interest rate forwards and swaps - dna elearning initiative - printout.pdf · 2012-01-17 ·...

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Interest Rate Forwardsand Swaps


1


Outline

PART ONE

! Chapter 1: interest rate forward contracts and their

pricing and mechanics

2


Outline

PART TWO

! Chapter 2: basic and customized swaps and their pricing and mechanics

3


Outline

PART THREE

! Chapter 3: Cross-currency swaps

! Chapter 4: Swap revaluation and trading strategies

4


Outline

PART FOUR

! Chapter 5: Quiz

5


Chapter 1: Forward Rate Agreements

! Use of interest rate forwards

! Determination of forward rate

! Mechanics of FRA

! Arbitraging incorrect FRA pricing

6


! Assume your bank on Jan 1, 2009 posts zero-coupon rates from 3 months to 5 years equal to the $ Libor rates

! 3-mo Libor is 3%; 6-mo Libor is 3.5%

! Depositor wants to know rate you would offer on 3-month deposit starting in 3 months, i.e. on April 1, 2009

! Usually denoted as 3x6 forward rate agreement, or FRA

Quoting a Forward Rate

7


Calculating FRA rate

! Done via no-arbitrage exercise

! First you borrow for 3 months and invest for 6 months

No-arbitrage pricing exercise

0 3 MO 6 MO

3.50%

3.00% ?

8


!Investment grows to (1+3.5%x181/360) = 1.0176 after 6 months

!Borrowing grows to (1 + 3% x 90/360) = 1.0075 after 3 months

!To break even you would need your borrowing in the next 3 months (91 days) to grow

by 1.0176/1.0075 = 1.0100

!This implies a borrowing rate for the 3x6 period of 3.9648%, calculated as (1.0100 - 1)

x (360/91)

!Can think of FRA rate of 3.9648% as the rate that makes an investor who invests for 6

months achieve same return as one who invests for 3 months and then again for 3 more

months at this forward rate

Calculating FRA rate

9


General FRA Equation

)(][][

3601

3601

3601 }{

FSS

LL

DaysDaysR

DaysR

F "#

$

"$

%"

)()()(360

1360

1360

1FsSLL DaysFDaysRDaysR "

$""

$%"

$

or

WhereRL = spot Libor for the long periodRS = spot Libor for the short periodF = forward LiborDAYSL = number of days in the long periodDAYSS = number of days in the short periodDAYSF = number of days covered by the forward period

10


! Customer does not actually need to place funds with bank quoting the FRA.

Customer places funds anywhere she likes, and collects or makes payment under

the FRA which brings her yield back to the FRA quoted rate all-in

! Example: if Libor resets 1% below FRA rate, customer collects annualized 1% from

bank, and vice-versa if Libor resets 1% above FRA rate

FRA Mechanics (1)

11


FRA Settlement Diagram

Customer Bank

3.9648%

Libor

Libor

12


! Libor resets at 3.75%, for period in question

! Client deposit is for $100 (so this is FRA notional amount)

! Interest earned on deposit is $100 x (3.75% x 91/360) = $0.9479

! Under the FRA, bank pays the depositor $100 x (3.9648% - 3.75%) x 91/360 = $0.0543

! Sum of these two amounts gives customer $1.0022, which is exactly equivalent to an annualized 3.9648% for a 91 day period

FRA Settlement Example

13

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Amount of net settlement will be known as soon as Libor setting takes place, at the beginning of the period in question.

Question is whether to wait until end of this period to pay it, or pay some discounted amount upfront.

FRA convention is to settle net payment as soon as it is possible to calculate it; this reduces counterparty and operational risk in the system; net payment due is discounted at current Libor, known as the settlement rate, i.e.

$0.0543/(1 + 3.75% x 91/360) = $0.0538

We can therefore write the formula for a FRA's net settlement amount as follows:

Notional x (Contract rate - Settlement Rate) x (Days/360)

(1 + Settlement Rate x Days/360)

FRA Mechanics (2)

14


3-month Libor FRA Table

starts in 21 months, ends in 24 months21x24




Period CoveredFRA

15


6-month Libor FRA Table


!!

!!




Period CoveredFRA

16


! We know from before that 6-mo. Libor is 3.50%; 12-mo. Libor is 4.00%

! As before, we can set

(1+ 4% x 365/360) = (1 + 3.5% x 181/360) x (1 + F x 184/360);

! This leads to 6x12 FRA rate = 4.41%

! You should confirm this on Worksheet FRA

Pricing 6x12 FRA

17


! Assume dealer is quoting 3x6 FRA at 4.50%

! You borrow at 3.50% for 6 months, invest at 3% for 3 months and lock in 4.5% investment rate for next 3 months; at maturity your profit per dollar is $0.00136, calculated as

(1 + 3% x 90/360) x (1 + 4.50% x 91/360) – (1 + 3.5% x 181/360)

! This may seem small, but remember you can do this on large notional amounts. On $100MM, you would earn $136,309

! Conversely if dealer quotes FRA at 3.6%, you “invert” the trade: lend for 6 months at 3.5%, borrow for 3 months at 3%, and lock in the 3.6% rate at which you borrow again in 3 months’ time for 3 more months; at maturity your profit per dollar is $0.00093. You can check this calculation.

FRA Arbitrage

18


FRA Arbitrage with Bid-Offer

! Bank is quoting the following rates:

4.02%4.00%3x6 FRA

3.6%3.4%6-months

3.1%2.9%3-months

OfferBid

!Bank is quoting a FRA rate that appears higher than the correct one, while quoting 3-

and 6-month interest rates that straddle the rates we had seen before

!Again, you try to arbitrage the unusually high FRA rate by implementing the first

version of the arbitrage we showed previously: you borrow for 6 months, lend for 3

months, and lock in the FRA rate to lend again for 3 months in 3 months’ time

19


! Given the bid offer spread, the outcome of this trade per dollar is

(1 + 2.9% x 90/360) x (1 + 4% x 91/360) – (1 + 3.6% x 181/360),

which this time comes to a loss of $0.00067, or $66,558 if the notional is $100MM.

! If conversely you tried to lend for 6 months, borrow for 3 months, and lock in the FRA rate to borrow again for 3 months in 3 months’ time, the outcome per dollar would be

(1 + 3.4% x 181/360) – (1 + 3.1% x 90/360) x (1 + 4.02% x 91/360),

which again generates a loss, this time of $0.00090 per dollar, or $89,597 on a notional of $100MM

! Apparent error in the FRA rate was not large enough to enable you to overcome the cost disadvantages of paying the bid-offer spread on 3 separate instruments

FRA Arbitrage with Bid-Offer

20


! No arbitrage is ever completely risk-free

! Main residual risks of arbitrage we illustrated are:

! Risk of default by the entity or entities to whom you lend the money

! Risk of default by the FRA counterparty when the FRA is in your favor at

settlement

! Risk that the settlement payment under the FRA, since it is discounted and

paid upfront, cannot be reinvested for the forward period at a rate at least as

high as the one used for the discounting

Final Remarks on Arbitrage

21



(Part II)


(Part II)

22


Chapter 2: Interest Rate Swaps

IRS construction and pricing

Basic interest swap pricing and use for hedging interest rate risk

Use and pricing of customized swaps for more complex situations

23


Assume on Jan 1, 2009 a borrower has borrowed $100MM for 3 years at 3-mo. Libor flat.

With Libor at 3%, first interest payment will be $0.75MM.

Borrower has “exposure” to Libor for all 11 subsequent interest periods. A rising Libor will increase interest expense and reduce reported income.

Interest Rate Exposure (1)

24


! Assume FRA rates you are quoting are as shown in this table:


6.75%92Oct 1,201133x3612

6.50%92July 1,201130x3311

6.25%91Apr 1, 201127x3010

6.00%90Jan 1,201124x279

5.75%92Oct 1,2010 21x248

5.50%92July 1,201018x217

5.25%91Apr 1, 2010 15x186

5.00%90Jan 1, 201012x155

4.75%92Oct 1,20099x124

4.50%92July 1, 20096x93

4.00%*91April 1, 20093x62

3.00%90Jan 1,2009Spot1

Rate for clientNumber of Days in period

Period Start DateContractDesignation

PeriodNumber

*/ This 4% rate for the 3x6 FRA is our old 3.9684%, adjusted to reflect the bank’s bid-offer. Similarly, all other rates include the bank’s profit margin

25


! Borrower could hedge each reset using a series of FRAs, but would have unequal

and rising interest payments in each quarter.

! What the client needs is a smoother pattern of cash outflows – this is where the

interest rate swap comes in, as described in Worksheet Simplified IRS Pricing

! Swap fixed leg is single rate that makes all cash flows borrower pays have a PV

equal to PV of cash flows he would have paid under series of FRAs


26

Period Libor/FRA Net payment

1 3.00% $750,000

2 4.00% $1,000,000

3 4.50% $1,125,000

4 4.75% $1,187,500

5 5.00% $1,250,000

6 5.25% $1,312,500

7 5.50% $1,375,000

8 5.75% $1,437,500

9 6.00% $1,500,000

10 6.25% $1,562,500

11 6.50% $1,625,000

12 6.75% $1,687,500

Notional 100,000,000

Simplified IRS Pricing (1)

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Period

Libor/

FRA

Net

payment DFs PVNP

Adjusted

net

payment PVANP1 3.00% 750,000 0.9926 744,417 1,250,000 1,240,6952 4.00% 1,000,000 0.9827 982,729 1,250,000 1,228,4113 4.50% 1,125,000 0.9718 1,093,270 1,250,000 1,214,7454 4.75% 1,187,500 0.9604 1,140,465 1,250,000 1,200,4895 5.00% 1,250,000 0.9485 1,185,668 1,250,000 1,185,6686 5.25% 1,312,500 0.9362 1,228,823 1,250,000 1,170,3087 5.50% 1,375,000 0.9235 1,269,878 1,250,000 1,154,4348 5.75% 1,437,500 0.9105 1,308,786 1,250,000 1,138,0759 6.00% 1,500,000 0.8970 1,345,507 1,250,000 1,121,256

10 6.25% 1,562,500 0.8832 1,380,007 1,250,000 1,104,00611 6.50% 1,625,000 0.8691 1,412,258 1,250,000 1,086,35212 6.75% 1,687,500 0.8547 1,442,238 1,250,000 1,068,324

Totals 14,534,045 13,912,762

(Difference) 621,283

Notional

Fixed rate 5.0000%

Simplified IRS Pricing (2)

100,000,000



3-year Swap

But: (i) all settlements under IRS occur at end of period; and (ii) there are 12 settlements here, versus only one under FRA

Borrower Bank

5.22%

Libor

Libor

29


Approximate Swap Formula

! Previous approach ignored exact daycount to keep matters simple

! Worksheet IRS Correct Pricing shows correct calculation with appropriate

daycounts. The correct swap rate turns out to be 5.23%, only one basis point

away from our approximation

! Under the simplified approach, we effectively solved the following equation:

i

i

i

i

i

DFNDFF

N ""#"" $$##

12

1

12

1 4

Libor

4

30

Period

Libor/

FRA

Net

payment DFs PVNP

Adjusted net

payment PVANP1 3.00% 750,000 0.9926 744,417 1,250,000 1,240,6952 4.00% 1,011,111 0.9826 993,538 1,263,889 1,241,9233 4.50% 1,150,000 0.9714 1,117,166 1,277,778 1,241,2964 4.75% 1,213,889 0.9598 1,165,088 1,277,778 1,226,4085 5.00% 1,250,000 0.9479 1,184,936 1,250,000 1,184,9366 5.25% 1,327,083 0.9355 1,241,531 1,263,889 1,182,4107 5.50% 1,405,556 0.9226 1,296,718 1,277,778 1,178,8348 5.75% 1,469,444 0.9092 1,336,027 1,277,778 1,161,7639 6.00% 1,500,000 0.8958 1,343,654 1,250,000 1,119,712

10 6.25% 1,579,861 0.8818 1,393,181 1,263,889 1,114,54511 6.50% 1,661,111 0.8674 1,440,895 1,277,778 1,108,38112 6.75% 1,725,000 0.8527 1,470,940 1,277,778 1,089,586

Totals 14,728,091 14,090,488


Notional

Fixed rate 5.0000%

IRS Correct Pricing

100,000,000



! Equation explains why swap rate is often described as the time-weighted

average of the relevant Libors

"

"

#

#

$

#12

1

12

1

i

i

i

ii

DF

DFLibor

FSo

Approximate Swap Formula

32


Customized Swaps

! We will customize swaps with the following features:

! A profit margin for the dealer

! A notional that changes over time

! A forward start date

! Different settlement frequencies between the floating and fixed leg, and/or also

different daycounts

! A feature that fixes the Libor reset under the swap at the end rather than the

beginning of the period in question (“Libor-in-arrears”). Each of these except

the first is examined via its own worksheet

33


Amortizing Swap (1)

! Now assume loan amortizes in last year in four equal installments

! Swap needs to amortize in the same way, otherwise borrower will end up over-hedged in year 3

! Careful with Amortizations: first amortization here occurs at end of period 9, so notional at beginning of period 9 is still $100 MM. Similarly notional at beginning of period 12 is $25 MM, not zero.

34

Libor/

FRA

Notional (at

beg of

period)

Net

payment PVNP

Adjusted

net

payment PVANP3.00% 100,000,000 750,000 744,417 1,250,000 1,240,6954.00% 100,000,000 1,011,111 993,538 1,263,889 1,241,9234.50% 100,000,000 1,150,000 1,117,166 1,277,778 1,241,2964.75% 100,000,000 1,213,889 1,165,088 1,277,778 1,226,4085.00% 100,000,000 1,250,000 1,184,936 1,250,000 1,184,9365.25% 100,000,000 1,327,083 1,241,531 1,263,889 1,182,4105.50% 100,000,000 1,405,556 1,296,718 1,277,778 1,178,8345.75% 100,000,000 1,469,444 1,336,027 1,277,778 1,161,7636.00% 100,000,000 1,500,000 1,343,654 1,250,000 1,119,7126.25% 75,000,000 1,184,896 1,044,886 947,917 835,908 6.50% 50,000,000 830,556 720,448 638,889 554,190 6.75% 25,000,000 431,250 367,735 319,444 272,396

Totals 12,556,143 12,440,472


Fixed Rate 5.00%

Amortizing Swap



Amortizing Swap (2)

Reducing notionals in year 3 puts lower weights on higher Libors when curve is

upward sloping – so swap rate decreases. Swap rate would increase if curve is

inverted

Swap whose notional increases over time (“accreting swap”) has a higher rate than

a bullet swap in an upward sloping curve environment, and a lower rate in an

inverted curve environment. Such a swap is often used to hedge a construction or

project finance loan

These four results are summarized in this table:

36

Amortizing notional lower higher

Accreting notional higher lower

Libor spot/forward curve

Varrying Notional

Upward Downward



Forward-starting Swap

! Assume commitment for a 2-year loan has been signed but disbursement will not

take place until one year from now

! Forward-starting swap is priced simply by setting the notional to zero until the start

date.

! In upward sloping curve environment, this will increase swap rate versus bullet

swap; and vice versa in inverted curve environment

38

Libor/F

RA

Notional (at

beg of

period)

Net

payment DFs PVNP

Adjusted

net

payment PVANP

3.00% - - 0.9926 - - - 4.00% - - 0.9826 - - - 4.50% - - 0.9714 - - - 4.75% - - 0.9598 - - - 5.00% 100,000,000 1,250,000 0.9479 1,184,936 1,464,399 1,388,175 5.25% 100,000,000 1,327,083 0.9355 1,241,531 1,480,670 1,385,216 5.50% 100,000,000 1,405,556 0.9226 1,296,718 1,496,941 1,381,027 5.75% 100,000,000 1,469,444 0.9092 1,336,027 1,496,941 1,361,028 6.00% 100,000,000 1,500,000 0.8958 1,343,654 1,464,399 1,311,764 6.25% 100,000,000 1,579,861 0.8818 1,393,181 1,480,670 1,305,711 6.50% 100,000,000 1,661,111 0.8674 1,440,895 1,496,941 1,298,490 6.75% 100,000,000 1,725,000 0.8527 1,470,940 1,496,941 1,276,471

Totals 10,707,882 10,707,882

(Difference) -

Fixed Rate 5.86%

Forward-Starting Swap



Frequencies and Daycounts

! Often it is necessary to structure swap with each leg having a different frequency

and/or daycount

! For example borrower may want to swap 3-mo Libor-based loan into semi-annual

fixed with 30/360 daycount to netter compare all-in cost to an outstanding bond

40

Libor/F

RA

Notional (at

beg of period) Net payment PVNP

Adjusted net

payment PVANP

3.00% 100,000,000 750,000 744,417 - 4.00% 100,000,000 1,011,111 993,538 2,667,304 2,620,948 4.50% 100,000,000 1,150,000 1,117,166 - 4.75% 100,000,000 1,213,889 1,165,088 2,667,304 2,560,073 5.00% 100,000,000 1,250,000 1,184,936 - 5.25% 100,000,000 1,327,083 1,241,531 2,667,304 2,495,352 5.50% 100,000,000 1,405,556 1,296,718 - 5.75% 100,000,000 1,469,444 1,336,027 2,667,304 2,425,128 6.00% 100,000,000 1,500,000 1,343,654 - 6.25% 100,000,000 1,579,861 1,393,181 2,667,304 2,352,129 6.50% 100,000,000 1,661,111 1,440,895 - 6.75% 100,000,000 1,725,000 1,470,940 2,667,304 2,274,461

Totals 14,728,091 14,728,091

(Difference) -

Fixed Rate 5.33%

5.23% qtly

5.33% Effective

5.26% s.a.

5.34% Daycount adjustment

Frequency



Libor-in-arrears (1)

! Assume borrower is swapping from 5.53% fixed to floating to benefit from

anticipated decline in rates.

Borrower Bank

5.23%

Libor

5.53%

42


!Regular swap puts him into synthetic floating at Libor + 30bps

!Borrower would like spread over Libor reduced or eliminated

!Often this is done by asking borrower to sell some optionally to the bank, e.g. option to

extend swap, or to allow bank to increase notional on certain dates; but some borrowers

do not want to sell optionality

!LIA swap works exactly like a regular swap in all respects, except that fixing of Libor

for each period takes place two business days before the end of that period, instead of

the beginning


43



! If curve is upward sloping, this will increase the rate on the fixed leg of the swap

! Borrower saves money under LIA if Libor forwards turn out to have overstated

outcomes

! Accurate pricing needs a “convexity adjustment”, which is too complicated to

discuss here. Its impact is small but increases with long-dated swaps and in volatile

rate environments

44

Libor/FR

A

Notional (at

beg of period) Net payment PVNP

Adjusted net

payment PVANP

3.00% 100,000,000 1,000,000 992,556 1,391,082 1,380,726 4.00% 100,000,000 1,137,500 1,117,731 1,406,538 1,382,093 4.50% 100,000,000 1,213,889 1,179,231 1,421,995 1,381,395 4.75% 100,000,000 1,277,778 1,226,408 1,421,995 1,364,827 5.00% 100,000,000 1,312,500 1,244,182 1,391,082 1,318,674 5.25% 100,000,000 1,390,278 1,300,651 1,406,538 1,315,863 5.50% 100,000,000 1,469,444 1,355,660 1,421,995 1,311,884 5.75% 100,000,000 1,533,333 1,394,116 1,421,995 1,292,886 6.00% 100,000,000 1,562,500 1,399,639 1,391,082 1,246,088 6.25% 100,000,000 1,643,056 1,448,908 1,406,538 1,240,338 6.50% 100,000,000 1,725,000 1,496,314 1,421,995 1,233,479 6.75% 100,000,000 1,788,889 1,525,420 1,421,995 1,212,562 7.00% - - - -

Totals 15,680,816 15,680,816

(Difference) -

Fixed Rate 5.56%

LIA



IRS Replication (1)

! You have been asked to quote on a 5-year swap in which you will pay 6-mo

Libor and receive fixed semiannually

! You issue a 5-year bond and invest the proceeds in a 5-year floating rate note

that pays 6-month Libor flat

! Note that net initial proceeds are zero, and repayment of your investment’s

principal in 5 years’ time would repay the bond you have issued, so again there

will be no net cash flow for the principal

46


IRS Replication (2)

Dealer

6-month Libor

Fixed

6-month Libor

Couponson fixed rate bond

!To break even under the swap, you would set the fixed leg equal to the fixed coupon

you are paying on your bond

!In fact any IRS can be viewed as two bond positions, one long and one short: if you

are receiving fixed under the swap and paying floating , it is as if you have issued a

Libor-based liability and invested the proceeds in a fixed rate bond; and vice-versa

47


Basis Swap (1)

! Now assume your bank wants to borrow £100MM for 5 years but has no access to the £ funding market. The £/$ spot rate is 1.50

! Your bank issues a $150MM 5-year FRN which pays 6-month $ Libor and converts the proceeds into £100MM in the spot FX market

! Assume also another bank, in the UK, wants to borrow $150MM for 5 years but has no access to the $ funding market. This bank issues a £100MM 5-year FRN which pays 6-month £ Libor and converts the proceeds into USD 150MM

! Now the two banks enter into the swap described in this diagram:

48


Your Bank UK Bank

£ Libor on £100 MM semi-annually

$ Libor on $150 MM semi-annually

£ 100 MM bond

at £ Libor

£100 MM at maturity

$150 MM at maturity

$150 MM bond

at $ Libor

Basis Swap (2)

This kind of swap is called a Libor basis swap, and plays a critical role in funding

operations of large banks, and also in pricing cross-currency swaps for customers

Note very carefully that unlike an IRS, a basis swap includes an exchange of

principal at maturity, without which neither party would have been properly hedged

in our example

49


Libor Basis Swap Screen

8410-year

637-year

425-year

313-year

312-year

OfferBidTerm

GBP/USD 6 month Libor Basis Swap

50



(Part III)


(Part III)

51


Outline

PART THREE

! Chapter 3: Cross-currency swaps

! Chapter 4: Swap revaluation and trading strategies

52


Chapter 3: Cross-currency swaps

! Pricing a basic cross-currency swap

! Alternative pricing methods

53


Cross-Currency Swap

! Suppose you have issued a $150MM 3-year 5.5% bond but really needed £ to fund a UK expansion

! You convert the $ proceeds of your issuance into £, and seek to hedge your future $ coupon and principal obligations into £ by buying $ and selling £ under a series of derivative contracts

! Outright FX forwards would create liability in £ with uneven cash outflows since forward rates for successive dates will not be equal

! You would like your bank to make all payments required in $ under the bond, in return for you making to the bank, usually on the same dates, payments in £ whose profile is identical to that of a £ bond issue

54

$ Libor

Curve DFs $ Pmts

PV of

Pmts Days

£ Libor

Curve DFs

£

Pmts

PV of

Pmts

3.00% 0.99 2.06 2.05 91 5.00% 0.99 1.00 0.994.00% 0.98 2.06 2.03 90 4.90% 0.98 1.00 0.984.50% 0.97 2.06 2.00 92 4.80% 0.96 1.00 0.964.75% 0.96 2.06 1.98 92 4.70% 0.95 1.00 0.955.00% 0.95 2.06 1.95 91 4.60% 0.94 1.00 0.945.25% 0.94 2.06 1.93 90 4.50% 0.93 1.00 0.935.50% 0.92 2.06 1.90 92 4.40% 0.92 1.00 0.925.75% 0.91 2.06 1.88 92 4.30% 0.91 1.00 0.916.00% 0.90 2.06 1.85 91 4.20% 0.90 1.00 0.906.25% 0.88 2.06 1.82 91 4.10% 0.89 1.00 0.896.50% 0.87 2.06 1.79 92 4.00% 0.88 1.00 0.886.75% 0.85 152.06 129.65 92 3.90% 0.88 101.00 88.44

Totals 150.82 98.70

148.05

$ Fixed 5.50% £ Fixed 4.00%

$ Notional 150 £ Notional 100

PV Diff 2.77

FX Spot 1.5

Cross-Currency Swap



YourCompany Bank

£ Fixed% on £100 MM semi-annually

$5.5% on $150 MM semi-annually


$150 MM at maturity

$150 MM bond

at $5.5%

CCS Diagram

56


Basis Point Conversion

! You cannot simply add or subtract equal numbers of basis points to each leg after you have finished pricing the swap

! Suppose in other words that after you priced the previous swap, it turned out that the $ bond could not be issued at 5.50%, but rather at 5.75%

! Client would logically ask you now to reset the $ leg of the swap to this level, and would understand that the £ leg also would need to be revised upward

! An identical adjustment of 25 bps to the £ leg would be incorrect, because PVs must be equal, and these PVs are obtained by discounting off two different curves

57

Constructing/hedging CCS

► Most typically this is done using 3 swaps, two IRS and one Libor basis swap, as shown in this diagram:

Company

$ Bond at 5.50%

Bank

$Fixedon $150 MM

£Fixed on £100 MM

$150 MMat maturity


$Fixed on $150 MM

$L on $150 MM

x bps on £100 MM

$L on $150 MM


$150 MM at maturity

£L on £100 MM

£Fixed on £100 MM

£L+

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58


CCS Variations (1)

! Note that you can also keep one of the legs floating, by eliminating one of the

interest rate swaps

59

CCS Variations (2)

Company

$ Bond at 5.50%

Bank

$Fixedon $150 MM

£Fixedon £100 MM

$150 MMat maturity


$Fixed on $150 MM

£L+x bps on £100 MM


$150 MM at maturity


60


CCS Variations (3)

! Also frequencies and daycount for the two legs do not have to match – so we could have the $ leg based on quarterly under a 30/360 day convention, while the £ leg could be based on 6-month Libor and under an act/365 day basis

61


Chapter 4: Swap Revaluation & Trading Strategies

! IRS factor sensitivities

! Effect of parallel shift and pivot on swap value

! Use of IRS by traders

62

Period

Original

Curve

New

Curve

Net

payment DFs PVNP

Adjusted net

payment PVANP

1 4.00% 4.00% 1,000,000 0.9901 990,099 1,334,067 1,320,858 2 4.25% 4.25% 1,074,306 0.9796 1,052,363 1,348,890 1,321,339 3 4.50% 4.50% 1,150,000 0.9684 1,113,704 1,363,712 1,320,671 4 4.75% 4.75% 1,213,889 0.9568 1,161,477 1,363,712 1,304,832 5 5.00% 5.00% 1,250,000 0.9450 1,181,264 1,334,067 1,260,707 6 5.25% 5.25% 1,327,083 0.9326 1,237,683 1,348,890 1,258,020 7 5.50% 5.50% 1,405,556 0.9197 1,292,699 1,363,712 1,254,216 8 5.75% 5.75% 1,469,444 0.9064 1,331,887 1,363,712 1,236,053 9 6.00% 6.00% 1,500,000 0.8930 1,339,490 1,334,067 1,191,313

10 6.25% 6.25% 1,579,861 0.8791 1,388,863 1,348,890 1,185,815 11 6.50% 6.50% 1,661,111 0.8647 1,436,430 1,363,712 1,179,257 12 6.75% 6.75% 1,725,000 0.8501 1,466,382 1,363,712 1,159,260

Totals 14,992,343 14,992,343

(Difference) -

Notional

Fixed rate 5.34%

Shift 0.00%

Pivot 0.00%

Sensitivities

100,000,000


Libor Curve

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0 1 2 3 4 5 6 7 8 9 10 11 12 13L

ibo

rs

Period


Period

Original

Curve

New

Curve

Net

payment DFs PVNP

Adjusted net

payment PVANP

1 4.00% 5.02% 1,255,000 0.9876 1,239,445 1,334,067 1,317,532 2 4.25% 5.27% 1,332,139 0.9746 1,298,332 1,348,890 1,314,658 3 4.50% 5.52% 1,410,667 0.9611 1,355,742 1,363,712 1,310,616 4 4.75% 5.77% 1,474,556 0.9471 1,396,551 1,363,712 1,291,571 5 5.00% 6.02% 1,505,000 0.9331 1,404,250 1,334,067 1,244,760 6 5.25% 6.27% 1,584,917 0.9185 1,455,745 1,348,890 1,238,954 7 5.50% 6.52% 1,666,222 0.9034 1,505,342 1,363,712 1,232,041 8 5.75% 6.77% 1,730,111 0.8881 1,536,479 1,363,712 1,211,087 9 6.00% 7.02% 1,755,000 0.8728 1,531,701 1,334,067 1,164,325

10 6.25% 7.27% 1,837,694 0.8570 1,574,931 1,348,890 1,156,018 11 6.50% 7.52% 1,921,778 0.8409 1,615,937 1,363,712 1,146,685 12 6.75% 7.77% 1,985,667 0.8245 1,637,150 1,363,712 1,124,359

Totals 17,551,606 14,752,606

(Difference) (2,799,000)

Notional

Fixed rate 5.34%

Shift 1.02%

Pivot 0.00%

DO NOT MODIFY

Sensitivities (cont)

100,000,000


Libors

Libor Curve

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Period



Swap Revaluation

Fixed rate receiver has a gain when rates decline, since fixed payments are discounted at lower discount rates; it follows that receiving fixed under a swap is one way of taking a view on declining interest rates

Can reach same result by remembering that fixed rate receiver can be viewed as owning a fixed rate bond which he is funding at Libor; so he would have gain on his asset when rates decline, while liability’s value remains essentially the same

Price gain for the Receiver of a 1 bp drop in rates should be roughly equal to the price gain of owning a 3-year, $100MM quarterly-pay bond whose coupon is 5.34%

More generally DV01 of a swap is very nearly the same as DV01 of a fixed rate bond whose principal, coupon, maturity and daycount/frequency match those of the fixed leg of the swap

Swap also has convexity – like a bond – which means changes in its value are not linear relative to changes in rates

67

Inputs

Discount rate 5.34%

Tenor in years 3.00

Coupon 5.34%

Initial price of the bond $100,000,000

New price of the bond $100,027,555

DV01 of 3-year, quarterly-pay 5.34% bond



Effect of Curve Shift on Swap P&L

(20,000,000)

(15,000,000)

(10,000,000)

(5,000,000)

-

5,000,000

10,000,000

15,000,000

20,000,000

-6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00%

Shift (%)

PV

(U

SD

)

69



(Part IV)


(Part IV)

70

Question 1

On January 1, 2009, a borrower whose interest cost is Libor + 75 bps under a $100MM loan signs up for the 12x18 FRA. The rate under the FRA is 5%. 6-month Libor resets at the beginning of January 2010 at 6%. What is the exact all-in amount paid in that period by the borrower, taking into account both the loan and the FRA, and assuming the FRA settles at the end of that period?

a) $2,555,556

b) $2,890,972

c) $3,066,667

d) $3,450,000

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71

Question 1 Solution

► Under the loan, the borrower pays 100MM x 6.75% x 181/360, while under the FRA he receives 100MM x (6% - 5%) x 181/360

► So his all-in cost comes to 100MM x 5.75% x 181/360, which equals $2,890,972


72

Question 2

How much money could you make if you tried to arbitrage the following rates a dealer quotes you on Jan 1, 2009? Assume a $100MM notional for your arbitrage.

a) $15,779

b) $18,534

c) $19,621

d) $22,222

Bid Offer

3-month Libor 3.96% 4.00%

6-month Libor 4.28% 4.32%

3x6 Libor FRA 4.40% 4.45%



73

Question 2 Solution

► You borrow for 3 months at 4.00%, lock in a borrowing rate for the next 3 months of 4.45%, and invest the money for 6 months at 4.28%

► Per one dollar, your profit would be

(1 + 4.28% x 181/360) – (1 + 4.00% x 90/360) x (1 + 4.45% x 91/360)

which amounts to $0.00016 per dollar, or $15,779 per $100MM



74

Question 3

► You are asked on Jan 1, 2009 to price the fixed rate leg of an interest rate

swap having the terms below. You (the dealer) will be receiving the

floating payments, and will need to earn a profit. Please use the same

Libor curve we have used in the early part of this module.

a) 4.98%

b) 5.10%

c) 5.29%

d) 5.49%

Notional Initially $100,000,000, but amortizing in 2 equal repayments in the last 2 semesters

Tenor 2 years

Floating Rate Index 3-month Libor

Payment Frequency Quarterly for floating leg

Semi-Annual for fixed leg

Daycount Act/360 for floating leg

30/360 for fixed leg

Swap Start date July 1 2009

Dealer profit $125,000


75

Days

Libor/

FRA

Notional (at

beg of

period)

Net

payment DFs PVNP

Adjusted

net

payment PVANP

90 3.00% - 0.9926 - - - 91 4.00% - 0.9826 - - - 92 4.50% 100,000,000 1,150,000 0.9714 1,117,166 - 92 4.75% 100,000,000 1,213,889 0.9598 1,165,088 2,645,201 2,538,858 90 5.00% 100,000,000 1,250,000 0.9479 1,184,936 - 91 5.25% 100,000,000 1,327,083 0.9355 1,241,531 2,645,201 2,474,674 92 5.50% 100,000,000 1,405,556 0.9226 1,296,718 - 92 5.75% 100,000,000 1,469,444 0.9092 1,336,027 2,645,201 2,405,032 90 6.00% 50,000,000 750,000 0.8958 671,827 - 91 6.25% 50,000,000 789,931 0.8818 696,590 1,322,601 1,166,319 92 6.50% - 0.8674 - - - 92 6.75% - 0.8527 - - -

8,709,883 8,584,883

(Diff) 125,000

5.2904%

Qn 3 Solution



Question 4

You have entered into a 10-year interest rate swap in which you receive 6% fixed semi-annually and pay Libor semi-annually on a notional of $10MM. Right after you enter the swap, the whole Libor curve shift upwards by a parallel 100bps. Which of the following is most likely to be your approximate P&L? Do not use Excel to estimate your answer.

a) A gain of $700,000

b) A loss of $700,000

c) A gain of $300,000

d) A loss of $300,000

77


Question 4 Solution

! Since you are receiving the fixed payments and rates have risen, you should expect

a loss since the PV of the cash flows you are receiving will diminish, as we

discussed before

! The duration of the swap should pretty obviously be much closer to 7 than to 3,

since the swap we modeled earlier in the module had a 3-year maturity and a

duration of around 2.8

! A duration of 7 looks about right, meaning the swap would lose 7% of its notional

given the 1% shift in rates

! The actual duration of this swap is in fact around 7.1 It would be a good exercise

for you to confirm this

78


Question 5

Which of the following methods would not be a possible hedge for an interest rate swap you have entered into where you are paying fixed and receiving Libor?

a) Issue a fixed rate note and invest the proceeds in a floating rate note

b) Enter another interest rate swap with another counterparty under which you receive fixed and pay Libor

c) Enter into a series of FRAs in which you are paying the settlement rate and receiving the contract rate

d) Purchase a fixed rate bond and pledge it to a bank’s repo desk for funding at the Libor rate

79


Question 5 Solution

! You are paying fixed under the swap so you need to receive fixed under your hedge

and pay Libor

! The only hedge that fails to do this is the first one

80


Question 6

The Libor curve in USD is flat at 3%, while in GBP it is flat at 5%. On Jan 1, 2009, a borrower issues a GBP 200MM 3-year bond paying 7% quarterly, and wishes to swap it into a USD synthetic liability based on 3-month Libor. The FX spot rate is 1.75. The dealer would like to earn a profit margin of $500,000 on the swap. What is the all-in cost the dealer can quote the borrower?

a) Libor -10

b) Libor +107

c) Libor + 179

d) Libor + 203

81

$ Libor

Curve DFs $ Pmts

PV of

Pmts Days

£ Libor

Curve DFs

£

Pmts

PV of

Pmts

3.00% 0.99 4.40 4.37 91 5.00% 0.99 3.50 3.463.00% 0.99 4.40 4.34 90 5.00% 0.98 3.50 3.413.00% 0.98 4.40 4.30 92 5.00% 0.96 3.50 3.373.00% 0.97 4.40 4.27 92 5.00% 0.95 3.50 3.333.00% 0.96 4.40 4.24 91 5.00% 0.94 3.50 3.293.00% 0.96 4.40 4.21 90 5.00% 0.93 3.50 3.253.00% 0.95 4.40 4.17 92 5.00% 0.92 3.50 3.213.00% 0.94 4.40 4.14 92 5.00% 0.91 3.50 3.173.00% 0.93 4.40 4.11 91 5.00% 0.89 3.50 3.133.00% 0.93 4.40 4.08 91 5.00% 0.88 3.50 3.093.00% 0.92 4.40 4.05 92 5.00% 0.87 3.50 3.053.00% 0.91 354.40 323.58 92 5.00% 0.86 203.50 175.29

Totals 369.85 211.06 in £

369.35 in $

$ Fixed 5.03% £ Fixed 7.00%

$ Notional 350 £ Notional 200

PV Diff 0.50

FX Spot 1.75

Qn 6 Solution (1)


Period

Libor/F

RA

Net

payment DFs PVNP

Adjusted net

payment PVANP1 3.00% 750,000 0.9926 744,417 750,000 744,417 2 3.00% 758,333 0.9851 747,023 758,333 747,023 3 3.00% 766,667 0.9776 749,486 766,667 749,486 4 3.00% 766,667 0.9702 743,784 766,667 743,784 5 3.00% 750,000 0.9629 722,198 750,000 722,198 6 3.00% 758,333 0.9557 724,727 758,333 724,727 7 3.00% 766,667 0.9484 727,116 766,667 727,116 8 3.00% 766,667 0.9412 721,584 766,667 721,584 9 3.00% 750,000 0.9342 700,643 750,000 700,643

10 3.00% 758,333 0.9272 703,096 758,333 703,096 11 3.00% 766,667 0.9201 705,414 766,667 705,414 12 3.00% 766,667 0.9131 700,047 766,667 700,047

Totals 8,689,535 8,689,535

(Difference) -

NotionalFixed rate 3.0000%

Qn 6 Solution (2)

100,000,000


Question 6 Solution

Borrower

£7% on £200 MM

$5.03% on $350 MM

$Libor on $350 MM

$3% on $350 MM


$350 MM at maturity

£200 MM

Bond at 7%


84


Question 7

Today is January 1, 2009. The USD Libor curve is monotonically downward sloping, with 6-month Libor spot at 6% and the 5-year forward 6-month Libor at 4%. You have just priced a spot-starting bullet interest rate swap, and are now asked to price a swap which starts on January 1, 2010 and amortizes by 25% of its original notional amount on each anniversary. The fixed leg of the swap will be increased as a result of:

a) Both the delayed-start date and the amortization schedule

b) The delayed-start date but not the amortization schedule

c) The amortization schedule but not the delayed-start date

d) Neither the delayed-start date nor the amortization schedule

85


Question 7 Solution

! Since the curve is downward sloping, the highest values for Libor are the earlier

ones, and the lowest values are the last ones

! Postponing the start date of the swap removes the highest Libors from the

calculation of the swap rate, so reduces the swap rate

! Amortizing the swap notional towards the end puts lower weights on the last few

Libors, so increases the swap rate

! Therefore only the second effect increases the swap rate

86


Question 8

A company has just issued $100 million of fixed rate 5-year bonds at a price of par with a coupon of 6.00% (annual) and up-front fees of .50. The company wishes to swap the debt into £. The spot rate is 1.25. Annual swap rates for 5-years are:

$ interest rate swaps: 6.12 / 6.15

£ interest rate swaps: 7.98 / 8.02

£/$ LIBOR basis swaps: £ Libor + 1 / £ Libor + 4

Assuming the company pays full bid-offer spreads, what is the all-in cost of the dollar debt swapped into fixed £? (Be sure to use the all-in cost of the $ debt by taking into account the up-front fees paid on the offering.)

a) 8.06%

b) 8.42%

c) 8.44%

d) 8.46%

87

Inputs

Proceeds (99.50)

Tenor in years 5.00

Coupon 6%

YTM of the bond 6.12%

Bond YTM


Question 8 Solution

Company$6.12% Bond

$Libor

$6.12%

$Libor

£Libor + 4bps

$100 MM at maturity

£80 MM at maturity

£Libor

£8.02%


89

interest rate forwards and swaps - dna elearning initiative - printout.pdf · 2012-01-17 ·...

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