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 shortselling restrictions (perfect market assumption)
• Returns are normally distributed or investor’s
utility is a quadratic function in returns
CAPM Derivation
Return
m
Efficient
frontier
rf
Sp
A. For a well-diversified portfolio, the
equilibrium return is:
E(rp) = rf + [E(rm-rf)/sm]sp
• For the individual security, the
return-risk relationship is determined by using the
following (trick):
rp = wri + (1-w)rm
sp=[w2s2i +(1-w)2s2m+2w(1-w)sim]0.5
where sim is the covariance of asset i
and market (m) portfolio, and w is the weight.
drp/dw = ri -rm
dsp/dw =
2ws2i -2(1-w)s2m+2sim-4wsim
2sp
• dsp/dw
drp/dw
dsp/dw
=
w=0
=
w=0
sim - s2m
sm
ri -rm
(sim-s2m)/sm
The slope of this tangential portfolio
at M must equal to: [E(rm) -rf]/sm,
Thus,
ri -rm
(sim-s2m)/sm
Thus, we have CAPM as
ri = rf + (rm-rf)sim/s2m
= [rm-rf]/sm
Properties of SLM
If we express the return-risk relationship as beta,
then we have
ri = rf + E(rm -rf) bi
Return
SML
E(rm )
rf
beta=1
RISK
Zero-beta CAPM
• No Riskless Asset
Return
p
q
z
s2 p
E(ri) = E(rq) + [E(rp)-E(rq)]
covip -covpq
s2p -covpq
where p, q are any two arbitrary portfolios
CAPM and Liquidity
• If there are bid-ask spread (c) in trading
asset i, then we have:
• E(ri) = rf + bi[E(rm)-rf] + f(ci)
where f is a non-linear function in c (trading
cost).
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
Factor model(Single-index Model)
• We can summarize firm return, ri, is:
ri = E(ri)+mi + ei
where mi is the unexpected macro factor; ei is the firmspecific factor.
• Then, we have:
ri = E(ri) + biF + ei
where biF = mi, and E(mi)=0
• CAPM implies:
E(ri) = rf + bi(Erm-rf)
in ex post form,
ri =rf + bi(rm-rf) + ei
ri = [rf+bi(Erm-rf)]+bi(rm-Erm) + ei
ri = a + bRm + ei
Total variance:
s2i = b2is2m + s2(ei)
The covariance between any two stocks
requires only the market index because ei and
ej is assumed to be uncorrelated.
Covariance of two stocks is: cov(ri, rj) =bibjs2m
These calculations imply:
n estimates of return
n estimates of beta
n estimates of s2(ei)
1 estimate of s2m
In total =3n+1 estimates required
Price paid= idiosyncratic risk is assumed to be
uncorrelated
Index Model and Diversification
• ri = a + biRm +ei
• rp=ap +bpRm +ep
s2p=b2ps2m + s2(ep)
where:
s2(ep) = [s2(e1)+...s2(en)]/n
(by assumption only! Ignore covariance terms)
Market Model and Empirical Test
Form
• Index (Market) Model for asset i is:
• ri = a + biRm + ei
Excess return, i
slope=beta
=cov(i,m)s2m
Rm
R2 =coefficient of determination
= b2s2m/s2i
Arbitrage Pricing Theory (APT)
• APT - Ross (1976) assumes:
ri =E(ri) + bi1Fi+...+bikFk + ei
where:
bik =sensitivity of asset i to factor k
Fi = factor and E(Fi)=0
• Derivation:
w1+...+wn=0
(1)
rp =w1r1+...wnrn =0
(2)
• If large no. of securities (1/n tends to 0), we have:
Systematic + unsystematic risk=0
(sum of wibi) (sum of wiei)
That means:
w1E(r1)+...wnE(rn) =0 (no arbitrage condition)
Restating the above conditions, we have:
w1 + ...wn =0
(0)
w1b1k +...+wnbnk=0 for all k (1)
Multiply:
d0 to w1+...wn =0
d1 to w1d1b11+...wnd1bn1=0
dk to w1dkb1k+...wndkbnk=0
(0’)
(1-1)
(1-k)
Grouping terms vertically yields:
w1(d0+d1b11+d2b12+...dkb1k)+w2(d0+d1b21+d2b22+...dkb2k )+
wn(d0+d1bn1+d2bn2+...dkbnk)=0
E(ri) = d0 + d1bi1+...+dkbik
(APT)
If riskless asset exists, we have
rf =d0, which then implies:
APT:
E(ri) -rf = d1bi1 + ...+dkbik , and
di = risk premium
=Di -rf
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.
Illustration of APT
• Given:
• Asset Return
Two Factors
bi1
bi2
x
0.11
0.5 2.0
y
0.25
1.0 1.5
z
0.23
1.5 1.0
• D1=0.2; D2=0.08 and rf=0.1
E(ri)=rf + (Di-rf)bi1+ (D2-rf)bi2
E(rx)=0.1+(0.2-0.1)0.5+(8%-0.1)2=11%
E(ry)=0.1+(0.2-0.1)1+(8%-0.1)1.5=17%
E(rz)=0.1+(0.2-0.1)1.5+(8%-0.1)1=23%
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 wx=0;wy=1;wz=0
Risk factor 1= 1(1.0)=1
Risk factor 2= 1(1.5)=1.5
Original rp=(0.11+0.25+0.23)/3=19.67%
New rp=0(11%)+1(25%)+0(23%)=25%