ChapterMcGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.11Diversification and Risky Asset Allocation11-2Diversification• Intuitively, we all know that if you hold many investments• Through time, some will increase in value• Through time, some will decrease in value• It is unlikely that their values will all change in the same way• Diversification has a profound effect on portfolio return and portfolio risk.• But, exactly how does diversification work?11-3Diversification and Asset Allocation• Our goal in this chapter is to examine the role of diversification and asset allocation in investing.• In the early 1950s, professor Harry Markowitz was the first to examine the role and impact of diversification.• Based on his work, we will see how diversification works, and we can be sure that we have “efficiently diversified portfolios.” – An efficiently diversified portfolio is one that has the highest expected return, given its risk.– You must be aware that diversification concerns expected returns.11-4Expected Returns, I.• Expected return is the “weighted average” return on a risky asset, from today to some future date. The formula is:• To calculate an expected return, you must first:– Decide on the number of possible economic scenarios that might occur.– Estimate how well the security will perform in each scenario, and– Assign a probability to each scenario– (BTW, finance professors call these economic scenarios, “states.”) n1ssi,sireturnpreturn expected11-5Expected Risk Premium• Recall:• Suppose riskfree investments have an 8% return. If so,– The expected risk premium on Jpod is 12%– The expected risk premium on Starcents is 17%• This expected risk premium is simply the difference between– The expected return on the risky asset in question and– The certain return on a risk-free investmentRate RiskfreeReturn Expected Premium Risk Expected 11-6Calculating the Variance of Expected Returns• The variance of expected returns is calculated using this formula:• This formula is not as difficult as it appears. • This formula says is to add up the squared deviations of each return from its expected return after it has been multiplied by the probability of observing a particular economic state (denoted by “s”).• The standard deviation is simply the square root of the variance.Varianceσ Deviation Standard   n1s2ss2return expectedreturnpσVariance11-7Portfolios• Portfolios are groups of assets, such as stocks and bonds, that are held by an investor.• One convenient way to describe a portfolio is by listing the proportion of the total value of the portfolio that is invested into each asset.• These proportions are called portfolio weights.– Portfolio weights are sometimes expressed in percentages.– However, in calculations, make sure you use proportions (i.e., decimals).11-8Portfolios: Expected Returns• The expected return on a portfolio is a linear combination, or weighted average, of the expected returns on the assets in that portfolio.• The formula, for “n” assets, is:In the formula: E(RP) = expected portfolio returnwi = portfolio weight in portfolio asset iE(Ri) = expected return for portfolio asset i    n1iiiPREwRE11-9Variance of Portfolio Expected Returns• Note: Unlike returns, portfolio variance is generally not a simple weighted average of the variances of the assets in the portfolio.• If there are “n” states, the formula is:• In the formula, VAR(RP) = variance of portfolio expected returnps = probability of state of economy, sE(Rp,s) = expected portfolio return in state sE(Rp) = portfolio expected return• Note that the formula is like the formula for the variance of the expected return of a single asset.     n1s2Psp,sPREREpRVAR11-10Diversification and Risk, I.11-11Diversification and Risk, II.11-12Why Diversification Works, I.• Correlation: The tendency of the returns on two assets to move together. Imperfect correlation is the key reason why diversification reduces portfolio risk as measured by the portfolio standard deviation.• Positively correlated assets tend to move up and down together.• Negatively correlated assets tend to move in opposite directions.• Imperfect correlation, positive or negative, is why diversification reduces portfolio risk.11-13Why Diversification Works, II.• The correlation coefficient is denoted by Corr(RA, RB) or simply, A,B. • The correlation coefficient measures correlation and ranges from:From: -1 (perfect negative correlation)Through: 0 (uncorrelated)To: +1 (perfect positive correlation)11-14Why Diversification Works, III.11-15Why Diversification Works, IV.11-16Calculating Portfolio Risk• For a portfolio of two assets, A and B, the variance of the return on the portfolio is:Where: xA = portfolio weight of asset AxB= portfolio weight of asset Bsuch that xA+ xB= 1.(Important: Recall Correlation Definition!))RCORR(Rσσx2xσxσxσB)COV(A,x2xσxσxσBABABA2B2B2A2A2pBA2B2B2A2A2p11-17The Importance of Asset Allocation, Part 1.• Suppose that as a very conservative, risk-averse investor, you decide to invest all of your money in a bond mutual fund. Very conservative, indeed? Uh, is this decision a wise one?11-18Correlation and Diversification, I.11-19Correlation and Diversification, II.11-20Correlation and Diversification, III.• The various combinations of risk and return available all fall on a smooth curve.• This curve is called an investment opportunity set,because it shows the possible combinations of risk and return available from portfolios of these two assets.• A portfolio that offers the highest return for its level of risk is said to be an efficient portfolio.• The undesirable portfolios are said to be dominated orinefficient.11-21The Markowitz Efficient Frontier• The Markowitz Efficient frontier is the set of portfolios with the maximum return for a given risk AND the minimum risk given a return.• For the plot, the upper left-hand boundary is the Markowitz efficient frontier. • All the other possible combinations are inefficient. That is, investors would not hold these portfolios because they could get either– more return for a given level of risk, or– less risk for a given level of