MATLAB: Where can I find information on comparing Yield Quotations derived from the MATLAB Financial Toolbox to Yields from Other Vendors and Conventions

Financial Toolbox

I am comparing yields calculated from the BNDYIELD function to the ISMA yields. I am converting the semi-annual yield obtained from BNDYIELD to an annual yield that should match the ISMA yields. I am using the following transformation:
annual_yield =  (1 + semi_annual_yield / 2) ^ 2 - 1
For most bonds, this transformation produces results that match the ISMA yields. However, certain bonds do not. For instance, the DE0001135028 bond with a clean price of 106.127, 6% coupon, annual pay bond maturing 1/4/07 and settling 5/2/05, and Actual/Actual, the BNDYIELD function gives 2.219%. Bloomberg shows 2.215% for the same bond using US conventional yield and 2.228% using German conventional yield. I need more information on Yield Quotations derived from MATLAB in comparison to Yield Quotations from other sources.

Best Answer

In MATLAB 7.4 (R2007a), support for fixed-income computations that use International Securities Market Association (ISMA) day count conventions has been added for the MATLAB Financial Toolbox suite of fixed income related functions (e.g., BNDYIELD, BNDPRICE, CFAMOUNTS, etc.). Specifically, the following three new day count conventions have been added:
 actual/actual (ISMA)
 actual/360 (ISMA)
 actual/365 (ISMA)
The new day count conventions above should facilitate comparison of yields from MATLAB and other sources.
If you are using a prior release of MATLAB, the information below applies to comparing yields from other sources. Specifically, prior releases support only the Securities Industry Association (SIA) conventions documented in the following publication:
"Standard Securities Calculation Methods: Fixed Income Securities Formulas for Analytic Measures", Volume II, (1994); Securities Industry Association, Inc., New York, NY (ISBN 1-882936-02-7).
Although SIA conventions allow for a variety of common coupon payments per year (e.g., quarterly, semi-annually, annually) and day-count bases for the calculation of accrued interest, the calculation of "time factors", also referred to as "time to cash flow", is often a source of confusion when comparing yield quotes from various data vendors and other conventions used around the world.
As outlined in above SIA reference (see page 3, and pages 63-75), a "time factor" is the unit of time between settlement and a cash flow date used to discount, or compute the present value of, the given cash flow.
These time factors, according to SIA conventions, are always specified in semi-annual periods using an actual calendar. Of course, these time factors will generally have a fractional portion, but will nevertheless be expressed in semi-annual time periods. Again, note that time factors use an actual day count for the calculation, which may be distinct from the day count convention used to compute accrued interest.
Many vendors will provide yield quotations based on other conventions, such as the International Securities Market Association (ISMA), in which yields are quoted in annual periods. In short, ISMA conventions offer the convenience that one period is also one year.
To convert a semi-annual yield quotation to an effective annual yield quotation would involve the application of the following formula (see the above SIA publication, Formula 3, "Conversion of a Universally Comparable Semi-Annual Yield to an Effective Annual Cash Flow Yield" on page 74)
Y_Annual = (1 + Y_Semi/2) ^ 2 - 1)
This conversion is correct for SIA conventions.
Unfortunately, due to the semi-annual time factors calculated under SIA conventions, the annual effective yield conversion above does not always agree with an annual effective yield quotation based on annual time factors, such as is common in ISMA standards.
For reference, please see the first comment at the bottom of page 74 of the above SIA publication (paraphrasing or elaborating in brackets []):
"This formula may not produce the same results as calculating a Y_Annual using SIA Formula 2 [on page 72] with CPY = 1 [the number of compounding periods per year = 1]. The difference derives from the calculation of T [the time factors] in semi-annual periods vs. annual periods using an actual calendar".
Unfortunately, the SIA vs. ISMA yield discrepancy, although small in many cases, may in fact be somewhat pronounced in others (e.g., for bonds with only a few cash flows remaining).
Put another way, with differing conventions, effective yields based on 2 semi-annual periods are not always identical yields based on 1 annual period.
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