### Portfolio Example

Here's a simple example of two securities in a portfolio. The expected return for security A is 5% and 10% for security B. The standard deviation for A is 4%, and for B it is 8%. Notice that security A has a lower return than security B because it has less risk (lower standard deviation). Seven different portfolios of A and B are created. The expected return increases going from portfolio #1 , which has no money invested in security B, to portfolio #7, which has all money invested in security B. The standard deviation for each portfolio depends on the correlation coefficient for the two securities. Three standard deviations are calculated for each portfolio using different correlation coefficients for securities A and B (+1.0, 0.0, and -1.0). With a correlation coefficient of +1.0, the expected return and the standard deviation increases for each portfolio as more money is invested in security B. Not everyone will prefer the same portfolio. People who are more risk averse will prefer a portfolio with little or no security B in it, such as portfolios #1 or #2. On the other hand, people who are less risk averse will prefer a portfolio with a lot of security B in it, such as portfolios #6 or #7. With a correlation coefficient of 0, the expected return of each portfolio increases but the standard deviation initially decreases. Everyone would prefer portfolio #2 to portfolio #1 because portfolio #2 offers a higher expected return (6.25% vs 5%) and a lower standard deviation (3.61% vs 4%). Some people will prefer portfolio #2 to any other portfolio while others may still prefer portfolio #7. It depends on the degree to which a person is risk averse. With a correlation coefficient of -1.0, portfolio #3 is going to be preferred over either portfolios #1 and #2 because it has a higher expected return and lower standard deviation. The standard deviation of portfolio #3 is zero. This means that you could expect a positive return with absolutely no risk. This not a coincidence. If two securities are perfectly negatively correlated as with portfolio #3, there is always some combination of the two securities that yields a zero standard deviation. This is the perfect diversification, but there is one small problem. Perfect negative correlation is extremely rare in the real world, if it exists at all.

Portfolio example

 Security A Security B Expected return 5% 10% Standard deviation 4% 8%

Portfolio equations

Expected return = WArA + WBrB where "W" is the percentage of all money invested in a security (expressed as a decimal) and "r" is the security's expected return.

Standard deviation:

 Portfolio Percentage of money invested in security A WA Percentage of money invested in security B WB Portfolio's expected return Portfolio's standard deviation (correlation coefficient is +1.0) Portfolio's standard deviation (correlation coefficient is 0.0) Portfolio's standard deviation (correlation coefficient is -1.0) 1 1.0 (100%) 0 (0%) 5% 4% 4% 4% 2 .75 .25 6.25 5 3.61 1 3 .6667 .3333 6.67 5.33 3.77 0 4 .50 .50 7.5 6 4.47 2 5 .3333 .6667 8.33 6.67 5.5 4 6 .25 .75 8.75 7 6.08 5 7 0 1.0 10 8 8 8

The relationship between expected return and standard deviation is graphed below for the seven portfolios assuming three different degrees of correlation between securities A and B. Point A represents portfolio #1, and Point B represents portfolio #7. As you go from portfolio #1 to portfolio #7, you take a path from point A to point B. The three paths reflect the three different correlation coefficients assumed in the example.

When a financial security is added to a portfolio, its risk can no longer be measured by standard deviation because some of its risk has been eliminated. In a portfolio the variation in a security's return associated with company-specific factors can be offset by including securities with different company-specific factors. However, because the returns of all securities tend to be positively correlated, some risk remains. This is labeled market risk. In a portfolio the risk of a single security is measured by beta, which measures the correlation of a security's return with that of the entire market. A security with a beta equal to 1.0 means that its return is perfectly positively correlated with the return of the market as a whole. If the market return goes up or down by 6%, the security's return will go up and down by the same 6%. A beta greater than one means that the security's return fluctuates more than the market return. If the market return goes up by 4%, the security's return will go up by more than 4%. A beta of less than one means that, if the market return goes up by 10%, the security's return will go up by less than 10%. The required return of a security in a portfolio is equal to the risk-free return, usually measured by the return on 90-day Treasury bills, plus a risk premium, measured by the market return minus the risk-free return, multiplied by beta. This relationship is known as the Capital Asset Pricing Model:

Last Update: 6 September 2000