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Week 2

Week 2. Efficient Diversification. Diversification and Portfolio Risk. The risky portfolio of only one stock.

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Week 2

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  1. Week 2 Efficient Diversification

  2. Diversification and Portfolio Risk • The risky portfolio of only one stock. • Market (Systematic; Non-diversifiable) risk: Risk factors common to the whole economy (Risk from general economic conditions, such as business cycle, inflation, interest rate, exchange rate, etc.). • Unique (Unsystematic; Diversifiable) risk: Risk that can be eliminated by diversification (Risk from firm-specific factors, such as success in R&D, management style, philosophy, etc.). • Portfolio risk does fall with diversification, but the power of diversification to reduce risk is limited by common source of risk.

  3. Asset Allocation with Two Risky Assets • Covariance and Correlation • Key determinant of portfolio risk is the extent to which the returns on the two assets tend to vary either in tandem or in opposition. • How to measure the degree and direction of co-movement of returns on two assets. • Covariance: • Cov(r1,r2) =  p(s)[r1(s) – E(r1)][r2(s) – E(r2)] • A measure of the extent to which the returns tend to vary each other: Positive – Vary with each other; Negative – Vary inversely. • It is difficult to interpret: How strong.

  4. Asset Allocation with Two Risky Assets • Covariance and Correlation (continued) • Correlation: • 12 = Cov(r1,r2)/[s1 s2]. • A standardized measure of the extent to which the returns tend to vary each other. • Easier to interpret how strong the relationship is. • Range of correlations:-1.0 <r< 1.0 • If r = 1.0: Perfectly positively correlated. • If r = -1.0: Perfectly negatively correlated. • If r = 0: Unrelated (independent) to each other.

  5. Two-Risky-Assets Portfolio: Return • Rate of return on the portfolio [rp] rp = w1r1 + w2r2, where w1 = Proportion of funds in security 1 w2 = (1 - w1)= Proportion of funds in security 2 r1 = Rate of return of security 1 r2 = Rate of return of security 2 • Expected rate of return on the portfolio [E(rp)] E(rp)= w1E(r1 ) + w2E(r2 ), where E(r1 ) = Expected rate of return of security 1 E(r2 ) = Expected rate of return of security 2

  6. Two-Risky-Assets Portfolio: Risk • Variance of the rate of return on the portfolio [p2] p2 = (w11)2 + (w22)2 + 2w1w2 Cov(r1, r2) p2 = (w11)2 + (w22)2 + 2w1w2 12s1s2, where12 = Variance of rate of return on security 1 22 = Variance of rate of return on security 2 Cov(r1, r2) = Covariance between r1 and r2 12 = Correlation between r1 and r2 • Expected rate of return of the portfolio is simply a weighted average of returns on two securities. • However, standard deviation of the portfolio return is NOT a weighted average of standard deviations of returns on two securities unless 12 = 1.0.

  7. Two-Risky-Assets Portfolio: Risk-Return Trade-off • Two risky assets: Bond Fund (B) & Stock Fund (S). • E(rB )=10%, E(rS )=17%, B=12%, S=25%,BS=0 • 100% in bonds (wB=1.0, wS=0): E(rp)= 10%; p = 12% • 50% in bonds and 50% in stocks (wB=0.5, wS=0.5) E(rp)= 0.510% + 0.517% = 13.5%; p 2 = (0.5  12)2 + (0.5  25)2 + (2  0.5  0.5  0  12  25) = 192.25; p = 13.87% By shifting from 100% in bonds to 50% in each of bonds and stocks, the portfolio standard deviation is increased by only 1.87%, not by 6.5% to the average of the component standard deviations [(25+12)/2 = 18.5%]. • 100% in stocks (wB=0, wS=1.0): E(rp)= 17%; p = 25%

  8. Two-Risky-Assets Portfolio: Risk-Return Trade-off • Analyst can and must show investors the entire investment opportunity set, which is the set of all attainable combinations of risk and return offered by portfolios formed using available assets in differing proportion. • wB wS E(rp) sp 0.0 1.0 17.0 25.00 0.2 0.8 15.6 20.14 0.4 0.6 14.2 15.75 0.5 0.5 13.5 13.87 0.6 0.4 12.8 12.32 0.8 0.2 11.4 10.824 0.8127 0.1873 11.31 10.818 1.0 0.0 10.0 12.00

  9. Investment Opportunity Set for Bond & Stock Funds Stock (S) Portfolio Z Minimum Variance Portfolio (A) Bond (B) Portfolio

  10. Mean-Variance Criterion • Investors compare portfolios using a mean-variance criterion; Prefer higher return and lower risk. • Portfolio A dominates Portfolio B if all investors prefer A over B: This will be the case if it has higher mean return and lower variance – E(rA)  E(rB) and sA  sB. • The choice among the dominant portfolios is not as obvious, because higher expected return is accompanied by higher risk (The choice will depend on individual’s willingness to trade off risk against expected return). • The Choice? – E(rA) > E(rB) and sA > sB. • Low correlation aid diversification and high correlation results in a reduced effect of diversification.

  11. Diversification Benefits and Correlation • If BS = 1.0: No Diversification Benefits. • p2 = (wBB)2 + (wSS)2; p = wBB + wSS • Portfolio standard deviation is simply a weighted average of standard deviations of two securities. • No inefficient portfolio; choice among portfolios depends only on risk preference. • If –1.0 < BS < 1.0: Some Diversification Benefits. • When –1.0 < BS <0, Greater benefits than when 0< BS < 1.0. • If BS = -1.0: Greatest Diversification Benefits. • p2 = (wBB - wSS)2; p = ABS[wBB - wSS] • If wBB = wSS, p = 0. • wB = [S2 - BSsBsS]/[B2 + S2 - 2BSsBsS]

  12. Investment Opportunity Set: Varying Correlation Stock Fund r=-1.0 r=1.0 r=0 r=0.5 r=0.2 Bond Fund

  13. sB Bond E(rB) = .10 = .12 rBS = .2 sS Stock E(rS) = .17 = .25 Minimum Variance Portfolio sS2- sBsSrBS wB = sB2+sS2- 2sBsSrBS (.25)2 - (.25)(.12)(.2) wB= (.12)2 + (.25)2 - 2(.25)(.12)(.2) wB= .8706; wS=.1294

  14. Optimal Risky Portfolio with a Risk-Free Asset • Two risky funds (Bond & Stock) + Risk-free Asset. • E(rB )=10%; E(rS )=17%; B=12%; S=25%;BS=0.2; and rf = 8%. • Two possible CALs from rf. • CAL 1: Portfolio A (Minimum variance portfolio). • E(rA) = (.8706)(10)+(.1294)(17)=10.91% sA = [(.870612)2+(.129425)2+(2.8706.129412 250.2)] = 11.54% • SA = [E(rA)-rf]/sA = (10.91 – 8)/11.54 = 0.25 • CAL 2: Portfolio X (wB=0.65). • E(rX) = 12.45%; sX = 12.83% • SB = [E(rX)-rf]/sX = (12.45 – 8)/12.83 = 0.35 • Higher reward-to-variability ratio for CAL 2.

  15. Optimal Risky Portfolio with a Risk-Free Asset • The higher reward-to-variability of Portfolio X (CAL2) means that combinations of Portfolio X and a risk-free asset provide higher expected return for any level of risk: Portfolio X dominates Portfolio A. • The CAL, which is tangent with investment opportunity set will provide the highest feasible reward-to-variability ratio. • The tangency portfolio (O) is the optimal risky portfolio. • Investors will choose the optimal risky portfolio regard-less of their degree of risk aversion. • Investors differ only in their allocation of investment funds between the optimal risky portfolio an the risk-free asset.

  16. ALTERNATIVE CALs CAL (O) E(r) CAL (X) Portfolio O (Optimal Portfolio) CAL (A) Portfolio X Portfolio A (Minimum Variance Portfolio) rf=8% s

  17. Efficient Diversification with Many Risky Assets • Three Separate Steps for Asset Allocation • Identify the best possible (most efficient) risk-return combinations available from the set of risky assets. • Determine the optimal risky portfolio of risky assets. • Choose an appropriate complete portfolio based on investor’s risk aversion by mixing the risk-free assets with the optimal risky portfolio. • Efficient Frontier of Risky Assets • The optimal combinations with many risky assets result in the lowest level of risk for a given return (or the highest level of return for a given risk). • The optimal trade-off is described as the efficient frontier; These portfolios are dominant.

  18. The Efficient Frontier of Risky Assets and Choice of Optimal Risky Portfolio E(rp) Optimal CAL Optimal Risky Portfolio Efficient Frontier of Risk Assets Individual Assets rf Global Minimum Variance Portfolio sp

  19. Efficient Diversification with Many Risky Assets • Efficient Frontier of Risky Assets (continued) • In real world, various constraints may preclude a particular investor from choosing portfolios on the efficient frontier: Portfolio managers can tailor an efficient frontier to meet any particular objective. • Choosing the optimal risky portfolio • The CAL formed from the optimal risky portfolio will be tangent to the efficient frontier of risky assets; This CAL will dominates all other CALs. • Preferred Complete Portfolio • Separation theory: Portfolio choice can be separated into two independent tasks – (1) determination of the optimal risky portfolio, and (2) the personal choice of the best mix of risky portfolio and risk-free asset.

  20. Efficient Diversification with Many Risky Assets • Preferred Complete Portfolio (continued) • The best risky portfolio is the same for all investors regardless of risk-aversion: More (Less) risk-averse investor will invest more in risk-free (risky) asset. • In real world, the optimal risky portfolio for different clients also may vary because of portfolio constraints: But, a few portfolios may be sufficient to serve the demands of a wide range of investors. • If the optimal risky portfolio is the same for all clients, professional management is more efficient and less costly. • If different managers use different input data to develop different efficient frontiers, they will offer different optimal portfolios; Thus, security analysis is important in portfolio selection.

  21. A Single-Factor Model • Factor Models: Statistical models designed to estimate two components (systematic and firm-specific) of risk for a particular security or portfolio. • Ri = E(Ri) + ßiM + ei (Single Factor Model) Ri = Excess return on a security = (ri – rf) E(Ri) = Expected excess return at the start of the holding period M = Market or macroeconomic surprises during the holding period ßi =sensitivity of the security’s return to macroeconomic factor ei =impact of unanticipated firm-specific events. • One reasonable approach to measure the factor is to use the rate of return on a broad index of securities (like the S&P500) as a proxy for the common macro factor

  22. A Single-Index Model • Index Model: A model of stock returns using a market index to represent common (systematic) risk factor – Separates the realized rate of return on a security into systematic and firm-specific components. • Ri = ai + ßiRM + ei (Single-Index Model) Ri = Risk premium (Excess return) on a security = (ri – rf) ai= Stock’s excess return when market’s excess return is zero. RM = Risk premium (Excess return) on a market index = (rM – rf) ßiRM= Component of return due to movements in overall market ei = Component of return due to unexpected firm-specific events • Single-index model specifies the two sources of security risk: Systematic and Firm-specific Risks.

  23. Components of Risk • Market or systematic risk ( ßiRM): Risk attributable to the security’s sensitivity to movements in overall market. • Unsystematic or firm specific risk ( ei): Risk not related to the market index • Total risk = Systematic Risk + Unsystematic Risk si2 = bi2sm2 + s2(ei) where; si2= Variance of the excess return of a stock (total variance) bi2sm2= Variance attributable to the uncertainty common to the entire market s2(ei) = Variance attributable to firm-specific risk factors which are independent of market performance

  24. Security Characteristic Line (SCL) • SCL is a statistical representation of the single-index model, that is, the plot of a security’s excess returns as a function of the excess return of the market. • Single-index model can be estimated by the regression of the excess return of a security (Ri) on the excess return of (RM). • Regression line: E(RiRM) = ai + ßiRM • The greater the slope of the regression, the greater the security’s systematic risk, as well as its total variance. • The deviations of actual returns from the regression line measure the effects of firm-specific events.

  25. Graphical Representation of Single-Index Model Security Excess Returns (Ri) Security Characteristic Line . . . . . . . . . . . . . . . . . . . . . . . ßi . . . ai . . . . . . . . . . . . . . . . . . . . . . . . Market Excess Returns (RM)

  26. Examining Percentage of Variance • One way to measure the relative importance of systematic risk is to measure the ratio of systematic variance to total variance (r2) r2 = ßi2 sM2 / si2 = bi2sM2 / [bi2sM2 + s2(ei)] where r = Correlation between Ri and RM. • This measures the ratio of explained variance to total variance, that is, the proportion of total variance that can be attributed to market fluctuations. • A large (small) absolute correlation means systematic (firm-specific) variance dominates total variance. • At extreme, perfect correlation (positive or negative) means security returns are fully explained by market return (No firm-specific effects).

  27. Diversification and Single-Index Model • What are the systematic and unsystematic variances of portfolio which includes securities whose returns are given by single-index model? • The beta of the portfolio (bp) is the simple average of the individual security betas; the systematic variance is bp2sM2. • The systematic component of each security return, bi2sM2, is perfectly correlated with the systematic part of any other security’s return because of the common factor – No diversification effects on systematic risk: Average beta of component securities is relevant to portfolio systematic risk. • Because firm-specific effects are independent of each other, their risk effects are offsetting – Diversification effects on unsystematic risk by holding many securities. • For a well-diversified investor, only relevant risk of a security is the systematic risk of the security measured by bi.

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