Topic Six Valuing swap contracts How do you value a swap contract?

Topic Six

Valuing swap contracts

How do you value a swap contract?

6.2

Topic Six

Sub-Topics

Characteristics of swaps The comparative advantage theory of swaps Theoretical LIBOR/swap rates and curves Valuation of interest rate swaps Valuation of currency swaps

6.3

Topic Six

Student learning outcomesAfter completing this unit you should be able to do the following:

Describe the characteristics of swap contracts Define and give examples of various types of swaps: currency, interest

rate, commodity, equity Discuss the comparative advantage theory of swaps Explain the relevance of swaps and Eurodollar futures in determining

theoretical zero rate curves Construct a theoretical LIBOR/swap zero rate curve using the bootstrapping

method Calculate the value of an interest rate swap using bond prices Calculate the value of an interest rate swap using FRA prices Calculate the value of a currency swap using bond prices Calculate the value of a currency swap using forward prices Explain how credit risk arises in a swap

6.4

Topic Six

References

Hull, J.C. (2011). Fundamentals of Futures and Option Markets. New Jersey: Pearson Education. Chapter 7.

6.5

Characteristics of swaps

Describe characteristics of swaps

An agreement between two parties to exchange a series of future cashflows Typically one party makes floating payments, such as

an interest rate, a currency rate, an equity or a commodity price and the other makes payments that are either fixed or floating and whose value is determined by another factor

Swaps typically have a value of zero at initiation as, with the exception of currency swaps no cashflow is exchanged at initiation of the swap

6.6



On the settlement dates on which the cashflows are to be exchanged, typically one party makes a payment of the net amount owing to the other

It is rare for swaps to call for actual physical delivery of the underlying asset

6.7


Define different types of swaps

Interest rate swaps: one party pays a fixed rate and the other pays a floating rate with both sets of payments in the same currency

Commodity swaps: one party pays a fixed rate and the other pays a rate equal to the change in price of a commodity

Equity swaps: one party pays a fixed rate and the other pays a rate equal to the return on an equity index

Currency swaps: each party makes interest payments to the other in different currencies

Our Toy Company

What currency are their loans in?

What currency are their payment in?

What currency are their receipts in?

6.8

6.9


Describe characteristics of swaps Example 6.1

Microsoft and Intel have initiated a 3-year swap on March 5, 2007. Microsoft has a floating rate loan. Intel have a fixed rate loan. Microsoft agrees to pay Intel an interest rate of 5% pa, paid and

compounding semi-annually. The notional principal is $100 million. In return Intel agrees to pay Microsoft the 6-month LIBOR rate on the same notional principal.

Assume the 6-month LIBOR rates as quoted on Mar-5, 2007 and then after at six monthly intervals turn out to be 4.2%, 4.8%, 5.3%, 5.5%, 5.6%, 5.9% and 6.4%. Calculate the payments made/received by Microsoft.

6.10



Assume the 6-month LIBOR rates as quoted on Mar-5, 2007 and then after at six monthly intervals turn out to be 4.2%, 4.8%, 5.3%, 5.5%, 5.6%, 5.9% and 6.4%. Calculate the payments made/received by Microsoft.

Sept 5 cashflows: Microsoft pays Intel at the agreed fixed rate 5% pa

Intel would pay Microsoft at the LIBOR 6-mth rate as at Mar 5.

m5.2$205.0

000,000,100$

m1.2$2042.0

000,000,100$

6.11



---------Millions of Dollars---------

LIBOR FLOATING FIXED Net

Date Rate Cash Flow Cash Flow Cash Flow

Mar.5, 2007 4.2%

Sept. 5, 2007 4.8% +2.10 –2.50 –0.40

Mar.5, 2008 5.3% +2.40 –2.50 –0.10

Sept. 5, 2008 5.5% +2.65 –2.50 +0.15

Mar.5, 2009 5.6% +2.75 –2.50 +0.25

Sept. 5, 2009 5.9% +2.80 –2.50 +0.30

Mar.5, 2010 6.4% +2.95 –2.50 +0.45

6.12



Intel MS

LIBOR

5%

6.13



IBM and British Petroleum entered into a 5-year currency swap on February 1, 2007. IBM has agreed to pay a fixed rate of interest of 5% pa in sterling and will receive a fixed rate of interest of 6% pa in US dollars from British Petroleum. Interest payments are made annually and the principal amounts are USD18 million and STG10 million.

Calculate the payments made/received by IBM.

6.14



Calculate the payments made/received by IBM. The Feb 1 2007 cashflows

IBM pays BP USD18m and receives STG10m The Feb 1 2005 cashflows

IBM pays BP BP pays IBM

mSTGmSTG 5.005.010

mUSDmUSD 08.106.018

6.15



Year

Dollars Pounds$

------millions------

2007 –18.00 +10.002008 +1.08 –.50

2009 +1.08 –.50 2010 +1.08 –.50

2011 +1.08 –.50 2012 +19.08 -10.50

£

6.16



IBM BP

Sterling 5%

Dollars 6%

6.17



Swap counterparties usually deal through a financial intermediary

“Plain vanilla” fixed-for-floating swaps on interest rates are usually structured so that the financial institution earns about 3 to 4 basis points on a pair of offsetting transactions

Where previously the parties dealt directly and there was only one swap agreement, involving a financial intermediary means there are two swap agreements, one between each counterparty and the intermediary

6.18



F.I.

LIBOR LIBOR

4.985% 5.015%

Intel MS

6.19



F.I.

STG 5.015% STG 5%

USD 6% USD 6.015%

IBM BP

6.20



Many financial institutions act as market makers for swaps, entering into swaps without having an offsetting swap with another counterparty

Market Makers use bonds, FRAs, interest rate futures and other derivative instruments to hedge their exposure to swaps they enter into as principal

Market makers post bid – offer quotes for swaps with the bid-offer spread typically 3 – 4 basis points

Swap rate is the mid rate As the swap is worth zero at initiation Pfix = Pflt

6.21



A LIBOR-based floating rate cashflow on a swap payment date is calculated

where L is the principal, R is the relevant LIBOR rate and n is the number of days since the last payment

The fixed rate is usually quoted as actual/365 or 30/360, in which case to equate the fixed and floating rates either the fixed is multiplied by 360/365 or the floating by 365/360

360LRn

PaymentFloating

6.22

The comparative advantage theory of swaps

Discuss comparative advantage theory

The comparative advantage theory of swaps suggests that the attractiveness of swaps is due to the ability of a swap to reduce the cost of borrowing to each party to the swap, where each party has a comparative advantage in a different debt market Each party could borrow in the market in which they

have a comparative advantage and so minimise their cost of funds by borrowing in their lowest cost market

The parties could then enter into a swap so as to pay a fixed or floating rate as desired

6.23


Discuss comparative advantage theoryExample 6.3

Two companies, AAACorp and BBBCorp, both wish to borrow $10 million for five years. AAACorp can borrow fixed at 10.0% and floating at 6-mth LIBOR plus 0.3%, whereas BBBCorp can borrow fixed at 11.2% and floating at 6-mth LIBOR plus 1.00%. AAACorp has a AAA credit rating and wants to borrow floating, while BBBCorp has a BBB credit rating and wants to borrow fixed.

How might each company achieve their objectives and minimise their cost of borrowing via a swap agreement?



FIXED FLOATING

AAACorp 10.00 L+0.3

BBBCorp 11.2 L+1.0

_____ _____

1.2 0.7

BBBCorp has relative advantage in floating but wants fixed, AAACorp has relative advantage in fixed and wants floating

6.24

6.25



How might each company achieve their objectives and minimise their cost of borrowing via a swap agreement?

AAACorp agrees to pay BBBCorp at 6-mth LIBOR and receives from BBBCorp at fixed rate of 9.95% on $10m AAACorp interest rate cashflows:

It pays 10% pa to outside lenders It receives 9.95% pa from BBBCorp It pays LIBOR to BBBCorp

Overall 10% - 9.95% = .05% worse off L+.3 –L =.3 better off Total .3 -.05 = .25 better off



How might each company achieve their objectives and minimise their cost of borrowing via a swap agreement? BBBCorp interest rate cashflows:

It pays LIBOR + 1% to outside lenders It receives LIBOR from AAACorp It pays 9.95% to AAACorp

Overall 11.2 - 9.95% = 1.25% better off L- L+1.0 =1.0 worse off Total 1.25 -1.00 = .25 better off

6.26

6.27



AAA BBB

LIBOR

LIBOR+1%

9.95%

10%

6.28



AAA F.I. BBB10%

LIBOR LIBOR

LIBOR+1%

9.93% 9.97%

6.29



The spreads between the rates offered to AAACorp and BBBCorp are a reflection of the extent to which BBBCorp is more likely to default than AAACorp

The fact that the spread between the companies at the 5-year rate is larger than at the 6-month rate is indicative of the perception that the probability of default rises faster for a lower rated company than it does for a higher rated company

6.30



It should be noted that the fixed rate is fixed for 5-years, whereas the floating rate is reset every six months. Hence, every six months the lender to BBBCorp will reset the floating rate to match the current 6-month LIBOR rate and may review the margin charged over LIBOR to BBBCorp

BBBCorp appears in Example 6.3 to pay 10.95%, however it will do so only so long as it can continue to borrow at a 1% margin over LIBOR.

6.31



Should the margin charged to BBBCorp over LIBOR increase its comparative advantage will reduce and BBBCorp could end up paying a higher rate of interest under the swap as the margin is not “swapped”, only the LIBOR rate

AAACorp has fixed a rate of LIBOR + 0.05%, which is a good deal, though it has done so by accepting a counterparty risk with the financial intermediary, which it would not have borne if it had simply issued a bond

6.32

Theoretical LIBOR/swap rates and curves

Construct a LIBOR/swap zero rate

LIBOR is the rate of interest at which AA-rated banks borrow for period between 1 and 12 months from other banks

Forward LIBOR rate is derived from Eurodollar futures rates for a three-month period in the future

Swap rate is the average of the bid rate a market maker pays in exchange for receiving LIBOR and the offer rate the market maker is prepared to accept for paying LIBOR

6.33



LIBOR rates are used as proxies for risk-free rates when valuing derivatives, though they are not entirely risk-free

Swap rates similarly have a low credit risk A financial institution can earn the 5-year swap rate

on a notional principal by lending and re-lending the principal for 10 consecutive 6-month periods out to five years, then entering into a swap to exchange the LIBOR income for the 5-year swap rate

The credit risk on a 5-year swap corresponds to that of a series of 6-month LIBOR loans to AA-rated banks

6.34



Swap rates define a set of par yield bonds The value of a newly issued floating-rate bond that

pays 6-month LIBOR is always equal to its principal or par value, when the LIBOR/swap zero curve is used for discounting, as LIBOR constitutes both the rate of interest on the bond and the discount rate of the bond’s cashflow, hence Pflt = Ppar

For a newly issued swap where the fixed rate equals the swap rate, Pfix = Pflt, therefore if Pflt = Ppar, Pfix = Ppar

6.35



A LIBOR/swap zero curve can be constructed using LIBOR rates for the first 12-months, Eurodollar futures for the next 24-months and swap rates for maturities of greater than 36-months.

T

elnR

CeC...eCeCPV

e

i

T iRi

i

i

T iRii

TRTRT iRi

hence

111

111

111

6.36


Construct a LIBOR/swap zero rate Example 6.4

The 6-month, 12-month and 18-month LIBOR/swap zero rates have been determined as 4%, 4.5% and 4.8% pa with continuous compounding and the 2-year swap rate for a swap with semiannual payments, is 5%.

Calculate the 24-month LIBOR/swap zero rate.

6.37



Calculate the 24-month LIBOR/swap zero rate.

paor

T

eR

eee

CeCeCeCPV

e

i

T iRi

i

i

T iRii

TRTRT iRi

%95.4,0495.0.00.2

9057.0lnln

9057.05.102

5.25.25.2100

...

5.1048.00.1045.05.004.0

111

111

111

6.38



Eurodollar futures contracts maturing in March, June, September and December are sometimes used to help construct the LIBOR/swap zero curve, particularly for the maturities between 12-months and 36-months.

The Eurodollar futures rates are converted into forward rates for future 3-month periods using a convexity adjustment

ttrateFuturesrateForward 212

21

6.39



The annual volatility of the short-term interest rate is typically observed to be 1.2% and the price of an 8-year Eurodollar futures contract is quoted as 94.

Calculate the forward rate for the 3-month period 8 years hence.

6.40



Calculate the forward rate for the 3-month period 8 years hence.

05562.000475.04

360365

06.01ln4

00475.025.88012.021 2

6.41



If F1 is the forward rate calculated from the ith Eurodollar futures contract and Ri is the zero rate for a maturity Ti we have

so that

TTTRTR

Fii

iiiii

1

11

T

TRTTFR

i

iiiiii

1

11

6.42



The 400-day LIBOR zero rate has been calculated as 4.8% pa with continuous compounding and, from a Eurodollar futures quote, it has been calculated that the forward rate for a 91-day period beginning in 400 days is 5.30% pa with continuous compounding.

Calculate the 491-day LIBOR/swap zero rate.

6.43



Calculate the 491-day LIBOR/swap zero rate.

04893.0

491400048.091053.0

6.44

Valuation of an Interest Rate Swap

Calculate value of Interest Rate Swap

The value of an interest rate swap at initiation is zero After some time its value may rise or fall depending on

underlying prices and rates Interest rate swaps can be valued either:

as the difference between the value of a fixed-rate bond and the value of a floating-rate bond, or

as a portfolio of forward rate agreements (FRAs)

6.45



If we assume that a notional principal is exchanged at maturity, its net cashflows are unchanged as is its value

Hence a swap may be valued based on the value of a replicating portfolio of a long position and an offsetting short position in a fix rate bond and a floating rate bond

The fixed rate bond is valued in the usual way

The floating rate bond is valued based on the principal and the next floating payment, k*

ekLP t*r**flt

n

i

niRiifix eCP

1

6.46



The value of a interest rate swap where the holder has a long position in the fixed rate bond and a short position in the floating rate bond is given

The value of a interest rate swap where the holder has a short position in the fixed rate bond and a long position in the floating rate bond is given

PPV fltfixswap

PPV fixfltswap

6.47


Calculate value of Interest Rate Swap Example 6.7

A financial institution has agreed to pay 6-month LIBOR and receive 8% pa compounding semi-annually on a notional principal of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding are 10%, 10.5% and 11%, respectively for maturities 3-months, 9-months and 15-months. The 6-month LIBOR rate at the last payment date was 10.2% pa compounding semi-annually.

Calculate the value of the swap to the financial institution using bond pricing formulas.

6.48




Time Fixed CF Float CF Discount PV fixed PV float

0.25 4.0 105.1 0.9753 3.901 102.505

0.75 4.0 0.9243 3.697

1.25 104.0 0.8715 90.640

Total 98.238 102.505

6.49




mmmV swap 267.4$505.102$238.98$

6.50



A swap can be characterised as a portfolio of FRAs An FRA can be valued using the forward interest as an

estimate of the future spot interest rate Use the LIBOR/swap zero curve to calculate forward

rates for each of the LIBOR rates that will determine the swap cashflows

Calculate the swap cashflows on the assumption that the LIBOR spot rates will equal the forward rates

Discount these swap cashflows (using the LIBOR/swap zero curve) to obtain the swap value

6.51



A financial institution has agreed to pay 6-month LIBOR and receive 8% pa compounding semi-annually on a notional principal of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding are 10%, 10.5% and 11%, respectively for maturities 3-months, 9-months and 15-months. The 6-month LIBOR rate at the last payment date was 10.2% pa compounding semi-annually.

Calculate the value of the swap to the financial institution using FRA pricing formulas.

6.52



Calculate the value of the swap to the financial institution using FRA pricing formulas.

Time Fixed CF Float CF Net CF Discount PV Net CF

0.25 4 -5.100 -1.100 0.9753 -1.073

0.75 4 -5.522 -1.522 0.9243 -1.407

1.25 4 -6.051 -2.051 0.8715 -1.787

Total -4.267

6.53



How to calculate floating cash flows Step 1 calculate forward Step 2 convert to semi annual compounding Step 1 Using formula 4.4 R2*T2-R1*T1 = 10.5*.75 – 10*.25 = .05375 = .1075 T2-T1 .75-.25 .5 Step 2 convert continuous compounding to semi annual using

formula 4.3 Rm = m(e-rc/m -1)=2(e-.1075/2 -1) = .11044 per annum or .05522 for 6

months for the 9 monthly semi annual cash flow. Therefore the cash flow is 100 million * .05522 or 5.522 million

6.54

Valuation of an Currency Swap

Calculate value of Currency Swap

The value of an currency swap at initiation is zero After some time its value may rise or fall depending on

underlying prices and rates Currency swaps can also be valued either:

as the difference between the value of a domestic currency bond and the value of a foreign currency bond, or

as a portfolio of forward contracts

6.55



The value of a currency swap where the domestic currency is received and the foreign currency is paid by the holder is given

The value of a currency swap where the foreign currency is received and the domestic currency is paid by the holder is given

PSPV fdswap 0

PPSV dfswap 0

6.56


Calculate value of Currency Swap Example 6.9

Suppose the term structure of LIBOR/swap interest rates is flat in both the US and Japan. The Japanese rate is 4% pa and the US rate is 9% pa, both continuously compounding. A financial institution has entered into a currency swap in which it receives 5% pa in yen and pays 8% pa in dollars, once a year. The principals are USD10 million and JPY1,200 million. The swap will last for another 3 years and the spot JPY/USD exchange rate is 110.


6.57




Time $ Bond CF PV $ CF Y Bond CF PV Yen

1 0.8 0.7311 60 57.65

2 0.8 0.6682 60 55.39

3 0.8 0.6107 60 53.22

3 10.0 7.6338 1,200 1,064.30

Total 9.6439 1,230.55

6.58




mV swap 5430.1$5439.9110

55.230,1

6.59



Each exchange of payments can be considered a forward contract, hence a swap is replicated by a portfolio of forward contracts

Forward foreign exchange contracts can be valued assuming the forward foreign exchange rate is realised, i.e. the future spot rate is estimated to equal the current forward exchange rate for that maturity

The value of the swap is the sum of the values of the forward contracts underlying the swap

6.60



Suppose the term structure of LIBOR/swap interest rates is flat in both the US and Japan. The Japanese rate is 4% pa and the US rate is 9% pa, both continuously compounding. A financial institution has entered into a currency swap in which it receives 5% pa in yen and pays 8% pa in dollars, once a year. The principals are USD10 million and JPY1,200 million. The swap will last for another 3 years and the spot JPY/USD exchange rate is 110.

Calculate the value of the swap to the financial institution using forward pricing formulas.

6.61



Calculate the value of the swap to the financial institution using forward pricing formulas.

Time $ Bond CF Y Bond CF Forward $ value Yen Net CF PV

1 0.8 60 0.009557 0.5734 -0.2266 -0.2071

2 0.8 60 0.010047 0.6028 -0.1972 -0.1647

3 0.8 60 0.010562 0.6337 -0.1663 -0.1269

3 10.0 1,200 0.010562 12.6746 +2.6746 2.0417

Total 1,5430

6.62


Explain how credit risk arises in a swap

Contracts such as swaps that are private arrangements between two companies entail credit risks.

A financial institution has credit risk exposure only when the value of the swap is positive to the institution

Potential losses from defaults on a swap are much less than the potential losses from defaults on a loan with the same principal.

Potential losses from defaults on a currency swap are greater than that on an interest rate swap.

Topic Six Valuing swap contracts How do you value a swap contract?

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Transcript of Topic Six Valuing swap contracts How do you value a swap contract?