Transcript Lecture 3 - Wulin Suo's Homepage

Interest Rates
and Swaps
Wulin Suo
1
Term Structure Analysis
Wulin Suo
2
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


Application:




spot curve
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
3
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
4
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
5
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
6
Bootstrapping Method

Example:
Bond

Price
Y1
Y2
Y3
A
99.50
105
0
0
B
101.25
6
106
0
C
100.25
7
7
107
Write yT as the implied spot rate (or simply
zero rate) with maturity T:
100  C
P1 
1  y1
y1  105/ 99.50 1
1
P1
b(0,1) 

 0.9476
1  y1 100  C
7
Bootstrapping

…
Calculating y2 and b(0,2):
P2 
C2
100  C2
100  C2


C
b
(0,1)

2
1  y1 (1  y2 )2
(1  y2 )2
y2  5.32%

106
101.25  6  0.9476 
(1  y2 ) 2
1
b(0, 2) 
 0.9015
2
(1  y2 )
For y3 and b(0,3):
P3 
C3
C3
100  C3
100  C3



C
b
(0,1)

C
b
(0,
2)

3
3
1  y1 (1  y2 )2 (1  y3 )3
(1  y3 )3
y3  7.02%
b(0,3)  0.8159
Maturity
Implied Zero
Zero Rate
A
0.9476
5.53%
B
0.9015
5.32%
C
0.8159
7.02%
8
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
Pn  Cnb(0,1)  Cnb(0, 2) 
 (100  Cn )b(0, n)
P1  (100  C1 )b(0,1)

Step 1:
b(0,1) 

P1
100  C1
y1 
1
1
b(0,1)
Pn  Cnb(0,1)  Cnb(0, 2)   (100  Cn )b(0, n  1)
b(0, n) 
100  Cn
In general:
 1 
yn  

b
(0,
n
)


1/ n
1
9
Bootstrapping


Semi-annual coupon payments can be
handles similarly, and zero rates at semiannual intervals can thus be obtained
What the maturities not on the annual/semiannual intervals?


…
interpolation
Restriction on the zero price: for t1 < t2 < …
<tn,
b(0, t1 )  b(0, t2 ) 
 b(0, tn )
10
Par Bond Yield Curve
Question: Based on the zero curve, what is the
coupon rate that makes the bond trade at par?


1Y maturity:
100  X 1
100 
1  y1
2Y maturity:
100 

X1  5.53
X2
100  X 2

1  y1 (100  y2 )2
X 2  5.327
3Y maturity:
100  X 3b(0,1)  X 3b(0, 2)  (100  X 3 )b(0,3)

X 3  6.908
In general:
n100  X n b(0,1)  b(0,2) 
 b(0, n) 100b(0, n)
11
Zero Curve vs Par Yield

Upward sloping
12
Continuous Compounding

Zero rate with continuous compounding:
b(t, T )  e


 yT (T t )
yT  
1
ln b(t , T )
T t
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:
T
P  100b(t , T )   CT b(t , s)ds
t

Par yield is defined as
T
100  100b(t , T )  100 yT  b(t , s)ds
t
13
Continuous Compounding …

Instantaneous short rate is defined as
r (t )  limT

t

yT  
ln b(t , T ) |T t
T
If there is no uncertainty in the short rate, then
the following relationship must hold:
db(t , T )  b(t , T )r (t )dt
or
 T

b(t , T )  exp    r ( s)dt 
 t

14
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
(1  yT2 )T2 t  (1  yT1 )T1 t (1  ft (T1 , T2 ))T2 T1
 (1  yT2 )T2 t 
ft (T1 , T2 )  
T1 t 
(1

y
)


T1
1
T2 T1
1
15
Forward Rates …

If we know all the implied zeros, then
 b(t , T1 ) 
ft (T1 , T2 )  

b
(
t
,
T
)

2 

1
T2 T1
1
Implication for coupon bond pricing:
C
C
P


2
1  y1 (1  y2 )

100  C

(1  yn )n
coupons are reinvested at the forward rates
16
Forward Rates vs Zero Rates

If the zero curve is
curve is upward
sloping
ft (T1 , T2 )  yT2

If the zero curve is
curve is downward
sloping
ft (T1 , T2 )  yT2
17
Example

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%
f0(1,2):
(1  5.263%)(1  f0 (1, 2))  (1  5.409%)2

f0(2,3):
(1  4.409%)2 (1  f0 (2,3))  (1  5.567%)3

f0 (2,3)  5.88%
f0(3,4):
(1  5.567%)3 (1  f0 (3, 4))  (1  6.070%)4

f0 (1, 2)  5.56%
f0 (3, 4)  7.59%
f0(1,4):
(1  5.263%)(1  f0 (1, 4))3  (1  6.070%)4
f0 (1, 4)  6.34%
18
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
19
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
20
Treasury Futures
21
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: zerosum game
Underlying can be: interest rates, bills, notes, bonds, etc
22
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
23
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
100  (1  Yd )
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  93.50 # of days to maturity 
$1m  1 


100
360


24
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
25
Treasury Note and Bond Futures ...

There are some option embedded in the T-note and Tbond 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
26
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?
27
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.
28
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
40
7
100

 159.38

i
40
1.04
i1 (1  4%)
so the conversion factor for this bond is 1.5938
29
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:
36
7
100
7

 163.72
i
36
1.04
i1 1.04
1.04 1  1.9804%
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
30
Cheapest to Deliver Bond

Party with short position in the future contract receives
quoted future price  conversion factor + acc int

The cost of purchasing the bond:
quoted price + acc int

The cheapest to delivery bond is the one for which
quoted price - quoted futures price  conversion factor


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
31
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:
Pf  P  (1  y1 )
32
Spot-Forward Parity ...

In general, the spot-forward parity relation for a zero is
Pf  P  (1 yn )

n
Similarly, if the forward is written on a coupon bond and
the contract matures at t=2, consider the following
strategies
 buy one bond
( Pf  C) /(1  y2 )2
 borrow
 borrow
C /(1  y1 )
 the net result is the same as a long forward

C
C 
2
Pf   P 


(1

y
)
2
2 
1

y
(1

y
)

1
2

33
Spot-Forward Parity ...


In general
In fact, consider a forward contract for delivery of an
instrument which today has n+t periods until maturity
P

Pf  (P  PV (C))  (1  yn )n
CF1

1  y1

CFn

n
(1  yn )

CFnt
(1  ynt )nt
Assume the maturity of the contract is n

CFn 
CF1
n
Pf   P 
 
(1

y
)
n

1  y1
(1  yn ) n 

CFn 1
CFn t

 
1  n f1
(1  n ft )t
34
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 increases
CF > 0
when y decreases
it realizes a lose when cost of financing is high, and profit when
reinvestment is low

35
Futures vs. Forward Contracts ...

So everything else being equal, one prefers a long
position in a forward contract:
Pfut  Pfor

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
36
Eurodollar Futures Contracts
Floaters, Swaps
37
Eurodollar Futures Contract


Traded on IMM (Chicago), SIMMEX (Singapore), and the LIFFE
(London)
The futures price is quoted as
100  LIBOR
and the contract is settled in cash for a price equal to
# of days to mat
Pf  $10, 000  100  LIBOR f  
360
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
38
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
LIBOR f  6.05%
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
Pf  $10,000  93.95  1/ 4  $234,875
and I pay the cash price of
Pf  $10,000  93.50  1/ 4  $233,750
39
Eurodollar Futures Contract ...
The net cash flow is Pf  P  $1,125
 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
PVBP=-(Pf ' Pf )  10,000 100 1bp  1/ 4  $25
40
How to Calculate LIBOR Rate?

Let b(t,s) be the principal amount the LIBOR rate Lt ( ) , 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
1
b(t , s) 
1  Lt ( )  / 360
so
Lt ( ) 


360  1

1
  b(t , s) 
How is the LIBOR rate determined in the futures contract on
settlement date?
41
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 3and 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
 Rs  R f  
# of days to mat
Principal

360
1+RS  (# of days to mat)/360
42
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
(11.5%  11%) 

90
100, 000, 000

 121,506.68
360 1  11.5%  (90 / 360)
When the FRA is first initiated, what should the rate in the contract
be?
43
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
44
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 oneperiod spot rate y1 at the beginning of every period. Maturity value: M
 one period before maturity, the price of the floater
PF 

M  M  y1
1  y1
two periods before maturity
PF'  M  y1
PF 
M
1  y1
45
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:

PF 
M  Coupon
(1+y1/2 )1/ 2
D* 
1
1 y
in between: same as zero
46
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
c ' y1
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
47
Inverse-Floaters ...

Hence the price can be written as
PIV  PC  PF  PZ

Duration:
*
DIV



PC
P
P
 DC*  F  DF*  Z  DZ*
PIV
PIV
PIV
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
48
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

c  c ' k  y1
PIV  PC  k  PF  k  PZ
PC
k  PF
k  PZ
*
*
D   DC 
 DF 
 DZ*
P
P
P
*
49
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
50
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
51
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



52
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?
53
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,
V  PF  PC  M  PC
T
cB
M
PC  

i
T
(1

y
)
(1

y
)
i 1
i
T
54
Valuation ...

The swap rate when the contract is initiated is thus
cB 
1  1/(1  yT )T
T
1/(1  y )
t 1

PVBP of a swap:
t
t
PVBP=PVBPF  PVBPC
 ( DF*  PF  DC*  PC ) 
where

1
100, 00
DF*  1/(1  y1 )
In general, PVBP of a swap is negative for the buyer: the buyer
(paying fixed) gains when the interest rate increases
55
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
 1

PVBP  
100, 000  8.53 100, 000  /100, 000  75.59
 1.03

where 8.53 is the modified duration of a par bond with maturity T=10,
when the interest rate is 3%
56
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:
CS   cS'  y1   M
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
57
Other Types of Swaps






Amortizing & step-up swaps
Extendible & puttable swaps
Index amortizing swaps
Equity swaps
Commodity swaps
Differential swaps
Wulin Suo
58