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**Return risk and the Security market line**

Chapter 13 Return risk and the Security market line

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**Chapter Outline Expected Returns and Variances of a portfolio**

Announcements, Surprises, and Expected Returns Risk: Systematic and Unsystematic Diversification and Portfolio Risk Systematic Risk and Beta The Security Market Line (SML)

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Expected Returns (1) Expected returns are based on the probabilities of possible outcomes Expected means average if the process is repeated many times Expected return = return on a risky asset expected in the future

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**Expected Returns (2) Probability Expected return Stock A Stock B Boom**

0.2 20% 15% Normal 0.4 10% 8% Recession -5% 2% RA = RB = If the risk-free rate = 3.2%, what is the risk premium for each stock?

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**Variance and Standard Deviation (1)**

Unequal probabilities can be used for the entire range of possibilities Weighted average of squared deviations

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**Variance and Standard Deviation (2)**

Consider the previous example. What is the variance and standard deviation for each stock? Stock A Stock B

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**Portfolio = a group of assets held by an investor**

Portfolios The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets Portfolio = a group of assets held by an investor Portfolio weights = Percentage of a portfolio’s total value in a particular asset

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Portfolio Weights Suppose you have $ 20,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security? $5,000 of A $9,000 of B $5,000 of C $1,000 of D

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**Portfolio Expected Returns (1)**

The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio You can also find the expected return by finding the portfolio return in each possible state and computing the expected value

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**Expected Portfolio Returns (2)**

Consider the portfolio weights computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio? A: 19.65% B: 8.96% C: 9.67% D: 8.13% E(RP) =

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**Portfolio Variance (1) Steps:**

Compute the portfolio return for each state: RP = w1R1 + w2R2 + … + wnRn Compute the expected portfolio return using the same formula as for an individual asset Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

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**Portfolio Variance (2) Consider the following information**

Invest 60% of your money in Asset A State Probability A B Boom % 10% Recession % 30% What is the expected return and standard deviation for each asset? What is the expected return and standard deviation for the portfolio?

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Solution:

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**Another Way to Calculate Portfolio Variance**

Portfolio variance can also be calculated using the following formula: Correlation is a statistical measure of how 2 assets move in relation to each other If the correlation between stocks A and B = -1, what is the standard deviation of the portfolio?

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Solution:

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**Different Correlation Coefficients (1)**

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**Different Correlation Coefficients (2)**

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**Different Correlation Coefficients(3)**

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**Possible Relationships between Two Stocks**

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Diversification (1) There are benefits to diversification whenever the correlation between two stocks is less than perfect (p < 1.0) If two stocks are perfectly positively correlated, then there is simply a risk- return trade-off between the two securities.

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Diversification (2)

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**Expected vs. Unexpected Returns**

Expected return from a stock is the part of return that shareholders in the market predict (expect) The unexpected return (uncertain, risky part): At any point in time, the unexpected return can be either positive or negative Over time, the average of the unexpected component is zero Total return = Expected return + Unexpected return

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**Announcements and News**

Announcements and news contain both an expected component and a surprise component It is the surprise component that affects a stock’s price and therefore its return Announcement = Expected part + Surprise

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**Systematic Risk Risk factors that affect a large number of assets**

Also known as non-diversifiable risk or market risk Examples: changes in GDP, inflation, interest rates, general economic conditions

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**Unsystematic Risk Risk factors that affect a limited number of assets**

Also known as diversifiable risk and asset-specific risk Includes such events as labor strikes, shortages.

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**Returns Unexpected return = systematic portion + unsystematic portion**

Total return can be expressed as follows: Total Return = expected return + systematic portion + unsystematic portion

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**Effect of Diversification**

Portfolio diversification is the investment in several different asset classes or sectors Diversification is not just holding a lot of assets Principle of diversification = spreading an investment across a number of assets eliminates some, but not all of the risk

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**The Principle of Diversification**

Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns Reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another There is a minimum level of risk that cannot be diversified away and that is the systematic portion

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**Portfolio Diversification (1)**

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**Portfolio Diversification (2)**

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**Diversifiable (Unsystematic) Risk**

The risk that can be eliminated by combining assets into a portfolio If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away The market will not compensate investors for assuming unnecessary risk

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Total Risk The standard deviation of returns is a measure of total risk For well diversified portfolios, unsystematic risk is very small Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk

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**Systematic Risk Principle**

There is a reward for bearing risk There is no reward for bearing risk unnecessarily The expected return (and the risk premium) on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away

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**Measuring Systematic Risk**

Beta (β) is a measure of systematic risk Interpreting beta: β = 1 implies the asset has the same systematic risk as the overall market β < 1 implies the asset has less systematic risk than the overall market β > 1 implies the asset has more systematic risk than the overall market

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High and Low Betas

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**Total vs. Systematic Risk**

Consider the following information: Standard Deviation Beta Security A 20% Security B 30% Which security has more total risk? Which security has more systematic risk? Which security should have the higher expected return?

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Solution:

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Portfolio Betas Consider the previous example with the following four securities Security Weight Beta A B C D What is the portfolio beta?

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**Beta and the Risk Premium**

The higher the beta, the greater the risk premium should be The relationship between the risk premium and beta can be graphically interpreted and allows to estimate the expected return

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**Consider a portfolio consisting of asset A and a risk-free asset**

Consider a portfolio consisting of asset A and a risk-free asset. Expected return on asset A is 20%, it has a beta = 1.6. Risk- free rate = 8%.

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**Portfolio Expected Returns and Betas**

Rf

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**Reward-to-Risk Ratio:**

The reward-to-risk ratio is the slope of the line illustrated in the previous slide Slope = (E(RA) – Rf) / (A – 0) Reward-to-risk ratio = If an asset has a reward-to-risk ratio = 8? If an asset has a reward-to-risk ratio = 7?

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**The Fundamental Result**

The reward-to-risk ratio must be the same for all assets in the market If one asset has twice as much systematic risk as another asset, its risk premium is twice as large

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**Security Market Line (1)**

The security market line (SML) is the representation of market equilibrium The slope of the SML is the reward-to- risk ratio: (E(RM) – Rf) / M The beta for the market is always equal to one, the slope can be rewritten Slope = E(RM) – Rf = market risk premium

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**Security Market Line (2)**

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**The Capital Asset Pricing Model (CAPM)**

The capital asset pricing model defines the relationship between risk and return E(RA) = Rf + A(E(RM) – Rf) If we know an asset’s systematic risk, we can use the CAPM to determine its expected return

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CAPM Consider the betas for each of the assets given earlier. If the risk-free rate is 4.5% and the market risk premium is 8.5%, what is the expected return for each? Security Beta Expected Return A 3.6 B .7 C 1.7 D 1.9

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**Factors Affecting Expected Return**

Time value of money – measured by the risk-free rate Reward for bearing systematic risk – measured by the market risk premium Amount of systematic risk – measured by beta

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