International Finance: Putting Theory Into Practice

19.5. TECHNICAL NOTES 739

Technical Note 19.3 Aggregating the two efficiency conditions.
We want to aggregate, and obtain the world-market return, which is defined as

̃rw=Wca ̃rp+Wus ̃rp

∗

Wca+Wus (19.47)

withWcaandWusdefined as the invested wealths, both measured incad, of Canada and theus,
respectively. To build this world return into the model we multiply both sides of [19.31] byWca,
and [19.32] byWus. On the right-hand sides of the equations below we have immediately put these
factors inside the covariances. Next we sum the two equations, and lastly we divide by total world
wealth and use [19.47]:

WcaE( ̃rj−r) = λcov( ̃rj, Wcar ̃p)

WusE( ̃rj−r) = λcov( ̃rj, Wus ̃rp∗) +Wus(1−λ) cov( ̃rj,s)

(Wca+Wus)E( ̃rj−r) = λcov( ̃rj,(Wca ̃rp+Wusr ̃p∗)) +Wus(1−λ) cov( ̃rj,s)

⇒E( ̃rj−r) = λcov( ̃rj,r ̃w) +WcaW+usWus(1−λ) cov( ̃rj,s).

For ease of manipulation, in [19.33] we denoteWus/(Wca+Wus)(1−λ) =κ.