19.5. TECHNICAL NOTES 737
19.5 Technical Notes xx CONTENTS
Technical Note 19.1 The efficiency condition
Let the desirability of the portfoliopbe denoted byVp=V(E( ̃rp−r),var( ̃rp)). The optimum is
found by setting, for each risky assetj, the derivative ofVpw.r.t.xjequal to zero. The effect of a
small change inxjonVpworks through two channels: the expectation, and the variance; so below
we seexj’s effect onVpvia the mean, and similarlyxj’s effect onVpvia the variance. In the second
line we fill in the effect ofxjon mean and variance, Equations [19.16] and [19.17]:
0 = ∂x∂V
j
=∂∂VE()∂∂xE()
j
+∂∂Vvar()∂var()∂x
j
,= ∂∂VE()E( ̃rj−r) +∂var()∂V 2cov( ̃rj,r ̃p);
⇒0 = E( ̃rj−r)−λpcov( ̃rj, ̃rp), (19.40)whereλp:=− 2 ∂V/∂∂V/∂var()E().This is a positive number since a higher variance lowers the desirabilityV
while a higher expected return increases it. Crypto-mathematicians recognize this ratio of partial
derivatives as the implicit derivative (or marginal trade-off) of mean for variance in the chosen
solution.