Capital asset pricing models is very important

This is about capital asset pricing models in financial management

1. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 1

2. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 2
Try these problems
Ch 6
Problem 6
Problem 12
Problems 14-16 (see p 157)
Ch 7
Problem 1
Problem 5

4. Portfolio Diversification
and the
Capital Asset Pricing Model
Prof. Ian Giddy
New York University
New York University/ING Barings

5. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 5
Equity Risk and Return: Summary
Investors diversify, because you get a
better return for a given risk.
There is a fully-diversified “market
portfolio” that we should all choose
The risk of an individual asset can be
measured by how much risk it adds to
the “market portfolio.”

6. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 6
Capital Allocation Possibilities:
Treasuries or an Equity Fund?
r
f
=7%
E(r
P
)
=17%

P=27%
10%
P
Expected Return
Risk
7%
THE EQUITY FUND

7. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 7
We Can Buy Some T-bills and Some of
the Risky Fund...
C.A.L.
SLOPE=0.37
E(R)
SD
17%
14%
18.9% 27%
ONE PORTFOLIO:
30% Bills, 70% Fund
E(R)=.3X7+.7X17=14%
SD=.7X27=18.9%
r
f
=7%

8. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 8
...Or Buy Two Risky Assets
A
E(r)
B


9. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 9
Diversification
Asset F Asset G Portfolio of
Assets F and G
R
e
t
u
r
n
Time
R
e
t
u
r
n
Time
R
e
t
u
r
n
Time
kkk

10. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 10
Portfolio Return...
To compute the return of a portfolio: use the
weighted average of the returns of all
assets in the portfolio, with the weight given
each asset calculated as
(value of asset)/(value of portfolio).
The portfolio return E(R
p) is:
E(R
p) = (w
1k
1)+(w
2k
2)+ ... (w
nk
n) =  w
j k
j

where w
j = weight of asset j, k
j = return on asset j

11. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 11
...and Risk (Standard Deviation)
Portfolio return is the weighted average
of all assets’ returns,
But portfolio standard deviation is
normally less than the weighted average
of all assets’ standard deviations!
The reason: asset returns are
imperfectly correlated.

12. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 12
Measuring Portfolio Risk
The variance of a 2-asset portfolio is:
where w
A
and w
B
are the weights of A and B in the
portfolio.
P
2
A
2
A
2
B
2
B
2
A B A B A B = w + w + 2ww   

13. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 13
Return and Risk, Generalized
Portfolio return:
where w
i are the weights of each asset in the portfolio.
(Expected return is simply the weighted sum of the
individual asset returns.)
Portfolio variance:
When i = j, the term w
i
w
j
F
i
F
j
D
ij
becomes w
i
2
F
i
2
.
E(R) = wE(R)
p
i=1
n
i i
P
2
i=1
n
j=1
n
ijijij
= ww 

14. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 15
Covariance and Correlation
cov
,
[
,
()][
,
()]
KI
i
P
i
r
Ki
Er
K
r
Ii
Er
I
  

KI
KI
KI
,
cov
,

The correlation coefficient scales the
covariance to a value between -1 and +1:

15. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 19
Risk and Return of Stocks, Bonds and
a Diversified Portfolio
Rate of Return
State Prob.EquityBondPortfolio
Recession1/3-7%+17% +5%
Normal 1/3+12% +7% +9.5%
Boom 1/3+28% -3%+12.5%
Expected Return11% 7.0%9.0%
Variance 204.7%66.7%9.5%
Standard Deviation14.3%8.2%3.1%

16. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 20
The Correlation Between Stock and
Bond Returns
Covariance = 0.3333(-7-11)(17-7) +
0.3333(12-11)(7-7) +0.3333(28-11)(-3-7) = -
116.67
Correlation= -116.66 / 14.3(8.2)= -
0.99
     

pRER RER
s
s
n
se e sb b
1
, ,
() ()

cov
,eb
eb


17. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 21
Portfolio Return and Standard
Deviation
Given:
WS = 0.5 RS = 12%
S = 25%
WB = 0.5 RB = 9%
B = 12%
and

S,B = 0.2
Rp = 0.5(12)+0.5(9) = 10.5%

P = [(0.5)
2
(25)
2
+(0.5)
2
(12)
2
+2(0.5)(0.5)(25)(12)(0.2)]
1/2
= (156.25+36+30)
1/2
= (222.25)
1/2
= 14.91%

18. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 22
Attainable Set of Risk/Return
Combinations
A
E(r)
B

AB1

AB
0

AB
1


19. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 23
The Minimum-Variance Frontier of
Risky Assets

Efficient frontier
Individual
assets
Global minimum-
variance portfolio
E(r)

20. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 24
The Efficient Frontier of Risky Assets
with the Optimal CAL

Efficient frontier
CAL(P)
E(r)
r
f

21. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 28
The Capital Asset Pricing Model
(CAPM)
CAPM Says:
The total risk of a financial
asset is made up of two
components.
A. Diversifiable
(unsystematic) risk
B. Nondiversifiable
(systematic) risk
The only relevant risk is
nondiversifiable risk. 
CAL(P)
E(r)
r
f

22. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 31
Types of Risk
P
o
r
t
f
o
l
i
o
R
i
s
k

kp
Number of Securities (Assets) in Portfolio
1 5 10 15 20 25
}
}
{
TOTAL RISK
NONDIVERSIFIABLE RISK
DIVERSIFIABLE RISK

23. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 32
The Model: CAPM
The CAPM (Capital Asset Pricing Model) links
together nondiversifiable risk and return for all
assets:
A. Beta Coefficient (b) is a relative
measure of nondiversifiable risk; an index of
the change of an asset's return in response to
a change in the market return
B. Market Return (k
m) is the return on the
market portfolio of all traded securities

24. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 33
Security Market Line
Nondiversifiable Risk, 
0 .50 1.0 1.5 2.0 . . .
SML
}
Market Risk
Premium: 4%}
Asset Z’s Risk
Premium: 6%
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
R
z
=
R
m
=
R
F =
Required
Return, R(%)


R
F

m 
z

25. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 34
The Equation for the CAPM
R
j= R
F + 
j (R
m - R
F)
where:
R
j= Required return on asset j;
R
F = Risk-free rate of return

j= Beta Coefficient for asset j;
R
m = Market return
The term [
j(R
m - R
F)] is called the risk premium and (R
m-
R
F) is called the market risk premium

26. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 36
Silicon Graphics
Consider an investment in Silicon Graphics. It
has a Beta of 2.0 (riskier than the average
stock). If the T-bill rate is 5% and the S&P
return is 10%, what is the required return for
Silicon Graphics stock?
k
j =.05 + [2.0 x (.10-.05)]
=.05 + [2.0 x (.05)]
= .05 + .10
= .15 or 15%

27. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 37
Graphic Depiction of CAPM
REQUIRED
RETURN
15% R
j
with b
j
= 2.0
10% R
m
5% R
F

0 .50 1.0 1.5 2.0 . . .
Security
Market
Line
}
Market Risk
Premium: 5%}
Stock’s
Risk
Premium:
10%
Beta (Nondiversifiable Risk)
SML = R
j= .05 + 
j(.10-.05)
Given: R
F = 5%; R
m = 10%

28. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 38
Interpreting Beta
Market Beta = 1.0 = average level of
risk
A Beta of .5 is half as risky as average
A Beta of 2.0 is twice as risky as average
A negative Beta asset moves in opposite
direction to market

29. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 43
Finding Beta: Example
You have found the following data relative
to the stock of the Telmex Corp. and
current conditions:
Required/expected return = 20%
Market portfolio return =
11%
Risk premium for market portfolio= 6%
What is the Beta of Telmex stock?

30. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 44
Determine the Risk-Free Rate
Algebraic Solution Graphic Solution
R
m - R
F = .06
.11 - R
F = .06
R
F = .05
R
m
=11%
R
f
= 5%
} 6%
} 5%
1.0
Beta
SML
R
i
= R
f
+ 
i
(R
M
- R
f
)

31. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 45
Plug into SML Formula
.20 = .05 + [Beta x (.11 - .05)]
.15 = Beta x (.06)
.15
= Telmex Beta 2.5
.06

32. Portfolio Theory
Assignment
Prof. Ian Giddy
New York University
New York University/ING Barings

33. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 47
Try these problems
Ch 6
Problem 6
Problem 12
Problems 14-16 (see p 157)
Ch 7
Problem 1
Problem 5

34. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 48
BKM Chapter 6, Problem 6
Gold
Stocks
Optimal CAL
E(r)
P
A. If 
G,S
<+1, gold
is still an
attractive asset
to hold as part
of a portfolio.
E(r)
Optimal CAL
Gold
Stocks
P
B. If 
G,S
=+1, a portfolio
of stocks and bills only
dominates a portfolio
with gold in all
instances

35. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 49
BKM Chapter 6, Problem 12
R
A
-R
fSince B’s error is
small, diversification
effect is less than for
A, which has large
unsystematic risk.
R
M
-R
f
R
B
-R
f
R
M
-R
f
Stock A has a large
error term so would
be very risky if all
funds were in this
one basket.
A
B

36. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 50
BKM, Chapter 6, Problem 14
 Most diversification achieved with
1st 20 stocks
By choosing low-correlated
assets in the portfolio, risk may
not be affected significantly. But
would these be the best-return
stocks?
P
o
r
t
f
o
l
i
o
R
i
s
k

kp
Number of Securities (Assets) in Portfolio
1 5 10 15 20

37. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 51
BKM, Chapter 6, Problem 15
 The risk/number of stocks
relationship is nonlinear, so risk
increases as number of stock is
further reduced
P
o
r
t
f
o
l
i
o
R
i
s
k

kp
Number of Securities (Assets) in Portfolio
1 5 10 15 20

38. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 52
BKM, Chapter 6, Problem 16

Hennessy’s portfolio
E(r)
 Limiting Hennessy’s
holdings may have little
impact on the risk of the
total portfolio

39. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 53
BKM, Chapter 7, Problem 1
E(R
P) = R
f + [E(R
M) - R
f]
20 = 5 + (15-5)
 =15/10 = 1.5

40. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 54
BKM, Chapter 7, Problem 5
S&P
Return
Liberty
Travel
Nynex
Weak
market
5% 2% 3.5%
Strong
market
20% 32% 14%

41. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 55
BKM, Chapter 7, Problem 5
A) The beta is the change in the stock return
per change in the market return. Therefore:

Aggressive = (2-32)/(5-20) = 2.00
Defensive =
(3.5-14)/(5-20) = .70
B) The expected return is an average of the two
possible outcomes:E(R
Agg.) = .5(2+32) =
17%E(R
Def.) = .5(3.5+14) = 8.75%

42. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 56
BKM, Chapter 7, Problem 5
C/ The SML is determined by the market
expected return of .5(20+5) = 12.5%, with a
beta of 1, and the bill return of 8%.
Therefore, the equation for the security
market line is:
E(R) = 8 + (12.5 - 8)

43. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 57
BKM, Chapter 7, Problem 5
SML
E(r)
M
D
A
8%
12.5%
.71.0 2.0 
17%

44. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 58
BKM, Chapter 7, Problem 5
D/StockSML E(R)Analyst E(R)Alpha
Agg. 17% 17% 0
Def. 11.15% 8.75% -2.4%
SML
E(r)
M
D
A
8%
12.5%
.71.0 2.0

D

45. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 59
BKM, Chapter 7, Problem 5
E/ The hurdle rate is determined by the
project beta .7. The correct discount
rate is 11.15%, the fair return on the
defensive stock.

46. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 60
Equity Risk and Return: Summary
Investors diversify, because you get a
better return for a given risk.
There is a fully-diversified “market
portfolio” that we should all choose
The risk of an individual asset can be
measured by how much risk it adds to
the “market portfolio”
The CAPM tells us how the required
return relates to the relevant risk.

47. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 61

48. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 62
www.giddy.org

49. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 63

50. Copyright ©1997 Ian H. Giddy
Portfolio Diversification and the CAPM 64
www.giddy.org
Ian Giddy
NYU Stern School of Business
Tel [phone]; Fax [phone]
[email protected]
http://www.giddy.org