SSRN-id3230485 (2) The use of derivatives to hedge market risk in corpotare financing

The Use of Derivatives to Hedge Market Risk in Corporate Financing
The Use of Derivatives to Hedge Market Risk in
Corporate Financing
Javier Sánchez Verdasco a, 1
a
Affiliate professor Corporate Finance and Financial Markets at ESCP Europe, IEF, IEB
and IESIDE Business Institute. Managing Director at Incompany Formación en Finanzas.
ABSTRACT
In this tutorial article, the strategies available to hedge market risks arising
from different financing instruments are explained. Financial derivatives,
whether futures or options have been widely applied in companies to
mitigate or eliminate potential losses due to the uncertainty in interest or
foreign exchange currency rates. However, the mathematical complexity of
derivatives has sometimes been a barrier to non-highly specialised financial
managers in understanding their foundations, advantages and ways to apply
them in exposure reduction strategies. To address this issue, a practical
approach to the use of derivatives is presented in this article. The swap
valuation concepts and foundations of pricing are dealt with in the text, but
their formal valuation techniques are described separately in the appendices.
Explanations will be provided to calculate option premiums with the
extensively used and free downloadable software of John Hull (Derivagem),
but the mathematics involving the models used in option valuation will not
be shown as they are outside the scope of this paper.
Keywords: hedge; IRS; swap, corporate risk, swaption, CAP, Collar,
currency swap, exchange
JEL: A20,A22,A23,A33,G10,G11,G13,G15,G23,G32,M21
1
jsverdasco@incompany.es
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Contents
Abstract .................................................................................................................... 1
Introduction .............................................................................................................. 4
Types of Financial Risks .......................................................................................... 4
Financial Risks in financing structures ..................................................................... 4
What is a derivative? Futures and Options. .............................................................. 5
Hedging interest rate risk in financing ..................................................................... 6
Interest Rate Swap (IRS) .......................................................................................... 7
Definition.............................................................................................................. 7
Fix for Floating IRS (plain vanilla or coupon swap) ........................................... 8
Main features of IRS ............................................................................................ 8
How to use an IRS to hedge a bullet loan that pays a floating interest ................ 9
How to hedge the interest rate of a bond to be issued in the future (rate-lock) .. 10
Forward Swaps ....................................................................................................... 13
Hedging with interest rate options: Caps, Floors and Collars. Swaptions.............. 16
Hedging with CAPs ............................................................................................ 17
Hedging with COLLARs .................................................................................... 20
Swaptions ........................................................................................................... 22
Hedging Exchange Risk ......................................................................................... 25
Currency Forward............................................................................................... 25
Currency Swap ................................................................................................... 26
Hedging a fixed rate facility in a foreign currency ............................................. 26
Hedging a floating rate facility in a foreign currency......................................... 27
Equity Swaps .......................................................................................................... 28
Hedging on stock buys pending settlement ........................................................ 29
Hedging an equity position ................................................................................. 30
Appendix I. Zero-coupon yield calculation ............................................................ 31
Zero-coupon vs par yield rates ........................................................................... 31
How to derive the zero-coupon rates from the Swap Curve ............................... 32
The bootstrap method: .................................................................................... 32
The matrix approach: ...................................................................................... 33
Calculating zero rates (spot) in Excel ................................................................ 35
Appendix II. Forward Rates. Concept and calculation. ......................................... 36
Appendix III. Introduction to Interest Rate Swaps Valuation ............................... 38
Bond methodology .............................................................................................. 38
Forward Rates methodology .............................................................................. 40
Appendix IV. Introduction to Option Valuation.................................................... 41
Intrinsic and Time (or extrinsic) Value of an option .......................................... 41
Interest rate options valuation............................................................................ 43
Caps ................................................................................................................ 43
Swaptions ....................................................................................................... 46
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
References .............................................................................................................. 47
Figures
Figure 1. Coupon swap: receives floating pay fixed ................................................ 8
Figure 2. Purchase a call interest rate option as a part of a cap (1) ........................ 17
Figure 3. Purchase a call interest rate option as a part of a cap (2) ........................ 18
Figure 4. Sell a put option as a part of a floor ........................................................ 21
Tables
Table 1. Floating interest rate loan. Cash flow structure .......................................... 9
Table 2. Interest Rate Swap yield curve ................................................................. 10
Table 3. Hedged structure: floating rate loan + IRS. Cash flow structure.............. 10
Table 4. Interest Rate Par Swap and Zero-coupon yield curves ............................. 11
Table 5. Interest Rate Par Swap + 1% and Zero-coupon yield curves ................... 12
Table 6. Hedge structure: floating rate loan + forward swap (1)............................ 14
Table 7. Interpolated zero-coupon rates ................................................................. 15
Table 8. Hedge structure: floating rate loan + forward swap (2)............................ 16
Table 9. Several zero cost collar alternatives ......................................................... 22
Table 10. IRS Market rates EUR and USD ............................................................ 26
Table 11. Hedging a fixed rate bond in foreign currency through a ccy swap ....... 27
Table 12. Hedging a floating rate loan (bond) in foreign currency through a ccy
swap ........................................................................................................................ 28
Table 13. Equity swap ........................................................................................... 29
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Introduction
The effort that a company makes to correctly manage its operations can be
worthless if the financial risks cause a loss in business results. The key
challenge for the corporate risk manager is to determine the risks the
company is willing to run and the ones it wishes to mitigate through a
hedging strategy, frequently, by using financial derivatives (futures and
options).
Though financial risks arise not only from financing decisions but also from
export-import activities that involve exchange and commodity risks, this
article will deal just with the former by identifying interest and exchange
risks appearing in different financing instruments and showing how can they
be hedged using futures (or forwards) and options.
To avoid distraction from the core concepts related to hedging financing
facilities, some related notions that are convenient to know, but not strictly
necessary, have been separately explained in the annexes: the zero-coupon
yield curve calculations and the foundations of swap and options valuation.
Types of Financial Risks
Financial Risks in companies can be classified as credit, liquidity and
market risks. The former is related to the eventual impact of a client’s
insolvency on the business results. Liquidity risk refers to the difficulty to
commit to short-term payments or the lack of capacity to sell the assets of
the company at “a reasonable price”. Finally, market risks include those
exposing the firm to a potential loss due to changes in prices of the financial
market variables. Market risks are classified as interest, foreign exchange,
stock prices and commodity risks.
Financial Risks in financing structures
Among the above mentioned, the risks a financing structure can face are:
• Liquidity risk. To avoid it, the non-current assets have to be financed
with permanent resources. That is to say, with equity and long-term
debt, since if the assets were financed with short-term debt and could
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
not be rollover at maturity, the firm would incur suspension of
payments.
• Interest rate risk. Long-term loans frequently pay a floating interest
related to a market rate (e.g. 1-year LIBOR). The interest rate is
periodically revised in accordance with the reference rate, which
could be higher than previously.
Additionally, short-term loans, once they mature, need to be renewed
to maintain the working capital financing structure. Of course, the
new interest rate could be higher. In both cases, higher financial costs
could damage the P&L account of the company.
• Foreign Exchange risk (fx). When the headquarters of a company is,
for instance, in the eurozone and it is being financed in another
currency (e.g. USD or yen), the counter value in euros of both the
service of debt (principal and interest payment) and the remaining
amount pending to be paid increase when the foreign currency
appreciates. We will analyse this in greater detail in the epigraph
corresponding to foreign exchange hedging.
The objective of this paper is to show how to hedge the interest and fx risk
in financing structures using derivatives (futures and options).
What is a derivative? Futures and Options.
A derivative is a financial instrument whose price depends on another socalled underlying asset. Regarding the market risk variables, they are
classified as derivatives in commodities (oil), currencies (EUR/USD),
stocks (Microsoft) and stock indices (S&P 500) or interest rates (LIBOR 1
year, US 10-year Treasury Bonds). Other derivatives related to credit risk,
such as credit default swaps, exist but are not covered in this paper.
One of the characteristics of a derivative is that there is a commitment to
some conditions today that will apply or may apply at a future date. If the
buyer must commit to the agreement, the derivative is called a future, but
when he has the right but not the obligation, it is called an option.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
A future obliges a buyer and a seller to respectively buy or sell an
underlying asset at a determined price at a future date. In a more restricted
definition future is reserved to those deals transacted in exchange markets
(e.g. Chicago Mercantile Exchange), while the term forward is used for
operations in OTC 2 markets.
An option, however, gives the buyer the right to buy (call) or sell (put) the
underlying asset at the strike or exercise price. The option will be only
exercised if it is convenient for the option holder (the option buyer).
In the case of a 3-month future bought on STOCK1 at 20 €, if at the
expiration date the STOCK1 price is, for instance, 22 €, the result of buying
at 20 € and selling at 22 € will yield earnings of 2 €. Conversely, if the price
had been 17 €, a loss of 3 € would have arisen.
When the deal is contracted through an option, in the case the spot price at
expiration is 22 €, the buyer will exercise the option and obtains a payoff of
2 €. However, the option will not be exercised if the stock price is 17 € since
there is no gain in exercising at 20 € when the stock can be bought at 17 € in
the market. Differently from futures, the buyer will exercise only when the
payoff is positive, so the seller will require being compensated with an
option premium. This option premium paid by the buyer represents his
privilege with respect to the seller of having rights but no obligations.
Hedging interest rate risk in financing
Two main types of interest-bearing financing resources can be considered:
• Bank loans.
• Bond issues.
These instruments are subject to interest rate risks. It is important to note
that both long and short-term financial liabilities can be exposed to financial
risk. Long-term liabilities are frequently referenced to market rates (e.g. 32
OTC: Over the Counter refers to bilateral dealing and no clearing house intervenes to
regulate the market. The OTC markets are more flexible but less secure in terms of credit
risk than the exchange markets.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
month LIBOR) whose levels are unknown in advance. On the other hand,
the short-term financial debt has to be rolled over to meet the operating
working capital needs, again at unknown interest rates. In the case of an
increase in interest rates in a financing structure, financial expenses rise, and
earnings decrease.
As regards their nominal evolution, financing structures can be classified as:
• Bullet, the same nominal throughout the life of the loan or bond and a
unique principal payment at maturity
• Oscillating:
− Seasonal, such as a line of credit or discount on bills of exchange
financing whose nominal depends on the treasury needs of the
company.
− Amortizing, where the principal amount decreases throughout the
life of the loan. Examples are a mortgage or a car or machinery
loan.
− Accreting, when the nominal increases through the life of the loan.
This type of loans is related to construction projects (highways,
hospitals, power facilities) where increasing financial resources are
needed up to the end of the construction. Once the project is
finished, the loan generally follows a bullet or amortising pattern
of payments.
All hedging instruments presented below suit fixed or oscillating financing
structures.
Interest Rate Swap (IRS)
Definition
An interest rate swap (IRS) is a financial derivative instrument in which two
parties agree to exchange interest rate cash flows, based on a specified
notional amount from a fixed rate to a floating rate (or vice versa) or from
one floating rate to another.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Fix for Floating IRS (plain vanilla or coupon swap)
Coupon swaps involve an exchange of fixed rate for floating rate payments
on a notional amount during a period. For instance, a company may receive
annually during 5 years the 1 year LIBOR rate (whatever the rate is every
year) multiplied by a notional amount of 1 million USD and pay yearly to a
bank a fixed rate (the 5-year swap rate at inception of the deal) on the same
notional. Conversely, the company can receive the fixed rate and pay the
floating one (see figure 1).
Figure 1. Coupon swap: receives floating pay fixed
Fixed rate
Annual, during T years
Bank
Company
Floating rate
LIBOR
3M, 6M or 1Y
This type of IRS is the one used for hedging floating rate financial facilities.
Other IRS such as basis swaps, where an institution receives and pays
floating rates of different terms (e.g. receives EURIBOR 3M and pays
EURIBOR 6M), are related to positions on the slope of the interest yield
curve. These swaps are outside the scope of our objective.
Main features of IRS
• They are tailor-made contracts; therefore, the terms of the contract
(interest rate references, currency, notional amount, calculation basis,
maturity) are agreed by the two counterparties.
• In practice, maturities span between 1 and 30 years.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
• The amount is notional, that is to say, there is no real cash delivery,
neither at inception nor maturity. There are cash movements only on
interest rates differences on a fictitious nominal.
• Usually, netting applies: the cash flow movement is the difference
between the money to be paid and received.
• There is a credit risk, since one counterparty may not commit to its
obligation to the other party throughout the life of the swap.
How to use an IRS to hedge a bullet loan that pays a floating interest
Assume a floating rate loan with the following features (table 1):
Table 1. Floating interest rate loan. Cash flow structure
Loan
Rate of interest
Euribor 1 y + 1%
1.12%
1,000,000
Euribor 1 y (current)
Loan €
Years
Pay
1
2
3
Euribor 1.12% + 1%
Euribor??? +1%
Euribor??? +1%
To hedge an eventual increase in the 1-year Euribor interest rate, which
would cause an undesirable rise in financial expenses, we contract an IRS
receiving 1-year Euribor and paying the 3-year fixed rate. Assuming this
fixed rate is currently 1.87% in the market, as indicated in table 2:
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Table 2. Interest Rate Swap yield curve
Years
1
2
3
5
7
10
Swap
rates
1.12
1.50
1.87
2.46
2.91
3.34
The hedged financial structure (loan + IRS) will be:
Table 3. Hedged structure: floating rate loan + IRS. Cash flow structure
Loan
IRS
Loan +IRS
Euribor 1 y + 1%
Rate of interest
1.12%
1,000,000
Euribor 1 y (current)
Loan €
Receive Floating
Pay fixed
Fixed rate +
spread
Pay
Pay
Years
Pay
Receive
1
2
3
Euribor 1.12% + 1%
Euribor??? +1%
Euribor??? +1%
Euribor 1.12%
Euribor ???
Euribor ???
1.87%
1.87%
1.87%
2.87%
2.87%
2.87%
so that, whatever happens, the financial cost remains fixed at 2.87% for the
next 3 years (the 3-year swap rate + the spread over Euribor)
How to hedge the interest rate of a bond to be issued in the future (ratelock)
When a company plans to issue a fixed rate bond to meet a financing need
in the future, it is running an interest rate risk from the date the issue is
considered up to the moment it is settled. To hedge this risk, the company
can contract an IRS receiving floating and paying a fixed interest rate. This
IRS will be cancelled at the date the bond is issued. The result will be that
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
the cancellation value of the swap will compensate the interest rate variation
in the market, as shown below 3.
The zero-coupon interest rates derived from the par swap shown in table 2
are:
Table 4. Interest Rate Par Swap and Zero-coupon yield curves
Years
1
2
3
Par Swap
rates
1.120
1.500
1.870
Zero
Coupon
Rates
1.120
1.503
1.879
One of the methodologies used in IRS valuation is to consider that the swap
is composed in fact of two bonds: the first leg is a floating rate bond
(receiving in our example) and the second one is a fixed coupon rate bond
(paying). 4
The price of a floating rate bond at inception or re-pricing dates of the
variable interest rate is always 100%: if interest rates go up, the coupon
payments increase, but the zero-coupon rates used to discount the cash
flows also rise. Both effects offset each other, and the bond continues at par
value.
While a fixed rate bond worths also 100 at inception -when receiving the
market fixed interest rate and discounted with the market zero coupon ratesthe price of the bond will change when a movement in interest rates takes
place: the discount rates will change but not the coupon received, that is
fixed.
At inception, in our example:
3
See appendix I to understand the calculation of the zero-coupon yield from the par swap
yield curve and appendix II for an introduction on swap valuation)
4
See appendices II and III for further detail
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Receive floating leg =100
Paying fixed leg =100, as shown below:
1.87
1.87
1.87
+
+
1
2
3
(1 + 0.0112 ) (1 + 0.01503) (1 + 0.01879 )
100 =
The swap value is: +100-100 = 0
To intuitively show the effect of an interest rate variation in the price of an
IRS, let’s assume that one second after inception a 1% parallel upward
movement in the swap rates occurs. The new swap and zero-coupon rates
will be now:
Table 5. Interest Rate Par Swap + 1% and Zero-coupon yield curves
Zero
Coupon
Rates
Par Swap
rates
Years
1
2
3
2.120
2.500
2.870
2.120
2.505
2.884
Then, the new price will be:
Receive floating leg =100
Paying fixed leg = 97.15:
97.15 =
1.87
(1 + 0.0212 )
1
+
1.87
(1 + 0.02505)
2
+
101.87
(1 + 0.02884 )
3
Swap value: +100-97.15 = 2.85
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Therefore, the cancellation value of this IRS would be + 2.85. The intuition
behind the calculations is that the swap holder is quite comfortable paying a
fixed rate of 1.87 when under the new circumstances the fixed market rate to
be paid per year would be 1% higher. If the counterparty wants to convince
the swap holder to cancel the deal, a compensation for the present value of
the difference has to be offered:
2.85 =
1
(1 + 0.0212 )
1
+
1
(1 + 0.02505)
2
+
1
(1 + 0.02884 )
3
If the bond is now issued, the coupon to be paid will be 1% higher, but
through the IRS cancellation, 2.85% will be obtained. This amount is
precisely the present value of the extra-cost of the bond. The same applies
when, for instance, the issue of the bond is to be performed in 6 months.
Then, a forward swap will be contracted: a 3- year IRS starting in 6 months
time.
Forward Swaps
When plain vanilla IRS are used in hedging floating interest rate loans, the
IRS cash flow exchange takes place from inception, though, in fact, the first
loan payment does not incorporate any risk, since it is known from the
beginning. In our example (table 1), the first year Euribor has been already
established (at 1.12%) and only the second and subsequent cash flows of the
loan are unknown. Despite that, the plain vanilla IRS (exchange of cash
flows since the beginning) is, due to its operating simplicity, the most often
used instrument for hedging floating rate loans. In such a way, the firm will
fix its financial expenses at 2.87% for years 1, 2 and 3 (the 1.87% three-year
swap rate plus the 1% bank margin) 5.
Another alternative with an equal financial result would be:
1.
5
Leave the loan without any hedging instrument during the
first year, that is to say, 1.12% + bank’s margin = 2.12%.
Note that the net cash flow of the IRS at the first year-end is already known 1,000,000
(1.12% − 1.87%).
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
2.
Contract a forward swap, which is an IRS starting at a future
date with a determined maturity since that moment. In our
case, starting in 1-year time for 2 years.
The forward swap rate to be applied to those 2 remaining years is the one
making both alternatives financially equivalent:
FORWARD SWAP
IRS
1.12
x
x
1.87
1.87
+
+
= 1.87 1 +
+
=5.4327
1
2
3
2
3
(1 + 0.0112 ) (1 + 0.01503) (1 + 0.01879 ) (1 + 0.0112 ) (1 + 0.01503) (1 + 0.01879 )
Isolating the unknown forward rate, an interest rate 2.257% is obtained as
the fixed rate for an IRS starting in 1-year time for 2-years 6.
Thus, the cash flow structure of the loan + forward swap combined position
will be:
Table 6. Hedge structure: floating rate loan + forward swap (1)
Rate of interest
Euribor 1 Y (current)
Loan notional
Loan + FW
Swap
Forward Swap
Loan
Euribor 1 Y + 1%
1.12%
1,000,000
Receive Floating
Pay fixed
Fixed rate +
spread
Pay
Pay
Years
Pay
Receive
1
2
3
Euribor 1.12% + 1%
Euribor??? +1%
Euribor??? +1%
Euribor ???
Euribor ???
2.257%
2.257%
6
2.120%
3.257%
3.257%
The what if /goal seek Excel tool can also be used by forcing the left side of the equation
to be 5.4327 by changing “x”
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The forward swap concept above has been developed from the equivalence
with a plain vanilla IRS: the results of performing the hedge in one way or
the other are indistinct.
However, the use of forward swaps in practice is related to hedging floating
rate loans that will start in the future. For instance, if the loan starts in 6
months and:
•
•
Euribor 6 months = 0.9%
2.5 year interpolated par swap rate 7= (1.500% + 1.870%)/2 = 1.685%
The resulting zero-coupon yield curve will be:
Table 7. Interpolated zero-coupon rates
Years
0.5
1.0
1.5
2.0
2.5
3.0
Par Swap
rates
0.900
1.120
1.310
1.500
1.685
1.870
Zero
Coupon
Rates
0.900
1.120
1.312
1.503
1.691
1.879
Note 8
Then, the rate of the forward swap to be applied to a floating rate financial
facility starting in 6 months with a residual life of 2 years will be:
7
8
See table 7
For the sake of simplicity zero rates have been obtained through linear interpolation. More
sophisticated techniques are applied in financial institutions’ Treasury Desks (such as the
Spline Cubic Methodology). The results differ around 0.01%. Therefore, the linear
interpolation can be used as an acceptable reference in corporate hedging strategies.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
FORWARD SWAP
IRS
0.839
0.9 / 2
x
x
1.685
1.685
+
+
=
+
+
1,5
2,5
(1 + 0.009 × 0.5) (1 + 0.01312 )1,5 (1 + 0.01691)2,5
(1 + 0.009 × 0,5) (1 + 0.01312 ) (1 + 0.01691)
(1 + 6 month payment)2 = (1 + 0.01685) ⇒ effective 6 month rate = 0.839%
Note 9
Isolating the unknown, a forward swap rate 1.883% is obtained and, to see
what the rate of the financial structure is, the spread of 1% is added:
Table 8. Hedge structure: floating rate loan + forward swap (2)
Loan
Rate of interest
Euribor 1 Y +
Loan notional
Loan + FW
Swap
Forward Swap
1%
Receive Floating
Pay fixed
Fixed rate +
spread
Pay
Pay
1,000,000
Years
Pay
Receive
0.5-1.5
1.5-2.5
Euribor??? +1%
Euribor??? +1%
Euribor ???
Euribor ???
1.883%
1.883%
2.883%
2.883%
Hedging with interest rate options: Caps, Floors and Collars. Swaptions
Swaps protect the results of the firm from an upward movement in interest
rates, but they do not permit the firm to take advantage of a decrease in the
financial expenses in the event of a fall in interest rates. Options allow the
right to buy (call) or sell (put) an underlying asset if convenient for the
buyer 10. Then, by using options, it is possible:
9
The term (0.9/2)/(1+0.009 x 0.5) does not indicate any actual flow in this case. It is used
to calculate the forward swap rate to be applied from year 0.5 to 2.5 based on the present
value of both financial structures with different cash-flow distribution but an equal result.
10
See a more detailed description of financial options in Appendix III
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
• To limit the financial cost by exercising the option if rates go above
the protection level (exercise price or strike).
• To benefit from a decrease in interest rates without exercising the
option and consequently leaving the rate of the financial structure as
low as the market rate of the loan (if below the strike).
As the option buyer has only right but no obligation, the seller will ask him
to pay a premium.
Hedging with CAPs
Following the 3-year loan example (table 5), for the first year we do not
need to buy an option (the 1-year Euribor is already established), but to
hedge the 2 remaining years with options, the following steps will have to
take place:
• First, buy an option call at exercise price 2.257% (the forward swap
rate of the above example) expiring the first day of the 2nd year. On
that date, the decision of exercising the option or not will be taken.
The eventual option payoff will occur on the last day of the 2nd year.
Figure 2. Purchase a call interest rate option as a part of a cap (1)
Hedging the
interest rate for
the 2nd year
Option
premium = 0.044 %
The buyer (hedger)
will exercise the
option
2.257%
1 year Euribor rate
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
• Second, buy another call option, same exercise price, to hedge the 3rd
year, that is to say, one expiring on the first day of the 3rd year, with
an eventual payoff at the end of the 3rd year.
Figure 3. Purchase a call interest rate option as a part of a cap (2)
Hedging the
interest rate for
the 3rd year
Option
premium = 0.498 %
The buyer (hedger)
will exercise the
option
2.257%
1 year Euribor rate
Note 11
A caplet represents each of the interest rate call options composing a CAP,
which is a chain of caplets one after another. By adding these two caplets a
premium CAP= 0.044% + 0,498% = 0.5424% is obtained.
The CAP exercise price -level of protection- can be decided by the hedger.
Conversely, the IRS or the forward swap rate are defined based on the
market yield curve, therefore cannot be chosen. A strike equal to the
forward swap rate has been applied in this example to facilitate the
comparison between both alternatives.
11
Option premiums have been obtained using the zero-coupon rates shown in table 4 and a
flat volatility of 20%. Calculation has been performed using the software Derivagem
(John Hull) www.rotman.utoronto.ca/~hull/software/DG200.01.xls
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19
The Use of Derivatives to Hedge Market Risk in Corporate Financing
EXERCISE. Forward Swap vs CAP
1. Following the above example, compare the results of the 3-year
interest rate hedging obtained through:
•
•
A 2-year forward swap starting in 1-year time
A CAP for the 2nd and the 3rd years with a strike equal to the
forward swap rate
2. Indicate:
a) The maximum and minimum cost of each alternative.
b) In which circumstances one would be inclined to choose one
alternative or the other.
Solution
1.
HEDGING WITH FORWARD SWAP
Loan
Rate of interest
Forward Swap
Euribor 1 Y + 1%
Euribor 1 Y (current)
Loan notional
1.12%
Receive
Floating
Pay fixed
Loan + Fw
Swap
Fixed rate
+ spread
1,000,000
Years
Pay
1
Euribor 1.12% + 1%
Years
Pay
HEDGING WITH CAP
Pay
CAP
Loan + CAP
Pay 0.5424% for 2 options
(years 2 and 3)
Euribor <
Euribor >
Euribor <
Euribor >
2.257%
2.257%
2.257%
2.257%
Receive
Pay
Pay
2.120%
2.120%
Eur+1%+
premium
3.257% +
premium
Eur+1%+
premium
3.257% +
premium
No payments
2.120%
2
Euribor??? +1%
Euribor ???
2.257%
3.257%
Eur less
2.257%
3
Euribor??? +1%
Euribor ???
2.257%
3.257%
Eur less
2.257%
2.a)
In case of hedging with a CAP, if there is an increase in interest rates above
the strike, an annualised extra payment of 0.5424%/ 2 years = 0.2712% has
to be added. Consequently, the maximum annual cost will be 2.257%+1%
+0.2712%. However, if rates go to zero, the minimum yearly cost of the
financial structure will be just 0%+1%+0.2712%, thus taking advantage of
the decrease in interest rates.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
2.b)
Assuming that an interest rate hedging is necessary since its evolution is
uncertain and can negatively affect business results if we believe that:
 Rates can fall, then a CAP is preferable since we take advantage of
that decrease (though the option premium needs to be afforded).
 “Almost sure” rates will rise. Then we will contract a swap since the
payment of the CAP premium is avoided and the cost of the
resulting financial structure, if interest rates finally rise, is lower than
the one of the CAP.
Hedging with COLLARs
As a cap is composed of a chain of interest rate call options, a floor consists
of a chain of interest rate put options. In the case, for instance, of an
Insurance Company that needs to protect the interest rate proceeds from its
floating rate investments, buying a floor could be an adequate hedging
strategy. At the respective expiration dates of those put options two
scenarios could take place:
a) The Insurance Company will exercise the options of the floor if
interest rates are below the strike and thus receive the decrease
in interest rates multiplied by the nominal, so a balance between
the decrease on the proceeds from the investment and the
options pay-offs will happen.
b) No exercise will occur in case interest rates are above the strike,
and the investment proceeds will be higher the greater the
interest rates.
In the case of hedging financing structures, floors are part of a structure
called collar, which is the combination of buying a cap and selling a floor.
The advantage of buying a cap with respect to contracting an IRS is that
there is protection from interest rates increases, but it also permits benefiting
from a decrease in interest rates. The disadvantage is that a premium must
be paid to the seller. If someone pays me money for renouncing the benefit
of the decrease in interest rates from a determined level, I could pay the
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21
The Use of Derivatives to Hedge Market Risk in Corporate Financing
premium cap with this money; then I would not have any costs for the cap
protection.
In fact, by selling a floor, I renounce a decrease in interest rates: if rates are
below the strike the benefits of a decrease in interest rates “are transferred”
to the floor buyer who paid me a premium that allows me to buy the cap
protection.
Figure 4. Sell a put option as a part of a floor
The buyer will
exercise the
option
1,89%
Option premium=
0,17 %
1 year Euribor rate
Assume that a cap with a 3% strike costs 0.17%. We ask the options
calculator to tell us the floor level whose premium is 0.17%: the result is
1.89%. Then we buy a cap with a 3% strike and sell a floor with a 1.89%
strike, thereby obtaining a collar which allows paying an interest rate in the
range of (3%, - 1.89%) 12at zero cost. Other zero cost structures could be:
12
To which the margin the bank applies on the floating interbank rates has to be added.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Table 9. Several zero cost collar alternatives
EXERCISE PRICE
OPTION PREMIUM
CAP
FLOOR
Cap=Floor
Collar cost
2.50%
2.14%
0.38%
0
3.00%
4.00%
1.89%
1.54%
0.17%
1.09%
0
0
Forward swap 1-3
2.257%
Note that, when broadening the collar range significantly, the result is
practically a floating rate financial cost and if the range is too narrow it is
equivalent to contracting a forward swap paying a fixed rate. Obviously,
non-zero cost collars can be structured by renouncing a decrease in interest
rates from a lower interest rate level. Then, a lower floor premium will be
received so that only a part of the cap cost will be subsidised.
Swaptions
Buying a swaption consists of paying a premium to have the possibility but
not the obligation to enter into an IRS at a future date at a determined fixed
rate. Assume that:
1.
It is probable, but not sure that external financial resources are going
to be needed.
For instance, we are negotiating an acquisition whose success is
subject to the shareholders meeting or the antitrust commission
approval. Buying an option on an IRS, the financing cost can be
fixed to assess the feasibility of the project in case it finally takes
place.
At the exercise date of the option to enter in the IRS, it could be
that:
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
a) The underlying deal does not take place:
− If at that time, the market rate of the IRS is higher than that of
the exercise price, the swaption will be exercised, and
immediately cancelled, since a profit from the present value of
the difference between market and strike prices will be obtained
13
.
− Conversely, if it is lower, no exercise will occur, since entering
and cancelling the IRS will cause losses.
b) The underlying deal takes place:
− If the market swap rate is lower than the strike, the swaption
will not be exercised, and an IRS will be contracted at market
price with a more favourable rate.
− If the market swap rate is higher, then the option will be
exercised entering into an IRS at the strike defined when the
swaption was contracted, which guarantees a fixed financial
cost during the life of the IRS.
2.
It is certain that financial resources will be needed, but a scenario of
a decrease in interest rates is probable. Therefore, hedging from an
interest rate increase is convenient but, at the same time, benefitting
from a decrease in rates is desirable. In this case, a swaption is
contracted and, if on the exercise date:
− The market rate is higher than the strike, the swaption will be
exercised entering into an IRS at the strike fixed rate.
13
If, for instance, the exercise price of an option on a IRS with annual payments is a fixed
rate of 2% and the market rate is 3%, by entering into the IRS and immediately
cancelling it, the amount to be received will be the summation of the present value of
each annual difference (3%-2%) x Nominal, discounted at the respective zero-coupon
rates.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
− The market rate is inferior to the swaption strike, there will be
no gain in exercising the swaption, and an IRS at a cheaper
rate will be contracted.
EXERCISE. CAP vs Swaption
For a hedge starting in 1-year time, hedge period 2 years, strike 2.257%,
reference rate 1-year Euribor.
Which of the following products will have a higher option premium?
a. Buy a CAP
b. Buy a swaption
Solution
• The CAP premium will be higher since the CAP protects from
interest rates higher than the strike, receiving the difference between
the market and the strike rates multiplied by the nominal. No
negative cash flows will take place, in any case. Conversely, in the
case of a swaption, once it is exercised, entering into an IRS paying a
fixed rate, it could occur that the floating rate (receive) is lower than
the fixed rate (pay) and, consequently, payments must be transferred
to the counterparty.
• An alternative way to explain it, is that protection against an increase
in interest rates is achieved with the CAP but, at the same time, an
advantage from a decrease can occur. However, once the swaption is
exercised, the financial cost will be fixed at the strike price for the
rest of the life of the financing structure, with no benefit from an
eventual downward interest rate.
• Using Derivagem a CAP Premium of 0.544% is obtained, while that
of the swaption is 0.369%
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Hedging Exchange Risk
Exchange Risk can be classified into three categories:
1. Transactional Risk, referring to the committed cash to be paid or
received in a foreign currency at a future date. Typically, exportimport transactions in which oscillation in the rate of exchange can
produce significant variations in the counter value of revenues or
costs in the domestic currency.
Currency forwards or options are frequently used to hedge this risk
2. Economic Risk, unlike the Transactional Risk, it affects the
revenues and expenses not already committed to being received in a
foreign currency in the course of the enterprise operations.
The best hedging strategy in this case -and probably the only one
useful- is trying to find natural hedges by compensating revenues
and expenses in the foreign currency where the company operates.
3. Translation Risk may arise from losses related to the conversion of
foreign exchange denominated assets and liabilities into the
domestic currency.
Currency swaps are generally used to hedge this type of risk.
Currency Forward
It is a contract by which the exchange rate for a purchase or sale of a
currency at a future date can be locked to avoid the foreign exchange risk of
the transaction.
Currency forwards are over-the-counter (OTC) instruments, as they do not
trade on a centralised exchange. Also known as an “outright forward.” As
mentioned above, it is frequently used to hedge a cash flow to be paid or
received in a foreign currency to avoid exchange risk. Unlike the currency
swap that hedges a stream of cash flows -as will be seen later-, the currency
forward hedges a single flow. The EUR/USD Exchange forward formulas
are shown below:
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
< 1 year

 1 + i$
Fw = Spot 

1 + i €

> 1 year
n 

360 
n 

360 
Fw = Spot
(1 + i$ )n / 360
(1 + i€ )n / 360
Where i represents the interest rate of each currency at a tenor of n days.
Currency Swap
The currency swap (sometimes named cross currency swap) is an instrument
that allows hedging a stream of foreign currency cash flows in a single
contract. For instance, the principal and interest payments of a bond or loan
denominated in a foreign currency. In that case, an appreciation of the
foreign currency (depreciation of the local) will cause more interest and
principal to be paid in the equivalent local currency. If EUR/USD changes
from 1.20 to 1.10 USD per 1 EUR, fewer USD will be received per EUR,
and more local currency will be needed to pay the debt service.
Hedging a fixed rate facility in a foreign currency
Assume the exchange and interest rates are:
Table 10. IRS Market rates EUR and USD
EUR/USD
1.2
Years
PAR
SWAP
EUR
PAR
SWAP
USD
1
2
3
1.120%
1.500%
1.870%
0.590%
1.230%
1.830%
In the case of a 3-year 1.83% fixed coupon bond amounting to 1 million
USD, the USD cash flows can be converted into the equivalent EUR
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27
The Use of Derivatives to Hedge Market Risk in Corporate Financing
payments through a CCY swap according to the financial structure shown
below:
Table 11. Hedging a fixed rate bond in foreign currency through a ccy swap
EUR/USD =
1.20
BOND
USD
Pay interest
1.83%
Lend
(deposit to)
Year
0
1
2
3
CCY SWAP
USD
€
Receive interest Pay interest
1.83%
1.87%
1,000,000.0
-18,300.0
-18,300.0
-1,018,300.0
-1,000,000.0
18,300.0
18,300.0
1,018,300.0
Borrow
(deposit from)
833,333.3
-15,250.0
-15,250.0
-848,583.3
The process can be described as follows:
1. A 1,000,000 USD bond is issued to obtain financing resources.
2. That amount is lent (deposited), receiving every year the 3-year
USD market interest rate 1.83%.
3. The counter value in EUR at the spot EUR/USD 1.2 (833,333.3
EUR) is borrowed, paying 1.87% each year (the market interest rate
at that tenure).
4. At maturity, the principals plus the last coupon of the bond and the
USD and EUR legs of the CCY swap are paid.
As shown, the bond and the USD leg of the CCY swap offset each other,
leaving just the EUR leg fixed payments of the CCY swap remaining.
Hedging a floating rate facility in a foreign currency.
In case the financial facility is a floating loan or bond in a foreign currency,
the market risk exposure can be offset through a floating-for-fixed CCY
swap. Thus, the floating rate payments in the foreign currency will be
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
converted into fixed-rate cash flows in the local currency, eliminating both
the interest and exchange risks:
Table 12. Hedging a floating rate loan (bond) in foreign currency through a ccy swap
EUR/USD =
1.20
BOND (LOAN)
USD
Pay interest
Year
USD 1 year Libor
CCY SWAP
USD
€
Receive interest
Pay interest
floating
fixed
USD 1 year Libor
1.87%
Lend
(deposit
Borrow
to)
(deposit from)
0
1,000,000
-1,000,000
833,333.3
1
2
- Libor x 1 MM
- Libor x 1 MM
- Libor x 1 MM
-1,000,000.0
+ Libor x 1 MM
+ Libor x 1 MM
+ Libor x 1 MM
1,000,000.0
-15,250.0
-15,250.0
3
-848,583.3
In fact, this hedge is equivalent to:
• An IRS in USD, receiving floating and paying fixed interest, then
converting the flows of the financing structure from floating to fixed
USD eliminating the interest rate risk; and
• A fixed-for-fixed currency swap, lending USD and borrowing EUR,
thus hedging the exchange risk.
Equity Swaps
Equity swaps are characterised by an exchange of cash flows in which at
least one of them is an equity flow. They can be classified into two types:
1. Transfering the change in equity results to a third party.
Counterparty A makes payments to B by the positive or negative
returns on a determined stock or index (including dividends), while
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
B pays to A floating or fixed interest rate on the corresponding
principal amount.
2. Swapping the results of two different equity assets or indices. A and
B exchange returns based on two different stocks or indices, for
instance, A transfers to B the proceeds (positive or negative) on
IBEX 35 and receives from B that of the DAX.
As the second type refers to the field of asset management, it is out of the
scope of the present work. Therefore, our analysis will focus solely on the
first.
Hedging on stock buys pending settlement
Assume that a purchase of shares of a listed company has been agreed upon
but cannot be settled for formal or legal reasons or, just, because no
financing resources have yet been obtained.
To all effects, both the buyer and the seller consider that the positive or
negative proceeds from the stocks belong to the buyer. Both parties enter
into an equity swap with the following structure:
Table 13. Equity swap
"LEGAL" OWNER OF THE SHARES "A"
Shares
EQUITY SWAP
"A"
Gain
Pays gain + dividends "A" Receives int.
to "B"
payment from "B"
"A"
Loss
Receives loss
"A" Receives int.
payment from "B"
Gain or loss = positive or negative results due to changes in the share
price
The financial effect for the “B” counterparty of the equity swap is as if he
actually has the economic rights on the shares and he pays the cost of
financing the amount needed to purchase them. While, for “A”, it is as if he
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
sells the shares and places the proceeds in a deposit receiving an interest rate
on the corresponding principal amount.
Hedging an equity position
The equity swap is also used when the holder of an equity position wishes to
maintain the legal rights (i.e. voting capacity in stockholder or board of
directors’ meetings) but wants to transfer the economic results (change in
stock price plus dividends) to a counterparty, generally a financial
institution. Obviously, that decision will be taken when a downward
movement in stock returns is expected or unaffordable.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Appendix I. Zero-coupon yield calculation
Zero-coupon vs par yield rates
A zero rate (or spot rate), for maturity T, is the interest rate earned on an
investment that provides a payoff only at time T (no intermediate coupons
are paid). It is the rate used to discount cash flows to calculate the present
value of any investment. This curve cannot be directly observed in the
market since most of the bonds pay coupons periodically, instead of a
unique coupon at maturity.
The par yield rate for maturity T is the fixed periodical coupon rate that
causes the bond price to equal its face value (100%). In this case, the coupon
paid is always equal to the internal rate of return (IRR) of the bond, For
instance, for a 3-year bond, coupon 5 and IRR= 5%:
100 =
5
(1 + 0.05)
1
+
5
(1 + 0.05)
2
+
105
(1 + 0.05)
3
If the market return asked for an investment (IRR) is 5%, then the issuer of
the bond pays a coupon of 5 so as the price to be paid for the bond will be
100 (par value).
When the market rate (IRR) changes to a different rate different from the
fixed coupon, the price of the bond will not be 100 any more:
102.78 =
5
(1 + 0.04 )
1
+
5
(1 + 0.04 )
2
+
105
(1 + 0.04 )
3
The Swap Curve is a par yield curve at which top rated Banks (at least AA)
operate with Interest Rate Swaps (IRS) in the interbank market.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
How to derive the zero-coupon rates from the Swap Curve
The bootstrap method:
Bonds A, B and C have the following features:
BOND
Years to
maturity
Annual
coupon
Price
1
2
3
3%
4%
5%
100%
100%
100%
A
B
C
The below-shown equations must hold:
100 =
=
100
100 =
103
(I)
(1 + r1 )
1
4
(1 + r1 )
1
5
(1 + r1 )
1
+
+
104
(1 + r2 )
5
(1 + r2 )
(II)
2
2
+
105
(1 + r3 )
3
Where r1, r2, r3 are, respectively, the 1, 2 and 3 years zero rates.
From (I) we get r1 = 3%. Then “bootstrapping” in (II):
4
104
100 =
+
; r2 =
4.02%
1
2
(1 + 0.03) (1 + r2 )
Finally, substituting in (III):
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(III)
33
The Use of Derivatives to Hedge Market Risk in Corporate Financing
100 =
5
(1 + 0.031 )
1
+
5
(1 + 0.0402 )
2
+
105
(1 + r3 )
3
; r3 =5.07%
The matrix approach:
When calculating the zero rates yield curve, this method is not practical (due
to the number of steps involved). To simplify the calculations, the matrix
approach can be utilised to obtain the discount factors simultaneously and
their correspondent zero rates.
Discount factors are defined as:
=
d1
1
=
; d2
1
(1 + r1 )
1
=
; d3
2
(1 + r2 )
1
(1 + r3 )
3
Then, I, II and III can be rearranged this way:
100 = 103 d1
=
100
100 =
4 d1 + 104 d 2
5 d1 +
5 d 2 + 105 d3
0   d1  100 
103 0

  

 4 104 0   d 2  = 100 
 5
5 105   d3  100 

0 
103 0


4
104
0


 5

5
105


−1
0   d1  103 0
0 
103 0

  

4
104
0
d
4
104
0
=
2

  

 5
 d   5

5
105
5
105

 3  

0 
 d1  103 0
  

d
4
104
0
=
2
  

d   5

5
105
 3 

−1
100 


100


100 


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−1
100 


100


 100 


34
The Use of Derivatives to Hedge Market Risk in Corporate Financing
 d1   0.970873786 
  

 d 2  =  0.924197162 
 d   0.862139479 
 3 

Once the discount factors are known, the equivalent present value of any
stream of cash flows can be obtained. Therefore, the zero rates are not
necessary to carry out any valuation. Notwithstanding, those zero rates can
be derived as follows:
d1=
1/1
1
(1 + r1 )
1
1
1


− 1; r1=
; r1=   − 1; r1= 
 0.970873786 
 d1 
1
d=
; r=
2
2
 
2
(1 + r2 )
 d2 
1
1/2
1/3
1
d=
; r=
 
3
3
3
(1 + r3 )
 d3 
1
1/1
1


− 1; r=
2
 0.924197162 
3%
1/2
− 1; r=
4.02%
2
1/3
1


− 1; r=
3
 0.862139479 
− 1; r=
3
5.07%
Thus, obtaining the same zero rates than those calculated using the
“bootstrapping” method.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Calculating zero rates (spot) in Excel
1. Cash Flow Matrix
103
4
5
2. Inverse Matrix, Select output
range 3x3
0
104
5
0
0
105
=MINVERSE()
3. Maintaining the selected range, enter the function MINVERS
4. When we are asked the range, we select that of the matrix in point 1.
5. Ctrl + ↑ + enter
6. The Inverse Matrix is obtained
0.00970874
0
0
-0.0003734 0.009615
0
-0.0004445 -0.000458 0.00952
7. Multiply times the price vector, Select output range 3x1
0.00970874
0
-0.0003734 0.009615
0
0
ₓ
-0.0004445 -0.000458 0.00952
100
100 =
=MMULT()
100
8. Maintaining the selected range, enter the function MMULT
9. First matrix: Inverse ; second matrix: price vector
10. Ctrl + ↑ + enter
0.00970874
0
0
-0.0003734 0.009615
0
-0.0004445 -0.000458 0.00952
ₓ
Discount rates
100
0.970873786
100 = 0.924197162
100
0.862139479
11. To obtain zero rates
Discount
rates
Year
1
2
3
Zero rates
0.970874 +(1/0.9708738)^(1/1)-1
0.924197 +(1/0.9241972)^(1/2)-1
0.862139 +(1/0.8621395)^(1/3)-1
=
=
=
3.00%
4.02%
5.07%
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Appendix II. Forward Rates. Concept and calculation.
Let’s continue with the spot zero coupon yield curve obtained above:
4.02%
3%
??
0
2y
1y
What should be the interest to be applied for 1 year in 1-year time if we
commit the rate today? If the Expectations Theory holds:
(1 0.0402 )
(1 + r ) (1 + r ) =+
(1 r ) ; (1 + 0.03) (1 + r ) =+
1
0,1
1
2
1,2
1
1
0,2
1,2
2
; r1,2 =
5.05%
Similarly:
(1 0.0507 )
(1 + r ) (1 + r ) =+
(1 r ) ; (1 + 0.0402 ) (1 + r ) =+
2
0,2
1
2,3
3
0,3
1
2
2,3
3
; r2,3 =
7.20%
Therefore:
Zero spot rates
1 year
2 years
3 years
3.00%
4.02%
5.07%
Forward rates
1 year in 0 time
1 year in 1 year time
1 year in 2 years time
3.00%
5.05%
7.20%
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
It holds an equivalence between the zero rates and the forward rates. For
instance, to borrow or to invest at the 3-year zero rate is equivalent to a deal
at 1-year spot rate, and then at the forward rates of the second and the third
year:
(1 + r )(1 + r )(1 + r ) =(1 + r ) ; (1 + 0.03)(1 + 0.0505)(1 + 0.0720 ) =(1 + 0.0507 )
3
0,1
1,2
2,3
0,3
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Appendix III. Introduction to Interest Rate Swaps Valuation
Two main approaches are used for IRS valuation: The Bond and the
Forward Rates methodologies.
Bond methodology
As mentioned on page 11, this method consists of considering the two legs
of the swap as two bonds, floating and fixed coupon.
Let’s go back to the par yield and zero-coupon curves used in table 4 and
assume an IRS that receives floating and pays fixed. Following what we
learnt in Appendix II, we can easily derive the forward rates 14
Years t
0.5
Par Swap
rates
Zero
Coupon
Rates
Forward
Rates (t-1,t)
0.9
0.9
1
1.120
1.120
1.120
2
1.500
1.503
1.887
3
1.870
1.879
2.635
At inception:
Receive floating leg =100
The market floating coupons are the forward rates
1.12
1.887
+
+ 102.635 3
1
2
1
+
0.0112
1
+
0.01503
(
) (
) (1 + 0.01879 )
100 =
14
We include 0.5 year rates. Up to 1 year par and zero rates are the same
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Paying fixed leg =100, as shown below:
1.87
1.87
1.87
+
+
1
2
3
+
+
+
1
0.0112
1
0.01503
1
0.01879
)
) (
(
) (
100 =
The swap value is: +100-100 = 0
After 6 months a parallel movement of 1% (100 basis points) in the swap
curve takes place. The new curves are now 15:
Years t
0.5
1
1.5
2
2.5
3
Par
Zero
Swap Coupon
rates
Rates
1.900
1.900
2.120
2.120
2.310
2.312
2.500
2.505
2.685
2.695
2.870
2.884
Discount
factors
Forward
Rates (t-1,t)
0.991
0.979
0.966
0.952
0.936
0.918
2.120
2.515
2.891
3.271
3.648
The swap valuation using the bond method presents the results stated below:
Receive floating leg = 100.168
The market floating coupons are the forward rates:
100.1684 =
1.12
+
2.515
+
103.271
(1 + 0.019 × 0.5) (1 + 0.02312 )1.5 (1 + 0.02695)2.5
Beware that the first coupon was defined 6 months ago =1.12, the second
will be the forward rate r(0.5,1.5) and the third coupon, the principal plus r(1.5,2.5).
A short-cut to calculate the value of the floating leg is to assume that, at the
first coupon payment date, we will receive the first coupon and a price of
the bond =100. 16
15
Linear interpolation is used to calculate par and zero rates. Forward rates follow the
Apendix II formula. Consider compounded capitalisation for t > 1 year and a simple cap.
for t < 1 year.
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
100.1684 =
1.12 + 100
(1 + 0.019 × 0.5)
Paying fixed leg =100, as shown below:
98.9778 =
1.87
1.87
101.87
+
+
1.5
(1 + 0.019 × 0.5) (1 + 0.02312 ) (1 + 0.02695)2.5
The swap value is: +100.1684 - 98.9778= 1.1906
Forward Rates methodology
In this method, the floating leg cash flows to be received are the forward
rates 17. Once the fixed payments are subtracted from those floating
proceeds, the sum of the present value of that difference, using the zero
rates, is the swap value. In our example:
Years t
0.5
1.5
2.5
Receive
Pay fixed Rec-pay
float
1.120
2.515
3.271
1.87
1.87
1.87
-0.750
0.645
1.401
Discount
factor
Present value
0.991
0.966
0.936
-0.743
0.623
1.311
Swap value
1.1906
As expected, the value of the swap using the Bond or the Forward Rate
methodology is equivalent.
Since, from that moment, changes in coupons will be compensated with variations in
discount rates: the price of a floating rate bond is always 100 at inception or interest
revision dates.
16
Except for the first cash Flow which is known, as it has been already defined. In our
example 1.12
17
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Appendix IV. Introduction to Option Valuation
An option gives the holder the right, but not the obligation, to buy (call
option) or sell (put option) a given quantity of an asset in the future, at
prices agreed today (strike or exercise price), in exchange of the payment of
a premium.
Intrinsic and Time (or extrinsic) Value of an option
Intrinsic value is the difference between the exercise price of the option and
the spot price of the underlying asset. Let’s think about a call option giving
the right to buy an asset at 10 (strike) when its current price in the market is
12. In case the option expires now, the holder will make 12-10=2 18 since he
will be happy by exercising the right to buy at 10 and sell the asset one
second later at 12.
Premium
price
Intrinsic
Value
2
10
Strike (or
exercise price)
12
Price of the
underlying asset
in the martet
Using Derivagem, 19 if we input a time to maturity of one second:
1/365/24/60/60=0.0001 years, we obtain a premium price of 2.00001,
basically the intrinsic value. This value cannot be negative: no one exercises
the option when losing money.
And the seller, consequently, will lose 2.
Derivagem (John Hull) www.rotman.utoronto.ca/~hull/software/DG200.01.xls. Sheet
Equty_Index_Futures_Options
18
19
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If the time to maturity increases from 1 second to 1 year, then the seller will
run a higher risk: he not only can lose 2 but more than that. Consequently,
the seller will ask for a higher premium, now: 2.29. Thus, the option
premium can be broken down into 2.29 (total premium) = 2 (intrinsic) +
0.29 (time value).
Underlying Data
Underlying Type:
Stock Price:
Volatility (% per year):
Risk-Free Rate (% per year):
12.00
20.00%
1.00%
Calculate
Option Data
Option Type:
Imply Volatility
Time to Exercise:
Exercise Price:
Put
0.0027 (years)
10.00
Call
Price: 2.00027397
Assume know that the volatility (degree of oscillation of the asset price) is
higher than before. Again, the risk of the seller of the option is higher
demanding a higher premium. Let’s say that volatility increases from 20%
to 30%. The price then rises to 2.61; the time value is in this case 0.61. The
risk-free rate is not normally too relevant in the price: it is used to convert
the payoff at maturity in present value:
Underlying Data
Underlying Type:
Stock Price:
Volatility (% per year):
Risk-Free Rate (% per year):
12.00
30.00%
1.00%
Calculate
Option Data
Option Type:
Imply Volatility
Time to Exercise:
Exercise Price:
1.0000 (years)
10.00
Put
Call
Price: 2.61195476
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
For learning about the formulas used in option valuation, see Hull (2016).
Interest rate options valuation
Caps
As above mentioned on page 19, a Cap is a chain of interest rate call
options. We pay a premium to “buy” the interest rate at a certain level
(strike) to be entitled to receive the difference between the market rate and
the strike. Since in a floating rate loan, the second, third and subsequent
revisions of interest rate levels generate an interest rate risk, we can hedge
that risk by buying call options for those revisions. If interest rates rise
above the strike, we exercise the option receiving an amount that
compensates the higher rates of the loan.
While in other financial options the market price of the underlying asset can
be directly found in the market (i.e. the price of Microsoft shares, a barrel of
oil, a bond), the correspondent to the different call options composing a cap
are the forward rates. As seen in Appendix II, forward rates can be
calculated from zero rates. Let’s show the calculation of the Cap of page
17. 20
Swap / Cap Data
Underlying Type:
Settlement Frequency:
Principal :
Cap/Floor Start (Years):
Cap/Floor End (Years):
Cap/Floor Rate (%):
100
1.00
3.00
2.257%
Imply Breakeven Rate
20.00%
Imply Volatility
Term Structure
Time (Yrs) Rate (%)
1
1.120%
2
1.503%
3
1.879%
Pricing Model:
Volatility (%):
Floor
Cap
Price:
20
0.5424256
Derivagem DG200.01. Sheet Caps_and_Swap_OPtions
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Notice that zero coupon rates need to be input (last column). In fact, what
the calculator is doing is to compute the forward rates internally (they are
not shown on the screen)
To go one step forward in our analysis, we are going to decrease the strike
to 2%, introduce the forward rates (those not shown in the screen) and
disclose the results of the second year and third year caplets.
The current market interest rate is the forward rate r1,2 (1 year in 1 year
time): 1.887 (last column). Comparing 1.887-2.00 we obtain an intrinsic
value of zero: the market interest rate is below 2.00, so no proceeds from
exercising the option will be obtained. However, there is still some time
remaining to maturity, and the market rate could change to an “in the
money” position. That is why the option premium worths 0.1087, which
represents the time or extrinsic value.
Swap / Cap Data
Underlying Type:
Settlement Frequency:
Principal :
Cap/Floor Start (Years):
Cap/Floor End (Years):
Cap/Floor Rate (%):
100
1.00
2.00
2.000%
Imply Breakeven Rate
20.00%
Imply Volatility
Term Structure
Time (Yrs) Rate (%) Fw Rates
1
1.120%
2
1.503%
1.887%
3
1.879%
2.635%
Pricing Model:
Volatility (%):
Floor
Cap
Price:
0.1087852
Repeating the process for the option to hedge the third year, in which the
market rate is r2,3 = 2.635, the intrinsic value is now positive 2.635- 2 =
0.635. As the total option price is 0.679, the difference shows the time or
extrinsic value. 21
In the total option price there is also an effect of the present value of the payoff. In fact
the contribution of the 2.635-2=0.635 is 0.635/(1+0.01879)^3= 0.601
21
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Swap / Cap Data
Underlying Type:
Settlement Frequency:
Principal :
Cap/Floor Start (Years):
Cap/Floor End (Years):
Cap/Floor Rate (%):
100
2.00
3.00
2.000%
Imply Breakeven Rate
20.00%
Imply Volatility
Term Structure
Time (Yrs) Rate (%) Fw Rates
1
1.120%
2
1.503%
1.887%
3
1.879%
2.635%
Pricing Model:
Volatility (%):
Floor
Cap
Price:
0.6788437
The Cap price to receive the difference between the market and the 2%
strike rate is then the sum of the two caplets option premiums = 0.1088
+0.6788= 0.7876. This price is obtained directly considering a Cap starting
year 1 and finishing year 3, as shown below:
0
Underlying Type:
Settlement Frequency:
Principal :
Cap/Floor Start (Years):
Cap/Floor End (Years):
Cap/Floor Rate (%):
100
1.00
3.00
2.000%
Imply Breakeven Rate
20.00%
Imply Volatility
Term Structure
Time (Yrs) Rate (%)
1
1.120%
2
1.503%
3
1.879%
Pricing Model:
Volatility (%):
Floor
Cap
Price:
0.7876289
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
Swaptions
To calculate the premium of the Swaption of the exercise of page 24, we use
the same data than that of the Cap and change the underlying type from
Cap/Floor to Swap Option:
0
Underlying Type:
Settlement Frequency:
Principal :
Swap Start (Years):
Swap End (Years):
Swap Rate (%):
100
1.00
3.00
2.257%
Imply Breakeven Rate
20.00%
Imply Volatility
Term Structure
Time (Yrs) Rate (%)
1
1.120%
2
1.503%
3
1.879%
Pricing Model:
Volatility (%):
Rec. Fixed
Pay Fixed
Price:
0.3685026
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The Use of Derivatives to Hedge Market Risk in Corporate Financing
References
Buckley, A. (1986): Multinational Finance. Philip Allan.
Copeland, L. (1989): Exchange Rates and International Finance.
Addison-Wesley.
Döhring, B. (2008): Hedging and Invoicing strategies to reduce
exchange rate exposure: a euro-area perspective. European Economy
Economic Papers 299. January.
Eitemann, D.K. & Stonehill, A.I. (1989): Multinational Business
Finance (5th ed.). Addison-Wesley.
Fabozzi, F. (1990): The Handbook of Fixed Income securities. Irwin.
Hull, J. (2016): Options, Futures, and Other Derivatives, International
7th Edition
Chapter 7 (intermediate)
Chapters 28 and 32 (advanced)
Lessard, D.R. (1985): International Financial Management (2nd ed.).
Wiley.
Stern, J.M. & Chew, D.H. (1988): New Developments in International
Finance. Blackwells.
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