Document Remains Under Development *** WinORS Mean ...€¦ · WinORS Mean-Variance Efficient Set...

*** Document Remains Under Development ***

WinORS Mean-Variance Efficient Set See ARMDAT, chapter 7 for definitions and use of the equity

Browsing The Equity / Hedge Portfolio via the Web

After the equity/hedge portfolio has been posted to the web database, you may browse the current results for this portfolio. The web portfolio login

screen requires two entries: a) the username – email used to register the portfolio when posted to the web and b) the password (case sensitive). The

Online Now! browser screen appears as follows:

WinORS Mean-Variance Efficient Set Page: 2

Most Recent Release Date: 22-FEB-13 All Rights Reserved

View Web Portfolio From an Internet Browser

From any Internet connected browser, enter the following URL: www.nkd-

group.com . On the left-side menu, choose the link for Online Now! View Web Portfolio From Within WinORS

Choose the following menu sequence: TOOLS | BROWSE WEB PORTFOLIO.

Quadratic Optimization and the Markowitz Efficient Set

In this section we explain the process of producing the efficient set by solving the canonical form of the Markowitz quadratic optimization problem. The

canonical form of a constrained quadratic program is stated such that the objective function is comprised of squared and cross-product terms. There

are at least two constraints for this form of the efficient set problem. One constraint forces full investment (100% allocation). The other constraint

forces the portfolio diversification to achieve a portfolio required rate of return. Don’t worry; you will not be required to understand the intricacies of the method. However, you will be required to look at a tab that has the full

mathematical specification.

Begin by solving for the Markowitz efficient set (we note that more advanced users have the option of solving a number of alternative model specifications that are not discussed here). Choose the menu sequence as exemplified

below:

http://www.nkd-group.com/

http://www.nkd-group.com/



The result of executing the menu sequence is to produce the dialog box as displayed below. The default settings are adequate for a basic solution.

Tabbed Dialog Box

Notice the tabs at the top. The first tab is used to control the acquisition of period return data. The next four tabs (not all are shown) are used to set various versions of the efficient set generating model. The last tab is used to

set various options (although the default settings will generally be satisfactory for most users).



Returns from Equity Portfolio Tab

Portfolio metrics are always computed from a time-series of returns. The time series of returns are obtained from the actual prices of the securities

under consideration. Period Return calculations in WinORS are obtained as follows:

1. Historical data is extracted from the first available observation in year

2000 through the last available observation in the end year (e.g., 2012).

2. The extracted price data is converted to period returns (rit) by one of

two methods: a) normal (see equation a) or log differenced (see equation b):

a. rit = [(Pit – Pit-1) + Dit]/ Pit-1 b. rit = Ln[Pit ]– Ln[Pit-1] .

Di is the cash dividend for the ith firm paid during observation period t.

The Time Scale group determines whether the study will be done using daily or monthly period returns. The download algorithm always extracts daily

data. Hence, when a more granular time scale (monthly) is desired for your analysis you must indicate whether to multiply the data by 100 (for readability). No choice leaves the data in its original units.

The Type of Monthly option is used to select the method by which to convert

daily data to a monthly time dimension. The Decisions for Returns Tab group is used to control the tab insertion of

the period return data. Use Existing Data is an option if you have previously created a returns

tab. This choice avoids the time-consuming step of downloading and calculating new returns data.

Overwrite Data is the appropriate choice when you wish to calculate a

new set of period of returns and replace the existing returns tab (memory saving feature).

Insert new tab will always place the current calculation of returns on a new tab.

In most cases the user will choose to insert the data on a new tab. If this process has been done at an earlier step, the user has the ability to Use

Existing Data; a choice that will avoid the download of data unnecessarily (an informational dialog box will appear).

Coming down the dialog box the user encounters the informational statement (in red):

NOTE: Returns for the S&P 500 market index are appended automatically



Although returns for a market index are not required for the basic mean-

variance (Markowitz) efficient set, other calculations do require a market index (e.g., calculations for Vasicek beta). The market index is the real-time

market version of the S&P 500. The ticker symbol is GSPC. At this point it is possible to click on the Go Solve button to generate the

basic mean-variance efficient set. However, advanced users may find it desiireable to also set up one or more advanced portfolio modeling

alternatives. These are discussed in a later section. For now: Click on Go Solve.

The process starts by downloading data (time to completion is dependent on

the speed of your Internet connection). The process will end with the presentation of the following dialog box with the period returns in the background.

Observation

You will note that row 2 of the WinORS spreadsheet contains a 1-letter code. For all securities that will be “Included” in the solution you will observe the letter “I”

in the row 2 cell. The market index (the last column) will have the letter code “M”.

When 30-percent or more of the data is missing (controlled by the Yahoo finance database) row 2 will be

blank (no letter indicator). This means that the security will NOT be a component in the mean-variance optimization.

The Dialog Box

The check-box for Auto Load Efficient Frontier produces a graphical display of the efficient set upon the conclusion of all calculations.

The check-box for “Insert Optimal Percentage into Equity Portfolio Sheet” will allow you to indicate which portfolio on the efficient set you wish to insert

into the Equity Portfolio template in columns XX-XX. You may consider this an optimal reference portfolio. NOTE: this check box is “off” by default.



Simply click on OK to produce a chart display of the efficient set. Additionally, a new tab will be inserted to present the financial characteristics of each Markowitz corner portfolio.



Hint: Would you like to turn off the ticker symbols in the chart? Follow these steps:

1. Double click on chart 2. Select the series "Securities" -- the one with the triangle from the

Chart | Series tab 3. Switch to the "Series" tab -- the one next to the Chart tab 4. Choose the "Marks" tab

5. Choose the "Style" tab 6. Click "Visible" off

Insert an Optimal Portfolio into the Equity Portfolio Template

With a solution efficient set at hand you may now choose one of the

portfolios and its diversification pattern to insert into the equity template. Browse the efficient corner portfolios that define the efficient frontier. You

may find a portfolio with an expected rate of return that approaches the goal you have set for the managed portfolio. To track this particular portfolio as a

reference portfolio execute the following menu sequence (/AFPCE):



The following dialog box appears so that you may choose the desired reference portfolio to insert.



Note that the “Efficient Portfolio” to insert is keyed to the column the mouse pointer is in. In the above case that is column “I”, or, Markowitz efficient

portfolio #8. If this not the desired portfolio then iterate the “Efficient Portfolio” dialog box to the desired column letter.

The “Beta” reported on the Equity Template is the Yahoo Finance beta by default. You have the choice of replacing that beta with the beta internally

computed from your choice of market index over the time-frame of this analysis.

The “Overwrite” check-box controls whether you wish to replace a previously inserted optimal portfolio.

The “Price Now” check-box will automatically present the WinORS portfolio

price retrieval dialog box.

Click OK. After performing a price valuation update the enhanced Equity Portfolio Template will be displayed.

The Updated Equity Template

The security diversification allocation of the designated optimal portfolio has been inserted into column AB of the Equity Portfolio.



Reference portfolio effects are created in columns AC – AE. Using the starting value of the equity portfolio (column X) of $161,355.84, WinORS will

compute the share allocation of the optimal portfolio so that the starting value of the efficient portfolio closely matches that of the managed portfolio.

See the value in column AD of $161,355.84. The share distribution used to obtain this value is based on the values displayed in column AC.

Subsequent price updates over future time periods will lead to slight changes in the optimal diversification pattern. These value oriented changes can be

observed on a current basis by viewing column AE.

Advanced Quadratic Programming Models In this section the focus is on how to produce a new spreadsheet tab that presents the canonical statement of a chosen constrained quadratic

programming model used to produce and efficient set. The presentation begins with a demonstration of how to set up the standard mean-variance

model of Markowitz. Return to the Efficient Portfolio Dialog box: /AFPE.

Do not change the calculation of returns. That is, click the Yes button:

The Portfolio Model Selection dialog box appears next.



Immediately click on the tab that reads: Markowitz Models (QP). QP is

the abbreviation for the term quadratic programming.

To produce the canonical QP version of the mean-variance model, select the first check-box option: Estimated Correlations Constraint Quadratic Model (QEC). More advanced models are discussed in a later

chapter.

NOTE: the abbreviation QEC is important. This abbreviation will be referred to later when you must choose which solution to insert into the spreadsheet.

The Portfolio Solution Range dialog box appears next. You have seen this

dialog box earlier. The default selections have not changed; however, note that by leaving the “Auto Load Efficient Frontier” selection in the

on state you will generate a duplicate graph of the original

efficient frontier. Because an optimal portfolio has already been

inserted into the Equity Portfolio template, the default state of “off” for the insertion option is just fine.

Click OK.

To view the QEC model it is necessary to load the file from a disk file. Advanced

calculations are computed and stored on disk so that you can selectively choose

what to view and when. Execute the menu tree: /SC. The Solutions File List dialog box appears.

Open up the tree by clicking the + sign

indicator as follows:

Finance Methods

Portfolio Management Markowitz QP Models

Then select QEC (highlight and click OK). A new tab is inserted with a QEC label.



Suggestion: The variance-covariance matrix contains many small numbers. To see actual values increase the number of significant digits after the

decimal point.

Highlight the spreadsheet (/EA) Format cells by menu tree /OC followed by setting significant digits to

5 and clicking the radio button for “Scientific”

Produce a “True” QP Solution

To produce the Efficient Set using the canonical form of the problem you will

find that the software iterates down the efficient set starting from the MRP. Use the menu tree: /AFPQS. For now, accept the default entries on the range dialog box.

After some blinking and computing the following solution characteristics are

produced:

A new tab, QEC Portfolios, is inserted with a simulation of 10 portfolios

from the MRP to MVP. The Efficient Set produced by the simulation of the QP model is

appended to the efficient set graph. The two should be close, but owing to numerical calculations the two will not be exact.



The Simulated Portfolios form the QP QEC Model

The Efficient Set Charts with Both Computations

Actual Efficient Sets Zoom to Demonstrate Proximity

Efficient Frontier

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Adding Policy Constraints to the Efficient Portfolio Set

This section is intended for advanced user only.

A constraint measures by how much a decision maker faces a reduction in the freedom to exercise choice in the allocation of scarce resources. Policy

constraints are those limitations imposed by law, regulation, or managerial choice. In this section we learn how to state and incorporate policy constraints to limit the enumeration of the efficient frontier.

To implement policy constraints it is necessary to insert the Markowitz full-

specification model into the spreadsheet. To do this, execute the following menu sequence.

Adding a Specific Constraint

Begin by choosing the menu tree /AFPQP:



The dialog box for the policy constraint wizard appears as follows:

The policy constraint is appended to the next available row in the constraint section of the QP model formulation. Note how the constraint is given a

descriptive name in column A. In this case a Service sector constraint that sets the maximum percentage investment to 25% has been appended.



Below, it is possible to see the latter part of the spreadsheet. In this section both the sense of the constraint as well as the right-hand-side (RHS) are

displayed.

You may now solve the constrained QP problem.



Below is the dialog box for the menu option. The range and Inert On box

have been pre-set for best results. The SIMULATE input is an iterate control. This input is used to control the number of intermediate portfolios created between the maximum rate of return portfolio (MRP) and the minimum

variance portfolio (MVP).

The result of clicking on OK is to produced the constrained efficient portfolio solution. In part, it is displayed as a constrained efficient set. Note how the

blue line (constrained efficient set), is below and interior to the unconstrained efficient set. We know this must be the case as one of the service firms that was impacted by the constraint is EBAY. Hence, it is

impossible for the two efficient sets to have the starting point (MRP).



End.

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