Wulin Suo 1

Interest Rates

and Swaps

Term Structure Analysis

Wulin Suo 2

3

Term-Structure It refers to the relationships of YTM of default

free bonds and their maturities Spot rate of interest: YTM on pure discount bonds

spot curve Application:

it allows one to discount each cash flow separately – reasonable for default-free securities (e.g., T-securities)

Each cash flow is discounted by a rate corresponding to that maturity

It takes into consideration of reinvestment rate Disadvantage: ignores the liquidity of a specific

bond

4

Pure Discount Bond Zero-Coupon Bond (ZCB) price b(t,T): the

price at time t of a bond that pays $1 at maturity time T and nothing else

ZCBs are simply called zeros b(t,T) is essentially a discount factor:

for an amount of $C received at time T, one should pay Cb(t,T)

In realty, zero-coupon does exist STRPIS: Separate Trading of Registered Interest

and Principal Securities

5

ZCB Advantages of investing in zeros:

Assured growth (assuming one is holding to maturity)

Ideal to match liabilities Low initial investment Automatic compounding of interest A wide selection of issuers, and maturities

ranging from one year to 30 years Relatively high liquidity

6

Building a Zero Curve It is the zero curve implied by the prices of

coupon bonds traded in the market The spot rates may be different from those rates

implied the strips securities The implied zero curve can be used as reference

rate to check if the strips rates are out of line with the treasury security market

Unlike yield curve, which can be built the many actively traded T-securities

Zero curve are usually computed by using the bootstrapping method

7

Bootstrapping Method Example:

Write yT as the implied spot rate (or simply zero rate) with maturity T:

Bond Price Y 1 Y 2 Y 3

A 99.50 105 0 0

B 101.25 6 106 0

C 100.25 7 7 107

11

100

1

CP

y

1 105 / 99.50 1y

1

1

1(0,1) 0.9476

1 100

Pb

y C

8

Bootstrapping … Calculating y2 and b(0,2):

For y3 and b(0,3):

2 2 22 22 2

1 2 2

100 100(0,1)

1 (1 ) (1 )

C C CP C b

y y y

22

106101.25 6 0.9476

(1 )y

2 5.32%y 22

1(0,2) 0.9015

(1 )b

y

3 3 3 33 3 32 3 3

1 2 3 3

100 100(0,1) (0,2)

1 (1 ) (1 ) (1 )

C C C CP C b C b

y y y y

3 7.02%y (0,3) 0.8159b

Maturity Implied Zero Zero Rate

A 0.9476 5.53%

B 0.9015 5.32%

C 0.8159 7.02%

9

Bootstrapping … In general, assume that for each year n, there

is a coupon bond maturing in n years and paying an annual coupon of Cn, and a cash price of Pn

Step 1:

In general:

(0,1) (0,2) (100 ) (0, )n n n nP C b C b C b n

1 1(100 ) (0,1)P C b

1

1

(0,1)100

Pb

C

1

11

(0,1)y

b

(0,1) (0,2) (100 ) (0, 1)(0, )

100n n n n

n

P C b C b C b nb n

C

1/1

1(0, )

n

ny b n

10

Bootstrapping … Semi-annual coupon payments can be

handles similarly, and zero rates at semi-annual intervals can thus be obtained

What the maturities not on the annual/semi-annual intervals? interpolation

Restriction on the zero price: for t1 < t2 < … <tn,

1 2(0, ) (0, ) (0, )nb t b t b t

11

Par Bond Yield CurveQuestion: Based on the zero curve, what is the

coupon rate that makes the bond trade at par? 1Y maturity:

2Y maturity:

3Y maturity:

In general:

1

1

100100

1

X

y

1 5.53X

2 22

1 2

100100

1 (100 )

X X

y y

2 5.327X

3 3 3100 (0,1) (0,2) (100 ) (0,3)X b X b X b 3 6.908X

100 (0,1) (0,2) (0, ) 100 (0, )nn X b b b n b n

12

Zero Curve vs Par Yield

Upward sloping

13

Continuous Compounding Zero rate with continuous compounding:

yT is usually simply called zero yields or spot yields

Par bond yield: a bond is paying a coupon continuously at a rate of

CT, ie, over a small interval [s, s+dt], an amount of CTdt is paid:

Par yield is defined as

( )( , ) Ty T tb t T e 1ln ( , )Ty b t T

T t

100 ( , ) ( , )T

T

t

P b t T C b t s ds

100 100 ( , ) 100 ( , )T

T tb t T y b t s ds

14

Continuous Compounding … Instantaneous short rate is defined as

If there is no uncertainty in the short rate, then the following relationship must hold:

or

( ) lim ln ( , ) |T T tT tr t y b t TT

( , ) ( , ) ( )db t T b t T r t dt

( , ) exp ( )T

t

b t T r s dt

15

Forward Rates Forward rate is the rate observed now that

will be applied to period of time in the future we write the forward rate as ft(T1,T2) This rate can be achieved through Forward Rate

Agreements (FRAs) If we know all the zero rates, then

2 1 2 1

2 1

2 2 12

1

1

1 2

1

1 2

(1 ) (1 ) (1 ( , ))

(1 )( , ) 1

(1 )

T t T t T TT T t

T t T TT

t T tT

y y f T T

yf T T

y

16

Forward Rates … If we know all the implied zeros, then

Implication for coupon bond pricing:

coupons are reinvested at the forward rates

2 1

1

11 2

2

( , )( , ) 1

( , )

T T

t

b t Tf T T

b t T

21 2

100

1 (1 ) (1 )nn

C C CP

y y y

17

Forward Rates vs Zero Rates

If the zero curve is curve is upward sloping

If the zero curve is curve is downward sloping

21 2( , )t Tf T T y21 2( , )t Tf T T y

18

Example

f0(1,2):

f0(2,3):

f0(3,4):

f0(1,4):

Maturity zero price zero yield

1 0.9500 5.263%

2 0.9000 5.409%

3 0.8500 5.567%

4 0.7900 6.070%

20(1 5.263%)(1 (1,2)) (1 5.409%)f

0 (1,2) 5.56%f

2 30(1 4.409%) (1 (2,3)) (1 5.567%)f 0 (2,3) 5.88%f

3 40(1 5.567%) (1 (3,4)) (1 6.070%)f

0 (3,4) 7.59%f

3 40(1 5.263%)(1 (1,4)) (1 6.070%)f 0 (1,4) 6.34%f

19

Building Zero Curves

Bonds with some maturities may not exist in the market

Some bonds that have similar maturities and coupons are trading at quite different yields impossible to quantify liquidity

Which price should one use, bid or ask? it doesn’t really matter as long as one is

consistent

20

Building Zero Curves One way to overcome the lack of

information corresponding to some maturities is through interpolation interpolate yield curve interpolate zero price model implied zero rates directly

21

Treasury Futures

22

Futures and Forwards

Forward contract: agreement between the buyer and the seller to settle a trade at some pre-specified (forward price) at some future date

Futures contract: similar to forward contract, but are standardized and exchange traded

Forwards, futures, options, and swaps are all in zero net supply

If the long side gains, then the short side loses: zero-sum game

Underlying can be: interest rates, bills, notes, bonds, etc

23

Interest Rate Futures

Most actively traded futures in US: 3-month T-bills: $1 million face value, IMM of CME 3-month Eurodollar certificates of deposit: IMM of

CME, London International Financial Futures Ex. 20-year, 8% Treasury coupon bonds, $100,000 face

value; CBT 10-year, 8% Treasury note, $100,000 face value; CBT 5-year Treasury note, $100,000 face value; CBT Basket of 40 Muni bonds (index); CBT

24

Treasury Futures Contracts

The deliverable asset to the T-bill futures is a $1 million face value T-bill that had 90 days to maturity there is a cheapest-to-delivery option because the T-bill can a 90,

91, or 92 day T-bill the futures price is quoted as

where Yd is the discount rate on the T-bill Example: If the quoted futures price is 93.50, then the price paid

the long part at delivery is

100 (1 )dY

100 93.50 # of days to maturity$1 1

100 360m

25

Treasury Note and Bond Futures The deliverable assets to the 10- and 5-year T-note futures are,

respectively: A $100,000 face value note with maturity of 6.5 to 10 years from

the delivery date A $100,000 face value on-the-run note with original maturity of

less than 5.25 years, and maturity of at least 4.25 years from delivery date

The deliverable asset to the T-bond futures is a $100,000 face value bond with maturity (or earliest call date) of at least 15 years

26

Treasury Note and Bond Futures ... There are some option embedded in the T-note and T-

bond futures (for the short part) the delivery instrument is is not unique, and it can also

be a basket of qualified securities with total face value of $100,000. A conversion factor is used to determine quantity of the eligible securities to be delivered

time option: underlying can be delivered any day during the delivery month

wild-card option end-of-month option: contract cease to trade seven

business days prior to the last business day of the delivery month, although delivery can be made until the last business day

27

Treasury Note and Bond Futures ... Although most of the contracts are settled before

maturity, a significant amount is settled by delivery unlike futures on equity, which settled by cash, futures

on T-notes and T-bonds are settled by delivery how to close out a futures position before maturity?

Delivery tends to take place at the end of the month when the spot curve is upward sloped (why?). However, when the spot curve is downward sloping, the delivery pattern may be mixed

With all those options for the short side, does it mean the short side has an advantage?

28

Conversion Factor

For the futures contract on T-bonds Cash price received by party with short position =

Quoted futures price × Conversion factor + Accrued interest

The conversion factor for a bond is computed in the following way: it is the value the bond on the first day of the delivery month on the assumption that the interest rate for all maturities equals 8% per annum (with semiannual compounding). The bond maturity and the times to the coupon payment dates are rounded down to the nearest three months for the purpose of the calculation. If, after rounding, the bond lasts an exact number of half years, the first coupon is assumed to be paid in six months. Otherwise the first coupon is assumed to be paid after three months and accrued interest is subtracted.

29

Conversion Factor ...

Example: Consider a 14% coupon bond with 20 years and 2 months to maturity. For the calculation of the conversion factor, the bond is assumed to last exactly 20 years, and the first coupon is paid in six months time. The value of the bond is

so the conversion factor for this bond is 1.5938

40

401

7 100159.38

(1 4%) 1.04ii

30

Conversion Factor ...

Example: Consider a 14% coupon bond with 18 years and 4 months to maturity. For the purpose of calculating the conversion factor, the bond is assumed to have exactly 18 years and 3 months to maturity, and the first coupon is assumed to paid in three month. Discounting all the payments back to a point in time three months from today:

Interest rate for a three-month period:PV of the bond is 163.72/1.019804=160.55. Subtracting the accrued

interest of 3.5, it becomes 157.05, so the conversion factor is 1.5705

36

361

7 1007 163.72

1.04 1.04ii

1.04 1 1.9804%

31

Cheapest to Deliver Bond

Party with short position in the future contract receives

The cost of purchasing the bond:

The cheapest to delivery bond is the one for which

is the least This number is usually referred to as the basis of the

bond Obviously, during the delivery month, the basis of a bond

has to be positive

quoted price - quoted futures price conversion factor

quoted price + acc int

quoted future price conversion factor + acc int

32

Spot-Forward Parity Condition In the following, we assume there is only one delivery security and

hence the BAC should be zero to rule out the arbitrage opportunities Consider the case where the underlying instrument is a ZCB, and

the futures contract matures at T=1 Consider the following strategies

buy one bond: now –P; at T=1: P’ borrow Pf/(1+y1): now Pf/(1+y1), at T=1: -Pf

net: now Pf/(1+y1) – P, at T=1: P’- Pf

which is equivalent to a long forward, so the cost should be zero:

1(1 )fP P y

33

Spot-Forward Parity ... In general, the spot-forward parity relation for a zero is

Similarly, if the forward is written on a coupon bond and the contract matures at t=2, consider the following strategies buy one bond borrow borrow the net result is the same as a long forward

(1 )nf nP P y

22( ) /(1 )fP C y

1/(1 )C y

222

1 2

(1 )1 (1 )f

C CP P y

y y

34

Spot-Forward Parity ... In general

In fact, consider a forward contract for delivery of an instrument which today has n+t periods until maturity

Assume the maturity of the contract is n

( ( )) (1 )nf nP P PV C y

1

11 (1 ) (1 )n n tn n t

n n t

CF CFCFP

y y y

1

1

1

1

(1 )1 (1 )

1 (1 )

nnf nn

n

n n tt

n n t

CFCFP P y

y y

CF CF

f f

35

Futures vs. Forward Contracts Although the spot-forward parity relation is sometimes applied to

futures contract, this is incorrect: the daily resettlement of a futures contract leads to random cash flows throughout the life of the contract

For most of the futures contracts, the resulting cash flows are small and the maturity is short, so discounting CFs as it would be appropriate does not change the prices much

Long position is the futures contract:CF < 0 when y increasesCF > 0 when y decreases

it realizes a lose when cost of financing is high, and profit when reinvestment is low

36

Futures vs. Forward Contracts ... So everything else being equal, one prefers a long

position in a forward contract:

Exact relationship can be calculated when specifying an explicit model of interest rates

Note: under the risk-neutral probability, the forward price is expected future spot price, while the futures price is a martingale

fut forP P

37

Eurodollar Futures ContractsFloaters, Swaps

38

Eurodollar Futures Contract Traded on IMM (Chicago), SIMMEX (Singapore), and the LIFFE

(London) The futures price is quoted as

and the contract is settled in cash for a price equal to

where LIBOR is a money market rate quoted on an annualized basis The face amount of the contract is $1 million dollars Unlike other options, the underlying is based on an interest rate, not a

security price There are no flexibilities in Eurodollar futures contract

100 LIBOR

# of days to mat$10,000 100 LIBOR

360f fP

39

Eurodollar Futures Contract ... The most actively traded contracts are for three and six month LIBOR

Contracts are settled on the 2nd business day before the 3rd Wednesday of the maturity month

Example: Consider the three month Eurodollar futures price quoted on 01/02/87 for maturity 03/16/87 of 93.95, the implied LIBOR rate on the contract is

Suppose I take a short position in the contract. On 03/16/87 the 3-month LIBOR was 6.50 for a cash price of 93.50. Hence I receive the futures price

and I pay the cash price of

$10,000 93.95 1/ 4 $234,875fP

LIBOR 6.05%f

$10,000 93.50 1/ 4 $233,750fP

40

Eurodollar Futures Contract ...The net cash flow is

Obviously, in practice, this is the accumulated payment because the change is settled daily

The price sensitivity of the contract can be measured by its PVBP: given that the futures price is linear in the underlying LIBOR rate, we do not need to take derivatives. The change in the futures price for one basis point change in the LIBOR rate can be easily calculated.

In the case of the 3-month LIBOR contract

$1,125fP P

PVBP=-( ' ) 10,000 100 1 1/ 4 $25f fP P bp

41

How to Calculate LIBOR Rate? Let b(t,s) be the principal amount the LIBOR rate , at the date t

for the maturity date s is quoted on, and is the number of days between t and s, then we have

so

How is the LIBOR rate determined in the futures contract on settlement date?

( )tL

1( , )

1 ( ) / 360t

b t sL

360 1( ) 1

( , )tL b t s

42

FRAs

Then Forward Rate Agreement (FRA) market is the OTC equivalent of the exchanged-traded Eurodollar futures

The liquid and easily accessible sector of the FRA market is for 3- and 6-month LIBOR, 1-month forward they are referred to as 1x4 and 1x7 contracts, respectively

Contracts for delivery of 2, 3, 4, 5, and 6-month forward are also available

On the delivery date, the buyer of the contract receives

# of days to mat Principal

360 1+ (# of days to mat)/360s fS

R RR

43

FRAs ...

Consider a 1x4, $100 million FRA at 11%. In one month, if the three-month reference rate, say LIBOR, is above the forward rate, then the seller must pay the payer the discounted difference between the two rates times the principal $100 million. For example, if the 3-month rate is 11.5%, then the payment will equal

When the FRA is first initiated, what should the rate in the contract be?

90 100,000,000(11.5% 11%) 121,506.68

360 1 11.5% (90 / 360)

44

Floaters

Floating-rate notes, or floaters, are debt securities with coupons based on a short-term index, such as the prime rate or the 3-month T-bill rate, and that are reset for more than once a year

Big impetus to the market: $650 million issue of floating-rate notes issued on July 30, 1974 by Citicorp

Characteristics of the issue: coupon rate to be adjusted semi-annually (every june and Dec) at

100bp above T-bill rate Beginning on June 1976, and on every reset date thereafter, the

notes were puttable at par A floor of 7.7% on the coupon was established for the first year

45

Floaters ... This market mushroomed after 1982

at the end of 1992: 221 issues for 26.7 billion of floating-rate corporate debt outstanding in US

How to price such a debt? (ignore the option and credit risk) Consider a coupon bond whose coupon rate is set equal to the one-

period spot rate y1 at the beginning of every period. Maturity value: M one period before maturity, the price of the floater

two periods before maturity

1

11F

M M yP

y

'1

11F

F

P M yP M

y

46

Floaters ...

Hence, at any reset date, the price of the floater equals its maturity value. In between reset dates, the price is the floater is the same as a zero with maturity value M+Coupon, for example, ½ period before maturity

Duration at reset:

in between: same as zero

1/ 21/2

Coupon

(1+y )F

MP

* 1

1D

y

47

Inverse-Floaters

An inverse floater is a bond whose coupon payment is inversely related to some index level of interest rate

Consider a coupon bond whose coupon rate is set equal to

A long position in an inverse floater generates the same cash flows as being long in a coupon bond with c=c’ being short in a simple floater with c=y1

being long in a zero

where all bonds have the same maturity and principal

1'c y

48

Inverse-Floaters ...

Hence the price can be written as

Duration:

Note that the Macaulay duration of an inverse floater can exceed the time to maturity. The value of the bond is negatively affected by interest rate through two sources: coupon rate, and discount rate

In general, an inverse floater comes from a floor on the coupon payment to prevent the coupon from fall below zero

IV C F ZP P P P

* * * *C F ZIV C F Z

IV IV IV

P P PD D D D

P P P

49

The General Case In general, we can set the coupon rate equal to

A long position in such a bond generates the same cash flow as being long in a coupon bond with c=c’ being long in k simple floaters with c=y1

being short in a k zeros

where all bonds have the same maturity and principal Similarly

1'c c k y

IV C F ZP P k P k P

* * * *C F ZC F Z

P k P k PD D D D

P P P

50

Adjustable-Rate Notes

Adjustable-rate notes, or variable-rate notes, are debt securities with coupon based on a longer-term index

For example, the base rate may be the 2-year treasury yield. The coupon is reset every two years to reflect the new level of the treasury security

A plain vanilla adjustable-rate note trades at par at reset, and trades like a coupon bond with maturity equal to the time until the next reset between reset dates

51

Swaps

Swap is the largest derivatives market in the world: current size ~ $6 trillion in notational amount

Swap: two parties exchange payments Types of swaps:

interest rate, fixed to floating: 2 to 5 years maturity, same currency

basis, floating to floating: 2 to 7 years maturity, same currency, both parties pay floating, different indices

currency, fixed to fixed: 2 to 10 years maturity, different currencies

currency, fixed to floating: 2 to 10 years maturity, different currencies

52

Swaps ... currency, floating to floating: 2 to 10 years maturity, different

currencies yield curve. floating to floating: same currency, both parties pay

floating rates corresponding to different points on the yield curve other types of swaps: equity swaps, credit swaps, volatility

swaps, etc index amortizing swaps: the notional amount is amortized over

time, and the interest rate payments decrease over time mainly used for hedging risks for MBS

Terms of the swap are agreed upon today, but payments start some time in the future. Like forwards, the contract has no value when first initiated

Market instruments related to swaps: interest rate caps/floors swaptions

53

Swaps ... Why enter a swap?

relative advantage in the fixed rate as opposed to the variable rate tax advantages possibility to exchange fixed with variable payments on illiquid assets

(and vice versa) bet on or hedge against interest rate risk

Swap positions can be closed in the same way as forwards and futures contracts, by taking an offsetting position in the same contract

For interest rate swaps, notional principal is never exchanged (while currency swaps do) credit risk?

54

Valuation

We illustrate how to value a swap when the underlying is interest rate For valuation purpose, a swap can be treated as

a package of FRAs, or the buyer long a floater, and short a fixed rate bond

the seller has the opposite position It can be easily treated as a package of FRAs For the 2nd approach, assume that the notional amount is M. The

buyer’s position then has a value

However,

F C CV P P M P

1 (1 ) (1 )

TB

C i Ti i T

c MP

y y

55

Valuation ...

The swap rate when the contract is initiated is thus

PVBP of a swap:

where

In general, PVBP of a swap is negative for the buyer: the buyer (paying fixed) gains when the interest rate increases

1

1 1/(1 )

1/(1 )

TT

B Tt

tt

yc

y

F C

* *

PVBP=PVBP PVBP

1( )

100,00F F C CD P D P

*11/(1 )FD y

56

Valuation ...

Example: Assume a flat spot curve, with y1=3%. Consider a swap with a notional amount M=$100,000 and maturity T=10. The pvbp for the swap is the beginning is thus

where 8.53 is the modified duration of a par bond with maturity T=10, when the interest rate is 3%

1100,000 8.53 100,000 /100,000 75.59

1.03PVBP

57

Swaps ...

The floating rate paid by the seller may be flat, or it may be a spread over a short-term rate (LIBOR)

Assume the seller to pay the spread:

The value of the swap for the buyer is the price of a floater plus the price of an annuity, and minus a coupon bond

Price sensitivity can be considered similarly

'1S SC c y M

Wulin Suo 58

Other Types of Swaps

Amortizing & step-up swaps Extendible & puttable swaps Index amortizing swaps Equity swaps Commodity swaps Differential swaps