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 bond’s coupon rate exceeds market yields, the bondholder will choose to extend

If the bond’s 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 firm’s 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 firm’s 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 bond’s yield-to-maturity.

Bond Pricing
Example

If F = $100,000; T = 8 years; the coupon rate is 10%, and the bond’s 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)


	A borrowing arrangement where the borrower issues an IOU to the investor.


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


	Callable Bonds
	Convertible Bonds
	Puttable Bonds
	Floating Rate Bonds
	Preferred Stock


	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.


	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.


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


	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 firm’s financial condition.
	The risk to the firm is that the yield spread is fixed over the life of the bond.


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


	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.


	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


	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.


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


	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.


	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 bond’s yield-to-maturity.


	If F = $100,000; T = 8 years; the coupon rate is 10%, and the bond’s yield-to-maturity is 8.8%, the bond's price is computed as:



	= $106,789.52


	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.


	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.


	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%


	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%.


	If a bond is selling at par, yield = coupon rate. If the bond sells at a discount, yield > coupon rate. Hence the following approximation:


	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.


	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%


	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.


	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


	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)