Bonds and Their Valuation

Introduction • Corporations typically have two sources of financing: bonds and

stocks.

• Business actually follows a cycle.

• During economic recession, investors usually invest in bonds, and during economic boom, investors usually invest in stocks.

• During recession, people don’t spend money. They save. Thus economy is depressed. Companies lose money and some even go bankrupt. Their stocks are highly unattractive. The government has to step in, and use monetary policy to boost economy.

• To boost spending, the government LOWERS the interest rate, so that people will take their money out of the bank and spend them.

• When people spend money, the economy improves. Corporations grow, and investors will buy stocks because they are attractive.

Introduction • More money is pushed into the system, and eventually, the supply

will be more than demand, thus the value of money goes down. Products will become more expensive and inflation rises. The government has to step in and use monetary policy to control the level of inflation

• To fight inflation, the government RAISES the interest rate, so people would rather put their money in the bank rather than spend it. When savings increase and spending decrease, economy is once again depressed.

• For investors, when interest rates are raised, bond prices go down, making them attractive.

Bonds • A bond is a long-term debt instrument

• A bond is an IOU – a promise to pay both interest and principal.

• Issuance of bonds is a type of financing.

• Interest is usually semiannual.

• Principal is paid at maturity.

• May be new issues or outstanding bonds/seasoned issues.

• 4 main types of bonds, and they differ with respect to expected return and degree of risk.

Kinds of Bonds Kinds of Bonds

What they are Exposure to Default Risk

Expected Return Other Risks

Treasury Bonds

Bonds issued by the federal government, a.k.a. Government

bonds

None (T-bond decline

when interest rates rise)

Small May be subject to other risks

(liquidity risk, maturity risk)

Corporate Bonds

Bonds issued by corporation

Present Larger than T-bonds

May be subject to other risks

Municipal Bonds

Bonds issued by state and local governments

Present Lower than corporate bonds with the same

default risk

May be subject to other risks

Foreign Bonds

Bonds issued by foreign governments

or by foreign corporations.

Foreign Corps –Present

Foreign Government – Some

Depends May be subject to

other risks, including

currency risk.

Key Features of a Bond • Par Value - face value of a bond.

• Coupon interest rate – stated annual interest rate on a bond.

• Maturity Date – the date when the par value of bonds should be repaid.

• Issue Date – the date when the bonds are issued.

• Call Provisions – provision that the issuer has the right to redeem bonds under specified terms before the normal maturity date

• Sinking Fund Provisions – provision in a bond contract that requires the issuer to retire a portion of the bond issue each year.

Par Value • Face value

• Assume $1,000 (multiples of 1000 also okay)

Coupon interest rate • Stated Annual Interest rate on a bond

• Eg: If you hold $10,000 nominal of a bond described as a 4.5% loan stock, you will receive $450 per year (in two installments of $225)

• Generally fixed and paid every six months

• Multiply by par to get $ of interest

• Coupon interest payment = Coupon Payment/Par Value

Common Bonds • Fixed Rate Bond

– A bond whose interest rate is fixed for its entire life.

• Floating Rate Bond – Its interest rate fluctuates with shifts in general level of interest rates.

– A bond whose interest is pegged to a benchmark, such as a T-bill rate, and adjusted periodically.

– Prices of floating rate bond remain relatively stable because neither a capital gain nor a capital loss occurs as market interest rates go up or down.

• Zero Coupon Bond (Discount bond/deep discount bond) – Pays no annual interest but is sold at a discount below par.

– Compensation is not in the form of interest, but in the form of capital appreciation.

• Original Issue Discount Bond – Any bond originally offered at a price below its par value.

Sample Problem:

• You have the choice of choosing between a one-year zero-coupon bond with a face value of $1,000, which can be purchased for $952.38 or a one-year 4% semi-annual coupon bond trading at its face value of $1,000. Which bond will you choose and why?

Maturity

• Years until the bond has to be repaid.

• Maturity declines

• Original Maturity – the number of years to maturity at the time a bond is issued.

Issue Date

• Date when the bond was issued.

Call Provisions • Allows the issuer to pay back the bond before maturity – it allows them to

refinance the bonds

• Call Premium – amount that is paid by the issuer to the bondholder for the right to call the bonds before maturity.

• Issuers will call the bond, when interest rate is declining.

• Subjects the bondholder to reinvestment risk.

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

• Deferred call – bonds cannot be called until after several years after issuance.

• Declining call premium – the value of call premium decreases as the bond nears maturity.

Sinking Fund Provision • A provision in a bond contract that requires the issuer to retire a portion of the

bond issue each year.

• May be advantageous to the bondholder as it creates liquidity.

• May de disadvantageous to the bondholder in times of declining interest rate.

Sinking funds are generally handled in 2 ways

• Call x% at par per year for sinking fund purposes – Lottery (at par value - $1,000 per bond)

• Buy bonds on open market

The Company will choose the lowest cost way.

If the bonds sell for more than $1,000 in the open market or when interest rate is falling, the issuer will call the bonds.

If the bonds are selling for less than $1,000, or when interest rate is rising, the issuer will buy the bonds in the open market.

Other Features

• Convertible Bond

– Bondholder’s option to convert bonds for common stock of the issuing firm (Convert during boom).

• Warrant

– Long-term option to buy a stated number of common stock at a specified price (Buy when specified price is less than market price).

• Putable Bond

– Bondholder’s option to sell the bond back prior to maturity at a prearranged price (BH put when interest rates are rising).

• Income Bond

– A bond that pays interest only if interest is earned. They are riskier than regular bonds.

• Indexed (Purchasing Power) Bond

– Interest payments are based on an inflation index to protect the holder from inflation.

– Interest paid rises automatically when inflation rate rises.

Discount versus Premium Bonds

• Discount Bonds

– A bond that is issued for less than its par value.

– A bond that is currently trading for less than its par value in the secondary market.

• Premium Bonds

– A bond that is issued for more than its par value.

– A bond that is currently trading for more than its par value in the secondary market.

Bond Valuation

• The value of any financial asset is the PV of the cash flows it is expected to generate.

• There are mainly two cash flows of a bond: Principal and Interest

• Bond Value = PV of interest + PV of principal

• Pbond = INT (PVIFA) + M (PVIF)

© 1998 The Dryden Press

Financial Asset Values

PV =

CF

1 + k . . . +

CF

1 + k

1 n

1

2

2 1

CF

k n .

0 1 2 n

k

CF1 CFn CF2 Value

...

+ + +


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

• It is also called the bond’s required rate of return (yield) or the effective rate.

• Ex: GM bond must yield 9% but a Lady Luck bond must yield 18%


What’s the value of a 10-year, 10% coupon bond if kd = 10%? (assume annual compounding)

V

k k B

d d

$100 $1 ,

1

000

1 1 10 10 . . . +

$100

1 + k d

100 100

0 1 2 10

10%

100 + 1,000 V = ?

...

= $90.91 + . . . + $38.55 + $385.54

= $1,000.

+ + + +

Changes over time

• Generally, the coupon rate is set to equal required yield when bond issued. So they generally sell at par $1,000

– If coupon is set too low (sell at a discount), you aren’t willing to pay $1,000 for the bond (coz the PV is lower than par)

– If the coupon is set too high (sell at a premium), the demand for the bond will be high (coz the PV is higher than par)

• Over time the required yield changes (changes in market interest rates, inflation, recession, etc)

• Thus the price of the bond will change

Bond Terminology

Discount Par Premium

Price of Bond Sold for less than face value

Sold for equal to face value

Sold for more than face value

Investor Requirement

Higher return than coupon rate

Equal to coupon rate

Less return than coupon rate

• Find the present value of a 4 year 10%, 1,000 bonds with a discount rate of 10%, assuming semi-annual payments.

– N = 8; I = 5%; PV = ?; PMT = 50; FV = 1,000

– PV of Principal: 1,000 x 0.676839362 = 676.84

– PV of Interest : (1,000 x 5%) x 6.463212759 = 323.16

– PV of Principal and Interest = 676.84 + 323.16 = 1,000

Par Bonds

• If expected inflation remains the same after 1 year.

• Find the present value of a 3 year 10%, 1,000 bonds with a discount rate of 10%, assuming semi-annual payments. – N = 6; I = 5%; PV = ?; PMT = 50; FV = 1,000

– PV of Principal: 1,000 x 0.746215397 = 746.22

– PV of Interest : (1,000 x 5%) x 5.075692067 = 253.78

– PV of Principal and Interest = 746.22 + 253.78 = 1,000

Interest paid Interest expense Amortization Balance 1,000.00

1 50.00 50.00 - 1,000.00 2 50.00 50.00 - 1,000.00

3 50.00 50.00 - 1,000.00

4 50.00 50.00 - 1,000.00 5 50.00 50.00 - 1,000.00

6 50.00 50.00 - 1,000.00

Discount Bonds

• If expected inflation rose by 2%, causing k = 12% after 1 year.


– PV of Principal: 1,000 x 0.7049054 = 704.91

– PV of Interest : (1,000 x 5%) x 4.917324326 = 245.87

– PV of Principal and Interest = 704.91 + 245.87 = 950.78

Interest expense Interest paid Amortization Balance (Increasing) (Same) (Increasing) 950.78

1 57.05 50.00 7.05 957.82

2 57.47 50.00 7.47 965.29

3 57.92 50.00 7.92 973.21

4 58.39 50.00 8.39 981.60 5 58.90 50.00 8.90 990.50

6 59.43 50.00 9.43 999.93

Premium Bonds

• If expected inflation fell by 2%, causing k = 8% after 1 year.


– PV of Principal: 1,000 x 0.790314526 = 790.31

– PV of Interest : (1,000 x 5%) x 5.242136857 = 262.11

– PV of Principal and Interest = 790.31 + 262.11 = 1,052.42

Interest expense Interest paid Amortization Balance (Decreasing) (Decreasing) 1,052.42

1 42.10 50.00 (7.90) 1,044.52 2 41.78 50.00 (8.22) 1,036.30

3 41.45 50.00 (8.55) 1,027.75 4 41.11 50.00 (8.89) 1,018.86

5 40.75 50.00 (9.25) 1,009.61

6 40.38 50.00 (9.62) 1,000.00

Premium and Discount Bonds

• @ K of 12%: Price = $950.78 (Less than par -called a discount bond)

• @ K of 8%: Price = $1,052.42 (More than par-called a premium bond)

• Note there is an INVERSE relationship between yield and price

• Because the coupon rate and the par value can’t be changed, the price must change


M

Bond Value ($)

Years remaining to Maturity

1052.42

1,000

950.78

3 0

kd = 8%.

kd = 12%.

kd = 10%.

Bond Yields • Yield to Maturity (YTM) or Redemption Yield

– The rate of return earned on a bond if it is held to maturity.

– The discount rate which forces the PV of all the cash flows to equal the price paid for the bond

– Solving for k (or for the i)

– Equal to ERR only if

• The probability of default is zero

• The bond cannot be called.

• Yield to Call (YTC) – The rate of return earned on a bond if it is called before its maturity date.

• Current Yield (CY) – The annual interest payment on a bond divided by the bond’s current

price.

Yield to Maturity

• Suppose your 7% $1,000 par value bond is selling for $950 and it matures in 4 years. What is the YTM?

• a. 6%; b. 6.5%; c. 7%; d. 8%; e. 8.5% • PMT (1+YTM)^-1 + PMT (1+YTM)^-2….. + PMT (1+YTM)^-n

Par(1+YTM)^-n = PV

• 70 (1+YTM)^-1 + 70 (1+YTM)^-2 + 70 (1+YTM)^-3 + 70 (1+YTM)^-4 + 1,000 (1+YTM)^-4 = 950

• YTM will never be A,B,C because it is sold at a discount. So you have to do trial and error to get YTM.

Yield to Call

• A 2-year, 10% semiannual coupon, $1,000 par value bond is selling for $1,135.90 with a 2.95% YTM. It can be called after 1 year at $1,050. Compute its YTC.

• a. 2.58% b. 5.17% c. 10.15% d. 12.3% e. 14.24%

• YTC: N = 4; I = ?; PV = 1135.90; PMT = 50; FV = 1050

• PMT (1+YTC/2)^-1 + PMT (1+YTC/2)^-2….. + PMT (1+YTC/2)^-n FV(1+YTC/2)^-n = PV

• 50 (1+YTC/2)^-1 + 50 (1+YTC/2)^-2 + 50 (1+YTC/2)^-3 + 50 (1+YTC/2)^-4 + 1050 (1+YTC/2)^-4 = 1,135.90

Current Yield or Interest Yield

• If 1,000 Allied bonds with a 10% coupon were currently selling for $985, and interest is compounded annually, what is the current yield?

• How about if it is compounded semi-annually? • Current Yield = Annual Interest Payment/Bond’s Current Price

• Current Yield = Annual PMT / PV

• Current Yield = 100 / 985 = 10.15%

Solving for Total Yield

• Total Yield = Current Yield + Capital Gains Yield

• You have a bond priced at (with a carrying value of) $95.75 and has an annual coupon of $5.10. If the bond is a 10-year bond with nine years remaining and you were only planning to hold it for one year and the price of your bond fell to $87.34, what would be your net gain or loss for that period?

• your actual return for the period would be - 3.45% (-$3.31/$95.75) because although you gained $5.10 in PMT, your capital loss was $8.41.

• CY = PMT/PV = 5.1/95.75 = 5.33%

• CGY = (P1 – Po) / Po = (87.34 – 95.75) / 95.75 = - 8.78%

• Total ROR or Total Yield = 5.33% + - 8.78% = - 3.45%

• You can also use the Holding Period Return Formula to solve for Total 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?

The semiannual bond’s EFF% is:

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

semiannual bond.

EFFi

mNom

m

%.

.

1 1 1

0 10

21 10 25%

2

.

Risks Associated with a Bond

• Interest Rate Risk – The risk that interest rate will increase, thus the bond’s value will

decrease.

– For bonds that are similar with regards to all respects, except that one has a longer maturity, the bond with the longer maturity is typically exposed to more risk from a rise in interest rates.

– 14 year 1000 par bonds that yielded 10% vs. a 15 year 1000 par bonds that yielded 10%. If interest rise to 15%, you are stuck with 100 interest for only 14 years, and u can reinvest 1000 par bonds in 15% for a year and get 150 on the 15th year

– But if you have 15 years, then you are stuck with 100 interest for 15 years.

Risks Associated with a Bond

• Reinvestment Rate Risk – The risk that interest rate will decrease and the issuer will call the

bonds, therefore, you are forced to reinvest at a lower rate.

– Riskier for callable bonds, or normal bonds with shorter maturity.

• CONCLUSION – Long-term bonds = high interest rate risk, low reinvestment risk

– Short-term bonds = low interest rate risk, high reinvestment risk

Default Risk • A big factor in corporate bonds

• The higher the probability of default, the lower the price of the bond, the higher the required yield.

• Default risk will depend on the type of the bond issued.

– Mortgage Bonds

• Have a specific asset pledged as collateral.

• Lower required yields.

– Debentures

• Unsecured bonds

• No lien or claim against specific property

• Higher required yields.

• Subordinated debentures – has claims on assets only after senior debt has been paid off in times of bankruptcy.

Stronger companies can easily issue debentures. They don’t need to pledge specific property because they already established their reputation and have gained the confidence of investors; but definitely not small companies.


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

• Is measured by bond ratings, such as Moody’s, S&P, and Fitch Investors Service

• Bonds can be rated as Investment Grade or Junk.

What factors affect default risk and bond ratings?

• Financial performance

– Debt ratio

– TIE, FCC ratios

– Current ratios

• Provisions in the bond contract

– Secured vs. unsecured debt

– Senior vs. subordinated debt

– Sinking fund provisions

– Debt maturity

• Other factors

– Earnings stability

– Regulatory environment

– Potential product liability

– Accounting policies

Bond Ratings

• Changes in ratings will have large impact on bond prices.

• Higher ratings = higher prices = lower yields. Cheaper to raise capital

• Downgrades have the opposite effect

• Junk bonds= low ratings

Exercises:

• A bond matures in 12 years and pays an 8 percent annual coupon. The bond has a face value of $1,000 and currently sells for $985. What is the bond’s current yield and yield to maturity?

Exercises:

• The current price of a 10-year, $1,000 par value bond is $1,158.91. Interest on this bond is paid every six months, and the nominal annual yield is 14 percent. Given these facts, what is the annual coupon rate on this bond?

Exercises:

• Matteo Toys has bonds outstanding that have a 9 percent annual coupon and a face value of $1,000. The bonds will mature in 10 years, although they can be called before maturity at a call price of $1,050. The bonds have a yield to call of 6.5 percent and a yield to maturity of 7.4 percent. How long until these bonds may first be called?

Exercises:

• Recently, Ohio Hospitals Inc. filed for bankruptcy. The firm was reorganized as American Hospitals Inc., and the court permitted a new indenture on an outstanding bond issue to be put into effect. The issue has 10 years to maturity and a coupon rate of 10 percent, paid annually. The new agreement allows the firm to pay no interest for 5 years. Then, interest payments will be resumed for the next 5 years. Finally, at maturity (Year 10), the principal plus the interest that was not paid during the first 5 years will be paid. However, no interest will be paid on the deferred interest. If the required annual return is 20 percent, what should the bonds sell for in the market today?

Bonds and Their Valuation

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