740 CHAPTER 19. SETTING THE COST OF INTERNATIONAL CAPITAL
Technical Note 19.4 Identifyingλandκ.
Write the equation in matrix form,
E( ̃rj−r) = [cov( ̃rj,rw) , cov( ̃rj, ̃s)]»
λ
κ- . (19.48)
To identifyλandκwe write this for two benchmarks, the world market portfolio with returnrw
and theusdT-bill with returnr∗+ ̃s;
»
E( ̃rw−r)
r∗+ E( ̃s)−r
- =
»
var( ̃rw) cov( ̃rw, ̃s)
cov( ̃rw, ̃s) var( ̃s)- ×
»
λ
κ- ; (19.49)
⇒»
λ
κ- =
»
var( ̃rw) cov( ̃rw, ̃s)
cov( ̃rw,s ̃) var( ̃s)- − 1 »
E( ̃rw−r)
r∗+ E( ̃s)−r
. (19.50)
This can be substituted back into [19.48]. Now the covariance matrix of ( ̃rw, ̃s) premultiplied by the
vector of covariances ofrjwith these same variables ( ̃rw, ̃s) is the row vector of multiple regression
coefficients ofrjonto ( ̃rw,s ̃)—a generalisation ofb= cov( ̃y,x ̃)×var( ̃x)−^1 in ̃y=a+bx ̃+ ̃e:
E( ̃rj−r) = [cov( ̃rj,rw) , cov( ̃rj, ̃s)]»
var( ̃rw) cov( ̃rw, ̃s)
cov( ̃rw, ̃s) var( ̃s)- − 1 »
E( ̃rw−r)
r∗+ E( ̃s)−r
= [βj,w;s,γj,s;w]»
E( ̃rw−r)
r∗+ E( ̃s)−r- . (19.51)