Capital asset pricing model, risk return, beta, variance, standard deviation, assumption of model, implication of the model

1. Lecture 6
Capital Asset Pricing model
By Muhammad Shafiq
forshaf@gmail.com
http://www.slideshare.net/forshaf

2. An index of systematic risk.
It measures the sensitivity of a stock’s returns to
changes in returns on the market portfolio.
The beta for a portfolio is simply a weighted average of
the individual stock betas in the portfolio.
Beta = How much systematic risk a particular asset has
relative to an average asset
What is Beta?

3. BETA…
• A measure of the volatility, or systematic risk, of a security or a portfolio in
comparison to the market as a whole.
• Beta is used in the capital asset pricing model (CAPM), a model that calculates the
expected return of an asset based on its beta and expected market returns.
• Also known as "beta coefficient."

4. Beta calculation
•Beta is calculated using regression analysis
•

5. VALUE OF BETA
• β= 1
• β <1
• β>1
• For example, if a stock's beta is 1.2, it's theoretically 20% more
volatile than the market.

6. Expected Return depends on
3 things
The time value of money (risk-free rate, Rf)
The reward for bearing systematic risk (market risk premium
The amount of systematic risk (Beta)

7. What is Covariance?
s jk = s j s k r jk
sj is the standard deviation of the jth asset in the
portfolio,
sk is the standard deviation of the kth asset in the
portfolio,
rjk is the correlation coefficient between the jth and
kth assets in the portfolio.

8. Correlation Coefficient
A standardized statistical measure of the linear
relationship between two variables.
Its range is from -1.0 (perfect negative correlation), through 0
(no correlation), to +1.0 (perfect positive correlation).

9. Systematic Risk is the variability of return on stocks
or portfolios associated with changes in return on
the market as a whole.
Unsystematic Risk is the variability of return on
stocks or portfolios not explained by general market
movements. It is avoidable through diversification.
Total Risk = Systematic Risk +
Unsystematic Risk
Total Risk = Systematic Risk +
Unsystematic Risk

10. Total Risk = Systematic Risk +
Unsystematic Risk
Total
Risk
Unsystematic risk
Systematic risk
STDDEVOFPORTFOLIORETURN
NUMBER OF SECURITIES IN THE PORTFOLIO
Factors such as changes in nation’s
economy, tax reform by the Congress,
or a change in the world situation.

11. CAPM is a model that describes the relationship between
risk and expected (required) return; in this model, a
security’s expected (required) return is the risk-free rate
plus a premium based on the systematic risk of the
security.
Works for both individual assets and portfolios
• A model that describes the relationship between risk and expected
return and that is used in the pricing of risky securities.
• default model for risk in equity valuation and corporate finance.
• The general idea behind CAPM is that investors need to be
compensated in two ways: time value of money and risk
Capital Asset Pricing Model (CAPM)

12. Empirical Tests of the CAPM
• Stability of Beta
• betas for individual stocks are not stable, but
portfolio betas are reasonably stable. Further, the
larger the portfolio of stocks and longer the period,
the more stable the beta of the portfolio
• Comparability of Published Estimates of Beta
• differences exist. Hence, consider the return interval
used and the firm’s relative size

13. 1. Capital markets are efficient.
2. Homogeneous investor expectations
over a given period.
3. Risk-free asset return is certain
(use short- to intermediate-term
Treasuries as a proxy).
4. Market portfolio contains only
systematic risk .
CAPM Assumptions

14. Assumptions
• Can lend and borrow unlimited amounts under the risk free rate of interest
• Individuals seek to maximize the expected utility of their portfolios
over a single period planning horizon.
• Assume all information is available at the same time to all investors
• The market is perfect: there are no taxes; there are no transaction
costs; securities are completely divisible; the market is competitive.
• The quantity of risky securities in the market is given.

15. Limitations
CAPM has the following limitations:
• It is based on unrealistic assumptions.
• It is difficult to test the validity of CAPM.
• Betas do not remain stable over time.

16. CONCLUSION
Research has shown the CAPM to
stand up well to criticism,
although attacks against it have
been increasing in recent years.
Until something better presents
itself, however, the CAPM
remains a very useful item in the
financial management tool kit.

17. Introduction: background of CAPM
• Article written by Harry Markowitz in 1952
• Attention was on “common practice of portfolio diversification”.
• He showed how exactly portfolio returns by choosing that do not
move exactly together (covariance).
• His work is considered the foundation for risk and returns
• Markowitz showed that for a given level of expected return and for a given
security universe, knowledge of the covariance and correlation matrices are
required.

18. Introduction…
• CAPM also describes how the betas relate to the expected rates of
return that investors require on their investments.
• The key insight of CAPM is that investors will require a higher rate of
return on investments with higher betas.

19. Implications and relevance of CAPM
• Investors will always combine a risk free asset with a market portfolio of risky assets.
• Investors will invest in risky assets in proportion to their market value..
• Investors can expect returns from their investment according to the risk. This implies a
liner relationship between the asset’s expected return and its beta.
• Investors will be compensated only for that risk which they cannot diversify. This is the
market related (systematic) risk

20. 20
Quadratic Programming
• The Markowitz algorithm is an application of quadratic programming
• The objective function involves portfolio variance
• Quadratic programming is very similar to linear programming

21. 21
Portfolio Programming
in A Nutshell
• Various portfolio combinations may result in a given return
• The investor wants to choose the portfolio combination that provides
the least amount of variance

22. Normal distribution(symmetrical bell-shaped graph)
• It define by two numbers
• Average or expected returns
• Standard Deviations
• So, if returns are normally distribution, investor considers 2 measures
• Expected returns
• SD Investment A & B, B is preferred over A, but C is preferred over A
Investment A
Ret:15%
SD=7.5
%
Investment CInvestment B
Ret:20%
SD=7.5
%
Ret:10%
SD=15%

23. Combining stock into portfolio
Suppose you want to invest in PTCL shares or in Telenor. You expect
that PTCL offers 13% and Telenor 19% expected returns. Past variability
depicts SD 33.5% for PTCL and 46% for Telenor. If you invest 58% in
PTCL and 42% in Telenor:
the expected returns for portfolio on weighted average will be
=15% (13+19/2)
Portfolio SD will be: 30.83
Portfolio variance= = X2
1 σ2
1 + X2
2 σ2
2 + 2( X1X2 P12 σ1 σ2)
Portfolio SD is square root of the variance
** the complete calculation is show in my lecture No.5

24. Combining stock into portfolio
• You could achieve risk and return though thick line by various
combinations of the two stock.
• Question is which is the best combination? (depend on investor’s
attitude)
• We know that gain from diversification depends on how highly the
stock are correlated

26. Risk and return
EXCESS RETURN
ON STOCK
Expected RETURN
ON MARKET PORTFOLIO
Beta =
Rise
Run
Narrower spread
is higher correlation
Characteristic Line

27. Introduction of Borrowing and landing
Suppose; you can lend or borrow at some risk free rate rf
for you invest in T-bill (lend money) and half in common stock portfolio
(S). You can obtain any combination of expected returns and risk with
straight line joining rf and (S) in the next slides figure. Since borrowing is a
negative lending, you can extend the range of possibilities to the right of S
by borrowing funds at an interest rat of rf and investing them your owns
money in portfolio S.
Suppose portfolio S has expected return of 15%; SD 16%. T-Bills offers 5% ;
SD 0 (on risk free rate). The returns will be:
r= (½*expected return on S)+(1/2*Interest rate)=10%
Sigma = (½*SD of S)+(1/2*SD of bills)=10%
We can solve (2*expeted return on S)-(1*Interest rate)=25%
• And the SD of your investment is a;
sigma= (2*SD of S)-(1* SD of bills)=32%

28. Security Market Line
Return
BETA
.
rf
Risk Free
Return =
Market Return = rm
Efficient Portfolio
1.0

29. Security Market Line
Return
BETA
rf
Risk Free
Return =
Market Return = rm
1.0
Security Market
Line (SML)

30. Portfolio Possibilities Combining the Risk-Free Asset and
Risky Portfolios on the Efficient Frontier
)E( ports
)E(R port
RFR
M
C
A
B
D

31. Portfolio Possibilities Combining the Risk-Free Asset
and Risky Portfolios on the Efficient Frontier
RFR
M

32. Testing the CAPM
Avg Risk Premium
1931-65
Portfolio Beta
1.0
SML
30
20
10
0
Investors
Market
Portfolio
Beta vs. Average Risk Premium

33. Testing the CAPM
Avg Risk Premium
1966-91
Portfolio Beta
1.0
SML
30
20
10
0
Investors
Market
Portfolio
Beta vs. Average Risk Premium

34. Review of CAPM
• Investor likes high expected return and low SD
• Common Stock portfolios that offer the highest expected return for a
given SD known efficient portfolio
• Investor attitude is important
• Best efficient portfolio depends on investor’s assessment of expected
returns, SD and correlations
• Do not look at the risk of individual asset rather go on portfolio risk

35. What if a stock did not lie on the security
market line
• Imagine that you encounter stock A which is lying beta.5 and away
from security line. You probably not as it is risky as well as giving you
less returns
• If beta is 1.5 for another security B, what would you do, if, it is away
from security line
expected risk premium on stock= beta*expected risk premium on market
r= rf= B(rm-rf)

36. Some alternative theories
• Arbitrage pricing theory (APT)

37. CHAPTER 9 – The Capital Asset
Pricing Model (CAPM)
9 - 37
Alternative Asset Pricing Models
The Arbitrage Pricing Theory – the Model
• Underlying factors represent broad economic forces which are inherently
unpredictable.
• Where:
• ERi = the expected return on security i
• a0 = the expected return on a security with zero systematic risk
• bi = the sensitivity of security i to a given risk factor
• Fi = the risk premium for a given risk factor
• The model demonstrates that a security’s risk is based on its sensitivity to
broad economic forces.
...11110 niniii FbFbFbaER [9-10]

38. CHAPTER 9 – The Capital Asset
Pricing Model (CAPM)
9 - 38
Alternative Asset Pricing Models
The Arbitrage Pricing Theory – Challenges
• Underlying factors represent broad economic forces
which are inherently unpredictable.
• Ross and Roll identify five systematic factors:
1. Changes in expected inflation
2. Unanticipated changes in inflation
3. Unanticipated changes in industrial production
4. Unanticipated changes in the default-risk premium
5. Unanticipated changes in the term structure of interest rates
• Clearly, something that isn’t forecast, can’t be used to
price securities today…they can only be used to explain
prices after the fact.

39. 39
Arbitrage Pricing Theory
• APT background
• The APT model
• Comparison of the CAPM and the APT

40. 40
APT Background
• Arbitrage pricing theory (APT) states that a number of distinct factors
determine the market return
• Roll and Ross state that a security’s long-run return is a function of changes in:
• Inflation
• Industrial production
• Risk premiums
• The slope of the term structure of interest rates

41. 41
APT Background (cont’d)
• Not all analysts are concerned with the same set of economic
information
• A single market measure such as beta does not capture all the information
relevant to the price of a stock

42. The APT Model
• General representation of the APT model:
1 1 2 2 3 3 4 4( )
where actual return on Security
( ) expected return on Security
sensitivity of Security to factor
unanticipated change in factor
A A A A A A
A
A
iA
i
R E R b F b F b F b F
R A
E R A
b A i
F i
    




%
%

43. APT
1 1 2 2 3 3
1 1 1 2 2 2 3 3 3
1 1 2 2 3 3 1 1 2 2 3 3
Fixed Random
(Notice that the security index "A" has been ign
( )
( ) [ ( )] [ ( )] [ ( )]
( ) ( ) ( ) ( )
R E R F F F
R E R R E R R E R R E R
R E R E R E R E R R R R
   
   
      
    
       
       
1 4 4 4 4 4 4 2 4 4 4 4 4 4 3 1 4 44 2 4 4 43
ored for clarity purposes)

44. 44
Replicating the Randomness
• Let’s try to replicate the random component of
security A by forming a portfolio with the following
weights:
1 1 2 2 3 3 1 2 3 f
1 2 3 f 1 1 2 2 3 3
Fixed Random
on , on , on , and finally 1- on R
We get the following return (for this portfolio of factors):
(1- )R
R R R
R R R R
     
      
 
      
1 4 44 2 4 4 43 1 4 44 2 4 4 43

45. 45
Key Point in Reasoning
• Since we were able to match the random
components exactly, the only terms that differ at
this point are the fixed components.
• But if one fixed component is larger than the other,
arbitrage profits are possible by investing in the
highest yielding security (either A or the portfolio
of factors) while short-selling the other (being
“long” in one and “short” in the other will assure
an exact cancellation of the random terms).

46. 46
•Therefore the fixed components MUST BE THE
SAME for security A and the portfolio of factors
created, otherwise unlimited profits would be
possible.
So we have:
1 1 2 2 3 3 1 2 3 f
f 1 1 f 2 2 f 2 3 f
( ) ( ) ( ) ( ) (1- )R
Rearranging terms yields:
( ) R [ ( ) R ] [ ( ) R ] [ ( ) R ]
E R E R E R E R
E R E R E R E R
     
  
     
      

47. 47
Comparison of the
CAPM and the APT
• The CAPM’s market portfolio is difficult to construct:
• Theoretically all assets should be included (real estate, gold, etc.)
• Practically, a proxy like the S&P 500 index is used
• APT requires specification of the relevant macroeconomic factors

48. 48
Comparison of the
CAPM and the APT (cont’d)
• The CAPM and APT complement each other rather than compete
• Both models predict that positive returns will result from factor sensitivities
that move with the market and vice versa

49. Arbitrage pricing theory (APT)
Basic concept of CAPM is to use of portfolio effiently
Stephen Rosss has it own theory in shape of APT
APT does not ask about which portfolio is efficient, instead, it starts by
assuming that each stock’s return depends partly on pervasive
macroeconomic influence or factors and partly on noise- events that
are unique to that company

50. Thank you very much for your time and
discussion