Capital Asset Pricing Model (CAPM) Assumptions Investors are price takers and have homogeneous...
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Transcript of Capital Asset Pricing Model (CAPM) Assumptions Investors are price takers and have homogeneous...
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
CAPM Derivation
rf
Efficientfrontier
m
Return
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)2s2
m+2w(1-w)sim]0.5
where sim is the covariance of asset iand market (m) portfolio, and w is the weight.drp/dw = ri -rm
2ws2i -2(1-w)s2
m+2sim-4wsim
2sp
dsp/dw =
• dsp/dw =sim - s2
m
smw=0
drp/dwdsp/dw
w=0
= 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
= [rm-rf]/sm
Thus, we have CAPM asri = rf + (rm-rf)sim/s2
m
Properties of SLM
If we express the return-risk relationship as beta, then we have
ri = rf + E(rm -rf) bi
rf
beta=1 RISK
E(rm )SML
Return
Zero-beta CAPM
• No Riskless Asset
p
z
q
Return
s2p
where p, q are any two arbitrary portfolios
E(ri) = E(rq) + [E(rp)-E(rq)]covip -covpq
s2p -covpq
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 returnsn =50 estimates of variancesn(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 firm-specific 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:s2
i = b2is2
m + 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) =bibjs2
m
These calculations imply:n estimates of returnn estimates of betan estimates of s2(ei)1 estimate of s2
mIn 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=b2
ps2m + 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
Rm
Excess return, i
slope=beta=cov(i,m)s2
m
R2 =coefficient of determination = b2s2
m/s2i
Arbitrage Pricing Theory (APT)• APT - Ross (1976) assumes:
ri =E(ri) + bi1Fi+...+bikFk + ei
where:bik =sensitivity of asset i to factor kFi = 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 (0’)d1 to w1d1b11+...wnd1bn1=0 (1-1)dk to w1dkb1k+...wndkbnk=0 (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 haverf =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 portfolio4. 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 zi.e., 1/3 each
Risk factor 1=(0.5+1.0+1.5)/3=1Risk factor 2=(2+1.5+1.)/3 =1.5
Assume wx=0;wy=1;wz=0Risk factor 1= 1(1.0)=1Risk 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%