Portfolio funding

Slide 1

11-1

Return, Risk, and The Security Market Line

We have to define risk and then discuss how to measure it. We then must quantify the relationship

between an asset’s risk and its required return.

There are two types of risk: systematic and unsystematic. This distinction is crucial because, as we

will see, systematic risk affects almost all assets in the economy, at least to some degree, while

unsystematic risk affects at most a small number of assets. We then develop the principle of

diversification, which shows that highly diversified portfolios will tend to have almost no

unsystematic risk.

Slide 2

11-2

• Expected returns are based on the probabilities of possible outcomes.

• In this context, “expected” means average if the process is repeated many times.

• The “expected” return does not even have to be a possible return.

Where:

pi = the probability of state “i” occurring

Ri = the expected return on an asset in state i

Expected Returns

 =

= n

i

ii RpRE 1

)(

Slide 3

11-3

Example: Expected Returns

Stock C Stock T

State (i) Probability (Pi) (Ri) (Ri) ___

Boom 0.3 0.15 0.25

Normal 0.5 0.10 0.20

Recession ??? 0.02 0.01

1.00

• E(RC) = .3(.15) + .5(.1) + .2(.02) = 0.099

• E(RT) = .3(.25) + .5(.2) + .2(.01) = 0.177 13-3

• Suppose you have predicted the following returns for stocks C and T in three possible states of the economy. What are the expected returns?

What is the probability of a recession? 1- 0.3 - 0.5 = 0.2

Or work in the percentage terms.

E(RC) = .3(15%) + .5(10%) + .2(2%) = 9.9%

E(RT) = .3(25%) + .5(20%) + .2(1%) = 17.7%

Slide 4

11-4

Variance and Standard Deviation

• Variance (2) and standard deviation () measure the volatility of returns

• Variance = Weighted average of squared deviations

• Standard Deviation = Square root of variance

 =

−= n

i

ii RERp 1

22 ))((σ

Variance measures the dispersion of points around the mean of a distribution. In this context, we

are attempting to characterize the variability of possible future security returns around the expected

return. In other words, we are trying to quantify risk and return. Variance measures the total risk

of the possible returns.

Slide 5

11-5

Variance and Standard Deviation • Consider the previous example. What are the variance

and standard deviation for each stock?

Stock C Stock T

State (i) Probability (Pi) (Ri) (Ri) ___

Boom 0.3 0.15 0.25

Normal 0.5 0.10 0.20

Recession 0.2 0.02 0.01

E(RC) = 0.099 E(RT) = 0.177

• Stock C ▪ 2 = .3(0.15-0.099)2 + .5(0.10-0.099)2 + .2(0.02-0.099)2 = 0.002029

▪  = 0.045 (=4.50%)

• Stock T ▪ 2 = .3(0.25-0.177)2 + .5(0.20-0.177)2 + .2(0.01-0.177)2 = 0.007441

▪  = 0.0863 (=8.63%) 13-5

Slide 6

11-6

Another Example

• Consider the following information:

State(i) Probability (Pi) ABC, Inc. Return

Boom .25 0.15

Normal .50 0.08

Slowdown .15 0.04

Recession .10 -0.03

• What is the expected return? • E(R) = .25(0.15) + .5(0.08) + .15(0.04) + .1(-0.03) = 8.05%

• What is the variance? • Variance = .25(.15-0.0805)2 + .5(0.08-0.0805)2 + .15(0.04-0.0805)2 +

• .1(-0.03-0.0805)2 = 0.00267475

• What is the standard deviation? • Standard Deviation = 5.17%

13-6

⚫ E(R) = .25(0.15) + .5(0.08) + .15(0.04) + .1(-0.03) = 0.0805 (=8.05%) ⚫ Variance = .25(.15-0.0805)2 + .5(0.08-0.0805)2 + .15(0.04-0.0805)2 + .1(-0.03-0.0805)2 =

0.00267475

⚫ Standard Deviation = 0.0517 (=5.17%)

You may experience confusion in understanding the mathematics of the variance calculation. You may

have the feeling that you should divide the variance of an expected return by (n−1). Note that the

probabilities account for this division. We divide by n−1 in the historical variance because we are looking

at a sample. If we looked at the entire population (which is what we are doing with expected values), then

we would divide by n (or multiply by 1 ⁄ n) to get our historical variance. This is the same as saying that the

“probability” of occurrence is the same for all observations and is equal to 1 ⁄ n.

Slide 7

11-7

Portfolios

• Portfolio = collection of assets

• An asset’s risk and return impact how the stock affects the risk and return of the portfolio

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

Each individual has their own level of risk tolerance. Some people are just naturally more inclined

to take risk, and they will not require the same level of compensation as others for doing so. Our

risk preferences also change through time. We may be willing to take more risk when we are young

and without a spouse or kids. But, once we start a family, our risk tolerance may drop.

Slide 8

11-8

Portfolio Expected Returns

• Expected return for an asset:

• The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio

• Weights (wj) = % of portfolio invested in each asset

 =

= m

1j

jjP )R(Ew)R(E

 =

= n

1i

iiRp)R(E

The expected return on a portfolio is the sum of the product of the expected returns on the

individual securities and their portfolio weights. Let wj be the portfolio weight for asset j and m

be the total number of assets in the portfolio.

You can also find the expected return by finding the portfolio return in each possible state and

computing the expected value as we did with individual securities.

Slide 9

11-9

Example: Portfolio Weights

• Suppose you have $15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?

▪ $2000 of C

▪ $3000 of KO

▪ $4000 of INTC

▪ $6000 of BP

▪ C: 2000/15000 = .133

▪ KO: 3000/15000 = .2

▪ INTC: 4000/15000 = .267

▪ BP: 6000/15000 = .4

13-9

Weights (wj) ?

C – Citigroup

KO – Coca-Cola

INTC – Intel

BP – BP

Note that the sum of the weights = 1 (=100%).

A portfolio is a collection of assets, such as stocks and bonds, held by an investor. Portfolios can

be described by the percentage investment in each asset. These percentages are called portfolio

weights.

Example: If two securities in a portfolio have a combined value of $10,000 with $6000 invested

in IBM and $4000 invested in GM, then the weight on IBM = 6000 ⁄ 10000 = .6 (=60%) and the

weight on GM = 4000 ⁄ 100000 = .4 (=40%) or we can simply calculate the weight on GM by 1 –

0.6 = 0.4 (=40%) since the sum of the weights equals 1 (=100%).

Slide 10

11-10

Expected Portfolio Returns

• Consider the portfolio weights computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio?

•

▪ C: 19.69%

▪ KO: 5.25%

▪ INTC: 16.65%

▪ BP: 18.24%

• E(RP) = .133(0.1969) + .2(0.0525) + .267(0.1665) + .4(0.1824) = 0.1541

13-10

▪ C: 2000/15000 = .133

▪ KO: 3000/15000 = .2

▪ INTC: 4000/15000 = .267

▪ BP: 6000/15000 = .4

Weights (wj) Expected Return E(Rj)

Slide 11

11-11

Portfolio Variance

• Compute portfolio return for each state:

RP,i = w1R1,i + w2R2,i + … + wmRm,i

• Compute the overall 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

The calculation of portfolio variance requires three steps:

1. Compute the portfolio return for each state of the economy.

2. Compute the overall expected return of the portfolio.

3. Compute portfolio variance and standard deviation using the same formulas as for an

individual asset.

Unlike expected return, the variance of a portfolio is NOT the weighted sum of the individual

security variances. Combining securities into portfolios can reduce the total variability of returns.

Slide 12

11-12

Example: Portfolio Variance

• Consider the following information on returns and probabilities:

▪ Invest 50% of your money in Asset A State Probability A B

Boom .4 30% -5%

Bust .6 -10% 25%

• What are the expected return and standard deviation for each asset?

• What are the expected return and standard deviation for the portfolio?

13-12

Slide 13

11-13

Example: Portfolio Variance

• Invest 50% of your money in Asset A State Probability A B

Boom .4 30% -5%

Bust .6 -10% 25%

• What are the expected return and standard deviation for each asset?

Asset A: E(RA) = .4(0.3) + .6(-0.1) = 0.06 (=6%)

Variance(A) = .4(0.3-0.06)2 + .6(-0.1-0.06)2 = 0.0384

Std. Dev.(A) = 0.196 (=19.6%)

Asset B: E(RB) = .4(-0.05) + .6(0.25) = 0.13 (=13%)

Variance(B) = .4(-0.05-0.13)2 + .6(0.25-0.13)2 = 0.0216

Std. Dev.(B) = 0.147 (=14.7%)

• What are the expected return for the portfolio? E(Rp) = .5(0.06) + .5(0.13) = 0.095 (=9.5%)

13-13

Expected return and standard deviation for each asset

Or work in the percentage terms.

Asset A: E(RA) = .4(30%) + .6(-10%) = 6%

Variance(A) = .4(30%-6%)2 + .6(-10%-6%)2 = 384

Std. Dev.(A) = √384 =19.6% Asset B: E(RB) = .4(-5%) + .6(25%) = 13%

Variance(B) = .4(-5%-13%)2 + .6(25%-13%)2 = 216

Std. Dev.(B) = √216 =14.7%

Expected return for the portfolio (1)

The expected return on a portfolio is the sum of the product of the expected returns on the

individual securities and their portfolio weights. Weights (wj) = % of portfolio invested in each

asset. There are two stocks in your portfolio and you invest 50% of your money in Asset A.

What percent are you investing in Asset B? 50% (= (100% - 50%)). Thus, WA = 0.5, and WB = 0.5

Or work in the percentage terms.

E(Rp) = .5(6%) + .5(13%) = 9.5%

Slide 14

11-12

Example: Portfolio Variance

• Consider the following information on returns and probabilities:

▪ Invest 50% of your money in Asset A State Probability A B Portfolio

Boom .4 30% -5% 12.5% = RP,Boom Bust .6 -10% 25% 7.5% = RP,Bust

E(RA) = 6% E(RB) = 13% E(Rp)

• What are the expected return and standard deviation for the portfolio?

13-12

(1)

(2)

To compute the standard deviation for the portfolio, we need to follow three steps (See Slide 11)

1. Compute the portfolio return for each state of the economy.

There are two stocks in your portfolio and you invest 50% of your money in Asset A. What

percent are you investing in Asset B? 50% (= (100% - 50%)). Thus, WA = 0.5, and WB = 0.5

Portfolio return in boom: RP,Boom = 0.5(0.3) + 0.5(-0.05) = 0.125 (=12.5%)

Portfolio return in bust: RP,Bust = 0.5(-0.1) + 0.5(0.25) = 0.075 (=7.5%)

Or work in the percentage terms.

RP,Boom = 0.5(30%) + 0.5(-5%) = 12.5%

RP,Bust = 0.5(-10%) + 0.5(25%) =7.5%

2. Compute the overall expected return of the portfolio.

In Slide 13, we computed E(Rp) = .5(6%) + .5(13%) = 9.5% - (1)

You can also find the expected return for the portfolio by finding the portfolio return in each

possible state and computing the expected value as we did with individual securities.

E(Rp) = .4(12.5%) + .6(7.5%) = 9.5% - (2)

Slide 15

11-14

Example: Portfolio Variance

▪ Invest 50% of your money in Asset A State Probability A B Portfolio

Boom .4 30% -5% 12.5%

Bust .6 -10% 25% 7.5%

• What are the expected return and standard deviation for the portfolio?

Portfolio return in boom (RP,Boom) = .5(.3) + .5(-.05) = .125

Portfolio return in bust (RP,Bust) = .5(-.1) + .5(.25) = .075

Expected return for portfolio E(RP) = .5(.06) + .5(.13) = .095 or

.4(.125) + .6(.075) = .095

Variance of portfolio = .4(.125-.095)2 + .6(.075-.095)2 = .06

Standard deviation = .0245

13-14

3. Compute portfolio variance and standard deviation using the same formulas as for an individual

asset.

Expected return = .5(6%) + .5(13%) = 9.5% or

Expected return = .4(12.5%) + .6(7.5%) = 9.5%

Variance of portfolio = .4(12.5%-9.5%)2 + .6(7.5%-9.5%)2 = 6%

Standard deviation = 2.45%

Unlike expected return, the variance of a portfolio is NOT the weighted sum of the individual

security variances. Combining securities into portfolios can reduce the total variability of returns.

• Note that the variance is NOT equal to .5(384) + .5(216) = 300 and • Standard deviation is NOT equal to .5(19.6) + .5(14.7) = 17.17%

To compute variance of portfolio, follow the direction of (2) in the previous slide.

Slide 16

11-15

• Realized returns are generally not equal to expected returns.

• There is the expected component and the unexpected component.

▪ At any point in time, the unexpected return can be either positive or negative.

▪ Over time, the average of the unexpected component is zero.

Expected vs. Unexpected Returns

Expected vs. Unexpected Returns

Total return = Expected return + Unexpected return

Total return differs from expected return because of surprises, or “news.” This is one of the

reasons that realized returns differ from expected returns.

Slide 17

11-16

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

• This is very obvious when we watch how stock prices move when an unexpected announcement is made or earnings are different than anticipated

13-16

Announcement—the release of information not previously available. Announcements have two

parts: the expected part and the surprise part. The expected part is “discounted” information used

by the market to estimate the expected return, while the surprise is news that influences the

unexpected return.

Slide 18

11-17

Announcements and News

• On November 17, 2004 it was announced that K- Mart would acquire Sears in an $11 billion deal. Sears’ stock price jumped from a closing price of $45.20 on November 16 to a closing price of $52.99 (a 7.79% increase) and K-Mart’s stock price jumped from $101.22 on November 16 to a closing price of $109.00 on November 17 (a 7.69% increase). Both stocks traded even higher during the day.

• Why the jump in price?

13-17

It is easy to see the effect of unexpected news on stock prices and returns. Consider the following

two cases:

(1) On November 17, 2004 it was announced that K-Mart would acquire Sears in an $11 billion

deal. Sears’ stock price jumped from a closing price of $45.20 on November 16 to a closing price

of $52.99 (a 7.79% increase) and K-Mart’s stock price jumped from $101.22 on November 16 to

a closing price of $109.00 on November 17 (a 7.69% increase). Both stocks traded even higher

during the day. Why the jump in price? Unexpected news, of course.

(2) On November 18, 2004, Williams-Sonoma cut its sales and earnings estimates for the fourth

quarter of 2004 and its share price dropped by 6%. There are plenty of other examples where

unexpected news causes a change in price and expected returns.

Slide 19

11-18

Efficient Markets

• Efficient markets are a result of investors trading on the unexpected portion of announcements

• The easier it is to trade on surprises, the more efficient markets should be

• Efficient markets involve random price changes because we cannot predict surprises

13-18

Slide 20

11-19

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, etc.

Risk consists of surprises. There are two kinds of surprises: Systematic Risk & Unsystematic

Risk

• Systematic risk is a surprise that affects a large number of assets, although at varying degrees. It is sometimes called market risk.

• Example: Changes in GDP, interest rates, and inflation are examples of systematic risk.

Slide 21

11-20

Unsystematic Risk

• = Diversifiable risk

• Risk factors that affect a limited number of assets

• Also known as unique risk or asset-specific risk.

• Risk that can be eliminated by combining assets into portfolios

• Examples: labor strikes, part shortages, etc.

• Unsystematic risk is a surprise that affects a small number of assets (or one). It is sometimes called unique or asset-specific risk.

• Example: Strikes, accidents, and takeovers are examples of unsystematic risk.

Slide 22

11-21

Diversification

• Portfolio diversification is the investment in several different asset classes or sectors

• Diversification is not just holding a lot of assets

• For example, if you own 50 Internet stocks, you are not diversified

• However, if you own 50 stocks that span 20 different industries, then you are diversified

13-21

Portfolio diversification can substantially reduce risk without an equivalent reduction in expected

returns

• Reduces the variability of returns

Minimum level of risk that cannot be diversified away = systematic portion

Slide 23

11-22

• Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns.

• This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another.

• However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion.

The Principle of Diversification

Principle of Diversification – States that combining imperfectly correlated assets can produce a

portfolio with less variability than the typical individual asset.

The portion of variability present in a single security that is not present in a portfolio of securities

is called diversifiable risk. The level of variance that is present in portfolios of assets is non-

diversifiable risk.

Slide 24

11-23

Standard Deviations of Annual Portfolio Returns Table 13.7

A typical single stock on the NYSE has a standard deviation of annual returns around 49%, while

the typical large portfolio of NYSE stocks has a standard deviation of around 20%.

Slide 25

11-24

• The risk that can be eliminated by combining assets into a portfolio.

• Often considered the same as unsystematic, unique or asset-specific risk

• If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away.

Diversifiable Risk

When securities are combined into portfolios, their unique or unsystematic risks tend to cancel out,

leaving only the variability that affects all securities to some degree. Thus, diversifiable risk is

synonymous with unsystematic risk. Large portfolios have little or no unsystematic risk.

Slide 26

11-25

Total Risk = Stand-alone Risk

• Total risk = Systematic risk + Unsystematic risk

– The standard deviation of returns is a measure of total risk

• For well-diversified portfolios, unsystematic risk is very small

– Total risk for a diversified portfolio is essentially equivalent to the systematic risk

– The expected return (market required return) on an asset depends only on that asset’s systematic or market risk.

Systematic risk cannot be eliminated by diversification since it represents the variability due to

influences that affect all securities to some degree. Therefore, systematic risk and non-

diversifiable risk are the same.

Total risk = Non-diversifiable risk + Diversifiable risk

= Systematic risk + Unsystematic risk

Slide 27

11-26

Market Risk for Individual Securities

• Measured by a stock’s beta coefficient, j • Measures the stock’s volatility relative to the

market

While the standard deviation of returns is a measure of total risk, the beta coefficient measures

how much systematic risk an asset has relative to an asset of average risk.

Beta measures the volatility of an individual asset or portfolio relative to the market as a whole.

Slide 28

11-27

Measuring Systematic Risk

• How do we measure systematic risk? ▪ We use the beta coefficient

• What does beta tell us? ▪ A beta = 1 implies the asset has the same systematic

risk as the overall market

▪ A beta < 1 implies the asset has less systematic risk than the overall market

▪ A beta > 1 implies the asset has more systematic risk than the overall market

▪ Most stocks have betas in the range of 0.5 to 1.5

▪ Beta of a T-Bill = 0

13-27

Robert Hamada derived the following equation to reflect the relationship between levered and

unlevered betas (excluding tax effects):

L = U(1 + D/E)

where:

L = equity beta of a levered firm;

U = equity beta of an unlevered firm;

D/E = debt-to-equity ratio

Slide 29

11-28

Beta Coefficients for Selected Companies Table 13.8

Slide 30

11-29

• Consider the following information:

Standard Deviation Beta

Security C 20% 1.25

Security K 30% 0.95

• Which security has more total risk?

• Which security has more systematic risk?

• Which security should have the higher expected return?

Total vs. Systematic Risk

Security K has the higher total risk.

Security C has the higher systematic risk.

Security C should have the higher expected return.

• When securities are combined into portfolios, their unique or unsystematic risks tend to cancel out, leaving only the variability that affects all securities to some degree. Thus,

Total risk for a diversified portfolio is essentially equivalent to the systematic risk. The

expected return (market required return) on an asset depends only on that asset’s

systematic or market risk.

Slide 31

11-30

Portfolio Beta

βp = Weighted average of the Betas of the

assets in the portfolio

Weights (wj)= % of portfolio invested in asset j

 =

= n

j

jjp w 1



The beta of the portfolio is simply a weighted average of the betas of the securities in the

portfolio.

Slide 32

11-31

• Consider the previous example with the following four securities.

Security Weight Beta

C .133 1.685

KO .2 0.195

INTC .267 1.161

BP .4 1.434

• What is the portfolio beta?

• .133(1.685) + .2(.195) + .267(1.161) + .4(1.434) = 1.147

Example: Portfolio Betas

Which security has the highest systematic risk?

C

Which security has the lowest systematic risk?

KO

Is the systematic risk of the portfolio more or less than the market?

more

Slide 33

11-32

Example: Portfolio Expected Returns and Betas

0%

5%

10%

15%

20%

25%

30%

0 0.5 1 1.5 2 2.5 3

E x p

ec te

d R

et u

rn

Beta

Rf

E(RA)

A

There is a linear relationship between beta and expected return.

A riskless asset has a beta of 0.When a risky asset with >0 is combined with a riskless asset, the

resulting expected return is the weighted sum of the expected returns, and the portfolio beta is the

weighted sum of the betas. By varying the amount invested in each asset, we can get an idea of the

relation between portfolio expected returns and betas. This relationship is illustrated in this figure.

As can be seen, all of the risk-return combinations lie on a straight line. The equation for a line is:

Y = mx + b

where: y = expected return

x = beta

m = slope = risk-premium per unit of beta

b = y-intercept = risk-free rate

E(R) = slope (Beta) + y-intercept

The y-intercept is = the risk-free rate, so all we need is the slope

Slide 34

11-33

Reward-to-Risk Ratio: Definition and Example

• The reward-to-risk ratio is the slope of the line illustrated in the previous example

▪ Slope = (E(RA) – Rf) / (A – 0)

▪ Reward-to-risk ratio for previous example = (20 – 8) / (1.6 – 0) = 7.5

• What if an asset has a reward-to-risk ratio of 8 (implying that the asset plots above the line)?

• investors will want to buy the asset.

• What if an asset has a reward-to-risk ratio of 7 (implying that the asset plots below the line)?

• investors will want to sell the asset

13-33

The Reward-to-Risk Ratio is the expected return per unit of systematic risk. In other words, it is

the ratio of risk premium to systematic risk.

If the reward-to-risk ratio = 8, then investors will want to buy the asset. This will drive the price

up and the expected return down (remember time value of money and valuation). When will the

flurry of trading stop? When the reward-to-risk ratio reaches 7.5.

If the reward-to-risk ratio = 7, then investors will want to sell the asset. This will drive the price

down and the expected return up. When will the flurry of trading stop? When the reward-to-risk

ratio reaches 7.5.

Slide 35

11-34

Beta and the Risk Premium

• Risk premium = E(R ) – Rf • The higher the beta, the greater the risk

premium should be

• Can we define the relationship between the risk premium and beta so that we can estimate the expected return?

– YES!

The risk premium—the excess return of an asset above the risk-free rate.

Slide 36

11-35

Security Market Line

• The security market line (SML) is the

representation of market equilibrium

• The slope of the SML = reward-to-risk ratio = (E(RM) – Rf) / M

• But since the beta for the market is always

equal to one, the slope can be rewritten

• Slope = E(RM) – Rf = market risk premium

The line that gives the expected return/systematic risk combinations of assets in a well-functioning,

active financial market is called the security market line.

Market Portfolios: Consider a portfolio of all the assets in the market and call it the market portfolio.

This portfolio, by definition, has “average” systematic risk with a beta of 1. Since all assets must

lie on the SML when appropriately priced, the market portfolio must also lie on the SML. Let the

expected return on the market portfolio = E(RM). Then, the slope of the SML = reward-to-risk ratio

= [E(RM) − Rf] ⁄ M = [E(RM) − Rf] ⁄ 1 = E(RM) − Rf

Slide 37

11-36

Market Equilibrium

• In equilibrium, all assets and portfolios must have the same reward-to-risk ratio

• Each ratio must equal the reward-to-risk ratio for the market

M

fM

A

fA )RR(ER)R(E



− =

−

The basic argument is that since systematic risk is all that matters in determining expected return,

the reward-to-risk ratio must be the same for all assets. If it were not, people would buy the asset

with the higher reward-to-risk ratio (driving the price up and the return down).

The fundamental result is that in a competitive market where only systematic risk affects E(R),

the reward-to-risk ratio must be the same for all assets in the market. Consequently, the expected

returns and betas of all assets must plot on the same straight line.

E.g., Amazon (Asset j =A)

We can solve for 𝐸 𝑅𝐴

E RA Rf βA

E RM Rf

βM

[E RA Rf] x βM = βA x [E RM Rf ]

Since the market beta, βM = 1 (Wee Slide 28),

E RA Rf = βA [E RM Rf ]

E RA = Rf + βA [E RM Rf ]

Since E(Rf) = Rf,

E RA = Rf + βA [E RM Rf]

This is called the capital asset pricing model (CAPM)

Slide 38

11-37

SML and Equilibrium

The Capital Asset Pricing Model (CAPM):

E(Rj) = Rf + j (slope)

E(Rj) = Rf + j [E(RM) − Rf]

We can get an idea of the relationship between portfolio expected returns and betas.

Slide 39

11-38

The SML and Required Return

• The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM)

Rf = Risk-free rate (T-Bill or T-Bond)

RM = Market return ≈ S&P 500

RPM = Market risk premium = E(RM) – Rf E(Rj) = “Required Return of Asset j”

( ) ( ) jMfj

jfMfj

RPRRE

RRERRE





+=

−+=

)(

)()(

Slide 40

11-39

Capital Asset Pricing Model

• The capital asset pricing model (CAPM) defines the relationship between risk and return

E(Rj) = Rf + βj(E(RM) – Rf)

• If an asset’s systematic risk () is known, CAPM can be used to determine its expected return

Slide 41

11-40

Factors Affecting Required Return

• Rf measures the pure time value of money

• E(RM)-Rf measures the reward for bearing systematic risk

• j measures the amount of systematic risk

( ) jfMfj RRERRE −+= )()(

The CAPM states that the expected return for an asset depends on:

-The time value of money, as measured by Rf

-The reward per unit risk, as measured by E(RM) − Rf

-The asset’s systematic risk, as measured by 

Slide 42

11-41

Quick Quiz

Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%.

– What is the reward-to-risk ratio in equilibrium?

– What is the expected return on the asset?

• E(R) = 5% + (13% - 5%)* 1.2 = 14.6%

Reward-to-risk ratio = 13 – 5 = 8%

Expected return = 5 + 1.2(8) = 14.6%