Optimization and portfolio selection 171
Appendix: Formal definitions and procedures
Overview
Our portfolio selection is based on several criteria. Some subjective, but most
based on the results of an analysis of return data. Each fund has a style blend
described by a portfolio of passive indexes obtained by a quadratic fit of return
data. As a result, we are able to compare the fund with its style blend. We summa-
rize this comparison with a difference in risk-adjusted returns called the DTR α.
The DTR α is a measure of added value and is key to our final selection process.
However, before making this calculation, possible funds are carefully screened.
One part of the screening process is based on an analysis of risk and return sta-
tistics defined in terms of a DTR. The purpose of this appendix is to give formal
definitions to these statistics and explain how they are estimated.
The DTR
The DTR is the annualized rate of return required from investments to be able
to support expenditures during retirement. This rate of return depends on
many factors, including life expectancy, retirement contributions, and retire-
ment income from sources other than investments. As many of these factors are
not known, this rate can only be approximated. This is especially true if retire-
ment is many years in the future. We have developed a calculator to help an
investor determine a reasonable combination of retirement contribution level,
retirement age, and DTR. This calculator is available on the Sortino Investment
Advisors web site http://www.sortinoia.com.
About Example 3
The calculations for Example 3 are based on assuming a discrete probability
function of returns determined by the sample of returns. Here are the formal
definitions in this case.
The basic statistics — discrete version
Let p ( x ) be the discrete probability function of returns and T DTR.
Upside Probability∑xT px( )
Upside Potential∑xT ()()xTpx
Downside Deviation ∑xT ()()Txpx^2