Simplified comprehensive information on Bond valuation,

1. The Valuation of Bonds

2. Bond Values
 Bond values are discussed in one of two ways:
 The dollar price
 The yield to maturity
 These two methods are equivalent since a price
implies a yield, and vice-versa

3. Bond Yields
 The rate of return on a bond:
 Coupon rate
 Current yield
 Yield to maturity
 Modified yield to maturity
 Yield to call
 Realized Yield

4. The Coupon Rate
 The coupon rate of a bond is the stated rate of
interest that the bond will pay
 The coupon rate does not normally change
during the life of the bond, instead the price of
the bond changes as the coupon rate becomes
more or less attractive relative to other interest
rates
 The coupon rate determines the dollar amount of
the annual interest payment:

5. The Current Yield
 The current yield is a measure of the current
income from owning the bond
 It is calculated as:

6. The Yield to Maturity
 The yield to maturity is the average annual rate
of return that a bondholder will earn under the
following assumptions:
 The bond is held to maturity
 The interest payments are reinvested at the YTM
 The yield to maturity is the same as the bond’s
internal rate of return (IRR)

7. The Modified Yield to Maturity
 The assumptions behind the calculation of the YTM are
often not met in practice
 This is particularly true of the reinvestment assumption
 To more accurately calculate the yield, we can change
the assumed reinvestment rate to the actual rate at which
we expect to reinvest
 The resulting yield measure is referred to as the modified
YTM, and is the same as the MIRR for the bond

8. The Yield to Call
 Most corporate bonds, and many older government
bonds, have provisions which allow them to be called if
interest rates should drop during the life of the bond
 Normally, if a bond is called, the bondholder is paid a
premium over the face value (known as the call
premium)
 The YTC is calculated exactly the same as YTM, except:
 The call premium is added to the face value, and
 The first call date is used instead of the maturity date

9. The Realized Yield
 The realized yield is an ex-post measure of the
bond’s returns
 The realized yield is simply the average annual
rate of return that was actually earned on the
investment
 If you know the future selling price,
reinvestment rate, and the holding period, you
can calculate an ex-ante realized yield which can
be used in place of the YTM (this might be
called the expected yield)

10. Bond Valuation in Practice
 The preceding examples ignore a couple of
important details that are important in the real
world:
 Those equations only work on a payment date. In
reality, most bonds are purchased in between coupon
payment dates. Therefore, the purchaser must pay
the seller the accrued interest on the bond in addition
to the quoted price.
 Various types of bonds use different assumptions
regarding the number of days in a month and year.

11. Valuing Bonds Between Coupon Dates
 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).

12. 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.

13. Day Count Conventions
 Historically, there are several different assumptions that have been
made regarding the number of days in a month and year. Not all
fixed-income markets use the same convention:
 30/360 – 30 days in a month, 360 days in a year. This is used in the
corporate, agency, and municipal markets.
 Actual/Actual – Uses the actual number of days in a month and year.
This convention is used in the U.S. Treasury markets.
 Two other possible day count conventions are:
 Actual/360
 Actual/365
 Obviously, when valuing bonds between coupon dates the day count
convention will affect the amount of accrued interest.

14. The Term Structure of Interest Rates
 Interest rates for bonds vary by term to maturity,
among other factors
 The yield curve provides describes the yield
differential among treasury issues of differing
maturities
 Thus, the yield curve can be useful in
determining the required rates of return for loans
of varying maturity

15. Types of Yield Curves

16. Today’s Actual Yield Curve
Maturity YLD
PRIME 4.75%
DISC 1.25%
FUNDS 1.75%
90 DAY 1.71%
180 DAY 1.88%
YEAR 2.19%
2 YR 3.23%
3 YR 3.74%
4 YR 4.18%
5 YR 4.43%
7 YR 4.91%
10 YR 5.10%
15YR 5.64%
20 YR 5.76%
30 YR 5.61%
U.S. Treasury Yield Curve
24 April 2002
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
90
D
A
Y180
D
A
Y
Y
EA
R
2
Y
R
3
Y
R
4
Y
R
5
Y
R
7
Y
R
10
Y
R
15Y
R
20
Y
R
30
Y
R
Term to Maturity
Yield
Data Source: http://www.ratecurve.com/yc2.html

17. Explanations of the Term Structure
 There are three popular explanations of the term
structure of interest rates (i.e., why the yield
curve is shaped the way it is):
 The expectations hypothesis
 The liquidity preference hypothesis
 The market segmentation hypothesis (preferred
habitats)
 Note that there is probably some truth in each of
these hypotheses, but the expectations
hypothesis is probably the most accepted

18. The Expectations Hypothesis
 The expectations hypothesis says that long-term
interest rates are geometric means of the shorter-
term interest rates
 For example, a ten-year rate can be considered to
be the average of two consecutive five-year rates
(the current five-year rate, and the five-year rate
five years hence)
 Therefore, the current ten-year rate must be:
( ) ( ) ( )10 5
55
5
510 111 RRR t+++=+

19. The Liquidity Preference Hypothesis
 The liquidity preference hypothesis contends that
investors require a premium for the increased volatility
of long-term investments
 Thus, it suggests that, all other things being equal, long-
term rates should be higher than short-term rates
 Note that long-term rates may contain a premium, even if
they are lower than short-term rates
 There is good evidence that such premiums exist

20. The Market Segmentation Hypothesis
 This theory is also known as the preferred habitat
hypothesis because it contends that interest rates
are determined by supply and demand and that
different investors have preferred maturities
from which they do no stray
 There is not much support for this hypothesis

21. Bond Price Volatility
 Bond prices change as any of the variables
change:
 Prices vary inversely with yields
 The longer the term to maturity, the larger the change
in price for a given change in yield
 The lower the coupon, the larger the percentage
change in price for a given change in yield
 Price changes are greater (in absolute value) when
rates fall than when rates rise

22. Measuring Term to Maturity
 It is difficult to compare bonds with different
maturities and different coupons, since bond
price changes are related in opposite ways to
these variables
 Macaulay developed a way to measure the
average term to maturity that also takes the
coupon rate into account
 This measure is known as duration, and is a
better indicator of volatility than term to maturity
alone

23. Duration
 Duration is calculated as:
 So, Macaulay’s duration is a weighted average of
the time to receive the present value of the cash
flows
 The weights are the present values of the bond’s
cash flows as a proportion of the bond price

24. Notes About Duration
 Duration is less than term to maturity, except for
zero coupon bonds where duration and maturity
are equal
 Higher coupons lead to lower durations
 Longer terms to maturity usually lead to longer
durations
 Higher yields lead to lower durations
 As a practical matter, duration is generally no
longer than about 20 years even for perpetuities

25. Modified Duration
 A measure of the volatility of bond prices is the
modified duration (higher DMod = higher
volatility)
 Modified duration is equal to Macaulay’s
duration divided by 1 + per period YTM
 Note that this is the first partial derivative of the
bond valuation equation wrt the yield

26. Convexity
 Convexity is a measure of the curvature of the
price/yield relationship
 Note that this is the second partial derivative of
the bond valuation equation wrt the yield
Yield
D =Slope ofTangentLineMod
Convexity