l4b Risk Management Applications of Swaps and Swaptions

Copyright Prof Kian-Guan Lim SMU

RISK MANAGEMENT

APPLICATIONS OF SWAPS & SWAPTIONS

Professor Kian-Guan Lim

Singapore Management University

KEY POINTS HEADLINED IN RED


Firm issues $100 million 2-year FRN with semi-annual coupons reset to LIBOR + 2%. Lm is LIBOR that will be realized in m-months.

To eliminate variable interest rate risk or funding cost, firm enters into an IRS as 8% fixed-rate payer on notional of $100 million.

The net funding cost to the Firm is fixed 10% p.a. for 2 years.

Using interest rate swap to convert variable funding cost to fixed funding cost

100m ½ (L0+2) ½ (L6+2) ½ (L12+2) ½ (L18+2)+100m

IRS dealer pays firm LIBOR

½ L0 ½ L6 ½ L12 ½ L18

0 6 12 18 24 months

4% 4% 4% 4%


Firm issues $100 million 2-year Notes with semi-annual coupons fixed at 10%. Lm is LIBOR that will be realized in m-months.

To take a view on falling interest rates or funding cost, firm enters into an IRS as 7% fixed-rate receiver on notional of $100 million.

The net funding cost to the Firm is fixed LIBOR + 3% p.a. for 2 years. Firm gains if LIBOR falls below 7%.

Using interest rate swap to convert fixed funding cost to variable funding cost

100m 5% 5% 5% 5%

IRS dealer pays firm fixed 7%

½ L0 ½ L6 ½ L12 ½ L18

0 6 12 18 24 months

3.5% 3.5% 3.5% 3.5%


Using interest rate swaps to adjust bond portfolio duration

An IRS is a capital market instrument with its own modified duration Dm measure.

Need to define Dm carefully here.

Dm = - (B/B) yor % change in market value of bond per % change in the bond yield.

Dm > 0 for long position in bond since dB/dy < 0. However, the use of signed measure for duration allows for short position in bond to have negative duration if we employ B>0 for long position, and B<0 for short position.The latter is convenient since we can effectively reduce duration or sensitivity to zero by shorting the long position in bond.


Value of floating rate note/bond marks to par at each reset time Market value VFL of a FRN paying LIBOR is

100% of par at initiation and at each reset. Realized LIBOR rates

L0 L1 L2 L3 L4 LN+100

Pays L0xa0 L1xa1 L2xa2 L3xa3 LN-1xaN-1

accruals a0

Suppose at this reset time, VFL< 100, then arbitrageur could buy FRN. At the same time, sell $100 1-reset period FRN. At t=2, again sell $100 (use this to redeem the previous sold FRN). At t=2, use the N-period FRN L1xa1 to pay for the FR interest in the short 1-period position. Net gain at t=1 is $100 – VFL.

At non-reset time t*, VFL = 100(1+L0xa0)/(1+Lt*xat*)where 0<t*<a0, at* = a0 – t*, and Lt* is the new discount rate at t*. Here, VFLneeds not be 100.

t*at*


Modified duration of a FRN

Since the price of a N-period FRN converges to 100 at each reset, the Macaulay duration of such a FRN behaves like the duration of a 1-period FRN. Modified duration is Macaulay duration (1+y x a) or approximately 1-period.

In general, dVFL/dLt* = d{100(1+Ltxat)/(1+Lt*xat*)}/dLt*

- at*VFL, or Dm at* that is between 0 and a. Choose ½ a or half times the reset interval as an approximation.

Suppose each reset is 6 months e.g. a semi-annual floating rate coupon of a 1 yr or 2 yr or in general N yr bond. Then a long position in this FRN has an approximate duration of ½ x 0.5 or 0.25

Suppose each reset is 3 months for a FRN, then approximate duration is ½ x 0.25 = 0.125

VFLis full price Quoted price 100(1+Ltxat*) /(1+Lt*xat*)+accrued interest 100Lt(at-at*)


Modified duration of an IRS

A fixed payer position in an IRS is equivalent to short the fixed leg and long the floating leg.

In a 2-year semi-annual reset IRS, suppose the short fixed leg bond with price B has a duration of Dm= - 0.8.

The long FRN has Dm 0.25. Then a fixed payer position in the swap has dollar duration 0.25 x

100 – 0.8 x B. Divide this by 100 to obtain the swap duration of

0.25 – 0.8 x B/100Thus, the swap duration is sensitivity defined with respect to the notional principal.

In the above, suppose B=90. Then the swap duration is Dsw = 0.25 – 0.72 = - 0.47


Amount of IRS to use to adjust bond portfolio duration Firm has par value $200 million bonds with B=$180 m,

and modified duration 6.5. It wants to bring total duration (thus cash-flow risk) down to 3.5 in the wake of anticipated interest rate hikes.

Firm considers a 5-year fixed payer IRS with annual reset. Suppose the swap duration is – 4.5. Let the desired notional principal be P.

Total dollar duration = $180m x 6.5 + P x (-4.5) This equals the target dollar duration = $180m x 3.5 180m x 6.5 – 4.5 P = 180m x 3.5 P = $120m


Use of IRS to make risky arbitrage profit in structured notes Example of Inverse Floater: Firm issues inverse FRN that pays 8% -

LIBOR with maximum at 8% and minimum at 0%. Firm buys fixed rate note at 5%. Firm enters into a IRS paying LIBOR and receiving 4% fixed. All instruments have same notional and same resets and maturity.

Inflow (+)

Outflow (-)

Inverse Floater

Fixed Rate Note

IRS NET

LIBOR = L - (8% - L) + 5% - L + 4% 1%

Risk when L>8%, e.g. L=10%

0% + 5% - 6% -1%


Using equity swap to convert exposure to equity market risk into funding cost

Securities dealer carries a large inventory of market shares approximating the S&P 500 index

The dealer firm funds the holding by bank loan It is more concerned about meeting the funding cost, and does not

wish to be exposed to market risk, since it makes its earnings from buying and selling via the spread, and not from market movements.

It enters into an Equity Swap with an Insurance Company that prefers some equity returns.

Dealer pays % in S&P 500 index to Insurance Firm and receives in exchange LIBOR

Fixed tenor of N years with pre-specified reset times Note that the Dealer firm does not need to liquidate its share

holdings


Other types of equity swaps

Returns on single name swapped into returns on an index portfolio is an equity swap to reduce concentration risk

Returns on a domestic portfolio index swapped into an international portfolio index is an equity swap to diversify internationally without physically involved in the stock transactions

Returns on bonds index swapped into small caps funds index is an equity swap to allow a funds manager to reduce exposures to bonds and increase exposures to small cap equities as part of asset re-allocation


Equity swap has special problems

In an indexi to indexj swap, sometimes both changes can be in opposite directions so that one party pays on both legs to the other party

In an equity swap exchanging index risk with interest cost, the index may not track the portfolio exactly, resulting in tracking error risk


Using interest rate swaptions

To obtain the option to convert payments on future loans from floating to fixed (using payer swaption: enter a swap as fixed rate payer) or fixed to floating (using receiver swaption: enter a swap as fixed rate receiver)

To terminate an existing swap by exercising a swaption

Add or remove a call feature to a bond


Selling a receiver swaption to convert a callable bond into a non-callable bond

Callable bond issuer has the call option Call option will be exercised by issuer when interest rate falls and bond

price reaches the call value. Issuer will exercise to stop paying the higher fixed rate when interest rate has fallen too low. By calling, the issuer will refinance at a cheaper rate.

If issuer believes that the interest rate will not fall low enough, it may decide to cash-in and sell the call option feature.

It can do this by selling a receiver (or put) swaption (issuer pays fixed when swaption is exercised by investor)

When interest rate is high, investor will not exercise swaption. When interest rate becomes too low, investor will exercise swaption to

receive fixed and pay low floating. The issuer calls the bond. But its total exposure now includes this swaption that is exercised into a swap in which the issuer now pays high fixed rate and receives low floating. This is as if the bond was not called and the issuer continues to pay the high bond rate.


Buying a receiver swaption to convert a non-callable bond into a callable bond

Non-callable bond issuer anticipates falling interest rate and wants to have a call option on the bond

Call option will be exercised by issuer when interest rate falls and bond price reaches the call value. Issuer will exercise to stop paying the higher fixed rate when interest rate has fallen too low. By calling, the issuer will refinance at a cheaper rate.

Issuer can have this option by buying a receiver (or put) swaption (issuer receives fixed when it exercises the swaption)

When interest rate is low, issuer will exercise the swaption to receive fixed and pay low floating. The issuer continues to pay on the non-callable bond. However, the net effect is that the issuer effectively pays the lower floating, and it is as if it had called the bond. The swap tenor must equal the remaining tenor of the non-callable bond.


Black’s Model for European Eurodollar futures Options at CME

Not appropriate to assume that the futures price is lognormal since for short-term instruments, their value converges to par. Assume instead the yield is lognormally distributed.

Yield yt = 100 – index price Ft ; volatility of yt or of (100-Ft) is y2. Call is in-

the-money when index price Ft is higher, or yield (100-FT) is lower than a strike yield (100-X). Thus, it is like a put –> exercise call when yT is lower than 100-X.

Tσdd;Tσ

T/2σX)]- /(100)F-ln[(100d

)]N(-dF-100)N(-d X-100T)[P(0,c

y12

y

2y0

1

102

)]N(dX-100)N(d F-100T)[P(0,p 210

The European Eurodollar futures put option price is:


Caps and Floors A cap is a portfolio of call options or caplets on LIBOR.

Payoff at time tk+1 is N k max(Rk+1-K, 0) where N is the notional principal, k=tk+1-tk is the reset interval between last reset time tk and present reset time tk+1.

K is the cap rate, and Rk+1 is the current LIBOR rate at time tk+1

A floor is a portfolio of put options on LIBOR. Payoff at time tk+1 is N k max(K – Rk+1, 0)

Each cap or floor has a maturity of T years with 2xT semi-annual resets (or some with 4xT quarterly resets)


Black’s Model for Caps Each caplet is a call option on a future LIBOR rate with the payoff

occurring in arrears Using Black’s model we assume that the interest rate underlying

each caplet is lognormal The value of a caplet, for period (tk, tk+1) is

where Fk = forward interest rate for (tk, tk+1)

k = forward rate volatilityN = notional principal

K = cap ratek = tK+1 – tK

tσ -d=d and tσ

/2tσ/K)ln(Fd where

)]KN(d)N(d)[FtP(0,Nδ

k12

kk

k2kk

1

21k1kk


Theoretical Justification for Cap Model

rate.futureexpectedtheisF where

F][RE Since

K,0)][max(REN)tP(0,

is priceoption the

, tat time maturing bondcoupon -zero respect towith

measurey probabilit neutralrisk forward a Using

k

kk1k

k1k1k

1k


Swaptions A swaption or swap option is a contract that gives

the holder the right, not obligation, to exercise the option into an interest rate swap in the future before the swap option expiry.

A payer swaption provides the holder the right to pay a specified fixed rate and receive floating LIBOR when the interest rate swap is exercised.

A receiver swaption provides the holder the right to receive a specified fixed rate and pay a floating LIBOR when the interest rate swap is exercised.


Black’s Model for European Swaption

Assume that the swap rate is lognormali.e. dS = S dW

Consider a payer swaption which gives the holder the right to exercise the swap and pay fixed rate K on a n-year swap starting at time T. The payoff on each swap payment date is

where N is the notional principal, m is the payment frequency and sT is market n-year swap rate at time T.

0) K,max(sm

NT


Black’s Model for European Swaption

The Black model value of the swaption is

where s0 is the forward swap rate

is the swap rate volatility

tk is the time from today until the kth swap payment

and

)]N(dK)N(d [sAN 210

n m

1k

k )tP(0,m

1A

Tσdd;Tσ

T/2σ/K)ln(sdwhere 12

20

1


Swaptions and Bonds

An interest rate swap is essentially an exchange of a fixed-rate bond for a floating-rate bond.

Therefore, a payer swaption is an option for a holder to exchange a fixed-rate bond and receive a floating-rate bond.

Therefore, a receiver swaption is an option for a holder to exchange a floating-rate bond and receive a fixed-rate bond.


Swaptions and Bond Options When a swaption is exercised into a swap. At the start of

this swap, the floating-rate bond is at par value of 100%. Therefore a payer swaption (where the exercised swap is

for holder to pay fixed rate and receive floating rate) is a put option on a bond having similar reset frequency, maturity and fixed coupon rate, with a strike price of par 100%.

A receiver swaption (where the exercised swap is for holder to receive fixed rate and pay floating rate) is a call option on a bond having similar reset frequency, maturity and fixed coupon rate, with a strike price of par 100%.

l4b Risk Management Applications of Swaps and Swaptions

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