Bond Valuation 435

8/12/2019 Bond Valuation 435

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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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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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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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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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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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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

Bond Valuation 435

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