2. What is a Bond?
A bond is a tradable instrument that represents a
debt owed to the owner by the issuer. Most
commonly, bonds pay interest periodically
(usually semiannually) and then return the
principal at maturity.
4. Advantages of Bonds over Stocks
Bonds, while a more conservative investment
than stocks, can offer certain investors some very
attractive features:
Safety
Reliable income
Potential for capital gains
Diversification (especially for an otherwise all-equity
portfolio)
Tax advantages
5. Safety of Bonds
The safety of bonds derives mainly from two
things:
Bondholders are in line ahead of both preferred and
common stockholders for payment. Thus, if a firm
falls on hard times, it must first pay its bondholders
while stockholders may see dividends cut.
In the event that a company skips a payment or
violates covenants of the indenture, the creditors may
force it into bankruptcy to protect the value of their
investment. Stockholders have no such right.
6. Reliability of Income
Most bonds are “fixed-income” securities. As such, they
promise a fixed set of interest payments and the return of
the principal at maturity.
Investors can count on receiving their interest payments
in full and on time, except in the event of severe financial
distress. Common stockholders can never be sure of the
exact amount (and sometimes the exact timing) of
dividends.
Bonds that are callable (most corporates and some
Treasuries issued before 1985) do not offer as much
reliability, though it is still far better than stocks. As
interest rates decline, the probability of a call increases.
7. Potential for Capital Gains
Investors who do not hold a bond to maturity may enjoy
capital gains or suffer capital losses:
When interest rates fall, bond prices rise. Thus an investor who
buys when rates are high, and sells after rates fall will earn a
capital gain. The rate decrease may be due to general market
conditions or improvement in the company’s creditworthiness.
When interest rates rise, bond prices fall. Thus an investor who
buys when rates are low, and sells after rates rise will suffer a
capital loss. The rate increase may be due to general market
conditions or a decrease in the company’s creditworthiness.
All other things being equal, as the bond moves through time to
maturity, the price must move towards its face value. Thus,
bonds purchased at a discount will rise in price, and those
purchased at a premium will decline in price.
8. Diversification
Bonds, when added to an equity portfolio, can
lower risk while lowering returns slightly
(depending on the percentage of the portfolio
allocated to bonds).
While bond prices may be quite volatile, due to
the stability of the income that they provide bond
total returns tend to have low correlation with
stock returns.
The following slide shows the effect of adding
bonds to a stock portfolio.
10. Tax Advantages of Bonds
Some bonds are tax-advantaged:
Municipal bond income is free from federal income taxes, and
state income taxes in the state in which they were issued.
Income from U.S. Treasury issues are free from state and local
income taxes.
Income from Savings Bonds (Series EE and I) is free from state
and local income taxes, and federal income taxes on I Bonds
may be deferred for up to 30 years. Federal taxation may be
completely or partially eliminates when used for education.
Note that the above tax benefits are for income (interest
payments) only. Any capital gains are fully taxable at the local,
state, and federal levels.
11. Basic Bond Valuation
The intrinsic value of a bond, like stocks, is the
present value of its future cash flows.
Bonds, however, have much more predictable
cash flows and a finite life.
The cash flows promised by a bond are:
A series of (usually) constant interest payments
The return of the face value of the bond at maturity
12. Basic Bond Valuation (cont.)
The value of a bond is determined by four variables:
The Coupon Rate – This is the promised annual rate of interest. It is
normally fixed at issuance for the life of the bond. To determine the
annual interest payment, multiply the coupon rate by the face value of
the bond. Interest is normally paid semiannually, and the semiannual
payment is one-half the annual total payment.
The Face Value – This is nominally the amount of the loan to the
issuer. It is to be paid back at maturity.
Term to Maturity – This is the remaining life of the bond, and is
determined by today’s date and the maturity date. Do not confuse this
with the “original” maturity which was the life of the bond at issuance.
Yield to Maturity – This is the rate of return that will be earned on the
bond if it is purchased at the current market price, held to maturity, and
if all of the remaining coupons are reinvested at this same rate. This is
the IRR of the bond.
13. Basic Bond Valuation Example
Suppose that you are interested in purchasing a
3-year bond with a 10% semiannual coupon rate
and a face value of $1,000. If your required
return is 7%, what is the intrinsic value of this
bond?
Here is a timeline showing the cash flows:
1000
50 50 50 50 50 50
0 1 2 3 4 5 6
14. Basic Bond Valuation Example (cont.)
Note that the cash flows of the bond consist of:
An annuity, the interest payments, paid every six
months. This is calculated as:
CR FV 0.10 1000
Pmt 50
2 2
A lump sum which is the return of the face value of
the bond at the end of its life. This payment is made
at the same time as the last interest payment.
15. Basic Bond Valuation Example (cont.)
We can find the intrinsic value of these cash flows by finding the
present value of the interest payments and then adding the present
value of the face value:
1
1 6
1 0.07
1 N 1
1 kd FV 2 1000
VB Pmt N
50
kd 1 kd 0.07 1 0.07
2 2
Note that the first term is the present value of an annuity, and the
second is the present value of a lump sum
Do the math, and you’ll find that the bond is worth $1,079.93. Note
that this value must decline until it reaches $1,000 at maturity.
16. Bond Valuation Notes
A few things of note with regard to the example:
The interest is paid semiannually, so we first calculated the
annual interest and the divided it by two. If interest was paid,
say, quarterly, we would have divided the annual amount by
four.
Similarly, we must convert the number of years to maturity (3)
into the total number of periods (6).
Finally, we also must adjust your annual required return (7%) to
a semiannual return (3.5%).
These three variables must always be stated on a per period
basis.
Nearly all bonds (in the U.S.) pay interest more often than
annually. Most often this is semiannually, but it could also be
quarterly or monthly.
17. Valuing Bonds Between Coupon Dates
The bond valuation formula just presented has
one major flaw: It only works on a coupon date.
Since coupon dates (interest payment dates)
usually only occur twice per year, chances are (~
99.45%) you’ll buy (or sell) a bond between
coupon dates.
In this case, we must deal with accrued interest,
and the increase in the bond value since the last
coupon date.
18. Valuing Bonds Between Coupon Dates (cont.)
Imagine that we are halfway between coupon dates. We
know how to value the bond as of the previous (or next
even) coupon date, but what about accrued interest?
Accrued interest is assumed to be earned equally
throughout the period, so that if we bought the bond
today, we’d have to pay the seller one-half of the
period’s interest.
Bonds are generally quoted “flat,” that is, without the
accrued interest. So, the total price you’ll pay is the
quoted price plus the accrued interest (unless the bond is
in default, in which case you do not pay accrued interest,
but you will receive the interest if it is ever paid).
19. Valuing Bonds Between Coupon Dates (cont.)
The procedure for determining the quoted price
of the bonds is:
Value the bond as of the last payment date.
Take that value forward to the current point in time.
This is the total price that you will actually pay.
To get the quoted price, subtract the accrued interest.
We can also start by valuing the bond as of the
next coupon date, and then discount that value
for the fraction of the period remaining.
20. Valuing Bonds Between Coupon Dates (cont.)
Let’s return to our original example (3 years, semiannual
payments of $50, and a required return of 7% per year).
As of period 0 (today), the bond is worth $1,079.93. As
of next period (with only 5 remaining payments) the
bond will be worth $1,067.73. Note that:
1
1067 .73 1079 .93 1.035 50
P1 P0 Interest earned
So, if we take the period zero value forward one period,
you will get the value of the bond at the next period
including the interest earned over the period.
21. Valuing Bonds Between Coupon Dates (cont.)
Now, suppose that only half of the period has gone by. If
we use the same logic, the total price of the bond
(including accrued interest) is:
0.5
1079 .93 1.035 1098 .66
Now, to get the quoted price we merely subtract the
accrued interest:
QP 1098.66 25 1073.66
If you bought the bond, you’d get quoted $1,073.66 but
you’d also have to pay $25 in accrued interest for a total
of $1,098.66.
22. Bond Return Measures
There are three ways in which the expected
return of the bond is reported:
Current Yield (CY)
Yield to Maturity (YTM)
Yield to Call (YTC)
The current yield is simple, but inaccurate. The
yield to maturity (or yield to call) is much more
representative of the return you will receive, but
suffers from a problem of its own.
23. The Current Yield
The current yield on a bond is simply the annual interest
payment divided by its current price.
CR FV
CY
P0
For our example bond, the current yield is:
100
CY 0.0926
1079.93
Note that the current yield is ignoring the capital loss that
you will suffer over the remaining life of the bond (it
must sell for $1,000 at maturity), so it overstates the
expected return for bonds selling at a premium. For
discount bonds, the expected return is understated.
24. The Yield to Maturity
The yield to maturity gives the exact return that you will
actually earn under the following conditions:
You purchase the bond at today’s price
You hold the bond to maturity
You reinvest all interest payments at the same YTM
The last condition is the most difficult to achieve with
interest rates changing all the time. So, YTM is just an
estimate of your actual return.
However, the YTM does take into account the increase
or decrease in the price of the bond (capital gain or loss)
over the life of the bond.
25. The Yield to Maturity (cont.)
Suppose that we didn’t know that our required
return was 7% per year, but we did know that the
current bond price was $1079.93.
We could solve for the yield implied by that
price (i.e., the YTM).
Unfortunately, there is no closed-form solution
to the bond valuation equation, so we need to use
a trial and error algorithm to find the yield.
26. The Yield to Maturity (cont.)
Here is the bond valuation equation, slightly restated to
make the point:
1
1 N
1 YTM FV
PB Pmt N
YTM 1 YTM
Note that I have replaced the bond’s intrinsic value (VB)
with its price (PB), and its required return (kd) with its
yield (YTM).
Our problem now is to solve for that YTM given the
price.
27. The Yield to Maturity (cont.)
To find the YTM, we first make a guess at the yield. Say that we choose
10%. That gives us a price of $1,000 which is lower than the actual price.
To get the price to go up, we must lower our estimated yield.
Suppose we now try 5%. The price now is $1,137.70 which is too high. We
need to try a higher estimated yield.
Now, we know that the YTM must be between 5% and 10%, so let’s “split
the difference” and try 7.5%. We get $1,066.06. Close, but not close
enough.
We now know the YTM is between 5% and 7.5%, so choose 5.75%. We get
$1,115.59. We now know the answer is between 5.75% and 7.5%.
Next, try 6.625%. We get $1,090.48.
And so on. Keep splitting the difference until you arrive at the correct price.
The yield that achieves this is the YTM.
This is the type of process that your calculator goes through when solving
for the YTM (the “i” key). Eventually, you will find that the actual yield is
7%.
28. The Yield to Call
The yield to call (YTC) is exactly the same as
the YTM, except that it assumes that the bond
will be called at the next call date.
The only differences from calculating the YTM
are:
We need to change the number of periods until
maturity to the number of periods until it can be
called.
If a “call premium” is to be received, we must add
that premium to the face value of the bond.
29. Risks of Bonds
Bonds are generally less risky than stocks, but they do suffer from
several types of risk:
Credit risk – Risk of default. (See ratings on next slide)
Price risk – Risk of unexpected changes in rates, causing a capital loss.
Reinvestment risk – Risk that rates will fall and you will reinvest at a
lower rate.
Purchasing power risk – Risk that inflation will be higher than
expected.
Call risk – Risk that the bond will be called because of lower rates.
Liquidity risk – The risk that you will not be able to sell the bond at a
price near its full value.
Foreign exchange risk – Risk that a foreign currency will decline in
value, causing a decline in the value of your interest payments and
principal.
30. Bond Ratings
Credit risk is the most important source of risk for
owners of bonds. As a result, various rating agencies
(S&P, Moody’s, Fitch, and Dominion Bond) assign
grades to indicate the credit quality of various bond
issues. These ratings are similar to those provided for
insurance companies by A.M. Best.
As you should guess, yields on lower rated bonds will be
higher (more risk) than those on higher rated bonds (less
risk).
Bond ratings are a lot like grades: The agencies give A’s,
B’s, C’s, and D’s with various schemes to differentiate
within the category (i.e., AAA is better than AA).
31. Bond Ratings (cont.)
Bond Ratings by Agency
Moody's S&P Fitch DBRS DCR Definitions
Aaa AAA AAA AAA AAA Prime. Maximum Safety
Aa1 AA+ AA+ AA+ AA+ High Grade High Quality
Aa2 AA AA AA AA
Aa3 AA- AA- AA- AA-
A1 A+ A+ A+ A+ Upper Medium Grade
A2 A A A A
A3 A- A- A- A-
Baa1 BBB+ BBB+ BBB+ BBB+ Lower Medium Grade
Baa2 BBB BBB BBB BBB
Baa3 BBB- BBB- BBB- BBB-
Ba1 BB+ BB+ BB+ BB+ Non Investment Grade
Ba2 BB BB BB BB Speculative
Ba3 BB- BB- BB- BB-
B1 B+ B+ B+ B+ Highly Speculative
B2 B B B B
B3 B- B- B- B-
Caa1 CCC+ CCC CCC+ CCC Substantial Risk
Caa2 CCC - CCC - In Poor Standing
Caa3 CCC- - CCC- -
Ca - - - - Extremely Speculative
C - - - - May be in Default
- - DDD D - Default
- - DD - DD
- D D - -
- - - - DP
Source: http://www.bondsonline.com/asp/research/bondratings.asp
32. Bond Ratings (cont.)
Note from this and the next slide, that there is virtually no risk of default
within 1 year, and very little over longer periods, if you invest in investment
grade securities.
One you go below investment grade, however, the risk of default rises
dramatically.
33. Bond Ratings (cont.)
Default Rate by S&P Bond Rating
(15 Years)
60.00%
50.00%
Default Rate
40.00%
30.00%
20.00%
10.00%
0.00%
AAA AA A BBB BB B CCC
Default Rate 0.52% 1.31% 2.32% 6.64% 19.52% 35.76% 54.38%
S&P Bond Rating
34. Bond Ratings (cont.)
Who Rates Bonds?
Each company's share of the total global revenue in 2001 for credit rating agencies
Other
6%
Fitch
14%
Standard &
Poor's
42%
Moody's
38% Source: Wall Street Journal, 6 January 2003, p. C1 and M oody's
35. Corp. Bond Spreads Over Treasuries
This table Corporate (Industrials) Spreads over Treasuries (in basis points)
demonstrates that bond Rating 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 30 yr
Aaa/AAA 35 40 45 55 69 81 92
yields (spreads over Aa1/AA+
Aa2/AA
40
45
45
55
55
60
65
70
79
85
91
101
102
112
equivalent Treasuries) Aa3/AA-
A1/A+
50
60
60
70
65
85
80
100
95
116
111
132
123
148
increase as credit A2/A
A3/A-
70
80
80
95
100
110
115
125
136
152
155
170
171
193
ratings decline. Baa1/BBB+
Baa2/BBB
100
120
115
135
130
150
145
165
167
183
190
200
208
228
Baa3/BBB- 140 150 160 175 195 215 248
Note also that the Ba1/BB+ 275 300 325 350 375 425 475
Ba2/BB 300 325 350 400 450 525 600
spreads widen as Ba3/BB- 350 400 425 475 525 575 750
B1/B+ 450 475 500 575 650 700 825
maturity increases. B2/B
B3/B-
525
600
575
650
625
750
700
850
750
975
825
1075
975
1200
Caa/CCC 850 900 1050 1150 1250 1400 1600
Source: http://www.bondsonline.com/asp/corp/spreadind.html on 20 Nov 2001
36. Malkiel’s Bond Pricing Theorems
In 1961, Burton Malkiel published a paper where he
proved five important bond pricing theorems:
1. Bond prices move inversely to interest rates
2. Longer maturity bonds respond more strongly to a given change
in interest rates
3. Price sensitivity increases with maturity at a decreasing rate
4. Lower coupon bonds respond more strongly to a given change
in interest rates
5. Price changes are greater when rates fall than they are when
rates rise (asymmetry in price changes)
37. Duration
Two of Malkiel’s theorems relate directly to bond price
volatility.
He showed that the longer the term to maturity, the
greater the change in price, but the coupon rate also
affects volatility (lower coupons = more volatility).
These same observations led Frederick Macaulay to look
for a better measure of volatility than just the term to
maturity. In 1938, he discovered duration which
combines maturity and coupon rate to describe a bond’s
price volatility.
38. Duration (cont.)
Suppose that we have two bonds, identical except for their term to
maturity. Both bonds have coupon rates of 10% paid annually (for
simplicity), and face values of $1,000. Your required return is 10%.
Bond 1 has 5 years to maturity while Bond 2 has 10 years to
maturity.
The price of each bond is $1,000 (do the math).
Now, if your required return drops to 8%, which bond will increase
the most in value?
Bond 1 will be worth $1,079.85, and Bond 2 will be worth
$1,134.20
Bond 2 wins because of its longer maturity (see Malkiel’s theorem
2).
Note that had rates risen instead, then Bond 1 would lose less than
Bond 2 so Bond 1 would be favored.
39. Duration (cont.)
Now, suppose that our bonds both have 5 years to
maturity, but Bond 3 has a coupon rate of 7% and Bond 4
has a coupon rate of 10%.
With your required return at 10%, Bond 3 is worth
$886.28, and Bond 4 is worth $1,000 (do the math).
If your required return drops to 8%, which bond will
increase more in value?
Bond 3 will be worth $960.73 (an increase of 8.40%),
and Bond 4 will be worth $1,079.85 (an increase of
7.99%).
So Bond 3, with the lower coupon rate wins (see
Malkiel’s theorem 4).
40. Duration (cont.)
Finally, what if Bond 5 has a maturity of 5 years and a
5% coupon, while Bond 6 has a maturity of 10 years and
a 10% coupon?
Now we have a conflict. Bond 5 has a lower coupon
rate, but Bond 6 has a longer maturity. How do we know
which one will change more in price when rates drop to
8%?
First, note that at 10% Bond 5 is worth $810.46 and
Bond 6 is worth $1,000.
At 8%, Bond 5 is worth $880.22 (an 8.6% increase), and
Bond 6 is worth $1,134.20 (and increase of 13.42%).
Bond 6 wins because it has a longer duration.
41. Calculating Duration
Duration is a measure of the effective life of the
bond. It is a weighted-average term to maturity
where the weights are the present values of the
cash flows:
1 P 1 N CFt
DMac t
t
P0 kd P0 t 1 1 kd
Where DMac is the Macaulay duration, CFt is the
cash flow in period t, and P0 is the current price
of the bond.
42. Calculating Duration (cont.)
Let’s calculate the durations of Bond 5 and Bond 6 from
the last example, using a 10% required return:
50 50 50 50 1050
1
1 2
2 3
3 4
4 5
5
DMac,5 1.10 1.10 1.10 1.10 1.10 4.49
810.46
So, Bond 5 has a duration of 4.49 years. Similar
calculations will show that Bond 6 has a duration of 6.76
years.
The longer duration is the reason that Bond 6
outperformed Bond 5 when rates fell to 8%.
43. Properties of Duration
A longer term to maturity increases duration, all other
things being equal. However, duration increases with
term to maturity at a decreasing rate.
For coupon-bearing bonds, duration is always less than
term to maturity. For zero-coupon bonds it is exactly the
same as term to maturity.
Lower coupon rates lead to longer durations, all other
things being equal.
Higher yields lead to shorter durations.
When calculating duration for semiannual bonds, the
resulting duration will be in semiannual periods. You
must adjust this back to annual by dividing by 2.
44. The Use of Macaulay Duration
Macaulay duration is a very useful tool in bond
portfolio management:
When interest rates are expected to move, a portfolio
manager should change the duration of the portfolio
to take maximum advantage (or limit damage) of the
change.
Also, by using an “immunization” strategy, we can
eliminate both price risk and reinvestment risk to
guarantee a certain amount of funds will be available
to meet needs on a certain future date.
45. Modified Duration
Macaulay duration is a poor predictor of the
percentage change in a bond’s price when its
yield changes. We can improve this
approximation by calculating the modified
duration: D Mac
DMod
1 kd
Bond 5 has a modified duration of:
4.49
DMod ,5 4.08
1.10
46. Modified Duration (cont.)
Bond 6 has a modified duration of 6.14 years.
Note that we can calculate the approximate
percentage change in price of a bond as:
% P DMod kd
So, when rates drop by 2%, we would find that
Bond 5 should change by approximately:
% PBond 5 4.08 0.02 0.0816 8.16 %
% PBond 6 6.14 0.02 0.1229 12 .29 %
47. Modified Duration (cont.)
Let’s compare the approximations to the actual changes:
Bond 5: Actual = 8.6%, Approximate = 8.16%
Bond 6: Actual = 13.42%, Approximate = 12.29%
Why were the approximations so far off?
Mainly because of the huge (200bp) decline in rates. Duration is
technically only correct for very small changes in rates.
The relationship between price and yield is non-linear, but duration
measures linear changes.
If we used only a small change in rates (say, 0.1% or 1bp) we would be
much closer.
If we took account of the non-linear relationship (known as convexity
which is beyond the scope of this class) we would get closer still.
Note though, that the modified duration correctly tells us that Bond 6
will increase in value significantly more than Bond 5. This is really the
important result.
