negatively correlated), we have reduced the portfolio’s standard deviation to its lowest possible value for this weighting scheme: all the way down to 3%. At 0 (that is,...
Please join StudyMode to read the full document