Turn It On Again: Standard Deviation using Stats Worksheet on TI BA II Plus

Last week I taught our CIPM prep classes for the Principles and Expert Levels, and the commonly asked question about the fastest way to do standard deviation came up.

I will take this opportunity to refer to an old blog post (found here) that covered how to this using the statistics worksheet on the TI BA II  Plus calculator - one of the allowable calculators for the CIPM exams and CFA exams.

This post happens to be the most widely read post on my blog, probably because it is of interest beyond the CIPM exams - making it a "greatest hit" of a sort.

The post is useful for calculating:

• standard deviation of portfolio returns within a composite (internal dispersion)
• standard deviation of composite returns (external dispersion)
• tracking error

I caution candidates that I don't recommend you use the statistics worksheet for downside deviation calculations - I don't know of a way to easily deal with the target return (if someone does, please let me know).

Standard Deviation as Internal and External Dispersion: When Is It Required?

It has actually become apparent to me during the course of conducting GIPS verifications that there is some confusion as to when internal dispersion and external dispersion must be shown.  It occurred to me that CIPM candidates may have the same confusion.

Internal dispersion is a required presentation element of GIPS compliant composite presentations, as required by provision 5.A.1.i.  Provision 4.A.8 also requires compliant firms to disclose what measure of dispersion is shown.

External dispersion is a required presentation element of GIPS compliant compliant composite presentations for periods ending on or after January 1, 2011, for both the composite and the benchmark.  This is required by provision 5.A.2.  An additional risk measure (using the same periodicity) is required if firms feel that the three-year annualized ex-post standard deviation is not relevant or appropriate.

So what is the difference between internal dispersion and external dispersion, both in terms of what they measure and when they are required?

Internal dispersion is a measure of the range of returns within the composite within each annual period.  Firms are only required to show this when the composite has more than five portfolios (firms can, of course, opt to show internal dispersion when the composite has five or fewer portfolios).  The measure should only include portfolios that were in the composite for the full annual period.  Firms can choose what measure of internal dispersion they show (with the exception of real estate composites, where the required measure of dispersion is the range of returns; i.e., the difference between the high and low returns).  Some commonly used measures of internal dispersion are:

• standard deviation
• range of returns
• quartiles, quintiles, deciles, interquartile ranges

Standard deviation is probably the most commonly used measure of internal dispersion.

External dispersion is also a measure of the range of returns.  But in this case we are measuring the range of a given composite's past returns (over the most recent 36 months). It is meant to be a measure of the composite's risk.  External dispersion is required for any periods ending on/after January 1, 2011, regardless of the number of portfolios in the composite.  This is probably the single biggest misconception about the external dispersion that I have seen - many firms think they don't have to show it if the composite has five or fewer portfolios.  But even for a single portfolio composite, the external dispersion must be shown.  The only exception to this is for real estate composites (where section I.6 of the GIPS standards apply) and private equity composites (where section I.7 of the GIPS standards apply).

Also, external dispersion must be shown as the three-year ex-post annualized standard deviation (based on monthly returns).  So firms don't have a choice in what measure they show, unless they feel the standard deviation is not relevant or appropriate.  If firms feel this is the case, they must still show the three-year ex-post annualized standard deviation - and then they can show the risk measure they feel is more appropriate and/or relevant to their composite. 

One more point of comparison is the return that a reader of the presentation should look at in comparison to the dispersion measure.

• internal dispersion is a measure of the range of portfolio returns within the composite, and should be looked at in comparison to the composite's annual return for that period, which is really a weighted average of the portfolio returns in the composite.
• external dispersion is a measure of the range of composite returns over time, and should be looked at in comparison to the three-year annualized return for that period, which is really a geometric average of the composite returns over time.  Note that firms are not required to show the three-year annualized return, but it is recommended by GIPS provision 5.B.4

 Long post, but hope this adds some clarity on the subject!

Calculating Standard Deviation Using the Stats Worksheet (TI BA II Plus)

CIPM candidates are required to be able to calculate standard deviations for several purposes, including:

• the dispersion of annual portfolio returns within a composite (internal dispersion)
• the variability of a composite's past 36 months of returns (external dispersion)
• the ex-post variability of a portfolio's historical returns (standard deviation)
• the ex-post variability of a portfolio's historical excess returns vs. a benchmark (tracking error)

Calculating standard deviation can be, however, quite tedious with a calculator. If one does the calculation "long hand" style, it requires the following steps:

1. Finding the average of all return observations under consideration.
2. Measuring the distance of each return observation from the average return.
3. Squaring the distances from the average return.
4. Summing the squares.
5. Dividing the sum by the number of observations.
6. Taking the square root.

In order to save time, candidates that are using the Texas Instruments BA II Plus calculator can make use of the calculator's "Statistics Worksheet," which can calculate standard deviation with just a few keystrokes.

For example, consider the following history of returns:

• January 2010: 7.22%
• February 2010: 5.19%
• March 2010: 8.88%
• April 2010: 1.13%
• May 2010: 17.5%
• June 2010: 3.70%
• July 2010: 2.50%
• August 2010: 0.55%
• September 2010: -5.17%
• October 2010: 3.33%
• November 2010: 1.07%
• December 2010: 8.25%
• January 2011: 5.45%
• February 2011: 2.27%
• March 2011: 8.00%

To access the calculator's statistics worksheet, type [2ND] [7]. Note that the "2nd" function of the [7] key is "DATA."

The statistics worksheet remembers any past entries until they are cleared, so you may need to clear previous entries. If this is the case, once you have accessed the statistics worksheet, type [2nd] [CE/C] (note that the 2nd function of the [CE/C] key is "CLR Work").

When the calculator is ready, you should see the following on the display: "X01 0."

The calculator can accept two series of data: an "X" series and a "Y" series. I suggest that you use the "X" series. So, X01 is the first item, X02 is the second, and so on. Thus, you will need to skip the prompts for "Y" values.

Given this, the following keystrokes are required to enter the above returns into the worksheet:

7.22 [ENTER][DOWN ARROW][DOWN ARROW]
5.19 [ENTER][DOWN ARROW][DOWN ARROW]
1.13 [ENTER][DOWN ARROW][DOWN ARROW]
17.5 [ENTER][DOWN ARROW][DOWN ARROW]
3.70 [ENTER][DOWN ARROW][DOWN ARROW]
2.50 [ENTER][DOWN ARROW][DOWN ARROW]
0.55 [ENTER][DOWN ARROW][DOWN ARROW]
-5.17 [ENTER][DOWN ARROW][DOWN ARROW]
3.33 [ENTER][DOWN ARROW][DOWN ARROW]
1.07 [ENTER][DOWN ARROW][DOWN ARROW]
8.25 [ENTER][DOWN ARROW][DOWN ARROW]
5.45 [ENTER][DOWN ARROW][DOWN ARROW]
2.27 [ENTER][DOWN ARROW][DOWN ARROW]
8.00 [ENTER][DOWN ARROW][DOWN ARROW]

At this point, you are ready to do statistical calculations, including standard deviation. To access these calculations, type [2ND][8] (note that the "2nd" function of the [8] key is "STAT." The calculator should respond with "LIN" which indicates that the calculator is in "linear regression mode. This is what you need. The calculator also does other regression modes (exponential, logarithmic, etc.). If something other than "LIN" appears, type [2ND][ENTER] until you see "LIN."

At this point you may use the [DOWN ARROW] and the [UP ARROW] to cursor through the statistical calculations.

• The first item is the number of observations in the worksheet (in the "X" series).
• The next item is the average observation (in the "X" series).
• The next item is the sample standard deviation (in the "X" series).
• The next item is the population standard deviation, which is the item you need (in the "X" series).

If you have entered the keystrokes correctly, you should see that the standard deviation is 4.94.

Reading the keystrokes in this post may make it sound somewhat difficult, but if you practice this method, I am sure you will find it much faster than calculating standard deviations "long hand."

Having said this, I do recommend you calculate it both ways (using the statistics worksheet and "long hand") for the maximum learning experience.

Happy calculating!