Moreover, numerical values produced in interim calculations may have been rounded off when produced in a table and as a result when an operation is performed on the values shown in a table, the result may appear to be off. Just be aware of this. Here is an example of a common situation that you may encounter when attempting to replicate results.

Suppose that a portfolio has four securities and that the market value of these four securities are as shown below:

Assume further that we want to calculate the duration of this portfolio. This value is found by computing the weighted average of the duration of the four securities. This involves three steps. First, compute the percentage of each security in the portfolio. Second, multiply the percentage of each security in the portfolio by its duration. Third, sum up the products computed in the second step.

Let’s do this with our hypothetical portfolio. We will assume that the duration for each of the securities in the portfolio is as shown below:

Using an Excel spreadsheet the following would be computed specifying that the percentage shown in Column (3) below be shown to seven decimal places:

I simply cut and paste the spreadsheet from Excel to reproduce the table above. The portfolio duration is shown in the last row of Column (5). Rounding this value (5.985343) to two decimal places gives a portfolio duration of 5.99.

There are instances in the book where it was necessary to save space when I cut and paste a large spreadsheet. For example, suppose that in the spreadsheet I specified that Column (3) be shown to only two decimal places rather than seven decimal places. The following table would then be shown:

Excel would do the computations based on the precise percent of the portfolio and would report the results as shown in Column (5) above. Of course, this is the same value of 5.985343 as before. However, if you calculated for any of the securities the percent of the portfolio in Column (3) multiplied by the duration in Column (4), you do not get the values in Column (5). For example, for Security 1, 0.16 multiplied by 9 gives a value of 1.44, not 1.46815 as shown in the table above.

Suppose instead that the computations were done with a hand-held calculator rather than on a spreadsheet and that the percentage of each security in the portfolio, Column (3), and the product of the percent and duration, Column (5), are computed to two decimal places. The following table would then be computed:

Note the following. First, the total in Column (3) is really 0.99 (99%) if one adds the value in the columns but is rounded to 1 in the table. Second, the portfolio duration shown in Column (5) is 5.92. This differs from the spreadsheet result earlier of 5.99.

Suppose that you decided to make sure that the total in Column (3) actually totals to 100%. Which security’s percent would you round up to do so? If security 3 is rounded up to 34%, then the results would be reported as follows:

In this case, the result of the calculation from a hand-held calculator when rounding security 3 to 34% would produce a portfolio duration of 6.

Another reason why the result shown in the book may differ from your calculations is that you may use certain built-in features of spreadsheets that we did not use. For example, you will see in this book how the price of a bond is computed. In some of the illustrations in this book, the price of one or more bonds must be computed as an interim calculation to obtain a solution. If you use a spreadsheet’s built-in feature for computing a bond’s price (if the feature is available to you), you might observe slightly different results.

Please keep these rounding issues in mind. You are not making computations for sending a rocket to the moon, wherein slight differences could cause you to miss your target. Rather, what is important is that you understand the procedure or methodology for computing the values requested.

In addition, there are exhibits in the book that are reproduced from published research. Those exhibits were not corrected to reduce rounding error.