Contents

• Method 1: – Pricing bond from its yield to maturity– Calculating yield from bond price

• Method 2: – Pricing bond from Duration– Pricing bond from Duration and Convexity

The Fundamentals of Bond ValuationThe present-value model

n

tn

p

tt

m i

P

i

CP

2

12)21()21(

2

Where:Pm=the current market price of the bondn = the number of years to maturityCi = the annual coupon payment for bond ii = the prevailing yield to maturity for this bond issuePp=the par value of the bond

Example

• 8% coupon bond matures in 20 years with a par value of $1000.

• If yield to maturity is 10%

• If yield to maturity is 8%

• If yield to maturity is 6%

• What is the bond price?

The Fundamentals of Bond Valuation

• If yield < coupon rate, bond will be priced at a premium to its par value

• If yield > coupon rate, bond will be priced at a discount to its par value

• Price-yield relationship is convex (not a straight line)

Computing Bond YieldsYield Measure PurposeNominal Yield Measures the coupon rate

Current yield Measures current income rate

Promised yield to maturity Measures expected rate of return for bond held to maturity

Promised yield to call Measures expected rate of return for bond held to first call date

Realized (horizon) yield Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.

Nominal Yield

Measures the coupon rate that a bond investor receives as a percent of the bond’s par value

Current YieldSimilar to dividend yield for stocksImportant to income oriented investors

CY = Ci/Pm where: CY = the current yield on a bond

Ci = the annual coupon payment of bond i

Pm = the current market price of the bond

Promised Yield to Maturity• Widely used bond yield figure

• Assumes– Investor holds bond to maturity– All the bond’s cash flow is reinvested at the

computed yield to maturitySolve for i that will equate the current price to all cash flows from the bond to maturity

n

tn

p

ti

m i

P

i

CP

2

12)21()21(

2

Promised Yield to CallPresent-Value Method

Where:

Pm = market price of the bond

Ci = annual coupon payment

nc = number of years to first call

Pc = call price of the bond

ncc

nc

tt

im i

P

i

CP

2

2

1 )1()1(

2/

Example

Calculating Future Bond Prices

Where:

Pf = estimated future price of the bond

Ci = annual coupon payment

n = number of years to maturity

hp = holding period of the bond in years

i = expected semiannual rate at the end of the holding period

hpn

phpn

tt

if i

P

i

CP

22

22

1 )21()21(

2/

Example

• Coupon rate 10%, 25-year bond with promised YTM = 12%, face value $1000

• Current price is • But you expect market YTM decline to 8% in 5

years, therefore you want to calculate its future price Pf at the end of year 5 to estimate your expected YTM

• Pf = ?• What’s the realized YTM over this 5 years

(assuming all cash flows are reinvested at the computed YTM)

The Duration Measure

• Since price volatility of a bond varies inversely with its coupon and directly with its term to maturity, it is necessary to determine the best combination of these two variables to achieve your objective

• A composite measure considering both coupon and maturity would be beneficial

The Duration Measure

Developed by Frederick R. Macaulay, 1938Where: t = time period in which the coupon or principal payment occurs, typically

every 6 month

Ct = interest or principal payment that occurs in period t

i = semi-annual yield to maturity on the bond

price

)(

)1(

)1( 1

1

1

n

tt

n

tt

t

n

tt

t CPVt

iC

iC

tD

Example

• Calculate the Macaulay duration for a 5-year bond with 4% coupon rate, the ytm is 8%.

Characteristics of Duration• Duration of a bond with coupons is always less

than its term to maturity because duration gives weight to these interim payments– A zero-coupon bond’s duration equals its maturity

• There is an inverse relation between duration and coupon

• There is a positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturity

• There is an inverse relation between YTM and duration

Modified Duration and Bond Price Volatility

An adjusted measure of duration can be used to approximate the interest-rate sensitivity of a bond

m

YTM1

durationMacaulay duration modified

Where:

m = number of payments a year

YTM = nominal YTM

Duration and Bond Price Volatility• Bond price movements will vary proportionally with

modified duration for small changes in yields

• An estimate of the percentage change in bond prices equals the change in yield time modified duration

iDP

P

mod100

Where:

P = change in price for the bond

P = beginning price for the bond

Dmod = the modified duration of the bond

i = yield change in basis points

Duration

• Duration approximation (the straight line) always understates the value of the bond; it underestimates the increase in bond price when the yield falls, and it overestimates the decline in price when the yield rises.

Bond Convexity

• Modified duration is a linear approximation of bond price change for small changes in market yields

• Price changes are not linear, but a curvilinear (convex) function

iDP

P

mod100

Yield

Price

Duration

Pricing Error from convexity

Duration and Convexity

Correction for Convexity

n

tt

t tti

C

iPConvexity

1

22

)()1()1(

1

Correction for Convexity:

])([21 2

mod iConvexityiDP

P

Modified Duration-Convexity Effects

• Changes in a bond’s price resulting from a change in yield are due to:– Bond’s modified duration– Bond’s convexity

• Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change

• Convexity is desirable and always a positive number

Convexity - example• 30-year maturity, 8% coupon bond, and sells at an

initial YTM of 8%, so it is a par bond. The modified duration of the bond at its initial yield is 11.26 years, and its convexity is 212.4. If the yield increases from 8% to 10%, the bond price will fall to:

• 1) The duration rule:• 2) The duration with convexity rule:

Convexity - example• 1) The duration rule:

• 2) The duration with convexity rule:

%52.222252.002.026.11*

xyDP

P

%27.181827.0

02.x4.212x5.002.0x26.11

)(x2

1

2

2*

yconvexityxyDP

P

Limitations of Macaulay and Modified Duration

• Percentage change estimates using modified duration only are good for small-yield changes

• Difficult to determine the interest-rate sensitivity of a portfolio of bonds when there is a change in interest rates and the yield curve experiences a nonparallel shift

• Initial assumption that cash flows from the bond are not affected by yield changes