Fixed Income concept

1. MGT 6769
Basics of Fixed Income Securities
Topic 2
Alex Hsu, Ph.D.

2. 2.1 DISCOUNT FACTORS
 2.1.1 Discount Factors across Maturities
 2.1.2 Discount Factors over Time

3. 2.1 DISCOUNT FACTORS
 The discount factor between two dates, t and T, provides the
term of exchange between a given amount of money at t
versus a (certain) amount of money at a later date T: Z(t,T)
 On August 10, 2006 the Treasury issued 182-day Treasury bills.
The issuance market price was $97.477 for $100 of face value
 That is, on August 10, 2006, investors were willing to buy for
$97.477 a government security that would pay $100 on February 8,
2007
 This Treasury bill would not make any other payment between the
two dates
 Thus, the ratio between purchase price and the payoff, 0.97477 =
$97.477/$100, can be considered the market-wide discount factor
between the two dates August 10, 2006 and February 8, 2007
 That is, market participants were willing to exchange 0.97477
dollars on the first date for 1 dollar six months later

4. 2.1.1 Discount Factors across
Maturities
 Z(t,T) records the time value of money between t and T
 At any given time t, the discount factor is lower, the longer the maturity T. That is
given two dates T1 and T2, with T1 < T2, it is always the case that
Z(t,T1) ≥ Z(t,T2)
 It is always the case that market participants prefer a $1 sooner than later
 On August 10, 2006 the U.S. government also issued 91-day bills with a
maturity date of November 9, 2006. The price was $98.739 for $100 of face
value
 Thus, denoting again t = August 10, 2006, now T1 = November 9, 2006, and
T2 = February 8, 2007, we find that the discount factor Z(t,T1) = 0.98739,
which is higher than Z(t,T2) = 0.97477

5. 2.1.1 Discount Factors across Time
 Discount factors give the current value (price) of
receiving $1 at some point in the future
 These values are not constant over time
 One of the variables that determines this value is
inflation:
 Higher expected inflation, makes less appealing
money in the future so discounts go down
 Inflation is not the only variable that explains
discount factors

8. 2.2 INTEREST RATES
 2.2.1 Discount Factors, Interest Rates, and
Compounding Frequencies
 2.2.1.1 Semi-annual Compounding
 2.2.1.2 More Frequent Compounding
 2.2.1.3 Continuous Compounding
 2.2.2 The Relation between Discount Factors
and Interest Rates

9. 2.2.1 Discount Factors, Interest Rates,
and Compounding Frequencies
 Interest rates are closely related to discount factors
and are more similar to the concept of return on an
investment
 Yet it is more complicated, because it depends on the
compounding frequency
 The compounding frequency of interest accruals
refers to the number of times within a year in which
interests are paid on the invested capital
 For a given interest rate, a higher compounding
frequency results in a higher payoff
 For a given payoff, a higher compounding frequency
results in a lower interest rate

10. 2.2.1.1 Semi-annual Compounding
 In semi-annual compounding bondholders receive a coupon payment twice
a year
 To obtain the semi-annual compounding rate we have:
 where Z(t,T) is a discount factor
 Let t = August 10, 2006, and let T = August 10, 2007 (one year later)
 Consider a year investment of $100 at t with semi-annually compounded interest
r = 5%
 This terminology means that after six months the investment grows to $102.5 =
$100 × (1 + 5%/2), which is then reinvested at the same rate for another six
months, yielding at T:
Payoff at T: $105.0625 = ($100) × (1 + r/2) × (1 + r/2) = ($100) × (1 + r/2)2
 Given that the initial investment is $100, there are no cash flows to the investor
during the period, and the payoff at T is risk free, the relation between money at t
($100) and money at T (= $105.0625 = payoff at T) establishes a discount factor
between the two dates:
.
 
   
1
2
1
, 2 1
,
n
T t
r t T
Z t T  
 
 
  
 
 
 
 2
2
/
1
1
.
.
100
$
)
,
(
r
T
at
payoff
T
t
Z




11. 2.2.1.1 Semi-annual Compounding
 Another example:
 On March 1, 2001 (time t) the Treasury issued a 52-week
Treasury bill, with maturity date T = February 28, 2002
 The price of the Treasury bill was $95.713
 As we have learned, this price defines a discount factor
between the two dates of Z(t,T) = 0.95713
 At the same time, it also defines a semi-annually
compounded interest rate equal to r2(t,T) = 4.43%
 In fact, $95.713 × (1 + 4.43% / 2)2 = $100
 The semi-annually compounded interest rate can be
computed from Z(t,T) = 0.95713 by solving for r2(t,T)
.
 
 
%
43
.
4
1
95713
.
0
1
2
1
,
1
2
,
2
1
2
1
2 






















T
t
Z
T
t
r

12. 2.2.1.2 More Frequent Compounding
 Let the discount factor Z(t,T) be given, and let
rn(t,T) denote the annualized n-times
compounded interest rate. Then rn(t,T) is defined
by the equation
Rearranging for Z(t,T), we obtain
 
   
1
1
, 1
,
n
n T t
r t T n
Z t T  
 
 
  
 
 
 
 
 
 
1
,
,
1
n T t
n
Z t T
r t T
n
 

 

 
 

13. 2.2.1.3 Continuous Compounding

    
,
, r t T T t
Z t T e 


14. 2.2.1.3 Continuous Compounding
 An example:
 Consider the earlier example in which at t we invest $100 to receive $105
one year later
 Recall that the annually compounded interest rate is r1(t,t+1) = 5%, the
 semi-annually compounded interest rate is r2(t,t+1) = 4.939%, and the
monthly
 compounded interest rate is r12(t,t+1) = 4.889%
 The following table reports the n−times compounded interest rate also for
more frequent compounding
 As it can be seen, if we keep increasing n, the n−times compounded
interest rate rn(t,t+1) keeps decreasing, but at an increasingly lower rate
 Eventually, it converges to a number, namely, 4.879%
 This is the continuously compounded interest rate
 Note that in this example, there is no difference between the daily
compounded interest rate (n = 252) and the one obtained with higher
frequency (n > 252)
 That is, we can mentally think of continuous compounding as the daily
compounding frequency

15. 2.2.1.3 Continuous Compounding

16. 2.2.2 The Relation between Discount Factors
and Interest Rates
 Note that independently of the compounding
frequency, discount factors are the same
 Thus some useful identities are:
 
 
,
, ln 1 n
r t T
r t T n
n
 
  
 
 
 
 
,
, 1
r t T
n
n
r t T n e
 
  
 
 
 

17. 2.3 THE TERM STRUCTURE OF
INTEREST RATES
 2.3.1 The Term Structure of Interest Rates
over Time

18. 2.3.1 The Term Structure of Interest Rates
over Time
 The term structure of interest rates, or spot
curve, or yield curve, at a certain time t defines the
relation between the level of interest rates and their
time to maturity T
 The term spread is the difference between long
term interest rates (e.g. 10 year rate) and the short
term interest rates (e.g. 3 month interest rate)
 The term spread depends on many variables:
expected future inflation, expected growth of the
economy, agents attitude towards risk, etc.
 The term structure varies over time, and may take
different shapes

19. 2.3.1 The Term Structure of Interest Rates
over Time
 An example:
 On June 5, 2008, the Treasury issued 13-week, 26-week and 52-
week bills at prices $99.5399, $99.0142, and $97.8716, respectively
 Denoting t = June 5, 2008, and T1, T2, and T3 the three maturity
dates, the implied discount factors are Z(t,T1) = 0.995399, Z(t,T2) =
0.990142, and Z(t,T3) = 0.978716
 The discount factor of longer maturities is lower than the one of
shorter maturities
 The question is then: How much lower is Z(t,T3), say, compared to
Z(t,T2) or Z(t,T1)?
 Translating the discount factors into annualized interest rates
provides a better sense of the relative value of money across
maturities
 In this case, the continuously compounded interest rates are:
    %
8444
.
1
25
.
0
995399
.
0
ln
, 1 


T
t
r     %
9814
.
1
5
.
0
990142
.
0
ln
, 2 


T
t
r
    %
1514
.
2
1
978716
.
0
ln
, 3 


T
t
r

23. 2.4 COUPON BONDS
 2.4.1 From Zero Coupon Bonds to Coupon
Bonds
 2.4.1.1 A No Arbitrage Argument
 2.4.2 From Coupon Bonds to Zero Coupon
Bonds
 2.4.3 Expected Return and the Yield to
Maturity
 2.4.4 Quoting Conventions
 2.4.4.1 Treasury Bills
 2.4.4.2 Treasury Coupon Notes and Bonds

24. 2.4.1 From Zero Coupon Bonds to Coupon
Bonds
 The price of zero coupon bonds (with a
principal value of $100) issued by the
government are equal to:
The subscript “z” is mnemonic of “Zero” coupon
bond
 This means that from observed prices for zero
coupon bonds we can compute the discount
factors
   
, 100 ,
z
P t T Z t T
 

25. 2.4.1 From Zero Coupon Bonds to Coupon
Bonds
 Consider a coupon bond at time t with coupon
rate c, maturity T and payment dates T1,T2,…,Tn
= T. Let there be discount factors Z(t,Ti) for each
date Ti. Then the value of the coupon bond can
be computed as:
also:
     
1
100
, , 100 ,
2
n
c n i n
i
c
P t T Z t T Z t T


   

     
1
, , ,
2
n
c n z i z n
i
c
P t T P t T P t T

  


26. 2.4.1 From Zero Coupon Bonds to Coupon
Bonds
 An example:
 Consider the 2-year note issued on t = January 3,
2006 discussed earlier
 On this date, the 6-month, 1-year, 1.5-years, and 2-
year discounts were Z(t,t+0.5) = 0.97862, Z(t,t+1) =
0.95718, Z(t,t+1.5) = 0.936826 and Z(t,t+2) = 0.91707
 Therefore, the price of the note on that date was
 which was indeed the issue price at t
   









4
1
997
.
99
$
91707
.
0
100
$
5
.
0
,
1875
.
2
$
,
i
n
c i
t
t
Z
T
t
P

27. 2.4.1.1 A No Arbitrage Argument
 In well functioning markets in which both the coupon
bond Pc(t,Tn) and the zero coupon bond Pz(t,Tn) are
traded in the market, if the previous relation does
not hold, an arbitrageur could make large risk-free
profits. For instance, if
then the arbitrageur can buy the bond for Pc(t,Tn)
and sell immediately c/2 units of zero coupon bond
with maturities T1,T2,…,Tn-1 and (c/2+1) of the zero
coupon with maturity Tn
This strategy leads instantly to a profit
     
1
, , ,
2
n
c n z i z n
i
c
P t T P t T P t T

  


28. 2.4.2 From Coupon Bonds to Zero Coupon
Bonds
 We can also go the other way around, with
enough coupon bonds we can compute the
implicit value of zero coupon bonds
 With sufficient data we can obtain the discount
factors for every maturity
 This methodology is called bootstrap
methodology

29. 2.4.2 From Coupon Bonds to Zero Coupon
Bonds
 Let t be a given date. Let there be n coupon bonds,
with coupon ci and maturities Ti. Assume that
maturities are regular intervals of six months. Then,
the bootstrap methodology to estimate discount
factors, for every i = 1,…,n is as follows:
1. The first discount factor Z(t,T1) is given by:
2. Any other discount factor Z(t,Ti) for i = 2,…,n is
given by:
.
 
 
 
1
1
1
,
,
100 1 / 2
c
P t T
Z t T
c

 
 
   
 
 
1
1
1
, / 2 100 ,
,
100 1 / 2
i
c i i j
j
i
P t T c Z t T
Z t T
c


  

 


30. 2.4.2 From Coupon Bonds to Zero Coupon
Bonds
 An example:
 On t = June 30, 2005, the 6-month Treasury bill, expiring on T1 =
December 29, 2005, was trading at $98.3607
 On the same date, the 1-year to maturity, 2.75% Treasury note,
was trading at $99.2343
 The maturity of the latter Treasury note is T2 = June 30, 2006
 We can write the value of the two securities as:
Pbill(t,T1) = $98.3607 = $100 × Z(t,T1)
Pnote(t,T2) = $99.2343 = $1.375 × Z(t,T1) + $101.375 ×
Z(t,T2)
 We have two equations in two unknowns:
Z(t,T1) = $98.3607 / $100 = 0.983607
.
    965542
.
0
375
.
101
$
983607
.
0
375
.
1
$
2343
.
99
$
375
.
101
$
,
375
.
1
$
2343
.
99
$
, 1
2 






T
t
Z
T
t
Z

31. 2.4.2 From Coupon Bonds to Zero Coupon
Bonds
 Another example:
 On the same date, t = June 30, 2005, the December
31, 2006 Treasury note, with coupon of 3%, was
trading at $99.1093
 Denoting by T3 = December 31, 2006, the price of this
note can be written as:
P(t,T3) = $1.5 × Z(t,T1) + $1.5 × Z(t,T2) +
$101.5 × Z(t,T3) = $99.1093

32. 2.4.3 Expected Return and the Yield to
Maturity
 Yield to maturity is a kind of weighted average of the spot
rates corresponding to the different cash flows paid by a bond
 The higher a specific cash flow, the higher the weight on that
spot rate
 YTM is a convenient summary measure, but it has limitations:
 You can only calculate it after you know a bond’s price.
 It only applies to a single bond
 IMPORTANT: Yield to maturity is often not a good way to
compare investment decisions.
 If the yield curve is not flat, bonds with different maturities or coupon rates will
almost always have different yields
 The YTMs of two fairly priced bonds will differ if they have different coupon rates
or maturities

33. 2.4.3 Expected Return and the Yield to
Maturity
 Use the next table in the following example:
 Columns 1 to 6 display coupon rates, maturities, and quotes of the latest issued
Treasury notes on February 15, 2008
 Column 7 shows the discount curve Z(0,T) obtained from the bootstrap procedure,
and Column 8 reports the continuously compounded spot rate curve r(0,T)
 On February 15, 2008, traders could buy or sell two Treasury securities with the same
maturity T = 9.5 years, but with very different coupon rates
 In particular, a T-note with coupon c = 4.750% and a T-bond with coupon c = 8.875%
were available
 Using the discount factors Z(0,T) we can determine the fair prices of the two securities
 The yield to maturity of the c = 4.75 T-note is 3.7548% and for the c = 8.875 T-bond is
3.6603%
 The bond with the higher coupon has lower yield to maturity y
   
    5267
.
141
5
.
9
,
0
100
,
0
2
875
.
8
8906
.
107
5
.
9
,
0
100
,
0
2
750
.
4
5
.
9
5
.
0
875
.
8
5
.
9
5
.
0
750
.
4




























Z
T
Z
P
Z
T
Z
P
T
c
T
c

34. 2.4.3 Expected Return and the Yield to
Maturity

35. 2.4.4.1 Quoting Conventions -Treasury Bills
 Treasury bills are quoted on a discount basis;
rather than quoting a price, Treasury dealers
quote the following:
where n is the number of calendar days between
t and T
 
100 , 360
100
bill
P t T
d
n

 

36. 2.4.4.2 Quoting Conventions - Treasury
Coupon Notes and Bonds
 Treasury notes and bonds present an additional
complication:
 Between coupon dates an interest accrues on the bond, if
a bond is purchased between the coupon dates, the buyer
is entitled to the portion of the coupon that accrues
between the purchase date and the next coupon date and
the seller to the portion of the coupon that accrued
between the last coupon and the purchase date
 It is market convention to quote these without any
inclusion of accrued interests, so:
Invoice (Dirty) Price = Quoted (Clean) Price + Accrued
Interest
 Accrued Interest is given by:
 Accrued Interest = Interest Due in Full Period ×
Number of Days Since Last Coupon Date
Number of Days between Coupon Payments

37. 2.4.4.2 Quoting Conventions - Treasury
Coupon Notes and Bonds
 Today is 12/10/2001
 Semi-annual coupon bond issued on 11/15/2000
 Face value $100 with coupon rate of 3.5%
 Quoted price is $96.15625
 Settlement is 1 day for Treasury
 What is the dirty price?

38. 2.5 FLOATING RATE BONDS
 2.5.1 The Pricing of Floating Rate Bonds
 2.5.2 Complications

39. 2.5.1 The Pricing of Floating Rate Bonds
 A semi-annual Floating Rate Bond with maturity T
is a bond whose coupon payments c(Ti) at dates T1
= 0.5, T2 = 1, T3 = 1.5,…, Tn = T are determined by
the formula:
where r2(t) is the 6-month Treasury rate at t, and s is
a spread
 Each coupon date is also called reset date as it is
the time when the new coupon is reset
 If the spread of a floating rate bond is equal to zero,
the ex-coupon price of a floating rate bond on any
coupon date is equal to the bond par value
   
 
2
100 0.5
i i
c T r T s
   

40. 2.5.1 The Pricing of Floating Rate Bonds

 
 
 
 
 
 
2
2
2 2
100 1 0.5 / 2
100 100 0.5 / 2
0.5 100
1 0.5 / 2 1 0.5 / 2
r
r
V
r r
 
 
  
 

41. 2.5.1 The Pricing of Floating Rate Bonds
 An example:
 Consider a one year, semi-annual floating rate bond where the coupon at
time t = 0.5 depends on today’s interest rate r2(0), which is known
 If today r2(0) = 2%, then c(0.5) = 100 × 2%/2 = 1, what about the coupon
c(1) at maturity T = 1? This coupon will depend on the 6-month rate at time t
= 0.5, which we do not know today
 This implies that we do not know the value of the final cash flow at time T =
1, which is equal to 100 + c(1)
 Consider an investor who is evaluating this bond, this investor can project
himself to time t = 0.5, six months before maturity, can the investor at time t
= 0.5 guess what the cash flow will be at time T = 1? Yes, because at time t
= 0.5 the investor will know the interest rate
 So, he can compute what the value is at time t = 0.5
 Suppose that at time t = 0.5 the interest rate is r2(0.5) = 3%, then the
coupon at
 time T = 1is c(1) = 100×r2(0.5)/2 = 1.5
 This implies that the value of the bond at time t = 0.5 is equal to
Present Value of (100 + c(1)) = (100 + 1.5) / (1 + 0.03 / 2) = 100
which is a round number, equal to par

42. 2.5.1 The Pricing of Floating Rate Bonds
 Example (cont’d):
 What if the interest rate at time t = 0.5 was r2(0.5) = 6%? In this case, the coupon
rate at time T = 1 is c(1) = 100 × r2(0.5)/2 = 103, and the value of the bond at t =
0.5 is
Present Value of (100 + c(1)) = (100 + 3) / (1 + 0.06 / 2) = 100
still the same round number, equal to par
 Indeed, independently of the level of the interest rate r2(T1), we find that the
value of bond at t = 0.5 is always equal to 100:
Present Value of (100 + c(1)) = (100 × (1 + r2(0.5) / 2)) / (1 + r2(0.5) / 2) =
100
 Even if the investor does not know the cash flow at time T = 1, because it
depends on the future floating rate r2(0.5), the investor does know that at time t =
0.5 the ex-coupon value of the floating rate bond will be 100, independently of
what the interest rate does
 But then, he can compute the value of the bond at time t = 0, because the
coupon at time T1 = 0.5 is known at time t = 0 as it is given by c(0.5) = 100 ×
r2(0)/2 = 101; thus the value at time t = 0 is:
Present Value of (100 + c(0.5)) = (100 + 1) / (1 + 0.02 / 2) = 100

43. 2.5.2 Complications
 Complication #1: What if spread (s) isn’t zero?
 Effectively the spread is a fixed payment on the bond so
we can value it separately:
Price with spread = Price of no spread bond +
 Complication # 2: How do we value a floating rate
bond outside of reset dates?
 We know that the bond will be worth 100(1+c(ti)/2) at the
next reset date, note that c(ti) is known
 All we need to do is to apply the appropriate discount
 This leads to the following general formula
 
0.5
0,
n
t
s Z t

 

44. 2.5.2 Complications
 Let T1,T2,…,Tn be the floating rate reset dates
and let the current date t be between time Ti and
Ti+1: Ti < t < Ti+1. The general formula for a semi-
annual floating rate bond with zero spread s is:
where Z(t,Ti+1) is the discount factor from t to
Ti+1. At reset dates, Z(Ti,Ti+1) = 1/[1 + r2(Ti) / 2],
which implies:
     
1 2
, , 100 1 / 2
FR i i
P t T Z t T r T

   
 
 
 
, 100
FR
P t T 

45. 2.5.2 Example
 What is the price of a 0.75-year floating rate
bond that pays a semiannual coupon equal to
floating rate plus 2% spread? We know the
following:
 a. There is a zero coupon bond Pz(0, 0.25) =
99.70.
 b. There is a zero coupon bond Pz(0, 0.50) =
99.20.
 c. There is a coupon bond paying 3% quarterly
P(0, 0.75) = 101.7380.
 d. The interest rate from 3 months ago was 4%.