BMGT343 Chapter 6 Efficient Diversification Diversification and Portfolio Risk Two broad sources of uncertainty o Risk that has to do with general economic conditions cycle inflation interest exchange rates o Firm s research development management style philosophy Diversifying into many more securities continues to reduce exposure to firm specific factors so portfolio volatility should continue to fall The reduction of risk to very low levers due to independent risk sources is sometimes called insurance principle The risk that means even after diversification is called market risk systematic risk or nondiversifiable risk Risk that can be eliminated by diversification is called unique risk firm specific risk nonsystematic risk diversifiable risk Asset Allocation with Two Risky Assets 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 Portfolio risk depends on the correlation between the returns of the assets in the portfolio Expected return on each fund equals the probability weighted averages of the outcomes in the four scenarios The variance is the probability weighted average across all scenarios of the squared deviation between the actual return of the fund and its expected return The portfolio return is the weight average of the returns on each fund with weights equal to the proportion of the portfolio invested in each fund The low risk of the portfolio is due to the inverse relationship between the performances of the stock and bond funds In a mild recession stock fare poorly but this is offset by a large positive return of the bond fund In the boom scenario bond prices fall but stocks do very well o o The covariance is calculated in a manner similar to the variance instead of measuring the typical difference of an asset return from its expected value we wish to measure the extent to which the variation in the returns on the two assets tend to reinforce or offset each other If we compute the probability weighted average of the products across all scenarios we obtain a measure of the average tendency of the asset returns to vary in tandem Since this is a measure of the extent to which the returns tend to which the returns tend to vary with each other co vary it is called covariance o Cov rS rB p i rS i rS rB i rB Indexed by I ranges from 1 to S number of scenarios P i probability of each scenario Correlation coefficient is the covariance divided by the product of the standard deviations of the returns on each fund o Correlation coefficient SB Cov rs rb S B Correlations can range from values of 1 to 1 1 indicates perfect negative correlation strongest possible tendency for two returns to vary inversely o o 1 indicates perfect positive correlation two returns move in the same direction o 0 indicates that returns on two assets are unrelated 3 Rules ofTwo Risky Assets Portfolios o rP wBrB wRrS Rule 1 rate of return on the portfolio is a weighted average of the returns on the component securities with the investment proportions as weights Rule 2 The expected rate of return on the portfolio is a weighted average of the expected returns on the component securities with the same portfolio proportions as weights o E rP wBE rP wSE rP Rule 3 The variance of the weight of return on the two risky asset portfolio is o 2 P wB B 2 wS S 2 2 wB B wS S PBS where PBS is he correlation coefficient between the returns on the stock and bond funds The variance of the portfolio is a sum of the contributions of the component security variance plus a term that involves the coefficient between returns on the component securities Investment opportunity set set of available portfolio risk return combinations o This is the set of all attainable combinations of risk and return offered by portfolios formed using the available assets in The Mean Variance Criterion differing proportions investors desire portfolios that lie to the northwest o High expected returns north o Low volatility west Portfolio A dominates portfolio B if all investors prefer A over B o E rA E rB o A B Portfolios that lie below the minimum variance portfolio in the figure can therefore be rejected out of hand as inefficient Perfect portfolio correlation is the only case in which there is no benefit from diversification o Whenever p 1 the portfolio standard deviation is less than the weighted average of the standard deviation of the components of the component securities o There are benefits to diversification whenever asset returns are less than perfectly positively correlated Whenever correlation is negative there will be even greater diversification benefits The Optimal Risky Portfolio with a Risk free Asset the slope of the CAL is the Sharpe ratio of the risky portfolio that is the ratio of excess return of standard deviation o SP E rP rf P o This is the rate at which the investor can increase expected return by accepting higher portfolio volatility Optimal risky portfolio the best combination of risky asset to be mixed with safe assets to form the complete portfolio Steepest capital allocation line Portfolio results in the highest return per unit of risk Efficient Diversification with Many Risky Assets 1 we identify the best possible or most efficient risk return combinations available from the universe of risky assets 2 we determine the optimal portfolio of risky assets by finding the portfolio that supports the steepest CAL 3 we choose an appropriate complete portfolio on CAL0 based on the investor s risk aversion by mixing the risk free with the optimal risky portfolio efficient frontier graph representing a set of portfolios that maximizes expected return at each level of portfolio risk separation property property that implies portfolio choice can be separated into independent tasks o o determination of optimal risky portfolio which is purely a technical problem personal choice of the best mix of the risky portfolio and the risk free asset A Single Index Asset Market systematic risk is largely macroeconomic affecting all securities while firm specific risk factors affect only one particular firm or industry index models are statistical models designed to estimate these two components of risk for a particular security or portfolio o index model model that relates stock return to returns on a broad market index as well as firm specific influences excess return rate of return in excess of the risk free rate o Ri excess return o Ri ri rf rate of