Ralph Vince - Portfolio Mathematics

The Geometry of Mean Variance Portfolios 283

Recall that the tangent portfolio is found by taking the portfolio along
the constrained efficient frontier (arithmetic or geometric) that has the
highest Sharpe ratio, which is Equation (8.01). Now we lever this portfolio
up, and we multiply the weights of each of its components by a variable
namedq, which can be approximated by:

q=(E−RFR)/V (8.13)

where: E=The expected return (arithmetic) of the tangent

portfolio.

RFR=The risk-free rate at which we assume you can borrow

or loan.

V=The variance in the tangent portfolio.

Equation (8.13) actually is a very close approximation for the actual
optimalq.
An example may help illustrate the role of optimalq. Recall that our
unconstrained geometric optimal portfolio is as follows:

Component Weight

Toxico 1.025955

Incubeast .4900436

LA Garb .4024874

This portfolio, we found, has an AHPR of 1.245694 and variance of
.2456941. Throughout the remainder of this discussion we will assume for
simplicity’s sake an RFR of zero. (Incidentally, the Sharpe ratio of this port-
folio, (AHPR−(1+RFR))/SD, is .49568.)
Now, if we were to input the same returns, variances, and correlation co-
efficients of these components into the matrix and solve for which portfolio
was tangent to an RFR of zero when the sum of the weights is constrained
to 1.00 and we do not include NIC, we would obtain the following portfolio:

Component Weight

Toxico .5344908

Incubeast .2552975

LA Garb .2102117

This particular portfolio has an AHPR of 1.128, a variance of .066683,
and a Sharpe ratio of .49568. It is interesting to note thatthe Sharpe ratio