Derivatives Ch 10

Nature of Swaps

A swap is an agreement to exchange cash flows at specified future times according to certain specified rules

An Example of a “Plain Vanilla” Overnight Indexed Swap

• Deal entered into on March 8, 2022 where Apple agrees to receive 3-month SOFR & pay a fixed rate of 3% per annum every 3 months for 2 years on a notional principal of $100 million

• Next slide illustrates cash flows that could occur (Day count conventions are not considered)

Cash Flows to Apple for One Outcome
(See Table 7.1)

 

Date	SOFR Rate (%)	Floating Received (‘000s)	Fixed Paid (‘000s)	Net cash flow (‘000s)
June 8, 2022	2.20	550	750	-200
Sept 8, 2022	2.60	650	750	-100
Dec. 8, 2022	2.80	700	750	-50
Mar. 8, 2023	3.10	775	750	+25
June 8, 2023	3.30	825	750	+75
Sept 8, 2023	3.40	850	750	+100
Dec 8, 2023	3.60	900	750	+150
Mar 8, 2024	3.80	950	750	+200

Determination of Risk-Free Interest Rates

• OIS rates out to one year define zero rates because they typically involve a single exchange

• OIS rate for contracts lasting longer than one year define par yield

• The bootstrap method can be used to determine the zero curve

Bootstrap Example (Table 7.3)

 

OIS Maturity	OIS Rate	Compound. Freq. for OIS rate	Zero rate  (cont comp.)
1 month	1.8%	Monthly	1.7987%
3 months	2.0%	Quarterly	1.9950%
6 months	2.2%	Semiannually	2.1880%
12 month	2.5%	Annually	2.4693%
2 years	3.0%	Quarterly	2.9994%
5 years	4.0%	Quarterly	4.0401%

Zero Rate Given by Bootstrap Method (Figure 7.2)

Typical Uses of an
Interest Rate Swap

• Converting a liability from

  • fixed rate to floating rate 

  • floating rate to fixed rate 

• Converting an investment from 

  • fixed rate to floating rate

  • floating rate to fixed rate 

OIS Between Apple and Citigroup (Figure 7.1)

Apple Transforms a Liability from Floating to Fixed
(Figure 7.3)

Interest Rate Swap Between Citigroup and Intel

Intel Transforms a Liability from Fixed to Floating (Figure 7.4)

Apple Transforms an Asset from Fixed to Floating (Figure 7.5)

Intel Transforms an Asset from Floating to Fixed (Figure 7.6)

Quotes By a Swap Market Maker (Table 7.4)

 

Maturity	Bid (%)	Ask (%)	Swap Rate (%)
2 years	2.97	3.00	2.985
3 years	3.05	3.08	3.065
4 years	3.15	3.19	3.170
5 years	3.26	3.30	3.280
7 years	3.40	3.44	3.420
10 years	3.48	3.52	3.500

Day Count

• A day count convention is specified for fixed and floating payments

• For example, SOFR is likely to be actual/360 in the U.S. 

• The fixed rate might be quoted with actual/365 or 30/360

Confirmations

• Confirmations specify the terms of a transaction

• The International Swaps and Derivatives has developed Master Agreements that can be used to cover all agreements between two counterparties

• CCPs are used for standard swaps between two financial institutions

The Comparative Advantage Argument (Table 7.5)

• AAACorp wants to borrow floating

• BBBCorp wants to borrow fixed

A Swap where Companies Trade Directly with Each Other (Figure 7.7)

The Swap when a Financial Institution (F.I.) is Involved
(Figure 7.8)

Criticism of the Comparative Advantage Argument

• The 4.0% and 5.2% rates available to AAACorp and BBBCorp in fixed rate markets are 5-year rates

• The rates available in the floating rate market are 3-month rates

• BBBCorp’s fixed rate depends on the spread above floating it borrows at in the future

Valuation of an Interest Rate Swap

• Initially interest rate swaps are worth close to zero

• At later times they can be valued as a portfolio of forward rate agreements (FRAs)

• The procedure is to

  • Calculate floating forward rates

  • Calculate the swap cash flows that will occur if floating forward rates are realized

  • Discount these swap cash flows at OIS rates

Example (Example 7.1)

• Swap involves paying 3% per annum and receiving SOFR every six months on $100 million

• Swap has 1.2 years remaining (exchanges in 0.2, 0.7, and 1.2 years)

• Risk-free rate for 0.2, 0.7, and 1.2 years are 2.8%, 3.2% and 3.4%, respectively (continuously compounded)

• Rate observed for last 0.3 years is 2.3% continuously compounded

Example continued

• Floating rate for the exchange at 0.2 years is assumed to be 0.6×2.3%+0.4×2.8% or 2.50% (cont comp) or 2.516% (sa)

• Forward rate for 0.2 to 0.7 years is 3.36% (cont comp) or 3.388% (sa)

• Forward rate for 0.7 to 1.2 years is 3.68% (cont comp) or 3.714% (sa) 

Calculations ($ million)

 

Time  (yrs)	Fixed cash flow	Floating cash flow	Net cash flow	Discount factor	PV of net cash flow
0.2	−1.5000	+1.258	−0.242	0.9944	−0.241
0.7	−1.5000	+1.694	+0.194	0.9778	+0.190
1.2	−1.5000	+1.857	+0.357	0.9600	+0.343
					+0.292

Value of swap is $0.292 million

Value Changes Through Time

• To party paying fixed

  • How is swap value expected to change through time when term structure is upward sloping?

  • How is swap value expected to change through time when term structure is downward sloping?

An Example of a Fixed-for-Fixed Currency Swap (Figure 7.10)

   Five year agreement by BP to 

• Pay 3% on a US dollar principal of $15,000,000 

• Receive 4% on a sterling principal of £10,000,000

Exchange of Principal

• In an interest rate swap the principal is not exchanged

• In a currency swap the principal is exchanged at the beginning and the end of the swap

The Cash Flows (Table 7.6)

 

Date	Dollar Cash Flows (millions)	Sterling cash flow (millions)
Feb 1, 2022	+15.00	−10.00
Feb 1, 2023	−0.45	+0.40
Feb 1, 2024	−0.45	+0.40
Feb 1, 2025	−0.45	+0.40
Feb 1, 2026	−0.45	+0.40
Feb 1, 2027	−15.45	+10.40

Typical Uses of a
Currency Swap

• Conversion from a liability in one currency to a liability in another currency

• Conversion from an investment in one currency to an investment in another currency

Comparative Advantage May Be Real Because of Taxes

•  General Electric wants to borrow AUD

•  Quantas wants to borrow USD

Borrowing costs after adjusting for the differential impact of taxes could be:

	USD	AUD
General Electric	5.0%	7.6%
Quantas	7.0%	8.0%

Valuation of Fixed-for-Fixed Currency Swaps

Fixed for fixed currency swaps can be valued either using forward rates or as the difference between 2 bonds

Currency Swap Example

• All Japanese interest rates are 1.5% per annum (cont. comp.)

• All USD interest rates are 2.5% per annum (cont. comp.)

• 3% is received in yen; 4% is paid in dollars. Payments are made annually

• Principals are $10 million and 1,200 million yen

• Swap will last for 3 more years

• Current exchange rate is 110 yen per dollar

Valuation in Terms of Forward Rates (Example 7.2)

 

Time	DollarCash Flow	Yen cash flow	Forward rate	Dollar value of yen cash flow	Net cash flow	Present value
1	−0.4	+36	0.009182	0.3306	−0.0694	−0.0677
2	−0.4	+36	0.009275	0.3339	−0.0661	−0.0629
3	−10.4	+1236	0.009368	11.5786	+1.1786	+1.0934
Total						+0.9629

Valuation in Terms of Bonds  (Example 7.3)

 

Time	Cash Flows ($ millions)	PV  ($ millions)	Cash flows (millions of yen)	PV ( millions of yen)
1	0.4	0.3901	36	35.46
2	0.4	0.3805	36	34.94
3	10.4	9.6485	1,236	1,181.61
Total		10.4191		1,252.01

Value = 1,252.01/110−10.4191 = +0.9629 millions of dollars 

Other Currency Swaps

• Fixed-for-floating: equivalent to a fixed-for-fixed currency swap plus a fixed for floating interest rate swap

• Floating-for-floating: equivalent to a fixed-for-fixed currency swap plus two floating interest rate swaps 

Swaps & Forwards

• A swap can be regarded as a convenient way of packaging forward contracts

• When a swap is initiated the swap has zero value, but typically some forwards have a positive value and some have a negative value

Credit Risk

• When derivatives transactions with a counterparty are cleared bilaterally, they are netted

• There is exposure if the net value of outstanding transactions is greater than the collateral posted

Credit Default Swaps: A Quick First Look

• Notional principal (e.g. $100 million) and maturity (e.g. 5 yrs) specified

• Protection buyer pays a fixed rate (e.g. 150 bp) on the notional principal (the CDS spread) 

• If the reference entity (a country or company) defaults protection seller buys bonds issued by the reference entity for their face value and the spread payments stop. Total face value of bonds bought equals notional principal

Other Types of Swaps

• Amortizing/ step up

• Compounding swap

• Quanto (diff swap)

• Equity swap

• Extendible or puttable swap

• Commodity swap

• Volatility swap