1. CHAPTER 7 Capital Asset Pricing and Arbitrage Pricing Theory

2. 7.1 THE CAPITAL ASSET PRICING MODEL

3. 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

4. Assumptions Individual investors are price takers (perfect market)
Single-period investment horizon
No limit for borrowing or lending
No taxes nor transaction costs
Information is costless and available to all investors
Investors are rational mean-variance optimizers
Homogeneous expectations

5. Example of one factor model: Capital Asset Pricing Model (CAPM)
Since every body knows that a considerable extent of risk (i.e., unique risk) can be diversified away by making a portfolio of only around securities, so all the investors assume a diversified portfolio. Risk premium is then due only for market risk, represented by beta (rather than total risk represented by sigma).
Rj-rf=f(βj)
The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio
Beta risk measures the volatility of security return in response to the volatility of the market return.
βj= COV(Rj,Rm)/ σ2Rm
In an equilibrium situation expected return of each security is composed of the risk free return and the risk premium. E(Rj)=Rf+[E(Rm)-Rf]βj

6. Calculation of Expected Return (CAPM)
Economic state
Probability
Return j
Market Return
Depression
0.25 5 7
Normal
0.5 10
Inflation 20 15
COV(Rj,Rm)=∑(Rj-Rj)(Rm-Rm)(Pi)=15.6
σ2Rm=∑(Rm-Rm)2(Pi)=8.25
βj= COV(Rj,Rm)/ σ2Rm=15.6/8.25=1.9
E(Rj)=Rf+(Rm-Rf) βj=5%+(10.5%-5%)(1.9)=15.45%

7. Work Sheet of CAPM βj= COV(Rj,Rm)/ σ2Rm=15.6/8.25=1.9
Economic State
Probability
Return J
Market Return
ExptdValue j
Expd. Value M
Dev (Rj)
[Rj-E(Rj)]
Dev (Rm)
[Rm-E(Rm)] x
Dev(Rm)
Dev(Rj) X Pi
Depression
0.25 5 7
1.25
1.75
-6.25
-3.5
21.875
3.0625
Normal
0.5 10
-1.25
-0.5
0.625
0.3125
0.125
Inflation 20 15
3.75
8.75
4.5
39.375
5.0625
 Sum
11.25
10.5
15.6
8.25
βj= COV(Rj,Rm)/ σ2Rm=15.6/8.25=1.9
E(Rj)=Rf+(Rm-Rf) βj=5%+(10.5%-5%)(1.9)=15.45%

8. Security Market Line (SML)
Conservative
Return
Aggressive
Note: This point does not refer to borrowing. It is investment in a security which assumes more risk and return compared to market average. It is optimum for aggressive people. Concentrate on the IC curve.
Normal
SML
E(Rj)
E(Rm)
Rf=5%
Systematic Risk /Beta Risk
βm=1
βj=1.9

9. Monthly rates of return Monthly Excess ReturnsGM
Monthly rates of return
Monthly Excess Returns
Month GM
S&P500
T-bills 1
-24.57
a-2.19
0.5
-25.07
-2.69 2
-17.79
2.39
0.49
-18.28
1.9 3
-1.94
-1.63
0.51
-2.45
-2.14 4
22.94
6.07
0.52
22.42
5.55 5
-7.14
-5.35
-7.66
-5.87 6
-4.42
-0.49
-4.94
-1.01 7
-20.32
-8.01
0.53
-20.85
-8.54 8
2.9
0.41
2.4
-0.09 9
5.42
3.46
0.44
4.98
3.02 10
-0.71
-9.23
0.42
-1.13
-9.65 11
-2.76
-6.42
0.38
-3.14
-6.8 12
5.71
7.68
0.33
5.38
7.35 13
3.81
0.31
3.5
0.2 14
13.09
-2.5
0.3
12.79
-2.8 15
-1.17
-1.07
-1.47
-1.37 16
-13.92
-6.41
0.29
-14.21
-6.7 17
-21.64
-8.17
0.22
-21.86
-8.39 18
-3.68
1.81
0.18
-3.86
1.63 19
20.28
7.52
0.16
20.12
7.36 20
-2.21
0.76
0.14
-2.35
0.62 21
5.23
-1.56
5.09
-1.7 22
3.6
-2.08
0.15
3.45
-2.23 23
14.1
3.67
13.95
3.52 24
6.12
-6.14
5.97
-6.29 25
-3.12
-0.91
-3.27
-1.06 26
-14
-7.25
-14.14
-7.39 27
-12.91
-7.9
-13.05
-8.04 28
2.81
2.67
0.35 29
-18.72
-11
-18.86
-11.14 30
-14.52
8.64
0.13
-14.65
8.51 31
19.4
0.1
19.3
5.61 32
-7.15
-6.03
-6.13 33
-1.44
-2.74
-1.54
-2.84 34
-7.05
-1.8 35
-0.44
0.84
-0.54
0.74 36
7.23
8.1
7.13 37 -2
0.09
-2.09 38
1.13
0.08
1.82
1.05 39
3.97
1.62
3.89
1.54 40
9.8
1.79
9.72
1.71 41
-0.41
-1.19
-1.27 42
4.25
5.5
4.17 43
0.26
0.71
0.63 44
24.82
5.08
24.74 45
-6.97
1.73
1.65 46
1.22
-3.22
1.14 47
-1.81
-1.64
-1.89
-1.72 48
0.36
-1.68
0.28
-1.76 49
-4.28
1.21
-4.37
1.12 50
2.64
1.8
0.11
2.53
1.69 51
-7.41
-3.43
-7.52
-3.54 52
-4.24
0.23 53
2.83
0.94
2.69
0.8 54
-9.25
1.4
-9.4
1.25 55
3.86
-0.08
3.68 56
3.25
0.19
3.62
3.06 57
-8.11
-2.53
-8.31
-2.73 58
-3.15
1.89
-3.37
1.67 59
-17.56
-1.91 60
-12.15
-3.65
0.24
-12.39
-3.89

10. SUMMARY OUTPUT OF REGRESSION
R Square
0.33
ANOVA df SS MS F
Significance F
Regression 1
2126
2126.
28.35
1.72E-06
Residual 58
4351
75.02
Total 59
6478
Coefficients
Standard
Error
t Stat
P-value
Lower
95%
Upper
95.0%
Intercept
-1.1
1.125
-0.9
0.3
-3.3
1.1
-3.36
X Variable 1
1.323
0.248
5.3
1.72E-6
0.8
1.82
1.8

11. Calculation of beta Covariance (Rm,Rq)= 0.003155434 Var (Rm)=
Month
Closing DGEN Values
Market Return (Rm)
Closing Share Price
Returns of Qassem (Rq)
Jul-04
14.80
Aug-04
15.30
Sep-04
18.50
Oct-04
19.40
Nov-04
17.30
Dec-04
22.10
Jan-05
Feb-05
17.50
Mar-05
16.90
Apr-05
May-05
14.40
Jun-05
14.10
Jul-05
12.90
Aug-05
14.00
Sep-05
15.80
Oct-05
14.70
Nov-05
Dec-05
11.80
Jan-06
11.30
Feb-06
10.10
Mar-06
11.1
Apr-06
10.4
May-06
10.80
Jun-06
10.60
Jul-06
10.30
Aug-06
12.50
Sep-06
-0.056
Oct-06
11.10
Nov-06
8.6
Dec-06
10.2
Jan-07
11.3
Feb-07 12
Mar-07
11.5
Apr-07
10.9
May-07
10.6
Jun-07
Jul-07
11.6
Aug-07
12.3
Sep-07
15.9
Oct-07
18.1
Nov-07
21.1
Dec-07
20.8
Jan-08
25.2
Feb-08
29.50
Mar-08
32.50
Apr-08
34.80
May-08
34.10
Jun-08
38.10
Covariance (Rm,Rq)=
Var (Rm)=
Beta
0.675

12. Calculation of CAPM Return
Covarience (Rm,Rq)
Var Rm
Beta
Average Monthly Return 
Annualized Market Return Rm %
Risk-free Rate Rf   7%
ke   %

13. CML vs. SML
CML assumes that all investors may not hold diversified portfolio, and as such the concerned risk is total risk represented by sigma or standard deviation. SML assumes that all investors hold diversified portfolio, as such there is only systematic risk concerned, represented by beta.
In case of CML, risk premium is the price for total risk. In case of SML, risk premium is the price for systematic risk.
Among all the points of CML, there is only one point that belongs to the market which is optimum portfolio and the rest of the points are derived by means of borrowing or lending at the risk free rate. All the points of SML are available from the market that refers to risk-return of different stocks, and different portfolios including the market portfolio.

14. THE CAPM AND THE REAL WORLD (Limitations of CAPM)
Empirical Estimates are different from theoretical expectation. Endogenous risk free rate is higher than the true rate, and endogenous market risk premium is lower than the true one.
Research shows that other factors affect returns
January Effect (April Effect)
Small Firm Effect
Returns are related to factors other than market returns like book value relative to market value
Choice of market portfolio
Thin trading problem

15. 7.5 FACTOR MODELS AND THE ARBITRAGE PRICING THEORY

16. Arbitrage Pricing Theory
Arbitrage arises if an investor can construct a zero investment portfolio with a sure profit.
Since no investment is required, an investor can create large positions to secure large levels of profit.
In efficient markets, profitable arbitrage opportunities will quickly disappear.

17. Equilibrium rate of return
Role of Arbitrage: Simultaneous buying and selling of securities k k S1 S1 S2
9 %
8 %
7 % D D2 D1 Q Q
Market A
Market B

18. Factor Models: Announcements, Surprises, and Expected Returns
The return on any security consists of two parts.
First, the expected returns
Second, the unexpected or risky returns
A way to write the return on a stock in the coming month is:

19. Factor Models: Announcements, Surprises, and Expected Returns
Any announcement can be broken down into two parts, the anticipated (or expected) part and the surprise (or innovation):
Announcement = Expected part + Surprise.
The expected part of any announcement is the part of the information the market uses to form the expectation, R, of the return on the stock.
The surprise is the news that influences the unanticipated return on the stock, U.

20. Risk: Systematic and Unsystematic
A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree.
An unsystematic risk is a risk that specifically affects a single asset or small group of assets.
Unsystematic risk can be diversified away.
Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation, foreign exchange rates, etc.
On the other hand, announcements specific to a single company are examples of unsystematic risk.

21. Risk: Systematic and Unsystematic
We can break down the total risk of holding a stock into two components: systematic risk and unsystematic risk: 2
Total risk 
Nonsystematic Risk: 
Systematic Risk: m n

22. Systematic Risk and Betas
For example, suppose we have identified three systematic risks: inflation, GNP growth, and the foreign exchange spot rate S($=Tk).
Our model is:
F = the surprise in the factor.

23. Systematic Risk and Betas: Example
Suppose we have made the following estimates:
bI = -2.30
bGNP = 1.50
bS = 0.50
Finally, the firm was able to attract a “superstar” CEO, and this unanticipated development contributes 1% to the return.

24. Systematic Risk and Betas: Example
We must decide what surprises took place in the systematic factors.
If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then:
FI = Surprise in the inflation rate = actual – expected
= 8% – 3% = 5%

25. Systematic Risk and Betas: Example
If it were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then:
FGNP = Surprise in the rate of GNP growth
= actual – expected = 1% – 4%
= – 3%

26. Systematic Risk and Betas: Example
If it were the case that the dollar-Taka spot exchange rate, S($=Tk), was expected to increase by 10%, but in fact remained stable during the time period, then:
FS = Surprise in the exchange rate
= actual – expected = 0% – 10% = – 10%

27. Systematic Risk and Betas: Example
Finally, if it were the case that the expected return on the stock was 8%, then:

28. Relationship Between b & Expected Return
SML
Expected return D A B C b

29. The Capital Asset Pricing Model and the Arbitrage Pricing Theory
APT applies to well diversified portfolios and not necessarily to individual stocks.
With APT it is possible for some individual stocks to be mispriced - not lie on the SML.
APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio.
APT can be extended to multifactor models. For example, there may be GNP growth beta, inflation beta, foreign exchange beta, P:E beta and so on.