term flat price differently from most financial professionals.
The price with the accrued interest included increases until
the coupon is paid. When the coupon is paid, the accrued interest equals zero and both prices are identical. Repeated over
several coupon payments, this leads to a saw-toothed shape. By
comparison, the price without accrued interest produces a relatively flat plot. Examining the plots, it seems far more natural
to refer to the price without accrued interest as the flat price.
The reason financial professionals have switched the meaning of the term flat price remains unclear. Perhaps it’s because
the saw-toothed plot isn’t flat. We are more confident as to why
actuaries haven’t updated their use of the term—it’s because of
the paucity of available books on mathematical interest theory.
In many ways, Kellison’s text is an updated version of Butcher
and Nesbitt’s text (with the exception of the use of the word
flat price). The other authors may have used Kellison as their
One thing is clear: When evaluating the price of a bond between coupon dates, it’s important to remember that the dirty
price of any publicly traded bond is calculated using compound
interest and the clean price is found by subtracting the amount
of accrued interest calculated assuming linearity. When referring to the price, with or without accrued interest, it’s best to
avoid ambiguity and not use the wording flat price.
Actuaries who work with fixed-income securities, or who
have their master’s in business administration or are CFAs,
should already be aware of the discrepancy. Most of today’s texts
mention the term in passing only and favor other available terms,
such as clean price/dirty price or full price/price.
To eliminate confusion, we would recommend that the
(fixed-income) use of the term flat be used only in reference to
distressed bonds that are trading flat. If the term continues to
be used to refer to accrued interest in actuarial texts, it would
be best if it were mentioned in passing only or in a footnote that
explains both possible meanings. Additionally, if they haven’t
already, the Society of Actuaries and the Casualty Actuarial Society should remove the term flat price from all future exams
to avoid questions that may be considered ambiguous.
micHAel BerGer is an economic/actuarial consultant in the
legal services industry and a recent graduate of the University
of Waterloo, ontario, Canada. roBert l. Brown is a
professor of actuarial science at the University of Waterloo.
Broverman, S.A., Mathematics of Investment and Credit, 4th Edition,
Winsted, Conn.: Actex Publications, 2008.
Butcher, M.V., and Nesbitt, C.J., Mathematics of Compound Interest, Ann
Arbor, Mich.: Ulrich’s Books Inc., 1971.
CFA Institute, Level 1, 2006, Vol. 4, Fixed Income, Derivatives, Alternative
Investments and Portfolio Management, Boston: Pearson Custom Publishing, 2005. Daniel, J. W., and Vaaler, L.J.F., Mathematical Interest Theory, 1st Edition, New Jersey: Prentice Hall, 2006. Fabozzi, F.J. (editor), The Handbook of Fixed Income Securities,
7th Edition, New York: McGraw-Hill, 2005.
Harvey, C.R., Financial Glossary, Bloomberg LP,
Kellison, S.G., The Theory of Interest, 3rd Edition, Boston: Irwin/McGraw-
Macaulay, F.R., Some Theoretical Problems Suggested by the Movements of
Interest Rates in the United States Since 1856, New York: National Bureau
of Economic Research, 1938.
Securities Industry and Financial Market Association, Practice Guidelines
for Trading in Distressed Bonds,
Ruckman, C., and Francis, J., Financial Mathematics: A Practical Guide
for Actuaries and Other Business Professionals, 2nd Edition, Farmington,
Conn.: BPP Professional Education.
This article is solely the opinion of its authors. It does not express the official
policy of the American Academy of Actuaries; nor does it necessarily reflect
the opinions of the Academy’s individual officers, members, or staff.
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