Department of Banking and Finance SPRING 2007-08 Efficient Diversification Efficient Diversification by Asst. Prof. Sami Fethi.

Department of Banking and Finance SPRING 2007-08 Efficient Diversification Efficient Diversification by Asst. Prof. Sami Fethi

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 2 Diversification and Portfolio risk Recall: portfolio is a collection of assets and risk is the chance of financial loss. What are the sources of risk affecting a portfolio? 1) The first type risk is associated with general economic conditions such as the business cycle, the inflation rate, interest rate, exchange rate and so forth. None of them are predicted with certainty so all conditions affect a company’s the rate of return.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 3 Diversification and Portfolio risk 2) The second one is the firm-specific factors that affect a firm without noticeably affecting other firms. If you have one stock in your portfolio, this means that you cannot reduce risk factor. However, you need to consider a diversification strategy such as naïve diversification half of your portfolio in a company and leaving the other half in an other company. This precaution reduce portfolio risk.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 4 Diversification and Portfolio risk For instance, If you invest half of your risky portfolio in Mobil company and leaving the other half in Dell company, what happens to portfolio risk? Assume that if computer prices increases, this helps Dell company and when oil prices fall, this hurts Mobil company. The two effects are offsetting which stabilizes portfolio return.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 5 Diversification and Portfolio risk When all risk is firm-specific, diversification can reduce risk to low level. This reduction of risk to very low levels because of independent risk sources is called the insurance principle. When common sources of risk affect all firms, even extensive diversification cannot eliminate risk. Graphically, as portfolio standard deviation falls, the number of securities increases, but it is not reduced to zero.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 6 Diversification and Portfolio risk The risk that remains even after diversification is called market risk. This risk is attributable to market-wide risk sources. They are also called systematic or non-diversifiable risk. The risk that can be eliminated by diversification is called unique risk, firm-specific risk, non- systematic risk, or diversifiable risk. It is important to note that portfolio risk decreases as diversification increases.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 9 Asset allocation with two Risky Assets Portfolio risk depends on the correlation between the returns of the assets in the portfolio. Asset allocation across the three key asset classes: stocks, bonds, and risk-free money market securities. Example 1: suppose there are three possible scenarios for an economy: a recession period, a normal growth period, and a boom period. The stock fund will have a rate of return of –11% in recession, 13% in normal period and 27% in boom period.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 10 Example 1 cont.. Suppose that a bond fund will provide ROR of 16% in the recession, 6% in the normal period and –4% in the boom period. What is the expected or mean return for both stock and bond funds? The expected return on each fund equals the probability-weighted average of outcomes in the scenarios. The variance is the probability-weighted average across all scenarios of the squared deviation between the actual returns of the fund and its expected return.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 12 Example 2 Suppose, we form a portfolio with 60% invested in the stock fund and 40% in the bond fund. Calculate portfolio return in recession portfolio return in each scenario is the weighted average of the returns on the two funds. Calculate portfolio return in recession = 0.60 (-11%) + 0.40 (16%) -0.20%

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 13 Covariance and correlation If we compute the probability-weighted average of the products across all scenarios, we obtain a measure of the extent to which the returns tend to vary with each other, that is to co-vary, it is called the covariance. The negative value for the covariance indicates that the two asset vary inversely, that is when one assets performs well, the other tends to perform poorly. It is really difficult to interpret the magnitude of covariance. An easier statistics to interpret is the correlation coefficient.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 14 Covariance and correlation The correlation coefficient is simply defined as the covariance divided by the product of the standard deviation of the returns on each fund. Correlation coefficient (ρ)= covariance/ σ STOCK σ BOND Correlation can range from values of –1 to 1. Correlation of zero indicate that the returns on the two assets are unrelated to each other. Positive correlated shows two series move in the same direction while negative moves in opposite directions.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 15 Let us have the following table which shows covariance between the returns of the stock and bond funds:

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 16 Let us have the following table which shows covariance between the returns of the stock and bond funds:

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 17 Correlation coefficient The correlation coefficient is simply defined as the covariance divided by the product of the standard deviation of the returns on each fund. Correlation coefficient (ρ)=covariance/ σ STOCK σ BOND =-114/(14.92x7.75) i.e.,(-21) 2 x(0.3)+(3) 2 x(0.4)+(17) 2 x(0.3)=SQRT of VAR=14.92 i.e.,(-10) 2 x(0.3)+(0) 2 x(0.4)+(-10) 2 x(0.3)=SQRT of VAR=7.75 =-0.99 This confirms the overwhelming tendency of the returns on the stock and bond funds to vary inversely in the scenario analysis.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 18 Example 3 The rates of return of the bond portfolio in the three scenarios based on the previous table are 10% in a recession, 7% in a normal period, and 2% in a boom. The stock returns in the three scenarios are –12% (recession), 10% (normal), and 28% (boom). What are the covariance and correlation coefficient between the rates of return on the two portfolios?

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 20 r p = W 1 r 1 + W 2 r 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 r 1 = Expected return on Security 1 r 2 = Expected return on Security 2 Two-Security Portfolio: Return W i  i=1n = 1

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 21  p 2 = w 1 2  1 2 + w 2 2  2 2 + 2W 1 W 2 Cov(r 1 r 2 )  1 2 = Variance of Security 1  2 2 = Variance of Security 2 Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 22 E(r p ) = W 1 r 1 + W 2 r 2 Two-Security Portfolio  p 2 = w 1 2  1 2 + w 2 2  2 2 + 2W 1 W 2 Cov(r 1 r 2 )  p = [w 1 2  1 2 + w 2 2  2 2 + 2W 1 W 2 Cov(r 1 r 2 )] 1/2

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 23 Example 4 Suppose that for some reason you are required to invest 50% of your portfolio in bonds and 50% in stocks. r 1 =6%, r 2= 10%, σ 1 =12%, σ 2 =25%, w 1 =0.5, and w 2 =1-0.5=0.5. A) If the standard deviation of your portfolio is 15%, what must be the correlation coefficient between stock and bond returns? B) What is the expected rate of return on your portfolio?

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 24 Example 4 A) A)  p 2 = w 1 2  1 2 + w 2 2  2 2 + 2W 1 W 2  12 15 2 = (0.5x12) 2 + (0.5x25) 2 +2 (0.5x12) 15 2 = (0.5x12) 2 + (0.5x25) 2 +2 (0.5x12) (0.5x25) (0.5x25)  12  12 =0.21183 B) E (r p ) = W 1 E (r 1 )+ W 2 E (r 2 ) B) E (r p ) = W 1 E (r 1 )+ W 2 E (r 2 ) = (0.5x6)+ (0.5x10) = (0.5x6)+ (0.5x10) = 8% = 8%

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 25 Covariance  1,2 = Correlation coefficient of returns  1,2 = Correlation coefficient of returns Cov(r 1 r 2 ) =    1  2  1 = Standard deviation of returns for Security 1  2 = Standard deviation of returns for Security 2  1 = Standard deviation of returns for Security 1  2 = Standard deviation of returns for Security 2

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 26 Correlation Coefficients: Possible Values If  = 1.0, the securities would be perfectly positively correlated If  = - 1.0, the securities would be perfectly negatively correlated Range of values for  1,2 -1.0 <  < 1.0

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 27  2 p = W 1 2  1 2 + W 2 2    + 2W 1 W 2 r p = W 1 r 1 + W 2 r 2 + W 3 r 3 Cov(r 1 r 2 ) + W 3 2  3 2 Cov(r 1 r 3 ) + 2W 1 W 3 Cov(r 2 r 3 ) + 2W 2 W 3 Three-Security Portfolio

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 28 r p = Weighted average of the n securities r p = Weighted average of the n securities  p 2 = (Consider all pair-wise covariance measures)  p 2 = (Consider all pair-wise covariance measures) In General, For an n-Security Portfolio:

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 29  = 0 E(r)  = 1  = -1  =.3 13% 8% 12%20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS The figure shows the opportunity set with perfect positive correlation. No portfolio can be discarded as inefficient and the choice among portfolios depends only on risk preference. Diversification in the case of perfect positive correlation is not effective. Perfect positive correlation is the only case in which there is no benefit from diversification. In the case of negative correlation, there are benefits to diversification.  =.3  = 1  = 0)  =.3 is a lot better than  = 1 and quite a bit worse than (  = 0) zero correlation.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 30 TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS The figure shows the opportunity set with perfect positive correlation. No portfolio can be discarded as inefficient and the choice among portfolios depends only on risk preference. Diversification in the case of perfect positive correlation is not effective. Perfect positive correlation is the only case in which there is no benefit from diversification. In the case of negative correlation, there are benefits to diversification.  =.3  = 1  = 0)  =.3 is a lot better than  = 1 and quite a bit worse than (  = 0) zero correlation.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 31 Portfolio Risk/Return Two Securities: Correlation Effects Relationship depends on correlation coefficient -1.0 <  < +1.0 The smaller the correlation, the greater the risk reduction potential If  = +1.0, no risk reduction is possible

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 32 Minimum Variance Combination Investment opportunity set E(r) 10% 7%7%7%7% 11%16% St. Dev Stock.. portfolio Z portfolio Z. The mean variance portfolio The mean variance portfolio. Bonds Bonds 26%31% 6%6%6%6% A mean-var criterion indicates higher mean return and lower var. In this case, the stock fund dominates portfolio Z so has higher expected return and lower volatility. If portfolios lie below the min-var portfolio, they can be rejected as inefficient. This is valid for the case of zero correlation between the funds.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 33 1 1 1 1 2 2 - Cov(r 1 r 2 ) W1W1 W1W1 = = + + - 2Cov(r 1 r 2 ) 2 2 W2W2 W2W2 = (1 - W 1 ) Minimum Variance Combination-Example  2 2 2 E(r 2 ) =.14 =.20 Sec 2 12 =.2 E(r 1 ) =.10 =.15 Sec 1        2 Suppose, we invest some proportions in both stocks and in bonds and the other relevant input data as follows. Compute the proportions of the funds and the portfolio variance.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 34 W1W1W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(.2) W1W1W1W1 =.6733 W2W2W2W2 = (1 -.6733) =.3267 Example cont….

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 35 r p =.6733(.10) +.3267(.14) =.1131 p = [(.6733) 2 (.15) 2 + (.3267) 2 (.2) 2 + 2(.6733)(.3267)(.2)(.15)(.2)] 1/2 p = [.0171] 1/2 =.1308 Example cont….    

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 36 W1W1W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(-.3) W1W1W1W1 =.6087 W2W2W2W2 = (1 -.6087) =.3913 Minimum Variance Combination-example2:  = -.3

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 37 r p =.6087(.10) +.3913(.14) =.1157 p = [(.6087) 2 (.15) 2 + (.3913) 2 (.2) 2 + 2(.6087)(.3913)(.2)(.15)(-.3)] 1/2 p = [.0102] 1/2 =.1009 Minimum Variance: example2:  = -.3 cont…..    

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 38 Example 4-2 Suppose, you invest 50% in both stocks and in bonds and the other relevant input data as follows: st.dev B =12, st.dev S =25. (a) Compute the correlation coefficient between stock and bond returns if st.dev p =15. (a) Compute the correlation coefficient between stock and bond returns if st.dev p =15. (b) What is the expected ROR on your portfolio if expected RORs for stock and bond are 6 and 10 respectively. (b) What is the expected ROR on your portfolio if expected RORs for stock and bond are 6 and 10 respectively. (c) Are you likely to be better or worse off if the correlation coefficient between stock and bond returns is 0.22 compared to part (a). (c) Are you likely to be better or worse off if the correlation coefficient between stock and bond returns is 0.22 compared to part (a).

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 39 Example 4-2 cont..  2 1 BSBS 2 B 2 B 2 S 2 SP )r,r(Covww2ww  15 2 = [(0.5x12) 2 + (0.5  25) 2 + 2  (0.5x12)  (0.5  25 )]ρ SB ρ SB = 0.2138 E(r p ) = (0.5  6%) + (0.5  10%)  8% Smaller correlation implies greater benefits from diversification so there will be lower risk. r p = W 1 r 1 + W 2 r 2

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 40 Example 5 There are three mutual funds such as a stock fund, a long-term government fund and a T-bill money market fund and this yields a rate of 5.5%. The probability distributions of risky funds are: The correlation between the fund returns is 0.15. E.Returnst.dev stock fund15%32% Bond fund923

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 41 Example 5 cont.. Tabulate the investment opportunity set of the two risky funds (i.e., construct the covariance matrix ). What are the expected return, standard deviation and minimum variance portfolio?

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 42 Example 5 cont.. The parameters of the opportunity set are: E(r S ) = 15%, E(r B ) = 9%,  S = 32%,  B = 23%,  = 0.15,r f = 5.5% From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r S, r B ) =  S  B ]: BondsStocks Bonds 529.0 110.4 Stocks 110.41024.0

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 43 Example 5 cont.. The minimum-variance portfolio proportions are: w Min (B) = 0.6858 The mean and standard deviation of the minimum variance portfolio are: E(r Min ) = (0.3142  15%) + (0.6858  9%)  10.89%

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 44 Example 5 cont.. = [(0.3142 2  1024) + (0.6858 2  529) + (2  0.3142  0.6858  110.4)] 1/ 2 = 19.94%

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 45 Example 6 First draw the diagrams by using the figures of stock 1 versus stock 2. Second match up the diagrams (A-E) to the following list of correlation coefficients by choosing the correlation that best describes the relationship between the returns on the two stocks =-1, 0, 0.2, 0.5, 1.0.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 46 Example 6 cont.. Diagram A shows exact conflict and in this case cc is zero. Diagram B shows perfect positive correlation and cc is 1.0. Diagram C shows perfect negative correlation and cc is - 1.0. Diagram D and Diagram E show positive correlation but Diagram D is tighter. Therefore D is associated with a correlation of 0.5 and E is associated with a correlation of 0.2.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 47 Example 6 cont.. Diagram A shows exact conflict and in this case cc is zero. Diagram B shows perfect positive correlation and cc is 1.0. Diagram C shows perfect negative correlation and cc is -1.0. Diagram D and Diagram E show positive correlation but Diagram D is tighter. Therefore D is associated with a correlation of 0.5 and E is associated with a correlation of 0.2.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 48 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

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 49 E(r) The minimum-variance frontier of risky assets Efficientfrontier Globalminimumvarianceportfolio Minimumvariancefrontier Individualassets St. Dev. Efficient frontier represents the set of portfolios that offers the highest possible expected rate of return for each level of portfolio standard deviation. These portfolios may be viewed as efficiently diversified. Expected return- standard deviation combinations for any individuals asset end up inside the efficient frontier, because single-asset portfolios are inefficient- they are not efficiently diversified. The real choice is among portfolios on the efficient frontier above the minimum-variance portfolio.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 50 Efficient frontier Efficient frontier represents the set of portfolios that offers the highest possible expected rate of return for each level of portfolio standard deviation. These portfolios may be viewed as efficiently diversified. Expected return-standard deviation combinations for any individuals asset end up inside the efficient frontier, because single-asset portfolios are inefficient- they are not efficiently diversified. The real choice is among portfolios on the efficient frontier above the minimum-variance portfolio.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 51 Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless assets will dominate

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 53 Dominant CAL with a Risk-Free Investment (F) CAL(P) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) /   E(R P ) - R f ) /  P   E(R A ) - R f ) /    Regardless of risk preferences combinations of P & F dominate

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 54 Example 7 The correlation coefficient between X and M is – 0.2. Weight in M and X are 0.26 and 0.74. Find the optimal risky portfolio (o) and its expected return and standard deviation. Find the slope of the CAL generated by T-bills and portfolio o. Calculate the composition of complete portfolio (an investor consider 22.22% of complete portfolio in the risky p) o and the remainder in T-bills.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 55 E(r) CAL (Global minimum variance) CAL (X) CAL (M) 15 10 5% 2035 50 G M X  11.28 17.59 CAL (O) O

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 56 Example 7 In this case, you need to generate data to find out mean and st.dev for optimal risky portfolio (i.e., 11.28 and 17.59) and weight in X and M are (i.e., 0.26 and 0.74) respectively. The slope of CAL is (11.28-5)/17.59=0.357 The mean of the complete portfolio 0.22x11.28+ 0.7778x5=6.40% and its standard deviation is 0.22x17.59=3.91%. The composition of the complete portfolio is 0.22x0.26 (optimal pf calculated by using data for x) =0.06 (6%) in X. In M, 0.22x0.74 (optimal pf calculated by using data for M) =0.16 (16%) in M and 78% in T-bills.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 57 Single Factor Model r i = E(R i ) + ß i F + e ß i = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 58 Single Index Model Risk Prem Market Risk Prem or Index Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero market’s excess return is zero ß i (r m - r f ) = the component of return due to movements in the market index movements in the market index (r m - r f ) = 0 e i = firm specific component, not due to market movements movements    e rrrr i fm i i fi    

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 59 Let: R i = (r i - r f ) R m = (r m - r f ) R m = (r m - r f ) Risk premium format R i =  i + ß i (R m ) + e i Risk Premium Format

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 60 Estimating the Index Model Excess Returns (i) SecurityCharacteristicLine.................................................................................................... Excess returns on market index R i =  i + ß i R m + e i

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 61 Components of Risk Market or systematic risk: risk related to the macro economic factor or market index Unsystematic or firm specific risk: risk not related to the macro factor or market index Total risk = Systematic + Unsystematic

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 62 Measuring Components of Risk  i 2 =  i 2  m 2 +  2 (e i ) where;  i 2 = total variance  i 2  m 2 = systematic variance  2 (e i ) = unsystematic variance

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 63 Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk =  2 ß i 2  m 2 /  2 =  2  i 2  m 2 /  i 2  m 2 +  2 (e i ) =  2 Examining Percentage of Variance

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 64 Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 66 Example 8 Calculate the slope and intercept of characteristic lines for ABC and XYZ using the variances and co-variances concepts. What is the characteristic line of XYZ and ABC? Does ABC or XYZ have greater systematic risk? What percentage of variance of XYZ is firm specific risk

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 67 Example 8 The slope of coefficient for ABC ß ABC =cov (R ABC, R MRK )/var (R MRK ) =773.31/669.01=1.156 The intercept for ABC α ABC = AV.(R ABC )- ß ABC x AV.(R MRK ) =15.20-1.156 x 9.40=4.33 The security the characteristic line of ABC is R ABC =4.33 + 1.156 R MRK

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 68 Example 8 The slope of coefficient for XYZ ß XYZ =cov (R XYZ, R MRK )/var (R MRK ) =396.78.31/669.01=0.58 The intercept for XYZ α XYZ = AV.(R XYZ )- ß XYZ x AV.(R MRK ) =7.64-0.582 x 9.40=3.93 The security the characteristic line of XYZ is R XYZ =3.93 + 0.582 R MRK

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 69 Example 8 The beta coefficient of ABC is 1.15 greater than XYZ’s 0.58 implying that ABC has greater systematic risk. The regression of XYZ on the market index shows an R square of 0.132, the percent of unexplained variance (non-systematic risk) is 0.868 or 86.8%.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 70 Example 9 A pension fund manager is considering three mutual funds such as a stock fund, a long-term government and corporate bond fund, and a T-bill money market fund that yields a sure rate of 4.5%. The probability distributions of the risky funds as follows: (note: The correlation between the fund return is 0.18). Expected return Standard deviation Stock fund (S) 18%34% Bond fund (B) 1226

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 71 Example 9 a) Tabulate and draw the investment opportunity set of the two risky funds. b) Use investment proportions for the stock fund of 0 to 100% in increments of 20%. c)What expected return and the standard deviation does your graph show for the minimum variance portfolio? d)Draw a tangent from the risk-free rate to the opportunity set e) What is the reward-to-variability ratio of the best feasible CAL? f) What is the equation of the CAL? What is the standard deviation of your portfolio if it yields an expected return of 15%? g) What is the proportion invested in the T-bill fund and each of the two risky funds?

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 73 Example 9-Answer-b and c % in stocks% in bondsExp. returnStd dev. 00.00100.00 12.00 2 6.00 20.0080.00 13.20 23 34.0066.00 14.04 22.34minimum variance 40.0060.00 14.40 22.46 60.0040.00 15.60 24.50 70.8029.20 16.20 26.54tangency portfolio 80.0020.00 16.80 28.59 100.0000.00 18.00 34.00 E(r p ) = W 1 r 1 + W 2 r 2 = 1 (12) + (0) (18) = 12  p = [w 1 2  1 2 + w 2 2  2 2 + 2W 1 W 2 Cov(r 1 r 2 )] 1/2 = 1(26) 2 + (0)(34) 2 + 2(0)...= 26

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 74 Example 9-Answer-c wMin(B) = 0.66 The minimum-variance portfolio proportions are: The mean and standard deviation of the minimum variance portfolio are: = [(0.34 2  1156) + (0.66 2  529) + (2  0.34  0.66  159.12)]1/2= 22.34% E(rMin) = (0.34  18%) + (0.66  12%) = 14.04%

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 76 Example 9-Answer-e and f The reward-to-variability ratio of the optimal CAL is: The equation for the CAL is: Setting E(rC) equal to 15% yields a standard deviation of 23.75%.

Investment Management © 2007/08 Sami Fethi, EMU, All Right Reserved. Ch 6: Efficient Diversification 77 Example 9-Answer-g The mean of the complete portfolio as a function of the proportion invested in the risky portfolio (y) is: E(rC) = (l - y)rf + yE(rP) = rf + y[E(rP) - rf] = 4.5 + y(16.20- 4.5) Setting E(rC) = 15%  y = 0.89 (89% in the risky portfolio) 1 - y = 0.11 (11.00% in T-bills) From the composition of the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.89  0.7080 = 0.63 Proportion of bonds in complete portfolio = 0.89  0.2920 = 0.25

Department of Banking and Finance SPRING 2007-08 Efficient Diversification Efficient Diversification by Asst. Prof. Sami Fethi.

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Department of Banking and Finance SPRING 2007-08 Efficient Diversification Efficient Diversification by Asst. Prof. Sami Fethi.

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