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CHAPTER 6 Bond Valuation Models

Key features of BondsBond valuationMeasuring yieldAssessing risk

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Key Features of a Bond

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

Coupon interest rate: stated interest rate. Multiply by par value to get dollar interest payment. Generally fixed.

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Maturity: years until bond must be repaid. Declines over time.

Issue date: date when bond was issued.

Call provisionCall Protection periodCall Premium

Default Risk Special Features

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MINICASE 6

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SIMPLE?

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Value = + . . . .

ABSOLUTELY FUNDAMENTAL.

How can we value assets on the basis of expected future cash flows?

CF1

(1 + r)1

CF2

(1 + r)2

CFn

(1 + r)n

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The discount rate r is the opportunity cost of capital and depends on: riskiness of cash flows. general level of interest rates.

• Inflation• Supply and Demand of Money and Credit• Production opportunities• Time profile of consumption

How is the discount rate determined?

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Again, the discount rate r is the opportunity cost of capital. i.e. the rate of return that could be earned on alternative investments of similar risk.

Remember:

ri = r* + IP + LP + MRP + DRP for debt securities.

How is the discount rate determined?

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The coupon payments (an annuity).A lump sum (the maturity, or par, value to

be received in the future).

Value =Present Value of the interest annuity and the Present value of the maturity value.

The cash flows of a bond consist of:

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0 10

1001,000

1 2

100 100

Find the value of a 10-year annual coupon bond, when rd = 10%.

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Enter:

Solve for PV = $1,000, or 100% of Par.Computer:

Two Ways to Solve

Using tables:

Value = INT(PVIFA10%,10)+ M(PVIF10%,10).

Calculator:

N I/YR PV PMT FV10 10 100 1000

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10 10 100 1000N I/YR PV PMT FV

-1,000

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

$ 614.46 385.54

$1,000.00

PV annuity PV maturity value Value of bond

===

INPUTS

OUTPUT

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0 20

1001,000

1 2

100 100

Find the value of a similar 20-year annual coupon bond, when rd = 10%.

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Rule: When the required rate of return (rd) equals the coupon rate, the bond value (or price) equals the:

?

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PAR VALUE

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N I/YR PV PMT FV

What would the value of the bonds be if inflation rose causing kd= 13%?

Solution: ?

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10-year, 10% coupon bond, rd =13%

N PV FV

10 13 100 1000

Solution: -837.21 or 83.72% of par.

I/YR PMT

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N I/YR PV PMT FV

What if it were a 20 year, 10% annual coupon bond, with rd = 13%?

20-year, 10% coupon bond

20 13 100 1000

Solution: -789.25, or 78.925% of Par

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When rd rises above the coupon rate, bond values fall below par.

They sell at a discount: A Discount Bond

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N I/YR PV PMT FV

What would the value of the bonds be if rd = 7%?

10-year bond

10 7 100 1000

Solution: -1210.71, (Price 121.07)

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N I/YR PV PMT FV

What would the value of the bonds be if rd = 7%?

20-year bond

20 7 100 1000

Solution: -1317.82, (Price 131.78)

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When rd falls below the coupon rate,bond values rise above par.They sell at a premium: A PREMIUM BOND.

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What would happen over time to the value of a 10 % coupon, 30 bond if

(1)interest rates stayed at 10%?(2)If interest rates immediately rose to

13% and stayed there?(3)If interest rates immediately fell to

7.5% and stayed there?

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If interest rates stay at 10%

Value of 30 year bond?$1000.Value of 10 year bond?$1000Value of 1 year bond?$1000

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If interest rates rise to 13% and stay there?

Value of 30 year bond?$775.13Value of 10 year bond?$837.21Value of 1 year bond?$973.45

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If interest rates fell to 7% and stayed there?

Value of 30 year bond?$1372.27Value of 10 year bond?$1210.71Value of 1 year bond?$1028.04

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Value of 30 year, 10% coupon bond over time:

13721211

1000

837 775

M

rd = 10%

rd = 7%

rd = 13%

30 20 10 0Years to Maturity

NOTE: Not symmetric

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Value of 30 year, 10% coupon bond over time:

13721211

1000

837 775

M

rd = 10%

rd = 7%

rd = 13%

30 20 10 0Years to Maturity

BULLET-VIC

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What if 100 year bond?

$769.23 if k = 13%$1000 if k = 10%$1428.07 if k = 7%

Cf9-23

Cf9-24

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Summary

If rd remains constant:At maturity, the value of any bond

must equal its par value.Over time, the value of a premium

bond will decrease to its par value.Over time, the value of a discount

bond will increase to its par value.A par value bond will stay at its par

value.

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Value of 30 year, 10% coupon bond over time:

13721211

1000

837 775

M

rd = 10%

rd = 7%

rd = 13%

30 20 10 0Years to Maturity

ACTUAL BOND PRICE PROFILE

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What is “yield to maturity”?

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

That discount rate which equates PV (Bond’s CF) to its price.

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What is the YTM on a 10-year, 9% annual coupon, $1,000 par value bond

that sells for $887?

90 90 90

0 1 9 10rd=?

1,000PV1

. . .PV10

PVM

887 Find rd that “works”!

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First, what does the fact that this is a discount bond tell you about the

relationship between rd and the bond’s coupon rate?

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INPUTS

OUTPUT

Find rd

VB = + . . . + +INT

(1 + rd)1

INT(1 + rd)N

M(1 + rd)N

887 = + . . . + + .90

(1 + rd)1

90(1 + rd)10

1,000(1 + rd)10

10 -887 90 1000N I/YR PV PMT FV 10.91

Computer: (e-1)

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Find YTM if price were $1,134.20.

Sells at a premium. Because coupon = 9% > rd = 7.08%, bond’s value > par.

Computer:

INPUTS

OUTPUT

10 -1134.2 90 1000N I/YR PV PMT FV

7.08

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If coupon rate > rd, premium.

If coupon rate < rd, discount.

If coupon rate = rd, par bond.

If rd rises (falls), price falls

(rises).

At maturity, price = par.

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Definitions

Current yield = .

Cap gains yld = .

= YTM = + .

Annual coupon pmtCurrent price

Change in price Beg of period price

Exp totalreturn

ExpCurr yld

Exp capgains yld

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Find current yield and capital gains yield for a 9%, 10-year bond when the

bond sells for $887 (and YTM = 10.91%.)

Current yield =

= 0.1015 = 10.15%.

$90 $887

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Remember: YTM = Current yield + Capital gains yield. Therefore, Shortcut Answer:Cap gains yield = YTM - Current yld.

= 10.91% - 10.15%= 0.77%.

Could also find values in Years 0 and 1,get difference, and divide by value inYear 0. Same answer.

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Proof:

Value of a 9 year, 9% coupon bond if YTM = 10.91?

$893.87(P1 - P0)/ P0 = ($893.87 - $887)/$887

= .0077 = .77%

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Current yield = = 7.94%.

Capital gains yield = 7.08% - 7.94% = -0.85%.

Current yield = = 7.94%.

Capital gains yield = 7.08% - 7.94% = -0.85%.

$90 $1,134.20 $90 $1,134.20

10-year, 9% coupon bond withprice = $1,134.20 (YTM = 7.08%)

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9 year, 9% coupon bond, YTM = 7.08%;

Price = $1124.52

(P1 - P0)/P0 = 1124.52 - 1134.20

1134.20

= -0.85%

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What is interest rate (price) risk? Does a 1-yr or 10-yr 10% bond have

more price risk?

Price risk: Rising rd causes bond’s price to fall.

rd 1-year Change 10-year Change

5% $1,048 $1,386

10% 1,000 4.8% 1,000 38.6%

15% 956 4.4% 749 25.1%

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1-year

10-year

kd

Value

0

500

1000

1500

0% 5% 10% 15%

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DURATION a measure of price risk

Duration = Sum( witi)

where wi =PV ( CFi) / sum (PV ( CFi))

= PV ( CFi) / Price

ti = time to the ith cash flow

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DURATION

e.g. [PV(CF1)/P] x 1 +

[PV(CF2)/P] x 2 + [PV(CF3)/P] x 3 + ...

+[PV(CFn)/P] x n

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What is the duration of a 1 year 10% coupon bond, rd = .10

0 1

1100

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What is the duration of a 2 year 10% coupon bond, rd = .10

Cash flows: 0 1 2 100 1100

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What is the duration of a 2 year 10% coupon bond, rd = .10

Duration = [((100/(1.1))/1000)*1] + [((1100/(1.1)2)/1000)*2] =

Greater Interest rate risk than a 1 year bond.

1.91

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DURATION

WHAT IS DURATION OF A 2 YEAR ZERO COUPON BOND?

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What is the duration of a 2 year zero coupon bond?

0 2

[(1000/(1 + r)2)/(1000/(1 + r)2)] x 2 = 2

1000

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WHO CARES?

Who Cares?

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WHAT IS REINVESTMENT RATE RISK?

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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 at par having a 10% interest rate.

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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 price risk, low reinvestment rate risk.

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

TANSTAAFL

Nothing is riskless!

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CASE HANDOUT!

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SEMI-ANNUAL BONDS

Return to consideration of 10% coupon bonds (6-9).

Virtually all coupon bonds issued in the U.S. have semi-annual coupons.

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Semiannual Bonds

1.Multiply years by 2 to get periods = 2n.2.Divide nominal rate by 2 to get periodic rate = rd/2.3.Divide annual INT by 2 to get PMT = INT/2.

INPUTS

OUTPUT

2n rd/2 OK INT/2 OK

N I/YR PV PMT FV

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Find the value of 10-year, 10% coupon, semiannual bond if rd = 10%; if rd =

13%; if rd = 7%.

INPUTS

OUTPUT

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For 10%; 2(10) 10/2 100/220 5.0 50 1000N I/YR PV PMT FV

Find the value of 10-year, 10% coupon, semiannual bond if rd = 10%;

INPUTS

OUTPUT

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For 10%; 2(10) 10/2 100/220 5.0 50 1000N I/YR PV PMT FV

-1000

Find the value of 10-year, 10% coupon, semiannual bond if rd = 10%;

INPUTS

OUTPUT

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For 13%; 2(10) 13/2 100/220 6.5 50 1000N I/YR PV PMT FV

-834.72

Find the value of 10-year, 10% coupon, semiannual bond if rd = 13%;

INPUTS

OUTPUT

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For 7%; 2(10) 7/2 100/220 3.5 50 1000N I/YR PV PMT FV

-1213.19

Find the value of 10-year, 10% coupon, semiannual bond if rd = 7%.

INPUTS

OUTPUT

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Compare these results with the similar results for Annual Bonds.

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10% Annual and Semiannual Bonds

ANNUAL kd 1 yr 10 yr 30 yr

7% 1211 10% 1000 13% 837

Semiannual kd 1 yr 10 yr 30 yr

7% 1213 10% 1000 13% 835

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Spreadsheet Functions for Bond Valuation

See Ch 06 Mini Case.xls for details.PRICEYIELD

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You could buy, for $1,000, either a 10%, 10-year, annual payment bond or an

equally risky 10%, 10-year semiannual bond. Which would you prefer?

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You could buy, for $1,000, either a 10%, 10-year, annual payment bond or an

equally risky 10%, 10-year semiannual pmt. bond. Which would you prefer?

The semiannual bond’s EFF% is:

10.25% > 10% EFF% on annual bond, so buy semiannual bond.

EFF% = (1 + ) - 1

= (1 + ) - 1 =

10.25%.

miNom

m2.10

2

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If $1,000 is the proper price for the semiannual bond, what is the proper price for the annual payment bond?

Semiannual bond has kNom = 10%, with EFF% = 10.25%. Should earn same EFF% on annual payment bond, so:

10 10.25 100 1000 N I/YR PV PMT FV

-984.80

INPUTS

OUTPUT

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At a price of $984.80, the annual and semi-annual bonds would be in equilibrium, because investors would earn EFF% = 10.25% on either bond.

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00 11 22 33 . . .. . . 88

100 100 100 . . . 100100 100 100 . . . 100

What is the cash flow stream of aperpetual bond with an annual

coupon of $100?

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A perpetuity is a cash flow stream of equal payments at equal intervals into infinity.

Vperpetuity = .PMT

r

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V10% = = $1000.

V13% = = $769.23.

V7% = = $1428.57.

V10% = = $1000.

V13% = = $769.23.

V7% = = $1428.57.

$1000.10$1000.10

$1000.13$1000.13

$1000.07$1000.07

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YIELD TO CALL

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A 10-year, 10% semiannual coupon,$1,000 par value bond is selling for$1,135.90 with an 8% yield to maturity.It can be called after 5 years at $1,050.

What’s the bond’s nominal yield tocall (YTC)?

INPUTS

OUTPUT

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0 1 2 3 4 5 0 50 50 50 50 50 50 50 50 50 (50+ 1050)

12 3 4 5

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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.

Later

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A 10-year, 10% semiannual coupon,$1,000 par value bond is selling for$1,135.90 with an 8% yield to maturity.It can be called after 5 years at $1,050.

What’s the bond’s nominal yield tocall (YTC)?

10 -1135.9 50 1050N I/YR PV PMT FV 3.765 x 2 = 7.53%

INPUTS

OUTPUT

Question j

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kNom = 7.53% is the rate brokers would quote. Could also calculate EFF% to call:

EFF% = (1.03765)2 - 1 = 7.672%.

This rate could be compared to monthly mortgages, etc.

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Bond brokers quote (annual) nominal rates. This is o.k. if you are comparing

semi-annual rates, as you would be with bonds.

But, if you are comparing returns of instruments having different payment patterns, e.g. MBS’s, you should use

EAR.

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

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

Coupon rate = 10% vs. YTC = rd = 7.53%. Could raise money by selling new bonds which pay 7.53%.

Could thus replace bonds which pay $100/year with bonds that pay only $75.30/year.

Investors should expect a call, hence YTC = 7.5%, not YTM = 8%.

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In general, if a bond sells at a premium, then (1) coupon > rd, 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. Why?

Bond may be called

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 Measureof Default Risk

Investment Grade Junk Bonds

Moody’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, FCC ratiosCurrent ratios

(More…)

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

(More…)

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Other factorsEarnings stabilityRegulatory environmentPotential product liabilityAccounting policies

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What’s a sinking fund?

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.

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

2. Buy bonds on open market.

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

Sinking funds are generally handledin 2 ways

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Bankruptcy

Two main chapters of Federal Bankruptcy Act:Chapter 11, ReorganizationChapter 7, Liquidation

Typically, company wants Chapter 11, creditors may prefer Chapter 7.

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If company can’t meet its obligations, it files under Chapter 11. That stops creditors from foreclosing, taking assets, and shutting down the business.

Company has 120 days to file a reorganization plan.Court appoints a “trustee” to

supervise reorganization. Management usually stays in

control.

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Company must demonstrate in its reorganization plan that it is “worth more alive than dead.”

Otherwise, judge will order liquidation under Chapter 7.

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If the company is liquidated, here’s the payment priority:1. Secured creditors from sales of

secured assets.2. Trustee’s costs3. Wages, subject to limits4. Taxes5. Unfunded pension liabilities6. Unsecured creditors7. Preferred stock8. Common stock

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In a liquidation, unsecured creditors generally get zero. This makes them more willing to participate in reorganization even though their claims are greatly scaled back.

Various groups of creditors vote on the reorganization plan. If both the majority of the creditors and the judge approve, company “emerges” from bankruptcy with lower debts, reduced interest charges, and a chance for success.

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Enough for now on bonds?

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Week 2s Capital Gains Yield

Change in price Price

I.e. P1 – P0

P0

Next Period: (P2 – P1)/ P1