Capital asset pricing models is very important

This is about capital asset pricing models in financial management

1. Capital Asset Pricing Model
(CAPM)
Assumptions
•Investors are price takers and have homogeneous
expectations
•One period model
•Presence of a riskless asset
•No taxes, transaction costs, regulations or short-
selling restrictions (perfect market assumption)
•Returns are normally distributed or investor’s
utility is a quadratic function in returns

2. CAPM Derivation
r
f
Efficient
frontier
m
Return
S
p
A. For a well-diversified portfolio, the
equilibrium return is:
E(rp) = rf + [E(r
m
-r
f
)/s
m
]s
p

3. •For the individual security, the
return-risk relationship is determined by using the
following (trick):
r
p = wr
i + (1-w)r
m
s
p=[w
2
s
2
i +(1-w)
2
s
2
m+2w(1-w)s
im]
0.5
where s
im is the covariance of asset i
and market (m) portfolio, and w is the weight.dr
p/dw = r
i -r
m
2ws
2
i -2(1-w)s
2
m+2s
im-4ws
im
2s
p
ds
p
/dw =

4. •ds
p/dw =
s
im
- s
2
m
s
mw=0
dr
p
/dw
ds
p
/dw
w=0
=
r
i
-r
m
(s
im
-s
2
m
)/s
m
The slope of this tangential portfolio
at M must equal to: [E(r
m) -r
f]/s
m,
Thus,
r
i
-r
m

(s
im
-s
2
m
)/s
m
= [r
m-r
f]/s
m
Thus, we have CAPM as
r
i
= r
f
+ (r
m
-r
f
)s
im
/s
2
m

5. Properties of SLM
If we express the return-risk relationship as beta,
then we have
r
i
= r
f
+ E(r
m
-r
f
) b
i
r
f
beta=1 RISK
E(r
m
)
SML
Return

6. Zero-beta CAPM
•No Riskless Asset
p
z
q
Return
s
2
p
where p, q are any two arbitrary portfolios
E(r
i
) = E(r
q
) + [E(r
p
)-E(r
q
)]
cov
ip -cov
pq
s
2
p
-cov
pq

7. CAPM and Liquidity
•If there are bid-ask spread (c) in trading asset i,
then we have:
•E(r
i) = r
f + b
i[E(r
m)-r
f] + f(c
i)
where f is a non-linear function in c (trading cost).

8. Single-index Model
•Understanding of single-index model sheds light
on APT (Arbitrage Pricing Theory or multiple
factor model)
•suppose your analyze 50 stocks, implying that
you need inputs:
n =50 estimates of returns
n =50 estimates of variances
n(n-1)/2 = 50(49)/2=1225 (covariance)
•problem - too many inputs

9. Factor model(Single-index Model)
•We can summarize firm return, r
i, is:
r
i = E(r
i)+m
i + e
i
where m
i is the unexpected macro factor; e
i is the firm-specific factor.
•Then, we have:
r
i
= E(r
i
) + b
i
F + e
i
where b
iF = m
i, and E(m
i)=0
•CAPM implies:
E(r
i) = r
f + b
i(Er
m-r
f)
in ex post form,
r
i =r
f + b
i(r
m-r
f) + e
i
r
i = [r
f+b
i(Er
m-r
f)]+b
i(r
m-Er
m) + e
i
r
i = a + bR
m + e
i

10. Total variance:
s
2
i = b
2
is
2
m + s
2
(e
i)
The covariance between any two stocks
requires only the market index because e
i

and e
j is assumed to be uncorrelated.
Covariance of two stocks is: cov(r
i, r
j) =b
ib
js
2
m
These calculations imply:
n estimates of return
n estimates of beta
n estimates of s
2
(e
i
)
1 estimate of s
2
m
In total =3n+1 estimates required
Price paid= idiosyncratic risk is assumed to
be uncorrelated

11. Index Model and Diversification
•r
i
= a + b
i
R
m
+e
i
•r
p=a
p +b
pR
m +e
p
s
2
p=b
2
ps
2
m + s
2
(e
p)
where:
s
2
(e
p) = [s
2
(e
1)+...s
2
(e
n)]/n
(by assumption only! Ignore covariance terms)

12. Market Model and Empirical Test
Form
•Index (Market) Model for asset i is:
•r
i = a + b
iR
m + e
i
Rm
Excess return, i
slope=beta
=cov(i,m)s
2
m
R2 =coefficient of determination
= b
2
s
2
m
/s
2
i

13. Arbitrage Pricing Theory (APT)
•APT - Ross (1976) assumes:
r
i =E(r
i) + b
i1F
i+...+b
ikF
k + e
i

where:
b
ik
=sensitivity of asset i to factor k
F
i
= factor and E(F
i
)=0
•Derivation:
w
1
+...+w
n
=0(1)
r
p =w
1r
1+...w
nr
n =0(2)
•If large no. of securities (1/n tends to 0), we have:
Systematic + unsystematic risk=0
(sum of w
ib
i) (sum of w
ie
i)

14. That means:
w
1
E(r
1
)+...w
n
E(r
n
) =0 (no arbitrage condition)
Restating the above conditions, we have:
w
1
+ ...w
n
=0 (0)
w
1b
1k +...+w
nb
nk=0 for all k (1)
Multiply:
d
0 to w
1+...w
n =0 (0’)
d
1
to w
1
d
1
b
11
+...w
n
d
1
b
n1
=0 (1-1)
d
k to w
1d
kb
1k+...w
nd
kb
nk=0 (1-k)
Grouping terms vertically yields:
w
1(d
0+d
1b
11+d
2b
12+...d
kb
1k)+w
2(d
0+d
1b
21+d
2b
22+...d
kb
2k )+
w
n(d
0+d
1b
n1+d
2b
n2+...d
kb
nk)=0
E(r
i) = d
0 + d
1b
i1+...+d
kb
ik (APT)

15. If riskless asset exists, we have
r
f =d0, which then implies:
APT:
E(r
i) -r
f = d
1b
i1 + ...+d
kb
ik , and
d
i
= risk premium
=D
i -r
f

16. APT is much robust than CAPM for several
reasons:
1. APT makes no assumptions about
the empirical distribution of asset
returns;
2. APT makes no assumptions on
investors’ utility function;
3. No special role about market portfolio
4. APT can be extended to multiperiod
model.

17. Illustration of APT
•Given:
•Asset Return Two Factors
bi1bi2
x 0.110.52.0
y 0.25 1.01.5
z 0.231.51.0
•D
1=0.2; D
2=0.08 and r
f=0.1
E(r
i)=r
f + (D
i-r
f)b
i1+ (D
2-r
f)b
i2
E(r
x)=0.1+(0.2-0.1)0.5+(8%-0.1)2=11%
E(r
y)=0.1+(0.2-0.1)1+(8%-0.1)1.5=17%
E(r
z)=0.1+(0.2-0.1)1.5+(8%-0.1)1=23%

18. Suppose equal weights in x,y and z
i.e., 1/3 each
Risk factor 1=(0.5+1.0+1.5)/3=1
Risk factor 2=(2+1.5+1.)/3 =1.5
Assume w
x
=0;w
y
=1;w
z
=0
Risk factor 1= 1(1.0)=1
Risk factor 2= 1(1.5)=1.5
Original r
p=(0.11+0.25+0.23)/3=19.67%
New r
p=0(11%)+1(25%)+0(23%)=25%