Portfolios Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08% The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: rP wB rB wS rS 5% 50% ( 7%) 50% (17%)
Portfolios Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012 Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.E (rP ) wB E (rB ) wS E (rS ) 9% 50% (11%) 50% (7%)
Portfolios Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08% The variance of the rate of return on the two risky assets portfolio is σP 2 (wB σ B ) 2 (wS σ S ) 2 2(wB σ B )(wS σ S )ρ BS where BS is the correlation coefficient between the returns on the stock and bond funds.
Portfolios Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08% Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than either stocks or bonds held in isolation.
As long at the correlation coefficient is < 1, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities. In the above case: SD of portfolio= 3.08% Weighted average of SD = 14.31%*0.5 + 0.0816*0.5 = 0.07155 + 0.0408 = 0.11235 = 11.235% This difference is due to the negative correlation between the two securities.
return Portfolios with Various Correlations 100% Since any probable correlation of = -1.0 stocks securities X and Y will range between – 1.0 and + 1.0, the triangle in the above figure specifies the limits to diversification. The risk- = 1.0 return curves for any correlations = 0.2 within the limits of – 1.0 and + 1.0, 100% will fall within the triangle. bonds Relationship depends on correlation coefficient -1.0 < < +1.0 If = +1.0, no risk reduction is possible If = –1.0, complete risk reduction is possible
Portfolio Risk Depends on Correlation between Assets 12 Investing wealth in more than one security reduces portfolio risk. This is attributed to diversification effect. However, the extent of the benefits of portfolio diversification depends on the correlation between returns on securities. When correlation coefficient of the returns on individual securities is perfectly positive then there is no advantage of diversification. The weighted standard deviation of returns on individual securities is equal to the standard deviation of the portfolio. Diversification always reduces risk provided the correlation coefficient is less than 1.
The Efficient Set for Many Securities return Individual Assets PConsider a world with many risky assets; we can stillidentify the opportunity set of risk-return combinationsof various portfolios.
The Efficient Set for Many Securities return minimu m variance portfolio Individual Assets PThe section of the opportunity set above theminimum variance portfolio is the efficient frontier.
Investment Opportunity Set: The n-Asset Case 15 An efficient portfolio is one that has the highest expected returns for a given level of risk. The efficient frontier is the frontier formed by the set of efficient portfolios. All other portfolios, which lie outside the efficient frontier, are inefficient portfolios.
Efficient Portfolios of risky securities 16An efficientportfolio is one thathas the highestexpected returns fora given level of risk.The efficient frontieris the frontier formedby the set of efficientportfolios. All otherportfolios, which lieoutside the efficientfrontier, areinefficientportfolios.
Diversification and Portfolio Risk Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns. This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another. However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion.
RISK DIVERSIFICATION: SYSTEMATIC AND UNSYSTEMATIC RISK 18 When more and more securities are included in a portfolio, the risk of individual securities in the portfolio is reduced. This risk totally vanishes when the number of securities is very large. But the risk represented by covariance remains. Risk has two parts: 1. Diversifiable (unsystematic) 2. Non-diversifiable (systematic)
Systematic Risk 19 Systematic risk arises on account of the economy-wide uncertainties and the tendency of individual securities to move together with changes in the market. This part of risk cannot be reduced through diversification. It is also known as market risk. Investors are exposed to market risk even when they hold well-diversified portfolios of securities. Risk factors that affect a large number of assets Includes such things as changes in GDP, inflation, interest rates, etc.
Unsystematic Risk 21 Unsystematic risk arises from the unique uncertainties of individual securities. It is also called unique risk. These uncertainties are diversifiable if a large numbers of securities are combined to form well- diversified portfolios. Uncertainties of individual securities in a portfolio cancel out each other. Unsystematic risk can be totally reduced through diversification.
Hence…Total Risk Total risk = systematic risk + unsystematic risk The standard deviation of returns is a measure of total risk. For well-diversified portfolios, unsystematic risk is very small. Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk.
Since the systematic risk can’t be diversified, the investor will require compensation for bearing this risk. Diversified portfolios with no unsystematic risk, move with the market
Portfolio Risk and Number of Stocks, p355 In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk, VARIANCE Portfolio risk Non diversifiable risk; Systematic Risk; Market Risk; COVAR n
Optimal Portfolio with a Risk-Free Asset return 100% stocks rf 100% bondsIn addition to stocks and bonds, consider a world thatalso has risk-free securities like T-bills.
Riskless Borrowing and Lending return 100% stocks Balanced fund rf 100% bondsNow investors can allocate their money across the T-bills and a balanced mutual fund.