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### 9 year, 9% coupon bond, YTM = 7.08%;Price = $1124.52

CHAPTER 6 Bond Valuation Models

- Key features of Bonds
- Bond valuation
- Measuring yield
- Assessing risk

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.

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

- Issue date: date when bond was issued.
- Call provision
- Call Protection period
- Call Premium
- Default Risk
- Special Features

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

Value = + . . . .

ABSOLUTELY FUNDAMENTAL.

How is the discount rate determined?

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?

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.

The cash flows of a bond consist of:

- 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.

Two Ways to Solve

Using tables:

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

Calculator:

Enter:

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

Computer:

10 10 100 1000

N

I/YR

PV

PMT

FV

PV annuity

PV maturity value

Value of bond

$ 614.46

385.54

$1,000.00

=

=

=

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

INPUTS

10 10 100 1000

N I/YR PV PMT FV

-1,000

OUTPUT

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

?

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

20-year, 10% coupon bond

20 13 100 1000

N

I/YR

PV

PMT

FV

Solution: -789.25, or 78.925% of Par

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

- They sell at adiscount: A Discount Bond

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

10-year bond

10 7 100 1000

N

I/YR

PV

PMT

FV

Solution: -1210.71, (Price 121.07)

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

20-year bond

20 7 100 1000

N

I/YR

PV

PMT

FV

Solution: -1317.82, (Price 131.78)

When rd falls below the coupon rate,

bond values rise above par.

They sell at apremium: A PREMIUM

BOND.

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?

If interest rates stay at 10%

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

If interest rates rise to 13% and stay there?

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

If interest rates fell to 7% and stayed there?

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

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

rd = 7%

1372

1211

1000

837

775

rd = 10%

M

rd = 13%

30 20 10 0

Years to Maturity

NOTE: Not symmetric

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

rd = 7%

1372

1211

1000

837

775

rd = 10%

M

rd = 13%

30 20 10 0

Years to Maturity

BULLET-VIC

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.

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

rd = 7%

1372

1211

1000

837

775

rd = 10%

M

rd = 13%

30 20 10 0

Years to Maturity

ACTUAL BOND PRICE PROFILE

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.

What is the YTM on a 10-year, 9% annual coupon, $1,000 par value bond that sells for $887?

0

1

9

10

rd=?

90

90

90

1,000

PV1

.

.

.

PV10

PVM

Find rd that “works”!

887

First, what does the fact that this is a discount bond tell you about the relationship between rd and the bond’s coupon rate?

Find rd

INT

(1 + rd)1

INT

(1 + rd)N

M

(1 + rd)N

VB = + . . . + +

90

(1 + rd)1

90

(1 + rd)10

1,000

(1 + rd)10

887 = + . . . + + .

10 -887 90 1000

N I/YR PV PMT FV

10.91

Computer: (e-1)

INPUTS

OUTPUT

Find YTM if price were $1,134.20.

10 -1134.2 90 1000

N I/YR PV PMT FV

7.08

INPUTS

OUTPUT

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

Computer:

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.

Definitions

Annual coupon pmt

Current price

Current yield = .

Cap gains yld = .

= YTM = + .

Change in price

Beg of period price

Exp total

return

Exp

Curr yld

Exp cap

gains yld

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

Current yield =

= 0.1015 = 10.15%.

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 in

Year 0. Same answer.

Proof:

- Value of a 9 year, 9% coupon bond if YTM = 10.91?
- $893.87
- (P1 - P0)/ P0 = ($893.87 - $887)/$887 = .0077 = .77%

10-year, 9% coupon bond with

price = $1,134.20 (YTM = 7.08%)

$90

$1,134.20

Current yield = = 7.94%.

Capital gains yield = 7.08% - 7.94%

= -0.85%.

(P1 - P0)/P0 = 1124.52 - 1134.20

1134.20

= -0.85%

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%

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

DURATION

- e.g. [PV(CF1)/P] x 1 +
- [PV(CF2)/P] x 2 + [PV(CF3)/P] x 3 + ... +[PV(CFn)/P] x n

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

- Cash flows: 0 1 2 100 1100

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

DURATION

- WHAT IS DURATION OF A 2 YEAR ZERO COUPON BOND?

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

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.

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.

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

- Short-term bonds: Low price risk, high reinvestment rate risk.
- TANSTAAFL
- Nothing is riskless!

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.

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.

2n rd/2 OK INT/2 OK

N I/YR PV PMT FV

INPUTS

OUTPUT

Find the value of 10-year, 10% coupon, semiannual bond if rd = 10%; if rd = 13%; if rd = 7%.

INPUTS

OUTPUT

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

For 10%; 2(10) 10/2 100/2

20 5.0 50 1000

N I/YR PV PMT FV

INPUTS

OUTPUT

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

For 10%; 2(10) 10/2 100/2

20 5.0 50 1000

N I/YR PV PMT FV

-1000

INPUTS

OUTPUT

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

For 13%;2(10) 13/2 100/2

20 6.5 50 1000

N I/YR PV PMT FV

-834.72

INPUTS

OUTPUT

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

For 7%; 2(10) 7/2 100/2

20 3.5 50 1000

N I/YR PV PMT FV

-1213.19

INPUTS

OUTPUT

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

10% Annual and Semiannual BondsSpreadsheet Functions for Bond Valuation

- See Ch 06 Mini Case.xls for details.
- PRICE
- YIELD

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?

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:

EFF% = (1 + ) - 1

= (1 + ) - 1 = 10.25%.

iNom

m

m

2

.10

2

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

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:

INPUTS

10 10.25 100 1000

N I/YR PV PMT FV

-984.80

OUTPUT

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.

What is the cash flow stream of a

perpetual bond with an annual

coupon of $100?

8

0

1

2

3

. . .

100 100 100 . . . 100

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

PMT

r

Vperpetuity = .

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 to

call (YTC)?

INPUTS

OUTPUT

LaterHow 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.

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 to

call (YTC)?

10 -1135.9 50 1050

N I/YR PV PMT FV

3.765 x 2 = 7.53%

INPUTS

OUTPUT

Question j

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.

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.

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%.

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.

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.

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

What factors affect default risk and bond ratings?

- Financial performance
- Debt ratio
- TIE, FCC ratios
- Current ratios

(More…)

Provisions in the bond contract

- Secured versus unsecured debt
- Senior versus subordinated debt
- Guarantee provisions
- Sinking fund provisions
- Debt maturity

(More…)

Other factors

- Earnings stability
- Regulatory environment
- Potential product liability
- Accounting policies

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.

Sinking funds are generally handledin 2 ways

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.

Bankruptcy

- Two main chapters of Federal Bankruptcy Act:
- Chapter 11, Reorganization
- Chapter 7, Liquidation
- Typically, company wants Chapter 11, creditors may prefer Chapter 7.

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.

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

Otherwise, judge will order liquidation under Chapter 7.

If the company is liquidated, here’s the payment priority:

1. Secured creditors from sales of secured assets.

2. Trustee’s costs

3. Wages, subject to limits

4. Taxes

5. Unfunded pension liabilities

6. Unsecured creditors

7. Preferred stock

8. Common stock

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.

Week 2s Capital Gains Yield

- Change in price
- Price
- I.e. P1 – P0
- P0
- Next Period: (P2 – P1)/ P1

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