Chapter  

7:  Capital  Asset  Pricing  Model  and  

Arbitrage  Pricing  Theory
In  Chapter  6,  we  showed  how  a  single-­factor  model  could  be  used  to  estimate  
the  2  components  of  risk  (“market”  and  “firm-­specific”)  for  a  security.    

In  Chapter  7,  we’ll  discuss  the  most  commonly  used  single-­factor  model

§ Capital  Asset  Pricing  Model  (CAPM):  a  model  that  relates  the  

required  rate  of  return  for  a  security  to  its  risk  as  measured  by  beta  

Then  we’ll  discuss  how  CAPM  can  be  used  to  calculate  risk  premiums,  
alpha,  and  underpriced  and  overpriced  securities.

Then  we’ll  briefly  introduce

• Multi-­factor  pricing  models
• Arbitrage  pricing  theory  (APT)

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 Capital  Asset  Pricing  Model  (CAPM)
• Equilibrium  model  that  underlies  all  modern  financial  theory

• Derived  using  principles  of  diversification  with  simplified  assumptions

• Markowitz,  Sharpe,  Lintner  and  Mossin  are  researchers  credited  with  its  
development  (early  1970’s)

CAPM  is  a  theoretical  economic  model  that  requires  these  assumptions:

– Individual  investors  are  price  takers
– Single-­period  investment  horizon
– Investments  are  limited  to  traded  financial  assets
– No  taxes  nor  transaction  costs
– Information  is  costless  and  available  to  all  investors
– Investors  are  rational  mean-­variance  optimizers
– Homogeneous  expectations

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 Capital  Asset  Pricing  Model  (CAPM)
Resulting  Equilibrium  Conditions

• All  investors  will  hold  the  same  portfolio  for  risky  assets  (market  portfolio)
• Market  portfolio  contains  all  securities  and  the  proportion  of  each  security  is  
its  market  value  as  a  percentage  of  total  market  value
• Risk  premium  on  the  market  depends  on  the  average  risk  aversion  of  all  
market  participants
• Risk  premium  on  an  individual  security  is  a  function  of  its  covariance  with  
the  market

CAPM  implies  the  expected-­return-­beta  relationship

E(ri )  =    rf +    βi [E(rM  )  – rf ]                                [Text,  eqn.  7.2,  p  194]

The  rate  of  return  on  any  asset  exceeds  the  risk-­free  rate  by  a  risk  premium  
equal  to  the  asset’s  systematic  risk  measure  (its  beta)  times  the  risk  
premium  of  the  (benchmark)  market  portfolio.
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 Capital  Asset  Pricing  Model  (CAPM)
CAPM  expected-­return-­beta  relationship

E(ri )  =    rf +    βi [E(rM  )  – rf ]                                [Text,  eqn.  7.2,  p  194]

The  rate  of  return  on  any  asset  exceeds  the  risk-­free  rate  by  a  risk  premium  
equal  to  the  asset’s  systematic  risk  measure  (its  beta)  times  the  risk  
premium  of  the  (benchmark)  market  portfolio.

Your  text  (page  194):  The  of  the  expected-­return-­beta  relationship  of  the  CAPM  
makes  a  powerful  economic  statement.    Consider  2  stocks:

s b
Stock  A                40%              0.5
Stock  B              15%              1.5

CAPM  states  that  Stock  B  should  be  priced  such  that  its  risk  premium  will  be  3x  
that  of  Stock  A’s  risk  premium.    Only  systematic  risk  matters  for  prices.  
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 CAPM  Portfolio  Beta  Coefficients  
The  beta  of  any  set  of  securities  is  the  weighted  average  of  the  individual  
securities’  betas.

b p = w 1 b 1 + w2 b 2 + ... + wn b n
N
= å
j =1
w j b j

Example:      What  is  the  beta  of  a  portfolio  made  up  of:
• 25%  of  Stock  H  that  has  beta  =  2
• 45%  of  Stock  A  that  has  beta  =  1,  and
• 30%  of  Stock  L    that  has  beta  =    0.5

• Beta  of  the  portfolio  =  (.25)(2)  +  (.45)(1)  +  (.30)(0.5)    =        __________

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Figure  7.2  The  security  market  line  (SML)  and  a  positive  alpha  stock:
-­ The  SML  is  a  graphical  representation  of  the  expected-­return-­beta  
relationship  of  the  CAPM.
-­ Alpha  is  the  abnormal  rate  of  return  on  a  security  in  excess  of  what  would  be  
predicted by  an  equilibrium  model  such  as  the  CAPM.

In this example, a stock with

a beta of 1.2 has an E(r) =
6%+1.2(14%-6%)=15.6%.

If its actual E(r) is 17%,

then the stock’s a = 1.4%

How is the SML different

from the CAL?

1) SML relates E(r) to b

2) CAL relates E(r) to s

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Figure  6.12  Scatter  Diagram  and  Security  
Characteristic  Line  for  Dell  Stock

How is the SML different

from the SCL in Ch 6?
1) SML relates E(r) to b
2) SCL relates security’s
excess return to the market
portfolio’s excess return

6-­7
 Estimating  the  CAPM  beta  of  individual  stocks
• Use  (monthly)  historical  data  on  T-­bills,  S&P  500  and  individual  securities
• Use  Ordinary  Least  Squares  (OLS)  Regression  to  regress  risk  premiums  for  
individual  stocks  against  the  risk  premiums  for  the  S&P  500
• Slope  is  the  beta  for  the  individual  stock

6-­8
Table  7.1  Monthly  Return  Statistics  for  T-­bills,  S&P  500  
and  General  Motors

*Note: In your text (9h ed. ) the firm in this example is Google
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 Figure  7.3  Cumulative  Returns  for  
T-­bills,  S&P  500  and  GM  Stock

GM is a cyclical stock --- its returns tend to be above or below

the S&P 500 depending on the business (economic) cycle. It
looks like it moves more with the business cycle than does the
S&P 500. What do you expect for GM’s beta?

*Note: In your text (9th ed. ) the firm in this example is Google
6-­10
 Table  7.2  Security  Characteristic  
Line  for  GM:  Summary  Output  from  Excel  OLS

*Note: In your text (9th ed. ) the firm in this example is Google
6-­11
 Limitations  of  the  CAPM
• The  CAPM  is  based  on  expected  conditions,  but  we  only  have  
historical  data  to  use  to  estimate  beta.  

• Timeframe  (frequency  of  returns  and  historical  time  period  used)  for  
the  regression  of  the  historical  data  greatly  impacts  our  estimate  of  
beta.

• As  conditions  change,  future  volatility  may  differ  from  past  volatility.    

Volatility  here  just  means  magnitude  of  changes  in  prices.

• Where  does  our  forecast  of  the  risk-­free  rate  (  r  RF ) and  the  required  
rate  of  return  for  the  market  ( r  m )  come  from?

• Alternative  “multi-­factor”  models  have  been  developed  to  try  to  

address  some  of  these  limitations.    “Multi-­factor”  means  that  there  are  
more  explanatory  variables  on  the  right-­hand  side  of  the  regression.

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 Fama  French  Three-­Factor  Model
• Model  developed  in  1996  by  2  professors,  Eugene  Fama  and  
Kenneth  French.    
• Kenneth  French  is  still  at  Dartmouth  but  runs  a  multi-­million  dollar  
consulting  firm  largely  based  on  the  model.
• Data  for  the  monthly  factors  for  the  model  available  for  free  at:  
mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

• Model  specification  is:

ri -­ rf =  ai +  βM (rM -­ rf)  +  βHML (rHML)  +  βSMB (rSMB)  +  ei

• First  factor  is  the  market’s  return  (like  CAPM)

• The  2nd Factor  is  Size  aka  Market  Cap  (Small  minus  Big,  or  “SMB”).    
• The  3rd Factor  is  Book  value  relative  to  Market  value  (High  minus  
Low,  or  “HML”)  
• The  basic  rational  is  that  stocks  of  small  firms  historically  earn  
higher  returns  and  carry  higher  risk.    Similarly,  stocks  of  companies  
with  lower  book  to  market  values  (“Value  Stocks”)    do  better.
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 Arbitrage  Pricing  Theory  (APT)
• Arbitrage  – creation  of  riskless  (aka  risk-­free)  profits  by  taking  
positions  that  take  advantage  of  security  “mispricing”

• If  two  portfolios  are  mispriced,  the  investor  could  buy  the  low-­priced  
portfolio  and  sell  the  high-­priced  portfolio

• Arbitrage  arises  if  an  investor  can  construct  a  zero  beta  investment  
portfolio  with  a  return  greater  than  the  risk-­free  rate

• In  efficient  markets,  profitable  arbitrage  opportunities  will  quickly  

disappear

• Arbitrage  Pricing  Theory  is  a  theory  of  risk-­return  relationships  

derived  from  no-­arbitrage  considerations  in  large  capital  markets  -­-­-­
in  other  words,  if  there  are  arbitrage  opportunities  do  not  exist,  what  
relationships  between  risk  and  return  have  to  hold?

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 Arbitrage  Pricing  Theory  (APT)  -­ continued

The  result:  For  a  well  diversified  portfolio

Rp =  βpRS (Excess  returns)    
(rp,i – rf)  =  βp(rS,i – rf)    

and  for  an  individual  security

(rp,i – rf)  =  βp(rS,i – rf)  +  ei

For  example,  for  4  systematic  factors:

(rp,i – rf)  =  βp,1(r1,i – rf)    +  βp,2(r2,i – rf)  +  βp,3(r3,i – rf)  +  βp,4(r4,i – rf)  +  ei

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 Arbitrage  Pricing  Theory  (APT)  -­ continued
Arbitrage  Pricing  Theory  posits  a  single-­factor  security  market

Rp =  ai +  βp RM    +  ep

• Rp is  the  excess  return  of  a  well-­diversified  portfolio.  

• ai is  the  excess  return  of  the  portfolio  when  RM =  0.
• βp is  the  portfolio’s  responsiveness  to  a  macroeconomic  factor
• RM is  our  “single  factor”.
• ep is  the  portion  of  excess  return  that  results  from  nonsystematic  risk

• By  definition,  well-­diversified  means  the  portfolio  has  negligible  firm-­

specific  risk.
• Therefore: Rp =  ai +  βp RM

No  arbitrage  opportunities  mean  that  ai =  0.    (If  βp =  0,  Then  Rp =  0.)
Thus,  APT  implies  the  same  expected-­return-­beta  relationship  as  CAPM.
E(ri )  =    rf +    βi [E(rM  )  – rf ]

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 APT  and  CAPM  Compared
• APT  applies  to  well  diversified  portfolios  and  not  necessarily  to  
individual  stocks

• With  APT  it  is  possible  for  some  individual  stocks  to  be  mispriced  –
i.e.,  they  may  not  lie  on  the  SML

• APT  is  more  general  in  that  it  gets  to  an  expected  return  and  beta  
relationship  without  the  assumptions  of  the  CAPM  theory

• APT  can  be  extended  to  multifactor  models.

• Hedge  funds  use  multifactor  models  to  search  for  arbitrage  

opportunities:
– As  they  take  positions,  prices  move  toward  equilibrium  to  
eliminate  the  arbitrage  opportunity
– Sometimes  their  models  “get  equilibrium  wrong”  and  lose  money  
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