Chapter 7

Valuation and Characteristics of Bonds

Chapter 7 Topic Overview

Bond Characteristics Annual and Semi-Annual Bond Valuation Finding Returns on Bonds Reading Bond Quotes Bond Risk and Other Important Bond

Valuation Relationships

Bond Characteristics

Par Value = stated face value that is the amount the issuer must repay.

Coupon Interest Rate Coupon = Coupon Rate x Par Value Maturity Date = when the par value is

repaid. This makes a bond’s cash flows look like

this:

Characteristics of Bonds

Bonds pay fixed coupon (interest) payments at fixed intervals (usually every 6 months) and pay the par value at maturity.

00 1 1 2 . . .2 . . . nn

$I $I $I $I $I $I+$M$I $I $I $I $I $I+$M

Types of Bonds

Debentures: unsecured debt = bonds. Subordinated Debentures Mortgage Bonds Zero Coupon Bonds: no coupon

payments, just par value. Convertible Bonds: can be converted into

shares of stock.

Types of Bonds(cont.)

Indexed Bonds: coupon payments and/or par value indexed to inflation. TIPs: Indexed US Treasury coupon bond,

fixed coupon rate, par value indexed. I-Bonds: Indexed US Treasury zero coupon

bond. Junk bonds: speculative or below-

investment grade bonds; rated BB and below. High-yield bonds.

Types of Bonds(cont.)

Eurobonds - bonds denominated in one currency and sold in another country. (Borrowing overseas).

example - suppose Disney decides to sell $1,000 bonds in France. These are U.S. denominated bonds trading in a foreign country. Why do this? If borrowing rates are lower in France, To avoid SEC regulations.

The Bond Indenture

The bond contract between the firm and the trustee representing the bondholders.

Lists all of the bond’s features: coupon, par value, maturity, etc. Lists restrictive provisions which are

designed to protect bondholders. Describes repayment provisions.

Value

Book Value: value of an asset as shown on a firm’s balance sheet; historical cost.

Liquidation value: amount that could be received if an asset were sold individually.

Market value: observed value of an asset in the marketplace; determined by supply and demand.

Intrinsic value: economic or fair value of an asset; the present value of the asset’s expected future cash flows.

Security Valuation

In general, the intrinsic value of an asset = the present value of the stream of expected cash flows discounted at an appropriate required rate of return.

Can the intrinsic value of an asset differ from its market value?

Valuation

Ct = cash flow to be received at time t. k = the investor’s required rate of return. V = the intrinsic value of the asset.

V = V = t = 1t = 1

nn

$Ct

(1 + k)t

Bond Valuation

Discount the bond’s cash flows at the investor’s required rate of return. the coupon payment stream (an annuity). the par value payment (a single sum).

00 1 1 2 . . .2 . . . nn

$I $I $I $I $I $I+$M$I $I $I $I $I $I+$M

Bond Valuation

Vb = $It (PVIFA kb, n) + $M (PVIF kb, n)

$It $M

(1 + kb)t (1 + kb)nVVbb = + = +

nn

t = 1t = 1

Bond Valuation Example #1

Duff’s Beer has $1,000 par value bonds outstanding that make annual coupon payments. These bonds have an 8% annual coupon rate and 12 years left to maturity. Bonds with similar risk have a required return of 10%, and Moe Szyslak thinks this required return is reasonable.

What’s the most that Moe is willing to pay for a Duff’s Beer bond?

0 1 2 3 . . . 12

1000 80 80 80 . . . 80

P/Y = 1 12 = N

10 = I/Y 1,000 = FV80 = PMT

CPT PV = -$863.73

Note: If the coupon rate < discount rate, the bond will sell for less than the par value: a discount.

Let’s Play with Example #1

Homer Simpson is interested in buying a Duff Beer bond but demands an 8 percent required return.

What is the most Homer would pay for this bond?

0 1 2 3 . . . 12

1000 80 80 80 . . . 80

P/Y = 1 12 = N

8 = I/Y 1,000 = FV80 = PMT

CPT PV = -$1,000

Note: If the coupon rate = discount rate, the bond will sell for its par value.

Let’s Play with Example #1 some more. Barney (belch!) Barstool is interested

in buying a Duff Beer bond and demands on a 6 percent required return.

What is the most Barney (belch!) would pay for this bond?

0 1 2 3 . . . 12

1000 80 80 80 . . . 80

P/Y = 1 12 = N

6 = I/Y 1,000 = FV80 = PMT

CPT PV = -$1,167.68

Note: If the coupon rate > discount rate, the bond will sell for more than the par value: a premium.

Bonds with Semiannual Coupons

Double the number of years, and divide required return and annual coupon by 2.

VVBB = = II/2/2(PVIFA(PVIFAkkbb/2,2N/2,2N) + M(PVIF) + M(PVIFkkbb/2,2N/2,2N))

Semiannual Example

A $1000 par value bond with an annual coupon rate of 9% pays coupons semiannually with 15 years left to maturity. What is the most you would be willing to pay for this bond if your required return is 8% APR?

Semiannual coupon = 9%/2($1000) = $45Semiannual coupon = 9%/2($1000) = $45 15x2 = 30 remaining coupons15x2 = 30 remaining coupons

0 1 2 3 . . . 30

1000 45 45 45 . . . 45

P/Y = 1 15x2 =30 = N

8/2 = 4 = I/Y 1,000 = FV

90/2 = 45 = PMTCPT PV = -$1,086.46

Finding a bond’s rate of return?

Expected Return In the marketplace, we know a bond’s current

price(PV), but not its return. Yield to Maturity (YTM) = the rate of return the

bond would earn if purchased at today’s price and held until maturity.

Annual Actual Return Current Yield + Capital Gains Yield I/P0 + (P1 – P0)/P0 = (P1 – P0 + I)/P0

Yield To Maturity

The expected rate of return on a bond. The rate of return investors earn on a

bond if they hold it to maturity.

$It $M

(1 + kb)t (1 + kb)nPP00 = + = +

nn

t = 1t = 1

Yield to Maturity Example

$1000 face value bond with a 10% coupon rate paid annually with 20 years left to maturity sells for $1091.29.

What is this bond’s yield to maturity?

0 1 2 3 . . . 20

1000-1091.29 100 100 100 . . . 100

P/Y = 1 -1091.29 = PV

20 = N1,000 = FV100 = PMT

CPT I/Y = 9% = YTM

Let’s try this together.

Imagine a year later, the YTM for the bond on the previous slide fell to 8%.

What is the bond’s expected price? What is the holding period return, if we sell

the bond at this time assuming we bought the bond a year earlier?

PMT =100, FV = 1000

Reading Corporate Bond Quotes

Cur Net

Bonds Yld Vol. Close Chg.

IBM 6 ½ 28 6.6 14 98 1/4 -2 1/8

Most info is expressed as % of par value. Par value = 100.

For IBM, 6.5% annual coupon rate, matures in year 2028, Price is 98.25% of par value.

YTM Estimate for IBM Bond

Assuming $1000 Par (or Face) Value and semi-annual coupons

Price = 98.25% (1000) = 982.50, INT/2=1000(6.5%)/2= 32.50, FV = 1000

Assuming N = 26 (2028-2002): YTM?

982.50 =32.50(PVIFAYTM/2,2N)+1000(PVIFYTM/2,2N) Calculator Solution: -982.50 = PV,1000 = FV, 32.50

= PMT, 2N = 2(26) = 52 = N, CPT I/Y I/Y=YTM/2=3.32% YTM(APR) = 2(3.32%)= 6.64%

The Financial Pages: Treasury Bonds

Maturity Ask

Rate Mo/Yr Bid Asked Chg Yld

6 Feb 26 104:25 104:26 -15 5.63

What is the yield to maturity for this Treasury

bond? (assume (2026-2002) 24x2 = 48 half

years)

P/Y = 1, N = 48, FV = 1000,

PMT = 1000(6%/2) = 30,

PV = - 1,048.125 (104.8125% of par) Solve: I/Y = ytm/2 = 2.816%, YTM = 5.63%

Bond Valuation: What have we learned? 5 Important Relationships

Our Example 1: Duff’s Beer bonds 12-year bond

kb=6%, V = $1,167.68

kb=8%, V = $1,000

kb=10%, V = $863.73

These values illustrate the First & Second Important Relationships

First Relationship: Bond Prices and Interest Rates have an inverse relationship!

Bond Values for 8% Annual Coupon Bonds

0200400600800

100012001400

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

Required Return

($)M

ark

et V

alu

e

12-yr Bond

Second Important Relationship

From example 1: The coupon rate was 8%kb=6%, V = $1,167.68

kb=8%, V = $1,000

kb=10%, V = $863.73 When required rate = coupon rate

Bond Value = Par Value (M) When required rate > coupon rate

Bond Value < Par Value (M) When required rate < coupon rate

Bond Value > Par Value (M)

Bond Value Changes Over Time

Returning to the original example #1, where k = 10%, N = 12, INT(PMT) = $80, M(FV) = $1000, & V = $863.73.

What is bond value one year later when N = 11 and k is still = 10%?

VB = $80(PVIFA10%,11) + $1000(PVIF10%,11) = $870.10

What is the bond’s return over this year? (Proof of YTM = Expected Ret.)

Total Rate of Return = Current Yield + Capital Gains Yield (C.G.Y)

Beg. V = 863.73, End V = 870.10 Current Yield = Annual Coupon (INT) divided by

Beginning Bond Value Current Yld = $80/863.73 = 9.26% C.G.Y.=(870.10-863.73)/863.73= 0.74% Total Return = 9.26% + 0.74% = 10%

Third Relationship: Market Value approaches par value as maturity date approaches.

Bond Values Over Tim e

$870.10$ 8 6 3 .7 3

$ 1 ,1 6 7 .6 8

$800.00

$900.00

$1,000.00

$1,100.00

$1,200.00

12 10 8 6 4 2 0

Tim e to Maturity

Bo

nd

Va

lue

k = 1 0 %k = 8 %k = 6 %

Fourth Relationship: Interest Rate Risk

Measures Bond Price Sensitivity to changes in interest rates.

Long-term bonds have more interest rate risk than short-term bonds.

Interest Rate Risk Example

Recall from our earlier example (#1), the 12-year, 8% annual coupon bond has the following values at kd = 6%, 8%, & 10%. Let’s compare with a 2-yr, 8% annual coupon bond.

12-year bond 2-year bondkb=6%, V = $1,167.68 V = $1,036.67

kb=8%, V = $1,000 V = $1,000

kb=10%, V = $863.73 V = $965.29

Bond Price Sensitivity Graph

Bond Values for 8% Annual Coupon Bonds

0250500750

1000125015001750

0% 5% 10% 15%

2-yr Bond12-yr Bond30-yr Bond

Other Bond Risks

Reinvestment Rate Risk = opposite of interest rate risk, greater for short-term bonds, risk that income from bonds will fall.

Default Risk = measured by bond ratings = ability of issuer to fulfill debt obligations Aaa, AAA, best rating, lowest default risk

Fifth Relationship

In addition to length of time to maturity, the pattern ( and size) of cash flows affects a bond’s price sensitivity to changes in interest rates.

Duration measures and illustrates this relationship.

Duration

Weighted average time to maturity. Higher (longer) duration means greater

bond price sensitivity to changes in interest rates.

Duration Formula

PkCn

tt

t

b

t

Duration0

1 )1(=

t = year the cash flow is to be received, n = the number of years to maturity, Ct = the cash flow to be received at year t, kb = the bondholder’s required return, P0 = the bond’s present value (or today’s price).

Duration Example

Krusty Burger and Burns Power bonds both have 3 years to maturity, $1,000 par value, and a required return of 8 percent.

However, Krusty Burger makes annual coupon payments of 8%, while Burns Power is a zero coupon bond.

What is the duration of each bond?

Suggested Duration Calculation Steps

First, calculate today’s value of the bond. Second, find the PV today of each time weighted

bond CF (CF x time period the CF occurs). Third, add up all the time weighted PVs

Note: The CF and NPV calculator functions can be used to do steps 2 and 3.

Fourth, divide sum of time weighted PVs by today’s bond value = duration.

Krusty Burger Duration

Since Krusty’s required return and coupon rate are equal, today’s value = $1,000.

80 = PMT, 1000 = FV, 8 = I/Y, 3 = N, CPT PV = $1000

t C t x C PV(tC)1 80 80 C01 742 80 160 C02 1373 1080 3240 C03 2572NPV: I = 8, CPT NPV = 2783Krusty Burger Duration = 2783/1000 = 2.783

Burns Power Duration

Today’s Burns Power Bond Value: 0 = PMT, 1000 = FV, 8 = I/Y, 3 = N, CPT PV = $793.83

t C t x C PV(tC)1 0 0 C01 02 0 0 C02 03 1000 3000 C03 2381.50NPV: I = 8, CPT NPV = 2381.50Burns Power Duration = 2381.50/793.83 = 3.00NOTE: Duration for zero coupon bond = time to

maturity.

Duration Example Conclusion

Krusty Burger Duration = 2.783Burns Power Duration = 3.000 Burns Power bonds are more sensitive to

changes in interest rates. This is good if interest rates go down, but bad if

interest rates go up! From this example, you can see for bonds with

the same time to maturity, lower coupon rate bonds have more interest rate risk.