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Portfolio Balancing V3
- 1. Introduction to Portfolio Diversification and Balancing Ingemar Hulthage Ingemar_Hulthage @Yahoo.com AAII Computerized Investment Group February 2, 2008
- 6. Trading off risk and return - Preferences
- 9. Notation
- 18. Estimated variance of a portfolio - ( With correlations)
- 21. Minimum Variance Portfolio Efficient Frontier ↑ Minimum Variance portfolio
- 22. Portfolios with the Risk Free security
- 23. Efficient Portfolio Efficient Portfolio Efficient Frontier
Hinweis der Redaktion
- Note that uncertainty refers to deviations from the estimated return. Hence an estimated upward trend doesn’t contribute to the uncertainty. It’s deviations from the expected trend that creates uncertainty.
- Point out the Pareto Optimal securities in the previous slide. Pareto optimality guides the choice between some alternatives, but not all. If the plot included all possible portfolios the Pareto optimal portfolios would define what’s called the Efficient Frontier.
- The portfolio value is, of course, a dollar amount. Returns and standard deviations are in principle dollar amounts too. However, it’s convenient to express allocations, returns and standard deviations as percentages. The portfolio size then disappears out of most equations. Note however, that I’ll use fractions in all formulas and calculations, unless otherwise noted.
- The regular bold symbols are matrices. The ‘T’ indicates a transposed matrixs.
- It’s property of standard deviations, that they don’t add. Instead it’s their squares, the variances, that add. The standard deviation is always positive, hence there is a one to one correspondence to variance. I use the word variance sometimes because it’s shorter. If one doesn’t diversify (n=1) there is no reduction in the portfolio standard deviation.
- I have assumed that all standard deviations are the same and that every security is correlated with every other by some amount.
- Explain which expression is to be minimized and why it’s more complicated, even without correlations.
- Taking the derivative by each x i and lambda gives the first set of equations. The second set follows trivially. The first two sets have n+1 equations. Summing over all x i gives the last equation. Lambda can be solved and put back into the equation for x i . This gives the last equation. Note that it reduces to x i = 1/n if all sigmas are equal. The sum over all sigmas is essentially constant for large n, so that x i is essentially inversely proportional to the variance.
- This is the general expression for variance that needs to be minimized.
- The risk free investment is obviously the safest, if the return is acceptable. The risk free investment can be combined with a portfolio of securities. It turns out that such a combination has lower or equal variance to any portfolio on the Efficient Frontier.
- There is a point on the Efficient Frontier that has a tangent that goes through the risk free interest point. This tangent is called the Efficient Portfolio. The importance of this is that all the points on the line that defines Efficient Portfolios are Pareto Optimal. The points on the Efficient Frontier are not Pareto Optimal, except for the point that coincides with the tangent.