UCSC ECON 80H - Diversification and Risky Asset Allocation

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ChapterMcGraw-Hill/IrwinCopyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved.Diversification 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 highestexpected 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.”)• The next slide shows how the expected return formula is used when there are two states. – Note that the “states” are equally likely to occur in this example. – BUT! They do not have to be equally likely--they can have different probabilities of occurring.[]∑=×=n1ssi,sireturnpreturn expected11-5Expected Return, II.• Suppose:– There are two stocks:• Starcents• Jpod– We are looking at a period of one year.• Investors agree that the expected return:– for Starcents is 25 percent– for Jpod is 20 percent• Why would anyone want to hold Jpod shares when Starcents is expected to have a higher return?11-6Expected Return, III.• The answer depends on risk• Starcents is expected to return 25 percent• But the realized return on Starcents could be significantly higher or lower than 25 percent• Similarly, the realized return on Jpod could be significantly higher or lower than 20 percent.11-7Calculating Expected Returns11-8Expected 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 −=11-9Calculating 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 expectedreturnpσVariance11-10Example: Calculating Expected Returns and Variances: Equal State Probabilities(1) (3) (4) (5) (6)Return if Return ifState of State Product: State Product:Economy Occurs (2) x (3) Occurs (2) x (5)Recession -0.20 -0.10 0.30 0.15Boom 0.70 0.35 0.10 0.05Sum: E(Ret): 0.25 E(Ret): 0.20(1) (3) (4) (5) (6) (7)Return ifState of State Expected Difference: Squared: Product:Economy Occurs Return: (3) - (4) (5) x (5) (2) x (6)Recession -0.20 0.25 -0.45 0.2025 0.10125Boom 0.70 0.25 0.45 0.2025 0.10125Sum: 0.202500.45Calculating Expected Returns:Starcents: Jpod:(2)Probability ofState of Economy0.500.501.00Calculating Variance of Expected Returns:Starcents:(2)Probability ofState of Economy0.500.501.00 Sum = the Variance:Standard Deviation:Note that the second spreadsheet is only for Starcents. What would you get for Jpod?11-11Expected Returns and Variances, Starcents and Jpod11-12Portfolios• 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).11-13Portfolios: 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() ()[]∑=×=n1iiiPREwRE11-14Example: Calculating Portfolio Expected ReturnsNote that the portfolio weight in Jpod = 1 – portfolio weight in Starcents.(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)Starcents Starcents Jpod Jpod PortfolioReturn if Portfolio Contribution Return if Portfolio Contribution ReturnState of Prob. State WeightProduct:State WeightProduct: Sum: Product:Economy of State Occurs in Starcents:(3) x (4)Occurs in Jpod:(6) x (7) (5) + (8) (2) x (9)Recession 0.50 -0.20 0.50 -0.10 0.30 0.50 0.15 0.05 0.025Boom 0.50 0.70 0.50 0.35 0.10 0.50 0.05 0.40 0.200Sum: 1.00 0.225Calculating Expected Portfolio Returns:Sum is Expected Portfolio Return:11-15Variance 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,sPREREpRVAR11-16Example: Calculating Variance of Portfolio Expected Returns• It is possible to construct a portfolio of risky assets with zero portfolio variance! What? How? (Open this spreadsheet,


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