Portfolio Selection, Wealth Management And Market Risk

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Portfolio Selection, Wealth Management And Market Risk

  1. 1. THE PORTFOLIO SELECTION PROBLEM
  2. 2. INTRODUCTION <ul><li>THE BASIC PROBLEM: </li></ul><ul><ul><li>given uncertain outcomes, what risky securities should an investor own? </li></ul></ul>
  3. 3. INTRODUCTION <ul><li>THE BASIC PROBLEM: </li></ul><ul><ul><li>The Markowitz Approach </li></ul></ul><ul><ul><ul><li>assume an initial wealth </li></ul></ul></ul><ul><ul><ul><li>a specific holding period (one period) </li></ul></ul></ul><ul><ul><ul><li>a terminal wealth </li></ul></ul></ul><ul><ul><ul><li>diversify </li></ul></ul></ul>
  4. 4. INTRODUCTION <ul><li>Initial and Terminal Wealth </li></ul><ul><ul><ul><li>recall one period rate of return </li></ul></ul></ul><ul><ul><ul><li>where r t = the one period rate of return </li></ul></ul></ul><ul><ul><ul><li>w b = the beginning of period wealth </li></ul></ul></ul><ul><ul><ul><li>w e = the end of period wealth </li></ul></ul></ul>
  5. 5. INITIAL AND TERMINAL WEALTH <ul><li>DETERMINING THE PORTFOLIO RATE OF RETURN </li></ul><ul><ul><li>similar to calculating the return on a security </li></ul></ul><ul><ul><li>FORMULA </li></ul></ul>
  6. 6. INITIAL AND TERMINAL WEALTH <ul><li>DETERMINING THE PORTFOLIO RATE OF RETURN </li></ul><ul><li>Formula: </li></ul><ul><li>where w 0 = the aggregate purchase price at time t=0 </li></ul><ul><li> w 1 = aggregate market value at time t=1 </li></ul>
  7. 7. INITIAL AND TERMINAL WEALTH <ul><li>OR USING INITIAL AND TERMINAL WEALTH </li></ul><ul><li>where </li></ul><ul><li> w 0 =the initial wealth </li></ul><ul><li> w 1 =the terminal wealth </li></ul>
  8. 8. THE MARKOWITZ APPROACH <ul><li>MARKOWITZ PORTFOLIO RETURN </li></ul><ul><ul><li>portfolio return (r p ) is a random variable </li></ul></ul>
  9. 9. THE MARKOWITZ APPROACH <ul><li>MARKOWITZ PORTFOLIO RETURN </li></ul><ul><ul><li>defined by the first and second moments of the distribution </li></ul></ul><ul><ul><ul><li>expected return </li></ul></ul></ul><ul><ul><ul><li>standard deviation </li></ul></ul></ul>
  10. 10. THE MARKOWITZ APPROACH <ul><li>MARKOWITZ PORTFOLIO RETURN </li></ul><ul><ul><li>First Assumption: </li></ul></ul><ul><ul><ul><li>nonsatiation: investor always prefers a higher rate of portfolio return </li></ul></ul></ul>
  11. 11. THE MARKOWITZ APPROACH <ul><li>MARKOWITZ PORTFOLIO RETURN </li></ul><ul><ul><li>Second Assumption </li></ul></ul><ul><ul><ul><li>assume a risk-averse investor will choose a portfolio with a smaller standard deviation </li></ul></ul></ul><ul><ul><ul><li>in other words, these investors when given a fair bet (odds 50:50) will not take the bet </li></ul></ul></ul>
  12. 12. THE MARKOWITZ APPROACH <ul><li>MARKOWITZ PORTFOLIO RETURN </li></ul><ul><ul><li>INVESTOR UTILITY </li></ul></ul><ul><ul><ul><li>DEFINITION : is the relative satisfaction derived by the investor from the economic activity. </li></ul></ul></ul><ul><ul><ul><li>It depends upon individual tastes and preferences </li></ul></ul></ul><ul><ul><ul><li>It assumes rationality, i.e. people will seek to maximize their utility </li></ul></ul></ul>
  13. 13. THE MARKOWITZ APPROACH <ul><li>MARGINAL UTILITY </li></ul><ul><ul><li>each investor has a unique utility-of-wealth function </li></ul></ul><ul><ul><li>incremental or marginal utility differs by individual investor </li></ul></ul>
  14. 14. THE MARKOWITZ APPROACH <ul><li>MARGINAL UTILITY </li></ul><ul><ul><li>Assumes </li></ul></ul><ul><ul><ul><li>diminishing characteristic </li></ul></ul></ul><ul><ul><ul><li>nonsatiation </li></ul></ul></ul><ul><ul><ul><li>Concave utility-of-wealth function </li></ul></ul></ul>
  15. 15. THE MARKOWITZ APPROACH <ul><li>UTILITY OF WEALTH FUNCTION </li></ul>Wealth Utility Utility of Wealth
  16. 16. INDIFFERENCE CURVE ANALYSIS <ul><li>INDIFFERENCE CURVE ANALYSIS </li></ul><ul><ul><li>DEFINITION OF INDIFFERENCE CURVES : </li></ul></ul><ul><ul><ul><li>a graphical representation of a set of various risk and expected return combinations that provide the same level of utility </li></ul></ul></ul>
  17. 17. INDIFFERENCE CURVE ANALYSIS <ul><li>INDIFFERENCE CURVE ANALYSIS </li></ul><ul><ul><li>Features of Indifference Curves: </li></ul></ul><ul><ul><ul><li>no intersection by another curve </li></ul></ul></ul><ul><ul><ul><li>“ further northwest” is more desirable giving greater utility </li></ul></ul></ul><ul><ul><ul><li>investors possess infinite numbers of indifference curves </li></ul></ul></ul><ul><ul><ul><li>the slope of the curve is the marginal rate of substitution which represents the nonsatiation and risk averse Markowitz assumptions </li></ul></ul></ul>
  18. 18. PORTFOLIO RETURN <ul><li>CALCULATING PORTFOLIO RETURN </li></ul><ul><ul><li>Expected returns </li></ul></ul><ul><ul><ul><li>Markowitz Approach focuses on terminal wealth (W 1 ), that is, the effect various portfolios have on W 1 </li></ul></ul></ul><ul><ul><ul><li>measured by expected returns and standard deviation </li></ul></ul></ul>
  19. 19. PORTFOLIO RETURN <ul><li>CALCULATING PORTFOLIO RETURN </li></ul><ul><ul><li>Expected returns: </li></ul></ul><ul><ul><ul><li>Method One: </li></ul></ul></ul><ul><ul><ul><ul><ul><li>r P = w 1 - w 0 / w 0 </li></ul></ul></ul></ul></ul>
  20. 20. PORTFOLIO RETURN <ul><ul><li>Expected returns: </li></ul></ul><ul><ul><ul><li>Method Two: </li></ul></ul></ul><ul><ul><ul><li>where r P = the expected return of the portfolio </li></ul></ul></ul><ul><ul><ul><li>X i = the proportion of the portfolio’s initial value invested in security i </li></ul></ul></ul><ul><ul><ul><li>r i = the expected return of security i </li></ul></ul></ul><ul><ul><ul><li>N = the number of securities in the portfolio </li></ul></ul></ul>
  21. 21. PORTFOLIO RISK <ul><li>CALCULATING PORTFOLIO RISK </li></ul><ul><ul><li>Portfolio Risk: </li></ul></ul><ul><ul><ul><li>DEFINITION : a measure that estimates the extent to which the actual outcome is likely to diverge from the expected outcome </li></ul></ul></ul>
  22. 22. PORTFOLIO RISK <ul><li>CALCULATING PORTFOLIO RISK </li></ul><ul><ul><li>Portfolio Risk : </li></ul></ul><ul><ul><li>where  ij = the covariance of returns between security i and security j </li></ul></ul>
  23. 23. PORTFOLIO RISK <ul><li>CALCULATING PORTFOLIO RISK </li></ul><ul><ul><li>Portfolio Risk: </li></ul></ul><ul><ul><ul><li>COVARIANCE </li></ul></ul></ul><ul><ul><ul><ul><li>DEFINITION : a measure of the relationship between two random variables </li></ul></ul></ul></ul><ul><ul><ul><ul><li>possible values: </li></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>positive: variables move together </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>zero: no relationship </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>negative: variables move in opposite directions </li></ul></ul></ul></ul></ul>
  24. 24. PORTFOLIO RISK <ul><ul><ul><li>CORRELATION COEFFICIENT </li></ul></ul></ul><ul><ul><ul><ul><li>rescales covariance to a range of +1 to -1 </li></ul></ul></ul></ul><ul><ul><ul><li>where </li></ul></ul></ul>

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