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**Diversification and Risky Asset Allocation**

4/15/2017 Chapter 11 Diversification and Risky Asset Allocation Chapter 11 examines the role of diversification and asset allocation in investing. Diversification reduces risk in investing how it actually works how 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.

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Learning Objectives 1. 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.

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**Expected Return of an Asset**

4/15/2017 Expected Return of an Asset Expected 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” data What 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 = weights Prs = probability of a state Rs = return if a state occurs s = number of states

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**Expected Return & Risk Premium**

4/15/2017 Expected Return & Risk Premium 5 potential states More complicated than textbook For 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%

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**Measuring Variance and Standard Deviation**

4/15/2017 Measuring Variance and Standard Deviation Expected return on an asset ≠ all we need to know Need a measure of the risk associated with the expected return. = Chapter 1, variance and standard deviation = measures of risk Calculation 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.

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**Calculating Dispersion of Returns**

4/15/2017 Calculating Dispersion of Returns First 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 of Note that variance is measured in percent squared Standard deviation =10.95% = square root of variance Now 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%.

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**Calculating Dispersion of Returns**

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Portfolios Portfolios = 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 = proportions Sometimes expressed in percentages. In calculations, make sure you use proportions (i.e., decimals).

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4/15/2017 Portfolio Return The 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 portfolio Notice 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 i E(Ri) = Expected return on Security i

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**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 A Column 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 returns Summing Col 6 = 12.8% = expected portfolio return 7.0% = .60(-5%) + .40(25%)

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**Portfolio Return Alternate Method – Step 1**

4/15/2017 Portfolio Return Alternate Method – Step 1 Alternative approach Step 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.

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**Portfolio Return Alternate Method – Step 2**

4/15/2017 Portfolio Return Alternate Method – Step 2 Assume 60% in Stock A; 40% in Stock B With the expected return on each asset and the portfolio weights: Multiply each E(R) X wgt Sum results = portfolio expected return of 12.80%.

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**Portfolio Variance & Standard Deviation**

4/15/2017 Portfolio Variance & Standard Deviation Computing expected return on a portfolio is pretty straightforward Computing a portfolio’s variance is not so easy! Calculate the deviation of the portfolio’s return in a state from the portfolio expected return Square the deviations Multiple squared deviations by state probabilitiies Sum to find variance Take square root of the variance, to get standard deviation

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**Portfolio Variance & Standard Deviation**

4/15/2017 Portfolio Variance & Standard Deviation Table follows steps from previous slide Beginning 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 deviation Column 7 applies each state’s probability. Summing Column 7 = portfolio variance = Square root of variance = standard deviation = 3.4%.

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**Risk-Return Comparison**

4/15/2017 Risk-Return Comparison E(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?

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4/15/2017 Stocks A & B vs. States Graph = 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

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**Portfolio Returns: 60% A – 40% B**

4/15/2017 Portfolio Returns: 60% A – 40% B Red 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.

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**Diversification and Risk**

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**Diversification and Risk**

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**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.

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**Correlation Coefficient**

4/15/2017 Correlation Coefficient Correlation Coefficient = ρ (rho) Scales covariance to [-1,+1] r = -1.0 Two stocks can be combined to form a riskless portfolio r = +1.0 No risk reduction at all In general, stocks have r ≈ Risk is lowered but not eliminated Correlation coefficient scaled -1 and +1 Correlation coefficient = +1 = the returns on the two assets move EXACTLY together. Correlation coefficient = -1 = they move EXACTLY opposite Perfect 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

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4/15/2017 Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0) 25 15 -10 Stock W Stock M -10 Portfolio WM 25 25 15 15 Two perfectly negatively correlated stocks … W and M When combined – they flat line – all risk eliminated -10

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4/15/2017 Returns distribution for two perfectly positively correlated stocks (ρ = 1.0) Stock M 15 25 -10 Stock M’ 15 25 -10 Portfolio MM’ 15 25 -10 Two perfectly positively correlated stocks … No risk reduction

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**Why Diversification Works**

ρ = +1 ρ = -1 ρ = 0

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**Why Diversification Works**

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**Why Diversification Works**

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Covariance of Returns 4/15/2017 Measures how much the returns on two risky assets move together To better understand correlation, we need covariance This 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 number Positive or negative Will Not Be on a Test

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**Covariance vs. Variance of Returns**

4/15/2017 Covariance vs. Variance of Returns The first formula = covariance Second formula = familiar variance Notice similarity Will Not Be on a Test

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**Covariance Will Not Be on a Test Deviation = RA,S – E(RA)**

4/15/2017 Calculating covariance We already have the expected returns on A and B For each, calculate the deviation in each state from each asset’s expected return Multiply these deviations Multiply the result by each state’s probability and sum to arrive at covariance For our 2 stocks, covariance = … meaning? Deviation = RA,S – E(RA) Covariance (A:B) = Will Not Be on a Test

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Covariance Will Not Be on a Test

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**Correlation Coefficient**

4/15/2017 Correlation Coefficient Interpreting covariance is easier with the correlation coefficient. Scales covariance to a range of -1, +1 Correlation coefficient of A and B = -.967 Almost perfectly negatively correlated Will Not Be on a Test

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**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 A xB = portfolio weight of asset B such that xA + xB = 1

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**Portfolio Risk Example**

4/15/2017 Portfolio Risk Example Continuing our 2-stock example Equation 11.3 in the text = the 2-asset version Substituting into equation 11.3 will again result in a portfolio standard deviation 3.4%. 33

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**s of n-Stock Portfolio Subscripts denote stocks i and j**

4/15/2017 Subscripts denote stocks i and j ri,j = Correlation between stocks i and j σi and σj =Standard deviations of stocks i and j σij = Covariance of stocks i and j These are equivalent formulas for calculating portfolio variance (and standard deviation) using either covariance or the correlation coefficient. WHERE these metric fit in the risk calculation Either COV or correlation can REDUCE portfolio risk Will Not Be on a Test

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**Portfolio Risk-2 Risky Assets**

4/15/2017 Portfolio Risk-2 Risky Assets Applying the formula to our 2-stock portfolio … Same portfolio standard deviation found on slides 11 and 12 This formula is the general form … The text uses the 2-stock form

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**Diversification & the Minimum Variance Portfolio**

4/15/2017 Diversification & the Minimum Variance Portfolio Assume 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 text Correlation coefficient = 0.10 some combination of these two portfolios should significantly reduce the risk There is a minimum variance combination which will provide the lowest risk for a given return.

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**Correlation and Diversification**

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**Correlation and Diversification**

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**The Minimum Variance Portfolio**

4/15/2017 The Minimum Variance Portfolio Table 11.9 Stocks Bonds E(R) 12% 6% Std Dev 15% 10% Corr Coeff 0.10 100% Stocks Replicated from textbook Data 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% interval Graph details results Standard deviation on the X-axis Expected portfolio return on the Y-axis. The minimum variance portfolio (MVP) = Portfolio with the lowest standard deviation Portfolio combination positioned the furtherest to the left In 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.” MVP 100% Bonds

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**The Minimum Variance Portfolio**

4/15/2017 XA* = % 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 portfolio Equation = percentage to be invested in one of the two assets Using sample data: Minimum variance portfolio = % in stocks % in bonds Resulting in an expected portfolio return of 7.73% with a standard deviation of 8.69%.

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**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.

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**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 either More return for a given level of risk, or Less risk for a given level of return.

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**Markowitz Efficient Frontier**

4/15/2017 Markowitz Efficient Frontier Efficient Frontier Investment Opportunity Set =All possible combinations of risk and return available from portfolios of these two assets Efficient Portfolio = A portfolio that offers the highest return for its level of risk Inefficient Portfolio = Undesirable portfolios dominated by efficient portfolios Efficient Frontier = the set of portfolios with: The maximum return for a given risk AND The minimum risk given a return With a two-asset sample, all portfolio combinations are on the curved line. Red line = Markowitz Efficient Frontier. Lower black curve = inefficient Dominated by a portfolio on the efficient frontier with equal risk and higher return. Inefficient Frontier

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4/15/2017 Efficient Frontier If 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 portfolios Do not fall on the efficient frontier Dominated by a portfolio on the red line

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**The Importance of Asset Allocation**

Suppose we invest in three mutual funds: One that contains Foreign Stocks, F One that contains U.S. Stocks, S One that contains U.S. Bonds, B Figure 11.6 shows the results of calculating various expected returns and portfolio standard deviations with these three assets. Expected Return Standard Deviation Foreign Stocks, F 18% 35% U.S. Stocks, S 12 22 U.S. Bonds, B 8 14

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**Risk and Return with Multiple Assets**

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**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)

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