Return and Risk The Capital Asset Pricing Model (CAPM)

Return and RiskThe Capital Asset Pricing Model (CAPM)

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Important Security Characteristics

μ, which ____ σ2 and σ, which _____ Covariance and Correlation, which are _____

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μ, which is What we expect to make σ2 and σ, which _____ Covariance and Correlation, which are _____

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μ, which is What we expect to make σ2 and σ, which measure the risk Covariance and Correlation, which are _____

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μ, which is What we expect to make σ2 and σ, which measure the risk Covariance and Correlation, which are what

we are learning about now

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Statistics Review: Covariance

Covariance: measures how two variables move in relation to one anotherPositive: The two variables move up together or

down together Ex: Height and Weight

Negative: When one moves up the other moves down

Ex: Sleep and Coffee consumption

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Calculation

Cov (X,Y) = σXY = Σ pi * (Xi – μx) * (Yi – μY)

σXY = p1*(X1 – μX)*(Y1 – μY)+p2*(X2 – μX)*(Y2 – μY)+….+pN*(XN – μX)*(YN – μY)

Note that σXX = Σ pi * (Xi – μx) * (Xi – μX)Which implies?The covariance of an asset with itself is variance

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Covariance Strength

If the covariance of asset 1 and 2 is -1,000, and between asset 1 and 3 is 500. Which asset is more closely related to 1?

We don’t know Covariance gives the direction, BUT NOT

the strength of the relation.

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Correlation Coefficient

Correlation Coefficient: Measures the strength of the covariance relationshipThe correlation coefficient is a standardization of

covariance

ρ12 = σ12 / (σ1 * σ2)

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Correlation Coefficient Formula

ρ12 = σ12 / (σ1 * σ2)

ρ 12 – Correlation Coefficient

σ12 -

σ1 -

σ2 -

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ρ12 = σ12 / (σ1 * σ2)


σ12 – Covariance between 1 and 2

σ1 -

σ2 -

12


ρ12 = σ12 / (σ1 * σ2)



σ1 - Std Dev of asset 1

σ2 -

13


ρ12 = σ12 / (σ1 * σ2)



σ1 – Std Dev of asset 1

σ2 – Std Dev of asset 2

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Possible Correlation Coefficients ρ12 has to be between -1 and 1

+1 implies that the assets are perfectly positively correlated

One goes up 10% the other goes up 10%0 implies that the assets are not related

One goes up 10% the other does nothing-1 implies that the assets are perfectly negatively

correlated One goes up 10% the other goes down 10%

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Comparing Strength

When determining the strength of a correlation all we care about is the absolute value of the correlation coefficient

If ρ13 is -0.8, and ρ23 is 0.5, which asset is more correlated with 3?

Series 1

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Correlation Coefficient Example

σ13 is -1,000; σ23 is 500 σ1 is 10; σ2 is 1,000; σ3 is 250 Is 1 or 2 more strongly correlated with 3? ρ13 = σ13 / (σ1 * σ3) = -1000/(10*250) = -0.4 ρ23 = σ23 / (σ2* σ3) = 500/(1000*250) = 0.002 Since |ρ13| is greater than |ρ23|, the

relationship between 1 and 3 is stronger

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Portfolio Risk

So far we’ve been examining the risk of individual assets, but what happens when we combine individual assets into a portfolio?

DIVERSIFICATION

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Portfolio Illustration

Return on Security A:

Return onSecurity B:

Return onPortfolio of A & B

Time

Time

Time

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Diversification

Reduces risk by combining assets that are unlikely to all move in the same direction, without sacrificing expected return

Intuition: “Don’t put all your eggs in one basket” Stocks don’t move in exactly the same way. On a given

day, while Boeing may yield positive 1% return, Microsoft may have gone down 1%. So, if you had invested a dollar each in Boeing and Microsoft, you would not have lost any money.

Historical Non-Diversifiers

People on record for saying: “Put all your eggs in one basket and watch it closely”Mark Twain: Died pennilessAndrew Carnegie: known as “The Richest Man in

the World”

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Two Types of Risk Diversifiable/Unsystematic/Unique risk

Diversifiable risk affects individual or small groups of firms (industries)

Ex: Lawsuits, Strikes

Non-Diversifiable/Systematic/Market riskAffects all firms, economy wide risks

Ex: Business Cycle, Inflations Shocks

Which does σ measure?Total Risk: Unsystematic plus Systematic

How Diversification Works

Reduces/eliminates unsystematic risk. Why can’t we diversify away systematic risk?

It affects everythingNowhere to hide

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The Wonders of Diversifying

deviation

standardPortfolio

Unique risk

Market risk

Number ofsecurities

5 10

Total risk = Unique risk + market risk

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Portfolio Variances FormulasTwo-Stock Portfolio σp

2 = w12σ1

2 + w22σ2

2 + 2w1w2σ12

σp2 = w1

2σ12 + w2

2σ22 + 2w1w2ρ12σ1σ2

Three-Stock Portfolio σp

2 = w12σ1

2 + w22σ2

2 + w32σ3

2 + 2w1w2σ12 + 2w1w3σ13 + 2w2w3σ23

σp2 = w1

2σ12 + w2

2σ22 + w3

2σ32 + 2w1w2ρ12σ1σ2

+ 2w1w3ρ13σ1σ3 + 2w2w3ρ23σ2σ3

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Portfolio Covariance Matrix

Stock 1

Stock 1 Stock 2

Stock N

Stock 3

Stock 2

Stock 3 Stock N

Var (1,1)

Var (2,2)

Var (3,3)

Var (N,N)

Cov (1,2)

Cov (1,3)

Cov (1,N) Cov (2,N) Cov (3,N)

Cov (N,3)

Cov (N,2)

Cov (N,1)

Cov (2,3)

Cov (2,1) Cov (3,1)

Cov (3,2)

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Portfolio Variance Example

Two stocks A and B have expected returns of 10% and 20%.

In the past, A and B have had std dev of 15% and 25%, respectively, with a correlation coefficient of 0.2.

You decide to invest 30% in A and the rest in B. Calculate the portfolio return and portfolio risk. Has

diversification been of any use? Explain.

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Calculations wA = 30%wB =

A = 10% B = 20%

A = 15% B = 25% AB = 0.2

Portfolio Return = Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

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Calculations wA = 30%wB = 70%

A = 10% B = 20%

A = 15% B = 25% AB = 0.2

Portfolio Return = Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

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A = 10% B = 20%

A = 15% B = 25% AB = 0.2

Portfolio Return = 0.3*0.10 + 0.7*0.20 = 17% Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

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Calculations wA = 30% wB = 70% A = 10% B = 20% A = 15% B = 25% AB = 0.2 Portfolio Return = 17% Portfolio Variance =

0.32*0.152+0.72*0.252+2*0.3*0.7*0.2*0.15*0.25=358 %2

Portfolio Standard Deviation = Weighted Average Standard Deviation

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A = 10% B = 20%

A = 15% B = 25% AB = 0.2

Portfolio Return = 17% Portfolio Variance =358%2

Portfolio Standard Deviation = 18.9% Weighted Average Standard Deviation

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A = 10% B = 20%

A = 15% B = 25% AB = 0.2

Portfolio Return = 17% Portfolio Variance =358%2

Portfolio Standard Deviation = 18.9% Weighted Average Standard Deviation

0.3*0.15+0.7*0.25= 22%

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Remarks on Diversification

Diversification reduces the p from 22% to 18.9%

What happens if AB = 1?There is no diversification benefit

What happens as AB approaches -1?Diversification increases

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Which Stock do you Prefer? Stock A :

= 10%; = 2%

Stock B : = 10%; = 3%

Stock C : = 12%; = 2%

Alone C, but if we have other investments we need correlation

Which stock do you prefer?

0.00

0.05

0.10

0.15

0.20

0.25

0 5 10 15 20

Returns

Pro

babi

lity Stock A

Stock B

Stock C

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Fundamental Premise of Portfolio Theory Rational investors prefer the highest expected

return at the lowest possible risk How can investors lower risk without

sacrificing return?

Diversification

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Possible two asset portfolios

What are the possible portfolios we can create using only stocks and bonds The correlation coefficient is -0.99Stock: std dev = 14.3% expected return = 11.0%Bond: std dev = 8.2% expected return = 7.0%

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Possible Portfolios

Some portfolios are better: why? which ones? They offer a higher return for the same risk

Portfolo Risk and Return Combinations

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

Por

tfol

io R

etur

n

100% stocks

100% bonds

Efficient Portfolios

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Efficient Portfolio

Efficient Portfolios Offers the highest return for a given level of riskOffers the lowest risk for a given level of return

Can these same risk and return pairing be achieved with a single stock?NO

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Including More Assets

In the real world there are more than two assets

The efficient frontier is the outer most envelope of possible portfolios, given the universe of available assetsForm every possible portfolio of the various assets

and the efficient frontier are the portfolios on the edge

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retu

rn

P

Possible Portfolios

Including More Assets

Possible Efficient Portfolios

Efficient Portfolios

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Including a Risk Free Asset

How will the ability to lend and borrow at the risk free rate affect our possible risk-return combinations?

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Risk Free and Risky Assets

Start with our risky assets Add a risk free asset Reform our possible portfolios

retu

rn

P

rf

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Risk Free and Risky Assets

Start with our risky assets Add risk free assets, and form portfolios

What are the new efficient portfolios? What assets comprise the new efficient portfolios?

Risk Free & an Efficient Portfolio

retu

rn

P

rf

CML

Capital Market Line

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Which Efficient Portfolio? M, the MARKET PORTFOLIO

It offers the best risk return trade off SHARPE RATIO is the “price” of risk: (ri - rf) / i

Measure the return per unit of riskA higher Sharpe Ratio implies that we receive more

return per unit of risk

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Why only two assets?

Holding any risky asset other than M offers a lower risk adjusted returnThe investor is bearing more risk than necessaryThe investor is accepting too low a return

Rf is how we adjust for our risk tolerance

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Investing on the CML: In Two Easy Steps1. Find M

2. Determine your level of risk aversion Risk aversion determines location on the CML The more risk averse the investor the safer the

portfolio Closer to the risk free

Where on the CML to invest? Why do people move along

the CML?Risk Aversion

Risk averse investors hold more?Risk Free Asset

Risk tolerant investors hold more?Market Portfolio

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rf

M

CML

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How do we Move Along the CML? We move along the CML by altering our holdings of the

risk free asset Risk Averse investors buy T-Bills

Guaranteed returnThis is similar to lending money

Less Risk Averse investors short T-BillsThey borrow T-Bills, sell them, and use the proceeds to buy

more of the market portfolio They must return the T-Bill plus interest at some future point

This is similar to borrowing money

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Where are we Buying/Shorting T-Bills

rf

retu

rn

MBuy T-B

ills

Short T-B

ills

Risk & Pricing: Aside Investor A is undiversified, while B has a diversified

portfolio. Both investors want to buy a share in Facebook. Who gets the share? Which investor has a higher discount rate?

Assume: A’s discount rate is 20% and B’s is 10% & Facebook is expected to pay a constant $20 dividend Which investor offers a higher price for the share? (Hint: The price

offered is the PV of expected cash flows) B = 20/.1 = $200 A = 20/.2 = $100

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A

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Risk compensation

In a diversified portfolio, risk depends exclusively on the underlying securities exposure to systematic risk

Since unique risk can be diversified away the market will not compensate an investor for holding it Within the market the undiversified investor will always loss to the

diversified investor

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Measuring Systematic Risk

BETA (β): is how we measure systematic risk exposureβ – measures the sensitivity of the stock return to the

market return (ex S&P 500)

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β Formulas

βi = i,m / m2

βi = ρim m i / m2

βi = (ρim i )/ m

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βi = (ρim i )/ m Interpretation

i – Measures asset ‘i’ total risk ρim – Measures the proportion of i’s total risk

that is systematic m – Measures the total market risk

Which is??? ρimi– Measures the systematic risk of asset ‘i’ So βi:

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i – Measures asset ‘i’ total risk ρim – Measures the proportion of i’s total risk

that is systematic m – Measures the total market risk

Which is??? Systematic Risk ρimi– Measures the systematic risk of asset ‘i’ So βi:

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i – Measures asset ‘i’ total risk

ρim – Measures the proportion of i’s total risk that is systematic

m – Measures the total market risk

Which is??? Systematic Risk ρimi– Measures the systematic risk of asset ‘i’

So βi: is the ratio of the asset’s systematic risk to the systematic risk of the marketAn asset’s marginal contribution to the risk of

the portfolio

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Market β

βm = m,m / m2

βm = m2 / m

2

βm = 1

What is the β of the risk free asset? rf,m = 0 so βrf = 0

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Notes on β

β – tells us how sensitive a stock is to market movements“Average Stock” has a β of 1Stocks with β > 1 amplify market movementsStocks with 0 < β < 1 reduce market movementsStocks with negative β?Move opposite the market

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Portfolio β

The weighted average of the component stocks’ β

Example: You invested 40% of your money in asset A, βA is 1.5 and the balance in asset B, βB is 0.5. What is the portfolio beta?

βPort = 0.4*1.5 + 0.6*0.5 = 0.9

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CAPM and β CAPM states that expected returns are proportional to an

investment’s systematic riskA stock’s expected risk premium varies in proportion to it’s β

E(Ri) = Rf + βi (RM - Rf)

Security Market Line (SML) is the graphical representation of this relation

Stock’s risk premium

CAPM Assumptions Investors all have the same expectations Investors are risk averse and utility-maximizing Investors only care about mean and variance

Expected return and risk Perfect markets

No taxesNo transactions costsUnlimited borrowing and lending at the risk-free rate

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Security Market Line

What is the slope of the SML?


β

Rf

RM

1

Exp

ecte

d R

etur

n

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Security Market Line

What is the slope of the SML? The market risk premium, rm - rf


β

Rf

RM

1

Exp

ecte

d R

etur

n

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Putting Stocks in the SML

In equilibrium, all stocks should lie on the SML. This means that all stocks are correctly pricedA stock under the SML is: Under or Over pricedA stock above the SML is: Under or Over priced

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Dis-Equilibrium SML

A

B

C

E(r

i)

β

SML

A: is over-priced, people won’t want it. Demand falls so the price falls

C: is under-priced, people really want it. Demand rises so the price rises

B: is correctly priced

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CAPM Formula

CAPM: E(Ri) = Rf + βi (RM - Rf)

Market Risk Premium: (RM - Rf)

Asset i’s Risk Premium: βi (RM - Rf)

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Example

Rf = 5%

Historical average risk premium is 8.4% β = 1.5, return = β = 1, return = β = 0.5, return = β = 0, return = β = -1, return =

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Example

Rf = 5%

Historical average risk premium is 8.4% β = 1.5, return = 0.05 + 1.5 *0.084 = 0.176 β = 1, return = β = 0.5, return = β = 0, return = β = -1, return =

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Example

Rf = 5%

Historical average risk premium is 8.4% β = 1.5, return = 0.05 + 1.5 *0.084 = 0.176 β = 1, return = 0.05 + 1 *0.084 = 0.134 β = 0.5, return = β = 0, return = β = -1, return =

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Example

Rf = 5%

Historical average risk premium is 8.4% β = 1.5, return = 0.05 + 1.5 *0.084 = 0.176 β = 1, return = 0.05 + 1 *0.084 = 0.134 β = 0.5, return = 0.05 + 0.5 *0.084 = 0.092 β = 0, return = β = -1, return =

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Example

Rf = 5%

Historical average risk premium is 8.4% β = 1.5, return = 0.05 + 1.5 *0.084 = 0.176 β = 1, return = 0.05 + 1 *0.084 = 0.134 β = 0.5, return = 0.05 + 0.5 *0.084 = 0.092 β = 0, return = 0.05 + 0 *0.084 = 0.050 β = -1, return =

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Example

Rf = 5%

Historical average risk premium is 8.4% β = 1.5, return = 0.05 + 1.5 *0.084 = 0.176 β = 1, return = 0.05 + 1 *0.084 = 0.134 β = 0.5, return = 0.05 + 0.5 *0.084 = 0.092 β = 0, return = 0.05 + 0 *0.084 = 0.050 β = -1, return = 0.05 + -1 *0.084 = -0.034

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Example 1

The stock market moves up by 10%. Assume stock A has a beta of 1.5, stock B has a beta of 0.5 and stock C has a beta of -0.5.

Predict A, B, and C’s response to the market?

A: 15%

B: 5%

C: -5%

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Example 2

βA is 1.5, and A is 20% βB is 2, and B is 15%; Which stock has a higher expected return?

Stock B

Why We Care

Basic investment ruleBig rewards are accompanied by large risks

Explains the risk return trade off

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Documents

Return and Risk The Capital Asset Pricing Model (CAPM)