8 - 1 copyright © 2002 by harcourt, inc.all rights reserved. chapter 8 bonds and their valuation...

8 - 1

Copyright © 2002 by Harcourt, Inc. All rights reserved.

CHAPTER 8Bonds and Their Valuation

Key features of bondsBond valuationMeasuring yieldAssessing risk

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Key Features of a Bond

1. Par value: Face amount; paid at maturity. Assume $1,000.

2. Coupon interest rate: Stated interest rate. Multiply by par to get $ of interest. Generally fixed.

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3. Maturity: Years until bondmust be repaid. Declines.

4. Issue date: Date when bondwas issued.

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How does adding a “call provision” affect a bond?

Issuer can refund if rates decline. That helps the issuer but hurts the investor.

Therefore, borrowers are willing to pay more, and lenders require more, on callable bonds.

Most bonds have a deferred call and a declining call premium.

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Provision to pay off a loan over its life rather than all at maturity.

Similar to amortization on a term loan.

Reduces risk to investor, shortens average maturity.

But not good for investors if rates decline after issuance.

What is a sinking fund?

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1. Call x% at par per year for sinking fund purposes.

2. Buy bonds on open market.

Company would call if kd is below the coupon rate and bond sells at a premium. Use open market purchase if kd is above coupon rate and bond sells at a discount.

Sinking funds are generally handled in 2 ways

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Financial Asset Values

PV =

CF

1+k... +

CF

1+k1 n

12

21

CF

kn .

0 1 2 nk

CF1 CFnCF2Value

...

+ ++

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The discount rate (ki) is the opportunity cost of capital, i.e., the rate that could be earned on alternative investments of equal risk.

ki = k* + IP + LP + MRP + DRP.

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V

k kB

d d

$100 $1,

1

000

11 10 10 . . . +

$100

1+ kd

= $90.91 + . . . + $38.55 + $385.54= $1,000.

++++

100 100

0 1 2 1010%

100 + 1,000V = ?

...

What’s the value of a 10-year, 10% coupon bond if kd = 10%?

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The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10:

1 PVA = PMT * (1- (1+k)n ) kPV = FV(1/(1+k)n)

1 V bond = PMT * (1- (1+kd)n + M(1/(1+kd)n) kd

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1 V bond = PMT * (1- (1+kd)n + M(1/(1+kd)n) kd

1 V bond = 100 * (1- (1+.1)10 +1000(1/(1+.1)10) .1PV annuity $614.46PV maturity value 385.54V bond = $1000.00 When kd = coupon interest rate, bond trades @ par value.

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10 10 100 1000N I/YR PV PMT FV

-1,000

INPUTS

OUTPUT

With annual compounding, P/Y = 1, and End mode. Use all TVM keys in the calculation - you have an annuity and a future cash flow outside the annuity.

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Most bonds pay interest semiannually,so m = 2. You must divide k by 2, multiplyn by 2. The payment will be $50 every 6 months.If the market rate of interest increased to 12%: 1 V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2) kd /2 1 V bond = 50 * (1- (1+.12/2)10*2 +1000(1/(1+.12/2)10*2) .12/2

V bond =50(11.4699)+1000(.3118) = 885.30This bond trades at a discount to par.

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20 12 50 1000N I/YR PV PMT FV

-885.30

When kd rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount.

What would happen if expected inflation rose by 2%, causing k =12%?.

INPUTS

OUTPUT

P/Y=2, End mode

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If the market rate of interest decreased to 8%: 1 V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2) kd /2 1 V bond = 50 * (1- (1+.08/2)10*2 +1000(1/(1+.08/2)10*2) .08/2

V bond =50(13.5903)+1000(.4564) = 1135.92This bond trades at a premium to par.

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What would happen if inflation fell, and kd declined to 8%?

20 8 50 1000N I/YR PV PMT FV

-1,135.90

Price rises above par, and bond sells at a premium, if coupon > kd.

INPUTS

OUTPUT

P/Y=2, End mode

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The bond was issued 20 years ago and now has 10 years to maturity. What

would happen to its value over time if the required rate of return remained at

10%, or at 13%, or at 7%?

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M

Bond Value ($)

Years remaining to Maturity

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

kd = 7%.

kd = 13%.

kd = 10%.

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At maturity, the value of any bond must equal its par value.

The value of a premium bond would decrease to $1,000.

The value of a discount bond would increase to $1,000.

A par bond stays at $1,000 if kd remains constant.

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What is the “yield to maturity”?

YTM is the rate of return earned on a bond held to maturity. Also called the “promised yield.”

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What’s the YTM on a 10-year, 9% semiannual coupon, $1,000 par value

bond that sells for $885.45?

0 1 9 20

45 45 45

kd=?

1,000PV1 . . .PV20

PVM

885.45 Find kd that “works”!

...

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If you don’t have a financial calculator, need to usetrial and error. Know the price of the bond hasdropped to $885.45, so yield must have gone up. Try 11%, was a 9% coupon bond. 1 V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2) kd /2 1 885.45 = 45 * (1- (1+.11/2)10*2 +1000(1/(1+.11/2)10*2) .11/2

885.45 =45(11.9504)+1000(.3427) = 880.47Vbond needs to be higher, so yield needs to be lower, but not much lower. Without a financial calculator,an answer of 9% < k < 11% would be sufficient.

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20 -885.45 45 1000N I/YR PV PMT FV 10.91

Find kd

INPUTS

OUTPUT

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If coupon rate < kd, discount.

If coupon rate = kd, par bond.

If coupon rate > kd, premium.

If kd rises, price falls.

Price = par at maturity.

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If the price of the bond has increased to $1142.12, yield (market rate of interest) must have gone down. Try 7%, was a 9% coupon bond. 1 V bond = PMT * (1- (1+kd/2)n*2+ M(1/(1+kd /2)n *2) kd /2 1 1142.12 = 45 * (1- (1+.07/2)10*2 +1000(1/(1+..07/2)10*2) .07/2

1142.12 =45(14.2124)+1000(.5026) = 1142.16Vbond is approximately equal, so the YTM is 7% .

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Find YTM if price were $1,142.12.

20 -1142.12 45 1000N I/YR PV PMT FV

7.0

Sells at a premium. Because coupon = 9% > kd = 7.0%, bond’s value > par.

INPUTS

OUTPUT

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Definitions

Current yield = .

Capital gains yield = .

= YTM = + .

Annual coupon pmtCurrent price

Change in priceBeginning price

Exp totalreturn

Exp Curr yld

Exp capgains yld

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Current yield =

= 0.1015 = 10.15%.

Find current yield and capital gains yield for a 9%, 10-year bond when the bond sells for $887 and YTM = 10.91%.

$90 $887

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YTM = Current yield + Capital gains yield.

Cap gains yield = YTM – Current yield = 10.91% – 10.15% = 0.76%.

Could also find value in Years 1 and 2,get difference, and divide by value inYear 1. Same answer.

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What’s interest rate (or price) risk? Does a 1-year or 10-year 10% bond

have more risk?

kd 1-year Change 10-year Change

5% $1,048 $1,386

10% 1,000+4.8%

-4.4%1,000

+38.6%

-25.1%15% 956 749

Interest rate risk: Rising kd causes bond’s price to fall.

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0

500

1,000

1,500

0% 5% 10% 15%

1-year

10-year

kd

Value

.

..

. ..

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What is reinvestment rate risk?

The risk that CFs will have to be reinvested in the future at lower rates, reducing income.

Illustration: Suppose you just won $500,000 playing the lottery. You’ll invest the money and live off the interest. You buy a 1-year bond with a YTM of 10%.

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Year 1 income = $50,000. At year-end get back $500,000 to reinvest.

If rates fall to 3%, income will drop from $50,000 to $15,000. Had you bought 30-year bonds, income would have remained constant.

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Long-term bonds: High interest rate risk, low reinvestment rate risk.

Short-term bonds: Low interest rate risk, high reinvestment rate risk.

Nothing is riskless!

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Do all bonds of the same maturityhave the same price and reinvestment

rate risk?

No, low coupon bonds have lessreinvestment rate risk but more price risk than high coupon bonds.

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A 10-year, 10% semiannual coupon,$1,000 par value bondis selling for $1,135.90 with an 8% yield to maturity.It can be called after 4 years at $1,050.

What’s the bond’s nominal yield to call (YTC)? The price ofthe bond is now $1135.90, need to find YTC. Try 6%. 1 V bond = PMT * (1- (1+kd/2)n*2+CP(1/(1+kd /2)n *2) kd /2 1 1135.90 = 50 * (1- (1+.06/2)4*2 +1050(1/(1+.06/2)4*2) .06/21135.90 =50(7.01969)+1050(.7894) = 1179.85Need lower V bond = so YTC must be greater than 6%.

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With a financial calculator, you are justSolving for the interest rate, using the Call premium as the future value and the Time to possible call as n.

8 -1135.9 50 1050N I/YR PV PMT FV 3.568 x 2 = 7.137%

INPUTS

OUTPUT

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kNom = 7.137% is the rate brokers would quote. Could also calculate EFF% to call:

EFF% = (1.03568)2 – 1 = 7.26%.

This rate could be compared to monthly mortgages, etc.

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If you bought bonds, would you be more likely to earn YTM or YTC?

Coupon rate = 10% vs. YTC = kd = 7.137%. Could raise money by selling new bonds which pay 7.137%.

Could thus replace bonds that pay $100/year with bonds that pay only $71.37/year.

Investors should expect a call, hence YTC = 7.1%, not YTM = 8%.

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In general, if a bond sells at a premium, then (1) coupon > kd, so (2) a call is likely.

So, expect to earn:YTC on premium bonds.YTM on par & discount bonds.

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Disney recently issued 100-year bonds with a YTM of 7.5%--this represents the promised return. The expected return was less than 7.5% when the bonds were issued.

If issuer defaults, investors receive less than the promised return. Therefore, the expected return on corporate and municipal bonds is less than the promised return.

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Bond Ratings Provide One Measure of Default Risk

Investment Grade Junk BondsMoody’s Aaa Aa A Baa Ba B Caa C

S&P AAA AA A BBB BB B CCC D

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What factors affect default risk and bond ratings?

Financial performanceDebt ratioTIE ratioCurrent ratio

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Provisions in the bond contractSecured vs. unsecured debtSenior vs. subordinated debtGuarantee provisionsSinking fund provisionsDebt maturity

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Other factorsEarnings stabilityRegulatory environmentPotential product liabilityAccounting policies

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Top Ten Largest U.S. Corporate Bond Financings, as of July 1999

Issuer

Ford Motor Co.

AT&T

RJR Holdings

WorldCom

Sprint

Date

July 1999

Mar 1999

May 1989

Aug 1998

Nov 1998

Amount

$8.6 billion

$8.0 billion

$6.1 billion

$6.1 billion

$5.0 billion

8 - 1 copyright © 2002 by harcourt, inc.all rights reserved. chapter 8 bonds and their valuation...

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