bond valuation 435
TRANSCRIPT
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The Valuation of Bonds
Timothy R. Mayes, Ph.D.
FIN 4600: Chapter 12
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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
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Bond Yields
There are several ways that we can describe the
rate of return on a bond:
Coupon rate Current yield
Yield to maturity
Modified yield to maturity
Yield to call
Realized Yield
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The Coupon Rate
The coupon rate of a bond is the stated rate ofinterest that the bond will pay
The coupon rate does not normally changeduring the life of the bond, instead the price ofthe bond changes as the coupon rate becomesmore or less attractive relative to other interestrates
The coupon rate determines the dollar amount ofthe annual interest payment:
Annual Pmt Coupon Rate Face Value
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The Current Yield
The current yield is a measure of the current
income from owning the bond
It is calculated as:
CY Annual Pmt
Face Value
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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 bonds
internal rate of return (IRR)
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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
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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
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The Realized Yield
The realized yield is an ex-post measure of thebonds returns
The realized yield is simply the average annualrate of return that was actually earned on theinvestment
If you know the future selling price,
reinvestment rate, and the holding period, youcan calculate an ex-ante realized yield which canbe used in place of the YTM (this might becalled the expected yield)
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Calculating Bond Yield Measures
As an example of the calculation of the bond return
measures, consider the following:
You are considering the purchase of a 2-year bond (semiannual
interest payments) with a coupon rate of 8% and a current priceof $964.54. The bond is callable in one year at a premium of
3% over the face value. Assume that interest payments will be
reinvested at 9% per year, and that the most recent interest
payment occurred immediately before you purchase the bond.
Calculate the various return measures. Now, assume that the bond has matured (it was not called).
You purchased the bond for $964.54 and reinvested your
interest payments at 9%. What was your realized yield?
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Calculating Bond Yield Measures (cont.)
0 1 2 3 4
401,000
40 40 40-964.54
0 1 2
40
1,030
40-964.54Timeline
if called
Timeline
if not called
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Calculating Bond Yield Measures (cont.)
The yields for the example bond are:
Current yield = 8.294%
YTM = 5% per period, or 10% per year Modified YTM = 4.971% per period, or 9.943% per
year
YTC = 7.42% per period, or 14.84% per year
Realized Yield: if called = 7.363% per period, or 14.725% per year
if not called = 4.971% per period, or 9.943% per year
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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 additionto the quoted price.
Various types of bonds use different assumptions
regarding the number of days in a month and year.
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Valuing Bonds Between Coupon Dates (cont.)
Imagine that we are halfway between coupon dates. Weknow how to value the bond as of the previous (or nexteven) coupon date, but what about accrued interest?
Accrued interest is assumed to be earned equallythroughout the period, so that if we bought the bondtoday, wed have to pay the seller one-half of the
periods interest.
Bonds are generally quoted flat, that is, without theaccrued interest. So, the total price youll pay is thequoted price plus the accrued interest (unless the bond isin default, in which case you do not pay accrued interest,
but you will receive the interest if it is ever paid).
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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 thenext coupon date, and then discount that value
for the fraction of the period remaining.
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Valuing Bonds Between Coupon Dates (cont.)
Lets 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:
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.
50035.193.107973.1067 1
P1
P0
Interest earned
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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:
Now, to get the quoted price we merely subtract the
accrued interest:
If you bought the bond, youd get quoted $1,073.66 but
youd also have to pay $25 in accrued interest for a total
of $1,098.66.
66.1098035.193.1079 5.0
66.10732566.1098 QP
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Day Count Conventions
Historically, there are several different assumptions that have beenmade regarding the number of days in a month and year. Not allfixed-income markets use the same convention:
30/36030 days in a month, 360 days in a year. This is used in the
corporate, agency, and municipal markets. Actual/ActualUses 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.
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The Term Structure of Interest Rates
Interest rates for bonds vary by term to maturity,
among other factors
The yield curve provides describes the yielddifferential 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
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Types of Yield Curves
Rising Declining
Flat Humped
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Todays 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%
90DAY
180DAY
Y
EAR
2YR
3YR
4YR
5YR
7YR
1
0YR
15YR
2
0YR
3
0YR
Term to Maturity
Yield
Data Source: http://www.ratecurve.com/yc2.html
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Explanations of the Term Structure
There are three popular explanations of the termstructure of interest rates (i.e., why the yieldcurve is shaped the way it is): The expectations hypothesis
The liquidity preference hypothesis
The market segmentation hypothesis (preferredhabitats)
Note that there is probably some truth in each ofthese hypotheses, but the expectations hypothesisis probably the most accepted
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The Expectations Hypothesis
The expectations hypothesis says that long-terminterest rates are geometric means of the shorter-term interest rates
For example, a ten-year rate can be considered tobe the average of two consecutive five-year rates(the current five-year rate, and the five-year ratefive years hence)
Therefore, the current ten-year rate must be:
10 5555
510 111 RRR t
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The Expectations Hypothesis (cont.)
For example, if the current five-year rate is 8% and the
expected five-year rate five years from now is 10%, then
the current ten-year rate must be:
In an efficient market, if the ten-year rate is anything
other than 8.995%, then arbitrage will bring it back into
line
If the ten-year rate was 9.5%, then people would buy ten-
year bonds and sell five-year bonds until the rates came
back into line
10 5510 10.108.11 Rt
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The Expectations Hypothesis (cont.)
The ten-year rate can also be thought of a seriesof five two-year rates, ten one-year rates, etc.
Note that since the ten-year rate is observable,we normally would solve for an expected futurerate
In the previous example, we would usually solve
for the expected five-year rate five years fromnow:
5
5
5t
10
10t55t
R1
R1R1
55
10
08.1
08995.110.1
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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
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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 thatdifferent investors have preferred maturities
from which they do no stray
There is not much support for this hypothesis
D
S D
S
Banks
Insurance
Companies
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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
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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 tothese 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
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Duration
Duration is calculated as:
So, Macaulays duration is a weighted average ofthe time to receive the present value of the cash
flows The weights are the present values of the bonds
cash flows as a proportion of the bond price
D
Pmt t
iBond ice
tt
t
N
11
Pr
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Calculating Duration
Recall our earlier example bond with a YTM of5% per six-months:
Note that this is 3.77 six-month periods, which isabout 1.89 years
0 1 2 3 4
401,000
40 40 40-964.54
D
40
1051
40
1052
40
1053
1040
1054
964 54
363676
964 54 377
2 3 4. . . .
.
.
. .
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Notes About Duration
Duration is less than term to maturity, except forzero coupon bonds where duration and maturityare equal
Higher coupons lead to lower durations
Longer terms to maturity usually lead to longerdurations
Higher yields lead to lower durations As a practical matter, duration is generally no
longer than about 20 years even for perpetuities
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Modified Duration
A measure of the volatility of bond prices is themodified duration (higher DMod = highervolatility)
Modified duration is equal to Macaulaysduration divided by 1 + per period YTM
Note that this is the first partial derivative of thebond valuation equation wrt the yield
D D
i
Mod
1
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Why is Duration Better than Term?
Earlier, it was noted that duration is a bettermeasure than term to maturity. To see why, lookat the following example:
Suppose that you are comparing two five-yearbonds, and are expecting a drop in yields of 1%almost immediately. Bond 1 has a 6% couponand bond 2 has a 14% coupon. Which would
provide you with the highest potential gain ifyour outlook for rates actually occurs? Assumethat both bonds are currently yielding 8%.
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Why is Duration Better than Term? (cont.)
Both bonds have equal maturity, so a superficial
investigation would suggest that they will both
have the same gain. However, as well see bond2 would actually gain more.
81.308.1
11.4D
11.471.1159
508.1
1000t
08.1
120
D
98.308.130.4
D
44.415.920
508.1
1000t
08.1
60
D
2,Mod
5
5
1tt
2
1,Mod
5
5
1tt
1
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Why is Duration Better than Term? (cont.)
Note that the modified duration of bond 1 is longer than
that of bond two, so you would expect bond 1 to gain
more if rates actually drop.
Pbond 1, 8%= 920.15; Pbond 1, 7%= 959.00; gain = 38.85
Pbond 2, 8%= 1159.71; Pbond 2, 7%= 1205.01; gain = 45.30
Bond 1 has actually changed by less than bond 2. What
happened? Well, if we figure the percentage change, we
find that bond 1 actually gained by more than bond 2. %Dbond 1 = 4.22%; %Dbond 2 = 3.91% so your gain is
actually 31 basis points higher with bond 1.
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Why is Duration Better than Term? (cont.)
Bond price volatility is proportionally related to the
modified duration, as shown previously. Another way to
look at this is by looking at how many of each bond you
can purchase.
For example, if we assume that you have $100,000 to
invest, you could buy about 108.68 units of bond 1 and
only 86.23 units of bond 2.
Therefore, your dollar gain on bond 1 is $4,222.14 vs.$3,906.15 on bond 2. The net advantage to buying bond
1 is $315.99. Obviously, bond 1 is the way to go.
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Convexity
Convexity is a measure of the curvature of theprice/yield relationship
Note that this is the second partial derivative ofthe bond valuation equation wrt the yield
Yield
D = Slope of Tangent LineMod
Convexity
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Calculating Convexity
Convexity can be calculated with the following
formula:
For the example bond, the convexity (per period)
is:
C
i
CF
i t t
V
t
t
t
N
B
1
1 12
2
1
C
40 1 1
105
40 2 2
105
40 3 3
105
1040 4 4
105
105
964 54
17 82073
11025
964 54
1616393
964 5416758
2
1
2
2
2
3
2
4
2
. . . .
.
.
, .
.
.
, .
..
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Calculating Convexity (cont.)
To make the convexity of a semi-annual bond
comparable to that of an annual bond, we can
divide the convexity by 4 In general, to convert convexity to an annual
figure, divide by m2, where m is the number of
payments per year
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Calculating Bond Price Changes
We can approximate the change in a bonds price
for a given change in yield by using duration and
convexity:
If yields rise by 1% per period, then the price of
the example bond will fall by 33.84, but the
approximation is:
D D DV D i V C V iB Mod B B 05 2.
DVB 359 0 01 964 54 0 5 16 75 964 54 0 01 34 63 081 33822. . . . . . . . . .
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Solved Examples (on a payment date)
Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6
Term (years) 2 3 4 5 6 7
Yield 3% 5% 7% 9% 11% 13%
Coupon 100 80 60 40 20 0
Face Value 1000 1000 1000 1000 1000 1000
Value 1133.94 1081.70 966.13 805.52 619.25 425.06
Duration 1.91 2.79 3.67 4.58 5.62 7.00
Mod. Duration 1.86 2.66 3.43 4.20 5.06 6.19
Convexit 5.33 9.88 15.54 22.46 31.24 43.86