Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

172 Optimizing Optimization

Upside Potential RatioUpside Potential/Downside Deviation



   





()()

()()

xTpx

Txpx

xT

xT

∑

∑

2

The basic statistics — continuous version

Rather than using a discrete probability function and following a procedure
similar to that of Example 3 to estimate our statistics, we fit a 3  parameter
lognormal distribution by using a bootstrap procedure on historical data. First,
we give the general formulas for a probability density function and then we
will consider the special case of the lognormal. A more expansive justification
and further details for using the lognormal is given in Chapter 4 of Managing
Downside Risk in Financial Markets.
Let f ( x ) be the probability density function of returns and T  DTR. As
explained in the next section, we use a 3  parameter lognormal density function,
and in that case there are analytic expressions for each of the following integrals.

Upside Probability 

f()xdx

∫xT

Upside Potential∫xT ()()xTxdxf

Downside Deviation 

()()Tx xdx

xT

(^2) f
∫
Upside Potential RatioUpside Potential/Downside Deviation

Estimating the probability of returns with a

3  parameter lognormal

The formulas for the lognormal are not pretty. They are generally described in
terms of the parameters for the underlying normal. So, with the logarithmic
translations, they can get a bit involved. The resulting basic formulas are col-
lected together below. So for now, we will not concern ourselves with these
technical details. What we will do is describe the essentials.
First , what are the three parameters? There is some choice in selecting the
parameters. We made these selections so that the meanings of the parameters
would be easily understood in terms of the annual returns. The parameters are