Interest Rate Swap
Valuation, Forward Rate Derivation, and Yield Curves
for FAS 133 and IAS 39 on Accounting for Derivative Financial Instruments
Bob Jensen at Trinity University
Short-Cut Method for Interest Rate Swaps
Yield Curve and Forward Rate Calculations
Example 5 from Appendix B of FAS 133
Excerpts from the IAS 39 November 2001 Implementation and Guidance Supplement
Settlement Exit Value Amortization Rate Accounting for
Custom Interest Rate Swaps Having No Market Trading
Casting out on the Internet often results in a catch
Concept of Fair Value --- http://www.trinity.edu/rjensen/acct5341/speakers/133glosf.htm#FairValue
Note the book entitled PRICING DERIVATIVE SECURITIES, by T W Epps (University of Virginia, USA) The book is published by World Scientific --- http://www.worldscibooks.com/economics/4415.html
- Introduction and Overview
- Mathematical Preparation
- Tools for Continuous-Time Models
- Pricing Theory:
- Dynamics-Free Pricing
- Pricing Under Bernoulli Dynamics
- Black-Scholes Dynamics
- American Options and 'Exotics'
- Models with Uncertain Volatility
- Discontinuous Processes
- Interest-Rate Dynamics
- Computational Methods:
- Solving PDEs Numerically
- Computer Programs
From PwC Flashline, December 12, 2013
Derivative valuation—The transition to OIS discounting
Derivative pricing practices have evolved in recent years as market participants refine their pricing approaches to capture the elements underlying the pricing of derivative transactions in a changing market. One area that has continued to evolve relates to pricing assumptions on collateralized derivatives. Following the lessons learned during the financial crisis, many market participants recognized that the funding advantages from collateral that may be rehypothecated have value that should be considered in derivative pricing.
The incorporation of these funding advantages has had a broad impact on derivative pricing as a result of the increasingly common use of collateral. The increased use of collateral has been driven by an increased focus in the OTC market on credit risk and funding risk management, as well as by the migration of derivative activity to clearing houses where transactions are typically fully collateralized. As a result, certain collateralized derivatives may be presumed to require valuation based on discounting at the Overnight Indexed Swap (“OIS”) rate.
The derivative pricing changes also impact uncollateralized transactions as market conventions for the way prices are quoted for reference instruments, such as interest rate swaps, have changed.
This Dataline addresses some of the key financial reporting implications relating to these evolving pricing conventions ---
National Professional Services Group
Derivative pricing practices have evolved in recent years as market participants refine their pricing approaches to capture the elements underlying the pricing of derivative transactions in a changing market .
One area that has continued to evolve relates to pricing assumptions on collateralized derivatives. For many years market participants utilized collateral on bilateral over-the-counter (“OTC”) derivative transactions as a means of mitigating the credit risk of their counterparties. Following the lessons learned during the financial crisis, many market participants recognized that the funding advantages from collateral that may be rehypothecated has value that should be considered in derivative pricing.
The incorporation of these funding advantages has had a broad impact on derivative pricing as a result of the increasingly common use of collateral on derivative transactions . The increased use of collateral has been driven by an increased focus in the OTC market on credit risk and funding risk management , as well as by the migration of derivative activity to clearing houses where transactions are typically fully collateralized. As a result, certain collateralized derivatives may be presumed to require valuation based on discounting at the Overnight Indexed Swap (“OIS”) rate .
The derivative pricing changes also impact uncollateralized transactions as market conventions for the way prices are quoted for ref erence instruments , such as interest rate swaps , have changed.
This Dataline addresses some of the key financial reporting implications relating to these evolving pricing convention s . Companies with derivatives , whether bilateral OTC contracts or cleared transactions, should continue to focus on developments in market conventions when pricing those derivative s . Derivative pricing may re quire changes to valuation models or model inputs to ensure that valuations are performed consistent with market practices .
Developments in market conventions may lead to required changes in controls and processes relating to derivative valuations. Given the continuing level of change in the market related to derivative pricing, companies should implement appropriate procedures and controls so that their valuation approach and the related documentation and disclosures are periodically updated and remain consistent with current market practice.
Bob Jensen's threads on valuing interest rate swaps ---
"Detecting price artificiality and manipulation in futures markets: An
application to Amaranth," by Atanu Saha and Hans-Jürgen Petersen, Journal
of Derivatives & Hedge Funds (2012) 18, 254–271 ---
In this article we propose a general method to test whether economic data support the claim of futures market manipulation. We examine the question of whether or not Amaranth manipulated the market for natural gas futures using three alternative methods. The first is our contribution to the existing body of literature on the analysis of manipulation claims. The subsequent two have previously been discussed in the literature. All three methods yield the same result: economic data on futures prices and Amaranth's trades do not support the claim that Amaranth manipulated the natural gas futures market in 2006.
Continued in article
Bob Jensen's threads on how to value interest rate swaps ---
Bob Jensen's free tutorials on accounting for derivative financial
instruments and hedging activities ---
Especially note the FAS 133 and IAS 39 Glossary at
This is more than a glossary.
Valuation and Pricing of Interest Rate
Begin Here --- http://en.wikipedia.org/wiki/Interest_rate_swap
Then read below:
US$148 / £98 / S$196
US$111 / £73.50 / S$147
US$111 / £74 / S$147
US$83.25 / £55.50 /
Yield Curve and Forward Rate Calculations
Yield Curve Definition =
You might try generating it from the Eurodollars futures market --- http://secure.webstation.net/~ftsweb/texts/bondtutor/chap5.7.htm
You have to delve into finance textbooks for technical explanations of yield curve derivation.
For online sources, I recommend that you type in "deriving a yield
curve" in the Exact Phrase box at http://www.google.com/advanced_search?hl=en
finance textbooks cover the Bloomberg Terminal in theory but do not get into the
practical applications via a Bloomberg Terminal --- http://www.bondmarkets.com/newsletters/1996/pn6631.shtml
college programs have Bloomberg Terminals that allow students to perform
real-world simulations --- http://ej.iop.org/links/q23/gIcAWELLlUo32pyrPtDlKg/qf3_6_m01.pdf
Example 5 from Appendix B in FAS 133
This example is accompanied by an Excel workbook file 133ex05a.xls that can be downloaded from http://www.cs.trinity.edu/~rjensen/
Working Paper 305
paper was published in ”An Explanation of Example 5, Cash Flow Hedge
of Variable-Rate Interest Bearing Asset in SFAS 133,”
by Carl M. Hubbard and Robert E. Jensen, Derivatives Report, April
2000, pp. 8-13.
Corrections and Explanations of Example 5 in FAS 133
(March 10, 2000 Version)
Carl M. Hubbard and Robert E. Jensen
San Antonio, Texas
In 1998, the Financial Accounting Standards Board (FASB) issued the derivatives and hedge accounting FAS 133 (or FAS 133) standard that will be one of the most costly and confusing of all FASB standards to implement.[i] This paper is the second of two papers that are intended to help readers cope with the two most difficult illustrations in FAS 133. The first paper on Example 2, which dealt with a fair value interest rate swap, was published in Derivatives Report, November 1999, pp. 6-11. With regard to Example 5 on pages 72 – 76 of FAS 133, which is supposed to demonstrate the mechanics of accounting for a cash flow interest rate swap, we contend that information necessary for understanding Example 5 was omitted and that the table on page 75 of FAS 133 contains a repeating error that that confounds attempts to understand the development of the example. In this paper we discuss and explain the table on page 75 of FAS 133 and correct the Interest accrued errors. We also supply yield curves that are consistent with swap values in the table on page 75 and demonstrate the calculation of expected swap cash flows from forward rates that are derived from the yield curves.
The data omitted from Example 5 are the yield curves that were used to calculate forward rates that in turn are used to calculate expected swap cash flows. The swap values at the reset dates in the table on page 75 are the present values of future expected swap cash flows that are discounted at rates in the unknown yield curves. This is unfortunate omission in FAS 133. The flat yield curve assumption for Example 2 allows readers to follow the example without information beyond that provided in our earlier paper, but the upward sloping yield curve assumption in Example 5 requires disclosure of the yield curve at each reset date in order to verify the swap values and understand the example. Our discussion of Example 5 begins with Paragraph 133 on page 73 of FAS 133. A companion paper will focus on Example 5 beginning in Paragraph 131.
Introduction to Example 5 in FAS 133
Example 5 focuses on an application of FAS 133 by XYZ Company that has entered into an effective, receive fixed/pay variable interest rate swap that extends over eight quarters. In the swap contract XYZ receives a fixed (6.65%) rate and pays a variable LIBOR rate on a notional principal amount of $10 million. This swap hedges the company’s expected cash flows from $10 million of notional principal that earns a floating annual rate of LIBOR + 2.25%. All payments and reset dates are quarterly beginning July 1, 20X1. Since XYX has entered into a receive fixed/pay variable swap, XYZ obviously is concerned that LIBOR rates would decline and thus reduce the income from the floating rate investment. Exhibit 1 below summarizes the facts assumed in the interest rate swap in Example 5.
Exhibit 1: Terms of the Interest Rate Swap and Corporate Bonds in Example 5
Interest Rate Swap
Trade date and borrowing date July 1, 20X1 July 1, 20X1
Termination date June 30, 20X3 June 30, 20X3
Notional amount $10,000,000 $10,000,000
Fixed interest rate 6.65% Not applicable
Variable interest rate 3-month US$ LIBOR 3-month US$ LIBOR
Settlement dates and interest End of each calendar End of each calendar
payment dates quarter quarter
Reset dates End of each calendar End of each calendar
quarter through quarter through
March 31, 20X3 March 31, 20X3
The above Example 5 has been modified somewhat by the March 3, 2000 FASB Exposure Draft No. 207-A containing several proposed amendments to FAS 133. However, the numerical outcomes and the Page 75 answers were not revised from the original Example 5 in FAS 133. We still see the need for proposing some corrections and explanations of the Page 75 results.
the Errors and Explaining the Table
The Interest Accrued amounts on Page 75 of FAS 133 are not compatible with the swap value and LIBOR rates at the reset dates. Either the Interest accrued amounts or the swap values are incorrect. For the reader’s convenience, the original table from page 75 of FAS 133 is reproduced in Exhibit 2 below. Our corrected version of the table is presented in Exhibit 3. Readers may download an Excel workbook, best read in Excel, with cell comments that compare the FASB's original Page 75 of FAS 133 with our corrected table from http://www.trinity/edu/rjensen/caseans/133ex05d.htm.
Insert Exhibits 2 and 3
Accepting the swap values as correct does less damage to the table, thus we assume the Interest accrued amounts must be corrected. Our Exhibit 3 reports the corrected Interest accrued amounts for each quarter using the LIBOR rates listed on page 74 of FAS 133. Since the initial present value of expected swap cash flows must be zero, the Interest accrued as of 9/30/X1 is correctly given as zero in the table on page 75 of FAS 133, in Exhibit 2, and Exhibit 3. By 9/30/X1 the LIBOR has changed, and the present value of expected swap cash flows beyond 9/30/X1 is revised to $24,850. The Interest accrued on $24,850 on 12/31/X1 is $350 or 0.0563/4 x $24,850. Thus in Exhibit 3 the Interest accrued amount for 12/31/X1 is corrected to show the $350 amount. Since the swap values are assumed to be correct in the original table, Effect of change in rates is adjusted also by the correction.
Again on 12/31/X1 the present value of expected swap cash flows is recalculated using an unspecified yield curve and is $73,800. The Interest accrued on $73,800 on 3/31/X2 is 0.0556/4 x $73,800 or $1,026, not $1,210 as presented in the original table. As before, the entry for Effect of change in rates is also adjusted. Because of changes in interest rates and because of the passage of time, the present values of expected swap cash flows change each quarter. Each quarter’s Interest accrued in Exhibit 3 is recalculated using the reset LIBOR at the beginning of the quarter, and as seen in Exhibit 3 each quarter’s entry for Effect of change in rates is also corrected.[ii]
The actual swap payments (receipts) as of the reset dates for the eight quarters are shown in Paragraph 138, page 76 of FAS 133. The payment (receipt) is equal to the variable LIBOR rate paid in the swap less the fixed rate received times the notional principle, or on 9/30/X1 (0.0556 – 0.0665)/4 x $10 million = ($27,250), a receipt. On 12/31/X1 the swap payment (receipt) is (0.0563 – 0.0665)/4 x $10 million = ($25,500). The swap cash flows are calculated in that same manner for each quarter throughout the life of the swap. Each quarter’s Effect of change in rates is equal to the current period’s recalculated value of the swap less the previous period’s swap value less the current period’s Interest accrued less the current period’s swap payment (receipt). In other words Effect of change in interest rates is the balancing item.
As shown in Exhibit 3, the effective swap terminates on 6/30/X3 with a zero value, and the 3/31/X3 present value of the one remaining swap cash flow is amortized by the swap payment (receipt) on 6/30/X3, the Interest accrued on the 3/31X3 value of the swap, and an accumulated rounding error, if any. We believer, therefore, that our Exhibit 3 is the correct presentation of entries related to Example 5.
the Yield Curves
The underlying swap yield curves are unfortunately not disclosed by the FASB in Example 5. The swap values at the reset dates are the present values of the future expected swap cash flows. The cash flows that are discounted to the present value at each reset date cannot be known without the yield curves and the forward rates that are used to calculate the future quarterly expected cash flows. Furthermore, the discount rates that are used to calculate the values of the swap at the reset dates are the zero-coupon LIBOR’s that comprise the yield curves on the reset dates. Thus, in this example where an upward sloping yield curve is assumed, we must be given the yield curves at each reset date in order to replicate the calculations in the example. If we cannot easily replicate the example, it ceases to be an example in a pedagogical sense.
Nevertheless, we have discovered how the swap values on Page 75 of FAS 133 may have derived. In order to see how we derive yield curves that provide the same swap values given in the table on page 75 of FAS 133, the reader may wish to download the Excel workbook in Excel at http://www.trinity.edu/rjensen/caseans/133ex05a.xls and study the spreadsheet called "Effective." For this paper, we derived a yield curve at each reset date that provides the FASB’s swap values in the table on page 75 of FAS 133. We have some concerns as to whether the FASB’s swap values are theoretically sound, but that issue is reserved for another paper.
A yield curve is the graphic or numeric presentation of bond equivalent yields to maturity on debt that is identical in every aspect except time to maturity. In developing a yield curve, default risk and liquidity, for example, are the same for every security whose yield is included in the yield curve. Thus yields on U. S. Treasury issues are normally used to plot Treasury yield curves. The relationship between yields and time to maturity is often referred to as the term structure of interest rates. Similarly, an unknown set of estimated LIBOR yield curves underlie the FASB swap valuations calculated in Exhibit 3. Other than providing the assumption that the yields in the yield curves are zero-coupon rates, the FASB offers no information that would allow us to derive the yield curves or calculate the swap values in Example 5.
The typical yield curve gradually increases relative to years to maturity. That is, historically, short-term rates are somewhat lower than longer-term rates. In a recession with deflation or disinflation the entire yield curve shifts downward as interest rates generally fall and rotates counter-clockwise indicating that short-term rates have fallen to much lower levels than long-term rates. In rapid economic expansion accompanied by inflation, interest rates tend to rise and yield curves shift upward and rotate clockwise indicating that short-term rates have increased more than long-term rates. [iii]
The different shapes of the yield curve described above complicate the calculation of the present value of an interest rate swap and require the calculation and application of implied forward rates to calculate future expected swap cash flows. Fortunately Example 2 assumes that a flat yield curve prevails at all levels of interest rates. A flat yield curve means that as interest rates rise and fall, short-term and long-term rates move together in lock step, and future cash flows are all discounted at the same current discount rate. The cash flows and values in Example 5, however, are developed from the prevailing upward sloping yield curve at each reset date.
After reviewing the cases in Teets and Uhl, we used the tool Goal Seek in Excel to derive upward sloping yield curves and swap values at the reset dates that equal the FASB's values in the table on Page 75 of FAS 133.[iv] Exhibit 5 below shows the results of our calculations of the expected swap cash flows at each reset date and the value of the swap in Example 5 at each reset date. In order to see how we believe the swap values in the original table were derived for Example 5, the reader may wish to download the Excel workbook using Excel at http://www.trinity.edu/rjensen/caseans/133ex05a.xls and select “Effective.”
Insert Exhibit 4 here
In the development of Exhibit 4 we began with the derivation of a yield curve of LIBOR rates that provide a zero present value of future expected swap cash flows at the initiation of the swap. The only rate given in Example 5 for that first yield curve is 5.56% for the first quarter. Since the yield curve is upward sloping, we calculated a trial yield curve that begins at 5.56% and increases x% each quarter in the future. By supplying a value for x, we derived the trial yield curve and then calculated forward rates from that yield curve. The forward rates in this example are the future expected spot LIBOR rates that are implied by the zero-coupon rates in a yield curve. For example, assume that we have a two-year investment horizon and that the one-year LIBOR is 6.0% and the two-year LIBOR is 6.5%. If the LIBOR market is in equilibrium with respect to current rates and future expected rates, the 6.5% two-year rate must be the geometric mean of the one-year rate of 6.0% and the expected one-year rate beginning one year from now, which is 7.0%.
A forward rate is calculated in the following manner:
f(t) = [1 + r(t)]t/[1 + r(t-1)]t-1 – 1 (1)
in which f(t) is the forward rate for time period t, r(t) is the multi-period yield that spans t periods, and r(t-1) is the yield for an investment of t-1 periods. In the example above, 6.5% is r(t) and 6.0% is r(t-1). Thus, f(2), the forward LIBOR for year 2, is calculated as follows
f(2) = (1.065)2/1.06 – 1 = 0.07 or 7.0% (2)
Having calculated a forward rate for each quarter from the rates in the trial yield curve, we then asked Excel to give us the value of x, the slope of the upward sloping yield curve, that would provide a yield curve with forward rates that would calculate future expected swap cash flows whose present value is zero. The resulting yield curve, quarterly equivalent rates, forward rates, expected swap cash flows, and present values of cash flows as of 7/1/X1 are shown in the Panel 1 of Exhibit 4. A seen in Panel 1 of Exhibit 4, the first derived yield curve starts at 5.56% for the period ending 9/30/X1 and ends with a LIBOR of 6.68% for Eurodollar deposits maturing on 6/30/X3.
Future expected swap cash flows in Exhibit 4 are equal to the fixed rate received in the swap (0.0665/4) less the calculated forward rate times the notional principal. On 7/1/X1 in Example 5, the first quarter’s LIBOR of 5.56%/4 or 1.390% is also the first quarter’s forward rate, and the first period’s swap cash flow is (0.0665/4 - 0.0139) x $10 million = $27,250. The second quarterly forward rate is 1.470%, and the second quarter’s expected swap cash flow is (0.0665/4 - 0.0147) x $10 million = $19,265. The third quarterly forward rate is 1.550%, and so forth. When the thus calculated, the expected swap cash flows are discounted to the present value using the yields in the yield curve as discount rates. On 7/1/X1 the present value of the first swap cash flow of $27,250 is $27,250/1.0139 or $26,876. The present value of the second swap cash flow of $19,265 is $19,265/(1.01432) or $18,725, and so forth through the remaining six quarters. As shown in Exhibit 4, the 7/1/X1 the initial sum of the present values of the eight expected swap cash flows is zero.
At the first reset date of 9/30/X1 in Example 5 the spot LIBOR increases to 5.63%. We derived the LIBOR yield curve for 9/30/X1 using Goal Seek in the same manner as described above. We asked Excel to calculate a value of x, the slope of the new yield curve beginning at 5.63%, that would give us a yield curve whose forward rates would provide swap cash flow calculations whose present value discounted at the yields in the yield curve equals $24,850. The results of those calculations are presented in the Panel 2 of Exhibit 4. We repeated that yield curve, forward rate, cash flow, and present value derivation process in Excel for each of the remaining reset dates in Example 5. The results of those derivations are given in Panels 3 through 8 of Exhibit 4. Since the interest rate swap is assumed to be effective, the concluding swap value on 6/30/X3 is zero.
Example 5 of FAS 133 is supposed to provide an example of accounting entries for a receive fixed/pay variable interest rate swap that that effectively hedges the variable interest income from an investment. However, the table on page 75 of FAS 133 is incorrect, and the information provided in Example 5 on the derivation of swap values is incomplete. The page 75 table reports Interest accrued amounts that are inconsistent with the given LIBOR rates and swap values. The LIBOR rates in the upward sloping yield curves that were used to derive forward rates, future expected swap cash flows, and swap values at the reset dates are not given in the example. Our objective in this paper was to correct the errors in the page 75 table, explain the entries in the corrected table, and then provide yield curves that are consistent with the swap values given in Example 5. We explain the derivation of trial yield curves, forward rates, swap cash flows, and swap values in Example 5. The corrected table in Exhibit 3 combined with the yield curve data and forward rates in Exhibit 4 enable a reader to understand the derivation of the cash flows, swap values, and other accounting entries that are the subjects of Example 5.
Readers may download an Excel workbook demonstrating our calculations from http://www.trinity.edu/rjensen/caseans/133ex05a.xls
We want to acknowledge the help from two individuals who independently found a calculation error in our first round of calculations. Thanks go to Peter van Amson from BankWare Inc and Dr. Walter R. Teets from Gonzaga University. Dr. Teets and his co-author, Robert Uhl, provide a free book online at at http://www.gonzaga.edu/faculty/teets/index0.html.
Peter van Amson sent us a corrected version of our own spreadsheet. He also recommended the following references:
For W.R.T. swaps the standard text used in practice is the Hull Book. Options, Futures and Other Derivatives, John C. Hull it has a fairly straight forward valuation of swaps. For an "advanced" actually just more mathematical treatment of the problem I recommend Interest Rate Option Models, Ricardo Rebonato.
The Hull reference is as follows: John C. Hull, Options, Futures, and Other Derivatives (Prentice-Hall, 1999, ISBN: 0130224448)
The Rebonato referernce is as follows: Ricardo Rebonato, Interest Rate Option Models (John Wiley & Sons, Wiley Finance, 1998, ISBN 0-471-96569-3)
[i] The reader may learn more about FAS 133 and other FASB standards at http://www.rutgers.edu/Accounting/raw/fasb/st/stpg.html. Also note the March 3, 2000 FASB Exposure Draft No. 207-A containing several proposed amendments to FAS 133 at http://www.rutgers.edu/Accounting/raw/fasb/draft/amend133_ED.pdf.
Accrued interest on the present value of future expected cash flow
from the swap as of the end of the previous period for the eight
quarters is calculated as follows:
($Swap value)(LIBOR/4) = ($0)(0.0556/4) = $0
($24,850)(0.0563/4) = $350
($73,800)(0.0556/4) = $1,026
($85,910)(0.0547/4) = $1,175
(-$42,820)(0.0675/4) = ($723)
(-$33,160)(0.0686/4) = ($569)
(-$21,850)(0.0697/4) = ($381)
($1,960)(0.0657/4) = $32
[iii] See Chapter 6 “The Term Structure of Interest Rates” in James C. Van Horne Financial Market Rates and Flows, 5th Edition. Upper Saddle River, NJ: 1998 for an excellent discussion of yield curves.
[iv] See “J. Adams and Company: Accounting for Interest Rate Swaps in an Upward Sloping Yield Curve Environment” in Introductory Cases on Accounting for Derivative Instruments and Hedging Activities, 1998, by Walter R. Teets and Robert Uhl. This case and other cases are available free online at http://www.gonzaga.edu/faculty/teets/index0.html.
Excerpts from the IAS 39 Supplement
Supplement to the
Accounting for Financial Instruments - Standards, Interpretations, and Implementation Guidance
In July 2001, IASB issued a consolidated document that includes all questions and answers approved in final form by the IAS 39 Implementation Guidance Committee as of 1 July 2001, including the fifth batch of proposed guidance (issued for comment in December 2000). The Q&A respond to questions submitted by financial statement preparers, auditors, regulators, and others and have been issued to help them and others better understand IAS 39 and help ensure consistent application of the Standard.
There is also a publication, Accounting for Financial Instruments - Standards, Interpretations and Implementation Guidance, which is available from IASB Publications. This book contains the current text of IAS 32 and IAS 39, SIC Interpretations related to the accounting for financial instruments as well as the IAS 39 Implementation Guidance Questions and Answers.
In November 2001, the IGC issued a document with the final versions of 17 Q&A and two illustrative examples that were issued in draft form for public comment in June 2001. That document replaces pages 477-541 in the publication Accounting for Financial Instruments - Standards, Interpretations, and Implementation Guidance, which was published in July 2001. Draft Questions 10-22, 18-3, 38-6, 52-1, and 112-3 were eliminated in the final document, primarily because the issues involved are being addressed in the Board’s current project to amend IAS 39. In November 2001, the IASB issued the following free document:
Supplement to the
Accounting for Financial Instruments - Standards, Interpretations, and Implementation Guidance
Implementation Guidance IAS 39 Implementation Guidance
The yield curve provides the foundation for computing future cash flows and the fair value of such cash flows both at the inception of, and during, the hedging relationship. It is based on current market yields on applicable reference bonds that are traded in the marketplace. Market yields are converted to spot interest rates (‘‘ spot rates’’ or ‘‘ zero coupon rates’’) by eliminating the effect of coupon payments on the market yield. Spot rates are used to discount future cash flows, such as principal and interest rate payments, to arrive at their fair value. Spot rates also are used to compute forward interest rates that are used to compute variable and estimated future cash flows. The relationship between spot rates and one- period forward rates is shown by the following formula:
Spot --- forward relationship
Also, for purposes of this illustration, assume the following quarterly-period term structure of interest rates using quarterly compounding exists at the inception of the hedge.
Yield curve at inception -- (beginning of period 1)
The one-period forward rates are computed based on the spot rates for the applicable maturities. For example, the current forward rate for Period 2 calculated using the formula above is equal to [1.04502/1.0375] - 1 = 5.25%. The current one-period forward rate for Period 2 is different from the current spot rate for Period 2, since the spot rate is an interest rate from the beginning of Period 1 (spot) to the end of Period 2, while the forward rate is an interest rate from the beginning of Period 2 to the end of Period 2.
In this example, the enterprise expects to issue a 100,000 one-year debt instrument in three months with quarterly interest payments. The enterprise is exposed to interest rate increases and would like to eliminate the effect on cash flows of interest rate changes that may occur before the forecasted transaction occurs. If that risk is eliminated, the enterprise would obtain an interest rate on its debt issuance that is equal to the one-year forward coupon rate currently available in the marketplace in three months. That forward coupon rate, which is different from the forward (spot) rate, is 6.86%, computed from the term structure of interest rates shown above. It is the market rate of interest that exists at the inception of the hedge, given the terms of the forecasted debt instrument. It results in the fair value of the debt being equal to par at its issuance.
At the inception of the hedging relationship, the expected cash flows of the debt instrument can be calculated based on the existing term structure of interest rates. For this purpose, it is assumed that interest rates do not change and that the debt would be issued at 6.86% at the beginning of Period 2. In this case, the cash flows and fair value of the debt instrument would be as follows at the beginning of Period 2:
Issuance of fixed rate debt
Since it is assumed that interest rates do not change, the fair value of the interest and principal amounts equals the par amount of the forecasted transaction. The fair value amounts are computed based on the spot rates that exist at the inception of the hedge for the applicable periods in which the cash flows would occur had the debt been issued at the date of the forecasted transaction. They reflect the effect of discounting those cash flows based on the periods that will remain after the debt instrument is issued. For example, the spot rate of 6.38% is used to discount the interest cash flow that is expected to be paid in Period 3, but it is discounted for only two periods because it will occur two periods after the forecasted transaction occurs.
The forward interest rates are the same as shown previously, since it is assumed that interest rates do not change. The spot rates are different but they actually have not changed. They represent the spot rates one period forward and are based on the applicable forward rates.
The objective of the hedge is to obtain an overall interest rate on the forecasted transaction and the hedging instrument that is equal to 6.86%, which is the market rate at the inception of the hedge for the period from Period 2 to Period 5. This objective is accomplished by entering into a forward starting interest rate swap that has a fixed rate of 6.86%. Based on the term structure of interest rates that exist at the inception of the hedge, the interest rate swap will have such a rate. At the inception of the hedge, the fair value of the fixed rate payments on the interest rate swap will equal the fair value of the variable rate payments, resulting in the interest rate swap having a fair value of zero. The expected cash flows of the interest rate swap and the related fair value amounts are shown as follows:
At inception of the hedge, the fixed rate on the forward swap is equal to the fixed rate the enterprise would receive if it could issue the debt in three months under terms that exist today.
Measuring hedge effectiveness
If interest rates change during the period the hedge is outstanding, the effectiveness of the hedge can be measured in a number of ways.
Assume that interest rates change as follows immediately prior to the issuance of the debt at the beginning of Period 2:
Yield curve -- Rates increase 200 basis points
Under the new interest rate environment, the fair value of the pay-fixed at 6.86%, receive-variable interest rate swap which was designated as the hedging instrument would be as follows:
Fair value of interest rate swap
In order to compute the effectiveness of the hedge, it is necessary to measure the change in the present value of the cash flows or the value of the hedged forecasted transaction. There are at least two methods of accomplishing this measurement.
Method A -- Compute change in fair value of debt
Under Method A, a computation is made of the fair value in the new interest rate environment of debt that carries interest that is equal to the coupon interest rate that existed at the inception of the hedging relationship (6.86%). This fair value is compared with the expected fair value as of the beginning of Period 2 that was calculated based on the term structure of interest rates that existed at the inception of the hedging relationship, as illustrated above, to determine the change in the fair value. Note that the difference between the change in the fair value of the swap and the change in the expected fair value of the debt exactly offset in this example, since the terms of the swap and the forecasted transaction match each other.
Method B -- Compute change in fair value of cash flows
Under Method B, the present value of the change in cash flows is computed based on the difference between the forward interest rates for the applicable periods at the effectiveness measurement date and the interest rate that would have been obtained had the debt been issued at the market rate that existed at the inception of the hedge. The market rate that existed at the inception of the hedge is the one-year forward coupon rate in three months. The present value of the change in cash flows is computed based on the current spot rates that exist at the effectiveness measurement date for the applicable periods in which the cash flows are expected to occur. This method also could be referred to as the "theoretical swap" method (or "hypothetical derivative" method) because the comparison is between the hedged fixed rate on the debt and the current variable rate, which is the same as comparing cash flows on the fixed and variable rate legs of an interest rate swap.
As before, the difference between the change in the fair value of the swap and the change in the present value of the cash flows exactly offset in this example, since the terms match.
There is an additional computation that should be performed to compute ineffectiveness prior to the expected date of the forecasted transaction that has not been considered for purposes of this illustration. The fair value difference has been determined in each of the illustrations as of the expected date of the forecasted transaction immediately prior to the forecasted transaction, that is, at the beginning of Period 2. If the assessment of hedge effectiveness is done before the forecasted transaction occurs, the difference should be discounted to the current date to arrive at the actual amount of ineffectiveness. For example, if the measurement date were one month after the hedging relationship was established and the forecasted transaction is now expected to occur in two months, the amount would have to be discounted for the remaining two months before the forecasted transaction is expected to occur to arrive at the actual fair value. This step would not be necessary in the examples provided above because there was no ineffectiveness. Therefore, additional discounting of the amounts, which net to zero, would not have changed the result.
Under Method B, ineffectiveness is computed based on the difference between the forward coupon interest rates for the applicable periods at the effectiveness measurement date and the interest rate that would have been obtained had the debt been issued at the market rate that existed at the inception of the hedge. Computing the change in cash flows based on the difference between the forward interest rates that existed at the inception of the hedge and the forward rates that exist at the effectiveness measurement date is inappropriate if the objective of the hedge is to establish a single fixed rate for a series of forecasted interest payments. This objective is met by hedging the exposures with an interest rate swap as illustrated in the above example. The fixed interest rate on the swap is a blended interest rate composed of the forward rates over the life of the swap. Unless the yield curve is flat, the comparison between the forward interest rate exposures over the life of the swap and the fixed rate on the swap will produce different cash flows whose fair values are equal only at the inception of the hedging relationship. This difference is shown in the table below:
If the objective of the hedge is to obtain the forward rates that existed at the inception of the hedge, the interest rate swap is ineffective because the swap has a single blended fixed coupon rate that does not offset a series of different forward interest rates. However, if the objective of the hedge is to obtain the forward coupon rate that existed at the inception of the hedge, the swap is effective, and the comparison based on differences in forward interest rates suggests ineffectiveness when none may exist. Computing ineffectiveness based on the difference between the forward interest rates that existed at the inception of the hedge and the forward rates that exist at the effectiveness measurement date would be an appropriate measurement of ineffectiveness if the hedging objective is to lock in those forward interest rates. In that case, the appropriate hedging instrument would be a series of forward contracts each of which matures on a repricing date that corresponds with the occurrence of the forecasted transactions.
It also should be noted that it would be inappropriate to compare only the variable cash flows on the interest rate swap with the interest cash flows in the debt that would be generated by the forward interest rates. That methodology has the effect of measuring ineffectiveness only on a portion of the derivative, and IAS 39 does not permit the bifurcation of a derivative for purposes of assessing effectiveness in this situation (IAS 39.144). It is recognized, however, that if the fixed interest rate on the interest rate swap is equal to the fixed rate that would have been obtained on the debt at inception, there will be no ineffectiveness assuming that there are no differences in terms and no change in credit risk or it is not designated in the hedging relationship.
November 23, 2003 message from Ira
I've just posted a new article on the Kawaller & Company web site -- "Swap Valuations: The Bank One Swaps Market Precedent". It was published in the latest issue of Risk Magazine, and it was co-authored with John Ensminger and Lou Le Guyader. It may be of particular relevance for those with a tax or regulatory orientation.
If you are interested, you can view it by clicking here: http://www.kawaller.com/pdf/Risk_tax_issues.pdf
(Or copy this link address and past it into the address field in you internet browser.)
You can also go to the Kawaller & Company website to find the following: " Other articles on derivatives, risk management, and related accounting " A schedule of upcoming conferences relating to these topics; and " A description of Kawaller & Co.'s services
Please feel free to contact me with any questions, comments, or suggestions.
Ira Kawaller Kawaller & Company, LLC http://www.kawaller.com
Casting out on the Internet often results in a catch
Over a year 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 [firstname.lastname@example.org]
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 435-437, 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
My original Example 5 solution is in the 133ex05a.xls Excel workbook at http://www.cs.trinity.edu/~rjensen/
Also see 133ex05.htm at http://www.cs.trinity.edu/~rjensen/
Return to Bob Jensen's main glossary on derivative financial instruments accounting --- http://www.trinity.edu/rjensen/acct5341/speakers/133glosf.htm