Presentation on theme: "Diversification and Risky Asset Allocation"— Presentation transcript:
1 Diversification and Risky Asset Allocation 4/15/2017Chapter 11Diversification and Risky Asset AllocationChapter 11 examines the role of diversification and asset allocation in investing.Diversification reduces risk in investinghow it actually workshow much diversification is needed to achieve an efficient investment portfolio.Efficiently diversified portfolio =highest expected return, given its risk.Review/expand our understanding of asset risk & return Portfolio risk and return.
2 Learning Objectives1. How to calculate expected returns and variances for a security.2. How to calculate expected returns and variances for a portfolio.3. The importance of portfolio diversification.4. The efficient frontier and the importance of asset allocation.
3 Expected Return of an Asset 4/15/2017Expected Return of an AssetExpected return = the “weighted average” return on a risky asset expected in the future.Chapter 1 = calculating an expected return based on historical data.Chapter 11 = “expectational” dataWhat return we expect to achieve given certain states of the economy.Associated with each state of the economy is the probability of that state occurring with the probabilities summing to 100%.“Expected” return= multiply the probability of a state times the return expected in that state and sum the results.Simple weighted average with the probabilities = weightsPrs = probability of a stateRs = return if a state occurss = number of states
4 Expected Return & Risk Premium 4/15/2017Expected Return & Risk Premium5 potential statesMore complicated than textbookFor each state, associated probability of occurrence and an expected return.Last column =result of multiplying each probability times the expected return.Summing last column = expected return of 15%Assume risk-free rate = 8%, then risk premium = 7%.Risk-free rate = 8%Risk Premium = 15% - 8% = 7%
5 Measuring Variance and Standard Deviation 4/15/2017Measuring Variance and Standard DeviationExpected return on an asset ≠ all we need to knowNeed a measure of the risk associated with the expected return.= Chapter 1, variance and standard deviation = measures of riskCalculation different with expectational data.Formulas on this slide = how to calculate variance and standard deviation from a probability distribution.Notice that to compute variance, we are applying each probability to the squared variation of each state’s return from the expected return.
6 Calculating Dispersion of Returns 4/15/2017Calculating Dispersion of ReturnsFirst step = calculating stock’s expected return (Cols 1-4)Col 5: deviation of each state’s return from the 15% expected return.Col 6: deviation squared.Col 7: probability (column 2) times the squared deviation (column 6).Summing column 7 = variance ofNote that variance is measured in percent squaredStandard deviation =10.95% = square root of varianceNow we have a better idea of the risk-return pattern for this stock with an expected return of 15% and a standard deviation of 10.95%.
8 PortfoliosPortfolios = groups of assets, such as stocks and bonds, that are held by an investor.Portfolio Description = list the proportion of the total value of the portfolio that is invested into each asset.Portfolio Weights = proportionsSometimes expressed in percentages.In calculations, make sure you use proportions (i.e., decimals).
9 4/15/2017Portfolio ReturnThe rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights.Moving from one asset to a portfolio of assets,Similar approach to arrive at the expected return on a portfolioNotice how similar the formula on this slide is to the previous one for the return on an individual asset.Difference =weights- xi =% of portfolio value invested in each asset.Expected portfolio return = weighted average of the expected returns of the assets in the portfolio, weighted by the proportion of the portfolio invested in that asset.xi = Proportion of funds in Security iE(Ri) = Expected return on Security i
10 Portfolio Return 7.0% = .60(-5%) + .40(25%) 4/15/2017 Add Asset B to our sample Asset (A)Form a portfolio of 60% A and 40% B.Columns 1 and 2 repeat the states and probabilities of each state of the economy.Column 3 repeats the expected returns for Asset AColumn 4 adds expected returns for Asset B.Column 5 shows the results of calculating the expected return of the portfolio in each state.For example, the expected portfolio return in State 1 equals (-5% x 60%) + (25% x 40%) to arrive at 7%.Column 6 = probability of each state x column 6 portfolio returnsSumming Col 6 = 12.8% = expected portfolio return7.0% = .60(-5%) + .40(25%)
11 Portfolio Return Alternate Method – Step 1 4/15/2017Portfolio Return Alternate Method – Step 1Alternative approachStep 1:Compute the expected return on each of the two assets.We already have the expected return on Asset A as 15%.Applying the same technique to Asset B’s data yields an expected return of 9.5%.We’ll complete the calculations on the next slide.
12 Portfolio Return Alternate Method – Step 2 4/15/2017Portfolio Return Alternate Method – Step 2Assume 60% in Stock A; 40% in Stock BWith the expected return on each asset and the portfolio weights:Multiply each E(R) X wgtSum results = portfolio expected return of 12.80%.
13 Portfolio Variance & Standard Deviation 4/15/2017Portfolio Variance & Standard DeviationComputing expected return on a portfolio is pretty straightforwardComputing a portfolio’s variance is not so easy!Calculate the deviation of the portfolio’s return in a state from the portfolio expected returnSquare the deviationsMultiple squared deviations by state probabilitiiesSum to find varianceTake square root of the variance, to get standard deviation
14 Portfolio Variance & Standard Deviation 4/15/2017Portfolio Variance & Standard DeviationTable follows steps from previous slideBeginning with the expected portfolio returns in each state we calculated previously, we calculate each one’s deviation from the portfolio expected return, which is shown in column 5.Column 6 squares that deviationColumn 7 applies each state’s probability.Summing Column 7 = portfolio variance =Square root of variance = standard deviation = 3.4%.
15 Risk-Return Comparison 4/15/2017Risk-Return ComparisonE(RA) > E(RB)But: Std Dev (A) > Std Dev (B)Portfolio of 60% A and 40% B:Expected return for the portfolio is a average of the two assets’ returns.Standard deviation of the portfolio is significantly lower than that of either stock!Made-up example to dramatically demonstrate the affects of diversification, and it does – but why does it work?
16 4/15/2017Stocks A & B vs. StatesGraph = expected returns for A and B in each state of the economy.Assume state 1 = “bust”; state 5 = “boom”Asset A moves WITH the economy – down in a bust, up in a boom.Asset B moves counter to the economy – up in a down economy, down in a boom.With A and B moving almost completely opposite to each other, when combined, one’s losses are offset by the other’s gains.This is exactly what you want in risk reduction
17 Portfolio Returns: 60% A – 40% B 4/15/2017Portfolio Returns: 60% A – 40% BRed line = expected returns on the portfolio (P)Combination of A and B results in some loss of return but a substantial reduction in risk.Clearly what drives this risk reduction is the fact that the returns on the two assets move in opposite directions.Covariance and Correlation measure the tendency of the returns on two assets to move together.
20 Why Diversification Works Correlation = The tendency of the returns on two assets to move together.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.
21 Correlation Coefficient 4/15/2017Correlation CoefficientCorrelation Coefficient = ρ (rho)Scales covariance to [-1,+1]r = -1.0 Two stocks can be combined to form a riskless portfolior = +1.0 No risk reduction at allIn general, stocks have r ≈Risk is lowered but not eliminatedCorrelation coefficient scaled -1 and +1Correlation coefficient = +1 = the returns on the two assets move EXACTLY together.Correlation coefficient = -1 = they move EXACTLY oppositePerfect for total risk elimination but not very realistic.Most stocks in the U.S. are positively correlated with a correlation coefficient between 0.35 and 0.67 – positively correlated but not perfectly.Will Not Be on a Test
22 4/15/2017Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)2515-10Stock WStock M-10Portfolio WM25251515Two perfectly negatively correlated stocks … W and MWhen combined – they flat line – all risk eliminated-10
23 4/15/2017Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)Stock M1525-10Stock M’1525-10Portfolio MM’1525-10Two perfectly positively correlated stocks …No risk reduction
27 Covariance of Returns4/15/2017Measures how much the returns on two risky assets move togetherTo better understand correlation, we need covarianceThis IS NOT in the text!Like correlation, covariance measures how much the returns on two risky assets move together … but no scaling … result can be any numberPositive or negativeWill Not Be on a Test
28 Covariance vs. Variance of Returns 4/15/2017Covariance vs. Variance of ReturnsThe first formula = covarianceSecond formula = familiar varianceNotice similarityWill Not Be on a Test
29 Covariance Will Not Be on a Test Deviation = RA,S – E(RA) 4/15/2017Calculating covarianceWe already have the expected returns on A and BFor each, calculate the deviation in each state from each asset’s expected returnMultiply these deviationsMultiply the result by each state’s probability and sum to arrive at covarianceFor our 2 stocks, covariance = … meaning?Deviation = RA,S – E(RA)Covariance (A:B) =Will Not Be on a Test
31 Correlation Coefficient 4/15/2017Correlation CoefficientInterpreting covariance is easier with the correlation coefficient.Scales covariance to a range of -1, +1Correlation coefficient of A and B = -.967Almost perfectly negatively correlatedWill Not Be on a Test
32 Calculating 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
33 Portfolio Risk Example 4/15/2017Portfolio Risk ExampleContinuing our 2-stock exampleEquation 11.3 in the text = the 2-asset versionSubstituting into equation 11.3 will again result in a portfolio standard deviation 3.4%.33
34 s of n-Stock Portfolio Subscripts denote stocks i and j 4/15/2017Subscripts denote stocks i and jri,j = Correlation between stocks i and jσi and σj =Standard deviations of stocks i and jσij = Covariance of stocks i and jThese are equivalent formulas for calculating portfolio variance (and standard deviation) using either covariance or the correlation coefficient.WHERE these metric fit in the risk calculationEither COV or correlation can REDUCE portfolio riskWill Not Be on a Test
35 Portfolio Risk-2 Risky Assets 4/15/2017Portfolio Risk-2 Risky AssetsApplying the formula to our 2-stock portfolio …Same portfolio standard deviation found on slides 11 and 12This formula is the general form …The text uses the 2-stock form
36 Diversification & the Minimum Variance Portfolio 4/15/2017Diversification & the Minimum Variance PortfolioAssume the following statistics for two portfolios, one of stocks and one of bonds:New set of sample data:2 portfolios - one of stocks and one bonds.= example in Chapter 11 in textCorrelation coefficient = 0.10 some combination of these two portfolios should significantly reduce the riskThere is a minimum variance combination which will provide the lowest risk for a given return.
39 The Minimum Variance Portfolio 4/15/2017The Minimum Variance PortfolioTable 11.9StocksBondsE(R)12%6%Std Dev15%10%Corr Coeff0.10100% StocksReplicated from textbookData in Table 11.9,Portfolio weights from 100% stocks to 100% bonds at 5% intervals.Expected portfolio return and portfolio standard deviation at every 5% intervalGraph details resultsStandard deviation on the X-axisExpected portfolio return on the Y-axis.The minimum variance portfolio (MVP)= Portfolio with the lowest standard deviationPortfolio combination positioned the furtherest to the leftIn some cases, more than one portfolio for a given standard deviation.For example, 15% stock,85% bond =return of 6.90%, standard deviation of 9.01%40% stock, 60% bond split =return of 8.40% with a standard deviation of 8.90%.The latter combination is clearly better – less risk, more return.This superior portfolio is considered “efficient.”MVP100% Bonds
40 The Minimum Variance Portfolio 4/15/2017XA* = %Putting % in stocks and % in bonds yields an E(R) = 7.73% and a standard deviation of 8.69% as the minimum variance portfolio. (Ex 11.7 p.366)Formula for the MVP of a two asset portfolioEquation = percentage to be invested in one of the two assetsUsing sample data:Minimum variance portfolio =% in stocks% in bondsResulting in an expected portfolio return of 7.73% with a standard deviation of 8.69%.
41 Correlation and Diversification 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 or inefficient.
42 The Markowitz Efficient Frontier The Markowitz Efficient frontier = 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 eitherMore return for a given level of risk, orLess risk for a given level of return.
43 Markowitz Efficient Frontier 4/15/2017Markowitz Efficient FrontierEfficient FrontierInvestment Opportunity Set =All possible combinations of risk and return available from portfolios of these two assetsEfficient Portfolio = A portfolio that offers the highest return for its level of riskInefficient Portfolio = Undesirable portfolios dominated by efficient portfoliosEfficient Frontier = the set of portfolios with:The maximum return for a given risk ANDThe minimum risk given a returnWith a two-asset sample, all portfolio combinations are on the curved line.Red line = Markowitz Efficient Frontier.Lower black curve = inefficientDominated by a portfolio on the efficient frontier with equal risk and higher return.Inefficient Frontier
44 4/15/2017Efficient FrontierIf portfolio = >2 assets not all portfolios will graph on the curved line.Some may fall inside the curve as shown by the white dots above.White dots = inefficient portfoliosDo not fall on the efficient frontierDominated by a portfolio on the red line
45 The Importance of Asset Allocation Suppose we invest in three mutual funds:One that contains Foreign Stocks, FOne that contains U.S. Stocks, SOne that contains U.S. Bonds, BFigure 11.6 shows the results of calculating various expected returns and portfolio standard deviations with these three assets.Expected ReturnStandard DeviationForeign Stocks, F18%35%U.S. Stocks, S1222U.S. Bonds, B814
47 Useful Internet Sites www.411stocks.com (to find expected earnings) (for more on risk measures)(also contains more on risk measure)(measure diversification using “instant x-ray”)(review modern portfolio theory)(check out the reading list)