Classical Portfolio Construction 253
The first four values, the values for X 1 through X 4 , tell us the weights
(the percentages of investable funds) that should be allocated to these in-
vestments to achieve this optimal portfolio with a 14% expected return.
Hence, we should invest 12.391% in Toxico, 12.787% in Incubeast, 38.407%
in LA Garb, and 36.424% in the savings account. If we are looking at investing
$50,000 per this portfolio mix:
Stock Percentage (∗ 50 , 000 =) Dollars to InvestToxico .12391 $6,195.50
Incubeast .12787 $6,393.50
LA Garb .38407 $19,203.50
Savings .36424 $18,212.00Thus, for Incubeast, we would invest $6,393.50. Now assume that In-
cubeast sells for $20 a share. We wouldoptimallybuy 319.675 shares
(6393.5/20). However, in the real world we cannot run out and buy frac-
tional shares, so we would say that optimally we would buy either 319
or 320 shares. Now, the odd lot, the 19 or 20 shares remaining after we
purchased the first 300, we would have to pay up for. Odd lots are usu-
ally marked up a small fraction of a point, so we would have to pay extra
for those 19 or 20 shares, which in turn would affect the expected return
on our Incubeast holdings, which in turn would affect the optimal port-
folio mix. We are often better off to just buy the round lot—in this case,
300 shares. As you can see, more slop creeps into the mechanics of this.
Whereas we can identify what the optimal portfolio is down to the fraction
of a share, the real-life implementation requires again that we allow for
slop.
Furthermore, the larger the equity you are employing, the more closely
the real-life implementation of the approach will resemble the theoretical
optimal. Suppose, rather than looking at $50,000 to invest, you were running
a fund of $5 million. You would be looking to invest 12.787% in Incubeast (if
we were only considering these four investment alternatives), and would
therefore be investing 5,000,000∗.12787=$639,350. Therefore, at $20 a
share, you would buy 639,350/20=31,967.5 shares. Again, if you restricted
it down to the round lot, you would buy 31,900 shares, deviating from the
optimal number of shares by about 0.2%. Contrast this to the case where
you have $50,000 to invest and buy 300 shares versus the optimal of 319.675.
There you are deviating from the optimal by about 6.5%.
The Lagrangian multipliers have an interesting interpretation. To begin
with, the Lagrangians we are using here must be divided by .5 after the