MSA 736Fixed Income & Derivatives II

Chapters 5Chapters 5

Swaps Markets and Swaps Markets and ContractsContracts

Important Concepts in Chapter 12

The concept of a swapThe concept of a swap Different types of swaps, based on underlying currency, Different types of swaps, based on underlying currency,

interest rate, or equityinterest rate, or equity Pricing and valuation of swapsPricing and valuation of swaps Strategies using swapsStrategies using swaps

Four types of swapsFour types of swaps CurrencyCurrency Interest rateInterest rate EquityEquity Commodity (not covered in this book)Commodity (not covered in this book)

Characteristics of swapsCharacteristics of swaps No cash up frontNo cash up front

Fixed rate is set such that market value of swap = 0Fixed rate is set such that market value of swap = 0 Notional principalNotional principal Settlement date, settlement periodSettlement date, settlement period Credit riskCredit risk

Swap Basics

Swap Contract Terminology Notional principalNotional principal: Amount used to calculate : Amount used to calculate

periodic paymentsperiodic payments Swap rateSwap rate: The fixed rate on a interest rate : The fixed rate on a interest rate

swapswap

Floating rateFloating rate: Usually US LIBOR: Usually US LIBOR

TenorTenor: Time period covered by swap: Time period covered by swap

Settlement datesSettlement dates: Payment due dates: Payment due dates

Plain Vanilla Interest Rate Swap

Fixed interest rate payments are exchanged for Fixed interest rate payments are exchanged for floating-rate paymentsfloating-rate payments

Notional amount is not exchanged at the Notional amount is not exchanged at the beginning or end of the swap (both loans are in beginning or end of the swap (both loans are in same currency and amount) same currency and amount)

Interest payments are netted - On settlement Interest payments are netted - On settlement dates, both interest payments are calculated and dates, both interest payments are calculated and only the difference is paid by the party owing only the difference is paid by the party owing the greater amount the greater amount

Plain Vanilla Interest Rate Swap – Key Point

Floating rate payments are typically made in Floating rate payments are typically made in arrears, payment is made at end of period arrears, payment is made at end of period based on beginning-of-period LIBOR! (The based on beginning-of-period LIBOR! (The floating rate at time 0 dictates the floating floating rate at time 0 dictates the floating payment at time 1!)payment at time 1!)

Thus: First net payment is known at swap Thus: First net payment is known at swap initiation!initiation!

Plain Vanilla Interest Rate Swap

FR FR = fixed rate or swap rate= fixed rate or swap rate

TT = # days in settlement period= # days in settlement period

NP NP = notional principal= notional principal

The settlement payment made by the fixed-rate payer at time t (if negative, the fixed-rate payer will receive this amount):

t-1

T= (Swap FR – Libor ) × × (NP)

360

Vanilla Interest Rate Swap: Example

A 2-year interest rate swap with semiannual-settlement, floating A 2-year interest rate swap with semiannual-settlement, floating payments based upon LIBOR, notional principal of $10 million payments based upon LIBOR, notional principal of $10 million with a swap rate (the fixed rate) of 6%. Thus, semiannual fixed with a swap rate (the fixed rate) of 6%. Thus, semiannual fixed payments are: (0.06 / 2) × $10 million = $300,000. Assume the payments are: (0.06 / 2) × $10 million = $300,000. Assume the following LIBOR term structure:following LIBOR term structure:

LIBOR tLIBOR t0 0 == 5%, t5%, t1 1 == 5.8%, t5.8%, t2 2 == 6.2%, t6.2%, t3 3 == 6.6% 6.6%

1st payment1st payment: : Fixed-rate payerFixed-rate payer pays $50,000 net pays $50,000 net

(0.06 – (0.06 – 0.05 0.05 )(180/360)(10 million) = $50,000)(180/360)(10 million) = $50,000

Again, first net payment is known at swap initiation (t=0)!Again, first net payment is known at swap initiation (t=0)!

LIBOR 5% at tLIBOR 5% at t00, 5.8% at t, 5.8% at t11, 6.2% at t, 6.2% at t22, 6.6% at t, 6.6% at t33

****************************************************************************************************

22ndnd payment payment: : Fixed rate payerFixed rate payer pays $10,000 net pays $10,000 net

(0.06 – (0.06 – 0.0580.058)(180/360)(10 million) = $10,000)(180/360)(10 million) = $10,000

33rdrd payment payment: : Floating rate payerFloating rate payer pays $10,000 net pays $10,000 net

(0.06 – (0.06 – 0.0620.062)(180/360)(10 million) = –$10,000)(180/360)(10 million) = –$10,000

44thth payment payment: : Floating rate payerFloating rate payer pays $30,000 net pays $30,000 net

(0.06 – (0.06 – 0.0660.066)(180/360)(10 million) = –$30,000)(180/360)(10 million) = –$30,000

Interest Rate Swap Example Continued

Pricing and Valuation of Interest Rate Swaps

How is the swap rate (the fixed rate) determined?How is the swap rate (the fixed rate) determined? What is the market value of an interest rate swap at a date What is the market value of an interest rate swap at a date

between the swap initiation date and the next settlement between the swap initiation date and the next settlement date?date?

We answer these by replication: An pay-fixed side of a We answer these by replication: An pay-fixed side of a swap can be replicated by issuing a fixed rate bond and swap can be replicated by issuing a fixed rate bond and using the proceeds to buy a LIBOR-based floating rate using the proceeds to buy a LIBOR-based floating rate bond; where the face value of the two bonds equals the bond; where the face value of the two bonds equals the notional principal; similarly, pay-floating side can be notional principal; similarly, pay-floating side can be replicated by issuing a floating rate bond and using the replicated by issuing a floating rate bond and using the proceeds to buy a fixed rate bond proceeds to buy a fixed rate bond

The Swap Fixed Rate

Idea: Swap fixed rate must be set so swap value at Idea: Swap fixed rate must be set so swap value at initiation is zero (must be 0 if no money changes initiation is zero (must be 0 if no money changes hands at swap inception)hands at swap inception)

Method: Value the swap as a combination of fixed-Method: Value the swap as a combination of fixed-rate bond and floating rate bond rate bond and floating rate bond

Key insight: At each settlement date, the market Key insight: At each settlement date, the market value of the floating rate bond will value of the floating rate bond will always reset always reset to parto par (interest rates “adjust” to market rates) (interest rates “adjust” to market rates)

The Swap Fixed Rate – by Replication

Consider the replication of a 4-year annual pay swap:Consider the replication of a 4-year annual pay swap: The fixed rate bond will pay coupon every year for The fixed rate bond will pay coupon every year for

3 years and the final coupon and return of principal 3 years and the final coupon and return of principal at year 4at year 4

The floating rate bond will pay a coupon every year The floating rate bond will pay a coupon every year for 3 years (unknown in years 2 and 3) and the final for 3 years (unknown in years 2 and 3) and the final coupon and return of principal at year 4coupon and return of principal at year 4

The Swap Fixed Rate by Replication

Recall that the swap rate is set such that no money Recall that the swap rate is set such that no money changes hands at initiationchanges hands at initiation

For this to hold, the market value of the fixed rate For this to hold, the market value of the fixed rate bond must equal the present value of the floating-bond must equal the present value of the floating-rate bondrate bond

For a floating rate bond at initiation or For a floating rate bond at initiation or settlement date, the bond’s rate is reset so that settlement date, the bond’s rate is reset so that the bond trades at parthe bond trades at par

The Swap Fixed Rate

Assume $1,000 par bondsAssume $1,000 par bonds At initiation, the floating rate is valued at $1,000 At initiation, the floating rate is valued at $1,000

(since the rate is reset on day 1)(since the rate is reset on day 1)

Since no money changes hands at swap Since no money changes hands at swap inception, this also means that the fixed rate inception, this also means that the fixed rate must be worth $1,000.must be worth $1,000.

Summary, set the floating rate bond value to the Summary, set the floating rate bond value to the fixed rate bond value and solve for the fixed fixed rate bond value and solve for the fixed coupon rate – this is the swap rate!coupon rate – this is the swap rate!

The Swap Fixed Rate by Replication

The coupon rate for a 4-period fixed-rate bond The coupon rate for a 4-period fixed-rate bond must be:must be:

1 2 3 4 4

C C C C $1,000$1,000

1 R 1 R 1 R 1 R 1 R

This “C” is

the swap

fixed rate!

The Swap Fixed Rate by Replication

Let ZLet Zii= 1/(1+R= 1/(1+Rii); Z); Zii is PV factor for the i is PV factor for the ithth period period We can write:We can write:

1000 = Z1000 = Z11C + ZC + Z22C+ ZC+ Z33C+ ZC+ Z44C++ ZC++ Z44(1000)(1000)

Simplify:Simplify:

1000 – Z1000 – Z44 =C(Z =C(Z11 + Z + Z22+ Z+ Z33+ Z+ Z44)) Solve for C:Solve for C:

The Swap Fixed Rate

Per $1 par, we get the formula for the swap Per $1 par, we get the formula for the swap rate:rate:

4

1 2 3 4

1 ZC

Z Z Z Z

Price of n-period $1 zero coupon bond

= n-period discount factor

Swap Fixed Rate: Example Calculate the swap rate and the fixed payment on a Calculate the swap rate and the fixed payment on a

1-year, quarterly settlement1-year, quarterly settlement swap with a notional swap with a notional principal of principal of $10MM$10MM

1/(1+0.06)

MaturityAnnualized

RateDiscount Factor (Zi)

90-day 4.5% 0.98888

180-day 5.0% 0.97561

270-day 5.5% 0.96038

360-day 6.0% 0.94340

1/(1+0.045(90/360))

The Swap Fixed Rate: Example

Applying the formula for C, we get the Applying the formula for C, we get the quarterly swap rate:quarterly swap rate:

1 0.94340

0.98888 0.97561 0.96038 0.94340

0.0146 1.46%

Note: This is a quarterly rate! Need to annualize!

The Swap Fixed Rate: Example

Quarterly fixed-rate Quarterly fixed-rate paymentpayment is: is:

$10M × 0.0146 = $146,000$10M × 0.0146 = $146,000

Annualizing the quarterly rate gives us the Annualizing the quarterly rate gives us the annual swap rate ofannual swap rate of 1.46% × 4 =1.46% × 4 = 5.84%5.84%

Notional

principal

Valuing a Swap at date after inception(Fixed-Rate Payer Side)

Swap value:Swap value: Difference in PV of fixed and floating payments Difference in PV of fixed and floating payments To calculate the value, we must find:To calculate the value, we must find:

PV of “replicating” fixed rate bondPV of “replicating” fixed rate bond PV of “replicating” floating rate bond:PV of “replicating” floating rate bond:

Fixed-rate payer swap value = PVFixed-rate payer swap value = PVfloatfloat – PV – PVfixedfixed

Received by fixed payer

Paid by the fixed payer

Valuing a Swap: Example Consider the same swap priced atConsider the same swap priced at 5.84%:5.84%:

Assume 240 days have passedAssume 240 days have passed 90-day LIBOR at last settlement (day 180) was 3.5%90-day LIBOR at last settlement (day 180) was 3.5% Note that the two remaining settlement dates are now 30 and Note that the two remaining settlement dates are now 30 and

120 days away120 days away

Maturity Annualized Rate Discount Factor(Zi)

30-day 3.0% 0.99751

120-day 4.0% 0.98684

Valuing a Swap: Example

Given that the swap rate is 5.84%, the quarterly Given that the swap rate is 5.84%, the quarterly payments payments from the fixed sidefrom the fixed side are: are:

The fixed-rate payer will pay $0.0146 per $ of NP The fixed-rate payer will pay $0.0146 per $ of NP at the two remaining settlement dates (30 and 120 at the two remaining settlement dates (30 and 120 days from now) days from now)

90$0.0584× = $0.0146

360

Per $ of notional principal

Valuing a Swap: Example

The replicating The replicating fixed rate bondfixed rate bond has a: has a:

Coupon payment of $0.0146 in 30 daysCoupon payment of $0.0146 in 30 days

Coupon and principal payment totaling Coupon and principal payment totaling $1.0146 ($1.000 + $0.0146) in 120 days$1.0146 ($1.000 + $0.0146) in 120 days

The value of the replicating bond is the The value of the replicating bond is the present present value of these two paymentsvalue of these two payments

Discounted using the 30- and 120-day Discounted using the 30- and 120-day discount factorsdiscount factors

Valuing a Swap: Example

PV of fixed rate bond!

240 270 360

$0.01456

$1.00125

$0.0146

× 0.99751

× 0.98684

$1.0146

$1.01581

Valuing a Swap: Example

Next, find PV of the replicating floating rate bond:Next, find PV of the replicating floating rate bond:

Now the Now the trick: We know the value of the FRN at all settlement trick: We know the value of the FRN at all settlement dates will be $1, so the value of the floating rate bond is the PV of dates will be $1, so the value of the floating rate bond is the PV of $1.00875 in 30 days$1.00875 in 30 days Next coupon + $1 par value = $1.00875Next coupon + $1 par value = $1.00875

90floating rate payment in 30 days = 0.035×

3$0.

60= 00875

From last settlement date (given)

Valuing a Swap: Example

PV of floating bond!

It doesn’t matter what the floating rate payment is on day 360 because the bond will

reprice to par on day 270

240 270 360

$1.00624

$1.00875

× 0.99751

$?.?????

Valuing a Swap: Example

Finally!Finally! We can calculate the value to the We can calculate the value to the pay-fixed sidepay-fixed side (per $ of notional principal):(per $ of notional principal):

fixed

floating fixed

Value

= PV – PV

= $1.00624 – $1.01581= –$0.00957

Valuing a Swap: Example

Assuming a $10M notional principal, the value Assuming a $10M notional principal, the value to the fixed rate payer is:to the fixed rate payer is:

Make sense? Yes, rates decreased and the pay-Make sense? Yes, rates decreased and the pay-fixed side lost in valuefixed side lost in value

$10M× –0.0095 –$957 = ,700

Equity Swaps

CharacteristicsCharacteristics One party pays the return on an equity, the other One party pays the return on an equity, the other

pays fixed, floating, or the return on another equitypays fixed, floating, or the return on another equity Rate of return is paid, so payment can be negativeRate of return is paid, so payment can be negative Payment is not determined until end of periodPayment is not determined until end of period

Equity Swaps (continued)

The Structure of a Typical Equity SwapThe Structure of a Typical Equity Swap Setttling party paying stock and receiving fixedSetttling party paying stock and receiving fixed

Example: IVM enters into a swap with FNS to pay Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be 90 days for one year. Net payment will be

period settlementover stock on Return

365or 360

Daysrate) (Fixed

principal) (Notional

period settlementover index stock on Return 360

90.034500)($25,000,0

Equity Swaps (continued)

The Structure of a Typical Equity Swap (continued)The Structure of a Typical Equity Swap (continued) The fixed payment will beThe fixed payment will be

• $25,000,000(.0345)(90/360) = $215,625$25,000,000(.0345)(90/360) = $215,625

The first equity payment is:The first equity payment is:

So the first net payment is IVM pays $285,657.So the first net payment is IVM pays $285,657.

282,501$12710.55

2764.900$25,000,00

Equity Swaps (continued) The Structure of a Typical Equity Swap (continued)The Structure of a Typical Equity Swap (continued)

If IVM had received floating, the payoff formula If IVM had received floating, the payoff formula would be:would be:

If the swap were structured so that IVM pays the If the swap were structured so that IVM pays the return on one stock index and receives the return on return on one stock index and receives the return on another, the payoff formula would be:another, the payoff formula would be:

period settlementover stock on Return

360

Days(LIBOR)

principal) (Notional

indexstock other on Return -index stock oneon Return principal) (Notional