1. 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

2. CAPM Derivation rfrf Efficient frontier m Return SpSp A. For a well-diversified portfolio, the equilibrium return is: E(rp) = rf + [E(r m -r f )/s m ]s p

3. For the individual security, the return-risk relationship is determined by using the following (trick): r p = wr i + (1-w)r m s p =[w 2 s 2 i +(1-w) 2 s 2 m +2w(1-w)s im ] 0.5 where s im is the covariance of asset i and market (m) portfolio, and w is the weight. dr p /dw = r i -r m 2ws 2 i -2(1-w)s 2 m +2s im -4ws im 2s p ds p /dw =

4. ds p /dw = s im - s 2 m s m w=0 dr p /dw ds p /dw w=0 = r i -r m (s im -s 2 m )/s m The slope of this tangential portfolio at M must equal to: [E(r m ) -r f ]/s m, Thus, r i -r m (s im -s 2 m )/s m = [ r m -r f ]/s m Thus, we have CAPM as r i = r f + (r m -r f )s im /s 2 m

5. Properties of SLM If we express the return-risk relationship as beta, then we have r i = r f + E(r m -r f ) b i rfrf beta=1RISK E(r m ) SML Return

6. Zero-beta CAPM No Riskless Asset p z q Return s2ps2p where p, q are any two arbitrary portfolios E(r i ) = E(r q ) + [E(r p )-E(r q )] cov ip -cov pq s 2 p -cov pq

7. CAPM and Liquidity If there are bid-ask spread (c) in trading asset i, then we have: E(r i ) = r f + b i [E(r m )-r f ] + f(c i ) where f is a non-linear function in c (trading cost).

8. 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

9. Factor model(Single-index Model) We can summarize firm return, r i, is: r i = E(r i )+m i + e i where m i is the unexpected macro factor; e i is the firm- specific factor. Then, we have: r i = E(r i ) + b i F + e i where b i F = m i, and E(m i )=0 CAPM implies: E(r i ) = r f + b i (Er m -r f ) in ex post form, r i =r f + b i (r m -r f ) + e i r i = [r f +b i (Er m -r f )]+b i (r m -Er m ) + e i r i = a + bR m + e i

10. Total variance: s 2 i = b 2 i s 2 m + s 2 (e i ) The covariance between any two stocks requires only the market index because e i and e j is assumed to be uncorrelated. Covariance of two stocks is: cov(r i, r j ) =b i b j s 2 m These calculations imply: n estimates of return n estimates of beta n estimates of s 2 (e i ) 1 estimate of s 2 m In total =3n+1 estimates required Price paid= idiosyncratic risk is assumed to be uncorrelated

11. Index Model and Diversification r i = a + b i R m +e i r p =a p +b p R m +e p s 2 p =b 2 p s 2 m + s 2 (e p ) where: s 2 (e p ) = [s 2 (e 1 )+...s 2 (e n )]/n (by assumption only! Ignore covariance terms)

12. Market Model and Empirical Test Form Index (Market) Model for asset i is: r i = a + b i R m + e i Rm Excess return, i slope=beta =cov(i,m)s 2 m R2 =coefficient of determination = b 2 s 2 m /s 2 i

13. Arbitrage Pricing Theory (APT) APT - Ross (1976) assumes: r i =E(r i ) + b i1 F i +...+b ik F k + e i where: b ik =sensitivity of asset i to factor k F i = factor and E(F i )=0 Derivation: w 1 +...+w n =0(1) r p =w 1 r 1 +...w n r n =0(2) If large no. of securities (1/n tends to 0), we have: Systematic + unsystematic risk=0 (sum of w i b i ) (sum of w i e i )

14. That means: w 1 E(r 1 )+...w n E(r n ) =0 (no arbitrage condition) Restating the above conditions, we have: w 1 +...w n =0 (0) w 1 b 1k +...+w n b nk =0 for all k (1) Multiply: d 0 to w 1 +...w n =0 (0’) d 1 to w 1 d 1 b 11 +...w n d 1 b n1 =0 (1-1) d k to w 1 d k b 1k +...w n d k b nk =0 (1-k) Grouping terms vertically yields: w 1 (d 0 +d 1 b 11 +d 2 b 12 +...d k b 1k )+w 2 (d 0 +d 1 b 21 +d 2 b 22 +...d k b 2k )+ w n (d 0 +d 1 b n1 +d 2 b n2 +...d k b nk )=0 E(r i ) = d 0 + d 1 b i1 +...+d k b ik (APT)

15. If riskless asset exists, we have r f =d0, which then implies: APT: E(r i ) -r f = d 1 b i1 +...+d k b ik, and d i = risk premium =D i -r f

16. 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.

17. Illustration of APT Given: Asset Return Two Factors bi1bi2 x 0.110.52.0 y 0.25 1.01.5 z 0.231.51.0 D 1 =0.2; D 2 =0.08 and r f =0.1 E(r i )=r f + (D i -r f )b i1 + (D 2 -r f )b i2 E(r x )=0.1+(0.2-0.1)0.5+(8%-0.1)2=11% E(r y )=0.1+(0.2-0.1)1+(8%-0.1)1.5=17% E(r z )=0.1+(0.2-0.1)1.5+(8%-0.1)1=23%

18. 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 w x =0;w y =1;w z =0 Risk factor 1= 1(1.0)=1 Risk factor 2= 1(1.5)=1.5 Original r p =(0.11+0.25+0.23)/3=19.67% New r p =0(11%)+1(25%)+0(23%)=25%