Working Paper 315

The Theory of Interest Rate Swap Overhedging

Bob Jensen at Trinity University

Introduction

The purpose of this paper is to define the concepts of underhedging and overhedging in interest rate swaps. The paper was inspired at a December 1999 workshop in New York City when Martin Klein of Lehman Brothers alleged that overhedging was an unresolved FAS 133 issue for some clients. This paper demonstrates how overhedged swaps might be accounted for under FAS 133/138 from the Financial Accounting Standards Board (FASB) and IAS 39 from the International Accounting Standards Committee (IASC). [i]

Fixed-Fixed Swap Leg Foreign Currency Overhedging

The most obvious case of underhedging or overhedging arises when both legs of the swap have fixed rates. For example, it is common when hedging foreign exchange (FX) to have both legs of the swap be fixed in terms of interest rates. However, the legs are then specified in different currencies. For example, suppose that XYZ Company borrows DM20 million German marks at a rate R=8.00%. In order to hedge its FX exposure, it negotiates a swap to pay out a fixed rate P=10.27% and receive a hedged fixed rate H=f(P)=10.20%. Our general definitions are as follows:

R=hedged item fixed rate=8.00% payable in GermanDM (marks) on the hedged item

P=swap rate payable fixed rate of the hedging swap=10.27% in US$

F(P)=fixed rate receivable in the swap that increases monotonically with negotiated P levels

H=f(P)=swap rate receivable fixed rate of the hedging swap=10.20% receivable in GermanDM

H-P=net swap fixed rate differential=? Not comparable since rates apply to different currencies

U=R-H = overhedging/underhedging rate=-2.20%

Underhedge Þ (R-H)>0 and H=f(P)<f(R) with swap value V(FX rates and H)

Matched Hedge Þ (R-H)=0 and H=f(P)=f(R) with swap value V(FX rates and H)

Overhedge Þ (R-H)<0 and H=f(P)>f(R) with swap value V(FX rates and H)

Only in a matched swap will there be zero FX speculation in foreign currency by XYZ Company. A matched swap arises when U=R-H=0.00%. When U>0, some portion of the FX risk is not hedged. When U<0, there is too much hedging in terms of FX risk. The swap is used to hedge the FX exposure on both the principal of DM20 million and the R=8.00% interest payable quarterly at R/4=2.00%. My Working Paper 288 case illustration of overhedging can be found at http://www.cs.trinity.edu/~rjensen/288wp.htm.

Recall that XYZ Company only needs (R/4)(DM20,000,000)=DM400,000 to hedge its quarterly interest payments on the hedged item. The swap hedge pays XYZ Company (H/4)(DM20,000,000)=DM510,000 each quarter. With U=0 with R=H, the firm would have instead had a perfectly matched foreign currency (FX) hedge with no FX speculation. But when U=R-H<0, the firm is overhedged since there will be an excess of German marks arriving from the swap vis-à-vis the German marks needed for the hedged item. This means that the firm has too many German marks and is put into a FX speculation at a quarterly rate of -U/4. Under these definitions with a U=8.00%-10.20%=-2.20% overhedging rate, a R=8.00% hedged item rate would be overhedged by a rate of 2.20%/4=0.0055% per quarter. This translates into a speculation (0.0055%)(DM20,000,000)=DM110,000 excess German marks per quarter subject to FX risk.

When U=R-H<0, the overhedging proportion is -U/R=(H/R-1). In the above example, the overhedging proportion is 0.2750%=-0.0220/0.0800. Hence 27.50% of swap value should be charged to earnings rather than other comprehensive income (OCI). This represents the excess FX speculation exposure generated by a P=10.27% German mark swap leg payable giving rise to a H=f(P)=10.20% leg receivable in German marks. In a perfectly matched hedge, P would only have to be set to generate a H=f(P)=R=8.00% rate payable on the hedged item. When U=0 or U>0, there is no overhedging and all changes in the swap’s value can be posted to OCI under FAS 133 rules for a cash flow hedge. However, when U<0 overhedging exists, as in the above example, the –U/R proportion of swap value changes must go to current earnings rather than OCI. Given the swap values derived in my Working Paper 288 case, the overhedging postings accumulated in retained earnings are as shown below:

Working Paper 288 Case With R=8.00% and H=10.20% on a DM20,000,000 Notional
http://www.cs.trinity.edu/~rjensen/288wp.htm.

Date

DM/$
Ex Post
FX Rate

Change in
Interest Rate
Swap Value in $

Change in
Hedge
OCI in $

Overhedge
Change in
RE in $

09/30/X1

12/31/X1

03/31/X2

06/30/X2

09/30/X2

12/31/X2

03/31/X3

06/30/X3

2.0225

2.0150

1.9875

1.9750

1.9685

1.9480

1.9325

1.9300

($3,891)

$8,887

$976

$3,799

$4,020

($2,054)

($4,090)

($7,646)

(2821)

6443

708

2754

2915

1489

2965

5543

$1,070

($2,444)

($268)

($1,045)

($1,106)

$565

$1,125

$2,103

When R>H and U=R-H>0, the payable leg rate P is set too low for the receivable leg hedge rate H to fully hedge the foreign currency risk of the hedged items interest payments of DM400,000 per quarter. Hence, R>H is an underhedge by U=R-H. You can view a case illustration of FX underhedging at http://www.cs.trinity.edu/~rjensen/287wp.htm. With underhedging, the entire in swap value can be charged to OCI.

When U=R-H>0, the firm is underhedged since there will be a shortage of German marks arriving from the swap vis-à-vis the German marks needed for the hedged item.. This means that the firm now has too few German marks and is put into a FX speculation at a quarterly rate of +U/4. When R=12.00%, the hedged note would be underhedged at an annual rate of 12.00%-10.20%-1.80%.. Under these definitions with a U=1.80%, the quarterly underhedging is 1.80%/4=0.45%. The firm needs (R/4)(DM20,000,000) =600,000DM for quarterly interest payments on the hedged item. In the swap hedge it is only receiving (H/4)(DM20,000,000)=510,000DM. This leaves XYZ Company exposed to FX risk on a shortage of (0.45%(DM20,000,000)=90,000DM every quarter. Since the entire hedge rate H goes toward hedging the R hedged item rate and R>H, all of the hedge receivable can be posted to OCI each quarter under FAS 133 rules.

Working Paper 287 Case With R=12.00% and H=10.20% on a DM20,000,000 Notional
http://www.cs.trinity.edu/~rjensen/287wp.htm.

Date

DM/$
Ex Post
FX Rate

Change in
Interest Rate
Swap Value in $

Change in
Hedge
OCI in $

Underhedge
Change in
RE in $

09/30/X1

12/31/X1

03/31/X2

06/30/X2

09/30/X2

12/31/X2

03/31/X3

06/30/X3

2.0225

2.0150

1.9875

1.9750

1.9685

1.9480

1.9325

1.9300

(3,891)

8,887

976

3,799

4,020

(2,054)

(4,090)

(7,646)

(2821)

6443

708

2754

2915

1489

2965

5543

It should be noted in passing that FX hedges such as the swap illustrated above are not eligible for the Shortcut Method in FAS 133. The Shortcut method is expressly limited to vanilla interest rate swaps. FX hedging requires quarterly hedge effectiveness testing even though there really is no change of hedge ineffectiveness in the above illustration.

Fixed-Variable Interest Rate Swap Leg Overhedging

I begin this section what is, arguably, the most difficult and the most misleading illustration in FAS 133 --- Example 5 beginning in Paragraph 131. Initially it should be noted that Example 5 qualifies for the FAS 133 “Shortcut Method” such that there is never a question of cash flow hedge ineffectiveness. All changes in swap value are posted in Other Comprehensive Income (OCI) and, thereby, do not affect current earnings. Hubbard and Jensen (2000) are critical of Example 5 due to some minor errors and some major deficiencies. [ii] However, for purposes of this paper, Example 5 will be taken as given in FAS 133.

Example 5 beginning in Paragraph 131 focuses on an application of FAS 133 accounting 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 (H=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 XYZ Company 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.

For notational purposes, let r=2.25% depict the incremental rate of the hedged item, i.e., the investment earns (LIBOR+R)/4 in quarterly interest payments. Let p=0.00% depict the incremental rate of the swap leg payable such that every quarter XYZ Company owes (LIBOR+0.00%)/4 on the swap. Hence, the definitions are as follows:

R=LIBOR+r=LIBOR+2.25% is the hedged item rate receivable in US$

P=LIBOR+p=LIBOR+0.00% is the swap payable variable rate on the hedge payable in US$

U=swap rate differential=r-p=2.25%-0.00%=2.25% payable in US$

H==f(LIBOR)+p=6.65% is the negotiated swap receivable rate based upon both p and LIBOR swap curve in US$

Although U>0 might be viewed as an underhedge and U<0 as an overhedge, the value of U really does not affect the degree of interest rate speculation in Example 5. Since the r and p rates are both payable in the same currencies, any differential will be washed out in the derivation of the H fixed rate receivable under the swap hedge. Suppose that when r=p and U=0 results in a negotiated U=f(p)=8.90% swap payment divided into quarterly installments. In this “pure” matched swap, the XYZ receives H=8.90% in the swap and pays out LIBOR+2.25%.

H=f(LIBOR)+p=6.65%+p=6.65%+2.25%=8.90%

H-P=(f(LIBOR)+p)-LIBOR-p=6.65%-LIBOR

The net swap cash flow rate each quarter would be as follows for the hedged item and the hedge combined:

(H-(LIBOR+p))/4=(8.90%-(LIBOR+2.25%))/4=(6.65%-LIBOR)/4

In the FAS 133 Example 5 we have r<p where r=2.25% and p=0.00% such that the net swap cash flow rate for the hedged item and hedge combined becomes:

(H-(LIBOR+p))/4=(6.65%-(LIBOR+0.00%))/4=(6.65%-LIBOR)/4

The outcome will always be identical since the swap receivable rate is H=f(LIBOR)=6.65% is constant. In an Example 5 swap, it is inconceivable that any other type of swap receivable rate could be negotiated. The reason is that the H hedging swap receivable rate and the P swap payable rate are both netted out in the same currencies. The 6.65% is derived from the initial yield curve based on LIBOR at the start of the swap agreement. Any differential in P=LIBOR+p levels in the swap will be automatically adjusted by the same amount in the swap receivable rate U=f(P). It is analogous to swapping something that varies in value X(t) over time. If you swap for $110 and pay X(t)+$100 it makes no difference vis-a-vis when you receive $210 and pay X(t)+$200. By netting the swap payments, you net out at X(t)+$10 in either case as long as your are paying and receiving in the same currencies.

In other words the hedging swap receivable rate H is a constant rate that is independent of the swap payment increment p. Any change in p during the swap negotiations is offset by H=f(LIBOR). Hence, underhedging or overhedging is washed out in the net settlements. Accordingly, I have no argument with the Paragraph 137 of Example 5 accounting that allows all changes in swap value to be posted to OCI.

The FASB provides no help in valuation of swaps. Jensen and Hubbard (1980) show how the following V(LIBOR) Example 5 values can be derived at http://www.trinity.edu/rjensen/caseans/133ex05.htm:

Date

V(LIBOR)
Swap Value in $

Hedge
OCI in $

Overhedge
Retained Earnings in $

09/30/X1

12/31/X1

03/31/X2

06/30/X2

09/30/X2

12/31/X2

03/31/X3

06/30/X3

24,850

73,800

85,910

(42,820)

(33,160)

(21,850)

1,960

(24,850)

(73,800 )

(85,910)

42,820

3,3160

2,1850

(1,960)

If r>p or r<p, the swap value V(LIBOR) should be identical to the value when r=p in a matched hedge. Hence the accounting will be no different no matter what value of p swap payable rate increment is negotiated in the interest rate swap.:

Fixed-Variable Swap Leg Foreign Cross-Currency Overhedging

The FAS 138 amendments of FAS 133 made selected compound derivative instruments eligible for FAS 133 hedge accounting treatment. This is the main reason why two FASB members dissented on issuing FAS 138. One such compound derivative that became eligible for FAS 133 hedge accounting is a cross-currency swap that simultaneously hedges both interest rate risk and FX risk. Paragraph 29 of FAS 138 reads as follows:

The Board’s decision to permit fair value hedge accounting for assets and liabilities denominated in a foreign currency relates to the ability of an entity to designate a compound derivative as a hedging instrument in a hedge of both interest rate risk and foreign exchange rate risk. An entity’s ability to use a compound derivative would achieve the same result that would be achieved prior to this amendment with the use of an interest rate derivative as a qualifying hedging instrument to hedge interest rate risk and an undesignated foreign currency derivative to hedge exchange rate risk. Permitting use of a compound derivative in a fair value hedge of interest rate risk and foreign exchange risk would result in the value of the foreign currency asset or liability being adjusted for changes in fair value attributable to changes in foreign interest rates before remeasurement at the spot exchange rate. The ability to adjust the foreign currency asset or liability for changes in foreign interest rates effectively eliminates any difference recognized currently in earnings related to the use of different measurement criteria for the hedged item and the hedging instrument. The Board concluded that in the situations in which fair value hedges would be used, remeasurement of the foreign-currency-denominated asset or liability based on the spot exchange rate would result in the same functional currency value that would result if the instrument was remeasured based on the forward exchange rate.

Two of the seven FASB members dissented from issuing FAS 138 primarily due to the to partial hedging of interest rate risk and compound hedging of joint interest rate and FX risks. You can read the following beginning on Page 25 of FAS 138 regarding the dissents of Messr.s Foster and Leisenring:

While Statement 133 gave wide latitude to management in determining the method for measuring effectiveness, it is clear that the hedged risk is limited to (a) the risk of changes in the entire hedged item, (b) the risk attributable to changes in market interest rates, (c) the risk attributable to changes in foreign currency exchange rates, and (d) the risk attributable to changes in the obligor’s creditworthiness. Those limitations were designed to limit an entity’s ability to define the risk being hedged in such a manner as to eliminate or minimize ineffectiveness for accounting purposes. The effect of the provisions in this amendment relating to (1) the interest rate that is permitted to be designated as the hedged risk and (2) permitting the foreign currency risk of foreign-currency-denominated assets and liabilities to be designated as hedges will be to substantially reduce or, in some circumstances, eliminate the amount of hedge ineffectiveness that would otherwise be reflected in earnings. For example, permitting an entity to designate the risk of changes in the LIBOR swap rate curve as the risk being hedged in a fair value hedge when the interest rate of the instrument being hedged is not based on the LIBOR swap rate curve ignores certain effects of basis risk, which, prior to this amendment, would have been appropriately required to be recognized in earnings. Messrs. Foster and Leisenring believe that retreat from Statement 133 is a modification to the basic model of Statement 133, which requires that ineffectiveness of hedging relationships be measured and reported in earnings.

Neither FAS 133 nor FAS 138 address the problem of overhedging and underhedging in compound derivative instruments such as cross-currency hedges with interest rate swaps. Overhedging and underhedging of risk can arise in compound hedges that simultaneously hedge interest rate risk and FX risk. In the FAS 133 Example 5 discussed above, overhedging and underhedging (when r¹p) did not matter since swap legs were all in US$ and any difference between. In a cross-currency swap, overhedging and underhedging will matter since the swap legs are in different currencies.

Consider Example 2 in Section 2 of the FASB document entitled “Examples Illustrating Applications of FASB Statement No. 138” that can be downloaded from http://www.rutgers.edu/Accounting/raw/fasb/derivatives/examplespg.html. In that example the hedged item rate is fixed at R=5.68% and the hedge receivable rate is fixed at a matching H=5.68%. Since both rates are fixed and payable in the same (Euro) currency, there is no overhedging or underhedging of FX risk in that example. The hedge is a cross-currency hedge only because the swap payable leg has a variable rate rather than a fixed rate payable in US$. The FX risk would be unmatched only if the example were changed to where R¹H. For example, H=6.00%, there would be an overhedge by R-H=0.32%. If H=6.00%, the underhedge would be 0.68%. The analysis of the impact on the OCI is identical to that illustrated in the Working Papers 287 and 288 discussed above. The main impact of the cross-currency phenomenon (of having the swap leg payable rate P be variable) is that the fixed swap H receivable rate negotiated at the beginning of the swap is not necessarily identical to the fixed rate negotiated when the swap payable rate varies each quarter with changes in interest rates. Once a fixed rate H is negotiated, the fixed hedged item rate R and the fixed hedge receivable rate H become unmatched whenever R¹H.

Cross-currency hedging is only slightly more complicated when the hedged item and the swap leg receivable both have variable rates measured in the same foreign currency. For example, Example 5 of FAS 133 is complicated by an FX risk where R=LIBOR+r and P=LIBOR+p are both computed in terms of German marks instead of US$. Assume the swap receivable leg remains fixed at H in US$. For example, suppose r=2.25% and p=0.00% are payable in German marks and H=6.65% is receivable in US$. Since r and p are still measured in an identical currency, the value of H should be adjusted automatically for any r-p difference such that the net swap payment varies only with LIBOR as shown below:

(H-(LIBOR+p))/4=(6.65%-(LIBOR+0.00%))/4=(6.65%-LIBOR)/4

As a result, any overhedging or underhedging in terms of r and p will not affect the value of the swap.

In summary, when the hedged item receivable rate R and hedge payable rate P are both variable rates applying to the same currency, there is no FX overhedging or underhedging risk that impacts swap valuation. If the hedged item rate R and the hedged item rate H are both fixed, there is overhedging if R<H and underhedging if R>H. in terms of fixed rates for R and H.. It really doesn’t matter for measurement of overhedging whether the opposite swap leg rate is fixed at P or varies with interest rates in a cross-currency swap. When there is overhedging, OCI must be reduced by impact of the overhedging if the swap is a cash flow hedge. If there is underhedging of a cash flow swap or if the swap is a fair value hedge, then OCI does not enter into the picture under FAS 133 rules.

THE REMAINDER OF THIS SECTION IS NOT FINISHED!

Conclusion

The FASB has always confined its examples to matching or underhedging derivative financial instruments that are not speculative. This paper illustrates what happens with overhedging and proposes a way of dealing with it as long as the hedge is not a compound hedge of more than one risk factor. With compound hedging, measuring of hedge ineffectiveness and overhedging becomes much more troublesome.

Working Paper 315

The Theory of Interest Rate Swap Overhedging

Fixed-Fixed Swap Leg Foreign Currency Overhedging

Fixed-Variable Interest Rate Swap Leg Overhedging

Fixed-Variable Swap Leg Foreign Cross-Currency Overhedging

Conclusion

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