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The Valuation of Bonds

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Bond Values

Bond values are discussed in one of two ways:

The price

The yield to maturity

These two methods are equivalent since a price

implies a yield, and vice-versa

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Bond Price Relations

Bond Relation 1: Relation between coupon rate,required rate (discount rate or YIELD), bond value

(price), and face value (principal):

bondpremiumFVRCIf

bondparFVRCIf

bonddiscountFVRCIf

F/CratecouponCLet

R

R

R

R

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Bond Price Relations (2)

Bond Relation 2: Inverse relation between bond price

(value) and rate of return (YIELD).

VRIf

VRIf

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Bond Price Relations

Bond Relation 2: Price-Yield Curve depicts the

inverse relation between V and R. The Price-Yield

curve for the 10-year, 9% coupon bond:

BV

R

8550.93

100

9%10%

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Bond Price Relations

Bond Relation 3: The greater a bonds maturity, the

greater its price sensitivity to interest rate changes.

Symbolically:

GreaterMGreater

R%

V%Let

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Bond Price Relations

Bond Relation 4: The smaller a bonds coupon rate, the

greater its price sensitivity to interest rate changes.

Symbolically:

GreaterCLower

R%

V%Let

R

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Bond risks

interest rate risk-- The major risk facing all bondholdersis interest rate risk. This is the effect on bond prices wheninterest rates change. The prices of outstanding bonds moveinversely with changes in interest rates.

default risk-- the possibility that the issuing firm will notbe able to pay, on a timely basis, the interest and/orprincipal. Measured by bond ratings.

reinvestment rate risk-- a major risk factor for bond

investors holding longer term bonds. Since standard yieldcalculations assume reinvestment at a certain rate, actualreturns will be lower if rates fall and investors are not ableto reinvest at the assumed rate.

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Bond risks

inflation risk-- The possibility of unexpected changes inprice levels.

maturityrisk-- the longer the term of the security, the

more risk there is in the investment. The source of this riskis twofold: (a) harder to forecast farther in the future, and

(b) longer term bonds are more volatile than shorter bonds.

call risk-- the possibility of a bond being called in.

Liquidity (marketability) risk-- many bonds trade less

frequently than stocks. A liquid security is liquid if it can be

sold easily without significant price effects

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US market - Fractions

The fractions of points differ among bonds.

Fractions are either in thirds, eighths, quarters, halves,or 64ths.

On a 100 basis, a 1/2 point is 0.50 and a 1/32 point is0.03125.

A price quote of 97-4/32 (97-4) is 97.125 for a bond

with a 100 face value.

Bonds expressed in 64ths usually are denoted in thefinancial pages with a plus sign (+); for example,100.2+ would indicate a price of 100 3/64.

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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 of

the bond changes as the coupon rate becomesmore or less attractive relative to other interestrates

The coupon rate determines the amount of the

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

If the current yield is less than the coupon rate,

the bond would be purchased at a premium Roughly you can think of holding cost

It is calculated as:

ValuePmtAnnualCY

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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 themodified YTM, and is the same as the MIRR for the

bond

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The Yield to Call

Most corporate bonds, and some 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, reinvestmentrate, and the holding period, you can calculate anex-ante realized yield which can be used in place

of the YTM (this might be called the expectedyield)

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Valuing Bonds Between Coupon Dates

Imagine that we are halfway between coupon dates.We know how to value the bond as of the previous (ornext even) coupon date, but what about accruedinterest?

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 theperiods interest.

Bonds are generally quoted flat, that is, without theaccrued interest. So, the total price youll pay is the

quoted price plus the accrued interest (unless thebond is in default, in which case you do not payaccrued interest, but you will receive the interest if it isever paid).

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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 yield

differential among treasury issues of differingmaturities

Thus, the yield curve can be useful in determining

the required rates of return for loans of varying

maturity

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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 thechange 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) whenrates fall than when rates rise

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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 cashflows

The weights are the present values of the bondscash flows as a proportion of the bond price

D

Pmt t

i

Bond ice

tt

t

N

11

Pr

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Modified Duration

A measure of the volatility of bond prices is themodified duration (higher DMod = higher volatility)

Modified duration is equal to Macaulays duration

divided by 1 + per period YTM

Note that this is the first partial derivative of thebond valuation equation wrt the yield

D

D

iMod

1

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Convexity Convexity is a measure of the curvature of the

price/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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Convexity and Bond Price

Convexity can be calculated with the followingformula:

We can approximate the change in a bonds price

for a given change in yield by using duration andconvexity:

Ci

CF

it t

V

tt

t

N

B

11 1

2

2

1

V D i V C V iB Mod B B 0 5 2.

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Observations on duration

When bond has coupons, dur is < time to maturity

Holding maturity and YTM constant, there is inverse

relation between coupons and dur

A bond without coupons, e.g., zero coupon bond has

duration = time to maturity

Generally a positive relation exists between dur and

time to maturity; duration expands with time to

maturity but at a decreasing rate, especially beyond

15 years.

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Observations on convexity The longer the maturity, usually the greater will be the

convexity, all else equal.

Holding yield and duration constant, the higher the

coupon rate, the greater the convexity.

As interest rates decrease, the convexity of a bondincreases, and vice versa.

Two bonds with the same duration but different

convexities will experience different price changes

when there is a significant change in interest rates. Asa result, convexity is said to be desirable when

interest rates are volatile. Note on prior slide that

Bond 1(with the greater convexity) has greater upside

potential when interest rates fall and less downsidemovement when rates rise