CHAPTER SEVEN: BOND ANALYSIS AND INVESTMENT pdf

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

CHAPTER SEVEN: BOND ANALYSIS

AND INVESTMENT

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

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:

The Current Yield

• The current yield is a measure of the current income from owning the bond

• It is calculated as:

Face Value



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)

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

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

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)

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 price of

$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?

Calculating Bond Yield Measures (cont.)

if called Timeline

if not called

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

Bond Valuation in Practice

• The preceding examples ignore a couple of

important details that are important in 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.

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).

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.

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:

• 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.

 1 035  50 93

1079 73

.

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, you’d get quoted $1,073.66 but you’d also have to pay $25 in accrued interest for a total of

 1 035  1098 66 93

.

1079 0.5 

66 1073 25

66



QP

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.

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

Types of Yield Curves

Today’s Actual Yield Curve

R

20 Y

R

30 Y R

Term to Maturity

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

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 year rates (the current year rate, and the five-year rate five years

five-hence)

• Therefore, the current ten-year rate must be:

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   5 5

10 1 08 1 10

1 t R 

The Expectations Hypothesis (cont.)

• The ten-year rate can also be thought of a

series of 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 future rate

• In the previous example, we would usually

solve for the expected five-year rate five years from now:

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

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

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

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

• 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

N

  1

1

Pr

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

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

Why is Duration Better than Term?

• Earlier, it was noted that duration is a better measure than term to maturity To see why, look at the following example:

• Suppose that you are comparing two five-year bonds, and are expecting a drop in yields of

1% almost immediately Bond 1 has a 6%

coupon and bond 2 has a 14% coupon Which would provide you with the highest potential gain if your outlook for rates actually occurs? Assume that both bonds are currently yielding 8%

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 we’ll see bond 2 would actually gain more

11 4

5 08 1

1000 t

08 1

120 D

98 3 08 1 30 4 D

44 4 15 920

5 08 1

1000 t

08 1

60 D

5 5

1 t

t

1 , Mod

5 5

1 t

t 1

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.

• % D bond 1 = 4.22%; % D bond 2 = 3.91% so your gain is actually

31 basis points higher with bond 1

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.

• 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 of Tangent Line Mod

Convexity

2 2

2 3

2

4

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

Calculating Bond Price Changes

• We can approximate the change in a bond’s 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 VB   DMod  D i  VB  0 5   C VB  D i 2

     

Solved Examples (on a payment date)

Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6