Investments Chapter 8: Mean-Variance Analysis
Applying the Expected Utility Criterion in Practice: I <ul><li>We generally cannot apply the Expected Utility Criterion in...
Applying the Expected Utility Criterion in Practice: II <ul><li>Solution:   </li></ul><ul><li>The Mean-Variance Framework ...
Fundamentals of Mean-Variance Analysis <ul><li>Assumptions :  </li></ul><ul><li>Investors care only about the mean and the...
Mean-Variance  Indifference Curves: I <ul><li>Represent the subjective trade-off that investors make between risk ( σ ²)  ...
Mean-Variance  Indifference Curves: II <ul><li>Basic characteristics (continued): </li></ul><ul><li>2. Indifference curves...
Mean-Variance Efficiency Criterion <ul><li>Investment  A  dominates investment  B  if  either  of the following conditions...
Computing the Mean and Variance of a Portfolio of Assets <ul><li>Direct Approach </li></ul><ul><li>Conceptually the simple...
Efficient Mean-Variance Frontier: I <ul><li>Investors will focus on the set of portfolios with the smallest variance for a...
Efficient Mean-Variance Frontier: II <ul><li>Using the mean-variance criterion, it can be shown that investors will choose...
Finding the Mean-Variance Frontier <ul><li>Involves solving a quadratic programming problem. </li></ul><ul><li>Is not an e...
Individual Securities <ul><li>The characteristics of individual securities that are of interest are the: </li></ul><ul><ul...
Expected Return, Variance, and Covariance <ul><li>Consider the following two risky asset world. There is a 1/3 chance of e...
Expected Return, Variance, and Covariance
Expected Return, Variance, and Covariance
Expected Return, Variance, and Covariance
Expected Return, Variance, and Covariance
Expected Return, Variance, and Covariance
Expected Return, Variance, and Covariance
The Return and Risk for Portfolios Note that stocks have a higher expected return than bonds and higher risk. Let us turn ...
The Return and Risk for Portfolios The rate of return on the portfolio is a weighted average of the returns on the stocks ...
The Return and Risk for Portfolios The  expected  rate of return on the portfolio is a weighted average of the  expected  ...
The Return and Risk for Portfolios The variance of the rate of return on the two risky assets portfolio is  where   BS  i...
The Return and Risk for Portfolios Observe the decrease in risk that diversification offers. An equally weighted portfolio...
The Efficient Set for Two Assets We can consider other portfolio weights besides 50% in stocks and 50% in bonds … 100% bon...
The Efficient Set for Two Assets We can consider other portfolio weights besides 50% in stocks and 50% in bonds … 100% bon...
The Efficient Set for Two Assets 100% stocks 100% bonds Note that some portfolios are “better” than others. They have high...
Two-Security Portfolios with Various Correlations  100% bonds return  100% stocks    = 0.2    = 1.0    = -1.0
Portfolio Risk/Return Two Securities: Correlation Effects <ul><li>Relationship depends on correlation coefficient </li></u...
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  1. 1. Investments Chapter 8: Mean-Variance Analysis
  2. 2. Applying the Expected Utility Criterion in Practice: I <ul><li>We generally cannot apply the Expected Utility Criterion in practice because: </li></ul><ul><li>1. We do not know the precise shape of the utility function. </li></ul><ul><li>2. We do not know the precise shape of the probability distribution. </li></ul>
  3. 3. Applying the Expected Utility Criterion in Practice: II <ul><li>Solution: </li></ul><ul><li>The Mean-Variance Framework </li></ul><ul><li>An approximation to the expected utility framework </li></ul>
  4. 4. Fundamentals of Mean-Variance Analysis <ul><li>Assumptions : </li></ul><ul><li>Investors care only about the mean and the variance of their portfolio. </li></ul><ul><li>Justification : </li></ul><ul><li>The return distribution can be approximated by a normal distribution </li></ul><ul><li>The utility function can be approximated by a quadratic function. </li></ul><ul><li>This makes them both functions of mean and variance only . </li></ul>
  5. 5. Mean-Variance Indifference Curves: I <ul><li>Represent the subjective trade-off that investors make between risk ( σ ²) and return ( E ( R )). </li></ul><ul><li>Basic characteristics: </li></ul><ul><li>1. All combinations of risk and return on an indifference curve provide an investor with the same level of utility. </li></ul>
  6. 6. Mean-Variance Indifference Curves: II <ul><li>Basic characteristics (continued): </li></ul><ul><li>2. Indifference curves are upward sloping. </li></ul><ul><li>3. Indifference curves are convex, curving towards the expected return axis. </li></ul><ul><li>4. Moving to a higher indifference curve increases an investor’s expected utility. </li></ul><ul><li>5. Like utility functions, indifference curves are subjective. </li></ul>
  7. 7. Mean-Variance Efficiency Criterion <ul><li>Investment A dominates investment B if either of the following conditions hold: </li></ul><ul><li>E ( R A ) ≥ E ( R B ) and σ ² A < σ ² B </li></ul><ul><li>or </li></ul><ul><ul><li>E ( R A ) > E ( R B ) and σ ² A ≤ σ ² B </li></ul></ul>
  8. 8. Computing the Mean and Variance of a Portfolio of Assets <ul><li>Direct Approach </li></ul><ul><li>Conceptually the simplest method, but requires full information about the returns of all the individual assets in all possible states-of-the-world. </li></ul><ul><li>Indirect Approach </li></ul><ul><li>More complicated, but only requires the means, variances and covariances of the individual assets. </li></ul>
  9. 9. Efficient Mean-Variance Frontier: I <ul><li>Investors will focus on the set of portfolios with the smallest variance for a given mean: the mean-variance frontier. </li></ul><ul><li>The mean-variance frontier can be divided into two parts: an efficient frontier and an inefficient frontier. </li></ul>
  10. 10. Efficient Mean-Variance Frontier: II <ul><li>Using the mean-variance criterion, it can be shown that investors will choose their portfolio from the efficient mean-variance frontier (for example portfolios p ’ and p ’’, but NOT p , in this graph): </li></ul>
  11. 11. Finding the Mean-Variance Frontier <ul><li>Involves solving a quadratic programming problem. </li></ul><ul><li>Is not an easy task, but the computations can be performed with standard spreadsheet software. </li></ul>
  12. 12. Individual Securities <ul><li>The characteristics of individual securities that are of interest are the: </li></ul><ul><ul><li>Expected Return </li></ul></ul><ul><ul><li>Variance and Standard Deviation </li></ul></ul><ul><ul><li>Covariance and Correlation </li></ul></ul>
  13. 13. Expected Return, Variance, and Covariance <ul><li>Consider the following two risky asset world. There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund. </li></ul>
  14. 14. Expected Return, Variance, and Covariance
  15. 15. Expected Return, Variance, and Covariance
  16. 16. Expected Return, Variance, and Covariance
  17. 17. Expected Return, Variance, and Covariance
  18. 18. Expected Return, Variance, and Covariance
  19. 19. Expected Return, Variance, and Covariance
  20. 20. The Return and Risk for Portfolios Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks.
  21. 21. The Return and Risk for Portfolios The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:
  22. 22. The Return and Risk for Portfolios The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.
  23. 23. The Return and Risk for Portfolios The variance of the rate of return on the two risky assets portfolio is where  BS is the correlation coefficient between the returns on the stock and bond funds .
  24. 24. The Return and Risk for Portfolios Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation.
  25. 25. The Efficient Set for Two Assets We can consider other portfolio weights besides 50% in stocks and 50% in bonds … 100% bonds 100% stocks
  26. 26. The Efficient Set for Two Assets We can consider other portfolio weights besides 50% in stocks and 50% in bonds … 100% bonds 100% stocks
  27. 27. The Efficient Set for Two Assets 100% stocks 100% bonds Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These compromise the efficient frontier .
  28. 28. Two-Security Portfolios with Various Correlations 100% bonds return  100% stocks  = 0.2  = 1.0  = -1.0
  29. 29. Portfolio Risk/Return Two Securities: Correlation Effects <ul><li>Relationship depends on correlation coefficient </li></ul><ul><li>-1.0 <  < +1.0 </li></ul><ul><li>The smaller the correlation, the greater the risk reduction potential </li></ul><ul><li>If  = +1.0, no risk reduction is possible </li></ul>
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