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**AN INTRODUCTION TO PORTFOLIO MANAGEMENT**

Chapter 7 AN INTRODUCTION TO PORTFOLIO MANAGEMENT

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Chapter 8 Questions What do we mean be risk aversion, and what evidence indicates that investors are generally risk averse? What are the basic assumptions behind the Markowitz portfolio theory? What do we mean by risk, and what are some of the measures of risk used in investments? How do we compute the expected rate of return for an individual risky asset or a portfolio of assets? How do we compute the standard deviation of rates of return for an individual risky asset? What do we mean by the covariance between rates of return, and how is it computed? What is the relationship between covariance and correlation?

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Chapter 8 Questions What is the formula for the standard deviation for a portfolio of risky assets, and how does it differ from the standard deviation of an individual risky asset? Given the formula for the standard deviation of a portfolio, why and how do we diversify a portfolio? What happens to the standard deviation of a portfolio when we change the correlation between the assets in the portfolio? What is the risk-return efficient frontier of risky assets? Is it reasonable for alternative investors to select different portfolios from the set of portfolios on the efficient frontier? What determines which portfolio on the efficient frontier is selected by an individual investor?

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**Background Assumptions**

As an investor you want to maximize the returns for a given level of risk. Your portfolio includes all of your assets, not just financial assets The relationship between the returns for assets in the portfolio is important. A good portfolio is not simply a collection of individually good investments.

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Risk Aversion Portfolio theory assumes that investors are averse to risk Given a choice between two assets with equal expected rates of return, risk averse investors will select the asset with the lower level of risk It also means that a riskier investment has to offer a higher expected return or else nobody will buy it

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**Are investors risk averse?**

The popularity of insurance of various types attests to risk aversion Yield on bonds increase with risk classifications from AAA to AA to A…., indicating that investors require risk premiums as compensation Experimental psychology also confirms that humans tend to be risk averse

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**Are investors always risk averse?**

Risk preference may have to do with amount of money involved - risking only small amounts. Trips to the casino might seem to refute risk aversion, but realize that gaming is best thought of as entertainment, not investing

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**Definition of Risk One definition: Uncertainty of future outcomes**

Alternative definition: The probability of an adverse outcome We will discuss several measures of risk that are used in developing portfolio theory

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**Markowitz Portfolio Theory**

Derives the expected rate of return for a portfolio of assets and an expected risk measure Markowitz demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk under reasonable assumptions The portfolio variance formula shows how to effectively diversify a portfolio

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**Markowitz Portfolio Theory**

Assumptions Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period. Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.

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**Markowitz Portfolio Theory**

Assumptions Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.

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**Markowitz Portfolio Theory**

Under these five assumptions, a single asset or portfolio of assets is efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

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**Alternative Measures of Risk**

Variance or standard deviation of expected return (Main focus) Based on deviations from the mean return Larger values indicate greater risk Other measures Range of returns Returns below expectations Semivariance – measures deviations only below the mean

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**Expected Rates of Return**

Individual risky asset Expected rates of return are calculated by determining the possible returns (Ri) for some investment in the future, and weighting each possible return by its own probability (Pi). E(R) = S Pi Ri

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**Expected Return Example**

Economic Conditions Probability Return Strong % Average % Weak % E(R) = .20(40%) (12%) (-20%) E(R) = 8%

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**Expected Rates of Return**

Portfolio Weighted average of expected returns (Ri) for the individual investments in the portfolio Percentages invested in each asset (wi) serve as the weights E(Rport) = S wi Ri

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**Expected Return Example**

Weight (%) Expected Return (Ri) 30% % 30% % 40% % E(R) = .30(10%) (15%) (18%) E(R) = 14.7%

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**Variance & Standard Deviation of Returns**

Individual Investment Standard deviation is the positive square root of the variance Both measures are based on deviations of each possible return (Ri) from the expected return (E(R)) Variance: s2 = SPi(Ri-E(R))2

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**Standard Deviation of Expected Returns**

Economic Conditions Probability Return Strong % Average % Weak % E(R) = 8% s2 = .20 (40-8) (12-8) (-20-8)2 s2 = 448 s = 21.2%

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**Variance & Standard Deviation of Returns**

Before calculating the portfolio variance and standard deviation, several other measures need to be understood Covariance Measures the extent to which two variables move together For two assets, i and j, the covariance of rates of return is defined as: Covij = E{[Ri,t - E(Ri)][Rj,t - E(Rj)]} Apply to Actual sample data

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**Variance & Standard Deviation of Returns**

Correlation coefficient Values of the correlation coefficient (r) go from -1 to +1 Standardized measure of the linear relationship between two variables rij = Covij/(sisj) Covij= covariance of returns for securities i and j si= standard deviation of returns for security i sj= standard deviation of returns for security j

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**Portfolio Standard Deviation Formula**

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**Portfolio Standard Deviation Calculation**

The portfolio standard deviation is a function of: The variances of the individual assets that make up the portfolio The covariances between all of the assets in the portfolio The larger the portfolio, the more the impact of covariance and the lower the impact of the individual security variance

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**Portfolio Standard Deviation Calculation**

To show the effect of different covariance, we assume a portfolio which composed by two assets, E(R1)=0.20, E(σ1)＝0.10; E(R2)=0.20, E(σ1)＝0.10 W1=0.5; w2=0.5 Now consider the following five correlation coefficients and the covariance they yield.

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Use formula Thus, in case a

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Incase b, Portfolios c, d and e are as follows:

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**For each Case, the expected return :**

Thus, A(0.1, 0.2), B(0.087, 0.2), C(0.071, 0.2), D(0.05, 0.2), E(0.0, 0.2)

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**Portfolio Standard Deviation Calculation**

To show the effect of different weight, we assume a portfolio which composed by two assets, E(R1)=0.10, E(σ1)＝0.07 E(R2)=0.20, E(σ1)＝0.10 Correlation=0

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**holding the correlation coefficient constant, and the weight change the weights as follows:**

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Use the formula E(Rport) = S wi Ri St.dev Expc.return

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**Implications for Portfolio Formation**

Assets differ in terms of expected rates of return, standard deviations, and correlations with one another While portfolios give average returns, they give lower risk Diversification works! Even for assets that are positively correlated, the portfolio standard deviation tends to fall as assets are added to the portfolio

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**Implications for Portfolio Formation**

Combining assets together with low correlations reduces portfolio risk more The lower the correlation, the lower the portfolio standard deviation Negative correlation reduces portfolio risk greatly Combining two assets with perfect negative correlation reduces the portfolio standard deviation to nearly zero

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Estimation Issues Results of portfolio analysis depend on accurate statistical inputs Estimates of Expected returns Standard deviations Correlation coefficients With 100 assets, 4,950 correlation estimates Estimation risk refers to potential errors

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Estimation Issues With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets Single index market model: bi = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market Rm = the returns for the aggregate stock market

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**Numerous portfolio combination of available assets**

Standard deviation of return (σ)

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**The Efficient Frontier**

The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return Frontier will be portfolios of investments rather than individual securities Exceptions being the asset with the highest return and the asset with the lowest risk

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**Efficient Frontier and Alternative Portfolios**

B A C Standard Deviation of Return

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**The Efficient Frontier and Portfolio Selection**

Any portfolio that plots “inside” the efficient frontier (such as point C) is dominated by other portfolios For example, Portfolio A gives the same expected return with lower risk, and Portfolio B gives greater expected return with the same risk Would we expect all investors to choose the same efficient portfolio? No, individual choices would depend on relative appetites return as opposed to risk

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**The Efficient Frontier and Investor Utility**

An individual investor’s utility curve specifies the trade-offs she is willing to make between expected return and risk Each utility curve represent equal utility; curves higher and to the left represent greater utility (more return with lower risk) The interaction of the individual’s utility and the efficient frontier should jointly determine portfolio selection The optimal portfolio has the highest utility for a given investor It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

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**Selecting an Optimal Risky Portfolio**

U3’ U2’ U1’ Y U3 X U2 U1

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**Investor Differences and Portfolio Selection**

A relatively more conservative investor would perhaps choose Portfolio X On the efficient frontier and on the highest attainable utility curve A relatively more aggressive investor would perhaps choose Portfolio Y

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