When using the worksheet, you may change any inputs. To do so, click inside the appropriate box, then make your changes. When finished, click any area outside the boxes on the form.

You may copy inputs from other sources such as spreadsheets,
word processing documents, and other worksheets in this series,
then paste the results into the appropriate boxes on this form.
To copy an area from an Excel spreadsheet, select it, then select
**E**dit **C**opy. In the browser,
select a position in the input box, then select **E**dit
**P**aste. To copy an area from a box in the browser
to an Excel spreadsheet, select the text in the browser and
select **E**dit **C**opy. In the Excel
spreadsheet, select a position, then select **E**dit
**P**aste. This brings each row into the spreadsheet
as text. To convert to a matrix, select the column in which the
information is located (the left-most one shown), then select **D**ata
**T**ext-to-columns. Choose **Delimited**
and **Spaces** as delimiters and the information
will appear in the requisite number of cells.

When you save a page on your own disk (using the browser's **F**ile
Save **A**s command), only the original material in
the form will be saved. There are two ways to save and retrieve
worksheet information. You can copy the information you wish to
save to some other document, such as a spreadsheet, word
processing document or text file. You can also load the source
(ws_***.htm) file in a word processor and edit it to include your
inputs. You will find the default information in blocks marked
TEXTAREA and in the VALUE attributes of INPUT tags. Simply
replace the default values with your information, then save the
page as a file on your disk under any desired name.

When you change an input, the output area will generally be
cleared to avoid having old outputs appear simultaneusly with new
inputs. To produce new outputs, click the **PROCESS**
button.

The **Inputs** box should contain all the needed
input information other than the investor's rsk tolerance. Each
row provides the inputs for a single decision variable (for
example, an asset class). The columns must be in the order
indicated below. Each must be given a short heading, although
these are not actually checked for content.

The first three columns indicate the **Minimum**
proportions that may be invested in the items, the **Initial
**portions invested, and the **Maximum**
proportions that may be invested. The initial proportions will
usually sum to 1.0 although this is not necessary. Each initial
proportion must lie within or at an end of the range given by the
minimum and maximum allowed values. The optimizer will consider
only portfolios that are **feasible **(with each
proportion greater than or equal to its minimum value and less
than or equal to its maximum value) and for which the proportions
sum to the same value as the sum of the proportions in the
initial portfolio.

The next two columns provide the **Expected Returns **(ExpRet)
and **Standard Deviations** (StdDev) for the
decision variables. These are usually stated in terms of return
per year (for example, 10.5 for 10.5% per year). The remaining
columns provide the **Correlation Coefficients** for
the variables. The order must be the same as that used for the
rows in the table.

The next input box provides the investor's **Risk
Tolerance.** This indicates the marginal rate of
substitution of variance for expected return in the investor's
utility function. It is normally positive, but a zero value is
allowed for the special case in which the objective is to find
the feasible portfolio with the smallest possible standard
deviation.

The final input box provides the **Marginal Utility
Cutoff**.** **This is used to terminate the
algorithm. When the marginal utility of the best feasible swap
falls below this amount, the program stops. The smaller this
amount, the more precise the solution but the longer the time
required to obtain it.

The worksheet uses the gradient quadratic programming method
of Sharpe ["An Algorithm for Portfolio Improvement," ,
in Advances in Mathematical Programming and Financial Planning,
K.D.Lawrence, J.B. Guerard, Jr., and Gary D. Reeves, Editors, JAI
Press, Inc., 1987, pp. 155-170] to find the feasible portfolio
with the maximum possible **Utility**:

U

_{p}= e_{p}- (sd_{p}/t)

where:

U

_{p}= the utility of the portfolioe

_{p}= the portfolio's expected returnsd

_{p}= the portfolio's standard deviation of returnt = the investor's risk tolerance

All output is provided in a single box. Two tables are included. The first table shows the Initial Portfolio, the Optimal Portfolio and the Change (Optimal - Initial). The second table shows the characteristics of the Initial and Optimal Portfolio and the associated changes in the level of each one(Optimal - Initial). Unless the risk tolerance is zero, the third line shows the utility of the portfolios and the change therein.

You may enter any desired text in this box to describe the source of the input data, etc..

This version January 2, 1997

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