Ralph Vince - Portfolio Mathematics

The Geometry of Leverage Space Portfolios 339

we are discussing the sum of thef values of the components. Sincefis
only applied on the active portion of our portfolio in a dynamic fractionalf
strategy, we can state that the hedge ratio of the portfolio,H, equals:

H=

(

∑n

i= 1

fi

)

∗

active$

total equity

(10.06a)

where: H=The hedge ratio of the portfolio.

fi=Thefvalue of theith component in the portfolio.

active$=The active portion of funds in an account.

Equation (10.06a) gives us the hedge ratio for a portfolio being traded
on a dynamic fractionalfstrategy. Portfolio insurance is also at work in a
static fractionalfstrategy, only the quotient active$/total equity equals 1,
and the value forf(the optimal f) is multiplied by whatever value we are
using for the fraction off. Thus, in a static fractionalfstrategy, the hedge
ratio is:

H=

(n

∑

i= 1

fi

)

∗FRAC (10.06b)

We can state that in trading an account on a dynamic fractionalfbasis,
we are performing portfolio insurance. Here, the floor is known in advance
and is equal to the initial inactive equity plus the cost of performing the
insurance. However, it is often simpler to refer to the floor of a dynamic
fractionalfstrategy as the initial inactive equity of an account.
We can state that Equation (10.06a) or (10.06b) equals the delta of the
call option of the terms used in portfolio insurance. Further, we find that this
delta changes much the way a call option, which is deep out of the money
and very far from expiration, changes. Thus, by using a constant inactive
dollar amount, trading an account on a dynamic fractional fstrategy is
equivalent to owning a put option on the portfolio which is deep in the
money and very far out in time. Equivalently, we can state that trading a
dynamic fractional fstrategy is the same as owning a call option on the
portfolio which doesn’t expire for a very long time and is very far out of the
money.
However, it is also possible to use portfolio insurance as a reallocation
technique to steer performance somewhat. This steering may be analogous
to trying to steer a tanker with a rowboat oar, but this is a valid reallocation
technique. The method initially involves setting parameters for the program.
First, you must determine a floor value. Once chosen, you must decide