24 It is easily shown that W= BEAxi (12) 1-BExis) hence E{R}= Exi) (13) BE Analogously,the gross return on the riskless asset can be written as 11 RyBE (14) Since we have assumed the growth rate of consumption and dividends to be log normally dis- tributed, e4,+1120 E{R}= e1-a4,+1/2-aPa (15) and In E,{R}=-In B+au,-1/202o:+ao? (16) where u=E(Inx),=Var(Inx)and Inx is the continuously compounded growth rate of consumption. 1 Similarly R,= Be-ou,+12a (17) and In R,=-InB+au,-1/2a2o2 (18) In E(R.}-In R,=ao? (19) From (11)it also follows that In E{R)-In R,=ao.R. (20) where o:.g Cov(Inx,In R) (21) 19See Appendix A in Mehra(2003)24 It is easily shown19 that w E x E x t t t t = - + - + - b b a a { } { } 1 1 1 1 1 (12) hence E R E x E x t et t t t t { } { } { } , + + + - 1 = 1 1 1 b a (13) Analogously, the gross return on the riskless asset can be written as R E x f t t t , { } + + - 1 = 1 1 1 b a (14) Since we have assumed the growth rate of consumption and dividends to be log normally dis￾tributed, E R e e t et x x x x { } , / + ( ) /( ) + - +- 1 = 1 2 1 121 2 2 2 m s am a s b (15) and ln { } ln / E Rt et x x x , +1 =- + - + 22 2 b am a s as 1 2 (16) where mx = E x (ln ) ,s x Var x 2 = (ln ) and ln x is the continuously compounded growth rate of consumption. Similarly R e f x x = - + 1 1 2 2 2 b am a s / (17) and ln ln / Rf =- + - b am a s x x 1 2 2 2 (18) \ ln { } ln ER R e - =f as x 2 (19) From (11) it also follows that ln { } ln ER Rf x R, e e - = as (20) where s xR e e , = Cov x R (ln ,ln ) (21) 19 See Appendix A in Mehra (2003)