# CHAPTER 8 Bonds and Their Valuation

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CHAPTER 8 Bonds and Their Valuation
Key features of bonds Bond valuation Measuring yield Assessing risk

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.

3. Maturity: Years until bond
must be repaid. Declines. 4. Issue date: Date when bond was issued.

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.

What is 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 handled in 2 ways
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.

Financial Asset Values
1 2 n k ... CF1 CFn CF2 Value CF CF CF PV = 1 + 2 + ... + n . ( ) 1 ( ) 2 ( ) n 1 + k 1 + k 1 + k

The discount rate (ki) is the opportunity cost of capital, i. e
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.

What’s the value of a 10-year, 10% coupon bond if kd = 10%?
100 1 2 10 10% ,000 V = ? ... ( ) V k B d = \$100 \$1 , 1 000 10 . + = \$ \$ \$385.54 = \$1,000.

The bond consists of a 10-year, 10% annuity of \$100/year plus a \$1,000 lump sum at t = 10:
PVA = PMT * (1- (1+k)n ) k PV = FV(1/(1+k)n) V bond = PMT * (1- (1+kd)n + M(1/(1+kd)n) kd

1 V bond = PMT * (1- (1+kd)n + M(1/(1+kd)n) kd V bond = 100 * (1- (1+.1) (1/(1+.1)10) .1 PV annuity \$614.46 PV maturity value V bond = \$ When kd = coupon interest rate, bond par value.

With annual compounding, P/Y = 1, and End mode
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. INPUTS N I/YR PV PMT FV -1,000 OUTPUT

Most bonds pay interest semiannually,
so m = 2. You must divide k by 2, multiply n 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 V bond = 50 * (1- (1+.12/2)10* (1/(1+.12/2)10*2) .12/2 V bond =50( )+1000(.3118) = This bond trades at a discount to par.

What would happen if expected inflation rose by 2%, causing k =12%?.
P/Y=2, End mode INPUTS N I/YR PV PMT FV OUTPUT When kd rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount.

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 V bond = 50 * (1- (1+.08/2)10* (1/(1+.08/2)10*2) .08/2 V bond =50( )+1000(.4564) = This bond trades at a premium to par.

What would happen if inflation fell, and kd declined to 8%?
P/Y=2, End mode N I/YR PV PMT FV -1,135.90 INPUTS OUTPUT Price rises above par, and bond sells at a premium, if coupon > kd.

The bond was issued 20 years ago and now has 10 years to maturity
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%?

Years remaining to Maturity
Bond Value (\$) 1,372 kd = 7%. 1,211 kd = 10%. M 1,000 837 kd = 13%. 775 Years remaining to Maturity

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.

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

What’s the YTM on a 10-year, 9% semiannual coupon, \$1,000 par value bond that sells for \$885.45?
1 9 20 45 kd=? 1,000 PV1 . PV20 PVM 885.45 Find kd that “works”! ...

If you don’t have a financial calculator, need to use
trial and error. Know the price of the bond has dropped 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 = 45 * (1- (1+.11/2)10* (1/(1+.11/2)10*2) .11/2 =45( )+1000(.3427) = Vbond 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.

Find kd INPUTS N I/YR PV PMT FV 10.91 OUTPUT

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.

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 = 45 * (1- (1+.07/2)10* (1/(1+..07/2)10*2) .07/2 =45( )+1000(.5026) = Vbond is approximately equal, so the YTM is 7% .

Find YTM if price were \$1, INPUTS N I/YR PV PMT FV 7.0 OUTPUT Sells at a premium. Because coupon = 9% > kd = 7.0%, bond’s value > par.

Definitions Annual coupon pmt Current price Current yield = Capital gains yield = = YTM = Change in price Beginning 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 = = = 10.15%.

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 in Year 1. Same answer.

What’s interest rate (or price) risk
What’s interest rate (or price) risk? Does a 1-year or 10-year 10% bond have more risk? Interest rate risk: Rising kd causes bond’s price to fall. kd 1-year Change 10-year Change 5% \$1,048 \$1,386 +4.8% -4.4% +38.6% -25.1% 10% 1,000 1,000 15% 956 749

Value . 10-year 1,500 . . . . 1,000 1-year . 500 kd 0% 5% 10% 15%

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%.

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

Do all bonds of the same maturity have the same price and reinvestment rate risk?
No, low coupon bonds have less reinvestment rate risk but more price risk than high coupon bonds.

A 10-year, 10% semiannual coupon,\$1,000 par value bond
is selling for \$1, 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 of the bond is now \$ , 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 = 50 * (1- (1+.06/2)4* (1/(1+.06/2)4*2) .06/2 =50( )+1050(.7894) = Need lower V bond = so YTC must be greater than 6%.

With a financial calculator, you are just
Solving for the interest rate, using the Call premium as the future value and the Time to possible call as n. INPUTS N I/YR PV PMT FV 3.568 x 2 = 7.137% OUTPUT

kNom = 7. 137% is the rate brokers would quote
kNom = 7.137% is the rate brokers would quote. Could also calculate EFF% to call: EFF% = ( )2 – 1 = 7.26%. This rate could be compared to monthly mortgages, etc.

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%.

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.

Disney recently issued 100-year bonds with a YTM of 7
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.

Bond Ratings Provide One Measure of 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 ratio Current ratio

Provisions in the bond contract
Secured vs. unsecured debt Senior vs. subordinated debt Guarantee provisions Sinking fund provisions Debt maturity

Other factors Earnings stability Regulatory environment Potential product liability Accounting policies

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 \$5.0 billion

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