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CHAPTER 6

CHAPTER 6. Efficient Diversification. Diversification and Portfolio Risk. Suppose your portfolio has only 1 stock, how many sources of risk can affect your portfolio? Uncertainty at the market level Uncertainty at the firm level Market risk Systematic or Nondiversifiable

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CHAPTER 6

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  1. CHAPTER 6 Efficient Diversification

  2. Diversification and Portfolio Risk • Suppose your portfolio has only 1 stock, how many sources of risk can affect your portfolio? • Uncertainty at the market level • Uncertainty at the firm level • Market risk • Systematic or Nondiversifiable • Firm-specific risk • Diversifiable or nonsystematic • If your portfolio is not diversified, the total risk of portfolio will have both market risk and specific risk • If it is diversified, the total risk has only market risk

  3. Diversification and Portfolio Risk

  4. Diversification and Portfolio Risk

  5. Figure 6.1 Portfolio Risk as a Function of the Number of Stocks

  6. Covariance and Correlation • Why the std (total risk) decreases when more stocks are added to the portfolio? • The std of a portfolio depends on both standard deviation of each stock in the portfolio and the correlation between them • Example: return distribution of stock and bond, and a portfolio consists of 60% stock and 40% bond state Prob. stock (%) Bond (%) Portfolio Recession 0.3 -11 16 Normal 0.4 13 6 Boom 0.3 27 -4

  7. Covariance and Correlation • What is the E(rs) and σs? • What is the E(rb) and σb? • What is the E(rp) and σp? • E(r) σ Bond 6 7.75 Stock 10 14.92 Portfolio 8.4 5.92

  8. Covariance and Correlation • When combining the stocks into the portfolio, you get the average return but the std is less than the average of the std of the 2 stocks in the portfolio • Why? • The risk of a portfolio also depends on the correlation between 2 stocks • How to measure the correlation between the 2 stocks • Covariance and correlation

  9. Covariance and Correlation • Prob rs E(rs) rb E(rb) P(rs- E(rs))(rb- E(rb)) 0.3 -11 10 16 6 -63 0.4 13 10 6 6 0 0.3 27 10 -4 6 -51 • Cov (rs, rb) = -114 • The covariance tells the direction of the relationship between the 2 assets, but it does not tell the whether the relationship is weak or strong • Corr(rs, rb) = Cov (rs, rb)/ σs σb = -114/(14.92*7.75) = -0.99

  10. Covariance Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of returns s1 = Standard deviation of returns for Security 1 s2 = Standard deviation of returns for Security 2

  11. Correlation Coefficients: Possible Values Range of values for r1,2 -1.0 <r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated If ρ = 0, no correlation

  12. Two Asset Portfolio St Dev – Stock and Bond

  13. Numerical Example: Bond and Stock Returns E(Bond) = 6% E(Stock) = 10% Standard Deviation Bond = 12% Stock = 25% Correlation Coefficient (Bonds and Stock) = 0

  14. Numerical Example: Bond and Stock Case 1: Weights Bond = .5 Stock = .5 What is the E(rp) and σp E(rp) = 8% σp = 13.87% • Average std = (25+12)/2 = 18.5 • By combining stocks, get average return, but the risk is lower than average

  15. Numerical Example: Bond and Stock Case 1: Weights Bond = .75 Stock = .25 What is the E(rp) and σp E(rp) = 7% σp = 10.96% • By combining you get higher return than bond but lower risk than bond • This is power of diversification

  16. Two Asset Portfolio St Dev – Stock and Bond • Std of the portfolio is always smaller than the weighted average of the 2 std in the portfolio. • Std of the portfolio is maximized when the correlation = 1, and is minimized when correlation = -1 • No diversification benefit when the correlation = 1

  17. Figure 6.3 Investment Opportunity Set for Stock and Bonds

  18. Figure 6.4 Investment Opportunity Set for Stock and Bonds with Various Correlations

  19. 6.3 THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET

  20. Extending to Include Riskless Asset • The optimal combination becomes linear • A single combination of risky and riskless assets will dominate

  21. Figure 6.5 Opportunity Set Using Stocks and Bonds and Two Capital Allocation Lines

  22. Dominant CAL with a Risk-Free Investment (F) CAL(O) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) / s [ E(RP) - Rf) / s P] > [E(RA) - Rf) / sA] Regardless of risk preferences combinations of O & F dominate

  23. Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

  24. Figure 6.7 The Complete Portfolio

  25. Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem

  26. 6.4 EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS

  27. Extending Concepts to All Securities • The optimal combinations result in lowest level of risk for a given return • The optimal trade-off is described as the efficient frontier • These portfolios are dominant

  28. Figure 6.9 Portfolios Constructed from Three Stocks A, B and C

  29. Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets

  30. Portfolio Selection • Asset allocation • Security selection • These two are separable!

  31. Asset Allocation • “Asset allocation accounts for 94% of the differences in pension fund performance” • Identify investment opportunities (risk-return combinations) • Choose the optimal combination according to investor’s risk attitude

  32. Optimal Portfolio Construction Step 1: Using available risky securities (stocks) to construct efficient frontier. Step 2: Find the optimal risky portfolio using risk-free asset Step 3: Now We have a risk-return tradeoff, choose your most favorable asset allocation

  33. Expected Portfolio Return, rp Efficient Set Feasible Set Risk, p Feasible and Efficient Portfolios

  34. The feasible set of portfolios represents all portfolios that can be constructed from a given set of stocks. • An efficient portfolio is one that offers: • the most return for a given amount of risk, or • the least risk for a give amount of return. • The collection of efficient portfolios is called the efficient set or efficient frontier.

  35. Expected Return, rp IB2 IB1 Optimal Portfolio Investor B IA2 IA1 Optimal Portfolio Investor A Risk p Optimal Portfolios

  36. What impact does rRF have on the efficient frontier? • When a risk-free asset is added to the feasible set, investors can create portfolios that combine this asset with a portfolio of risky assets. • The straight line connecting rRF with M, the tangency point between the line and the old efficient set, becomes the new efficient frontier.

  37. Efficient Set with a Risk-Free Asset Expected Return, rp Z . B M ^ . rM The Capital Market Line (CML): New Efficient Set . A rRF M Risk, p

  38. What is the Capital Market Line? • The Capital Market Line (CML) is all linear combinations of the risk-free asset and Portfolio M. • Portfolios below the CML are inferior. • The CML defines the new efficient set. • All investors will choose a portfolio on the CML.

  39. Expected Return, rp CML I1 . M . ^ rM C ^ rR C = Optimal Portfolio rRF Risk, p c M

  40. Disadvantages of the efficient frontier approach • The efficient frontier was introduced by Markowitz (1952) and later earned him a Nobel prize in 1990. • However, the approach involved too many inputs, calculations • If a portfolio includes only 2 stocks, to calculate the variance of the portfolio, how many variance and covariance you need? • If a portfolio includes only 3 stocks, to calculate the variance of the portfolio, how many variance and covariance you need? • If a portfolio includes only n stocks, to calculate the variance of the portfolio, how many variance and covariance you need? • n variances • n(n-1)/2 covariances

  41. Single index model

  42. Single index model When we diversify, all the specific risk will go away, the only risk left is systematic risk component Now, all we need is to estimate beta1, beta2, ...., beta n, and the variance of the market. No need to calculate n variance, n(n-1)/2 covariances as before

  43. Estimate beta • Run a linear regression according to the index model, the slope is the beta • For simplicity, we assume beta is the measure for market risk • Beta = 0 • Beta = 1 • Beta > 1 • Beta < 1

  44. Advantages of the Single Index Model • Reduces the number of inputs for diversification • Easier for security analysts to specialize

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