This is a study which has been written to give a slide-show tour of risk-return tradeoffs under Markwitz Portfolio Theory. The study examines seven two-asset portfolios consisting of asset A, a hypothetical one, plus another hypothetical asset (B,C,D,E,F,G or H) which exhibits a unique degree of covariability with asset A. These portfolios represent the following case situations:

CASE 1. Perfectly Positively Correlated Assets
CASE 2. Perfectly Negatively Correlated Returns
CASE 3. Uncorrelated Returns: Risky Assets
CASE 4. Uncorrelated Returns: Risk-Free Asset Included
CASE 5. Not Perfectly Correlated Assets
CASE 6. Not Perfectly Correlated Assets: Extension 1
CASE 7. Not Perfectly Correlated Assets: Extension 2

In each case examined, the study shows the expected return, the standard deviation, and the variance of the individual asset returns, as well as the following portfolio statistics:

1. the covariance of the asset returns;
2. the correlation coefficient of the asset returns;
3. the ratio of the standard deviations of the asset returns;
4. the asset weights minimizing the portfolio variance;
5. the risk/return measures for various asset mixes; and
6. the portfolio possibilities curve.

By examining the portfolio risk/return tradeoffs, the study illustrates the basic conclusion of the MPT: that while the expected return of a portfolio is a weighted average of the expected returns of the individual assets, the expected risk is generally not a weighted average. The study also demonstrates that the extent of risk reduction under MPT is a function of the relationship between the correlation coefficient and the ratio of the standard deviations of asset returns.

1. The expected return of a portfolio is a weighted average of the expected returns on the individual assets.

2. The risk of a portfolio depends on the following factors:

a. the individual asset risks (standard deviaitions);
b. the covariance (correlation) between their returns;
c. the weights (relative amounts) invested in each asset.

3. Positive investments in assets would reduce the portfolio risk to levels below that of the individual assets so long as the correlation between their returns is less than the ratio of the standard deviation of their returns, calculated by dividing the smaller standard deviation by the larger one.

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