RETURN RISK AND THE SECURITY MARKET LINEHTTP://WWW.QUANTFINANCEJOBS.COM

Chapter 13

Chapter OutlineExpected Returns and Variances

of a portfolioAnnouncements, Surprises, and

Expected ReturnsRisk: Systematic and

UnsystematicDiversification and Portfolio RiskSystematic Risk and BetaThe Security Market Line (SML)

Expected Returns (1)

Expected returns are based on the probabilities of possible outcomes

Expected means average if the process is repeated many times

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Expected return = return on a risky asset expected in the future

Expected Returns (2)

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• RA = 6% risk premium = 6 – 3.2=2.8%

• RB = 7% risk premium = 7 – 3.2=3.8%

• If the risk-free rate = 3.2%, what is the risk premium for each stock?

Probability Expected returnStock A Stock B

Boom 0.2 20% 15%Normal 0.4 10% 8%

Recession 0.4 -5% 2%

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 B6

0.2 15 3 12.80.4 8 3.2 0.40.4 2 0.8 10

mean 7 23.2 Var4.8166 std

0.2 20 4 39.20.4 10 4 6.40.4 -5 -2 48.4

mean 6 94 Var9.6954 std

Portfolios

The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets

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Portfolio = a group of assets held by an investor

Portfolio weights = Percentage of a portfolio’s total value in a particular asset

Portfolio WeightsSuppose 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 WA=.25

◦$9,000 of B WB=.45

◦$5,000 of C WC=.25

◦$1,000 of D WD=.058

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) = 11.77

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0.25 19.65 4.9130.45 8.96 4.0320.25 9.67 2.4180.05 8.13 0.407

11.77

Portfolio Variance (1)Steps:1. Compute the portfolio return for

each state:RP = w1R1 + w2R2 + … + wnRn

2. Compute the expected portfolio return using the same formula as for an individual asset

3. 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

informationInvest 60% of your money in Asset

A◦State Probability A B

◦Boom .5 70%10%

◦Recession .5 -20%30%

1. What is the expected return and standard deviation for each asset?

2. What is the expected return and standard deviation for the portfolio?

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Solution:1.

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prob A (R) B(R)0.5 70 100.5 -20 30

E ( R ) 25 20Var( R ) 2025 100std ( R ) 45 10

prob A (R) B(R) Port( R )0.5 70 10 460.5 -20 30 0

E ( R ) 25 20 23Var( R ) 2025 100 529std ( R ) 45 10 23

Another Way to Calculate Portfolio VariancePortfolio 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?

ULULULUULLP CORRxxxx ,22222 2

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Solution:Var=1058.1 (1458+32-431.87)

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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

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Total return = Expected return + Unexpected return

Announcements and NewsAnnouncements 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

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Announcement = Expected part + Surprise

Systematic RiskRisk 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 RiskRisk 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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ReturnsUnexpected 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

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Principle of diversification = spreading an investment across a number of assets eliminates some, but not all of the risk

The Principle of DiversificationDiversification 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 28

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 RiskThe 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 PrincipleThere 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 RiskBeta (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 34

High and Low Betas

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Portfolio BetasConsider the previous example

with the following four securities◦Security Weight Beta◦A .1333.69

◦B .2 0.64◦C .2671.64

◦D .4 1.79What is the portfolio beta? 1.773

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Beta and the Risk PremiumThe 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. 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

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Rf

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 ResultThe 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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CAPMConsider 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 35.1%B .7 10.45%C 1.7 18.85%D 1.9 20.65%

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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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