738 CHAPTER 19. SETTING THE COST OF INTERNATIONAL CAPITAL
Technical Note 19.2 Using Ito’s Lemma to transcribe the fc efficiency
condition.
Start by relating thecadreturn onjto theusdreturn: 1 + ̃rj= (1 + ̃r∗j)(1 + ̃s), with ̃s= ∆S/S
andSiscad/usd. Solve for ̃r∗jand Taylor-expand as follows:
r ̃j∗=1 + ̃1 + ̃rsj− 1 ≈ ̃rj− ̃s−[ ̃rj ̃s] + ̃s^2 , (19.41)A readily acceptable result of Ito’s Lemma is that, for shorter and shorter holding periods, products
of three or more returns become too small to matter. This, firstly, justifies the above second-order
expansion. It also means that if we consider covariances of twofcreturns we only need to look at
first-order terms:
cov( ̃rj∗, ̃r∗k) ≈ cov( ̃rj−s, ̃r ̃k−s ̃),
= cov( ̃rj,r ̃k)−cov( ̃rj, ̃s)−cov( ̃rk, ̃s) + var( ̃s). (19.42)because all the other terms would lead to products of three or four returns.
One often reads that also inside an expectation only the first-order terms matter, because
products of returns are second order of smalls. But this is patently wrong. Indeed, variances and
covariances of returns are averages of products of two returns, but this surely does not mean that
they can be set equal to zero. Now the expectation of, say, the third term is
E( ̃r∗j ̃s) = E( ̃r∗j) E( ̃s) + cov( ̃r∗j, ̃s). (19.43)If we let the periods over which one observes return become shorter and shorter, all means and
all (co)variances shrink roughly in proportion to the time interval ∆t, so they preserve the same
relative order of magnitude relative to each other. But this means that the product of two means,
E( ̃r∗j) E( ̃s), shrinks to zero much faster than the covariance. That is, the product of two means is
second order of smalls but the covariance is not:
E( ̃r∗j ̃s) ≈ cov( ̃rj, ̃s) , and (19.44)
E( ̃s^2 ) ≈ var( ̃s); (19.45)Using the above in Equation [19.41], we get the following translated expected return:^16
E( ̃r∗j)≈E( ̃rj)−E( ̃s)−cov( ̃r∗p∗, ̃s) + var( ̃s). (19.46)Our results [19.46] and [19.42] for the translated mean and variance imply that the efficiency
condition [19.27] translates into the first equation below. We next write that equation for the
special case where assetjis thehcrisk-free asset, and lastly we subtract:
E( ̃rj)− E( ̃s) −cov( ̃rj,s ̃) +var( ̃s) =λ[cov( ̃rj, ̃rp∗) −cov( ̃rj, ̃s) −cov( ̃rp∗,s ̃) +var( ̃s)]
r− E( ̃s) − 0 +var( ̃s) =λ[0 − 0 −cov( ̃rp∗,s ̃) +var( ̃s)]
E( ̃rj)−r −cov( ̃rj,s ̃) =λ[cov( ̃rj, ̃rp∗) −cov( ̃rj, ̃s)] ,which leads to [19.32].
(^16) Note, in passing, how we find back our earlier numerical result that covariance betweencad
asset return and thecad/usdexchange rate lowers the expectedusdreturn. We also discover that
exchange risk has its impact on the expected return too. So both the covariance and the variance
have both ’good’ and ’bad’ aspects.