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Efficient Diversification

Efficient Diversification. P.V. Viswanath For a First Course in INvestments. Learning Goals. Understanding the insurance principle. Systematic and Unsystematic Risk How to historical data to construct means and variances

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Efficient Diversification

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  1. Efficient Diversification P.V. Viswanath For a First Course in INvestments

  2. Learning Goals • Understanding the insurance principle. • Systematic and Unsystematic Risk • How to historical data to construct means and variances • How to construct the investment opportunity set with two risky assets. • How to construct the investment opportunity set with two risky portfolios and a riskfreeasset. • How to use the single-index model to find your optimal portfolio. • How to use the single-index model if you believe that you have found a positive-alpha security.

  3. Two-Security Portfolio: Return E( ) r1 rp r2 r1 r2 = W1+W2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 = Expected return on Security 1 = Expected return on Security 2

  4. Two-Security Portfolio Return r1 r2 E(rp) = W1r1 +W2r2 W1 = W2 = = = 0.6 0.4 9.28% 11.97% Wi = % of total money invested in security i E(rp) = 0.6(9.28%) + 0.4(11.97%) = 10.36%

  5. Combinations of risky assets When we put stocks in a portfolio, p < Why? When Stock 1 has a return E[r1] it is likely that Stock 2 has a return E[r2] so that rp that contains stocks 1 and 2 remains close to What statistics measure the tendency for r1 to be above expected when r2 is below expected? Covariance and Correlation (Wii) Averaging principle < > E[rp] n = # securities in the portfolio

  6. Portfolio Variance and Standard Deviation Variance of a Two Stock Portfolio:

  7. Expost Covariance Calculations If when r1 > E[r1], r2 > E[r2], and when r1 < E[r1], r2 < E[r2], then COV will be positive. If when r1 > E[r1], r2 < E[r2], and when r1 < E[r1], r2 > E[r2], then COV will be negative. Which will “average away” more risk?

  8. Covariance and Correlation The problem with covariance Covariance does not tell us the intensity of the co-movement of the stock returns, only the direction. We can standardize the covariance however and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the stocks move together. • Standardized covariance is called the correlation coefficient, r

  9. The Effects of Correlation & Covariance on Diversification Asset A Asset B Portfolio AB

  10. The Effects of Correlation & Covariance on Diversification Asset C Asset D Portfolio CD

  11. Naïve Diversification The power of diversification Most of the diversifiable risk eliminated at 25 or so stocks

  12. Two-Security Portfolio: Risk s12 = Variance of Security 1 s22 = Variance of Security 2 Cov(r1r2) = Covariance of returns for Security 1 and Security 2 sp2= W12s12 + W22s22 + 2W1W2 Cov(r1r2)

  13. Calculating Variance and Covariance Ex post 2ABC = ABC = 2XYZ = XYZ = 1.37387 / (10-1) = 0.15265 39.07% 1.57885 / (10-1) = 0.17543 41.88%

  14. COV(ABC,XYZ) = rABC,XYZ = rABC,XYZ = 0.533973 / (10-1) = 0.059330 0.059330 / (0.3907 x 0.4188) COV / (sABCXYZ) = 0.3626

  15. Ex ante Covariance Calculation Using scenario analysis with probabilities the covariance can be calculated with the following formula:

  16. Three-Security Portfolio n or Q = 3 s2p = W12s12 + W22s22 + W32s32 + 2W1W2 Cov(r1r2) + 2W1W3 Cov(r1r3) + 2W2W3 Cov(r2r3)

  17. TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS E(r) 13% r = -1 50%A 50%B r = 0 r = .3 8% r = +1 St. Dev 12% 20% WA = 0% WB = 100% WA = 100% WB = 0% Stock A Stock B

  18. Minimum Variance Combinations -1< r < +1 s 2 - Cov(r1,r2) 2 = W1 s 2 s 2 - 2Cov(r1,r2) + 2 1 = (1 - W1) W2 Choosing weights to minimize the portfolio variance

  19. Efficient frontier Individual assets Minimum variance frontier The minimum-variance frontier of risky assets E(r) Efficient Frontier is the best diversified set of investments with the highest returns Found by forming portfolios of securities with the lowest covariances at a given E(r) level. Global minimum variance portfolio St. Dev.

  20. EF including international & alternative investments 100% Stocks 100% Stocks 80% Stocks 20% Bonds 60% Stocks 40% Bonds 40% Stocks 60% Bonds Ex-Post 2000-2002 The EF and asset allocation E(r) Efficient frontier 20% Stocks 80% Bonds St. Dev.

  21. Efficient frontier for international diversification Figure 6.11

  22. CAL (P) CAL (A) E(rP&F) E(rP) CAL (Global minimum variance) E(rA) A G P P&F Alternative Capital Allocation Lines E(r) Efficient Frontier P F Risk Free s A

  23. The Capital Market Line or CML CAL (P) = CML E(r) Efficient Frontier E(rP&F) • The optimal CAL is called the Capital Market Line or CML • The CML dominates the EF P E(rP) E(rP&F) F Risk Free s P&F P P&F

  24. Dominant CAL with a Risk-Free Investment (F) CAL(P) = Capital Market Line or CML dominates other lines because it has the the largest slope Slope = (E(rp) - rf) / sp (CML maximizes the slope or the return per unit of risk or it equivalently maximizes the Sharpe ratio) Regardless of risk preferences some combinations of P & F dominate

  25. A=2 A=4 The Capital Market Line or CML CML E(r) Efficient Frontier E(rP&F) Both investors choose the same well diversified risky portfolio P and the risk free asset F, but they choose different proportions of each. P E(rP) E(rP&F) F Risk Free s P&F P P&F

  26. Practical Implications o • The analyst or planner should identify what they believe will be the best performing well diversified portfolio, call it P. • P may include funds, stocks, bonds, international and other alternative investments. o This portfolio will serve as the starting point for alltheir clients. o The planner will then change the asset allocationbetween the risky portfolio and “near cash” investments according to risk tolerance of client. o The risky portfolio P may have to be adjusted for individual clients for tax and liquidity concerns if relevant and for the client’s opinions.

  27. Systematic risk Systematic risk arises from events that effect the entire economy such as a change in interest rates or GDP or a financial crisis such as occurred in 2007and 2008. If a well diversified portfolio has no unsystematic risk then any risk that remains must be systematic. That is, the variation in returns of a well diversified portfolio must be due to changes in systematic factors.

  28. Individual Securities How do we measure a stock’s systematic risk? Systematic Factors Returns well diversifiedportfolio  Δ interest rates, Δ GDP, Δ consumer spending, etc. Returns Stock A

  29. Single Factor Model Ri = E(Ri) + ßiM + ei Ri = Actual excess return = ri – rf E(Ri) = expected excess return Two sources of Uncertainty M ßi ei = some systematic factor or proxy; in this case M is unanticipated movement in a well diversified broad market index like the S&P500 = sensitivity of a securities’ particular return to the factor = unanticipated firm specific events

  30. Single Index Model Parameter Estimation Risk Prem Market Risk Prem or Index Risk Prem αi = the stock’s expected excess return if the market’s excess return is zero, i.e., (rm - rf) = 0 ßi(rm - rf)= the component of excess return due to movements in the market index ei = firm specific component of excess return that is not due to market movements

  31. Risk Premium Format Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) The Model: Ri = ai + ßi(Rm)+ ei

  32. Estimating the Index Model Security Characteristic Line Scatter Plot Excess Returns (i) . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . . . . . . . . . . . . . . . . . Ri = ai + ßiRm + ei Slope of SCL = beta y-intercept = alpha

  33. Components of Risk Market or systematic risk: Unsystematic or firm specific risk: Total risk = risk related to the systematic or macro economic factorin this case the market index risk not related to the macro factor or market index Systematic + Unsystematic i2 = Systematic risk + Unsystematic Risk

  34. Comparing Security Characteristic Lines Describe •  •  • e for each.

  35. Measuring Components of Risk si2 = where; bi2sm2 + s2(ei) si2 = total variance bi2sm2 = systematic variance s2(ei) = unsystematic variance

  36. Total Risk = Systematic Risk / Total Risk = Examining Percentage of Variance Systematic Risk + Unsystematic Risk r2 ßi2 sm2 / si2 = r2 bi2sm2 / (bi2sm2 + s2(ei)) = r2

  37. Advantages of the Single Index Model Reduces the number of inputs needed to account for diversification benefits If you want to know the risk of a 25 stock portfolio you would have to calculate 25 variances and (25x24) = 600 covariance terms With the index model you need only 25 betas Easy reference point for understanding stock risk. βM = 1, so if βi > 1 what do we know? If βi < 1?

  38. Sharpe Ratios and Alphas When ranking portfolios and security performance we must consider both return & risk “Well performing” diversified portfolios provide high Sharpe ratios: Sharpe = (rp – rf) / p You can also use the Sharpe ratio to evaluate an individual stock if the investor does not diversify

  39. The Treynor-Black Model • Suppose an investor holds a passive portfolio M but believes that an individual security has a positive alpha. • A positive alpha implies the security is undervalued. Suppose it is Google. • Adding Google moves the overall portfolio away from the diversified optimum but it might be worth it to earn the positive alpha. • What is the optimal portfolio including Google? • What is the resulting improvement in the Sharpe ratio?

  40. The Treynor-Black Model Weight of Google in the optimal portfolio O: The improvement in the Sharpe ratio (S) over the Sharpe of the passive portfolio M can be found as: Notice that the improvement in the Sharpe ratio is a function of This ratio is called the “information ratio”

  41. For multiple stocks in the active portfolio: The optimal weight of each security in the active portfolio is found as: A larger alpha increases the weight of stock i and larger residual variance reduces the weight of stock i. The Treynor-Black Model

  42. If A stands for the “active portfolio,” the active portfolio’s alpha, beta and residual risk can be found from: The Treynor-Black Model &

  43. Treynor-Black Allocation CAL E(r) CML P A M Rf s

  44. Are Stock Returns Less Risky in the Long Run? Consider the variance of a 2-year investment with serially independent returns r1 and r2: The variance of the 2-year return is double that of the one-year return and σ is higher by a multiple of the square root of 2 )

  45. Are Stock Returns Less Risky in the Long Run? Generalizing to an investment horizon of n years and then annualizing: For a portfolio:

  46. The Fly in the ‘Time Diversification’ Ointment The annualized standard deviation is only appropriate for short-term portfolios The variance grows linearly with the number of years Standard deviation grows in proportion to

  47. The Fly in the ‘Time Diversification’ Ointment To compare investments in two different time periods: Examine risk of the total rate of return rather than average sub-period returns Must account for both magnitudes of total returns and probabilities of such returns occurring

  48. Problem 2 Criteria 1: Eliminate Fund B Criteria 2: Choose Fund D Lowest correlation, best chance of improving return per unit of risk ratio.

  49. Problem 3 Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = ii Cov = OPABC = iii. NP = [wOP2OP2 + wABC2ABC2 + 2 wOP wABC (CovOP , ABC)]1/2 = [(0.92 .02372) + (0.12 .02952) + (2  0.9  0.1  .00028)]1/2 = 2.2673%  2.27% (0.9  0.67) + (0.1  1.25) = 0.728% 0.40  .0237  .0295 = .00027966  0.00028

  50. Problem 3 Subscript OP refers to the original portfolio, GS to government securities, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = ii. Cov = OPGS = iii. NP = [wOP2OP2 + wGS2GS2 + 2 wOP wGS (CovOP , GS)]1/2 = [(0.92 0.02372) + (0.12 0) + (2  0.9  0.1  0)]1/2 = 0.9 x 0.0237 = 2.133%  2.13% (0.9  0.67%) + (0.1  0.42%) = 0.645% 0  .0237  0 = 0

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