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
+=
−+=
)(
)()(
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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
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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
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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%