Introduction to Bonds
Description and Pricing
P. V. Viswanath

What are bonds?
A borrowing arrangement where the borrower issues an IOU to the investor.

Types of Bond Issuers
Treasury Bonds and Notes
Corporate Bonds
Other Issuers
State and Local Governments (Munis)
Government Agencies

Varieties of Cashflow Patterns
Callable Bonds
Convertible Bonds
Puttable Bonds
Floating Rate Bonds
Preferred Stock

Callable Bonds
Allows the issuer to repurchase the bond at a specified price(call price) before the maturity date.
Frequently, there is a period of call protection, during which the bonds cannot be called.Such bonds are called deferred callable bonds.

Convertible Bonds
Bondholders get the option to exchange each bond for a specified number of shares of stock for each bond (conversion ratio).
Example:
Bond selling at 990; convertible into 40 shares.
Current share price = $20; value of converted shares = $800 (conversion value) -- conversion suboptimal
Conversion premium = $990 - 800 = $190.

Puttable Bonds
The put bond gives the investor the option to extend or retire the bond at the call date.
If the bondís coupon rate exceeds market yields, the bondholder will choose to extend
If the bondís coupon rate is too low, it will be optimal to retire the bond and reinvest the proceeds at current yields.

Floating Rate Bonds
Interest payments are tied to some measure of current market rates.E.g., the rate might be adjusted annually to Prime plus 2%.
However, payments are not adjusted for changes in the firmís financial condition.
The risk to the firm is that the yield spread is fixed over the life of the bond.

Preferred Stock
Preferred stock is equity, because:
dividends are paid at the firmís discretion
payment priority at liquidation is below all bonds.
It resembles debt because:
Dividend amounts are fixed, and potentially
adjustable like coupon payments.

Credit Risk
Corporate bonds are subject to potential default. Therefore, the promised yield is the maximum possible yield to maturity of the bond, not necessarily the actual yield to maturity.
To compensate investors for the possibility of bankruptcy, a corporate bond must offer a default premium, a differential between the promised yield and the expected yield to maturity.
The default premium depends on :
the probability of default
the likely loss in the event of default.

Calculation of default premium
A simple model
Assume
the probability of default, p, is constant from year to year,
the loss on a bond that defaults equals a proportion l of the bond's market price in the previous year
y = the promised yield-to-maturity
y* = the expected yield-to-maturity
d = y - y* = default premium

Risk Premium
Risky bonds may also have to provide a risk premium.
The risk premium is the difference between the expected yield and the yield on a comparable default-free T-bond.
A risk premium is needed only if the bond risk is not diversifiable.

Covenants in Bond Indentures
Sinking Funds
Limitations on Further Debt issuance
Subordination of Further Debt
Negative Pledge
Dividend Restrictions
Collateral
Restrictions on Sale and Leaseback

Bond Pricing
A T-period bond with coupon payments of $C per period and a face value of F.
The value of this bond can be computed as the sum of the present value of the annuity component of the bond plus the present value of the FV.

Bonds with semi-annual coupons
Normally, bonds pay semi-annual coupons:
The bond value is given by:
where the first component is, once again, the present value of an annuity, and y is the bondís yield-to-maturity.

Bond Pricing
Example
If F = $100,000; T = 8 years; the coupon rate is 10%, and the bondís yield-to-maturity is 8.8%, the bond's price is computed as:
†††††††††††††††††††††††††††††††††††† = $106,789.52

The Relation between Bond Prices and Yields
Consider a 2 year, 10% coupon bond with a $1000 face value.If the bond yield is 8.8%, the price is 50†††††† + 1000/(1.044)4 = 1021.58.
Now suppose the market bond yield drops to 7.8%.The market price is now given by 50††††††† + 1000/(1.039)4 = 1040.02.
As the bond yield drops, the bond price rises, and vice-versa.

Bond Prices and Yields
A Graphic View

Bond Yield Measurement
Definitions
Yield to Maturity
A measure of the average rate of return on a bond if held to maturity.To compute it, we define the length of a period as 6 months, and then calculate the internal rate of return per period.Finally, we double the six-monthly IRR to get the bond equivalent yield, or yield to maturity.
Effective Annual Yield
Take the six-monthly IRR and annualize it by compounding.

Bond Yield Measurement: Examples
An 8% coupon, 30-year bond is selling at $1276.76.First solve the following equation:
This equation is solved by r = 0.03.
The yield-to-maturity is given by 2 x 0.03 = 6%
The effective annual yield is given by (1.03)2 - 1 = 6.09%

Computing YTM by Trial and Error
A 3 year, 8% coupon, $1000 bond, selling for $949.22
Period Cash flow Present Value
††††††††††† 9% ††††††††† 11% ††††††††† 10%
1 40 $38.28 $37.91 $38.10
2 40 $36.63 $35.94 $36.28
3 40 $35.05 $34.06 $34.55
4 40 $33.54 $32.29 $32.91
5 40 $32.10 $30.61 $31.34
6 1040 $798.61 $754.26 $776.06
Total ††††††† $974.21 $925.07 $949.24
The bond is selling at a discount; hence the yield exceeds the coupon rate.At a discount rate equal to the coupon rate of 8%, the price would be 1000.Hence try a discount rate of 9%.At 9%, the PV is 974.21, which is too high. Try a higher discount rate of 11%, with a PV of $925.07, which is too low.Trying 10%, which is between 9% and 11%, the PV is exactly equal to the price.Hence the bond yield = 10%.

Computing YTM by Trial and Error:
A Graphic View

Approximate formula for yield-to-maturity
If a bond is selling at par, yield = coupon rate.If the bond sells at a discount, yield > coupon rate.Hence the following approximation:

Realized Compound Yield
The Yield-to-Maturity measure implicitly assumes that all coupons can be invested at the internal rate of return.
An alternative ex-post yield measure can be computed that takes the actual reinvestment rates.

Holding Period Return
The holding period return differs from the previous two measures in that it is computed for the actual period of time that the bond is held.
Example:
If a bond is purchased for $1100, pays a coupon of $120 at the end of the year, and is then sold for $1210, the holding period return = (120 + 1210-1100)/1100 = 20.91%

Time Pattern of Bond Prices
Bonds, like any other asset, represent an investment by the bondholder.
As such, the bondholder expects a certain total return by way of capital appreciation and coupon yield.
This implies a particular pattern of bond price movement over time.

Time Pattern of Bond Prices: Graphic View

Time Pattern of Bond Prices in Practice
Coupons are paid semi-annually.Hence the bond price would increase at the required rate of return between coupon dates.
On the coupon payment date, the bond price would drop by an amount equal to the coupon payment.
To prevent changes in the quoted price in the absence of yield changes, the price quoted excludes the amount of the accrued coupon.
Example: An 8% coupon bond quoted at 96 5/32 on March 31, 1996 would actually require payment of 961.5625 + 0.5(80/2) = $981.5625

Tax treatment of bonds
The price appreciation on original issue discount (OID) bonds is an implicit interest payment.Hence the IRS calculates a price appreciation and imputes taxable interest income.Additional gains or losses due to market yield changes are treated as capital gains.
Example: An 8% 30 yr. bond issued at 81.071% of par is sold for 820.95 at the end of the year.If interest income is taxed at 36% and capital gains at 28%, the tax paid equals 0.36(80 + 811.80 - 810.71) + 0.28(820.95-811.80) = $31.75.
(YTM at issue = 10%; yearend price using this YTM = 811.80)