Chapter 11: RETURN, RISK, AND THE CAPITAL ASSET

PRICING MODEL
11.1 Individual Securities
- Expected return:
+ return that an individual expects a stock to earn over next period
+ = average return
+ based on: detailed analysis of a firm’s prospect, computer,based model,
special/inside information
- Variance and SD: variance is most common way to asset volatility
- Covariance and correlation:
+ covariance: is a statistic measuring interrelationship between 2 securities
+ are building blocks to an understanding of beta coefficient
11.2 Expected Return, Variance, and Covariance
11.2.1 Expected Return and Variance

1. Calculate expected return:

−.20+.10+.30+.50
E ( R Super )= =.175∨17.5 %
4
05+.20−.12+.09
E ( R Slow )= =.055 ,∨5.5 %
4
2. Calculate deviation
3. Deviations: indications of dispersion of returns. However, some are -, som are + 
squared deviations
(1) (2) (3) (4)
State of Rate Deviati Square
Economy of on from d Value
Retur Expecte Of
n d Deviati
Return on
R R- [R-
E(R) E(R)]2
Supertech (Expeced return = .175)
Depressi -.20 -.375 .14062
on 5
Recessio .10 -.075 .00562
n 5
Normal .30 .125 .01562
5
Boom .50 .325 .10562
5
Slowpoke (Expected return = .055)
Depressi .05 -.005 .00002
 on 5
Recessio .20 .145 .02102
n 5
Normal -.12 -.175 .03062
5
Boom .09 .035 .00122
5
4. Calculate variance
2 .140625+ .005625+.015625+.105625
σ Super = =.066875
4
2 . 000025+.021025+.030625+.001225
σ Slow= =.013225
4
5. Calculate SD
σ Super =√ σ 2Super =.2586 ,∨25.86 %
σ Slow= √ σ 2Slow =.1150 ,∨11.50 %

2
Var ( R )=σ =∑ [ R−E ( R ) ]
2

SD ( R )=σ =√ Var ( R )
- E(R): expected return
- R: acual return
*Squaring the differences makes them all +. If not, we would get 0 because returns that
were above the mean would cancel the ones below.
11.2.2 Covariance and Correlation
Covariance/correlation:
+ measure relationship between return on one stock and return on another.
+ measure how 2 random variables are related
Ex11.1 Calculating Covariance and Correlation
Tiếp tục với vd trên
1. [RSuper − E(RSuper)] × [RSlow − E(RSlow)]
2. Calculate covariance (σ or cov)
σSuper,Slow = Cov( RSuper, RSlow) = −.0195/4 = −.004875
RA RA – RB RB – [RA – E(RA)]
E(RA) E(RB) * [RB –
E(RB)}
Depressi -.20 -.375 .05 -.005 .001875
on
Recessio .10 -.075 .20 .145 - .010875
n
Normal .30 .125 -.1 -.175 −.021875
2
Boom .50 .325 .09 .035 .011375

 if 2 returns are positively related to each other  positive covariance; và ngược lại
 2 returns are unrelated to each other, covariance formula will not =0 exactly. This is
deu to sampling error; randomness alone will make calculation positive or negative.
 unrelated  cov shoudle =0
σA,B = Cov(RA,B) = Expected value of {[RA − E(RA)] × [RB − E(RB)]
Cov(RA,RB) = Cov(RB,RA) or σAB = σBA
+ E(RA), E(RB): expected return
3. Calculate correlation (hệ số tương quan)
Cov ( R A , R B ) −.004875
ρ A ,B =Corr ( R A , RB ) = = =−.1639
(σ A × σ A ) .2586 ×.1150
Corr(RA,RB) = Corr(RB,RA) or ρAB = ρBA
+ σA, σB: SD
+ Corr(RA,RB) or ρAB: corelation
 corr>0: positively correlated
 corr<0: negatively correlated
 corr=0: uncorrelated
 corr nằm giữa [-1;1] due to the standardizing procedure of dividing by the product of
2 SD
Perfect positive correlation
Cov(RA,B) = 1
Both return on Security A and B are higher/lower than average at same time

Perfect negative correlation

Cov(RA,B) = -1
Securities has a higher-than-average return còn B thì lower; và ngược lại.

Zero correlation
Cov(RA,B) = 0
Return on Security A is completely unrelated to return on Security B.

11.3 The Return and Risk for Portfolios

“the best combination or portfolio of securities to hold”  investor: high expected return,
low SD of return  should consider:
1. relationship between expected returns on individual securities & expected returns on a
portfolio made up of these securities
2. relationship between SD of individual securities, corr vetween these securities, and SD
of portfolio made up of these securities
11.3.1 The Expected Return on a Portfolio
“The expected return on a portfolio is a weighted average of the expected returns on the
individual securities.”

Ex11.2 Portfolio Expected Returns

- Expected return on a portfolio of these two securities:
E( RP) = XSuper (.175) + XSlow (.055)
XSuper: % of portfolio in Supertech
XSlow: % of portfolio in Slow
If invest $60 in Supertech and $40 in Slowpoke
 E( RP) = .6 × .175 + .4 × .055 = .127 or 12.7%
 E(RP) = XAE(RA) + XBE(RB)
+ E(RA), E(RB): expected returns on the two securities
+ XA, XB: proportions of =total portfolio in assets A and B
+ X A + XB = 1
 if XA = XB  XA = XB = E(RP)
11.3.2 Variance and Standard Deviation of a Portfolio
11.3.2.1 The Variance
Variance of a portfolio composed of two securities, A and B, is:

 var of portfolio dêpnd on both var of individual securities and cov between 2 securities.
+ var of a security: measure variability of an indicidual security’s return
+ cor: measure relationship between 2 securities
> negative cor  decrease var of entire portfolio
> positive cor  increase var of entire portfolio
- one security goes up when other goes down (or cive versa)  2 securities offset each
other  a headge in finance  risk of entire portfolio will be low
- both securities rise/fall together  not hedge at all  risk of portfolio will be higher
Tiếp tục với vd. Tìm variance for 2 securities?

= .36 × .066875 + 2 × [.6 × .4 × (− .004875)] + .16 × .013225

= .023851
11.3.2.2 The Matrix Approach

được thể hiện dưới dạng ma trận như sau:

11.3.2.3 Standard Deviation of a Portfolio

σ P= √ Var ( portfolio)= √ . 023851=.1544∨15.44 %
11.3.2.4 The Diversification Effect
Weighted average of standard deviations = XAσB + XAσB = .6 × .2586 + .4 × .115 = .2012
Ta có: σA,B = ρA,B σAσB


= .36 × .066875 + 2 × [.6 × .4 × (− .1639) × .2586 × .115] + .16 × .013225 =.023851
 SD:
SD of portfolio = weighted average of SD when ρ = 1
 ρ < 1  SD of a portfolio of two securities < weighted average of SD of the individual
securities.
 diversification effect applies as long as there is less than perfect correlation (ρ < 1)
11.3.2.5 An Extension to Many Assets
As long as Corr between pairs of securities < 1, SD of an index < weighted average of SD
of the individual securities within the index.
11.4 The Efficient Set for Two Assets

+ box, or “□”: a portfolio uiwth 60% invested in Super and 40% in Slow
 2 securities có vô số portfolio
Portfolio 1: 90% Slowpoke + 10% Supertech (ρ = −.1639).
Portfolio 2: 50% Slowpoke + 50+ Supertech (ρ = −.1639).
Portfolio 3: 10% Slowpoke + 90% Supertech (ρ = −.1639).
Portfolio 1′: 90% Slowpoke + 10% Supertech (ρ = 1).
Point MV: minimum variance portfolio. This is the portfolio with the lowest possible
variance. By definition, the same portfolio also must have the lowest possible standard
deviation.
lưu ý:
1. - diversification effect occurs whenever correlation between 2 securities <1
- cannot choose between points on curve and points on straight line
2. - MV: minimum variance portfolio (lowest possible SD)
3. - opportunity set / feasible set: An individual contemplating an investment in a
portfolio by the curved line
- cann’t achieve any point above the curve because can’t increase return, decrease
SD
- cann’t achieve any point below the curve because can’t return decrease, increase
SD
- relative tolerant of risk: portfolio 3, could even choose end point, investing all his
money in Super
- less tolerance for rish: portfolio 2
- as little risk as possible: MV, portfolio with minimum variance or minimum SD
4. -
5. - No investor want to hold a portfolio with an expected return < MV. Ex: portfolio 1
- investor chỉ quan tâm các điểm từ MV đến Super. Các điểm đó gọi là efficient set /
efficient frontier
- the lower the correlation, the more the curve bends  diversification effect rises as ρ

declines
11.5 The Efficient Set for Many Securities
Figure 11.6 The Feasible Set of Portfolios Constructed from Many Securities
Chỉ chọn các điểm trong vùng mờ.
Không NĐT nào chọn danh mục nào bên dưới đường biên hiệu quả vì sẽ có expected
return thấp hơn. Ex: chọn R chứ không chọn W.
Các phép tính tập hợp danh mục hiệu quả trước đây rất tốn kém, giờ đã có các phần
mềm, các add-in  các trên bày trên dần có ý nghĩa
11.5.1 Variance and Standard Deviation in a Portfolio of Many Assets
Table 11.4 Matrix Used to Calculate the Variance of a Portfolio

Mỗi cặp chứng khoán sẽ xuất hiện 2 lần trong ma trận: 1 ở góc phía trên bên phải và 1 ở
phía dưới bên phải
Ô nằm trong đường chéo (chứa phương sai) = số lượng chứng khoán.
Ô nằm ngoài đường chéo (chứa hiệp phương sai) tăng lên nhanh chóng khi số lượng
chứng khoáng tăng.
Phương sai tỷ suất sinh lợi của danh mục bao gồm rất nhiều chứng khoán thì
phụ thuộc nhiều hơn vào các hiệp phương sai giữa các chứng khoáng riêng lẻ
hơn là phương sai của chúng.
Table 11.5 Number of Variance and Covariance Terms as a Function of the Number of
Stocks in the Portfolio
11.6 Diversification
11.6.1 The Anticipated and Unanticipated Components of News
Return gồm 2 thành phần:
1. normal/expected return:
+ the part of return that shareholders in the market predict/expect
+ depend on all of in4 shareholders have that bears on stock
+ use all of our understanding of what will influence stock in next year
2. uncertain/risky part: come from unexpected in4 revwaled within the year
+ News about Flyers’ research.
+ Government figures released for the gross national product (GNP).
+ Results of the latest arms control talks.
+ Discovery that a rival’s product has been tampered with.
+ News that Flyers’ sales figures are higher than expected.
+ A sudden drop in interest rates.
+ The unexpected retirement of Flyers’ founder and president.
Reuturn in the coming year: R = E(R) + U
+ R: actual total return in the year
+ E(R): expected part of return
+ U: unexpected part of return
Theo thời gian, giá trị tb của U sẽ =0  tb R = E(R)
11.6.2 Risk: Systematic and Unsystematic
Các vd về thông tin ở trên chia làm 2 loại:
+ systematic risk: any risk that affects a large number of assets, each to a greater
or lesser degree
+ firm-specific / idiosyncratic / unsystematic risk: a risk that specifically affects a
single asset or a small group of assets
Không thể phân biệt rạch ròi giữa systematic và unsystematic được.
Risk of stock’s return có 2 thành phần: R = E(R) + U = E(R) + m + ε
+ E(R): stock expeted return
+ m: systematic risk / market risk  influence all assets in market to some extent
+ ε: unsystematic risk
Rủi ro phi hệ thống của Flyers (εF) không ảnh hưởng gì đến rủi ro phi hệ thống của
General Electric (εGM)  εF không tương quan với εGM
11.6.3 The Essence of Diversification
Rủi ro phi hệ thống: Do ε của 2 chứng khoán không tương quan  chúng có thể âm hoặc
dương  bù trừ lẫn nhau  ε của toàn bộ danh mục < ε của từng chứng khoán riêng lẻ 
nếu thêm 1SL không giới hạn các chứng khoán vào danh mục thì ε của toàn danh mục sẽ
biến mất
Rủi ro hệ thống: không thể bị triệt tiêu bằng đa dạng hóa.
Figure 11.7 Portfolio Diversification

SD: total risk, risk of portfolio

 SD của danh mục giảm khi số lượng chứng khoáng tăng. Tuy nhiên, rủi ro của
danh mục không bao giờ =0. Còn rủi ro phi hệ thống có thể bị triệt tiêu bằng
đang dạng hóa.
Phần màu đậm hơn: “diversifiable risk” is the part that can be eliminated by
diversification.
Phần màu nhạt hơn: “nondiversifiable risk”- a minimum level of risk that cannot be
eliminated by diversifying.
 Diversification reduces risk, but only up to a point. Put another way, some risk is
diversifiable and some is not.
11.6.4 The Effect of Diversification: Another Lesson from Market History
Table 11.6 Standard Deviations of Annual Portfolio Returns
Benefit in terms of risk reduction from adding securities drops off as we add more and
more

11.7 Riskless Borrowing and Lending

Ex11.3 Riskless Lending and Portfolio Risk

Total investment = $1,000

Invest in Merville = $350
Invest in risk-free asset = $650

1. Expected return on total investment is a weight average of 2 returns:

E(RP) = 35 × .14 + .65 × .10 = .114, or 11.4%
2. Variance of portfolio:

Var (risk-free asset) = 0

 = .352 × .202 = .00490

3. SD of portfolio:

= .35 × .20 (11.12) = .07

Figure 11.8 Relationship between Expected Return and Risk for Portfolios Composed of
the Riskless Asset and One Risky Asset
 Không giống như TH 2 tài sản có rủi ro, tập hợp các cơ hội đầu tư trong TH này là đường
thẳng chứ không phải là đường cong.
4. (borrows $200 at the risk-free rate) + (original sum of $1,000) = $1,200
- Expected return: E(RP) = 1.20 × .14 + ( −.2 × .10) = .148, or 14.8%
- SD of portfolio: σP = 1.20 × .2 = .24, or 24%
Đường nét đứt: tập hợp các cơ hội đầu tư mà NĐT phải đi vay ở mức lãi suất cao hơn
mức lãi suất cho vay  TSSL kỳ vọng trên khoản đầu tư giảm
11.7.1 The Optimal Portfolio
Figure 11.9 Relationship between Expected Return and Standard Deviation for an
Investment in a Combination of Risky Securities and the Riskless Asset
Các khoản đầu tư được tổng kết như sau:

 No point on line I is optimal

 Bất cứ điểm nào đạt được trên I thì cũng có thể đạt được trên II với cùng độ lệch
chuyển nhưng TSSL kỳ vọng cao hơn
 Line II: efficient set of all assets, both risky and riskless
 Chỉ nên chọn trong đường biên hiệu quả của các chứng khoán rủi ro (XAY)
 Separation principle:
1. - Tính expected return, var of individual securities, cov between pairs of securities
- Tính efficient set of risky asset (XA)
- Xác định điểm A: tangency between risk-free rate & efficient set of risky assets
 A is determined solely from estimates of returns, variances, and covariances. No
personal characteristics (degree of risk aversion,...)
2. - Determine how to combine point A
His choice is determined by his internal characteristics (ability to tolerate risk,...)

11.8 Market Equilibrium

11.8.1 Definition of the Market Equilibrium Portfolio
- Homogeneous expectations: all investor posses the same estimates of expected
returns, var, cov  all investors would hold the portfolio of risky assets represented by
Point A.
- Market portfolio: all investor choose same portfolio, one can determine what that
portfolio is  a market-value-weighted portfolio of all existing securities
11.8.2 Definition of Risk When Investors Hold the Market Portfolio
Beta: best measure of risk of an individual security
Ex11.4 Beta

- market’s return in a bullish economy: 15% − (−5%) = 20%

- expected return on Jelco in a bullish economy: 20% − (−10%) = 30%
 Jelco’s responsiveness coefficient: 30%/20% = 1.5
Figure 11.10 Performance of Jelco, Inc., and the Market Portfolio
 - Beta = responsiveness coefficient = magnigication factor = 1.5 (độ dốc)
- Characteristic line: đường thẳng được nối bởi 2 điểm X
 return of Jelco are magnified 1.5 times over those of market
Thị trường tốt thì Jelco tốt hơn, thị trường xấu thì Jelco xấu hơn
 Jeco is more responsive  Jelco contributes more to the risk of a large, diversified
portfolio
- negative beta:
+ hedges / insureance policies
+ expected to do well when the market does poorly and vice versa
+  diversified portfolio actually reduces the risk
Table 11.7 Estimates of Beta for Selected Individual Stocks

Vd: De Pepper Snapple có beta=.49  1% thay đổi trong thị trường thì Dr Pepper
Snapple được kỳ vonngj sẽ thay đổi .49%
Beta measures responsiveness of a security to movements in market
portfolio.
11.8.3 The Formula for Beta
Cov ( Ri , R M )
β i= 2
σ ( RM )

+ Cov(Ri,RM): cov between return on asset i and return on market portfolio

+ σ2(RM): var of market
average beta across all securities, when weighted by the proportion of each security’s
market value to that of the market portfolio, is 1

+ Xi: proportion of Security i’s market value to that of the entire market
+ N: number of securities in market
 if weight all securities by their market values  result portfolio is market
11.8.4 A Test
1. What sort of investor rationally views the variance (or standard deviation) of
an individual security’s return as the security’s proper measure of risk?
rational, risk-averse investor. If investor can hold only one security  var of that
security’s return = var of portfolio’s return
2. What sort of investor rationally views the beta of a security as the security’s
proper measure of risk?
Nếu NĐT giữ một danh mục đa dạng hóa  vẫn xem var (SD) là thước đo rủi ro của danh
mục, k quan tâm đến var (SD) của chứng khoán riêng lẻ mà quan tâm đến mức độ đóng
góp của chứng khoán đó vào var của toàn danh mục beta: đo lường đóng góp rủi ro của
chứng khoán iêng lẻ vào phương sai danh mục  beta là thước đo phù hợp cho rủi
ro của chứng khoán riêng lẻ trong danh mục đa dạng hóa tốt
Beta đo lường rủi ro hệ thống của một chứng khoán  NĐT da dạng hóa quan tâm đến
rủi ro hệ thống của mỗi chứng khoán, bỏ qua rủi ro phi hệ thống của các chứng khoán
riêng lẻ vì rủi ro phi hệ thống đã bị triệt tiêu trong danh mục đa dạng tốt

11.9 Relationship between Risk and Expected Return (CAPM)

11.9.1 Expected Return on the Market
E(RM) = RF + Risk premium
Equation refers to the expected return on market, not actual return in a particular month
or year because actual return on market over a particular period can < R F or even
negative
11.9.2 Expected Return on an Individual Security

Figure 11.11 Relationship between Expected Return on an Individual Security and Beta of
the Security

- SLM (security market line): graphical depiction of CAPM (capital asset pricing model)
- beta = 0  expected return on a stock = risk-free rate
- beta = 1  expected return on a stock = expected return on market
CAPM:
- expected return on a security is linearly realted to its beta
- expeced return on a security is positively related to its beta:
+ β = 0  E(R) = RF beause a security with β = 0 has no relevant risk, its expected
return should = risk-free rate
+ β = 1  E(R) = E(RM) because beta of market portfolio is also =1
SML (security market line):
- upward-sloping khi E(RM) > RF
- begin at RF and rises to E(RM) khi beta=1
- RF: intercept
- E(RM) – RF: slope
Ex11.5
βA = 1.5
βB =.7
rF = .03
E(RM) - rF = .08
 E(RA), E(RB) ?
E(RA) = rF + βA(E(RM) – rF) = .03 + 1.5*.08 = .15
E(RB) = rF + βB(E(RM) – rF) = .03 + .7*.08 = .086
3 vấn đề của CAPM:
1. Linearity:
- Relationship between expected return and beta corresponds to a straight line
- Các điểm nằm dưới SML (S, T) bị overpriced  có expected return thấp  chúng sẽ
giảm giá đến khi nào chúng nằm trên đường SML
- Ngược lại, các điểm nằm trên SML bị underprice  có expected return cao  chúng sẽ
tăng giá đến khi nào chúng nằm trên đường SML
2. Portfolio as well as securtities
- Equation (11.15):

- Figure 11.11: ở trên

 2 cái này vẫn đúng đối với portfolio.
A portfolio formed by investing equally in our two securities from Example 11.5 
expected return on portfolio:
E(RP) = .5 × .15 + .5 × .086 = .118, or 11.8%
The beta of the portfolio is a weighted average of the betas of the two securities:
βP = .5 × 1.5 + .5 × .7 = 1.1
Under the CAPM, the expected return on the portfolio is:
E(RP) = .03 + 1.1 × .08 = .118, or 11.8%
 CAPM đúng cho cả portfolio và individual
3. A potential confusion:
- Thường nhầm lẫn Line II (F11.9) với SML (F11.1)
+ line II:
> trace efficient set of portfolios formed from both risky assets and riskless asset
> each point represents an entire portfolio
Point A: portfolio composed entirely of risky assets
Every other point represents a portfolio of securities in A combined with
riskless asset
 Axes on F11.9 are expected return on a portfolio and SD odd a portfolio 
individual securities don’t lie along line II
+ SML: relate expected return to beta
- 2 cái này khác nhau:
+ Horizotal axis của F11.11 là beta, còn của 11.9 là SD
+ SML áp dụng cho cả portfolio và individual securities. Còn line II chỉ áp dụng cho
efficient portlolios

Figure 11.9 Relationship between Expected Return and Standard Deviation for an
Investment in a Combination of Risky Securities and the Riskless Asset

Với giả định kỳ vọng thuần nhất (homogeneous expectations), điểm A trong 11.9 trở
thành market portfolio, còn line II trở thành CML (capital market line)