Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond

Bond Valuation I

What is a bond?

„ Bond is an I.O.U.

„ Bond is a borrowing agreement

„ Bond issuers borrow money from bond holders

„ Bond is a fixed-income security that typically pays periodic coupon payments, and a principal payment at maturity

Cash Flows of A Typical Bond

$C

1 2 3 t

$C $C $C

$F

Bond Valuation

„ Important characteristics of a bond:

¾ Face Value ($ F) Principal amount.

¾ Coupon ($ C) The periodic interest payment.

¾ Maturity (t periods) Number of periods to maturity.

¾ Required rate (r ) Given the risks associated with the bond (e.g., default risk, interest rate risk), the appropriate required rate.

Coupon Rate and Current Yield

„ Coupon rate is C/F

„ Current yield is the annual coupon of a bond expressed as a percent of its market price, P.

„ Current yield gives the return on your

investment only in terms of the coupon interest payments.

C Y C

= P

Cash Flows of A Typical Bond

$C

1 2 3 t

$C $C $C

$F

Bond Valuation

„ Given face value, coupon payments, number of periods to maturity and required rate of return, we can write down a bond’s intrinsic value as:

( )

( )

B r

F r r

V C

+ +

⎥ ⎦

⎢ ⎤

⎣

⎡

− +

= 1 1

1 1

Bond Valuation

„ In general, the price of a bond is the present value of its future cash flows, which typically consists of coupons and principal value.

„ Using the above principle, you should be able to value bonds with uneven cash flows or uneven interest rates.

Example of a Coupon Bond

„ A coupon bond promises to pay $100 at the end of each year for the next 28 years and to return the principal of $1000 after 28 years.

„ The appropriate required rate of return on the bond, given its risk characteristics, is 9.5%.

„ What is the bond’s intrinsic value?

Bond Valuation Example

„ The principal payment is worth:

„ The coupon payments are worth:

„ Thus, Bond Value =

Citicorp Example

„ Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

Citicorp Example

„ Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8%

¾ Maturity 10 years

„ Cash flows from the bond are:

$8

1 2 3 10

$8 $8 $8

$100

Citicorp Example

„ Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

„ If the appropriate discount rate is 6%, what is the value of the bond?

Citicorp Example

„ Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

„ Solution:

The Bond Price-Return Relationship

„ Prices and returns on bonds are inversely related

$0

$50

$100

$150

$200

0% 2% 4% 6% 8% 10% 12% 14% 16%

Rate

Price

( )^t ( r)^t

F r r

P C

+ +

⎥⎦

⎢ ⎤

⎣

⎡

− +

= 1 1

1 1

Discount and Premium Bonds

„ The face value serves as a benchmark price to define discount or premia on bonds:

¾ Premium Bond:

– P > Face Value and r < Coupon Rate

¾ At Par Bond:

– P = Face Value and r = Coupon Rate

¾ Discount Bond:

– P < Face Value and r > Coupon Rate

Bond Maturity and Price Convergence

„ As the maturity date approaches, the price of a bond approaches its face value.

Discount Bond Premium Bond

75 85 95 105 115 125

10 9 8 7 6 5 4 3 2 1 0

Years to Maturity

Price

Bond Prices and Returns:

Yield-to-maturity (YTM)

„ YTM is the required rate of return that equates the market price of a bond to its intrinsic value.

„ YTM reflects the market’s required rate of return given the bond’s features and given its perceived riskiness.

( )

( )

B r

F r r V C

P ⎥+ +

⎦

⎢ ⎤

⎣

⎡

− +

=

= 1 1

1 1

Citicorp Example

„ Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

„ Suppose the market price of the bond is

$107.02. Find the current yield and YTM of the bond.

Citicorp Example

„ Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

„ Suppose the market price of the bond is

$107.02. Find the current yield and YTM of the bond.

Citicorp Example

„ Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

„ Suppose the market price of the bond is

$107.02. YTM is 7% because we have:

Using Equations to Find YTM

„ You need to solve for r such that:

( ) (

)

100

$ 1 1 1 8

$

10

10 =

+ +

⎥⎦

⎢ ⎤

⎣

⎡

− +

r r r

Using Equations to Find YTM

„ You need to solve for r such that:

„ Trial and error gives us:

r = 6%: _$8

. .

$100

. $114.

0 061 1

1 06₁₀ 1 06₁₀ 72

⎡−

⎣⎢

⎤

⎦⎥+ =

( ) (

)

100

$ 1 1 1 8

$

10

10 =

+ +

⎥⎦

⎢ ⎤

⎣

⎡

− +

r r r

Using Equations to Find YTM

„ You need to solve for r such that:

„ Trial and error gives us:

r = 6%:

r = 8%: _$8

. .

$100

. $100.

0 081 1

1 08₁₀ 1 08₁₀ 00

⎡−

⎣⎢

⎤

⎦⎥+ =

$8

. .

$100

. $114.

0 061 1

1 06₁₀ 1 06₁₀ 72

⎡−

⎣⎢

⎤

⎦⎥+ =

( ) (

)

100

$ 1 1 1 8

$

10

10 =

+ +

⎥⎦

⎢ ⎤

⎣

⎡

− +

r r r

Using Equations to Find YTM

„ You need to solve for r such that:

„ Trial and error gives us:

r = 6%:

r = 8%:

r = 7%: $8

. .

$100

. $107.

0 071 1

1 07₁₀ 1 07₁₀ 02

⎡−

⎣⎢

⎤

⎦⎥+ =

$8

. .

$100

. $100.

0 081 1

1 08₁₀ 1 08₁₀ 00

⎡−

⎣⎢

⎤

⎦⎥+ =

$8

. .

$100

. $114.

0 061 1

1 06₁₀ 1 06₁₀ 72

⎡−

⎣⎢

⎤

⎦⎥+ =

( ) (

)

100

$ 1 1 1 8

$

10

10 =

+ +

⎥⎦

⎢ ⎤

⎣

⎡

− +

r r r

Using a Financial Calculator

to Find YTM

PMT=$8

n=10

FV=$100 PV=$107.02

Calculate r% 7.00%

Bond Risk

„ The coupon payment and face value of a bond are fixed.

„ However, bonds are not riskfree.

„ Default risk is risk from the possibility that the issuer will not be able to pay coupon interest and face value as promised.

Bond Risk

„ As with any asset, risk increases the required returns for bonds.

„ Bonds are rated by several rating agencies such as Moody’s, Standard and Poor and Fitch Investor Services.

„ Lower ratings by these agencies mean more risk and higher required returns.

Bond Ratings

Quality S&P Moody’s

Aaa High

Junk In Default Upper Medium

Medium Prime

Aa AA

Baa AAA

Ba, B, Caa, Ca, C A

BBB

D BB, B, CCC, CC,C

D

A

Bond Risk

„Interest rate risk is risk from interest

rate changes during a bond’s term.

„It is easier to quantify the interest rate

risk than the default risk

„Use duration

Interest Rate Risk

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00

Jan-34 Jan-40 Jan-46 Jan-52 Jan-58 Jan-64 Jan-70 Jan-76 Jan-82 Jan-88 Jan-94 Jan-00

Bond Valuation II

„ Relationship between yield to maturity and maturity

„ Information on expected future short term rates can be implied from yield curve

„ Three major theories are proposed to explain the observed yield curve

Overview of Term Structure

of Interest Rates

Yields

Years to Maturity Upward Sloping

Downward Sloping

Flat

Yield Curves

Yield Curve

U.S. Treasury Yield Curve (3/13/2002)

0 1 2 3 4 5 6 7

0.00 10.00 20.00 30.00 40.00

Years to Maturity

Yields (%)

Expected Interest Rates

in Coming Years (Table 15.1)

Expected One-Year Rates in Coming Years

Year Interest Rate

0 (today) 8%

1 10%

2 11%

3 11%

Pricing of Bonds

using Expected Rates

) 1 )...(

1 )(

1 (

1

2

1 n

n r r r

PV = + + +

PV_n= Present Value of $1 in n periods r₁= One-year rate for period 1 r₂= One-year rate for period 2 r_n= One-year rate for period n

Long-Term Rates and Bond Prices

using Expected Rates

Time to Maturity Price of Zero* Yield to Maturity

1 $925.93 8.00%

2 841.75 8.995

3 758.33 9.660

4 683.18 9.993

* $1,000 Par value zero coupon bond

Forward Rates

„ A forward rate is the inferred short-term rate for a future period that makes the expected total return of a long-term bond equal to that of rolling over short-term bonds.

„ Under the expectations hypothesis, the market’s expectation of future short-term interest rates may be inferred from forward rates.

Forward Rates

„ Example: Suppose that two-year maturity bonds offer yields to maturity of 5%, and three-year bonds have yields of 6%. What is the forward rate for the third year?

„ If we buy a three-year bond, in 3 years per dollar invested we will have ______________

Forward Rates

„ If we buy a two-year bond and re-invest all proceeds in a one-year bond in the third year at a rate of r₃ , in 3 years per dollar invested we will have ______________________

„ Since the two strategies must be equally attractive, ________________________

„ Therefore, _________________________

1 1) 1 (

) 1 ) ( 1

( ₋

+ −

= +

+ _n

n n n

n y

f y

f_n= one-year forward rate for period n y_n= yield for a security with a maturity of n

) 1 ( ) 1 ( ) 1

( 1 ¹ n

n n n

n y f

y = + +

+ ₋ ⁻

Forward Rates from

Observed Long-Term Rates

Example of Forward Rates

using Table 15.2 Numbers

4 yr = 9.993% 3yr = 9.660% f_n= ? (1.0993)⁴ = (1.0966)³(1+f_n)

(1.46373) / (1.31870) = (1+f_n) f_n= .10998 or 11%

„ Expectations

¾ Forward rates are expected future interest rates

„ Liquidity Preference

¾ Upward bias over expectations

„ Market Segmentation

¾ Preferred Habitat

Theories of Term Structure

The Bond Price-Return Relationship

„ Prices and returns on bonds are inversely related

$0

$50

$100

$150

$200

0% 2% 4% 6% 8% 10% 12% 14% 16%

Rate

Price

( )^t ( r)^t

F r r

P C

+ +

⎥⎦

⎢ ⎤

⎣

⎡

− +

= 1 1

1 1

Bond Duration and Interest Rate

Sensitivity

„ Duration measures the responsiveness of a bond’s price to interest rate changes.

Duration is computed as:

0

1(1 )

P r C t D Duration

n

t t

∑

= +

⋅

=

=

r D D Duration Modified

= +

= 1

*

Bond Duration and Interest Rate

Sensitivity

„ Duration gives the average time at which the bond’s cash flows are received.

„ Modified duration gives the approximate percentage change in a bond’s price for a 100 basis point change in interest rates:

r

D

r

D r

P

P

∆

×

−

=

+

× ∆

−

∆ =

*

1

Duration for Citicorp Bond

„ The Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

Duration for Citicorp Bond

„ The Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

„ Suppose market price equals $114.72 and the required rate is 6%. Then duration is given by:

Duration for Citicorp Bond

„ The Wall Street Journal reports the following details on a corporate bond:

¾ Issuer Citicorp

¾ Face Value $100

¾ Coupon 8% annual coupon

¾ Maturity 10 years

„ Suppose market price equals $114.72 and the required rate is 6%. Then duration is given by:

Duration t

years

t

=t

⋅

+ + ⋅

+ =

∑=₍ ^$8_{. ) (} ^$100_{. )}

$114. .

1 0 06 10 1 0 06

72 7 445

10 1

10

Duration of a Zero Coupon Bond

Similar to the Citicorp Bond

„ Consider the following zero-coupon bond which is similar to the Citicorp bond:

¾ Face Value $100

¾ Interest paid at maturity $105.45

¾ Maturity 10 years

„ A required rate of 6% gives P₀=$114.72 and a duration of:

Duration and Price Changes

„ Example: A bond selling for $1,184 has a yield to maturity of 6% and a duration of 7.37 years. If interest rates rise by .50%, what is the new price of the bond?

Bond Maturity and Interest Rate

Sensitivity

„ Long-term bonds have greater interest rate risk than short term bonds.

$0

$50

$100

$150

$200

$250

0% 2% 4% 6% 8% 10% 12% 14% 16%

Rate

Price

10 Year 20 Year 5 Year

Size of Coupon and Interest Rate

Sensitivity

„ Low coupon bonds have greater interest rate sensitivity than high coupon bonds.

$0

$50

$100

$150

$200

1% 3% 5% 7% 9% 11% 13% 15%

Rate

Price

Coupon Zero

Rules for Duration

„ Rule 1: The duration of a zero-coupon bond equals its time to maturity.

1 2 3 T

$1000

Rules for Duration

„ Rule 2: Holding maturity constant, a bond’s duration is higher when the coupon rate is lower.

C

1 2 3 T

C C C

$1000

Rules for Duration

„ Rule 3: Holding coupon rate constant, a bond’s duration generally increases with its time to maturity. Duration always increases with maturity for at par bond or premium bond.

C

1 2 3 T

C C C

$1000

Passive Bond Management

„ Two classes of passive bond management.

„ The first is an indexing strategy that attempts to replicate the performance of a given bond index.

„ The second class is immunization.

Portfolio Duration

„ The duration of a portfolio is the value- weighted average of the individual asset durations:

∑

= ⁿ

i i i

P wD

D

1

Active Bond Management

„ There are basically two ways to profit from active bond management:

nCorrectly forecast interest rate changes and trade accordingly.

oIdentify bonds which are mispriced, given their perceived level of default risk.

„ Empirical evidence suggests that prices in the bond market are extremely efficient.

Convexity

„ The curvature of the price yield curve is called the convexity of the bond. Convexity is the rate of change of the slope of the price- yield curve.

„ Higher convexity means that prices of bonds will increase more when yields decrease and fall less when yields increase.

( )

2

1 Convexity y y

D P

P=− ×∆ + × ×∆

∆