Conflicting Theories on
How to Derive the
Interest Rate Swap Values in Example
5 in FAS 133
Bob Jensen and Carl Hubbard at Trinity University
Introduction
We hope that at least one reader of this document will come forward with either a reconciliation of two swap valuation theories discussed in this document or a convincing argument that resolves our uncertainties about how the FASB should have valued the interest rate swap of Page 75 of FAS 133. Please send your good help to Bob Jensen at rjensen@trinity.edu.
FAS 133 and IAS 39 will soon require adjusting all derivative financial instruments to fair value at least quarterly. For some derivatives such as futures and options contracts, fair value can be obtained daily from the Internet or the back of The Wall Street Journal. Those contracts can be "markedtomarket," because they are actively traded in organized markets such as the Chicago Mercantile Exchange (CME), the Chicago Board of Trade (CBOT), the Chicago Board of Options Exchange (CBOE), the Philadelphia Exchange, etc.
For nearly 80% of the derivatives contracts, amounting to trillions of dollars, adjusting to fair value is not so simple. Most derivatives contracts, such as forward and swap contracts, are custom (unique) contracts that are not traded in organized markets. At the date of inception, the freshlyinked forward and swap contracts are valued at zero. Subsequently, they do take on positive or negative fair values that must be estimated at least quarterly in order to comply with FAS 133 and IAS 39.
There is no guidance from the FASB or the IASC on how to value custom derivative contracts such as forwards and swaps. The FASB provides two illustrations alluding to value derivations from current yield (swap) curves in Examples 2 and 5 in FAS 133. Figuring out how the FASB valued the Example 2 swaps is not very difficult since the yield curves are assumed to be linear. Example 5, however, has haunted Professor Jensen for over a year. His good friend on the FASB, Gerhard Mueller, put him in touch with technical staff of the FASB. Then he learned that the person that wrote Example 5 was no longer with the FASB. Remaining staff proved unable to give much guidance on how swap values in Example 5 were devised.
Professor Jensen introduced his good friend Professor Hubbard to what is described below as the Teets and Uhl theory of swap valuation. Professor Hubbard introduced Professor Jensen to a tool in MS Excel called "Goal Seek." With this convergence of theory with Excel tools, we think we can now explain to the world how the FASB derived the swap values for Example 5 that are given on Page 75 of FAS 133. The problem, however, is that Professor Hubbard and Professor Jensen do not agree on swap valuation theory. Professor Jensen leans toward what is termed below as Jarrow and Turnbull theory. Professor Hubbard likes the Teets and Uhl theory. Dr. Hubbard buys into the FASB outcomes in Example 5 after correcting for what we view are some minor errors by the FASB in calculating "Accrued Interest." Professor Jensen remains a skeptic.
This document tries to fill in some missing gaps in Example 5 of FAS 133 (See Issue 2 below). It also corrects what we think are some calculation errors in Example 5 (See Issue 1 below). We are left, however, with two conflicting theories of how to value the swap at each repricing date. Of course swap valuation is only a small part of the extreme complexity and controversy brought about by FAS 133 and IAS 39. For the bigger picture, we recommend to Internet links:
Professor Jensen also has some tutorials on a "secret" server that you can access by writing him at rjensen@trinity.edu.
Issues 1 and 2 in Example 5 of FAS 133
Example 5 is arguably the most difficult example in FAS 133. The main problem is that, in Paragraph 131, the FASB disclosed that the yield curves were assumed to be upward sloping in the calculations of swap values. However, these calculations are never explained, and the FASB does not supply the yield (swap) curves needed to derive the swap valuations in the example.
We earlier reported on the AECM and CPASL lists that there were two serious issues in the FASB's Example 5 commencing in Paragraph 131 in FAS 133 on Accounting for Derivative Financial Instruments and Hedging Activities. Paragraphs 133 and 134 read as follows:
133. On July 1, 20X1, XYZ Company invests $10,000,000 in variablerate corporate bonds that pay interest quarterly at a rate equal to the 3month US$ LIBOR rate plus 2.25 percent. The $10,000,000 principal will be repaid on June 30, 20X3.
134. Also on July 1, 20X1, XYZ enters into a twoyear receivefixed, payvariable interest rate swap and designates it as a cash flow hedge of the variablerate interest receipts on the corporate bonds. The risk designated as being hedged is the risk of changes in cash flows attributable to changes in market interest rates. The terms of the interest rate swap and the corporate bonds are shown below.
Interest Rate Swap  Corporate Bonds  
Trade date and borrowing date  July 1, 20X1  July 1, 20X1 
Termination date  June 30, 20X3  June 30, 20X3 
Notional amount  $10,000,000  $10,000,000 
Fixed interest rate  6.65%  6.65% 
Variable interest rate  3month US$ LIBOR  3month US$ LIBOR 
Settlement dates and interest payment dates  End of each calendar quarter  End of each calendar quarter 
Reset dates  End of each calendar quarter  End of each calendar quarter 
Reset rates = 3month LIBOR  
7/1/X1 
5.56% 
We previously announced two major issues with respect to the FASB's FAS 133 Page 75 accounting for the above Example 5. These were as follows:
Issue 1
The Interest Accrued solutions on Page 75 are not compatible with the Interest Rate Swap balances for each of the reset dates. Readers may download an Excel workbook with cell comments that compare the FASB's original Page 75 of FAS 133 with our proposed corrected Page 75. When you download the following Excel workbook, please compare the FASB Pg. 75 versus Jensen Pg. 75 spreadsheets. The Excel workbook is at http://www.trinity.edu/rjensen/caseans/133ex05a.xls. (Note that you should download this file and read it in Excel rather than opening it in your web browser. The reason is that you will probably want to see how the calculations are made rather than just see the tabular results.)Issue 2
The underlying swap yield curves are not disclosed by the FASB in Example 5. At last we have discovered how the swap values on Page 75 of FAS 133 are derived. This does not resolve Issue 1 above where we contend that the FASB made some calculation errors. In order to see how we think the FASB derived the swap values for Example 5, download the following Excel workbook and study the spreadsheet called "Effective." The Excel workbook is at http://www.trinity.edu/rjensen/caseans/133ex05a.xls. (Note that you should download this file and read it in Excel rather than opening it in your web browser. The reason is that you will probably want to see how the calculations are made rather than just see the tabular results.)
Some Unresolved Theory Considerations for Issue 2
We cite two references that we are comparing for valuing interest rate swaps. The first theory begins on Page 435 of Derivative Securities by Robert Jarrow and Stuart Turnbull (SouthWestern College Publishing, Year 2000, ISBN 0538877405). The second theory is in a free book on the Internet entitled Introductory Cases on Accounting for Derivative Instruments and Hedging Activities by Walter Teets and Robert Uhl. Sections of interest for interest rate swap valuation are Pages 1826 and 102110. You can download the Teets and Uhl book, Power Point files, and spreadsheets for free from http://www.gonzaga.edu/faculty/teets/index0.html.
The above references lead to widely different swap valuations for identical yield (swap) curves. Among your authors, Carl Hubbard prefers the Teets and Uhl valuation theory. Bob Jensen clings to the Jarrow and Turnbull theory. Instead of settling this conflict with swords or singleshot pistols at sunrise, we decided to appeal to our friends.
We were able to use the Jarrow and Turnbull valuation approach to derive the swap valuations in Example 2 of FAS 133. The FASB never explained how to value the Example 2 swap each quarter, but you can read Bob Jensen's derivation at http://www.trinity.edu/rjensen/caseans/133ex02a.htm . However, Example 2 assumes linear yield (swap) curves when valuing the swaps each quarter. The Jarrow and Turnbull approach that worked so successfully in Example 2 did not work in Example 5 where nonlinear yield (swap) curves were assumed but not disclosed by the FASB. You can read more about the Jarrow and Turnbull theory at http://www.trinity.edu/rjensen/caseans/133ex05jt.htm.
It is not possible, in our viewpoint, to derive the FASB valuations on Page 75 of FAS 133 using the Jarrow and Turnbull theory. However, when we resorted to the Teets and Uhl theory, we did manage by rather simple trial and error process (using the Goal Seek tools in Excel) to derive swap valuations that conform with the FASB's outcomes of Page 75 of FAS 133. In order to see how we think the FASB derived the swap values for Example 5, download the following Excel workbook and study the spreadsheet called "Effective." The Excel workbook is at http://www.trinity.edu/rjensen/caseans/133ex05a.xls. (Note that you should download this file and read it in Excel rather than opening it in your web browser. The reason is that you will probably want to see how the calculations are made rather than just see the tabular results.)
Suppose we put this in a concrete example by focusing on the Example 5 (Page 75 in FAS 133) swap valuation on September 30, 2001 when seven quarters remain on the swap hedge. Since the FASB did not provide the yield (swap) curves for any part of Example 5, we resorted to the Goal Seek tool in Excel to derive the swap values shown on Page 75 of FAS 133.
The main difference between the Jarrow & Turnbull versus the Teets and Uhl swap valuation lies in the assumption of the ex ante swap cash flows. Jarrow and Turnbull assume a fixed annuity of swap payments for the remaining swap settlements used in the fair value adjusted swap valuations. For example, it is assumed above that the $27,250 swap payment is the best predictor (on September 30) of the remaining seven swap settlements. Teets and Uhl would vary this cash flow based upon "implied forward rates." Their spreadsheet solution, however, does not generate swap cash flows that make sense to me. To see their calculation spreadsheet go to http://www.trinity.edu/rjensen/caseans/teets/xls/swapval.xls.
Since we were unable to get the Teets and Uhl spreadsheet to work for this illustration, we developed an adaptation of the Teets and Uhl approach. For the September 30, 2001 swap valuation we derived the following using the ex ante quarterly yield curve shown below for September 30:
Maturity  09/30/01 Ex Ante Yield Curve 
Yield Curve Quarterly 
t  Future Value Interest Factor 
Forward Rates 
Swap Cash Received 
Swap Cash Paid 
Estimated Unhedged Cash Flow 
Net Hedged Cash Flow 
Estimated Swap Net Settlement 
09/30/01 Swap Present Value 

07/01/01  
09/30/01  5.56%  1.390% 

1.0000  1.390%  $166,250  $139,000  $195,250  $222,500  $27,250  $0  
12/31/01  5.63%  1.408%  1.00  1.0141  1.408%  $166,250  $140,750  $197,000  $222,500  $25,500  $25,146  
03/31/02  5.78%  1.445%  2.00  1.0291  1.482%  $166,250  $148,171  $204,421  $222,500  $18,079  $17,555  
06/30/02  5.93%  1.482%  3.00  1.0451  1.556%  $166,250  $155,594  $211,844  $222,500  $10,656  $10,174  
09/30/02  6.08%  1.519%  4.00  1.0621  1.630%  $166,250  $163,017  $219,267  $222,500  $3,233  $3,030  
12/31/02  6.22%  1.556%  5.00  1.0803  1.704%  $166,250  $170,443  $226,693  $222,500  ($4,193)  ($3,853)  
03/31/03  6.37%  1.593%  6.00  1.0995  1.779%  $166,250  $177,869  $234,119  $222,500  ($11,619)  ($10,453)  
06/30/03  6.52%  1.630%  7.00  1.1198  1.853%  $166,250  $185,297  $241,547  $222,500  ($19,047)  ($16,750)  
Totals =  $49,860  $24,850 
This $24,850 swap value above is identical to the FASB's swap value for September 20 on Page 75 of FAS 133. However, this value differs from the Jarrow and Turnbull theory explained at http://www.trinity.edu/rjensen/caseans/133ex05jt.htm. If we assume an identical hypothetical nonlinear term structure yield (swap) curve in a Jarrow and Turnbull (2,000, pg. 436) valuation approach, we derive a $179,895 swap value as shown below:
Date  09/30/01 Ex Ante Yield Curve 
Quarterly Yield Curve 
Amount  Yield
Curve Present Value Factor 
Estimated Swap Payment 
Estimated Swap Value on 09/30/01 
07/01/01  
09/30/01  5.56%  1.3900%  $1  0.986291  $27,250  $26,876 
12/31/01  5.63%  1.4075%  $1  0.972433  $27,250  $26,499 
03/31/02  5.78%  1.4446%  $1  0.957885  $27,250  $26,102 
06/30/02  5.93%  1.4817%  $1  0.942864  $27,250  $25,693 
09/30/02  6.08%  1.5188%  $1  0.927401  $27,250  $25,272 
12/31/02  6.22%  1.5559%  $1  0.911526  $27,250  $24,839 
03/31/03  6.37%  1.5930%  $1  0.895269  $27,250  $24,396 
Sum = 
$179,677 
The above $179,677 swap valuation is not consistent with the Teets and Uhl swap valuation approach explained in the case entitled "J. Adams and Company Revisited: Accounting for Interest Rate Swaps in an UpwardSloping Yield Curve Environment" in pp. 102110 of Introductory Cases on Accounting for Derivative Instruments and Hedging Activities by Walter R. Teets and Robert Uhl. Readers may download the Teets and Uhl book for free from http://www.gonzaga.edu/faculty/teets/index0.html.
As pointed out above, the main difference between the Jarrow & Turnbull versus the Teets and Uhl swap valuation approaches lies in the assumption of the ex ante swap cash flows. Jarrow and Turnbull assume a fixed annuity of swap payments for the remaining swap settlements used in the fair value adjusted swap valuations. For example, it is assumed above that the $27,250 swap payment is the best predictor (on September 30) of the remaining seven swap settlements. Teets and Uhl would vary this cash flow based upon "implied forward rates." Our application of both theories is illustrated for the FAS 133 Example 5 at http://www.trinity.edu/rjensen/caseans/133ex05a.xls.
Our Challenge to You
We hope that at least one reader of this document will come forward with either a reconciliation of these two swap valuation theories or a convincing argument that resolves our uncertainties about how the FASB should have valued the interest rate swap of Page 75 of FAS 133.
Please send your good help to Bob Jensen at rjensen@trinity.edu.
March 26, 2005 Update
A long time ago I posted a dilemma regarding valuation of interest rate swaps when I attempted to devise a valuation scheme to add to Example 5 in Appendix B of FAS 133  http://www.trinity.edu/rjensen/caseans/133ex05d.htm
On March 24, 2005 I received the following message from a nice man that I do not know named Raphael Keymer [raph@gawab.com]
I think I have a solution to your dilema
I’m responding to a ‘dilema’ you posted on the internet concerning recalculation of example 5 for FAS 133. I’ve recalculated the example in question on your web page and believe I’ve resolved the difference.
The Jarrow and Turnbill method was not properly implemented
It turns out the implementation of the ‘Jarrow and Turnbill’ methodology was not correct. When it is properly implemented both valuation methodologies give the same value as the first method employed by you.
Corrections required for Jarrow and Turnbill
Only fixed cash flows on the swap are to be discounted at first
Per the example on pages 435437, the fixed payments for the swap are considered first, and then only are the floating payments considered. This is calculated (in the worked example) as the present value of the stream of future fixed cash flows. The original implementation used a stream of the latest net cash settlement on reset in place of the stream of fixed cash flows.
The present value of the floating cash flows needs to be calculated
The original implementation didn’t calculate the present value of future floating rate cash flows on the swap.
Interpretation of interest rates was inconsistent with Teets & Uhl
Calculation of discount factors is dependant on the interpretation of the time period related to interest rates. The original Jarrow and Turnbill implementation used interest rates for earlier periods than those used in the Teets & Uhl implementation. This ‘correction’ will have the least effect on valuation differences.
Revised calculations have been attached in a spreadsheet
I placed his attached spreadsheet at http://www.cs.trinity.edu/~rjensen/133ex05aSupplement.xls