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).

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!

An Internal Rate of Return exercise...

A CIPM Expert Level candidate sent me the following question, asking what are the proper keystrokes to obtain the solution:

The Millers deposited $50,000 into their account on 1 May 2005 and another $40,000 on 1 July 2005. The portfolio also received and reinvested dividends of $30,000 on 1 July, plus another $30,000 on 31 December. The Miller's investment adviser, Greenbush Investments, uses a daily pricing system that shows account values (inclusive of dividends and contributions) of $2,375,000 and $2,460,000 on 1 May and 1 July, respectively. The account was valued at $2,225,000 on 1 January 2005 and at $2,445,000 on 31 December 2005.

What is the annual internal rate of return?

First, I suggest that readers of this blog review my post from a few months ago suggesting a series of steps to solve internal rate of return problems. That post is here.

Next, we should identify the important information in this problem; i.e., the cash flows that must be entered into the calculator - and those that should be ignored. The important cash flows are:

• the initial market value of $2,225,000 on 1/1/2005
• the contribution of $50,000 on 5/1/2005
• the contribution of $40,000 on 7/1/2005
• the ending market value of $2,445,000 on 12/31/2005

You can ignore the dividends that are described because they are "reinvested" - this means they remain in the portfolio and are not an external cash flow. If they were "not reinvested," that would mean that they should be treated as withdrawals at the time of payment... but that is not the case here.

You can also ignore the other valuations that are given. With internal rate of return calculations, only the initial value and the ending value are needed.

In order to enter this into your financial calculator, you will need to evenly space the cash flows (in time) and "zero fill" the empty periods. In this problem, you can assume monthly occurring cash flows if you treat the initial market value as being for 12/31/2004, and the contributions as occurring on 4/30/2005 and 6/30/2005 (rather than 5/1/2005 and 7/1/2005).

The "zero filled" cash flows will be on the following dates: 1/31, 2/28, 3/31, 7/31, 8/31, 9/30, 10/31 and 11/30.

Thus, the following keystrokes may be used (TI BA II Plus calculator):

[CF][2nd][CLR WORK] Clears cash flow worksheet

-2225000[ENTER] Enters 2,225,000 as CF0

[down arrow] 0 [ENTER] Enters 0 as CF1

[down arrow] 3 [ENTER] The frequency of this flow is three times

[down arrow] -50000 [ENTER] Enters 50,000 as CF2

[down arrow] 1 [ENTER] The frequency of this flow is once

[down arrow] 0 [ENTER] Enters 0 as CF3

[down arrow] 1 [ENTER] The frequency of this flow is once

[down arrow] -40000 [ENTER] Enters 40,000 as CF4

[down arrow] 1 [ENTER] The frequency of this flow is once

[down arrow] 0 [ENTER] Enters 0 as CF5

[down arrow] 5 [ENTER] The frequency of this flow is five times

[down arrow] 2445000 [ENTER] Enters 2,445,000 as CF6

[down arrow] 1 [ENTER] The frequency of this flow is once

[IRR][CPT] Computes the IRR


At this point, the calculator should tell you the solution is 0.4636%. But, this is a monthly return, because the spacing of our cash flows was monthly. We now need to convert this to an annual return. To do this, do the following steps:

• divide by 100 (converting the percentage to a decimal)
  
• add 1 (creating a wealth relative)
• raise the result of the last step to the 12th power
• subtract 1
• multiply by 100

This should give you an annual return of 5.70%

Steps for calculating IRR

The following steps may be used to calculate internal rate of return using one of the financial calculators (HP 12C and TI BA II Plus):

1. Identify all external cash flows, including beginning and ending market values, and arrange them in chronological order.

2. Arrange the cash flows so that they are evenly spaced with respect to time. The financial calculators expect cash flows to occur at a regular frequency, so the cash flows entered must be evenly spaced, with zero cash flows used to fill in any missing data. I will describe how this can be done below.

3. Create the present value equation, equating the sum of the present value of outflows (beginning market value and any contributions) with the sum of the present value of in-flows (ending market values, and any withdrawals). I recommend ordering cash flows on each side of the equation in chronological order, from the first through to the last.

4. Identify simultaneously occurring cash flows. These are cash flows that occur on the same date, including both outflows and in-flows. These will be netted for purposes of entry into the financial calculator.

5. Using the appropriate keystrokes, enter the cash flows into the financial calculator’s cash flow worksheet, and compute the IRR.

6. The solution obtained in step 5 will be an internal rate of return for the interval of the cash flow spacing (step 2). Using exponents, convert the internal rate of return to a return for the desired period.

Principles Level: Setting up your calculator

(Note: this tip applies to Expert Level candidates as well, but I presume that if you passed your Principles Level exam you must have had the calculator set the way you expect it to work!)


I generally recommend that candidates make sure to check two settings in particular on their calculators prior to the exam:

- precision
- order of operations

With respect to precision, my recommendation is that you set the number of decimal places that your calculator uses to the maximum value possible. I think it is better for you to see the answer in as "pure" a form as possible, and then you can choose to round or truncate your result, as necessary. For the Texas Instruments BA II Plus, for example, you can set the number of decimal places anywhere from 0 to 9 places.

As for order of operations, keep in mind that there are two different methods:

- chain (essentially left to right)

- algebraic

The chain mode of operation means that the calculator will perform calculations as they are entered (i.e., a literal "left to right" mode).

The algebraic mode of operation uses the following hierarchy:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

People often use the acronym PEMDAS (Please Excuse My Dear Aunt Sally) to help them remember the hierarchy under the algebraic mode. Another (less witty) form of the mnemonic device is BEDMAS (Brackets Exponents Division Multiplication Addition Subtraction). Use whatever works!

To illustrate the difference between chain mode and algebraic mode, consider the following expression: 3 + 5 * 4.

Under chain mode, the solution is 32. (Obtained by first adding 5 to 3 to get 8, then multiplying 8 times 4).

Under algebraic mode, the solution is 23. (Obtained by multiplying 5 times 4, then adding 3).

I hesitate to give a recommendation with respect to mode of operations - you should use whatever makes sense for you. Personally, I can't see why anyone would use the chain mode unless that is the way their brain works - and if it works, I don't see a reason to change.

Most financial calculators default to the chain mode, whereas scientific calculators tend to default to the algebraic mode. All of the "allowed" calculators for your CIPM exams are financial, of course. Thus, given that (I think) most people tend to think and use in algebraic mode, you probably need to change your calculator's settings to get it to work the way you want it to on the exam.