Capital%20Asset%20Pricing%20Model%20(CAPM)

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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 investors
  utility is a quadratic function in returns

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CAPM Derivation
Return
Efficientfrontier
m
rf
Sp
A. For a well-diversified portfolio, the
equilibrium return is E(rp) rf
E(rm-rf)/smsp
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• For the individual security, the return-risk
  relationship is determined by using the following
  (trick)rp wri (1-w)rmspw2s2i
  (1-w)2s2m2w(1-w)sim0.5where sim is the
  covariance of asset iand market (m) portfolio,
  and w is the weight.

drp/dw ri -rm 2ws2i -2(1-w)s2m2sim-4wsim

2sp
dsp/dw
4
sim - s2m

• dsp/dw

sm
drp/dw

dsp/dw
w0
The slope of this tangential portfolio at M must
equal to E(rm) -rf/sm,Thus,
ri -rm
rm-rf/sm
(sim-s2m)/sm
Thus, we have CAPM as
ri rf (rm-rf)sim/s2m
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Properties of SLM
If we express the return-risk relationship as
beta, then we haveri rf E(rm -rf) bi
Return
SML
E(rm )
rf
beta1 RISK
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Zero-beta CAPM

• No Riskless Asset

s2p
where p, q are any two arbitrary portfolios
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CAPM and Liquidity

• If there are bid-ask spread (c) in trading asset
  i, then we have
• E(ri) rf biE(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 inputsn 50 estimates of returnsn 50
  estimates of variancesn(n-1)/2 50(49)/21225
  (covariance)
• problem - too many inputs

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Factor model(Single-index Model)

• We can summarize firm return, ri, isri
  E(ri)mi eiwhere mi is the unexpected macro
  factor ei is the firm-specific factor.
• Then, we haveri E(ri) biF eiwhere biF
  mi, and E(mi)0
• CAPM impliesE(ri) rf bi(Erm-rf)
• in ex post form,ri rf bi(rm-rf) eiri
  rfbi(Erm-rf)bi(rm-Erm) ei
• ri a bRm ei

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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) bibjs2mThese calculations imply n
estimates of return n estimates of beta n
estimates of s2(ei) 1 estimate of s2mIn total
3n1 estimates required Price paid
idiosyncratic risk is assumed to be uncorrelated
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Index Model and Diversification

• ri a biRm ei
• rpap bpRm eps2pb2ps2m s2(ep)wheres2(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

Excess return, i
slopebeta cov(i,m)s2m
Rm
R2 coefficient of determination b2s2m/s2i
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Arbitrage Pricing Theory (APT)

• APT - Ross (1976) assumesri E(ri)
  bi1Fi...bikFk ei wherebik sensitivity of
  asset i to factor kFi factor and E(Fi)0
• Derivationw1...wn0 (1)rp w1r1...wnrn
  0 (2)
• If large no. of securities (1/n tends to 0), we
  haveSystematic unsystematic risk0(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 havew1 ...wn 0
(0)w1b1k ...wnbnk0 for all k
(1) Multiplyd0 to w1...wn 0
(0) d1 to w1d1b11...wnd1bn10 (1-1) dk
to w1dkb1k...wndkbnk0 (1-k) Grouping
terms vertically yields w1(d0d1b11d2b12...dkb1
k)w2(d0d1b21d2b22...dkb2k )wn(d0d1bn1d2bn2
...dkbnk)0 E(ri) d0 d1bi1...dkbik
(APT)
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If riskless asset exists, we have rf 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 portfolio 4. 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
• D10.2 D20.08 and rf0.1
• E(ri)rf (Di-rf)bi1 (D2-rf)bi2
• E(rx)0.1(0.2-0.1)0.5(8-0.1)211
• E(ry)0.1(0.2-0.1)1(8-0.1)1.517
• E(rz)0.1(0.2-0.1)1.5(8-0.1)123

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Suppose equal weights in x,y and z i.e., 1/3
each Risk factor 1(0.51.01.5)/31 Risk factor
2(21.51.)/3 1.5 Assume wx0wy1wz0Risk
factor 1 1(1.0)1 Risk factor 2
1(1.5)1.5 Original rp(0.110.250.23)/319.67
New rp0(11)1(25)0(23)25