Fixed Income

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Fixed Income
• Goals
– Understand what bonds are, how to
value them, and be able to price them
– Examine the risks inherent in bonds and
how they vary
1
• Why do we care
Fixed Income
2
• Why do we care
– Bonds are a major source of financing all over the world
– Bonds are issued by corporations, government
agencies, and governments of all levels
– E.g. Railways, Housing Finance
Companies, Hydroelectric Power Projects – Damodar
River Valley, etc.
– Understanding risks of bonds allows us to make
informed investment decisions
Fixed Income
• Global Bond Market
– In 2011 Over $90 Trillion
vsOver $60 trillion for
stock market
capitalization
– Why so?
– India’s GDP?
– U.S. (over $39 trillion)is
the biggest player but
India is growing fast
3
63
90
0 20 40 60 80 100
Stock Mkt Capitalization
Bond Issuance
Fixed Income Vs. Equities (2011)
Fixed Income
4
• Why called Fixed Income
– Coupon
– Maturity Value
– Investors know in advance what they would be getting.
• Types of Bonds
– Govt. Bonds
– Sovereign Bonds
– State Govt. Bonds
– Govt. Related Agencies
– Corporate Bonds
– Money Market Instruments – CPs, CDs
Bond Basics
Corporate Bonds
Bond Basics
Money Market Instruments
Bond Basics
Government Securities
Bond Basics
Bonds
In the most basic sense, a bond is a loan from an
investor to a firm. For this reason, bonds are also
talked about as DEBT
Bonds can be valued as the combination of TWO STREAMS of
cash flows:
Annuity of promised interest payments (called coupon payments,
usually paid every 6 months (semiannually) or annually)
Final repayment of Principal (called the face value, usually
$1,000)
Bonds
Bond Features
• Coupon Payments
– Regular interest payments until the bond matures
• Semiannual / Annual
• How low can the coupon rate of a bond be?
– Bonds can actually have no coupons. In this case, the coupon rate is zero
and that type of bond is referred to as a zero coupon bond.
• Par Value (also called face value)
– Amount of money to be repaid at the end of the loan.
– The par value is the principal of the loan.
9
Bond Features
• Maturity
– Number of years from issue date until the principal is
repaid. The maturity date is the date on which the
principal is repaid.
• Coupon rate (also called the stated rate)
– Annual coupon divided by the face value of the bond
• Current yield
– Annual coupon payments divided by the current
price of the bond. Doesn’t always equal coupon
rate.
10
Bond Valuation
• How do we value bonds?
• Consider a different question:
– Suppose you were to receive annual payments of $100
for 20 years. In addition, you would also receive a lump
sum payment of $1,000 at the end of the 20th year.
How much would you pay for this investment? Suppose
the current interest rate on investments of similar risk
was 10%
11
0 1 2 20
100 100
100 +
1000
Notice that we can break the
problem into two parts, both
of whichweALREADY know
how to do. The annual
payments of $100 are an
annuity and the $1,000 at the
20thyear isjustalumpsum.
Draw a time line of the cash flows
Bond Valuation
– Suppose you were to receive annual payments of $100 for
20 years. In addition, you would also receive a lump sum
payment of $1,000 at the end of the 20th year. How much
would you pay for this investment? Suppose the current
return on investments of similar risk was 10%
12
PV Ann ( )
. ( . )
. = ÷

¸

(
¸
(
=
100
010
1
1
110
85136
20
PV lump sum ( )
( . )
. = =
1000
110
14864
20
Since both of those present
values are at the SAME
POINT IN TIME (time 0),
wecansimplyaddthemup.
851.36 +148.64 =1,000
The value of the investment
would be $1,000.
Present value of the $100 annuity
Present value of the $1,000 final payment
Bond Valuation
• The prior example was exactly the way we can
price bonds
• We can price bonds using discounted cash flow
valuation
– Cash flows of a bond are the coupon payments and the
repayment of principal (face value)
•Coupon payments are simply an annuity
•Face value is simply a lump sum
– Finding the value of these cash flows requires us to
discount them at the MARKET RATE OF INTEREST.
13
Cash Flow and Coupon Rates
• Example: Suppose a bond has five years to
maturity, an annual coupon of $100, and a face
value of $1,000. What are its cash flows?
• What is the coupon rate for this bond? What is its
price?
14
Year 0 1 2 3 4 5
Coupons
Face
Value



100 100 100 100 100
1000
Coupon Rate =
Annual Coupon
Face Value
=
100
1,000
= 10%
We can’t determine the price because we don’t know the market
interest rate at which to discount the cash flows.
How To Value Bonds
• The value (also called the price) of any bond will simply
be the present value of its promised cash flows, which
are coupon payments and face value.
Bond value =PV(Coupon payments) +PV(Face Value)
=PV (Annuity) +PV (Lump Sum)
15
Bond Value =
C
YTM
1 -
1
(1+YTM)
+
Face Value
(1+YTM)
t t

¸

(
¸
(
Annuity Present Value
Lump Sum
Present Value
Where C =per period couponpayment, YTM =yield to maturity
(YTM is just the per period rater), and t =number of periods
remaining until maturity.
Yield-to-Maturity (YTM) – What is it?
• Annual rate you would earn if you bought the bond today
AND held it until it matured.
*
• Required market interest rate that makes the discounted cash
flows of the bond equal to its price
• Usually determined as the market interest rate of bonds with
similar risk
• YTM is simply the interest rate (r) that we will use in the bond
valuation equation
• DOES NOTalways equal the bond’s coupon rate!!!!

*
Implicit assumption of YTM
– YTM implicitly assumes that any coupons you get from the bond are
reinvested at the YTM.
– Thus, it is RARELY the case that the return you earn from owning a bond
equals the YTM that prevailed when you bought it.
– Nothing we can do about the assumption, but important to keep in
mind.
16
Notes on the Bond Pricing Formula
• Semi annual coupons
– Many bonds pay coupons twice a year. To account for
this, halve the coupon rate and the YTM and double the
number of periods.
– Remember that when discounting cash flows, the
payment, the interest rate and the number of periods
MUST ALL MATCH. Thus, if we use semiannual
periods, we need both a semiannual payment and rate
• Finding YTM
– Trial and error (tedious), EXCEL, financial calculator
17
Bond Pricing, Example
• Suppose a company Issues $1,000 bonds with 5 years to
maturity. The semi-annual coupon is $50. Suppose the
market quoted yield-to-maturity for similar bondsis 10%
(compounded semiannually). What is the present value
(i.e. current market price) of the bond? What if the YTM
was 8%? What if the YTM was 12%?
• Steps to calculate bond price
– Calculate the present value of the Face amount
– Calculate the present value of the coupon payments
– Add the two components to get the price
18
Example
19
t t
YTM) + (1
Value Face
+
YTM) + (1
1
- 1
YTM
C
= Price
(
¸
(

¸

10 10
.05) + (1
1,000
+
0.05) + (1
1
- 1
0.05
50
= Price
(
¸
(

¸

1,000 613.91 386.09 = = +
1. Price if similar bonds have a 10% yield-to-maturity:
Remember that payment, time, and rate ALL must match.
Since we have a semiannual payment we NEED a
semiannual rate. What is the effective semiannual rate?
Notice that 5 years means 10 semiannual periods.
Example
20
t t
YTM) + (1
Value Face
+
YTM) + (1
1
- 1
YTM
C
= Price
(
¸
(

¸

10 10
.04) + (1
1,000
+
0.04) + (1
1
- 1
0.04
50
= Price
(
¸
(

¸

2. Price if similar bonds have an 8% yield-to-maturity:
1,081.11 675.56 405.55 = = +
10 10
.06) + (1
1,000
+
0.06) + (1
1
- 1
0.06
50
= Price
(
¸
(

¸

39 . 26 9 558.39 368.00 = = +
3. Price if similar bonds have a 12% yield-to-maturity:
Notice
the
impact
of
Change
in YTM
on Price
Why do Bond Prices Change When Rates Change? – An
Example
• Recall the bond when similar bonds had a 12% YTM. The bond was
priced at $926.39 with rates at 12%.
• Now consider a different bond than ours, Bond NEW, which is a
newly issued bond almost identical to the bond, except that it has
a coupon rate equal to the market rate of 12%. What is the only
difference between buying the OLD bond and the NEW bond?
– With a 12% coupon rate, the NEW bond would have annual coupons of $120
or semiannual coupons of $60.
• Recall that coupon rate is annual coupons/face value. Thus, if face value is
$1,000 (assumed since not stated) and the coupon rate is 12%, we can
determine annual coupons=face value * coupon rate. Annual coupons =
1,000*0.12 =120. Since coupons are paid semiannually, each coupon payment
will be half the $120 or $60.
– Essentially, the only difference between the bonds is that the NEW bond
gives investors $10 extra every six months for five years. What is the value
of getting an additional $10 every six months for five years?
21
Why do Bond Prices Change When Rates Change? – An
Example
22
10 10
50 1 1,000
Price = 1 - + 926.39
0.06 (1+0.06) (1+.06)
(
=
(
¸ ¸
10 10
60 1 1,000
Price = 1 - + 1,000
0.06 (1+0.06) (1+.06)
(
=
(
¸ ¸
10
10 1
1 - =73.61
0.06 (1+0.06)
(
(
¸ ¸
What is the value of $10 each
6 months for five years if the
market rate is 12%
compounded semiannually?
What price would you be willing to pay for the NEW bond?
Recall the value of the IPC bond when rates were 12%
Notice that the difference in prices between the two bonds is PRECISELY the
value of the extra amount of the additional coupon payments for the NEW
bond. You may get higher coupons with the NEW bond than the IPC bond,
but you pay a higher price for them.
Why do Bond Prices Change?
• Notice that with bonds (at least the typical fixed coupon bonds we
will mostly deal with), the cash flows are predetermined at the
time you buy the bond.
– Coupon payments will not change regardless of what happens to
rates
– The face value of the bond is set and doesn’t change.
– These are precisely the reason bonds are referred to as FIXED
INCOME because the payments investors can expect are known and
unchanging.
23
Fixed over the life of the bond
Constantly changing over life of the bond
t t
YTM) + (1
Value Face
+
YTM) + (1
1
- 1
YTM
C
= Price
(
¸
(

¸

Since the payments (numerators) don’t change, but rates and time
(denominators) change constantly, it must be that the bond price changes due to
fluctuations with rates and time.
Why do Bond Prices Change?
• From the formula, you can see that with rates in the
denominator, we would expect an inverse relationship between
changes in rates and changes in bond prices. This is the
fundamental thing to remember about bonds
• BOND PRICES AND YIELDS ARE INVERSELY RELATED. WHEN
RATES RISE, BOND PRICES FALL.
• This should not be surprising since bonds are simply the present
value of a stream of cash flows and as we saw with time value of
money, when rates increase, present values decrease.
24
Fixed over the life of the bond
Constantly changing over life of the bond
t t
YTM) + (1
Value Face
+
YTM) + (1
1
- 1
YTM
C
= Price
(
¸
(

¸

Bond Prices and Face Values
• When a bond’s price equals its face value, the bond is
said to sell AT PAR, and is called a PAR BOND.
• If the price of a bond is LESS than its face value, the bond
is said to sell at a discount to par and is known as a
DISCOUNT BOND
• If the price of the bond is MORE than its face value, the
bond is said to sell at a premium to par and is known as a
PREMIUM BOND.
• We saw in the features of bond prices that bonds have a
coupon rate, current yield, and a YTM. There are distinct
relationships among those three for the above classes of
bonds.
25
Par, Discount, and Premium Bonds
• Par Bonds
– Price =Face Value
– YTM =Coupon Rate
– Current yield =Coupon rate
• Discount Bonds
– Price <Face Value
– YTM >Coupon Rate
– Current yield >Coupon rate
• Premium Bonds
– Price >Face Value
– YTM <Coupon Rate
– Current yield <Coupon rate 26
YTM =10%, Price =$1000
100
1000
10% =
Coupon Rate
Current Yield
100
1000
10% =
YTM =12%, Price =$926.39
Coupon Rate
Current Yield % 80 . 10
39 . 926
100
=
% 10
1000
100
=
YTM =8%, Price =$1081.11
Coupon Rate
Current Yield % 25 . 9
11 . 1081
100
=
% 10
1000
100
=
Interest Rate Risk
• Time to Maturity
– The longer the time to maturity, the greater the
interest rate risk, all else equal. Why?
•Higher t in formula =>greater compounding effect =>small
changes in r, big changes in price
• Coupon Rate
– The lower the coupon rate, the greater the interest
rate risk, all else equal. Why?
•Low coupon =>more of the bond’s value comes from the
face amount
•Low coupon =>less reinvestment risk to offset the price risk
27
Interest Rate Risk Examples
• Time to Maturity
– Consider 2 bonds each with semi-annual coupons and
a coupon rate of 8%. One matures in 30 years and
the other in 2 years. What is the change in bond price
when rates on similar bonds rise from 8% to 9%?
• Coupon Rates
– Consider 2 bonds each with semi-annual coupons and
20 years to maturity. One has a coupon rate of
15%, the other is a zero coupon bond. What is the
change in bond price when rates on similar bonds rise
from 8% to 9%?
28
Varying Time to Maturity
29
Rates go from 8% to 9%, two 8% semi-annual coupon bonds.
One matures in 30 years, the other in 2 years.
4 4
40 1 1,000
1 - +
0.04 (1.04) (1.04)
(
(
¸ ¸
YTM=8%
YTM=9%
2 year
30 year
324.44 +675.56 =1,000
60 60
40 1 1,000
1 - +
0.045 (1.045) (1.045)
(
(
¸ ¸
825.52 +71.29 =896.81
4 4
(1.045)
1,000
+
(1.045)
1
- 1
0.045
40

(
¸
(

¸

143.50 +838.56 =982.06
% 79 . 1
1000
1000 06 . 982
÷ =
÷
60 60
(1.04)
1,000
+
(1.04)
1
- 1
0.04
40

(
¸
(

¸

904.94 +95.06 =1,000
% 32 . 10
1000
1000 81 . 896
÷ =
÷
For the same change in rates, the long bond falls by more than five times the
amount the short term bond declines.
Varying Coupon Rates
30
Rates go from 8% to 9%, two 20-year bonds. One has 15%
coupon rate, other is a zero coupon bond
YTM=8% YTM=9%
15%
coupon
zero
coupon
40 40
75 1 1,000
1 - +
0.045 (1.045) (1.045)
(
(
¸ ¸
1,380.12 +171.93 =1,552.05
40 40
0 1 1,000
1 - +
0.04 (1.04) (1.04)
(
(
¸ ¸
0 +208.29 =208.29
40 40
(1.04)
1,000
+
(1.04)
1
- 1
0.04
75

(
¸
(

¸

1,484.45 +208.29 =1,692.74
% 31 . 8
74 . 692 , 1
74 . 692 , 1 05 . 552 , 1
÷ =
÷
40 40
(1.045)
1,000
+
(1.045)
1
- 1
0.045
0

(
¸
(

¸

0 +171.93 =171.93
% 46 . 17
29 . 208
29 . 208 93 . 171
÷ =
÷
For the same increase in rates, the low coupon bond drops
by more than twice as much as the high coupon bond
31
Interest Rate Risk
Consider semi-annual bonds when YTM
changes from 8% to 9%. What is the effect on
the prices of the bonds?
Coupon
1 year
(T=2)
2 year
(T=4)
5 year
(T=10)
20 year
(T=40)
30 year
(T=60)
15% -0.92% -1.72% -3.62% -8.31% -9.64%
8 % -0.94% -1.79% -3.96% -9.20% -10.32%
Zero -0.95% -1.90% -4.68% -17.46% -25.01%


For a given coupon rate, as time gets longer, the price change is more
negative for the same 100 basis point increase in rates
For a given time to maturity, as coupon rates fall, the price change is more
negative for the same 100 basis point increase in rates
Interest Rate Risk - Summary
• Long maturity bonds are more sensitive to yield changes
than short maturity bonds
• Size of coupon payments also important
• How can we determine the relative importance of these
components?
• We need some measure which accounts for the fact that
we get money at different times
• We need a measure of the effective maturityof the bond
– The concept of duration provides a quantitative measure of the
interest rate risk of a bond.
.
32
Other Risks Faced by Bond Investors
• Credit risk
– Although the firm is legally obligated to repay its
debt, sometimes it is unable and declares
bankruptcy. If this happens, bondholders may
receive onlya portion (perhaps none) of the money
theyaredue.
• Call risk
– Some bonds are issued with call provisions which
allowthe firmto repurchase the debt at a specified
priceafter sometime.
• Firms would ONLY call a bond if the interest rate they are
payingon thebond is higher then thecurrent market rate.
Remember that if coupon rate is higher than the YTM, the
bondwouldbesellingat apremium.
33
Default Risk and Credit Ratings
• There are organizations that study bonds of
various issuers (corporations, state and local
governments) and assign a CREDIT RATING to
reflect the likelihood of default.
• Investment grade debt
– AAA - BBB ( S&P), Aaa- Baa (Moody’s)
34
Sovereign Credit Ratings
• Provided by Rating Agencies such as S&P.
Moody’s, Fitch, etc.
• India’s credit rating?
35
What Should You Be Able To Do?
• Price any bond
• Understand how interest rate changes effect
bonds depending on their characteristics
36

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