Bond Valuation

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Week Five

Key characteristics of three types of bonds

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Treasure bonds: issued by the government. No default risk. Price changes with market interest rate.

Corporate bonds: issued by corporations. Have default risk. Higher default risk, higher interest rate to pay.

Municipal bond: issued by local government. Have default risk. No tax on coupon to bond holders.

Key Features of a Bond

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

Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest. Generally fixed.Coupon interest rate = coupon payment / par value

Example: A 5 year $1000 face value, 5% coupon interest rate bond can give investors $50 annual coupon payment.

Key Features of a Bond

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Maturity: Years until bond must be repaid. Declines with time.

Issue date: Date when bond was issued.Default risk (credit risk): Risk that issuer

will not make interest or principal payments.

Discussion: What will happen if a firm cannot pay its obligations, including coupon payment?

Call Provision

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Callable bonds are bonds that can be called by the issuer after a certain amount of time—the call protection period—at a specified price—the call price—which is usually higher than the face value of the bond (par + call premium).

The call price is highest in the 1st year that the bond can be called, and decreases as the time to maturity decreases.

Bond issuers issue callable bonds to take advantage of possible decreases in future interest rates, but they have to pay the bond buyer a higher coupon rate to compensate the buyer for the bond possibly being called.

Bonds are called when interest rates decline, and, thus, the bondholder not only loses the interest that the bond was paying, but also the capital appreciation of increased bond prices, which are capped at the call price.

The bondholder will not be able to invest in another bond paying the same interest rate for the same credit risk.

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

Call Provision

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When interest rates decline, straightbond prices increase, but callablebonds are capped at the call price,because the issuer will call the bondand issue new bonds with a lowercoupon rate.

What’s a sinking fund?

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A means of repaying funds that were borrowed through a bond issue. The issuer makes periodic payments to a trustee who retires part of the issue by purchasing the bonds in the open market.

Rather than the issuer repaying the entire principal of a bond issue on the maturity date,  another company (trustee) buys back a portion of the issue annually and usually at a fixed par value or at the current market value of the bonds, whichever is less. 

Should interest rates decline following a bond issue, sinking-fund provisions allow a firm to lessen the interest rate risk of its bonds as it essentially replaces a portion of existing debt with lower-yielding bonds.

From the investor's point of view, a sinking fund adds safety to a corporate bond issue: with it, the issuing company is less likely to default on the repayment of the remaining principal upon maturity since the amount of the final repayment is substantially less.

It is a 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. Safer thus offered with lower coupon rate, all else equal.

Sinking funds are generally handled in 2 ways

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Call x% at par per year for sinking fund purposes.Call if rd is below the coupon rate and bond

sells at a premium.Buy bonds on open market.

Use open market purchase if rd is above coupon rate and bond sells at a discount.

Convertible bonds

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Convertible bonds allow the holders to convert into the common stock of the issuing company.

Convertible securities are characterized by either specifying the conversion ratio explicitly or by specifying the conversion price.

The conversion ratio is the number of shares of stock that can be converted for each convertible bond. As another way to calculate the conversion ratio, the conversion price is the specified stock price used in determining the conversion ratio.

Example: a convertible bond with a face value of $1,000, and that had a conversion ratio of 10, would be convertible into 10 shares of stock. Alternately, if the bond indenture specified a conversion price of $50 per share, then the bond could be converted into 20 shares of stock.

The conversion price is specified before the convertible bond is issued, and is always higher than the market value of the stock on the date of issue.

The current stock price determines whether the convertible security will be converted or not.

Most convertibles are issued deep out of the money, so the stock would have to appreciate considerably before it would be profitable to convert.

Convertible bonds have lower coupon rate than non-convertible bond.

Financial Asset Valuation

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PV =CF

... +CF1 N2

(1 + r)2

CF.

0 1 2 N

r

CF1 CFNCF2Value

...

+ +(1 + r)1 (1 + r)N

Value of a 10-year, 10% coupon bond if rd = 10%

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VB =$100 $1,000

. . . +$100

100 100

0 1 2 1010%

100 + 1,000V = ?

...

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

++(1 + rd)1 (1 + rd)N (1 + rd)N

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

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10 10 100 1000N I/YR PV PMT FV

-1,000

$ 614.46 385.54

$1,000.00

PV annuity PV maturity value Value of bond

===

INPUTS

OUTPUT

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Using Excel to Solve the Price of a Bond ProblemSuppose the settlement date of a bond you purchased is November 30, 2001; the maturity date of the bond is December 31, 2028; the bond has a coupon rate of 6.25% and interest is paid semi-annually; the face value of the bond is $1000; and actual days per month/year is used for the day-count basis (not 30/360). Suppose investors currently want an 8.3% return for this type of bond. What price should they be willing to pay?

1.go to the Function Wizard and find the function category of FINANCIAL and then PRICE. If you do not have the PRICE function, go to TOOLS--ADD-INS and click on ANALYSIS TOOLPAK.2.To use the PRICE function, you need to complete the followingSETTLEMENT, MATURITY, RATE, YIELD, REDEMPTION, FREQUENCY, BASIS

(1)SETTLEMENT is the settlement date. Type in DATE(2001, 11, 30) . Click the Tab key (not the <Enter> key). (2) MATURITY is the maturity date. Type in DATE(2028, 12, 31). Click the Tab key.(3) RATE is the coupon rate. You type in .0625. Click the Tab key. (4) YIELD is the desired yield to maturity (or current market rate of interest). Type in .083. Click the Tab key. (5) REDEMPTION is the redemption value per $100 of value. Type in 100 since you will not be given more than $100 per $100 of face value (even if the face value of the bond is $1000). Click the Tab key. (6) FREQUENCY is how often interest is paid; 2 for semi-annual and 1 for annual. You type in 2 (the default) since interest is paid semi-annually. Click the Tab key. (7) BASIS Type of day-count basis to use: 0 means 30/360, 1 is actual/actual, etc. Type in 1. Click the Tab key. (8) Click on FINISH.

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Using Excel to Solve the Price of a Bond Problem

The answer I got was 78.02187.This means that investors are only willing to pay me 78.02 per 100. For a $1000 bond (multiply by 10), investors will only pay me $780.22.   Investors are only willing to pay about $780 for a bond with a $1000 face value.   Why?

If you know the PRICE investors are willing to pay and want to calculate the desired Yield to Maturity, go to the Function Wizard and use the YIELD function. Everything is the same as above except, instead of (4) YIELD, you will see PR (PRICE). You have to type in the price per $100. Thus, if investors are paying, say, $850 for a $1000 bond, you type in 85.

What would happen if expected inflation rose by 3%, causing r = 13%?

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When rd rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount. It is called: a discount bond.

10 13 100 1000N I/YR PV PMT FV

-837.21

INPUTS

OUTPUT

What would happen if inflation fell, and rd declined to 7%?

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If coupon rate > rd, price rises above par, and bond sells at a premium. It is called a premium bond.

10 7 100 1000N I/YR PV PMT FV

-1,210.71

INPUTS

OUTPUT

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Discussion: Suppose the bond with 10% coupon rate 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%?

Bond Value ($) vs Years remaining to Maturity

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M

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

rd = 7%.

rd = 13%.

rd = 10%.

Or refer to Figure 4 -2 at p126.

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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 rd remains constant.

What’s “yield to maturity”?

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

It assumes the bond will not default and will not be called.

YTM changes daily.

YTM on a 10-year, 9% annual coupon, $1,000 par value bond selling for $887

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90 90 90

0 1 9 10rd=?

1,000PV1 . . .PV10

PVM

887 Find rd that “works”!

...

Find rd

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

10.91

V

INT MB =

(1 + rd)1 (1 + rd)N ... +

INT

887 90

(1 + rd)1

1,000(1 + rd)N

= +90

(1 + rd)N

+ +

+ +

INPUTS

OUTPUT

...

(1 + rd)N

General rules

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If coupon rate < rd, bond sells at a discount.

If coupon rate = rd, bond sells at its par value.

If coupon rate > rd, bond sells at a premium.

If rd rises, price falls.Price = par at maturity.

Find YTM if price were $1,134.20.

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

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

7.08

INPUTS

OUTPUT

Or, in Excel, rate(10, 90, -1134.2, 1000,0) or use yield function in Excel.

Definitions

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Current yield =

Capital gains yield =

= YTM = +

Annual coupon pmtCurrent price

Change in priceBeginning price

Exp totalreturn

Exp Curr yld

Exp capgains yld

9% coupon, 10-year bond, P = $887, and YTM = 10.91%

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Current yield =

= 0.1015 = 10.15%.

$90 $887

YTM = Current yield + Capital gains yield.

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Cap gains yield = YTM - Current yield

= 10.91% - 10.15% = 0.76%.

Cap gains yield could also find bond values in Years 1 and 2, get difference, and divide by value in Year 1. Same answer. ((P2-P1)/P1)

Semiannual Bonds

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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

Value of 10-year, 10% coupon, semiannual bond if rd = 13%.

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

-834.72

INPUTS

OUTPUT

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Callable bonds have two potential life spans, one ending at the original maturity date and the other at the "callable date."

At the callable date, the issuer may "recall" the bonds from its investors - the issuer retires (or pays off) the bond by returning the investors' money.

Whether or not this occurs depends on the interest rate environment.

Callable bonds represent a normal bond, but with an embedded call option.

To compensate investors for this uncertainty, an issuer will pay a slightly higher interest rate than a similar non-callable bond. Issuers may offer bonds that are callable at a price in excess of the original par value.

Callable Bonds and Yield to Call

Callable Bonds and Yield to Call

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Despite the higher cost to issuers and increased risk to investors, these bonds can be attractive to either party.

Investors like them because they give a higher-than-normal rate of return, at least until the bonds are called away.

Callable bonds are attractive to issuers because they allow them to reduce interest costs at a future date should rates decrease.

There is no free lunch, and the higher interest payments received for a callable bond come with the price of reinvestment-rate risk (of the income from the bond) and diminished price-appreciation potential.

Why should you buy callable bonds?As the purchaser of a bond, you are betting that interest rates will remain the same or increase. If this happens, you would receive the benefit of a higher-than-normal interest rate throughout the life of the bond.

Nominal Yield to Call (YTC)

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10 -1135.9 50 1050 N I/YR PV PMT FV

3.765 x 2 = 7.53%

INPUTS

OUTPUT

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.

Note that YTC < YTM.

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

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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%.

Callable bond general rule of thumb

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In general, if a bond sells at a premium, then coupon > rd, so a call is likely.

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

Bond Ratings Provide One Measure of Default Risk

Investment Grade Junk Bonds

Moody’s Aaa Aa A BaaBa

B Caa C

S&PAAA

AA A BBBBB

B CCC D

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What factors affect default risk and bond ratings?

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Financial performanceDebt ratioCoverage ratios, such as interest coverage ratio

or EBITDA coverage ratioCurrent ratios

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

Other factorsEarnings stabilityRegulatory environmentPotential product liabilityAccounting policies

Bond Ratings and Bond Spreads (YahooFinance, March 2008)

Long-term Bonds Yield Spread

U.S. Treasury 4.40%

AAA 5.52% 1.52%

AA 5.90% 1.54%

A 6.14% 1.74%

BBB 6.51% 2.11%

BB 7.09% 2.69%

B 7.61% 3.21%

CCC 9.01% 4.61%

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Bond Risk: Interest rate risk and reinvestment rate risk

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rd 1-year Bond price Change

10-year

5% $1,048 $1,386

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

15% 956 4.4% 749 25.1%

Interest rate risk: Rising rd causes bond’s price to fall more for 10 year 10% bonds than for a 1 year 10% bonds.

Bond price Change

Value

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0

500

1,000

1,500

0% 5% 10% 15%

1-year

10-year

rd

Take away: Increase in interest rate can hurt bondholders. The longer the maturity of the bond, the more its price drops with higher interest rate, and thus the higher the interest rate risk.

Bond Risk: reinvestment rate risk

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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%.

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.

Take away: Decrease in interest rate can hurt bondholder. When interest rate falls, a bondholder may suffer a reduction in income. Reinvestment rate risk is high on callable bonds and short maturity bonds.