Fixed Income Securities
Dr. Rong Chen
The Department of Finance
Xiamen University
Syllabus
Part I basic knowledge: Fixed-income instruments, prices
and yields
Part II Term structure: Empirical properties and classical
theories of the term structure & Deriving the zero-coupon
yield curve
Part III Hedging interest-rate risk with duration,
convexity and other ways
Part IV Investment strategies: passive, active and
performance measurement.
Part
Part
Part
Part
V: Swaps and futures
VI: Dynamic term structure modeling
VII: Interest-rate derivatives with options
VIII: Securitization
Copyright © Rong Chen, 2007, Finance Department, XMU
2
References
Lionel Martellini, Philippe Priaulet, Stephane Priaulet, 2003,
Fixed-income securities: valuation, risk management and
portfolio strategies, Wiley.
Suresh M. Sundaresan, 1997, Fixed income markets and
their derivatives, South-Western College Publishing
John Hull, 2006, options, futures and other derivatives, Prentice
Hall
Moorad Choudhry, 2005, Fixed-income securities and derivatives
handbook, Bloomberg
Bond markets, 2000, Analysis and strategies, Frank J.
Fabozzi ,4th edition, NJ :Prentice Hall
(美)布鲁斯·塔克曼(Bruce Tuckman)著,黄嘉斌译, 1999, 北京：宇航出版社
李奥奈尔·马特里尼, 菲利普·普里奥兰德著,肖军译, 2002, 固定收益证券：对利率风险
进行定价和套期保值的动态方法 ,,北京：机械工业出版社
谢剑平， 2003， 固定收益证券：投资与创新，人民大学出版社
薛立言 刘亚秋， 2006， 债券市场，东华书局
3
林清泉，2005，固定收益证券, 武汉大学出版社
Internet resources
http://www.chinabond.com.cn/chinabond/i
ndex.jsp中国债券信息网
http://www.chinamoney.com.cn/databas/n
ew/zaxiang/shouye/index.jsp中国货币网
http://bond.homeway.com.cn/和讯债券
Copyright © Rong Chen, 2007, Finance Department, XMU
4
Fixed income securities?
Relatively fixed cash flows
securities
— traditional fixed income instruments：
bonds/money-market instruments ( repo, Tbills)
—Interest rate derivatives: futures,
forwards, swaps, options (caps/ floors)
—Bonds with embedded options
securitization
Copyright © Rong Chen, 2007, Finance Department, XMU
5
Chapter 10
Interest Rate
Swaps
Contents
Definition
Pricing and quotes
Uses
Other nonplain vanilla swaps
Copyright © Rong Chen, 2007, Finance Department, XMU
7
10.1
DESCRIPTION OF
SWAPS
Definition
Plain vanilla interest rate swap
— a party agrees to pay cash flows equal to
interest at a predetermined fixed rate on a
notional principle for a number of years. In
return, it receives interest at a floating
rate on the same notional principal for the
same period of time.
—OTC derivative product
Copyright © Rong Chen, 2007, Finance Department, XMU
9
An Example of a “Plain Vanilla”
Interest Rate Swap
An agreement by Microsoft with Intel
to receive 6-month LIBOR & pay a fixed
rate of 5% per annum every 6 months
for 3 years on a notional principal of
$100 million
Next slide illustrates cash flows
Copyright © Rong Chen, 2007, Finance Department, XMU
10
Cash Flows to Microsoft
---------Millions of Dollars--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.5, 2001
4.2%
Sept. 5, 2001
4.8%
+2.10
–2.50
–0.40
Mar.5, 2002
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2002
5.5%
+2.65
–2.50
+0.15
Mar.5, 2003
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2003
5.9%
+2.80
–2.50
+0.30
Mar.5, 2004
6.4%
+2.95
–2.50
+0.45
Copyright © Rong Chen, 2007, Finance Department, XMU
11
Terminology and conventions
All swaps are traded under the legal terms and conditions
fixed by the International Swap Dealer Association
( ISDA )
Terms:
— Maturity
— Notional amount
— Fixed interest rate : fixed leg
— Floating interest rate : floating leg : LIBOR
— Payment dates
Trade date
Effective date: calculate the interest payment
Payment date
Day-count basis
Dollars and Euro: Acutal/360
Sterling: Actual/ 365
— only the difference between the two interest payments is
exchanged
Copyright © Rong Chen, 2007, Finance Department, XMU
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10.2
PRICES
Pricing of Interest Rate
Swaps
Valuation in Terms of bonds
— Interest rate swaps can be valued as the
difference between the value of a fixed-rate
bond and the value of a floating-rate bond :
Valuation in terms of forwards
—Alternatively, they can be valued as a portfolio
of forward rate agreements (FRAs)
Forward projection methods
—On the assumption that future floating rates
are equal to the forward rates
Copyright © Rong Chen, 2007, Finance Department, XMU
14
10.2.1
VALUATION IN
TERMS OF BONDS
Cash Flows to Microsoft
---------Millions of Dollars--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.5, 2001
4.2%
Sept. 5, 2001
4.8%
+2.10
–2.50
–0.40
Mar.5, 2002
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2002
5.5%
+2.65
–2.50
+0.15
Mar.5, 2003
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2003
5.9%
+2.80
–2.50
+0.30
Mar.5, 2004
6.4%
+2.95
–2.50
+0.45
Copyright © Rong Chen, 2007, Finance Department, XMU
16
Valuation in Terms of Bonds
If a principal payments are both
received and paid at the beginning and
the end of the swap, this swap can be
regarded as a portfolio of a fixed-rate
bond and a floating-rate bond.
Vswap  B fl  B fix
Vswap  B fix  B fl
Copyright © Rong Chen, 2007, Finance Department, XMU
17
Valuation in Terms of Bonds
(Cont.)
The fixed rate bond is valued in the
usual way n
B fix  L k Ti  Ti 1  P  t , Ti   LP  t , Tn 
i 1
The floating rate bond is valued by
noting that it is worth
par immediately

B

(
L

Lk
T

T
)
P
t
,
T



fl
1
0
after the next payment date 1 
Also called
method”
Copyright“zero-coupon
© Rong Chen, 2007, Finance Department,
XMU
18
Example 10.1
Suppose that a financial institution has
agreed to pay 6-month LIBOR and
receive 8% per annum (with semiannual
compounding) on a notional principal of
$100 million. The swap has a remaining
life of 1.25 years. The LIBOR rates with
continuous compounding for 3-month, 9mon and 15-mon maturities are 10%,
10.5% and 11%. The 6-mon LIBOR rate
at the last
was 10.2%
Copyrightpayment
© Rong Chen, 2007,date
Finance Department,
XMU
19
Solutions:
—
B fix  4e0.1*0.25  4e0.105*0.75  104e0.11*1.25  $98.24
B fl  (100  5.1)e
—
0.1*0.25
 $102.51
VSwap=98.24－102.51=-$4.27
Copyright © Rong Chen, 2007, Finance Department, XMU
20
10.2.2
VALUATION IN
TERMS OF FRAS
Valuation in Terms of FRAs
Each exchange of payments in an
interest rate swap is an FRA
The FRAs can be valued on the
assumption that today’s forward rates
are realized
Steps:
—Find forward rates
—Calculate cash flows of each FRA on the
assumption that the LIBOR rates will equal
the forward rates
—The sum of all the discounted cash flows is
the value
of© Rong
the
swap
Copyright
Chen,
2007, Finance Department, XMU
22
Cash Flows to Microsoft
---------Millions of Dollars--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.5, 2001
4.2%
Sept. 5, 2001
4.8%
+2.10
–2.50
–0.40
Mar.5, 2002
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2002
5.5%
+2.65
–2.50
+0.15
Mar.5, 2003
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2003
5.9%
+2.80
–2.50
+0.30
Mar.5, 2004
6.4%
+2.95
–2.50
+0.45
Copyright © Rong Chen, 2007, Finance Department, XMU
23
Example 10.1
Suppose that a financial institution has
agreed to pay 6-month LIBOR and
receive 8% per annum (with semiannual
compounding) on a notional principal of
$100 million. The swap has a remaining
life of 1.25 years. The LIBOR rates with
continuous compounding for 3-month, 9mon and 15-mon maturities are 10%,
10.5% and 11%. The 6-mon LIBOR rate
at the last
was 10.2%
Copyrightpayment
© Rong Chen, 2007,date
Finance Department,
XMU
24
 $1.07
1. 0.5 100  (0.08  0.102)e
0.105  0.75  0.10  0.25
 0.1075
2. (1)
0.5
(2) 2  (e0.1075 2 1)  0.11044
(3) 0.5 100  (0.08  0.11044)e0.1050.75  $1.41
3. (1) 0.111.25  0.105  0.75  0.1175
0.5
(2) 2  (e
 1)  0.12102
(3) 0.5 100  (0.08  0.12102)e0.111.25  $1.79
4. -1.07-1.41-1.79=-4.27
0.10.25
0.1175 2
Copyright © Rong Chen, 2007, Finance Department, XMU
25
Valuation in Terms of FRAs
The result agrees with the result of the
method in terms of bonds---the forward
rates are based on the term structure.
The zero value of a swap initially doesn’t
mean that each FRA is equal to zero.
Copyright © Rong Chen, 2007, Finance Department, XMU
26
10.2.3
FORWARD
PROJECTION
METHOD
Forward projection method
This method is also based on the
assumption that the future floating
rates of the floating leg are equal to
the forward rates.
Actually it is based on the idea of cash
n
flows n

Vswap  L   k Ti  Ti1  P  t , Ti    f  0,i1,i  Ti  Ti1  P  t , Ti  
 i1
i 1
Copyright © Rong Chen, 2007, Finance Department, XMU

28
Example 10.1
Suppose that a financial institution has
agreed to pay 6-month LIBOR and
receive 8% per annum (with semiannual
compounding) on a notional principal of
$100 million. The swap has a remaining
life of 1.25 years. The LIBOR rates with
continuous compounding for 3-month, 9mon and 15-mon maturities are 10%,
10.5% and 11%. The 6-mon LIBOR rate
at the last
was 10.2%
Copyrightpayment
© Rong Chen, 2007,date
Finance Department,
XMU
29
Vswap
n
 n

 L   k Ti  Ti1  P  t , Ti    f  0,i1,i  Ti  Ti 1  P  t ,Ti  
i 1
 i1


 100* .04e0.1*0.25  .04e0.105*0.75  .04e 0.11*1.25  .051e
0.1*0.25
 .05522e 0.105*0.75  .05875e 0.11*1.25

 4.27million
Copyright © Rong Chen, 2007, Finance Department, XMU
30
Forward projection method
This method agrees with the previous
two methods for plain vanilla swaps.
They are equivalence in essence.
— Forward projection method / FRA
method
—Forward projection method / bond
method
This method is more general than the
other two and is the standard pricing
Copyright © Rong Chen, 2007, Finance Department, XMU
approach
used by the market.
31
“Zero-coupon method”
When the difference between the
measurement date and the payment
date is equal to the maturity of the
reference index:
P  t , Ti 
— Forward price:
P  t , Ti－1 
—Forward
rate:
1  P  t , Ti1  
f

1 
 0,i 1,i 

Ti  Ti 1  P  t , Ti 


1  P  t , Ti1   P  t , Ti  


Ti  Ti 1 
P  t , Ti 

Copyright © Rong Chen, 2007, Finance Department, XMU
32
Vswap
n
 n

 L   k Ti  Ti 1  P  t , Ti    f  0,i1,i  Ti  Ti 1  P  t , Ti  
i 1
 i1

n
 n

1  P  t , Ti1   P  t , Ti  
 L   k Ti  Ti 1  P  t , Ti   

 Ti  Ti 1  P  t , Ti  
 i1
P  t , Ti 
i 1 Ti  Ti 1 



n
 n

 L   k Ti  Ti 1  P  t , Ti     P  t , Ti 1   P  t , Ti   
i 1
 i1

 n

 L   k Ti  Ti 1  P  t , Ti   P  t , T0   P  t , Tn  
 i1

 n

 L   k Ti  Ti 1  P  t , Ti   P  t , Tn    LP  t , T0 
 i1

Equivalent to the
bond method
Only dependent on
zero –coupon
bonds
Copyright © Rong Chen, 2007, Finance Department, XMU
33
Some understandings
Initially, the value of a swap should be
zero so that it is a fair deal. Later on,
prices can differ depending on the
evolution of the term structure.
The fixed leg has longer duration and
therefore is more sensitive to the change
of the interest rate than the floating leg.
The advantage of a plain vanilla swap
compared to a coupon-bearing bond is that
its price is very much lower than that of
the latter while it has almost the same
Copyright © Rong Chen, 2007, Finance Department, XMU
34
sensitivity to rate changes.
Quotes of swaps
The fixed rate which makes the initial value of
the swap equal to zero is swap rate. The
floating rate is usually LIBOR.
— e.g. The bid price quoted by the market maker is
6% to pay the fixed rate as the ask price to
receive the fixed rate 6.05%
A swap could also be quoted as a swap spread -- the difference between the fixed rate of the
swap and the treasury benchmark bond yield of
the same maturity.
—E.g. a 7 year 3-month Libor swap, 45-50: paying 45
points above the 7-year benchmark bond yield and
receiving
the 3-month Libor, or receiving fixed 5035
Copyright © Rong Chen, 2007, Finance Department, XMU
basis points above the 7-year bond yield and paying
10.3
USES OF SWAPS
Optimizing the financial
conditions of a debt :
Comparative Advantage
AAACorp wants to borrow floating
BBBCorp wants to borrow fixed
Fixed
Floating
AAACorp
10.00%
6-month LIBOR + 0.30%
BBBCorp
11.20%
6-month LIBOR + 1.00%
32
The Swap
9.95%
10%
AAA
BBB
LIBOR+1%
LIBOR
33
The Swap when a Financial
Institution is Involved
9.93%
9.97%
10%
AAA
F.I
.
LIBOR
BBB
LIBOR+1%
LIBOR
39
Converting the financial
conditions of a debt or an
asset
Fixed rate <-> floating rate: debts or
assets
E.g. To finance their needs, most firms
issue long-term fixed-coupon bonds
because of the large liquidity of these
bonds. Sometimes they hope to transform
their debts into a floating-rate debt.
E.g. To optimize the matching of assets
and liabilities: A bank has an asset of 4year bond with a semiannually-paid 7%
fixed rate and $10 million principal value
Copyright © Rong Chen, 2007, Finance Department, XMU
40
and a CD at the 6-month Libor rate+0.2%.
Creating new assets using
swaps
An asset swap:
—An investor believes CAD rates will rise over
the medium term. They would like to purchase
CAD 50million 5yr Floating Rate Notes. There
are no 5yr FRNs available in the market in
sufficient size. The investor is aware of XYZ
Ltd 5yr 6.0% annual fixed coupon Bonds
currently trading at a yield of 5.0%. The bonds
are currently priced at 104.38. The investor
can purchase CAD 50million Fixed Rate Bonds
in the market for a total consideration of CAD
51,955,000 plus any accrued interest. They can
then enter
a 5 year Interest Rate Swap
Copyright © Rong Chen, 2007, Finance Department, XMU
41
(paying fixed) with the Bank as follows:
Copyright © Rong Chen, 2007, Finance Department, XMU
42
Flexibility
Customised to match underlying
securities
Can be reversed at any time
Can be traded as a package or
separately
Copyright © Rong Chen, 2007, Finance Department, XMU
43
Hedging interest rate risk
using swaps
Duration hedge
Duration/ convexity hedge
Note that hedging interest-rate risk of
a bond portfolio with swaps is an
efficient way when they have exactly
the same default risk. If not, a default
risk still exists that is not hedged.
Copyright © Rong Chen, 2007, Finance Department, XMU
44
Duration hedge
$ DP
D P
N*  

$ DS
D S
*
P
*
S
Copyright © Rong Chen, 2007, Finance Department, XMU
45
Duration/ convexity hedge
N1  $ DS 1   N1  $ DS 2   $ DP
N1  $CS1   N1  $CS 2   $CP
Copyright © Rong Chen, 2007, Finance Department, XMU
46
10.4
NONPLAIN VANILLA
SWAPS
Accrediting, amortizing and
roller coaster swaps
Bullet swap: the notional principal
remains unchanged
Accrediting swap: the notional amount
increases overtime
Amortizing swap: the notional amount
decreases in a predetermined way over
the life of the swap.
roller coaster swaps: the notional
amount may rise or fall from one period
Copyright © Rong Chen, 2007, Finance Department, XMU
48
to another.
Basis swap
A basis swap is a floating-for-floating
interest-rate swap that exchange the
floating rates of two different market
or/and different maturities.
Copyright © Rong Chen, 2007, Finance Department, XMU
49
Constant maturity swap and
constant maturity treasury
swap
CMS: LIBOR—a particular swap rate
CMT: LIBOR—a particular treasury-bond
rate
CMS-CMT
—A firm pays quarterly to a bank the 10-year
CMT rate+20 bps and receives quarterly from
the bank the 10-year CMS rate.
—Assuming there is a positive correlation
between the evolution of the spread CMS-CMT
and the spread between the yield of risky
bonds and default-free treasury bonds. It is a
valid hedging
instrument
toDepartment,
the credit
spread.50
Copyright © Rong
Chen, 2007, Finance
XMU
Other swaps
Forward-starting swap: a swap starting in
the future
Inflation-linked swap: could be used by
issuers of inflation-linked bonds
Libor in arrears swap: the floating rate is
set and paid in arrears.
Yield-curve swap: bet on the difference
between interest rates at two points on a
given yield curve.
Zero-coupon swap: exchange a fixed or
floating index that delivers regular
couponsCopyright
for an
index that delivers only one51
© Rong Chen, 2007, Finance Department, XMU
coupon at the beginning or at the end of