Capital Asset Pricing Model (CAPM) Assumptions Investors are price takers and have homogeneous...
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![Page 1: Capital Asset Pricing Model (CAPM) Assumptions Investors are price takers and have homogeneous expectations One period model Presence of a riskless asset.](https://reader036.fdocuments.us/reader036/viewer/2022082710/56649dfe5503460f94ae6b69/html5/thumbnails/1.jpg)
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
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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
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• 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 =
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• 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
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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
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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
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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).
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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
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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
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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
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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)
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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
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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)
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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)
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If riskless asset exists, we haverf =d0, which then implies:
APT: E(ri) -rf = d1bi1 + ...+dkbik , and
di = risk premium =Di -rf
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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.
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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%
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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%