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%