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**Diversification and Portfolio Management (Ch. 8)**

05/10/06

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**How investors view risk and return**

Investors like return. They seek to maximize return. But investors dislike risk. They seek to avoid or minimize risk. Why? Because human beings possess the psychological trait of “risk aversion” which is a dislike for taking risks.

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**Implications of risk aversion**

The “risk-return tradeoff” - Risk averse investors require higher rates of return to induce them to invest in higher risk securities. The higher a security’s risk, the higher the return investors demand. Thus, the less they are willing to pay for the investment, i.e. as risk increase, P0 decreases. Risk averse investors will diversify their investments in order to reduce risk.

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Diversification Definition - An investment strategy designed to reduce risk by spreading the funds invested across many securities. It is holding a broad portfolio of securities so as “not to have all your eggs in one basket.” Since people hold diversified portfolios of securities, they are not very concerned about the risk and return of a single security. They are more concerned about the risk and return of their entire portfolio.

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**Two components of an asset’s risk (standard deviation)**

Unique Risk - Also called “diversifiable risk” and “unsystematic risk.” The part of a security’s risk associated with random outcomes generated by events specific to the firm. This risk can be eliminated by proper diversification. Market Risk – Also called “systematic risk.” The part of a security’s risk that cannot be eliminated by diversification because it is associated with economic or market factors that systematically affect most firms. Market risk reflects economy-wide sources of risk that affect most firms and, hence, the overall stock market.

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**The expected return on a portfolio of stocks**

Assume N stocks are held in the portfolio. Stock i is held in the proportion, wi Then the expected return on the portfolio of stocks is the weighted average of the individual stock expected returns:

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**The standard deviation of returns for a portfolio of stocks**

The standard deviation of returns for the portfolio of stocks is given by: where returni is the return of the portfolio in state i and n represents the number of states of the economy.

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**The standard deviation of returns for a portfolio of stocks**

We can also calculate the standard deviation of returns for the portfolio of stocks as: where ρij= the correlation coefficient for stocks i and j

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**Correlation coefficient**

The “Correlation Coefficient” is a measure of the extent that two variables move or vary together. It ranges between –1.0 and +1.0 Positive correlation: a high value on one variable is likely to be associated with a high value on the other. Negative correlation: a high value on one variable is likely to be associated with a low value on the other. No correlation: values of each are independent of the other

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**Correlation coefficient**

It is denoted by the Greek letter, “rho”: ρ If ρ = +1.0, perfect positive correlation If ρ = -1.0, perfect negative correlation If ρ = 0, uncorrelated or independent

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**How diversification reduces risk**

Combining stocks into a portfolio reduces the variability of possible returns as long as the returns on the individual stocks are not perfectly correlated, i.e. as long as their correlation coefficients are less than +1.0.

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**Portfolio risk falls as you add securities**

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**You can’t eliminate “market risk”**

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**This pattern occurs because of the two components of a stock’s risk**

Total Risk = Market risk + unique risk The unique risk is “diversified away” when individual stocks are combined in a portfolio. Only market risk remains. The amount of the market risk is determined by the market risk of the individual stocks in the portfolio.

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**How should we measure portfolio risk now?**

Since diversification eliminates unique risk and leaves market or non-diversifiable risk, the latter is the only relevant risk for a diversified investor. Therefore, the relevant measure of risk for a portfolio is a measure of the sensitivity of the portfolio’s returns to changes in the return on the “market portfolio”. This is known as the portfolio’s beta (β) By definition, the market portfolio has a beta of 1 and the risk-free asset has a beta of 0.

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How to interpret a beta If βi > 1, returns to stock i are amplified relative to the market. If βi is between 0 and 1.0, returns to stock i tend to move in the same direction as the market but not as far. If βi < 1(very rare), returns to stock i tend to move in the opposite direction as the market.

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**How to interpret a beta-cont’d**

A stock with β = 1 has average market risk. A well-diversified portfolio of such stocks tends to move by the same percentage as the overall market moves. A stock with β = +.5 has below average market risk. A well-diversified portfolio of these stocks tends to move half as far as the overall market moves.

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**Measuring individual security betas**

Security betas are estimated by running a regression between the historical returns on the security and the historical returns on the market portfolio over the same period of time. Typically, betas are estimated using 5 years of historical monthly returns. The slope of the regression represents the beta

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**General comments about risk**

Most stocks are positively correlated with the market (ρi,m 0.65). σ 0.35 for an average stock. Combining stocks in a portfolio generally lowers risk.

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**Calculating portfolio betas**

Assume N stocks are held in the portfolio. Stock i is held in the proportion, wi Then the portfolio beta is the weighted average of the individual stock betas:

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**Risk-return trade-off revisited**

For a diversified investor whose only concern is non-diversifiable risk, measured by beta, this investor will now want higher return for a security with a higher beta. This linear relationship between a security’s expected return and beta is formalized by the Security Market Line (SML).

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**The Security Market Line**

E(r) SML E(rm) rf BETA 1 where rf is the return on the risk-free security and E(rm) is the expected return on the market portfolio

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**The Security Market Line**

E(r) SML E(rA) E(rm) rf BETA 1 βA

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Reward-to-risk ratio The reward-to-risk ratio is calculated as the ratio of the excess return (beyond that of the risk-free return) that is required or expected for a particular security given its level of risk: Reward to risk ratio This excess return (E(rA) – rf) is referred to as the asset’s risk premium, which is the return investors require beyond that of the risk free rate for security A

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**Capital Asset Pricing Model (CAPM)**

The CAPM assumes that the reward-to-risk ratio of all securities are equal….giving us the following model to estimate the expected or required return on a stock: where rf is the return on the risk-free security and is often proxied by the 3-month U.S. Treasury Bill or Treasury Bond rate; E(rm) is the expected return on the market portfolio and is often proxied by the return on the S&P 500 index and; E(rm) - rf represents the market risk premium

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Jensen’s alpha (α) Using the CAPM, and assuming that securities are priced based on this model, one can measure whether a particular security performed better or worse than expected by this model, i.e., did the security provide a return greater than that required and expected by investors? This performance measure is called Jensen’s alpha and is calculated as follows: where kA represents the actual return achieved by security A and E(kA) represents the expected return based on CAPM.

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**The Security Market Line and Jensen’s alpha**

E(r) SML E(rA) E(rm) Jensen’s alpha rA rf BETA 1 βA

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CAPM assumptions The CAPM relies on historical data to calculate its inputs, thus it implicitly assumes that the past is a good measure of the future The model assumes no transaction costs, identical and complete information, rational investors and that securities that are mispriced will self-adjust. These are all efficient market assumptions.

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CAPM limitations Theoretically, the market portfolio should consist of all assets. Studies have shown that additional explanatory (risk) factors must be considered in explaining security returns.

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